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abstract: 'Accreting supermassive black holes (SMBHs) can exhibit variable emission across the electromagnetic spectrum and over a broad range of time-scales. The variability of active galactic nuclei (AGN) in the ultra-violet (UV) and optical is usually at the few tens of percent level over time-scales of hours to weeks[@Caplar2017_PTF_QSOs]. Recently, rare, more dramatic changes to the emission from accreting SMBHs have been observed, including tidal disruption events (TDEs)[@Gezari2012_TDE; @Arcavi2014_TDEs_He; @Holoien2014_TDE_AS14ae; @Holoien2016_TDE_AS14li], “changing look” AGN[@LaMassa2015_changing; @MacLeod2016_CLAGN; @Ricci2016_IC751_CLAGN; @Runnoe2016_changing], and other extreme variability objects[@Lawrence2016_PS1_slow_hypervar; @Graham2017_extreme_var]. The physics behind the “re-ignition”, enhancement, and “shut-down” of accretion onto SMBHs is not entirely understood. Here we present a rapid increase in ultraviolet-optical emission in the centre of a nearby galaxy marking the onset of sudden increased accretion onto a SMBH. The optical spectrum of this flare, dubbed [AT2017bgt]{}, exhibits a mix of emission features. Some are typical of luminous, unobscured AGN, but others are likely driven by Bowen fluorescence - robustly linked here, for the first time, with high-velocity gas in the vicinity of the accreting SMBH. The spectral features and increased UV flux show little evolution over a period of at least [14]{} months. This disfavours the tidal disruption of a star as their origin, and instead suggests a longer-term event of intensified accretion. Together with two other recently reported events with similar properties, we define a new class of SMBH-related flares. This has important implications for the classification of different types of enhanced accretion onto SMBHs.'
author:
- 'Benny Trakhtenbrot$^{1,2}$, Iair Arcavi$^{2,3,4}$, Claudio Ricci$^{5,6,7}$, Sandro Tacchella$^{8}$, Daniel Stern$^{9}$, Hagai Netzer$^{2}$, Peter G. Jonker$^{10,11}$, Assaf Horesh$^{12}$, Julian E. Mejía-Restrepo$^{13}$, Griffin Hosseinzadeh$^{3,4}$, Valentina Hallefors$^{3,4}$, D. Andrew Howell$^{3,4}$, Curtis McCully$^{3,4}$, Mislav Baloković$^{8}$, Marianne Heida$^{14}$, Nikita Kamraj$^{14}$, George B. Lansbury$^{15}$, [Ł]{}ukasz Wyrzykowski$^{16}$, Mariusz Gromadzki$^{16}$, Aleksandra Hamanowicz$^{16}$, S. Bradley Cenko$^{17,18}$, David J. Sand$^{19}$, Eric Y. Hsiao$^{20}$, Mark M. Phillips$^{21}$, Tiara R. Diamond$^{17}$, Erin Kara$^{17,18}$, Keith C. Gendreau$^{17}$, Zaven Arzoumanian$^{17}$, Ron Remillard$^{22}$'
title: A new class of flares from accreting supermassive black holes
---
Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland.
School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel.
Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA.
Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA.
Núcleo de Astronomía de la Facultad de Ingeniería, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago, Chile
Chinese Academy of Sciences South America Center for Astronomy and China-Chile Joint Center for Astronomy, Camino El Observatorio 1515, Las Condes, Santiago, Chile.
Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China.
Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138, USA.
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, MS 169-224, Pasadena, CA 91109, USA.
SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands.
Department of Astrophysics / Institute for Mathematics, Astrophysics and Particle Physics, Radboud University, P.O. Box 9010, 6500GL Nijmegen, The Netherlands.
Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel.
European Southern Observatory, Casilla 19001, Santiago 19, Chile
Cahill Center for Astronomy and Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA.
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK.
Warsaw University Astronomical Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland.
Astrophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA.
Joint Space-Science Institute, University of Maryland, College Park, MD 20742, USA.
Department of Astronomy and Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA.
Department of Physics, Florida State University, 77 Chieftan Way, Tallahassee, FL 32306, USA.
Carnegie Observatories, Las Campanas Observatory, Casilla 601, La Serena, Chile.
MIT Kavli Institute for Astrophysics and Space Research, 70 Vassar Street, Cambridge, MA 02139, USA.
#### {#sec:intro .unnumbered}
#### {#sec:obj_and_obs .unnumbered}
[AT2017bgt]{} was discovered by the All Sky Automated Survey for Supernovae (ASAS-SN[@Shappee2014_ASASSN_N2617]) as ASASSN-17cv on 2017 February 21 in the early-type galaxy 2MASX J16110570+0234002, at [z[=]{}0.064 $z{=}0.064$]{} (ref. ; see Methods §\[SM\_sec\_obs\_img\]). The long-term ASAS-SN optical data show that the total emission from the galaxy brightened by a factor of $\sim$50% over a period of about two months, with half of the rise occurring within three weeks (see Supplementary Fig. 1). Follow-up [*Swift*]{} observations show that the UV emission increased by a factor of $\sim$75 compared to [*GALEX*]{} data from 2004, reaching a luminosity of $\nu L_\nu ({\rm NUV}) \simeq 8.9{\times}10^{44}\,{\ifmmode {\rm erg\,s}^{-1} \else erg s$^{-1}$\fi}$, and that the X-ray emission increased by a factor of ${\sim}2{-}3$ compared to [*ROSAT*]{} data from 1990 August (see ‘Detection and photometric monitoring’ and ’Archival multi-wavelength data’ in Methods for details on all new and archival data). The archival X-ray luminosity, of $L(2{-}10\,{\ifmmode {\rm keV} \else keV\fi}) \simeq 7{\times}10^{42}\,{\ifmmode {\rm erg\,s}^{-1} \else erg s$^{-1}$\fi}$, and the archival UV to X-ray luminosity ratio are consistent with what is commonly observed in AGN (that is, a UV-to-X-ray spectral slope of $\alpha_{\rm ox} {\approx} -1.2$; see Methods §\[SM\_sec\_mw\_data\]). Archival detections in the radio (from 1998; obtained by the Very Large Array) and in the mid-infrared (from 2010; obtained with the [*Wide-field Infrared Survey Explorer*]{}) can be accounted for by star formation in the host galaxy. Thus, [AT2017bgt]{} experienced a dramatic increase in its UV emission, accompanied by a smaller increase in optical and X-ray emission, sometime between 2004 and 2017.
The X-ray spectral energy distribution of [AT2017bgt]{}, as determined from our new observations over the energy range of 0.3-24[ keV]{}, using /XRT, , and , is broadly consistent with an unobscured AGN, dominated by a power-law with a photon index of $\Gamma{\simeq}1.9$ (see ’New X-ray data’ in Methods and Supplementary Fig. 2). During the current UV-bright state, [AT2017bgt]{} exhibits UV-to-X-ray and UV-to-optical luminosity ratios that are much larger than what is typically seen in unobscured AGN. The ratio of monochromatic UV-to-X-ray luminosities, $L_{\nu}(2500\,{\rm \AA})/L_{\nu}(2\,{\ifmmode {\rm keV} \else keV\fi}) \approx 6{\times}10^4$, is higher than the norm by a factor of at least $\sim$50 (i.e., $\alpha_{\rm ox}{\simeq}-1.9$; see ref. ). The corresponding UV-to-optical ratio is $\gtrsim$3.5, which is higher than the norm by a factor of at least $\sim$5.5 (ref. ). Over the first [14]{} months after discovery the UV and optical flux of [AT2017bgt]{} have shown very limited variability (Fig. \[fig:img\_spec\_monitoring\]), with a mild decline of $\lesssim$0.7 mag (factor of 2) in the UV, and $\lesssim$0.2 mag in the optical (without removal of an unknown amount of host contamination). The X-ray emission from [AT2017bgt]{} has been roughly constant during this period (Fig. \[fig:img\_spec\_monitoring\]a).
We obtained repeated optical spectroscopy of [AT2017bgt]{}, starting two days after discovery. The spectra display many features typical of unobscured AGN, but also some features that have never been clearly identified in such systems before (see ’Optical spectroscopy’ in Methods and Fig. \[fig:opt\_spec\_comp\_SDSS\]). Among the features typical of AGN, the spectra exhibit prominent, symmetrical, single-peaked hydrogen Balmer emission lines, with full-width at half-maximum of $\fwhm {\approx} 2000\,{\ifmmode {\rm km\,s}^{-1} \else km\,s$^{-1}$\fi}$. In unobscured, broad-line AGN (i.e., quasars), such emission lines are thought to originate from partially ionized gas with densities of order $10^{10}\,{\ifmmode {\rm cm}^{-3} \else cm$^{-3}$\fi}$. [AT2017bgt]{} also exhibits weaker and narrower, ${\sim}500\,{\ifmmode {\rm km\,s}^{-1} \else km\,s$^{-1}$\fi}$-wide, forbidden emission lines of \[O[iii]{}\]$\lambda\lambda4959,5007$ and \[N [ii]{}\]$\lambda\lambda6548,6584$, which are also common in AGN. Indeed, these forbidden narrow lines are already present in the earliest optical spectra, obtained within days from the transient detection, and indicate the presence of an AGN-like ionizing source (see Supplementary Figs. 3 and 4, and ’Optical spectroscopy’ in Methods). Given that the light-travel time to the corresponding (narrow) line-emitting region is of order $\gg$100 years[@Bennert2002_NLR_RL; @Mor2009], and the fact that the archival UV and X-ray data are consistent with AGN-like continuum emission, the system was most likely harbouring an actively accreting supermassive black hole (SMBH) well before the optical flux increase that triggered the transient detection.
Assuming that the optical and X-ray continuum emission from [AT2017bgt]{} indeed originates from the vicinity of a standard accretion flow onto a SMBH, as in AGN, and using standard AGN scaling relations, we infer a mass accretion rate of $\dot{M} {\sim} 0.04{-}0.11 \,{\ifmmode {\ifmmode M_{\odot} \else $M_{\odot}$\fi}\,{\rm yr}^{-1} \else ${\ifmmode M_{\odot} \else $M_{\odot}$\fi}\,{\rm yr}^{-1}$\fi}$; a broad line region (BLR) size of ${\hbox{$ {R_{\rm BLR}} $}}{\approx}20{-}30$ light-days (${\sim}2\times10^{-2}$ pc); and, combined with an line width of $\fwhb{\simeq}2,050,{\ifmmode {\rm km\,s}^{-1} \else km\,s$^{-1}$\fi}$, a SMBH mass of $\mbh {\approx} 1.8{\times}10^7\,{{\ifmmode M_{\odot} \else $M_{\odot}$\fi}}$, which in turn results in an Eddington ratio of $\lledd {\sim} 0.08{-}0.21$ (see ’Determination of key SMBH properties’ in Methods for details and Supplementary Fig. 5; key measured and derived properties are listed in Supplementary Table 1). The intense UV luminosity, on the other hand, could imply $\lledd {\sim} 1.4$, and further suggests that the line-emitting region does not necessarily follow standard AGN scaling relations (see ’Determination of key SMBH properties’ in Methods for details). Given the width of the Balmer lines and the high accretion rates suggested by the UV-based rough estimates of $\dot{M}$ and , the currently observed AGN-related properties of [AT2017bgt]{} are consistent with those of many narrow-line Seyfert 1 (NLSy1) galaxies.
Most importantly, the optical spectroscopy of [AT2017bgt]{} presents several strong emission features that are [*not*]{} seen in AGN (Fig. \[fig:opt\_spec\_comp\_SDSS\]), namely the strong emission line and the double-peaked emission feature near 4680Å, all with widths consistent with that of the broad component of the emission line. The redder of the two peaks near 4680 Å coincides with the transition, which in AGN typically exhibits line intensity ratios relative to of $F(\heii)/F(\hbeta) \leq 0.05$, but here is seen with $F(\heii)/F(\hbeta) \approx 0.5$ (see ref. ). The width and intensity of these spectral features remain roughly constant during our follow-up spectroscopic observations spanning over [14]{} months from discovery (Fig. \[fig:img\_spec\_monitoring\]c and Supplementary Fig. 3).
The second feature, centred at 4651.6 Å, cannot be associated with a second peak of emission, originating from a disk-like configuration of the -emitting gas, since the other optical and near-IR (NIR) emission lines, including several transitions, are single-peaked (the NIR helium lines are also exceptionally strong compared to typical AGN; see ’NIR spectroscopy’ in Methods and Supplementary Fig. 6). This feature may instead be associated with the emission line. Some recent studies have suggested that weak emission from this transition (sometimes noted as a “Wolf-Rayet” feature) may be present in the spectra of some TDEs[@Gezari2015_PS1_10jh; @Brown2018_iPTF16fnl; @Brown2017_AS14li_longterm], although significant line blending in those cases makes the identification less secure there. While the line is not seen in AGN spectra (Fig. \[fig:opt\_spec\_comp\_SDSS\]), it can be significantly enhanced by Bowen fluorescence, which would also produce and other O[iii]{} lines, as seen in our spectra. In Bowen fluorescence (BF), photons emitted from the Lyman$\alpha$-like transition of , at 303.783 Å, excite certain states of O[iii]{} and , due to the wavelength proximity between the corresponding energy transitions[@Bowen1928_BF; @WeymannWilliams1969_BF]. The excited O[iii]{} and states lead to a cascade of transitions, that may be observed as emission lines in the UV-optical regime. This process, and particularly strong , , and emission, is well established to occur in some planetary nebulae and X-ray binaries[@Schachter1989_BF_ScoX1; @KastnerBhatia1996_BF_compilation]. It is, however, generally not seen in AGN, and only a few Seyfert galaxies were reported to have narrow ($<$1,000 [\^[-1]{} kms$^{-1}$]{}-wide) BF lines[@WilliamsWeymann1968_BF; @Schachter1990_Bowen_NLR]. Here we present a case where BF emission lines from the BLR are associated with a (steadily) accreting SMBH, and specifically robust identification of broad O<span style="font-variant:small-caps;">iii</span> BF lines, and of the line (and indeed the BF cascade) in such a system, which were first predicted decades ago[@Netzer1985_HeII].
Comparing the $F(\heii)/F(\hbeta)$ and $F(\niii)/F(\heii)$ line intensity ratios seen in [AT2017bgt]{} to those predictions[@Netzer1985_HeII] suggests that the line-emitting gas is dense, with hydrogen number density of $n_{\rm H}\gtrsim 10 ^{9.5}\,{\ifmmode {\rm cm}^{-3} \else cm$^{-3}$\fi}$, and has a high abundance of metals – and in particular of nitrogen (likely exceeding 4 times solar). These gas densities are consistent with what is expected for the BLR in AGN, and high metallicities may be expected in extremely high luminosity and/or high AGN[@Shemmer2004]. Thus, [AT2017bgt]{} suggests that the key missing ingredient for broad BF lines in accreting SMBHs is extremely intense UV continuum emission. However, more detailed radiative transfer calculations are required to reproduce the line ratios seen in [AT2017bgt]{}, and to link them to the enhanced UV emission.
#### {#sec:discussion .unnumbered}
[AT2017bgt]{} joins two recently reported transient events in galaxy centres that also exhibit a prominent, broad and double-peaked emission feature near 4680 Å (Fig. \[fig:opt\_spec\_comp\_other\]). The first event was recently claimed to be a TDE in the active nucleus of the ultra-luminous infrared galaxy F01004-2237 (ref. ). The classification of this event as a TDE mainly relied on the association of this spectral feature with , a line also seen in a class of optical and UV-bright TDEs[@Gezari2012_TDE; @Arcavi2014_TDEs_He], though the line profile in F01004-2237 is different from that of known TDEs (see below). Another recently reported event, OGLE17aaj, shows hints of a persistent double-peaked emission feature on top of an otherwise normal optical AGN spectrum[@Wyrzykowski2014_OGLE_IV; @Gromadzki2017_OGLEaaj_Atel; @Wyrzykowski2017_OGLE_rep_Sept17] (M.G. et al., manuscript in preparation). In both these cases, the dramatic increase in optical continuum emission was followed by a period of rather persistent continuum and line emission, over time-scales of several months – similarly to [AT2017bgt]{}. Moreover, the width of the Balmer emission lines in all three sources classifies them as NLSy1 AGN, which are commonly thought to be powered by highly accreting SMBHs (in terms of ). Two other recently reported nuclear transients showed slowly-evolving light curves: PS16dtm, which was claimed to be a TDE in a NLSy1 AGN[@Blanchard2017_PS16dtm]; and PS1-10adi, which was interpreted as a likely peculiar kind of supernova[@Kankare2017_PS10adi]. However, these two events showed no BF-related lines in their spectra. Moreover, PS1-10adi (and similar events) show a significant flux decrease over periods of a few months, not seen in [AT2017bgt]{}. We therefore consider these two latter events to be unrelated to the new class identified here, that is: unobscured AGN-like spectra with extremely strong UV, , , and emission, and long-term persistence of these flare features. We caution that, given the data in hand, we cannot rule out the possibility that these events may all be driven by rather similar SMBH fueling mechanisms. Determining why some events show BF features and some do not requires more detailed modelling.
We propose that [AT2017bgt]{}, and likely similar events, are “rejuvenated” SMBHs that experienced a sudden increase in their UV-optical emission. This, in turn, enhanced the Ly$\alpha$ line and initiated the BF cascades of , , and O[iii]{}. The extreme UV and BF emission differentiates such events from “changing look” AGN. Given the likely light-travel timescales to the region emitting the broad Balmer lines (of order a few weeks), and the fact that they are seen in the earliest optical spectra we obtained, it is most likely that the BF emission features originate in a pre-existing BLR which was suddenly exposed to the intense ionizing UV emission whose spectral energy distribution is very different from those of normal AGN. It is therefore possible that, once the intense UV continuum emission would settle back to “normal AGN” levels, the BF features would disappear. This can be tested with on-going spectroscopic monitoring. Alternatively, the BF features may be related to a newly launched outflow, perhaps driven by the sudden increase in accretion rate (see ’Relevant mechanisms for the long-lived UV flare’ in Methods). In either case, the enhanced metallicity suggested by the strong line may or may not be related to the fast increase in UV luminosity.
The nature of the sudden UV and optical brightening event remains open. The evidence for AGN-like activity in [AT2017bgt]{} (and the other events in this proposed class), both before and after the UV-optical event, draws attention to processes related to (thin) accretion disks that feed SMBHs. Given the very little time evolution of the continuum and line emission in [AT2017bgt]{}-like events, as well as their spectral properties, they are unlikely to be be driven by TDEs (at least not ones similar to those reported thus far; see Fig. \[fig:heii\_spec\_comp\_tdes\]). We discuss the relevance of the TDE interpretation, and mention several other possibly relevant mechanisms, in the Methods (’Relevant mechanisms for the long-lived UV flare’). Regardless of the nature of the dramatic increase in UV flux, the properties of [AT2017bgt]{}, and other similar transient events in galaxy nuclei, demonstrate that prominent, a broad Bowen fluorescence emission feature may be confused with the feature commonly seen in TDEs. Such sources thus highlight the importance of long-term, multi-wavelength monitoring campaigns, in order to identify SMBH accretion that is driven purely by TDEs, and to distinguish it from other cases of SMBHs that experience a sudden enhancement, or indeed “re-ignition”, of their accretion. We suspect that the discovery of [AT2017bgt]{}-like events might have been hampered so far, due to their possible association with previously-known AGN (see discussion in ref. ). Long term monitoring of [AT2017bgt]{}-like events, and indeed the discovery of additional events of this sort, could help illuminate their nature, as well as their role in SMBH growth.
#### References
[35]{}
url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
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B.T. is a Zwicky Fellow. I.A. is an Einstein fellow. E.K. is a Hubble Fellow. We thank N. Caplar, J. Guillochon, Z. Haiman, E. Lusso, and K. Schawinski for useful discussions. We thank C. Tadhunter for providing the spectrum of the F01004-2237 transient and his helpful comments. Part of this work was inspired by discussions within International Team \#371, “Using Tidal Disruption Events to Study Super-Massive Black Holes”, hosted at the International Space Science Institute in Bern, Switzerland. We thank all the participants of the team meeting for their beneficial comments. Support for I.A. was provided by NASA through the Einstein Fellowship Program, grant PF6-170148. C.R. acknowledges support from the CONICYT+PAI Convocatoria Nacional subvencion a instalacion en la academia convocatoria a [n]{}o 2017 PAI77170080. P.G.J. acknowledges support from European Research Council Consolidator Grant 647208. A. Horesh acknowledges support by the I-Core Program of the Planning and Budgeting Committee and the Israel Science Foundation. G.H., D.A.H., and C.M. acknowledge support from NSF grant AST-1313484. M.B. acknowledges support from the Black Hole Initiative at Harvard University, which is funded by a grant from the John Templeton Foundation. G.B.L. acknowledges support from a Herchel Smith Research Fellowship of the University of Cambridge. [Ł]{}.W., M.G. and A.Hamanowicz acknowledge Polish National Science Centre grant OPUS no 2015/17/B/ST9/03167 to [Ł]{}.W. Research by D.J.S. is supported by NSF grants AST-1412504 and AST-1517649. E.Y.H. acknowledges the support provided by the National Science Foundation under Grant No. AST-1613472 and by the Florida Space Grant Consortium. This work makes use of observations from the Las Cumbres Observatory network. This publication also makes use of data products from the Wide-field Infrared Survey Explorer. [*WISE*]{} and [*NEOWISE*]{} are funded by the National Aeronautics and Space Administration.\
This work made use of data from the mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by the National Aeronautics and Space Administration. We thank the Operations, Software and Calibration teams for support with the execution and analysis of these observations. This research made use of the Data Analysis Software ([*NuSTAR*]{}DAS) jointly developed by the ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (USA).\
We thank the [*Swift*]{}, , and teams for scheduling and performing the target-of-opportunity observations presented here on short notice. The LRIS spectrum presented herein was obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. We recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.\
These results made use of the Discovery Channel Telescope at Lowell Observatory. Lowell is a private, non-profit institution dedicated to astrophysical research and public appreciation of astronomy and operates the DCT in partnership with Boston University, the University of Maryland, the University of Toledo, Northern Arizona University and Yale University. The upgrade of the DeVeny optical spectrograph has been funded by a generous grant from John and Ginger Giovale.\
The FLAMINGOS-2 spectrum was obtained at the Gemini Observatory under program GS-2017A-Q-33 (PI: Sand), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovaç [a]{}o (Brazil).
B.T. and I.A. led the data collection, analysis and interpretation, as well as the manuscript preparation. C.R. performed the analysis and modelling of archival and new X-ray data. S.T. performed the morphological and SED modelling of the host galaxy. D.S., M.B., M.H., N.K., and G.B.L. took part in obtaining and calibrating the Palomar and Keck spectra. H.N. contributed to the identification and interpretation of the Bowen fluorescence spectral features. P.G.J. and A. Horesh contributed to the interpretation of multi-wavelength data and to pursuing follow-up observations. J.E.M-R. contributed to the analysis of optical spectra. G.H., V.H., and C.M. contributed to collecting, calibrating and analysing the Las Cumbres Observatory and [*Swift*]{}/UVOT data. D.A.H. helped schedule and monitor the data from the Las Cumbres Observatory. [Ł]{}.W., M.G., and A. Hamanowicz contributed to NIR line identification and provided the optical spectrum of OGLE17aaj. S.B.C. provided the the DCT spectrum. D.S. provided the the Gemini-South/FLAMINGOS-2 NIR spectrum. E.Y.H., M.M.P., and T.D.D. provided the Magellan/FIRE NIR spectrum. E.K. contributed to the X-ray data analysis and interpretation. K.C.G., Z.A., and R.R. contributed to the data acquisition and calibration.
[**Supplementary information**]{} is available for this paper.\
[**Correspondence and requests for materials**]{} should be addressed to B.T. (email: benny@astro.tau.ac.il).
The authors declare that they have no competing financial interests.
{width="45.00000%"}
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{width="87.50000%"}
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#### Methods {#sec:methods}
Detection and Photometric Monitoring Observations {#SM_sec_obs_img .unnumbered}
=================================================
The source [AT2017bgt]{} (J2000.0 coordinates $\alpha=$ 16:11:05.696, $\delta=$ +02:34:00.52) was first detected by the ASAS-SN survey, using the ASASSN-Brutus telescope, on 2017 February 21, 15:07:12 (${\rm JD}=2457806.13$; see refs. ). The nuclear source was discovered at a host-galaxy-subtracted $V$-band Vega magnitude of $17.2 \pm 0.1$. The previous 5-$\sigma$ non-detection from February 13 was at a magnitude of 17.5.\
Supplementary Fig. 1 presents the long-term ASAS-SN $V$-band light-curve of the position of [AT2017bgt]{}, obtained using the ASAS-SN Light-Curve Server[@Shappee2014_ASASSN_N2617; @Kochanek2017_ASASSN_LC]. We note that the ASAS-SN $V$-band flux measurements are derived from apertures and calibration procedures that are significantly different from the /UVOT measurements presented in Fig. \[fig:img\_spec\_monitoring\] (and described below). In particular, the ASAS-SN Light-Curve Server uses a large aperture, with a radius of 16(compared to the 3.75 radius used for our new /UVOT data; see below). As such, they mainly illustrate the transient nature of [AT2017bgt]{}, and can otherwise provide only combined flux measurements of the nuclear source and its host galaxy (with possible additional blending from nearby sources).\
Evidently, a consistent trend of brightening began over one month prior to the detection of [AT2017bgt]{}. The automated ASAS-SN data suggest that the pre-event ASAS-SN $V$-band magnitude of the host galaxy was 15.85, from which we deduce that the nuclear event itself represented a brightening by a (peak) flux density corresponding to 16.7 magnitudes, or by $\nu L_\nu ({\rm opt.})\simeq 4.2\times10^{43}\,{\ifmmode {\rm erg\,s}^{-1} \else erg s$^{-1}$\fi}$. After reaching peak brightness, the transient began fading at a slow rate, dimming by roughly 0.3 mag at about 200 days after discovery (in the rest-frame). Later photometry, spanning $\sim$300-400 days after discovery, reveals brightness levels that are consistent with the peak. This limited evidence for any significant post-flare flux variability is consistent with what is seen in the /UVOT photometric monitoring observations (Fig. \[fig:img\_spec\_monitoring\]).
Shortly after detection, we initiated a monitoring campaign of [AT2017bgt]{} using the UV Optical Telescope[@Roming2005_Swift_UVOT] on board the Neil Gehrels observatory[@Gehrels2004_Swift], which covers the wavelength range $\lambda\sim1600-6000$ Å. The most relevant band for comparison with the archival [*GALEX*]{} data (see below) is UVM2, with $\lambda_{\rm eff}=2228$ Å. [AT2017bgt]{} was initially observed over 12 epochs, separated by 5.5-43 days. Four additional visits, separated by about a month, took place roughly a year after the transient detection (see Fig. \[fig:img\_spec\_monitoring\]b). For each of these visits, we measured the flux in all bands through circular apertures with a diameter of 7.5, to match the archival [*GALEX*]{} data (see below). We subtracted the sky flux using the same size aperture from an empty region near the transient. For host-galaxy subtraction, we assumed the archival [*GALEX*]{} flux measurement, discussed below. A more up-to-date measurement of the host contribution will only be available once the transient fades completely. Intrinsic UV luminosities were calculated by taking into account a Milky Way extinction with a colour excess of $E(B-V)=0.069$ and assuming a Cardelli et al. extinction law (ref. ; see also ref. ). This resulted in a correction of 0.6 magnitudes (or a factor of 1.74). We note that the Milky Way dust correction, and the host light subtraction were not applied to the UV-optical data shown in Fig. \[fig:img\_spec\_monitoring\], which instead shows the raw measured fluxes of [AT2017bgt]{}. Similarly to what is seen in the optical light-curve, the UV emission fades very slowly over the [14]{} months of our /UVOT monitoring, from about 14.9 to 15.6 AB magnitudes (i.e., about 50% drop in flux). After correcting for Milky Way extinction, this corresponds to a drop of $4.12\times10^{44}\,{\ifmmode {\rm erg\,s}^{-1} \else erg s$^{-1}$\fi}$ in luminosity, from $\nu L_\nu ({\rm NUV}){=}8.85{\times}10^{44}$ to $4.73{\times}10^{44}\,{\ifmmode {\rm erg\,s}^{-1} \else erg s$^{-1}$\fi}$. The fading seen in both the UV and optical emission is much slower than what is typically seen in TDEs. The only way in which the $\sim$0.7 magnitude decline in the UV light can be accounted for with a TDE light-curve of the form $L\propto(t-t_{\rm D})^{-5/3}$, is if the disruption time $t_{\rm D}$ was about two years before the discovery of [AT2017bgt]{} (observed frame, thus about 23 months in rest-frame), and is therefore inconsistent with the data.
Archival multi-wavelength data {#SM_sec_mw_data .unnumbered}
==============================
The host galaxy of [AT2017bgt]{} is clearly detected in over 9 separate visits of the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS), in all the 5 filters ($grizy$; with 9, 18, 26, 11, and 13 detections in each of these filters). The stacked images, which have “mean epochs” ranging 2011 August 17 to 2013 June 8, and are publicly available through the Pan-STARRS database, reveal an early-type host galaxy, with no obvious sign of a central point source.
A full `GALFIT` analysis[@Peng2010_GALFIT_v3] of the stacked $r$-band image, which has a mean epoch of 2011 August 17 (i.e., about 5.5 years before the transient detection), suggests a bulge-dominated morphology. For a single Sérsic fit, we find a Sérsic index of $n=5.0^{+0.9}_{-0.6}$. The uncertainties are estimated by varying the centre of the light profile within a box of 2$\times$2 pixels, following the approach described in ref. . Performing a bulge-disk decomposition with a fixed disk Sérsic index of $n_{\rm d}=1.0$ (pure exponential profile) gives a bulge-to-total ratio of $B/T=0.65^{+0.05}_{-0.06}$ with a bulge Sérsic index of $n_{\rm b}=3.7^{+0.8}_{-0.5}$. We further used `GALFIT` to assess the possible presence of a central point source. Integrating over the residuals of the disk+bulge fit provides an $r$-band magnitude of $r_{\rm AB}=20.4^{+0.6}_{-0.5}$. Alternatively, adding a central point source component to the `GALFIT` model results in $r_{\rm AB}=19.75^{+0.3}_{-0.4}$, again much fainter than the bulge and disk components, which now have $r_{\rm AB}=15.8$ and $16.9$, respectively (and $B/T=0.73^{+0.06}_{-0.08}$). We conclude that the publicly available archival Pan-STARRS data shows no indication of a central point source.
[AT2017bgt]{} is associated with the [*ROSAT*]{} source 1RXS J161105.2+023350. The archival X-ray detection was obtained as part of the [*ROSAT*]{} All Sky Survey (RASS)[@Voges2000_RASS], on 1990 August 9, when the source had a count rate of $(4.1\pm1.2)\times 10^{-2}\, {\rm ct\,s}^{-1}$. The [*ROSAT*]{} count rate can be converted into an observed $0.1-2.4$ [ keV]{} flux assuming that the photon index of the X-ray emission is $\Gamma=2$ within this band. This implies that the observed flux is $(5.4 \pm 1.5) \times 10^{-13}\,{\ifmmode {\rm erg\,cm}^{-2}\,{\rm s}^{-1} \else erg\,cm$^{-2}$\,s$^{-1}$\fi}$. The hydrogen line-of-sight column density of the Milky Way in the direction of the source is $5.34\times 10^{20}\,{\ifmmode {\rm cm}^{-2} \else cm$^{-2}$\fi}$, which provides an intrinsic $0.1-2.4\,{\ifmmode {\rm keV} \else keV\fi}$ flux of $F(0.1{-}2.4\,{\ifmmode {\rm keV} \else keV\fi}) {=} (1.1\pm 0.3) \times 10^{-12}\,{\ifmmode {\rm erg\,cm}^{-2}\,{\rm s}^{-1} \else erg\,cm$^{-2}$\,s$^{-1}$\fi}$. The corresponding archival intrinsic luminosity in the same band is $L(0.1{-}2.4\,{\ifmmode {\rm keV} \else keV\fi}){\approx} 10^{43}\,{\ifmmode {\rm erg\,s}^{-1} \else erg s$^{-1}$\fi}$, and in the 2–10keV range is $L(2{-}10\,{\ifmmode {\rm keV} \else keV\fi}){\simeq} 5.3{\times}10^{42}\,{\ifmmode {\rm erg\,s}^{-1} \else erg s$^{-1}$\fi}$ (all luminosities and related quantities were calculated assuming a cosmological model with dark energy and matter densities of $\Omega_{\Lambda}=0.7$ and $\Omega_{\rm M}=0.3$, and a Hubble constant of $H_{0}=70\,{\ifmmode {\rm km\,s}^{-1} \else km\,s$^{-1}$\fi}\,{\rm Mpc}^{-1}$). Considering the relation between $0.5-2\,{\ifmmode {\rm keV} \else keV\fi}$ emission and star formation[@Ranalli2003_Lx_SFR], a star formation rate of $\simeq 840\,{\ifmmode {\ifmmode M_{\odot} \else $M_{\odot}$\fi}\,{\rm yr}^{-1} \else ${\ifmmode M_{\odot} \else $M_{\odot}$\fi}\,{\rm yr}^{-1}$\fi}$ would be required to fully account for the X-ray emission in [AT2017bgt]{}. This extremely high value, which is two orders of magnitude larger than what we deduce from the archival data for the host galaxy of [AT2017bgt]{} (see below), means that [*ROSAT*]{} detected an AGN in this object.
The position of [AT2017bgt]{} was observed in the 1.4 GHz band, as part of the FIRST radio survey, on 1998 July 17. The reported peak and integrated flux densities are $S_{\rm p} = 1.22\pm0.15$ mJy, and $S_{\rm int} = 0.99\pm0.15$ mJy (ref. ). This translates to a monochromatic radio luminosity of $\nu L_\nu({\rm 1.4\,GHz}) {\simeq} 1.7 \times 10^{38}\,{\ifmmode {\rm erg\,s}^{-1} \else erg s$^{-1}$\fi}$ Such a low radio luminosity can originate purely from SF, with a corresponding star formation rate of about ${\rm SFR}{\simeq}7\,{\ifmmode {\ifmmode M_{\odot} \else $M_{\odot}$\fi}\,{\rm yr}^{-1} \else ${\ifmmode M_{\odot} \else $M_{\odot}$\fi}\,{\rm yr}^{-1}$\fi}$ (ref. ), which is consistent with the UV-based SFR estimate derived from archival [*GALEX*]{} data (see below). Indeed, the large majority of sources with such low radio luminosities are star-forming galaxies[@HeckmanBest2014_ARAA; @Padovani2016_radio_rev]. However, the aforementioned archival X-ray detection suggests that this may not be the case.
The area around [AT2017bgt]{} was also observed in the UV as part of the [*GALEX*]{} All-sky Imaging (AIS) survey, on 2004 May 17. The catalogued measured brightness of the corresponding source (GALEX J161105.7$+$023359) is $m_{\rm AB}({\rm NUV})=19.63 \pm 0.10$ in the NUV band ($\lambda_{\rm eff}=2274.4$Å), and $m_{\rm AB}({\rm FUV})= 19.60 \pm 0.18$ in the FUV band ($\lambda_{\rm eff}=1542.3$Å). These are measured through a circular aperture with a diameter of 7.5, which is preferable since it can be consistently applied to other imaging data for this source. After correcting for Milky-Way extinction, the NUV brightness translates to a luminosity density of $L_\nu({\rm NUV}) = 8.8\times 10^{27}\,{\rm erg\,s}^{-1}\,{\rm Hz}^{-1}$. Associating this luminosity with star formation in the host galaxy would translate to a star formation rate of ${\rm SFR}\simeq1.2\,{\ifmmode {\ifmmode M_{\odot} \else $M_{\odot}$\fi}\,{\rm yr}^{-1} \else ${\ifmmode M_{\odot} \else $M_{\odot}$\fi}\,{\rm yr}^{-1}$\fi}$ (for a Salpeter initial mass function; ref. ), which is within the range expected for a galaxy with the stellar mass of the host (see below; ref. ). Alternatively, the UV continuum emission observed with [*GALEX*]{} can also be interpreted as originating from accretion onto the SMBH (i.e., an AGN). Assuming no variability between the [*ROSAT*]{} and [*GALEX*]{} observations, we can calculate a UV-to-X-ray spectral slope, commonly defined through $F_\nu \propto \nu^{\alpha}$: $$\alpha_{\rm ox} \equiv \frac{\log(F_\nu[2\,{\ifmmode {\rm keV} \else keV\fi}]/F_\nu[2500\,{\rm \AA}])}{\log(\nu[2\,{\ifmmode {\rm keV} \else keV\fi}]/\nu[2500\,{\rm \AA}])} \approx -1.22 .
\label{SM_eq_aox}$$ This value is in excellent agreement with what is expected from AGN with similar (UV) luminosities[@Just2007_Xray_hiL; @Lusso2016_Lx_Luv].
Combining the [*GALEX*]{} data with the Pan-STARRS stacked optical images and 2MASS near-IR images, we obtain a UV-optical-near-IR spectral energy distribution (SED) of the host galaxy. All fluxes have been measured with the same aperture. The SED is fit assuming the stellar population synthesis templates of ref. , and an exponentially declining star formation history ($\mathrm{SFR}\propto\exp(-t/\tau)$, where $\tau$ is a model parameter denoting a typical timescale). This resulted in a stellar mass of $\mstar=2.7\times10^{10}\,{{\ifmmode M_{\odot} \else $M_{\odot}$\fi}}$, a star formation rate of ${\rm SFR}=4.0\,{\ifmmode {\ifmmode M_{\odot} \else $M_{\odot}$\fi}\,{\rm yr}^{-1} \else ${\ifmmode M_{\odot} \else $M_{\odot}$\fi}\,{\rm yr}^{-1}$\fi}$, and a dust attenuation in the $V$-band of $\mathrm{A}_{\rm V}=0.5$. Since the stellar mass does not depend on the UV-blue emission, we obtain an essentially identical mass estimate when refitting the SED but ignoring the [*GALEX*]{} data (i.e., if the archival UV detection is due to SMBH activity). However, the corresponding SFR drops to ${\rm SFR}\simeq0.02\,{\ifmmode {\ifmmode M_{\odot} \else $M_{\odot}$\fi}\,{\rm yr}^{-1} \else ${\ifmmode M_{\odot} \else $M_{\odot}$\fi}\,{\rm yr}^{-1}$\fi}$. We finally note that the area around [AT2017bgt]{} was observed with the [*The Wide-field Infrared Survey Explorer (WISE)*]{}[@Wright2010_WISE], during 2010 February and August. The host galaxy is clearly detected in multiple scans, with $w(1-4)$ Vega magnitudes of (12.283,11.857,8.881,6.511), and no clear evidence for variability. These suggest mid-IR colours that are consistent with star-forming galaxies[@Wright2010_WISE; @Stern2012_MIR_AGN_WISE].
New X-ray data {#SM_sec_Xrays .unnumbered}
==============
[**[*Swift*]{}/XRT observations and data.**]{} Our monitoring campaign of [AT2017bgt]{} provided X-ray measurements with the X-Ray Telescope (XRT; ref. ), covering the energy range $0.3-7\,{\ifmmode {\rm keV} \else keV\fi}$. The individual effective exposure times ranged between ${\sim}870{-}2400$ s. The [*Swift*]{}/XRT data obtained in each of the 16 visits was analysed using the <span style="font-variant:small-caps;">xrtpipelinev0.13.4</span> within <span style="font-variant:small-caps;">heasoftv6.24</span> following standard guidelines, to provide a time-series of observed X-ray fluxes and SEDs.
The XRT SEDs were fitted with a simple power-law model. This resulted in photon indices in the range $\Gamma {\simeq} 1.9-2.5$ – consistent with what is seen in low-redshift AGN[@Trakhtenbrot2017_BASS_GamX_LLedd]. The X-ray flux in the 0.1–2.4keV band measured in these /XRT observations, $F(0.1-2.4\,{\ifmmode {\rm keV} \else keV\fi}) {=} (1.02{-}1.66) \times 10^{-12}\,{\ifmmode {\rm erg\,cm}^{-2}\,{\rm s}^{-1} \else erg\,cm$^{-2}$\,s$^{-1}$\fi}$, is ${\approx} 2-3$ times brighter than that recorded by [*ROSAT*]{} in the same energy range ($5.39{\times} 10^{-13}\,{\ifmmode {\rm erg\,cm}^{-2}\,{\rm s}^{-1} \else erg\,cm$^{-2}$\,s$^{-1}$\fi}$), back in 1990. The new XRT measurements translate to (0.1–2.4) [ keV]{} luminosities of ${\approx} (1.0-1.6) {\times} 10^{43}\,{\ifmmode {\rm erg\,s}^{-1} \else erg s$^{-1}$\fi}$. By assuming power-law continuum X-ray emission, we obtain 2–10[ keV]{} luminosities in the range $L(2{-}10\,{\ifmmode {\rm keV} \else keV\fi}) {\approx} (3.5-15.5) {\times} 10^{42}\,{\ifmmode {\rm erg\,s}^{-1} \else erg s$^{-1}$\fi}$. We adopt a nominal value of $L(2{-}10\,{\ifmmode {\rm keV} \else keV\fi})=1.2{\times}10^{43}\,{\ifmmode {\rm erg\,s}^{-1} \else erg s$^{-1}$\fi}$, corresponding to the 2017 April 24 /XRT observation (see Supplementary Table 1). Combining the series of /XRT and UVOT measurements, we obtain UV-to-X-ray spectral slopes in the range $\alpha_{\rm ox}{\sim}(-2){-}(-1.7)$, with $\alpha_{\rm ox}=-1.8$ being both the value obtained for the April 24 observation, and the median value.
We have also constructed a stacked X-ray SED from the first 12 epochs of /XRT observations, corresponding to an effective exposure time of 20ks. While a model including a power-law and a blackbody component can adequately describe this stacked SED (resulting in a Cash statistic of $C-{\rm stat}=314.6$ for 318 degrees of freedom), it shows clear residuals around the FeK$\alpha$ line region. Adding a Gaussian component (’<span style="font-variant:small-caps;">zgauss</span>’ in `XSPEC`) with the redshift fixed to systemic ($z{=}0.064$) improves the fit ($C-{\rm stat}=306.5$ for 316 d.o.f.), and results in a photon index of $\Gamma{=}2.20{\pm}0.17$ and a blackbody temperature of $k_{\rm B}T {=} 112^{+2}_{-3}$eV. The emission line has an energy consistent with the FeK$\alpha$ fluorescent feature ($E{=}6.41^{+0.39}_{-0.15}$[ keV]{}), and is remarkably strong for an unobscured AGN, with an equivalent width of ${\rm EW}=900^{+494}_{-443}\,$eV. This is comparable with some of the strongest FeK$\alpha$ lines known to date, for example, the NLSy1 galaxies IRAS13224$-$3809 (${\rm EW}\simeq2.4\,{\ifmmode {\rm keV} \else keV\fi}$; ref. ) and 1H0707$-$495 (${\rm EW}{\simeq} 0.97\,{\ifmmode {\rm keV} \else keV\fi}$; ref. ), and the obscured AGN IRAS00521$-$7054 (${\rm EW}\simeq 0.80\,{\ifmmode {\rm keV} \else keV\fi}$; ref. ). We note, however, that the line strength significantly decreases in the higher-flux observations of [AT2017bgt]{}. Indeed, the emission line is not robustly detected in a stacked XRT SED that includes all epochs during which the flux was higher than the median, that is $F(2-10\rm\,keV){\gtrsim}10^{-12}\,{\ifmmode {\rm erg\,cm}^{-2}\,{\rm s}^{-1} \else erg\,cm$^{-2}$\,s$^{-1}$\fi}$. It is also not seen in the higher quality and much broader energy coverage X-ray spectrum we obtain using other facilities (see below).
[***NuSTAR* observations and data.**]{} We observed [AT2017bgt]{} in the 3-24[ keV]{} energy range using the Nuclear Spectroscopic Telescope Array (; ref. ), on 2018 June 25 (starting UT16:11:09), for a total of $\sim$44 ks. The [*NuSTAR*]{} data were processed using the [*NuSTAR*]{} Data Analysis Software <span style="font-variant:small-caps;">nustardas</span>v1.7.1 within <span style="font-variant:small-caps;">heasoft</span>v6.24, adopting the latest calibration files[@Madsen2015_NuSTAR_calib]. A circular region of 50was used for source extraction, while the background was extracted from an annulus centred on the X-ray source, with an inner and outer radii of 60 and 120, respectively.
[***NICER* observations and data.**]{} The Neutron star Interior Composition ExploreR (; ref. ), mounted aboard the International Space Station, observed [AT2017bgt]{} on 2018 June 25 (ObsIDs: 1100440108, 1100440109; starting UT15:53:07 and 23:35:46, respectively) for the entirety of the observation, but was heavily affected by high background rates, leaving 18 ks of Good Time Intervals. The data were processed using the standard data analysis software (’DAS’) v[2018-03-01\_V003]{} and were cleaned using standard calibration with ’[nicercal]{}’ and standard screening with ’[nimaketime]{}’. To filter out high background regions, we made a cut on magnetic cut-off rigidity ([COR\_SAX]{} $>$ 2) and selected events that were detected outside the SAA. We selected events that were not flagged as “overshoots” or “undershoots” (EVENT FLAGS$=$bxxxx00). We also omitted forced triggers. We required pointing directions to be at least 30 above the Earth limb and 40 above the bright Earth limb. The cleaned events use standard “trumpet” filtering to eliminate additional known background events (using the tool [nicermergeclean]{}). We estimated in-band background from the $13{-}15\,{\ifmmode {\rm keV} \else keV\fi}$ and trumpet-rejected count-rates, and used this to select the appropriate background model from observations of a blank field. robustly detected [AT2017bgt]{} above the background level in the energy range $0.3{-}2\,{\ifmmode {\rm keV} \else keV\fi}$.
[**Combined X-ray spectral analysis.**]{} The combined, quasi-simultaneous $0.3-24\,{\ifmmode {\rm keV} \else keV\fi}$ spectrum, including the and data of [AT2017bgt]{}, and a special short (400 s) /XRT observation taken on 2018 June 26 (UT11:13:14), is shown in Supplementary Fig. 2. We fitted this spectrum with a spectral model consisting of a primary power-law and a blackbody component to account for the soft excess[@GierlinskiDone2004_soft_excess; @Crummy2006_soft_excess], as well as Galactic absorption (i.e., <span style="font-variant:small-caps;">tbabs(zpo+zbb)</span> in `XSPEC`). This provides a good fit to the data, with $C$-Stat/$\chi^2{=}855$ for 918 degrees of freedom, and results in a blackbody temperature of $k_{\rm B} T {=} 145 {\pm} 4$ eV and a photon index of $\Gamma=1.94\pm0.07$. This slope is in very good agreement with that obtained by fitting the data alone ($\Gamma=1.95^{+0.09}_{-0.09}$). We also considered a cross-calibration constant ($C$) between the different instruments, which was fixed to 1 for Focal Plan Module A (FPMA), and was left free to vary for the other instruments. For /FPMB and /XRT we obtained $C=1.07^{+0.09}_{-0.08}$ and $C=1.03^{+0.38}_{-0.30}$, respectively, while for we found a larger value ($C=1.48_{-0.18}^{+0.21}$), possibly due, at least in part, to variability, since the /and observations were not completely overlapping. Both of these measurements are consistent with what is found in local AGN[@Winter2012_X_SEDs_WAs; @Ricci2017_BASS_Xray_cat]. We adopt the spectral parameters obtained from our fit to the combined, high-quality X-ray spectrum. We note that this combined spectrum shows no clear sign of a FeK$\alpha$ emission line, which is rather expected given that it was obtained when [AT2017bgt]{} was in a relatively high-flux state (see above).\
Optical spectroscopy. {#SM_sec_obs_spec .unnumbered}
=====================
[**Optical spectroscopic observations.**]{} Shortly after obtaining our classification spectrum, we initiated an intensive follow-up campaign using the Las Cumbres Observatory network of telescopes[@Brown2013_LCOGT]. The spectra were obtained using the twin FLOYDS spectrographs mounted on the Las Cumbres Observatory 2-meter telescopes in the Haleakala, Hawaii,USA, and Siding Spring, Australia, observatories. These spectra cover a wavelength range of $\lambda{=}3,200{-}10,000$ Å with a spectral resolution of $R {\equiv} \Delta\lambda/\lambda {=} 400{-}700$, and were obtained using 2slits and 45-minute-long observations. The spectra were reduced using the PyRAF-based `floydsspec` pipeline (<https://github.com/svalenti/FLOYDS_pipeline/>).
Additional, higher-resolution and higher-S/N optical spectra were obtained on several occasions during the first six months of monitoring [AT2017bgt]{} through the spring and summer of 2017. These include a March 3 spectrum taken with the DoubleSpec instrument on the Hale 200-inch telescope at the Palomar observatory (CA, USA); a March 16 spectrum taken with the DeVeny instrument on the Discovery Channel Telescope, at the Lowell observatory (AZ, USA); an April 28 spectrum taken with the LRIS instrument on the W. M. Keck telescope (HI, USA); and another Palomar/DoubleSpec spectrum taken on July 28. These spectra correspond to 10, 23, 66, and 157 days after the transient discovery, and are shown in Supplementary Fig. 3. All these optical spectra were reduced following standard procedures. The higher-resolution spectra show the same features as seen in the ones obtained with the Las Cumbres Observatory, specifically the AGN-like Balmer, \[O[iii]{}\], and \[N[ii]{}\] emission lines, as well as the prominent, broad, and double-peaked emission feature near 4640 Å.
To correct for the varying observing conditions during spectroscopic observations, we scaled all of our spectra to match the optical /UVOT photometry, linearly interpolating the UVOT $V$-band data to the spectroscopic epochs and using standard synthetic photometry. All our spectra are publicly available on the Weizmann Interactive Supernova data REPository (WISeREP[@YaronGalYam_2012_WISeREP]).
[**Spectral analysis of the $\mathbf{\hbeta}$ complex.**]{} We decomposed the spectral region of the optical spectrum of [AT2017bgt]{}, taken with the instrument mounted on the , following the methodology of ref. , with necessary adaptations. Specifically, we allowed for a much stronger broad emission line, and for an additional broad emission line to account for the blue peak seen in the optical spectra. Each of these broad emission features is modelled by two Gaussian profiles, with widths forced to match those fit for the broad line, and within the range $\fwhm= 2,000 - 10,000$ or $20,000\,{\ifmmode {\rm km\,s}^{-1} \else km\,s$^{-1}$\fi}$. The centres of these Gaussians are free to vary by $\pm1,500\,{\ifmmode {\rm km\,s}^{-1} \else km\,s$^{-1}$\fi}$ relative to those of the broad profile. This procedure resulted in a satisfactory fit, shown in Supplementary Fig. 5, which indicates that the rest-frame equivalent widths of the broad and emission lines are ${\rm rEW}=27.7$ and 24.0 Å, respectively. We stress that the spectral decomposition procedure accounts for the much weaker, blended emission from ionized iron (see ref. for details).
[**Composite spectrum construction.**]{} The SDSS composite shown in Fig. \[fig:opt\_spec\_comp\_SDSS\] was constructed from 1068 quasars at $0.24 \leq z \leq 0.75$, drawn from the large sample analysed in ref. . The redshift range was chosen to assure that the spectral complexes surrounding both the and lines is within the SDSS coverage. We further selected only those quasars for which the width of the broad component of the line is in the range $1,800 {\leq} \fwhb {\leq} 2,200\,{\ifmmode {\rm km\,s}^{-1} \else km\,s$^{-1}$\fi}$, comparable with what is measured in [AT2017bgt]{}. The composite was then constructed by computing the geometrical mean in each 1 Å (rest-frame) bin.
[**Strong emission line ratio diagnostics.**]{} We used the four high-resolution spectra discussed above (Supplementary Fig. 3) to measure strong emission line ratio diagnostics, and specifically [5007 $\lambda5007$]{}/ vs./ narrow line ratios. Supplementary Fig. 4a presents the measured line ratios, in addition to widely used classification schemes[@Kewley2001_BPT; @Kauffmann03_AGN_HOSTS; @Schawinski2007_feedback_ETs] – that is, the so-called Baldwin, Phillips & Terlevich (BPT) diagram[@BPT1981]. The emission line ratios we measure, and specifically those measured from the earliest high-resolution spectrum of [AT2017bgt]{} taken 10 days after discovery (in the rest-frame), are consistent with being driven by ionizing radiation from a “composite” emission source, which includes radiation from an accreting SMBH (that is, an AGN) in addition to young stars (that is, star-forming regions). The light-curve of the / line ratio, which includes all the available spectra where we could perform our spectral decomposition (that is., among the Las Cumbres Observatory spectra), is shown in Supplementary Fig. 4b. The light-curve shows essentially no real variability in the line ratio, which is expected given the long recombination time-scale in the low-density narrow-line emission region (of order 100 years; ref. ) and the long light travel time-scales ($\gg$100 years; refs. [@Bennert2002_NLR_RL; @Mor2009]).
We caution that, with a broad line width of about 2000[\^[-1]{} kms$^{-1}$]{}, and given the limited spectral resolution of the Las Cumbres data, it becomes challenging to properly decompose the narrow line emission from the entire line profile. This is the reason why many measurements are so close to the ${\ifmmode \left[{\rm O}\,\textsc{iii}\right] \else [O\,{\sc iii}]\fi}/\hbeta > 1$ constraint that is inherent to our spectral decomposition procedure.
Near-IR spectroscopy. {#SM_sec_obs_spec_nir .unnumbered}
=====================
We obtained two NIR spectra of [AT2017bgt]{}. The first spectrum was obtained on 2017 April 21 (i.e., 2 months after the transient detection), with the FIRE instrument on the Magellan-Baade telescope at the Las Campanas observatory[@Simcoe2008_FIRE]. The second spectrum was obtained on May 31 (99 days after detection), with the FLAMINGOS-2 instrument on the Gemini-South telescope[@Eikenberry2004_FLAMINGOS2]. Both spectra are shown in Supplementary Fig. 6, compared to a NIR composite spectrum of 27 quasars, taken from ref. . They exhibit a number of broad hydrogen emission lines (for example, ${\rm Pa}\,6\,\lambda 10938.095$, ${\rm Pa}\,7\,\lambda 10049.37$, ${\rm Pa}\,8\,\lambda9545.97$, ${\rm Pa}\,9\,\lambda 9229.01 $,${\rm Pa}\,11\,\lambda 8438.00$), as well as strong helium lines, such as $\hei\,\lambda 10829.894$ and $\heii\,\lambda 10123.61$. Importantly, in [AT2017bgt]{} all these emission lines are single-peaked, while the helium features are relatively strong. This is also the case for other helium emission lines across the optical regime, including He[ii]{} $\lambda3203$ (see Supplementary Fig. 3). This rules out the possibility that the blue peak within the broad optical emission feature (near 4640 Å) originates in a disk-like emitting region.
Determination of key SMBH properties {#SM_sec_BH_props .unnumbered}
====================================
The range of mass accretion rates is derived from bolometric luminosities, that are in turn estimated by assuming the X-ray and optical bolometric corrections of ref. . For the UV bolometric correction, we assume a value of 3.5, which reflects the range of values reported in the literature[@Runnoe2012; @Netzer2016_herschel_hiz], and the expectation that the bolometric correction at $\lambda_{\rm rest}{\simeq}2100$ Å (observed with /UVOT) would be larger than those calibrated for AGN continuum emission at 1450 Å (following a general AGN SED of $f_\nu \propto \nu^{-1/2}$; for example, ref. ). We stress that the UV-based bolometric luminosity, of $\approx2.9\times10^{45}\,{\ifmmode {\rm erg\,s}^{-1} \else erg s$^{-1}$\fi}$, is probably an overestimate, since the rest of the SED of [AT2017bgt]{} does not scale with the UV emission as in normal AGN[@Lusso2016_Lx_Luv]. To obtain mass accretion rates ($\dot{M}$), we finally assume a radiative efficiency of $\eta=0.1$, where $\eta \equiv \Lbol /\dot{M} c^2$ (and $c$ is the velocity of light).
The estimate of the size of the -emitting region, ${\hbox{$ {R_{\rm BLR}} $}}(\hbeta)$, and of the BH mass, , relies on the results of reverberation mapping experiments of broad-line AGN, which provide a way to link the observed optical continuum luminosity and the size of the line-emitting region and to obtain single-epoch, “virial” mass estimates[@Kaspi2000; @Shen2013_rev; @Peterson2014_review; @Mejia2016_XS_MBH]. For [AT2017bgt]{}, the working assumption of a virialized BLR (or -emitting region) can be justified by the (persistent) shapes of the broad Balmer emission lines, which resemble those of normal broad-line AGN. Moreover, the optical luminosities measured from our spectroscopy place [AT2017bgt]{} at the heart of the range of luminosities probed by reverberation mapping experiments of low-redshift AGN, and of the corresponding BLR size – luminosity relations[@Bentz2013_lowL_RL]. In what follows, we rely on the best-fit spectral model for the , / spectrum (described in ’Optical spectroscopy’ above). The monochromatic luminosity at 5100 Å, of $\lambda L_{\lambda}(5,100\,{\rm \AA}){=}\Loptfourthree{\times}10^{43}\,{\ifmmode {\rm erg\,s}^{-1} \else erg s$^{-1}$\fi}$, translates to an -emitting region size of ${\hbox{$ {R_{\rm BLR}} $}}=21.9$ or $28.4$ light-days, following the prescriptions given in refs. and , respectively. Combining the former value with an line width of $\fwhb{=}\hbfitfw\,{\ifmmode {\rm km\,s}^{-1} \else km\,s$^{-1}$\fi}$, and the mass prescription given in ref. , we obtain $\mbh{=}\Mseven{\times}10^{7}\,{{\ifmmode M_{\odot} \else $M_{\odot}$\fi}}$. The larger (latter) estimate would increase this estimate by about 0.1 dex, which is much smaller than the systematic uncertainties on , which are of order 0.3-0.4 dex (ref. ). Given the extremely intense UV continuum emission and the fact that the size of the -emitting region is driven by the ionizing UV (rather than optical) continuum, we further caution that the BLR size, and thus , could be considerably larger, perhaps by as much as a factor of ${\sim}4$ (see, for example, refs. ).
Relevant mechanisms for the long-lived UV flare {#SM_sec_mechanisms .unnumbered}
===============================================
Here We discuss some of the mechanisms that may be considered relevant for driving the intense UV brightening in [AT2017bgt]{} and the other events we associate with this proposed class. As noted in the main text, given the evidence for AGN-like activity in [AT2017bgt]{} (and the other events) both before and after the UV-optical event, we focus our brief discussion here on processes related to (thin) accretion disks that feed SMBHs. In this context, we note that the transient optical rise time of a few weeks is only slightly longer than the dynamical timescale in a thin disk, given our estimate ($\sim$4 days; see §\[SM\_sec\_timescales\] below). The longevity of the enhanced UV emission, of over a year, is considerably longer than the thermal timescale ($\sim$50 days, at most), and is instead starting to be comparable with the time-scales over which heating fronts may propagate through the disk ($\sim$3 years). [**Tidal disruption events.**]{} Some TDEs, which are generally thought to be powered by a newly formed, transient accretion disk, show a fast increase in UV continuum emission, accompanied by strong emission[@Gezari2012_TDE; @Arcavi2014_TDEs_He; @Holoien2016_TDE_AS15oi; @Holoien2016_TDE_AS14li; @Blagorodnova2017_TDE_iPTF16fnl; @Hung2017_TDE_iPTF16axa]. However, the temporal evolution of [AT2017bgt]{}, F01004-2237 and OGLE17aaj is much slower than what is seen in TDEs (see Fig. \[fig:img\_spec\_monitoring\] and Supplementary Fig. 1). In addition, the line seen in these three transients is much narrower than in TDEs (see Fig. \[fig:heii\_spec\_comp\_tdes\]). Although the TDE ASASSN-14li displays emission lines that are narrower than other TDEs (in a late-time spectrum, 86-days from discovery; ref. ), it still exhibits a very broad base of the line, not seen in our class of events. Tidal disruptions of giant stars can produce slowly-evolving light curves [@MacLeod2012_TDEs_giants], but in that case both the rise and decline of the light curve are expected to last several years, whereas the rise in our class of events is much more sudden. Recently, two events claimed to be TDEs were reported to have long-lived light curves. One is interpreted to be a TDE around an intermediate-mass BH[@Lin2018_TDE_IMBH] and the other is a dust-reprocessed TDE “echo” in a merging starburst galaxy[@Mattila2018_TDE_Arp299]. Neither of these cases relate to [AT2017bgt]{} (nor to the two sources we associate with it). Moreover, the strong [5007 $\lambda5007$]{} line emission seen in all three events suggests that the SMBHs were not completely dormant prior to the sudden (UV/optical) brightening, further weakening the case for a TDE. Given these stark differences, we conclude that a TDE is unlikely to be driving [AT2017bgt]{} (and by association, the F01004-2237 event and OGLE17aaj).
[**An interaction between an outflow and the BLR.**]{} A recently published model describes the interaction of an outflow launched from the vicinity of a SMBH with the BLR gas[@Moriya2017_BLR_winds_transients]. According to this model, an outflow with $v \lesssim 0.3\,c$ may account for the $>$1 year-long enhanced emission we observe in [AT2017bgt]{}. Such an outflow would, however, require a period of several months (${\hbox{$ {R_{\rm BLR}} $}}/ 0.1\,c$) to reach the BLR, and it is unclear how this mechanism would provide favourable conditions for extreme UV and BF emission. If the broad line is also emitted from an out-flowing region, then this may have implications for our estimates of and , as they assumed a virialized line emitting region (see ref. for a detailed discussion).
[**Other mechanisms.**]{} There are several other physical mechanisms that have been proposed to give rise to dramatic increase in (UV-optical) emission from accreting SMBHs. Periodic tidal interactions between the SMBH and an orbiting binary pair of stars may account for recurring “outbursts” of accretion-driven X-ray and optical emission, separated by several years, like those observed in some AGN[@Campana2015_IC3599; @Grupe2015_IC3599]. In each interaction episode some of the gas in the star envelope is accreted onto the SMBH through a transient disk, perhaps masquerading as a TDE[@MetzgerStone2017_TDE_impostors]. A stellar origin for the gas may explain the high metallicity and density of the BF line-emitting gas in [AT2017bgt]{}. However, it is unclear how this scenario would explain the persistence of the current episode of enhanced accretion (and UV emission) lasting over a year or, more importantly, the emission line profiles, which indicate an extended distribution of highly-ionized gas. Some models suggest that binary SMBHs may also exhibit sudden enhancement of SMBH accretion, accompanied by increased UV emission, driven by occasional shocks and events of drastic angular momentum loss in the dense circum-binary accretion flow[@Farris2015_BBHs_Acc]. As these episodes are expected to last over periods that are several times the binary orbital period, the prolonged enhanced UV emission can be accounted for, unless the binary separation is extremely tight (that is $a\lesssim 0.005$ pc). Given the data in hand, however, it is impossible to determine the real state of accretion onto the SMBH in [AT2017bgt]{} over the periods required to test any of these scenarios, and particularly between the archival detections (in the X-rays and UV) and the recent increase in UV-optical flux.
Typical timescales in thin accretion disks around SMBHs {#SM_sec_timescales .unnumbered}
=======================================================
For the sake of completeness, we recall here several key timescales in geometrically-thin, optically-thick accretion disks around AGN (and SMBHs). These timescales are derived in many references (for example, refs. ). Here we follow the recent discussion of these timescales in the context of changing look AGN, provided in ref. . The timescales are commonly parametrized in terms of the BH mass, (which for [AT2017bgt]{} we estimate to be ${\sim}1.4{\times}10^7\,{{\ifmmode M_{\odot} \else $M_{\odot}$\fi}}$); the distance from the BH, in terms of the gravitational radius, $r/r_{\rm g}$ (where $r_{\rm g} \equiv G\mbh/c^2$); the disk scaled height $h/r$, which in thin disks is of order $\sim$0.05; and the pseudo-viscosity parameter $\alpha$, which in thin disks is of order $\sim$0.03. The dynamical timescale is the typical timescale for (azimuthal) Keplerian motion of the disk material around the BH, and is given by: $$\begin{aligned}
t_{\rm dyn} &\simeq 14\,{\rm hours}
\left(\frac{\mbh}{10^7\,{{\ifmmode M_{\odot} \else $M_{\odot}$\fi}}}\right)
\left(\frac{r}{100\,r_{\rm g}}\right)^{3/2}
\label{SM_eq_t_dyn}\end{aligned}$$ The thermal timescale is the typical timescale for the disk cooling or heating, and thus further depends on the viscosity parameter: $$\begin{aligned}
t_{\rm th} &= t_{\rm dyn}/\alpha
{\simeq} 19\,{\rm days}
\left(\frac{\mbh}{10^7\,{{\ifmmode M_{\odot} \else $M_{\odot}$\fi}}}\right)
\left(\frac{r}{100\,r_{\rm g}}\right)^{3/2}
\left(\frac{\alpha}{0.03}\right)^{-1} .
\label{SM_eq_t_th}\end{aligned}$$ Cooling or heating fronts may travel throughout the disk on longer timescales, accounting for the disk geometry: $$\begin{aligned}
t_{\rm front} &= t_{\rm th}/(h/r)
{\simeq} 380\,{\rm days} \nonumber \\
& \quad \left(\frac{\mbh}{10^7\,{{\ifmmode M_{\odot} \else $M_{\odot}$\fi}}}\right)
\left(\frac{r}{100\,r_{\rm g}}\right)^{3/2}
\left(\frac{\alpha}{0.03}\right)^{-1}
\left(\frac{h/r}{0.05}\right)^{-1} .
\label{SM_eq_t_front}\end{aligned}$$ Finally, the viscous timescale, over which material travels radially from a radius $r$ to the BH, is yet longer: $$\begin{aligned}
t_{\rm \nu} &= t_{\rm front}/(h/r)
{\simeq} 21\,{\rm years} \nonumber \\
& \quad \left(\frac{\mbh}{10^7\,{{\ifmmode M_{\odot} \else $M_{\odot}$\fi}}}\right)
\left(\frac{r}{100\,r_{\rm g}}\right)^{3/2}
\left(\frac{\alpha}{0.03}\right)^{-1}
\left(\frac{h/r}{0.05}\right)^{-2} .
\label{SM_eq_t_visc}\end{aligned}$$ We recall that the radius within the disk that is relevant for the continuum emission is both wavelength- and accretion rate- dependent, so that the dynamical timescale can be expressed as: $t_{\rm dyn} {\simeq} 0.5\,{\rm sec}\, \dot{M}^{1/2} \lambda^{2}$ (with $\dot{M}$ given in [[yr]{}\^[-1]{} ${\ifmmode M_{\odot} \else $M_{\odot}$\fi}\,{\rm yr}^{-1}$]{} and $\lambda$ in Å). Given the derived properties of [AT2017bgt]{}, the near-UV data (with $\lambda_{\rm Eff} \simeq 2200$ Å) would correspond to ${\sim}300\,r_{\rm g}$.
#### Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request. All of our spectra are publicly available on the Weizmann Interactive Supernova data REPository (WISeREP)[@YaronGalYam_2012_WISeREP]. The data used to prepare Supplementary Fig. 1 are available from the ASAS-SN Light Curve Server (<https://asas-sn.osu.edu/>).
#### Additional References
[100]{}
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*et al.* A dust-enshrouded tidal disruption event with a resolved radio jet in a galaxy merger. ** ****, ().
*et al.* . ** ****, ().
, & . ** ****, ().
& . ** ****, ().
, , & . ** ****, ().
, , & . ** ****, ().
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, & ** (, ), edn.
** (, , ).
*et al.* . ** ****, ().
|
---
abstract: 'During the last years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wave function statistics. In the first part of this review we give a detailed introduction to the spectral theory of quantum graphs and discuss exact trace formulae for the spectrum and the quantum-to-classical correspondence. The second part of this review is devoted to the spectral statistics of quantum graphs as an application to quantum chaos. Especially, we summarise recent developments on the spectral statistics of generic large quantum graphs based on two approaches: the periodic-orbit approach and the supersymmetry approach. The latter provides a condition and a proof for universal spectral statistics as predicted by random-matrix theory.'
author:
- |
Sven Gnutzmann$^{1,2,*}$ and Uzy Smilansky$^{2,3}$\
$^1$ Freie Universität Berlin,\
Germany\
$^2$ The Weizmann Institute of Science,\
Rehovot, Israel\
$^3$ School of Mathematics, Bristol University,\
Bristol, United Kingdom\
$^*$ new address:\
School of Mathematical Sciences, University of Nottingham,\
Nottingham, United Kingdom
title: |
**Quantum Graphs:\
Applications to Quantum Chaos\
and Universal Spectral Statistics**
---
[100]{}
O. Agam, B.L. Altshuler, and A.V. Andreev, *Spectral statistics: From disordered to chaotic systems*, [Phys. Rev. Lett.]{} **75**, 4389 (1995). O. Agam, A.V. Andreev, and B.D. Simons, *Quantum chaos: a field theory approach*, [Chaos, Solitons & Fractals]{} **8**, 1099 (1997). M. Aizenman, R. Sims, and S. Warzel, *Stability of the Absolutely Continuous Spectrum of Random Schroedinger Operators on Tree Graphs*, math-ph/0502006 (2005). M. Aizenman, R. Sims, and S. Warzel, *Absolutely continuous spectra of quantum tree graphs with weak disorder*, Commun. Math. Phys. **264**, 371 (2006). M. Aizenman and S. Warzel, *Persistence under Weak Disorder of AC Spectra of Quasi-Periodic Schroedinger operators on Trees Graphs*, Moscow Math. Jour **5**, 499 (2005). M. Aizenman, R. Sims, and S. Warzel, *Fluctuation based proof of the stability of ac spectra of random operators on tree graphs*, math-ph/0510069 (2005). E. Akkermans, A. Comtet, J. Desbois, G. Montambaux, and Ch. Texier, *Spectral determinant on quantum graphs*, [Annals of Physics]{} **284**, 10 (2000). S. Alberverio and K. Pankrashkin, *A remark on Krein’s resolvent formula and boundary conditions*, J. Phys. A **38**, 4859 (2005).
I.L. Aleiner and A.I. Larkin, *Divergence of classical trajectories and weak localization*, Phys. Rev. B [**54**]{}, 14423 (1996). I.L. Aleiner and A.I. Larkin, *Role of divergence of classical trajectories in quantum chaos*, [Phys. Rev. E]{} **55**, R1243 (1997). I.L. Aleiner and A.L. Larkin, *Divergence of classical trajectories and quantum chaos*, [Chaos, Solitons & Fractals]{} **8**, 1179 (1997). S. Alexander, *Superconductivity of networks. A percolation approach to the effects of disorder*, Phys. Rev. B **27**, 1541 (1985). F. Ali Mehmeti, *A characterisation of generalized $C^\infty$-notion on nets*, Integral. Equ. Operator Theory **9**, 753 (1986).
A. Altland, M.R. Zirnbauer, *Random Matrix Theory of a Chaotic Andreev Quantum Dot*, Phys. Rev. Lett. **76**, 3420 (1996). A. Atland and M.R. Zirnbauer, *Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures*, Phys. Rev. B **55**, 1142 (1997). A. Atland, B.D. Simons, D. Taras-Semchuk, *Field Theory of Mesoscopic Fluctuations in Superconductor/Normal-Metal Systems*, Advances in Physics **49**, 321 (2000). A. Altland, B.D. Simons, M.R. Zirnbauer, *Theories of Low-Energy Quasi-Particle States in Disordered d-Wave Superconductors*, Physics Reports **359**, 283 (2002). P.W. Anderson, *New method for scaling theory of localization. II. Multichannel theory of a “wire” and possible extension to higher dimensionality*, Phys. Rev. B **23**, 4828 (1981). A.F. Andreev, *The thermal conductivity of the intermediate state in superconductors.*, Sov. Phys. JETP **19**, 1228 (1964). A.V. Andreev and B. Altshuler, *Spectral Statistics beyond Random Matrix Theory*, Phys. Rev. Lett. **75**, 902 (1995). N. Argaman, F.M. Dittes, E. Doron, J.P. Keating, A.Yu. Kitaev, M. Sieber, and U. Smilansky, *Correlations in the actions of periodic orbits derived from quantum chaos*, [Phys. Rev. Lett.]{} **71**, 4326 (1993). Y. Avishai and J.M. Luck, *Localization problem of a two-dimensional lattice in a random magnetic field*, Phys. Rev. B **45**, 1074 (1992). J.E. Avron, P. Exner, and Y. Last, *Periodic Schrödinger operators with large gaps and Wannier-Stark ladders*, Phys. Rev. Lett. **72**, 896 (1994). J.E. Avron, *Adiabatic Quantum Transport*, in E. Akkermans and G. Montambaux, J.L. Pichard and J. Zinn-Justin (editors), *Mesoscopic Quantum Physics*, Proceedings of the Les Houches Summer School LXI 1994, (North-Holland, 1995). F. Barra and P. Gaspard, *On the level spacing distribution in quantum graphs*, [J. Stat. Phys]{} **101**, 283 (2000). F. Barra and P. Gaspard, *Classical dynamics on graphs*, [Phys. Rev. E]{} **63**, 066215 (2001). F. Barra and P. Gaspard, *Transport and dynamics on open quantum graphs*, [Phys. Rev. E]{} **65**, 016205 (2002). C.W.J. Beenakker, *Random-matrix theory of quantum transport*, Reviews of Modern Physics **69**, 731 (1997). C.W.J. Beenakker, *Andreev billiards*, Lecture Notes in Physics **667**, 131 (2005). J. von Below, *A characteristic equation associated to an eigenvalue problem on $c^2$-networks*, Lin. Alg. Appl. **71**, 309 (1985).
J. von Below, *Sturm-Liouville eigenvalue problems on networks*, Math. Meth. Applied Sciences **10**, 383 (1988)
J. von Below, *Kirchhoff laws and diffusion on networks*, Alg. Appl. **21**, 692 (1989).
J. von Below, *Parabolic network equations*, 1st ed. (Tübingen, 1993).
J. von Below, *Can One Hear the Shape of a Network?*, in M. Dekker (editor), *Partial Differential Equations on Multistructures*, Lecture Notes in Pure and Applied Mathematics **219** (New York, 2001).
I. Bengtsson, Å. Ericsson, M. Kuś, W. Tadej, and K. Życzkowski, *Birkhoff’s polytope and unistochastic matrices, N=3 and N=4*, Commun. Math. Phys. **259**, 307 (2005). F. Bentosela, P. Duclos, and P. Exner, *Absolute continuity in periodic thin tubes and strongly coupled leaky wires*, Lett. Math. Phys. **65**, 75 (2003). D. Bercioux, M. Governale, V. Cataudella, and V. Marigliano Ramaglia, *Rashba-Effect-Induced Localization in Quantum Networks*, Phys. Rev. Lett. **93**, 056802 (2004).
G. Berkolaiko and J.P. Keating, *Two-point spectral correlations for star graphs*, [J. Phys. A]{} **32**, 7827 (1999). G. Berkolaiko, *Quantum Star Graphs and Related Systems*, PhD thesis, Bristol (2000). G. Berkolaiko, *Spectral gap of doubly stochastic matrices generated from equidistributed unitary matrices*, [J. Phys. A]{} **34**, L319 (2001). G. Berkolaiko, E.B. Bogomolny, and J.P. Keating, *Star graphs and Šeba billiards*, [J. Phys. A]{} **34**, 335 (2001). G. Berkolaiko, H. Schanz, and R.S. Whitney, *Leading off-diagonal correction to the form factor of large graphs*, Phys. Rev. Lett. **82**, 104101 (2002). G. Berkolaiko, J.P. Keating, and B. Winn, *Intermediate wave function statistics*, [Phys. Rev. Lett.]{} **91**, 134103 (2003). G. Berkolaiko, H. Schanz, and R.S. Whitney, *Form factor for a family of quantum graphs: an expansion to third order*, [J. Phys. A]{} **36**, 8373 (2003). G. Berkolaiko, *Form factor for large quantum graphs: evaluating orbits with time reversal*, [Waves in Random Media]{} **14**, S7 (2004). G. Berkolaiko, J.P. Keating, B. Winn, *No quantum ergodicity for star graphs*, Commun. Math. Phys. **250**, 259 (2004). G. Berkolaiko, R. Carlson, St. Fulling, and P. Kuchment (editors), *Quantum Graphs and Their Applications* Proceedings of the Joint Summer Research Conference in the Mathematical Sciences in Snowbird 2005, Contemp. Math. **415** (American Math. Society, Providence, 2006). M. V. Berry and M. Tabor, *Level clustering in the regular spectrum*, [Proc. R. Soc. Lond. A]{} **356**, 375 (1977). M.V. Berry, *Quantizing a classically ergodic system: Sinai’s billiard and the KKR method*, [Annals of Physics (New York)]{} **131**, 163 (1981). M.V. Berry, *Semiclassical theory of spectral rigidity*, [Proc. R. Soc. Lond. A]{} **400**, 229 (1985). M.V. Berry, *Some quantum-to-classical asymptotics*, in M.-J. Giannoni, A. Voros, and J. Zinn-Justin (editors), [*Chaos and Quantum Physics*, Les Houches Session LII 1989]{} (North-Holland, 1991). M.V. Berry and J.P. Keating, *A rule for quantizing chaos?* [J. Phys. A]{} **23**, 4839 (1990). R. Blümel, Y. Dabaghian, and R.V. Jensen, *Exact, convergent periodic-orbit expansions of individual energy eigenvalues of regular quantum graphs*, [Phys. Rev. E]{} **65**, 046222 (2002). R. Blümel, Y. Dabaghian, and R.V. Jensen, *Explicitly solvable cases of one-dimensional quantum chaggis*, [Phys. Rev. Lett.]{} **88**, 044101 (2002). E.B. Bogomolny and J.P. Keating, *Gutzwiller’s trace formula and spectral statistics: Beyond the diagonal approximation*, [Phys. Rev. Lett.]{} **77**, 1472 (1996). E.B. Bogomolny, B. Georgeot, M.-J. Giannoni, and C. Schmit, *Arithmetical chaos*, [Physics Reports]{} **291**, 219 (1997). O. Bohigas, M.-J. Giannoni, and C. Schmit, *Characterization of chaotic quantum spectra and universality of level fluctuation laws*, [Phys. Rev. Lett.]{} **52**, 1 (1984). J. Bolte and J. Harrison, *Spectral Statistics for the Dirac Operator on Graphs*, J. Phys. A **36**, 2747 (2003). J. Bolte and J. Harrison, *The spin contribution to the form factor of quantum graphs*, [J. Phys. A]{} **36**, L433 (2003). R.Bowen, *Equilibrium states and the ergodic theory of Anosov diffeomorphisms*, Lecture Notes in Mathematics **470** (Springer, 1975). P. Braun, S. Gnutzmann, F. Haake, M. Kuś, and K. Życzkowski, *Level Dynamics and Universality of Spectral Fluctuations*, Foundations of Physics **31**, 613 (2001). P. Brouwer, *On the Random-Matrix Theory of Quantum Transport* PhD thesis, Leiden (1997). J. Brüning, V. Geyler, and K. Pankrashkin, *Cantor and band spectra for periodic quantum graphs with magnetic fields*, math-ph/0511057 (2005).
R. Carlson, *Inverse eigenvalue problem on directed graphs*, Transactions of the American Mathematical Society **351**, 4069 (1999) R. Carlson, *Hill’s equation for a homogeneous tree*, Electronic Journal of Differential Equations **1997**, No. 23 (1997). R. Carlson, *Adjoint and self-adjoint differential operators on graphs*, Electronic Journal of Differential Equations **1999**, No. 6, (1999). R. Carlson, *Nonclassical Sturm-Liouville Problems and Schrödinger Operators on Radial Trees*, Electronic Journal of Differential Equations **2000**, No. 71 (2000). R. Carlson, *Spectral theory and spectral gap for periodic schrödinger operators on product graphs*, [Waves in Random Media]{} **14**, S29 (2004). G. Casati, F. Valz-Gris, and I. Guarneri, *On the connection between quantization of nonintegrable systems and statistical theory of spectra* Lettere al Nuovo Cimento **28**, 279 (1980).
C. Cattaneo, *The spectrum of the continuous Laplacian on a graph*, Monatsh. Math. **124**, 215 (1997).
J.T. Chalker and P.D. Coddington, *Percolation, quantum tunnelling and the integer Hall effect*, J. Phys. C **21**, 2665 (1988). T. Cheon and P. Exner, *An approximation to $\delta'$ couplings on graphs*, J. Phys. A **37**, L329 (2004). D. Cohen, H. Primack, and U. Smilansky, *Quantal-classical duality and the semiclassical trace formula*, [Annals of Physics (New York)]{} **264**, 108 (1998). D. Cohen, T. Kottos, and H. Schanz, *The conductance of a closed mesoscopic system*, J. Phys. A **264**, 11755 (2006). A. Comtet, J. Desbois, and S.N. Majumdar, *The local time distribution of a particle diffusing on a graph*, [J. Phys. A]{} **35**, L687 (2002). A. Comtet, J. Desbois, and Ch. Texier, *Functionals of the brownian motion, localization and metric graphs*, [J. Phys. A]{} **38**, R341 (2005). C.A. Coulson, Proceedings of the Physical Society London A, *Note on the Applicability of the Free-Electron Network Model to Metals*, **67**, 608 (1954).
D. M. Cvetković, M. Doob, and H. Sachs, *Spectra of Graphs* (Academic Press, New York, 1980).
P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, *Chaos: Classical and Quantum*, [ChaosBook.org]{} (Niels Bohr Institute, Copenhagen 2005). Yu. Dabaghian and R. Bl[ü]{}mel, *Explicit analytical solution for scaling quantum graphs*, Phys. Rev. E **68**, 055201(R) (2003). G. Dell’Antonio, P. Exner, and V. Geyler (guest editors) Special issue *Singular Interactions in Quantum Mechanics: Solvable Models*, J. Phys. A **38** Number 22 (2005); Section *Quantum Graphs: usual, fat, and leaky*, p. 4859-4932. J. Desbois, *Spectral determinant of Schrödinger operators on graphs*, [J. Phys. A]{} **33**, L63 (2000). J. Desbois, *Occupation times distribution for brownian motion on graphs*, [J. Phys. A]{} **35**, L673 (2002). T. Dittrich and U. Smilansky, *Spectral properties of systems with dynamical localization: II. Finite sample approach*, Nonlinearity **4**, 85 (1991).
T. Dittrich, B. Mehlig, H. Schanz, U. Smilansky, P. Pollner, and G. Vattay. *Spectral correlations in systems undergoing a transition from periodicity to disorder*, [Phys. Rev. E]{} **59**, 6541 (1999). F.J. Dyson, *Statistical theory of the energy levels of complex systems. I.*, [J. Math. Phys]{} **3**, 140 (1962). F.J. Dyson, *Statistical theory of the energy levels of complex systems. II.*, [J. Math. Phys]{} **3**, 157 (1962). F.J. Dyson, *Statistical theory of the energy levels of complex systems. III.*, [J. Math. Phys]{} **3**, 166 (1962). K. Efetov, *Supersymmetry in Disorder and Chaos*, (Cambridge University Press, Cambridge, 1997). W.D. Evans and M. Solomyak, *Smilansky’s model of irreverrsible quantum graphs I: The absolute continuous spectrum*, [J. Phys. A]{} **38**, 4611 (2005). W.D. Evans and M. Solomyak, *Smilansky’y model of irreverrsible quantum graphs II: The point spectrum*, [J. Phys. A]{} **38**, 7661 (2005). P. Exner and P. Šeba, *Free quantum motion on a branching graph*, Rep. Math. Phys. **28**, 7 (1989). P. Exner and E. Seresova, *Appendix resonances on a simple graph* J. Phys. A **27**, 8269 (1994). P. Exner, *Lattice Kronig-Penney models*, [Phys. Rev. Lett.]{} **74**, 3503 (1995). P. Exner and R. Gawlista, *Band spectra of rectangular graph superlattices*, Phys. Rev. B **53**, 7275 (1996). P. Exner, *Weakly coupled states on branching graphs*, Lett. Math. Phys. **38**, 313 (1996). P. Exner, *Contact interactions on graph superlattices*, [J. Phys. A]{} **29**, 87 (1996). P. Exner, *Magnetoresonances on a lasso graph*, Found. Phys. **27**, 171 (1997). P. Exner, *A duality between Schrödinger operators on graphs and certain Jacobi matrices*, Ann. Inst. H. Poincaré **66**, 359 (1997). P. Exner and T. Ichinose, *Geometrically induced spectrum in curved leaky wires*, J. Phys. A **34**, 1439 (2001). P. Exner and K. N[ě]{}mcov[á]{}, *Bound states in point-interaction star graphs*, [J. Phys. A]{} **34**, 7783 (2001). P. Exner and K. Yoshitomi, *Asymptotics of eigenvalues of the Schroedinger operator with a strong delta-interaction on a loop*, J. Geom. Phys. **41**, 344 (2002). P. Exner and K. N[ě]{}mcová, *Leaky quantum graphs: approximations by point-interaction Hamiltonians*, [J. Phys. A]{} **36**, 10173 (2003). P. Exner and S. Kondej, *Strong-coupling asymptotic expansion for Schroedinger operators with a singular interaction supported by a curve in ${{\ensuremath{\mathbbm{R}}}}^3$*, Rev. Math. Phys. **16**, 559 (2004). P. Exner and M. Tater, *Spectra of soft ring graphs*, [Waves in Random Media]{} **14**, S47 (2004). P. Exner and O. Post, *Convergence of spectra of graph-like thin manifolds*, J. Geom. Phys. **54**, 77 (2005). P. Exner and S. Kondej, *Scattering by local deformations of a straight leaky wire*, J. Phys. A **38**, 4865 (2005).
P. Exner, T. Ichinose, and S. Kondej, *On relations between stable and zeno dynamics in a leaky graph decay model*, [quant-ph/0504060]{} (2005). P. Exner and R.L. Frank, *Absolute continuity of the spectrum for periodically modulated leaky wires in $\mathbbm{R}^3$*, [math.SP/0508525]{} (2005). P. Exner and O. Turek, *Approximations of permutation-symmetric vertex couplings in quantum graphs*, in G. Berkolaiko, R. Carlson, St. Fulling, and P. Kuchment (editors), *Quantum Graphs and Their Applications* Proceedings of the Joint Summer Research Conference in the Mathematical Sciences in Snowbird 2005, Contemp. Math. **415** (American Math. Society, Providence, 2006). P. Exner, P. Hejcik, P. Šeba, *Approximations by graphs and emergence of global structures*, Rep. Math. Phys. **57**, 445 (2006). C. Flesia, R. Johnston, and H. Kunz, *Strong localization of classical waves: a numerical study*, Europhysics Letters **3**, 497 (1987). P.J. Forrester, *The spectrum edge of random matrix ensembles*, Nucl. Phys. B **402**, 709 (1993). L. Friedlander, *Absolute continuity of the spectra of periodic waveguides*, Contemp. Math. **339**, 37 (2003). L. Friedlander, *Extremal properties of eigenvalues for a metric graph*, Annales de l’Institut Fourier **55**, 199 (2005). L .Friedlander, *Genericity of simple eigenvalues for a metric graph*, Israel Journal of Mathematics **146**, 149 (2005).
S.A. Fulling, *Local Spectral Density and Vacuum Energy near a Quantum Graph Vertex*, in G. Berkolaiko, R. Carlson, St. Fulling, and P. Kuchment (editors), *Quantum Graphs and Their Applications* Proceedings of the Joint Summer Research Conference in the Mathematical Sciences in Snowbird 2005, Contemp. Math. **415** (American Math. Society, Providence, 2006).
Y.V. Fyodorov, *Basic features of Efetov’s supersymmetry approach*, in E. Akkermans and G. Montambaux, J.L. Pichard and J. Zinn-Justin (editors), *Mesoscopic Quantum Physics*, Proceedings of the Les Houches Summer School LXI 1994, (North-Holland, 1995) Y. Fyodorov, *Introduction to the random matrix theory: Gaussian unitary ensemble and beyond*, Lectures at the Isaak Newton Institute Summer School (Cambridge, 2004) [math-ph/0412017]{} (2004). P. Gaspard, *Chaos, Scattering, and Statistical Mechanics*, (Cambridge University Press, Cambridge 1998). N.I. Gerasimenko and B.S. Pavlov, *Scattering problems on non-compact graphs*, Theor. Math. Phys. **74**, 230 (1988). N.I. Gerasimenko, *Inverse scattering problems on noncompact graphs*, Theor. Math. Phys. **75**, 460 (1988). S. Gnutzmann, B. Seif, F. von Oppen, and M.R. Zirnbauer, *Universal spectral statistics of Andreev billiards: Semiclassical approach*, [Phys. Rev. E]{} **67**, 046225 (2003). S. Gnutzmann, U. Smilansky, and J. Weber, *Nodal counting on quantum graphs*, [Waves in Random Media]{} **14**, S61 (2004). S. Gnutzmann and B. Seif, *Universal spectral statistics in Wigner-Dyson, chiral, and Andreev star graphs. I. Construction and numerical results*, [Phys. Rev. E]{} **69**, 056219 (2004). S. Gnutzmann and B. Seif, *Universal spectral statistics in Wigner-Dyson, chiral, and Andreev star graphs. II. Semiclassical approach*, [Phys. Rev. E]{} **69**, 056220 (2004). S. Gnutzmann and A. Altland, *Universal spectral statistics in quantum graphs*, [Phys. Rev. Lett.]{} **93**, 194101 (2004). S. Gnutzmann and A. Altland, *Spectral correlations of individual quantum graphs*, Phys. Rev. E **72**, 056215 (2005). M.C. Goorden, *Superconductivity in Nanostructures: Andreev Billiards and Josephson Junction Qubits*, PhD thesis, Leiden (2005). T. Guhr, A. Müller-Groeling, and H.A. Weidenmüller, *Random matrix theories in quantum physics: Common concepts*, [Physics Reports]{} **299**, 189 (1998). B. Gutkin and U. Smilansky, *Can one hear the shape of a graph?* [J. Phys. A]{} **34**, 6061 (2001). M.C. Gutzwiller, *Periodic orbits and classical quantization conditions*, J. Math. Phys. **12**, 343 (1971). M.C. Gutzwiller, *Chaos in Classical and Quantum Mechanics*, (Springer, New York, 1990).
F. Haake, *Quantum Signatures of Chaos*, 2nd edition (Springer, Berlin, 2000). G. Hackenbroich and H.A. Weidenmüller, *Universality of random-matrix results for non-Gaussian ensembles*. Phys. Rev. Lett. **76**, 110 (1996). R.U. Haq, A. Pandey, and O. Bohigas, *Fluctuation properties of nuclear energy levels: Do theory and experiment agree?*, Phys. Rev. Lett. **48**, 1086 (1982). J.H. Hannay and A.M. Ozorio de Almeida, *Periodic orbits and a correlation function for the semiclassical density of states*, J. Phys. A **17**, 3429 (1984). M. Harmer, *Inverse scattering for the matrix Schrödinger operator and Schrödinger operator on graphs with general self-adjoint boundary conditions*, ANZIAM J. **44**, 161 (2002).
M. Harmer, *Inverse scattering on matrices with boundary conditions*, J. Phys. A **38**, 4875 (2005). S. Heusler, S. Müller, P. Braun, and F. Haake, *Universal spectral form factor for chaotic dynamics*, [J. Phys. A]{} **37**, L31 (2004). O. Hul, Sz. Bauch, P. Pako[ń]{}ski, N. Savytskyy, K. [Ż]{}yczkowski, and L. Sirko. *Experimental simulation of quantum graphs by microwave networks*, [Phys. Rev. E]{} **69**, 056205 (2004). Y. Imry, *Introduction to Mesoscopic Systems*, (Oxford University Press, New York, 1996). D.A. Ivanov, *The supersymmetric technique for random-matrix ensembles with zero eigenvalues*, J. Math. Phys. **43**, 126 (2002). M. Kac, *Almost periodicity and the representation of integers as sums of squares*, Am. J. Math. **62**, 122 (1940).
M. Kac, *Can one hear the shape of a drum?*, American Mathematical Monthly **73**, 1 (1966). L. Kaplan, *Eigenstate structure in graphs and disordered lattices*, [Phys. Rev. E]{} **64**, 036225 (2001). J.P. Keating and M.V. Berry., *False singularities in partial sums over periodic orbits*, [J. Phys. A]{} **20**, L1139 (1987).
J.P. Keating, *The cat maps: quantum mechanics and classical motion*, [Nonlinearity]{} **4**, 309 (1991). J.P. Keating, J. Marklof and B. Winn, *Value distribution of the eigenfunctions and spectral determinants of quantum star graphs*, Commun. Math. Phys. **241**, 421 (2003). R. Klesse, *Der Lokalisierungs-, Delokalisierungsübergang im Netzwerkmodell des Quanten-Hall-Effekts* PhD Thesis Köln 1996 (AWOS-Verlag, 1996). R. Klesse and M. Metzler, *Spectral Compressibility at the Metal-Insulator Transition of the Quantum Hall Effect*, Phys. Rev. Lett. **79**, 721 (1997). V. Kostrykin and R. Schrader, *Kirchhoff’s rule for quantum wires*, [J. Phys. A]{} **32**, 595 (1999). V. Kostrykin and R. Schrader, *Kirchhoff’s Rule for Quantum Wires. II: The Inverse Problem with Possible Applications to Quantum Computers*, Fortschritte der Physik **48**, 703 (2000). V. Kostrykin and R. Schrader. *The generalized star product and the factorization of scattering matrices on graphs*, [J. Math. Phys]{} **42**, 1563 (2001). V. Kostrykin and R. Schrader, *A random necklace model*, [Waves in Random Media]{} **14**, S75 (2004). V. Kostrykin and R. Schrader, *Generating Functions of Random Walks on Graphs*, math.CO/0404467 (2004) V. Kostrykin and R. Schrader, *Laplacians on Metric Graphs: Eigenvalues, Resolvents and Semigroups*, in G. Berkolaiko, R. Carlson, St. Fulling, and P. Kuchment (editors), *Quantum Graphs and Their Applications* Proceedings of the Joint Summer Research Conference in the Mathematical Sciences in Snowbird 2005, Contemp. Math. **415** (American Math. Society, Providence, 2006). V. Kostrykin and R. Schrader, *The inverse scattering problem for metric graphs and the traveling salesman problem*, math-ph/0603010 (2006). T. Kottos and U. Smilansky, *Quantum chaos on graphs*, [Phys. Rev. Lett.]{} **79**, 4794 (1997). T. Kottos and U. Smilansky, *Periodic orbit theory and spectral statistics for quantum graphs*, [Annals of Physics]{} **274**, 76 (1998). T. Kottos and U. Smilansky, *Chaotic scattering on graphs*, [Phys. Rev. Lett.]{} **85**, 968 (2000). T. Kottos and H. Schanz, *Quantum graphs: a model for quantum chaos*, [Physica E]{} **9**, 523 (2001). T. Kottos and U. Smilansky, *Quantum graphs: a simple model for chaotic scattering*, [J. Phys. A]{} **36**, 3501 (2003). T. Kottos and H. Schanz, *Statistical properties of resonance widths for open quantum graphs*, [Waves in Random Media]{} **14**, S91 (2004). T. Kottos, *Statistics of resonances and delay times in random media: Beyond random matrix theory*, J. Phys. A **3**, 10761 (2005). D. Kowal, U. Sivan, O. Entin-Wohlman, and Y. Imry, *Transmission through multiply-connected wire systems*, Phys. Rev. B **42**, 9009 (1990). P. Kuchment and H. Zeng, *Convergence of spectra of mesoscopic systems collapsing onto a graph*, J. Math. Anal. Appl. **258**, 671 (2001).
P. Kuchment, *Graph models of wave propagation in thin structures*, Waves in Random Media **12**, R1 (2003). P. Kuchment, *Quantum graphs: I. Some basic structures*, [Waves in Random Media]{} **14**, S107 (2004). P. Kuchment, *Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs*, [J. Phys. A]{} **38**, 4887 (2005). P. Kuchment (guest editor), Special Section *Quantum Graphs*, Waves in Random Media **14** Number 1, S1-S185 (2005) H. Kuhn, *Elektronengasmodell zur quantitativen Deutung der Lichtabsorption von organischen Farbstoffen I*, Helvetica Chimica Acta **31**, 1441 (1948) H. Kuhn, *Quantenmechanische Behandlung von Farbstoffen mit verzweigtem Elektronengas*, Helvetica Chimica Acta **32**, 2247 (1949). P. Kurasov and F. Stenberg, *On the inverse scattering problem on branching graphs*, J. Phys. A **35**, 101 (2002). P. Kurasov and M. Novaczyk, *Inverse spectral problems for quantum graphs*, J. Phys. A **38**, 4910 (2005). M.L. Mehta, *Random Matrices*, third edition (Academic Press, New York, 1991). J.A. Melsen, P.W. Brouwer, K.M. Frahm, and C.W.J. Beenakker, *Induced superconductivity distinguishes chaotic from integrable billiards*, [Europhys. Lett.]{} **35**, 7 (1996). R. Meyers, R.S. Strichartz, and A. Teplyaev, *Dirichlet forms on the Sierpinski gasket*, Pac. J. Math. **217**, 149 (2004). A.D. Mirlin, *Statistics of energy levels and eigenfunctions in disordered systems*, Phys. Rep. **326**, 259 (2000). A.D. Mirlin, *Statistics of energy levels and eigenfunctions in disordered and chaotic systems: Supersymmetry approach*, in G. Casati, I. Guarneri, and U. Smilansky (editors), *New Directions in Quantum Chaos*, Proceedings of the international School of Physics “Enrico Fermi", Course CXLIII (IOS Press, 2000). R. Mitra and S.W. Lee, *Analytical Techniques in the Theory of Guided Waves*, (Macmillan, New York 1971).
E.W. Montroll, *Quantum Theory on a Network. I. A Solvable Model Whose Wavefunctions Are Elementary Functions*, J. Math. Phys. **11**, 635 (1970). J. M[ü]{}ller and A. Altland, *Field theory approach to quantum interference in chaotic systems*, [J. Phys. A]{} **38**, 3097 (2005). S.Müller, *Classical basis for quantum spectral fluctuations in hyperbolic systems*, [Eur. Phys. J. B]{} **34**, 305 (2003). S. M[ü]{}ller, S. Heusler, P. Braun, F. Haake, and A. Altland *Semiclassical foundation of universality in quantum chaos*, [Phys. Rev. Lett.]{} **93**, 014103 (2004). S. M[ü]{}ller, S. Heusler, P. Braun, F. Haake, and A. Altland, *Periodic-orbit theory of universality in quantum chaos*, Phys. Rev. E **72**, 046207 (2005) B.A. Muzykantskii and D.E. Khmelnitskii, *Effective action in the theory of quasi-ballistic disordered conductors*, JETP Letters **62**, 76 (1995). T. Nagao and K. Slevin, *Laguerre ensembles of random matrices: Nonuniversal correlation functions*, J. Math. Phys. **34**, 2317 (1993). K. Naimark and M. Solomyak, *Eigenvalue estimates for the weighted Laplacian on metric trees*, Proc. London Math. Soc. **80**, 690 (2000). T. Nakayama, K. Yakubo, and R.L. Orbach, *Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations*, Rev. Mod. Phys. **66**, 381 (1994). S. Nicaise, *Spectre des réseaux topologiques finis*, Bull. Sc. Math. 2$^e$ Série **111**, 401 (1987).
S.P. Novikov, *Schrödinger operators on graphs and symplectic geometry*, in E. Bierstone, B. Khesin, A. Khovanskii, and J. Marsden (editors), *The Arnoldfest*, Proceedings of a Conference in Honour V.I. Arnold for His Sixtieth Birthday, (American Mathematical Society, 1999). A.M. Ozorio de Almeida, *Hamiltonian Systems: Chaos and Quantization*, (Cambridge University Press, Cambridge 1990). P. Pakoński, K. Życzkowski, and M. Kuś, *Classical 1d maps, quantum graphs and ensembles of unitary matrices*, [J. Phys. A]{} **34**, 9303 (2001). P. Pakonski, G. Tanner, and K. Życzkowski, *Families of line-graphs and their quantization* J. Stat. Phys. **111**, 1331 (2003). K. Pankrashkin, *Spectra of Schrödinger operators on equilateral quantum graphs*, Lett. Math. Phys. **77**, 139 (2006).
K. Pankrashkin, *Localization effects in a periodic quantum graph with magnetic field and spin-orbit interaction*, math-ph/0607034 (2006).
L. Pauling, *The diamagnetic anisotropy of aromatic molecules*, J. Chem. Phys. **4**, 673 (1936). Ph. Pechukas *Distribution of Energy Eigenvalues in the Irregular Spectrum* Phys. Rev. Lett. **51**, 943 (1983). V. Pivovarchik, *Inverse problem for the Sturm-Liouville equation on a simple graph*, SIAM J. Math. Anal. **31**, 801 (2001). J.R. Platt, *Classification of Spectra of Cata-Condensed Hydrocarbons*, J. Chem. Phys. **17**, 484 (1949). O. Post, *Branched quantum wave guides with Dirichlet boundary conditions*, J. Phys. A **38**, 4917 (2005). M. Puhlmann, H. Schanz, T. Kottos, and T. Geisel, *Quantum decay of an open chaotic system: A semiclassical approach*, [Europhys. Lett.]{} **69**, 313 (2005). M.J. Richardson and N.L. Balazs, *On the network model of molecules and solids*, Annals of Physics **73**, 308 (1972). K. Richter and M. Sieber, *Semiclassical Theory of Chaotic Quantum Transport*, Phys. Rev. Lett. **89**, 206801 (2002). J.-P. Roth, *Le spectre du Laplacien sur un graphe*, in G. Mokobodzki and D. Pinchon (editors), *Théorie du potentiel*, Proceedings of the Colloque Jacques Deny, Orsay 1983, Lecture Notes in Mathematics, (Springer, 1985). J.-P. Roth, *Spectre du Laplacien sur un graphe*, Comptes rendus de l’Académie des sciences Paris **296**, 793 (1983). J. Rubinstein and M. Schatzman, *Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum*, Arch. Ration. Mech. Anal. **160**, 271 (2001).
K. Ruedenberg and C.W. Scherr, *Free-Electron Network Model for Conjugated Systems. I. Theory*, J. Chem. Phys. **21**, 1565 (1953). D. Ruelle, *A measure associated with axiom A attractors*, Amer. J. Math. **98**, 619 (1976). Y. Saito, *The limiting equation for Neumann Laplacians on shrinking domains*, Electronic Journal of Differential Equations **2000**, No. 31 (2000). Y. Saito, *Convergence of the Neumann Laplacian on shrinking domains*, Analysis **21**, 171 (2001). H. Schanz and U. Smilansky, *Spectral Statistics for Quantum Graphs : Periodic Orbits and Combinatorics*, Philosophical Magazine B**80**, 1999 (2000). H. Schanz and U. Smilansky, *Periodic-orbit theory of Anderson localization on graphs*, [Phys. Rev. Lett.]{} **84**, 1427 (2000). H. Schanz and U. Smilansky, *Combinatorial identities from the spectral theory of quantum graphs*, Electronic J. Combin. **8**, R16 (2001). H. Schanz and T. Kottos, *Scars on quantum networks ignore the Lyapunov exponent*, [Phys. Rev. Lett.]{} **90**, 234101 (2003). H. Schanz, M. Puhlmann, and T. Geisel, *Shot noise in chaotic cavities from action correlations*, [Phys. Rev. Lett.]{} **91**, 134101 (2003). R. Scharf, B. Dietz, M. Kuś, F. Haake, and M.V. Berry, *Kramers’ degeneracy and quartic level repulsion*, [Europhys. Lett.]{} **5**, 383 (1988). A.G.M. Schmidt, B.K. Cheng and M.G.E da Luz, *Green function approach for general quantum graphs*, J. Phys. A **36**, L545 (2003). S. Severini and G. Tanner, *Regular quantum graphs*, [J. Phys. A]{} **37**, 6675 (2004). T. Shapira and U. Smilansky, *Quantum graphs which sound the same*, in F. Khanna and D. Matrasulov (editors), *Nonlinear Dynamics and Fundamental Interactions*, Proc. Nato Advanced Research Workshop, Tashkent 2004 (Springer, Berlin 2005) B. Shapiro, *Renormalization-Group Transformation for the Anderson Transition*, Phys. Rev. Lett. **48**, 823 (1982). T. Shirai, *The spectrum of infinite regular line graphs*, Trans. Amer. Math. Soc. **352**, 115 (2000).
M. Sieber, H. Primack, U. Smilansky, I. Ussishkin and H. Schanz, *Semiclassical Quantization of Billiards with Mixed Boundary Conditions*, J. Phys. A **28**, 5041 (1995). M. Sieber, and K. Richter, *Correlations between Periodic Orbits and their Rôle in Spectral Statistics*, Phys. Scr. T **90**, 128 (2001); M. Sieber, *Leading off-diagonal approximation for the spectral form factor for uniformly hyperbolic systems*, [J. Phys. A]{} **35**, L613 (2002). Ya.G. Sinai, *Gibbs measures in ergodic theory*, Russian Math. Surveys **27**, 21 (1972).
U. Smilansky and B. Verdene, *Action correlations and random matrix theory*, [J. Phys. A]{} **36**, 3525 (2003). U. Smilansky, *Irreversible quantum graphs*, [Waves in Random Media]{} **14**, S143 (2004). M. Solomyak, *On a differential operator appearing in the theory of irreversible quantum graphs*, [Waves in Random Media]{}, **14**, S173 (2004). M. Solomyak, *On the spectrum of the laplacian on regular metric trees*, [Waves in Random Media]{} **14**, S155 (2004). D. Spehner, *Spectral form factor of hyperbolic systems: leading off-diagonal approximation*, J. Phys. A **36**, 7269 (2003). H.-J. Stöckmann, *Quantum Chaos: An Introduction* (Cambridge University Press, Cambridge, 1999). G. Tanner, *Spectral statistics for unitary transfer matrices of binary graphs*, [J. Phys. A]{} **33**, 3567 (2000). G. Tanner, *Unitary-stochastic matrix ensembles and spectral statistics*, [J. Phys. A]{} **34**, 8485 (2001). G. Tanner, *The autocorrelation function for spectral determinants of quantum graphs*, [ J. Phys. A]{} **35**, 5985 (2002). D. Taras-Semchuk, A. Altland, *Quantum interference and the formation of the proximity effect in chaotic normal-metal/superconducting structures*, Phys. Rev. B **64**, 014512 (2001). A. Teplyaev, *Spectral analysis on infinite Sierpinski gaskets*, J. Funct. Anal. **159**, 537 (1998).
Ch. Texier and G. Montambaux, *Scattering theory on graphs*, J. Phys. A **34**, 10307 (2001). Ch. Texier, *Scattering theory on graphs II: the Friedel sum rule*, J. Phys. A **35**, 3389 (2002). Ch. Texier and G. Montambaux, *Weak localization in multiterminal networks of diffusive wires*, Phys. Rev. Lett. **92**, 186801 (2004). Ch. Texier and G. Montambaux *Dephasing due to electron-electron interaction in a diffusive ring*, Phys. Rev. B **72**, 115327 (2005). Ch. Texier and G. Montambaux, *Quantum oscillations in mesoscopic rings and anomalous diffusion*, J. Phys. A: **38**, 3455 (2005). M. Turek and K. Richter, *Leading off-diagonal contribution to the spectral form factor of chaotic quantum systems*, [J. Phys. A]{} **36**, L455 (2003). M. Turek, D. Spehner, S. Müller, and K. Richter, *Semiclassical form factor for spectral and matrix element fluctuations of multi-dimensional chaotic systems* Phys. Rev. E **71**, 016210 (2005) J.J.M. Verbaarschot, H.A. Weidenmüller, and M.R. Zirnbauer, *Grassmann variables in stochastic quantum physics: the case of compound-nucleus scattering*, Phys. Rep. **129**, 367 (1985). J.J.M. Verbaarschot and I. Zahed, *Spectral density of the QCD Dirac operator near zero virtuality*, Phys. Rev. Lett. **70**, 3852 (1993). J.J.M. Verbaarschot, *Spectrum of the QCD Dirac operator and chiral random matrix theory*, Phys. Rev. Lett. **72**, 2531 (1994). J. Verbaarschot, *The spectrum of the Dirac operator near zero virtuality for $N_c = 2$ and chiral random matrix theory*, Nucl. Phys. B **426**, 559 (1994). E. Wigner, *On the distribution of the roots of certain symmetric matrices*, [Annals of Mathematics]{} **67**, 325 (1958). B. Winn, *The Laplacian on a graph and quantum chaology*, PhD thesis, Bristol (2003). T. Yukawa, *New Approach to the Statistical Properties of Energy Levels*, Phys. Rev. Lett. **54**, 1883 (1985) J. Zakrzewski, K. Dupret, and D. Delande, *Statistical Properties of Energy Levels of Chaotic Systems: Wigner or Non-Wigner?*, Phys. Rev. Lett. **74**, 522 (1995). M.R. Zirnbauer, *Supersymmetry for systems with unitary disorder: Circular ensembles*, [J. Phys. A]{} **29**, 7113 (1996). M.R. Zirnbauer, *Riemannian symmetric superspaces and their origin in random matrix theory*, J. Math. Phys. **37**, 4986 (1996). M.R. Zirnbauer, *The color-flavor transformation and a new approach to quantum chaotic maps*, in: D. DeWit, A.J. Bracken, M.D. Gould, and P.A. Pearce (editors), *13th International Congress of Mathematical Physics*, (International Press, Cambridge, 1998). M.R. Zirnbauer, *Pair correlations of quantum chaotic maps from supersymmetry*, in: I.V. Lerner, J.P. Keating, and D.E. Khmelnitskii (editors), *Supersymmetry and Trace Formulae: Chaos and Disorder* (Kluwer/Plenum, New York, 1993). K. Życzkowski, M. Ku[ś]{}, W. S[ł]{}omczyński, and H.-J. Sommers, *Random unistochastic matrices*, [J. Phys. A]{} *36*, 3425 (2003).
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abstract: 'We study several methods for full or partial sharing of the decoder parameters of multilingual NMT models. We evaluate both fully supervised and zero-shot translation performance in 110 unique translation directions using only the WMT 2019 shared task parallel datasets for training. We use additional test sets and re-purpose evaluation methods recently used for unsupervised MT in order to evaluate zero-shot translation performance for language pairs where no gold-standard parallel data is available. To our knowledge, this is the largest evaluation of multi-lingual translation yet conducted in terms of the total size of the training data we use, and in terms of the diversity of zero-shot translation pairs we evaluate. We conduct an in-depth evaluation of the translation performance of different models, highlighting the trade-offs between methods of sharing decoder parameters. We find that models which have task-specific decoder parameters outperform models where decoder parameters are fully shared across all tasks.'
author:
- Chris Hokamp
- John Glover
- |
Demian Gholipour\
Aylien Ltd.\
Dublin, Ireland\
`<first-name>@aylien.com`
bibliography:
- 'acl2019.bib'
title: 'Evaluating the Supervised and Zero-shot Performance of Multi-lingual Translation Models'
---
Introduction
============
Multi-lingual translation models, which can map from multiple source languages into multiple target languages, have recently received significant attention because of the potential for positive transfer between high- and low-resource language pairs, and because of possible efficiency gains enabled by translation models which share parameters across many languages [@dong-etal-2015-multi; @ha-multilingual-2016; @firat-etal-2016-multi; @johnson-google-2016; @blackwood-etal-2018-multilingual; @sachan-neubig-2018-parameter; @aharoni-etal-2019-massively]. Multi-lingual models that share parameters across languages can also perform zero-shot translation, translating between language pairs for which no parallel training data is available [@gnmt2016; @ha-multilingual-2016; @johnson-google-2016].
Although multi-task models have recently been shown to achieve positive transfer for some combinations of NLP tasks, in the context of MT, multi-lingual models do not universally outperform models trained to translate in a single direction when sufficient training data is available. However, the ability to do zero-shot translation may be of practical importance in many cases, as parallel training data is not available for most language pairs [@gnmt2016; @johnson-google-2016; @aharoni-etal-2019-massively]. Therefore, small decreases in the performance of supervised pairs may be admissible if the corresponding gain in zero-shot performance is large. In addition, zero-shot translation can be used to generate synthetic training data for low- or zero- resource language pairs, making it a practical alternative to the bootstrapping by back-translation approach that has recently been used to build completely unsupervised MT systems [@firat-etal-2016-multi; @artetxe2018iclr; @lampleCDR18; @lample2018phrase]. Therefore, understanding the trade-offs between different methods of constructing multi-lingual MT systems is still an important line of research.
Deep sequence-to-sequence models have become the established state-of-the-art for machine translation. The dominant paradigm continues to be models divided into roughly three high-level components: *embeddings*, which map discrete tokens into real-valued vectors, *encoders*, which map sequences of vectors into an intermediate representation, and *decoders*, which use the representation from an encoder, combined with a dynamic representation of the current state, and output a sequence of tokens in the target language conditioned upon the encoder’s representation of the input. For multi-lingual systems, any combination of embedding, encoder and/or decoder parameters can potentially be shared by groups of tasks, or duplicated and kept private for each task.
![The decoder component of the transformer model [@vaswani-transformer:2017]. All parameters may be shared across all target tasks, or a unique set of decoder parameters can be created for each task (outer dashed line). Alternatively, we can create unique attention parameters for each task, while sharing the final feed-forward layers (inner dotted lines). The possibility of including an embedding for the target task is visualized at the bottom of the diagram. Illustration modeled after @sachan-neubig-2018-parameter.[]{data-label="fig:transformer_decoder"}](vis/lang_embeddings.pdf){width="45.00000%"}
Our work builds upon recent research on many-to-one, one-to-many, and many-to-many translation models. We are interested in evaluating many-to-many models under realistic conditions, including:
1. A highly imbalanced amount of training data available for different language pairs.
2. A very diverse set of source and target languages.
3. Training and evaluation data from many domains.
We focus on multi-layer transformer models [@vaswani-transformer:2017], which achieve state-of-the-art performance on large-scale MT and NLP tasks [@devlin2018bert; @bojar-EtAl:2018:WMT1]. The decoder component of the transformer is visualized in figure \[fig:transformer\_decoder\]. We study four ways of building multi-lingual translation models. Importantly, all of the models we study can do zero-shot translation: translating between language pairs for which no parallel data was seen at training time. The models use training data from 11 distinct languages[^1], with supervised data available from the WMT19 news-translation task for 22 of the 110 unique translation directions[^2]. This leaves 88 translation directions for which no parallel data is available. We try to evaluate zero-shot translation performance on all of these additional directions, using both gold parallel data, and evaluations based on pivoting or multi-hop translation.
#### Target Language Specification
Although the embedding and encoder parameters of a multi-lingual system may be shared across all languages without any special modification to the model, *decoding* from a multi-lingual model requires a means of specifying the desired output language. Previous work has accomplished this in different ways, including:
- pre-pending a special target-language token to the input [@gnmt2016]
- using an additional embedding vector for the target language [@lample2019cross]
- using unique decoders for each target language [@luong2016; @firat-etal-2016-multi]
- partially sharing some of the decoder parameters while keeping others unique to each target language [@sachan-neubig-2018-parameter; @blackwood-etal-2018-multilingual].
However, to the best of our knowledge, no side-by-side comparison of these approaches has been conducted. We therefore train models which are identical except for the way that decoding into different target languages is handled, and conduct a large-scale evaluation. We use only the language pairs and official parallel data released by the WMT task organisers, meaning that all of our systems correspond to the constrained setting of the WMT shared task, and our experimental settings should thus be straightforward to replicate.
Multi-Task Translation Models
=============================
This section discusses the key components of the transformer-based NMT model, focusing on the various ways to enable translation into many target languages. We use the terms source/target ***task*** and ***language*** interchangeably, to emphasize our view that multi-lingual NMT is one instantiation of the more general case of multi-task sequence to sequence learning.
Shared Encoders and Embeddings
------------------------------
In this work, we are only interested in ways of providing target task information to the model – information about the source task is never given explicitly, and encoder parameters are always fully shared across all tasks. The segmentation model and embedding parameters are also shared between all source and target tasks (see below for more details).
Multi-lingual Decoder Configurations
------------------------------------
Figure \[fig:transformer\_decoder\] visualizes the decoder component of the transformer model, with dashed and dotted lines indicating the parameter sets that we can replicate or share across target tasks.
### Target Task Tokens (<span style="font-variant:small-caps;">Prepend</span>)
@gnmt2016 showed that, as long as a mechanism exists for specifying the target task, it is possible to share the decoder module’s parameters across all tasks. In the case where all parameters are shared, the decoder model must learn to operate in a number of distinct modes which are triggered by some variation in the input. A simple way to achive this variation is by pre-pending a special “task-token” to each input. We refer to this method as **<span style="font-variant:small-caps;">Prepend</span>**.
### Task Embeddings (<span style="font-variant:small-caps;">Emb</span>)
An alternative to the use of a special task token is to treat the target task as an additional input feature, and to train a unique embedding for each target task [@lample2019cross], which is combined with the source input. This technique has the advantage of explicitly decoupling target task information from source task input, introducing a relatively small number of additional parameters. This approach can also be seen as adding an additional token-level *feature* which is the same for all tokens in a sequence [@sennrich-haddow:2016:WMT]. We refer to this setting as **<span style="font-variant:small-caps;">Emb</span>**.
### Task-specific Decoders (<span style="font-variant:small-caps;">Dec</span>)
In general, any subset of decoder parameters may be replicated for each target language, resulting in parameter sets which are specific to each target task. At one extreme, the entire decoder module may be replicated for each target language, a setting which we label **<span style="font-variant:small-caps;">Dec</span>** [@dong-etal-2015-multi].
### Task-specific Attention (<span style="font-variant:small-caps;">Attn</span>)
An approach somewhere in-between <span style="font-variant:small-caps;">Emb</span> and <span style="font-variant:small-caps;">Dec</span> is to partially share some of the decoder parameters, while keeping others unique to each task. Recent work proposed creating unique attention modules for every target task, while sharing the other decoder parameters [@sachan-neubig-2018-parameter; @blackwood-etal-2018-multilingual]. The implementation of their approaches differ significantly – we propose to create completely unique attention parameters for each task. This means that for each of our 11 languages, we have unique context- and self-attention parameters in each layer of the transformer decoder. We refer to this setting as **<span style="font-variant:small-caps;">Attn</span>**.
Experiments
===========
All experiments are conducted using the transformer-base configuration of @vaswani-transformer:2017 with the relevant modifications for each system discussed in the previous section. We use a shared sentencepiece[^3] segmentation model with 32000 pieces. We use all available parallel data from the WMT19 news-translation task for training, with the exception of `commoncrawl`, which we found to be very noisy after manually checking a sample of the data, and `paracrawl`, which we use only for <span style="font-variant:small-caps;">en-fi</span> and <span style="font-variant:small-caps;">en-lt</span>[^4].
We train each model on two P100 GPUs with an individual batch size of up to 2048 tokens. Gradients are accumulated over 8 mini-batches and parameters are updated synchronously, meaning that our effective batch size is $2 * 2048 * 4 = 16384$ tokens per iteration. Because the task pair for each mini-batch is sampled according to our policy weights and (fixed) random seed, and each iteration consists of 8 unique mini-batches, a single parameter update can potentially contain information from up to 8 unique task pairs. We train each model for 100,000 iterations without early stopping, which takes about 40 hours per model. When evaluating we always use the final model checkpoint (i.e. the model parameters saved after 100,000 iterations). We use our in-house research NMT system, which is heavily based upon OpenNMT-py [@opennmt].
The sampling policy weights were specified manually by looking at the amount of available data for each pair, and estimating the difficulty of each translation direction. The result of the sampling policy is that lower resource language pairs are upsampled significantly. Table \[tab:dataset-information\] summarizes the statistics for each language pair. Note that the data in each row represents a *pair* of tasks, i.e. the total number of segments seen for <span style="font-variant:small-caps;">en-cs</span> is split evenly between <span style="font-variant:small-caps;">en</span>$\rightarrow$<span style="font-variant:small-caps;">cs</span>, and <span style="font-variant:small-caps;">cs</span>$\rightarrow$<span style="font-variant:small-caps;">en</span>. Because we train for only 100,000 iterations, we do not see all of the available training data for some high-resource language pairs.
With the exception of the <span style="font-variant:small-caps;">Prepend</span> system, the input to each model is identical. Each experimental setting is mutually exclusive, i.e. in the <span style="font-variant:small-caps;">Emb</span> setting we do not prepend task tokens, and in the <span style="font-variant:small-caps;">Attn</span> setting we do not use task embeddings.
Figure \[fig:training-dev-progress\] plots the validation performance during training on one of our validation datasets. The language embeddings from the <span style="font-variant:small-caps;">Emb</span> system are visualized in figure \[fig:language-emb-visualization\].
![Word-level accuracy on WMT EN-DE 2014 dev set as training progresses. The model which has a <span style="font-variant:small-caps;">DE</span>-specific decoder achieves the highest accuracy on this dev set.[]{data-label="fig:training-dev-progress"}](vis/dev_accuracy_by_model.pdf){width="50.00000%"}
![Language embeddings of the <span style="font-variant:small-caps;">Emb</span> system projected with UMAP [@mcinnes2018umap-software].[]{data-label="fig:language-emb-visualization"}](vis/umap_projected_lang_embeddings.png){width="50.00000%"}
Results
-------
We conduct four different evaluations of the performance of our models. First, we check performance on the 22 supervised pairs using dev and test sets from the WMT shared task. We then try to evaluate zero-shot translation performance in several ways. We use the TED talks multi-parallel dataset [@Ye2018WordEmbeddings] to create gold sets for all zero-shot pairs that occur in the TED talks corpus, and evaluate on those pairs. We also try two ways of evaluating zero-shot translation without gold data. In the first, we do round-trip translation $\textsc{SRC}\rightarrow \textsc{Pivot} \rightarrow {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\textsc{SRC}}}]{\kern.1pt\mathchar"0362\kern.1pt} {\rule{0ex}{\textheight}} }{\textheight}}{2.4ex}}\stackon[-6.9pt]{\textsc{SRC}}{\tmpbox}}$, and measure performance on the $({\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\textsc{SRC}}}]{\kern.1pt\mathchar"0362\kern.1pt} {\rule{0ex}{\textheight}} }{\textheight}}{2.4ex}}\stackon[-6.9pt]{\textsc{SRC}}{\tmpbox}}, \textsc{SRC})$ pair – this method is labeled <span style="font-variant:small-caps;">Zero-Shot Pivot</span>. In the second, we use parallel evaluation datasets from the WMT shared tasks (consisting of $(\textsc{SRC}, \textsc{REF})$ pairs), and translate $\textsc{SRC} \rightarrow \textsc{Pivot} \rightarrow {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\textsc{TRG}}}]{\kern.1pt\mathchar"0362\kern.1pt} {\rule{0ex}{\textheight}} }{\textheight}}{2.4ex}}\stackon[-6.9pt]{\textsc{TRG}}{\tmpbox}}$, then measure performance on the resulting $ ({\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\textsc{TRG}}}]{\kern.1pt\mathchar"0362\kern.1pt} {\rule{0ex}{\textheight}} }{\textheight}}{2.4ex}}\stackon[-6.9pt]{\textsc{TRG}}{\tmpbox}}, \textsc{REF}) $ pairs (see below for more details), where the pivot and target language pair is a zero-shot translation task – this method is labeled <span style="font-variant:small-caps;">Zero-Shot Parallel Pivot</span>[^5]
Table \[tab:evaluation-datasets\] lists the WMT evaluation dataset that we use for each language pair. In the <span style="font-variant:small-caps;">Zero-Shot Pivot</span> setting, the reference side of the dataset is used as input.
Table \[tab:global-bleu\] shows global results for all parallel tasks and all zero-shot tasks, by system. Global scores are obtained by concatenating the segmented outputs for each translation direction, and computing the BLEU score against the corresponding concatenated, segmented reference translations. The results in table \[tab:global-bleu\] are thus *tokenized* BLEU scores.
Parallel Tasks
--------------
Table \[tab:parallel-tasks\] lists results for all supervised task pairs from WMT 2019. For each pair, we report BLEU scores on de-tokenized output, and compute scores using sacrebleu [^6]. Therefore, we expect BLEU scores to be equivalent to those used in the WMT automatic evaluation.
We note that across all but the lowest-resource tasks, the model with a unique decoder for each language outperforms all others. However, for <span style="font-variant:small-caps;">en$\rightarrow$gu</span> and <span style="font-variant:small-caps;">en$\rightarrow$kk</span>, the lowest-resource translation directions, the unique decoder model fails completely, probably because the unique parameters for <span style="font-variant:small-caps;">kk</span> and <span style="font-variant:small-caps;">gu</span> were not updated by a sufficient number of mini-batches (approximately 15,600 for <span style="font-variant:small-caps;">en$\rightarrow$gu</span> and 14,800 for <span style="font-variant:small-caps;">en$\rightarrow$kk</span>).
Zero-shot Translation Tasks
---------------------------
In order to test our models in the zero-shot setting, we first create a multi-parallel dataset from the from the TED Talks multi-parallel corpus [@Ye2018WordEmbeddings], which has recently been used for the training and evaluation of multi-lingual models. We filter the dev and test sets of this corpus to find segments which have translations for all of <span style="font-variant:small-caps;">en, fr, ru, tr, de, cs, lt, fi</span>, and are at least 20 characters long, resulting in 606 segments. Because this corpus is pre-processed, we first de-tokenize and de-escape punctuation using `sacremoses`[^7]. We then evaluate zero-shot translation for all possible pairs which do not occur in our parallel training data, aggregate results are shown in the second row of table \[tab:global-bleu\].
We then adapt an evaluation technique that has recently been used for unsupervised MT – we translate from the source language into a pivot language, then back into the source language, and evaluate the score of the resulting source-language hypotheses against the original source [@lampleCDR18]. This technique allows us to evaluate for all possible translation directions in our multi-directional model.
Aware of the risk that the model simply copies through the original source segment instead of translating, we assert that at least 95% of pivot translations’ language code is correctly detected by `langid`[^8], and pairs which do not meet this criteria for any system are removed from the evaluation for all systems (not just for the system that failed). For all models except <span style="font-variant:small-caps;">Emb</span> only <span style="font-variant:small-caps;">ru$\rightarrow$kk$\rightarrow$ru</span> <span style="font-variant:small-caps;">fi$\rightarrow$lt$\rightarrow$fi</span>, and <span style="font-variant:small-caps;">zh$\rightarrow$gu$\rightarrow$zh</span> failed this test, but for the <span style="font-variant:small-caps;">Emb</span> model 31 of the 110 translation directions failed (see tables \[tab:failed-pivot-tasks\] and \[tab:pivot-translation-full-results\](in appendix)[^9]. This result indicates that models which use language embeddings may have a more “fuzzy” representation of the output task, and are much more prone to copying than other approaches to multi-lingual MT.
Finally, we conduct the <span style="font-variant:small-caps;">Zero-Shot Parallel Pivot</span> evaluation using the same datasets in table \[tab:evaluation-datasets\] by translating from <span style="font-variant:small-caps;">EN</span> (or <span style="font-variant:small-caps;">DE</span> in the case of <span style="font-variant:small-caps;">\*-FR</span>) to each possible pivot language, and then from the pivot language into the target language. Compared to the <span style="font-variant:small-caps;">Zero-Shot Pivot</span> setting, this evaluation should help to protect against the risk of copying, because source and reference segments are not from the same language. Aggregate results for this setting are shown in the third row of table \[tab:global-bleu\], full results in table \[tab:parallel-pivot-translation-full-results\] in appendix.
Discussion
----------
Our results show that a models with either (1) a completely unique decoders for each target language or (2) unique decoder attention parameters for each target language clearly outperform models with fully shared decoder parameters.
As shown in table \[tab:global-bleu\], the <span style="font-variant:small-caps;">Zero-Shot Pivot</span> evaluation is the outlier in our results, with the <span style="font-variant:small-caps;">Emb</span> system outperforming the others. Even for the languages which passed the language identification filter used in this evaluation, we suspect that some copying is occurring for the <span style="font-variant:small-caps;">Emb</span> system, because of the mismatch in results between the <span style="font-variant:small-caps;">Zero-Shot Pivot</span> task and the <span style="font-variant:small-caps;">Supervised, Zero-Shot TED</span>, and <span style="font-variant:small-caps;">Zero-shot Parallel Pivot</span> tasks (see table \[tab:global-bleu\]). Since the ranking of the models according to the <span style="font-variant:small-caps;">Zero-Shot Parallel Pivot</span> evaluation is well aligned with the <span style="font-variant:small-caps;">Zero-Shot TED</span> and <span style="font-variant:small-caps;">Supervised</span> evaluations which use gold parallel evaluation data, we believe that this method is effective for zero-shot evaluation of translation quality for language pairs where no gold data is available.
It is plausible that the language-independence of encoder output could be correlated with the amount of sharing in the decoder module. Because most non-English target tasks only have parallel training data in English, a unique decoder for those tasks only needs to learn to decode from English, not from every possible source task. However, our results show that the <span style="font-variant:small-caps;">Attn</span> model, which partially shares parameters across target languages only slightly outperforms the <span style="font-variant:small-caps;">Dec</span> model globally, because of the improved performance of the <span style="font-variant:small-caps;">Attn</span> model on the lowest-resource tasks (Table \[tab:parallel-tasks\], Table \[tab:pivot-translation-full-results\] (in appendix)). Thus we conclude that multi-lingual encoders still learn to share information across languages, even when trained using decoders that are unique to each target task.
Related Work
============
@dong-etal-2015-multi [@firat-etal-2016-multi; @ha-multilingual-2016; @johnson-google-2016] and others have shown that multi-way NMT systems can be created with minimal modification to the approach used for single-language-pair systems. @johnson-google-2016 showed that simply prepending a target-task token to source inputs is enough to enable zero-shot translation between language pairs for which no parallel training data is available.
Our work is most similar to @sachan-neubig-2018-parameter, where several different strategies for sharing decoder parameters are investigated for one-to-many translation models. However, their evaluation setting is constrained to one-to-many models which translate from English into two target languages, whereas our setting is more ambitious, performing multi-way translation between 11 languages. @blackwood-etal-2018-multilingual showed that using separate attention parameters for each task can improve the performance of multi-task MT models – this work was the inspiration for the <span style="font-variant:small-caps;">Attn</span> setting in our experiments.
Several recent papers focus specifically upon improving the zero-shot performance of multi-lingual MT models [@chen-etal-2017-teacher; @arivazhagan2019missing; @gu2019improved; @lu-etal-2018-neural; @Maruan-consistency; @sestorain2019zeroshot].
Concurrently with this work, [@aharoni-etal-2019-massively] evaluated a multiway MT system on a large number of language pairs using the TED talks corpus. However, they focus upon <span style="font-variant:small-caps;">EN-\*</span> and <span style="font-variant:small-caps;">\*-EN</span>, and do not test different model variants.
Conclusions and Future Work
===========================
We have presented results which are consistent with recent smaller-scale evaluations of multi-lingual MT systems, showing that assigning unique attention parameters to each target language in a multi-lingual NMT system is optimal when evaluating such a system globally. However, when evaluated on the individual task level, models which have unique decoder parameters for every target task tend to outperform other configurations, except when the amount of available training data is extremely small. We have also introduced two methods of evaluating zero-shot translation performance when parallel data is not available, and we conducted a large-scale evaluation of translation performance across all possible translation directions in the constrained setting of the WMT19 news-translation task.
In future work, we hope to continue studying how multi-lingual translation systems scale to realistic volumes of training data and large numbers of source and target tasks.
[^1]: <span style="font-variant:small-caps;">cs</span>, <span style="font-variant:small-caps;">de</span>, <span style="font-variant:small-caps;">en</span>, <span style="font-variant:small-caps;">fi</span>, <span style="font-variant:small-caps;">fr</span>, <span style="font-variant:small-caps;">gu</span>, <span style="font-variant:small-caps;">kk</span>, <span style="font-variant:small-caps;">lt</span>, <span style="font-variant:small-caps;">ru</span>, <span style="font-variant:small-caps;">tr</span> and <span style="font-variant:small-caps;">zh</span>
[^2]: Note we do not consider auto-encoding, thus the number of translation directions is $11^{2}-11=110$.
[^3]: <https://github.com/google/sentencepiece>
[^4]: Turkish (<span style="font-variant:small-caps;">tr</span>) is included from the 2018 language pairs because the task-organizers suggest the possibility of using <span style="font-variant:small-caps;">tr</span> data to improve <span style="font-variant:small-caps;">kk</span> performance
[^5]: For the <span style="font-variant:small-caps;">Zero-Shot Pivot</span> and <span style="font-variant:small-caps;">Zero-shot</span> parallel pivot evaluations we use the first 1000 segments of each dataset, because we need to translate twice for every possible pair.
[^6]: `BLEU+case.mixed+ lang.<src-lang>-<trg-lang>+ numrefs.1+smooth.exp+tok.<trg-lang>+ version.1.2.19`
[^7]: https://github.com/alvations/sacremoses
[^8]: https://github.com/saffsd/langid.py
[^9]: We conduct round trip translation on all 110 directions, but we only use directions that are (1) not available in the parallel training data, and (2) pass the language identification test to compute the global zero-shot translation performance.
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---
abstract: 'We study the evolution of the entanglement of noninteracting qubits coupled to reservoirs under monitoring of the reservoirs by means of continuous measurements. We calculate the average of the concurrence of the qubits wavefunction over all quantum trajectories. For two qubits coupled to independent baths subjected to local measurements, this average decays exponentially with a rate depending on the measurement scheme only. This contrasts with the known disappearance of entanglement after a finite time for the density matrix in the absence of measurements. For two qubits coupled to a common bath, the mean concurrence can vanish at discrete times. Our analysis applies to arbitrary quantum jump or quantum state diffusion dynamics in the Markov limit. We discuss the best measurement schemes to protect entanglement in specific examples.'
author:
- 'S. Vogelsberger'
- 'D. Spehner'
title: Average entanglement for Markovian quantum trajectories
---
Introduction
============
Entanglement is a key resource in quantum information. It can be destroyed or sometimes created by interactions with a reservoir. When the two non-interacting parts of a bipartite system are coupled to independent baths, entanglement typically disappears after a finite time [@Diosi04; @Dodd04; @Eberly04; @Almeida07]. This phenomenon, called “entanglement sudden death” (ESD), occurs for certain initial states only or for all entangled initial states, depending on whether the system relaxes to a steady state belonging to the boundary of the set of separable states (e.g., to a separable pure state for baths at zero temperature) or to its interior (e.g., to a Gibbs state at positive temperature) [@Terra_Cunha07]. A quantum state lies on this boundary if it is separable and an arbitrarily small perturbation makes it entangled; this is the case, for example, for a pure separable state. When the two parts of the system are coupled to a common bath, sudden revivals of entanglement may take place after the state has become separable [@Braun02; @Ficek06; @Mazzola09].
In this article we consider the loss of entanglement between two non-interacting qubits coupled to one or two baths monitored by continuous measurements. Because of these measurements, the qubits remain at all times in a pure state ${| \psi(t) \rangle}$, which evolves randomly. To each measurement result (or “realization”) corresponds a quantum trajectory $t \in \real_+ \mapsto {| \psi(t) \rangle}$ in the Hilbert space $\complex^4$ of the qubits. In the Born-Markov regime, the dynamics is given by the quantum jump (QJ) model [@Dalibard92; @Carmichael] or, in the case of homodyne and heterodyne detections, by the so-called quantum state diffusion (QSD) models [@Carmichael; @Wiseman93; @Gisin92]. We study how the entanglement of the qubits evolves in time by calculating the average $\overline{C_{\psi(t)}}$ of the Wootters concurrence of ${| \psi (t) \rangle}$ over all quantum trajectories; $\overline{C_{\psi(t)}}$ differs in general from the concurrence $C_{\rho(t)}$ of the density matrix $\rho(t)= \overline{{| \psi(t) \rangle \langle \psi(t) |}}$ (here and in what follows the overline denotes the mean over all quantum trajectories) [@Nha04; @Carvalho]. For two qubits coupled to [*independent*]{} baths, we find that $$\label{eq-main_result}
\overline{C_{\psi(t)}} = C_0 \,e^{-\kappa t}$$ where $C_0 = C_{\psi(0)}$ is the initial concurrence and $\kappa\geq 0$ depends on the measurement scheme but not on the initial state ${| \psi(0) \rangle}$. In particular, if $C_0>0$ and $t_{\rm ESD} \in ]0,\infty[$ is the time at which entanglement disappears in the density matrix (assuming that this time is finite), then $C_{\rho(t)}=0$ at times $t\geq t_{\rm ESD}$ whereas $\overline{C_{\psi(t)}}$ can only vanish asymptotically. The continuous measurements on the two baths thus protect on average the qubits from ESD. Of course, this does not mean that [*all*]{} random wavefunctions ${| \psi(t) \rangle}$ remain entangled at all times. But in some cases, such as for pure dephasing or for infinite temperature baths, one can find measurement schemes such that $\kappa =0$; then, for all trajectories, if the qubits are maximally entangled at $t=0$ they remain maximally entangled at all times. We show that the best measurement scheme to protect entanglement is in general given by homodyne detection with appropriately chosen laser phases. Related strategies using quantum Zeno effect [@Maniscalco08], entanglement distillation [@Mundarain09], quantum feedback [@CarvalhoPRA07], and encoding in qutrits [@Mascarenhas10] have been proposed. It is assumed in this work that the measurements on the baths are performed by perfect detectors. The impact of detection errors has been studied in [@Mascarenhas10].
When the qubits are coupled to a common bath, we find that $\overline{C_{\psi(t)}}$ has a more complex time behavior than in (\[eq-main\_result\]). It may vanish at finite discrete times, and, for some initial states, be equal to $C_{\rho(t)}$. It is worthwhile to stress that the formula (\[eq-main\_result\]) is valid provided not only each qubit is coupled to its own bath, but also the baths are monitored independently from each other by the measurements. This means that the measurements are performed [*locally*]{} on each bath. Instead of looking for the measurement scheme maximizing the average concurrence $\overline{C_{\psi(t)}}$ of the two qubits in order to obtain the best entanglement protection, it is also of interest to find a way to perform the measurements such that $\overline{C_{\psi(t)}}$ is minimum and coincides with the concurrence $C_{\rho(t)}$ of the density matrix. This problem has been studied numerically in Ref. [@Carvalho] and analytically in [@Viviescas-arXiv] for specific models of couplings with the two baths. Our result (\[eq-main\_result\]) implies that for any Markovian dynamics, if the two qubits are initially entangled and ESD occurs for the density matrix $\rho(t)$, a scheme with the aforementioned property must necessarily involve measurements of non-local (joint) observables of the two baths. In the models studied in [@Carvalho; @Viviescas-arXiv], non-local measurements are indeed used in order to obtain an optimal scheme satisfying $\overline{C_{\psi(t)}}=C_{\rho(t)}$.
The paper is organized as follows. We briefly recall in Sec. \[sec-entanglement\_measure\] the definition of the concurrence of pure and mixed states and review the quantum jump unraveling of a Lindblad equation for the density matrix in Sec. \[sec-QJM\]. We treat the simple and illustrative case of two two-level atoms coupled to independent baths at zero temperature in Sec. \[sec-photon\_counting\], before showing formula (\[eq-main\_result\]) in Sec. \[sec-proof\_main\_result\] for a general quantum jump dynamics. The QSD unravelings are considered in Sec. \[sec-QSD\]; we obtain the average concurrence for such unravelings as limits of the concurrence for QJ dynamics (corresponding to homodyne and heterodyne detections with intense laser fields). Section \[sec-common\_bath\] is devoted to the evolution of the entanglement of two qubits coupled to a common bath at zero temperature. The main conclusions of the work are given in Sec. \[sec-conclusion\].
Entanglement measures for quantum trajectories {#sec-entanglement_measure}
==============================================
The entanglement of formation of a bipartite quantum system ${\cal S}$ in a pure state ${| \psi \rangle}$ is defined by means of the von Neumann entropy $E_\psi = - {\operatorname{tr}}(\rho_A \ln \rho_A)=-{\operatorname{tr}}(\rho_B \ln \rho_B)$ of the density matrices $\rho_A = {\operatorname{tr}}_{B}( {| \psi \rangle \langle \psi |})$ and $\rho_B = {\operatorname{tr}}_{A}( {| \psi \rangle \langle \psi |})$ of the two subsystems $A$ and $B$ composing ${\cal S}$ [@Bennett96]. If ${\cal S}$ is in a mixed state, $E_\rho$ is the infimum of $\sum p_k E_{\psi_k}$ over all convex decompositions $\rho = \sum_k p_k {| \psi_k \rangle \langle \psi_k |}$ of its density matrix (with $p_k \geq 0$ and $\| \psi_k\|=1$). When $A$ and $B$ have two-dimensional Hilbert spaces, $E_\rho = f ( C_\rho)$ is related to the concurrence [@Wootters98] $C_\rho$ by a convex increasing function $f: [0,1]\rightarrow [0,\ln (2)]$; $\rho$ is separable if and only if $C_\rho=0$, [i.e.]{}, $E_\rho=0$. For a pure state [@Wootters98], $$\label{eq-def-concurrence}
C_\psi = | \langle \sigma_{y} \otimes \sigma_{y} T \rangle_{\psi} |$$ where $\sigma_y = {{\rm{i}}}({| \!\! \downarrow \rangle \langle \uparrow \!\! |}- {|\!\! \uparrow \rangle \langle \downarrow \!\! |})$ is the $y$-Pauli matrix, $T: {| \psi \rangle}
= \sum_{s,s'} c_{s s'} {| s,s' \rangle} \mapsto
\sum_{s,s'} c_{s s'}^\ast {| s,s' \rangle}$ the anti-unitary operator of complex conjugation in the canonical basis $\{{| s,s' \rangle} = {| s \rangle} \otimes {| s' \rangle} ; s,s' =\uparrow,\downarrow\}$ of $\complex^2 \otimes \complex^2$, and $\langle \cdot \rangle_{\psi} = {\langle \psi |} \cdot {| \psi \rangle}$ the quantum expectation in state ${| \psi \rangle}$.
For quantum trajectories, one has always $\overline{E_{\psi (t)}} \geq E_{\rho (t)}$, this inequality being strict excepted if the decomposition $$\label{eq-def_rho}
\rho (t) = \overline{{| \psi(t) \rangle \langle \psi (t) |}}
= \int {{\rm{d}}}p [\psi] \,{| \psi(t) \rangle \langle \psi (t) |}$$ realizes the infimum defining $E_{\rho (t)}$. Thanks to the convexity of $f$, $\overline{E_{\psi(t)}} \geq f(\overline{C_{\psi(t)}} )$. Thus equation (\[eq-main\_result\]) shows that for independent baths and if $C_0>0$, $\overline{E_{\psi(t)}}\geq f( C_0 {e}^{-\kappa t})>0$ whatever the measurement scheme.
It is legitimate to ask which entanglement measure should be averaged since, for example, $\overline{E_{\psi(t)}}=E_0$ could be constant and $\overline{C_{\psi(t)}}$ time-decreasing if $E_0 \not= 0, \ln 2$. The concurrence is a natural candidate as it corresponds for pure states to the supremum over all self-adjoint local observables $J_A$ and $J_B$ with norms less than one of the modulus of the correlation between $J_A$ and $J_B$, $$\begin{aligned}
\nonumber
C_{\psi (t)}
& = &
\sup_{\|J_A\|, \|J_B\| \leq 1}
\bigl| \langle J_A \otimes J_B \rangle_{\psi(t)}
\\
& & - \langle J_A \otimes 1_B\rangle_{\psi (t)} \langle 1_A \otimes J_B \rangle_{\psi (t)} \bigr|\,.\end{aligned}$$ Moreover, $\overline{C_{\psi (t)}}$ is easy to calculate in the Markov regime and gives a lower bound on $\overline{E_{\psi (t)}}$.
Quantum jump model {#sec-QJM}
==================
Let us briefly recall the QJ dynamics [@Dalibard92; @Breuer; @Knight98]. As a result of a measurement on a particle (e.g. a photon) of the bath scattered by the qubits, the qubits wavefunction suffers a quantum jump $$\label{eq-jump_dyn}
{| \psi(t) \rangle}
\longrightarrow
{| \psi_{\text{jump}}^{(m,i)} \rangle} = \frac{J_m^i {| \psi(t) \rangle}}{\| J_m^i {| \psi(t) \rangle}\| }$$ where the jump operator $J_m^i$ is related to the particle-qubits coupling and the indices $m,i$ label all possible measurement results save for the most likely one, which we call a “no detection”. In the weak coupling limit, the probability that a measurement in the small time interval $[t, t+{{\rm{d}}}t]$ gives the result $(m,i)$ is very small and equal to ${{\rm{d}}}p_m^i (t) = \gamma_m^i \| J_m^i {| \psi(t) \rangle}\|^2 {{\rm{d}}}t$. The jump rate $\gamma_m^i$ does not depend on ${| \psi(t) \rangle}$ and is proportional to the square of the particle-qubit coupling constant. In the no-detection case the wavefunction of the qubits evolves as $$\label{eq-no_jump}
{| \psi(t+{{\rm{d}}}t) \rangle}
\! = \!
\frac{e^{-{{\rm{i}}}H_{\rm{eff}} {{\rm{d}}}t } {| \psi(t) \rangle}}
{\| e^{-{{\rm{i}}}H_{\rm{eff}}{{\rm{d}}}t } {| \psi(t) \rangle}\|}
,
H_{\rm{eff}}\!=\! H_0-\!\frac{{{\rm{i}}}}{2} \sum_{m,i} \gamma_m^i J_m^{i \dagger} J_m^i$$ where $H_0$ is the Hamiltonian of the qubits. The probability that no jump occurs in the time interval $[t_0,t]$ is $p_{\text{nj}} (t_0,t)=\| e^{-{{\rm{i}}}H_{\rm{eff}} (t-t_0)} {| \psi (t_0) \rangle}\|^2 $. (This is proven as follows: as $p_{\text{nj}} (t_0,t) -p_{\text{nj}} (t_0,t + {{\rm{d}}}t) = \sum_{m,i} {{\rm{d}}}p_m^i (t)
p_{\text{nj}} (t_0,t)$, one has $\partial \ln p_{\text{nj}}(t_0,t) /\partial t
=$ $ -
\sum_{m,i}
\gamma_{m}^{i} \| J_m^i {| \psi(t) \rangle} \|^2
=
(\partial /\partial t) \ln \| e^{-{{\rm{i}}}H_{\rm{eff}} (t-t_0)} {| \psi (t_0) \rangle}\|^2$ by (\[eq-no\_jump\]).) It is not difficult to show [@Dalibard92] that the density matrix $\rho(t) =\overline{{| \psi(t) \rangle \langle \psi (t) |}}$ satisfies the Lindblad equation $$\label{eq-Lindblad}
\frac{{{\rm{d}}}\rho}{{{\rm{d}}}t} =
- {{\rm{i}}}[ H_0, \rho ]
+ \sum_{m,i} \gamma_m^i \Bigl( J_m^i \rho J_m^{i \dagger}
- \frac{1}{2} \bigl\{ J_m^{i \dagger} J_m^i , \rho \bigr\} \Bigr)$$ where $\{ \cdot , \cdot \}$ denotes the anti-commutator. It is known that many distinct QJ dynamics unravel the same master equation (\[eq-Lindblad\]) [@Breuer]. For two qubits coupled to independent reservoirs $R_A$ and $R_B$, the jump operators are [*local*]{}, [i.e.]{}, they have the form $$\label{eq-jump_op}
J_m^A \otimes 1_B
\quad , \quad
1_A \otimes J_{m}^B$$ depending on whether the measurements are performed on $R_A$ or $R_B$. Here $J_m^i$ are $2 \times 2$ matrices.
The aforementioned absence of ESD for the mean concurrence of two qubits coupled to independent baths can be traced back to the existence of trajectories for which ${| \psi(t) \rangle}$ remains entangled at all times. Actually, for a trajectory without jump, ${| \psi_{\rm{nj}} (t) \rangle} \propto {e}^{-{{\rm{i}}}t H_{\rm eff}}{| \psi (0) \rangle}$, see (\[eq-no\_jump\]). By (\[eq-jump\_op\]) and since the qubits do not interact with each other, $e^{-{{\rm{i}}}t H_{\rm eff}}$ is the tensor product of two local operators acting on each qubit. If ${| \psi_{\rm{nj}}(t) \rangle}$ would be separable at a given time $t$ then, by reversing the dynamics ([i.e.]{}, by applying ${e}^{{{\rm{i}}}t H_{\rm eff}}$ to ${| \psi_{\rm{nj}}(t) \rangle}$) one would deduce that ${| \psi (0) \rangle}$ is separable. Hence $C_{\psi_{\rm{nj}}(t)}>0$ if $C_0>0$. But the no-detection probability between times $0$ and $t$ is nonzero and thus $\overline{C_{\psi(t)}}>0$ at all times. Note that this argument does not apply if non-local observables of the two baths are measured or if the two qubits are coupled to a common bath, since then the jump operators are non-local.
Photon counting {#sec-photon_counting}
===============
Let us illustrate the random dynamics described previously on a simple and experimentally relevant example [@Haroche]. Each qubit is a two-level atom coupled resonantly to the electromagnetic field initially in the vacuum (zero-temperature photon bath). The two atoms are far from each other and thus interact with independent field modes. Two perfect photon counters $D_i$ make a click when a photon is emitted by qubit $i$ ($i=A,B$), whatever the direction of the emitted photon. Doing the rotating wave approximation, the jump operators are $J^i_{-} = \sigma_{-}^i= {| \!\! \downarrow \rangle \langle \uparrow \!\! |}$. For simplicity we take $H_0=0$. By (\[eq-no\_jump\]), if no photon is detected in the time interval $[0,t]$ the qubits state is $${| \psi (t) \rangle}
= {{\cal N}}(t)^{-1}\sum_{s,s'=\uparrow ,\downarrow} c_{s s'} \,{e}^{-\gamma_{s s'} t/2}\,{| s,s' \rangle}$$ with $\gamma_{\uparrow \uparrow} = \gamma_A + \gamma_B$, $\gamma_{\uparrow \downarrow} = \gamma_A$, $\gamma_{\downarrow \uparrow} = \gamma_B$, $\gamma_{\downarrow \downarrow} = 0$ ($\gamma_i$ being the jump rate for detector $D_i$), $c_{s s'} = {\langle s,s' | \psi(0) \rangle}$, and ${{\cal N}}(t)^2 =
\sum_{s,s'} |c_{s s'} |^2 {e}^{-\gamma_{s s'} t}$. The concurrence (\[eq-def-concurrence\]) of ${| \psi (t) \rangle}$ is $C(t) = C_0 \,{{\cal N}}(t)^{-2} e^{-(\gamma_A+\gamma_B) t/2}$ with $C_0 = 2 | c_{\uparrow \uparrow} c_{\downarrow \downarrow}
- c_{\uparrow \downarrow} c_{\downarrow \uparrow}|$. If a photon is detected at time $t_j$ by, say, the photon counter $D_A$, the qubits are just after the jump (\[eq-jump\_dyn\]) in the separable state ${| \psi(t_j +) \rangle}
\propto$ ${|\!\! \downarrow \rangle}\otimes ( c_{\uparrow \uparrow} e^{-\gamma_{\uparrow \uparrow} t_j/2} {{| \!\! \uparrow \rangle}}
+ c_{\uparrow \downarrow} e^{-\gamma_{\uparrow \downarrow} t_j/2} {|\!\! \downarrow \rangle})$. Since neither a jump nor the inter-jump dynamics can create entanglement (the jump operators (\[eq-jump\_op\]) being local), ${| \psi(t) \rangle}$ remains separable at all times $t \geq t_j$, even if more photons are subsequently detected. Thus $C(t) =0$ if at least one photon is detected in the time interval $[0,t]$. Averaging over all realizations of the quantum trajectories and using the probability $p_{\text{nj}} (0,t)={{\cal N}}(t)^2$ that no photon is detected in $[0,t]$, one finds $\overline{C(t)} =C_0 e^{-(\gamma_A + \gamma_B) t/2}$.
This argument is easily extended to baths at positive temperatures by adding two jump operators $J_{+}^i=\sigma_{+}^i$ with rates $\gamma_+^i \leq \gamma_{-}^i$. The mean concurrence is then $
\overline{C(t)} =C_0 e^{-(\gamma_+^A + \gamma_-^A + \gamma_+^B + \gamma_-^B ) t/2}
$. It is compared in Fig. \[fig-concurrence\_2\_baths\_T>0\] with the concurrence of the density matrix obtained by solving the master equation (\[eq-Lindblad\]), which shows ESD for all initial states.
![\[fig-concurrence\_2\_baths\_T>0\] (Color online) Concurrences of two qubits coupled to independent baths at positive temperature as a function of $\gamma t$ for $\gamma_{+}^i = \gamma_{-}^i/2=\gamma$ and ${| \psi(0) \rangle}=({| \!\! \uparrow \uparrow \rangle}-{{\rm{i}}}{|\!\! \downarrow \downarrow \rangle})/\sqrt{2}$: (2a) $C_{\rho(t)}$ for the density matrix (blue dashed line); (2b) $C_{\psi (t)}$ for a single trajectory (black dotted line); (2c) $\overline{C_{\psi (t)}}$ averaged over 1500 trajectories and from Eq. (\[eq-main\_result\]) (red solid lines); (2d) $\overline{C_{\psi (t)}}$ for the best measurement scheme (see text). ](fig12.eps){width="1\columnwidth"}
General quantum jump dynamics {#sec-proof_main_result}
=============================
We now consider a general QJ dynamics with jump operators given by (\[eq-jump\_op\]). The Hamiltonian of the qubits has the form $H_0 = H_{A} \otimes 1_B + 1_A\otimes H_{B}$. Let $K= K_A \otimes 1_B + 1_A \otimes K_B$ with $$K_i=\frac{1}{2} \sum_m \gamma_{m}^{i}\, J_m^{i \dagger} J_m^i\,,$$ $\gamma_{m}^{i}$ being the jump rates for the detector $D_i$ ($i=A,B$). We first assume that no jump occurs between $t$ and $t+{{\rm{d}}}t$. By expanding the exponential in (\[eq-no\_jump\]), one gets $$\begin{aligned}
\label{eq-C_nj}
& & C (t+{{\rm{d}}}t)
=
p_{\text{nj}} (t,t+{{\rm{d}}}t)^{-1}
\bigl| \bigl\langle \sigma_{y} \otimes \sigma_{y} T \bigr\rangle_{\psi (t)}
\\
& &
{\nonumber}+{{\rm{i}}}{{\rm{d}}}t \bigl\langle H_{\rm{eff}}^\dagger \sigma_{y} \otimes \sigma_{y} T
+ \sigma_{y} \otimes \sigma_{y} T H_{\rm{eff}} \bigr\rangle_{\psi (t)} + {{\cal O}}({{\rm{d}}}t)^2\bigr| \end{aligned}$$ where $p_{\text{nj}} (t,t+{{\rm{d}}}t)=
\langle 1 - 2 K {{\rm{d}}}t +{{\cal O}}({{\rm{d}}}t)^2 \rangle_{\psi (t)}$ is the probability that no jump occurs between $t$ and $t + {{\rm{d}}}t$. Now, for any local operator $O_i$ acting on qubit $i$, one has $$\bigl\langle O_i \sigma_{y} \otimes \sigma_{y} T \bigr\rangle_{\psi (t)}
\!\!=\! \bigl\langle \sigma_{y} \otimes \sigma_{y} T O_i^\dagger \bigr\rangle_{\psi (t)}
\!\!=\! \frac{ {\cal{C}} (t)}{2}{{\operatorname{tr}}}_{\complex^2} (O_i)$$ with $$\label{eq-D(t)}
{\cal{C}} (t) = \langle \sigma_{y} \otimes \sigma_{y} T \rangle_{\psi (t)}
\!= \!2 \bigl(
c_{\uparrow \downarrow}^\ast (t) c_{\downarrow \uparrow}^\ast (t)
- c_{\uparrow \uparrow}^\ast(t) c_{\downarrow \downarrow}^\ast (t)
\bigr)$$ and $ c_{s s'} (t) = {\langle s,s' | \psi (t) \rangle}$. Since $C(t) = | {\cal{C}} (t)|$ and $H_{\rm{eff}}=\sum_i (H_i - {{\rm{i}}}K_i)$, this gives $$\label{eq-conc_increment_no_jump}
C (t+{{\rm{d}}}t) \,p_{\text{nj}} (t,t+{{\rm{d}}}t)
= C (t) \Bigl( 1 - {\operatorname{tr}}_{\complex^4} (K) \frac{{{\rm{d}}}t}{2} + {{\cal O}}({{\rm{d}}}t^2) \Bigr)
\,.$$ If detector $D_i$ gives the result $m$ in the time interval $[t,t + {{\rm{d}}}t]$, the concurrence is by virtue of (\[eq-jump\_dyn\]) $$\label{eq-conc_increment_jump}
C_{\text{jump}}^{(m,i)} (t+{{\rm{d}}}t)
=
\frac{\gamma_{m}^{i} \,{{\rm{d}}}t}{{{\rm{d}}}p_{m}^{i} (t)} C(t)
\bigl| {\det}_{\complex^2} (J_m^i) \bigr|
\,,$$ where we have used the identity $$\langle O_i^\dagger \sigma_y \otimes \sigma_y T O_i \rangle_{\psi (t)}
= {\cal{C}}(t) {\det}_{\complex^2}(O_i^\dagger )$$ valid for any local operator $O_i$ acting on qubit $i$. Collecting the previous formulas and using the Markov property of the jump process, one gets $\overline{C(t+{{\rm{d}}}t)} = \overline{C(t)} ( 1 - \kappa_{\rm{QJ}} \,{{\rm{d}}}t + {{\cal O}}({{\rm{d}}}t^2))$ with $$\label{eq-kappa}
\kappa_{\rm{QJ}}
=
\frac{1}{2} {\operatorname{tr}}_{\complex^4} ( K ) -
\sum_{m,i} \gamma_{m}^{i}
| {\det}_{\complex^2} (J_m^i) | \,.$$ Letting ${{\rm{d}}}t$ go to zero, one obtains ${{\rm{d}}}\overline{C(t)}/{{\rm{d}}}t = - \kappa_{\rm{QJ}} \,\overline{C(t)}$. The solution (\[eq-main\_result\]) of this differential equation has the exponential decay claimed previously. To show that $\kappa_{\rm{QJ}} \geq 0$, let $2\theta_m^i$ be the argument of $\det_{\complex^2} (J_m^i)$. We write $\kappa_{\rm{QJ}}= \sum_{m,i} \gamma_m^i
[ {{\operatorname{tr}}}_{\complex^2} (J_m^{i \dagger} J_m^i ) - 2 \Re \{ {e}^{-2 {{\rm{i}}}\theta_m^i}\,
{\det}_{\complex^2} (J_m^i)\}]/2$ as $$\label{eq-kappa_m}
\kappa_{\rm{QJ}}
=
\sum_{m,i} \frac{\gamma_m^i}{2}
\Bigl(
\bigl|
{\langle \uparrow \!\! |}\tilde{J}^i_{m} {| \!\! \uparrow \rangle}- {\langle\downarrow \!\! |}\tilde{J}^{i \dagger}_{m} {|\!\! \downarrow \rangle}\bigr|^2
+
\bigl|
{\langle \uparrow \!\! |}2 \Re \tilde{J}^i_{m} {|\!\! \downarrow \rangle}\bigr|^2
\Bigr)$$ with $\tilde{J}^i_{m}= {e}^{-{{\rm{i}}}\theta_m^i} J_m^i$ and $2 \Re\tilde{J}^i_{m}= \tilde{J}^i_{m}+\tilde{J}^{i \dagger}_{m}$. Thus $\kappa_{\rm{QJ}}$ is non-negative.
Note that $\kappa_{\rm{QJ}} =0$ if all matrices $J_m^i$ are self-adjoint and traceless (then $\theta_m^i=\pi/2$ and $\Re \tilde{J}^i_{m}=0$). We show in Fig. \[fig-concurrence\_2\_baths\_deph\] the concurrence of the density matrix given by solving (\[eq-Lindblad\]) for a pure dephasing with $J^i = {e}^{{{\rm{i}}}\pi/4} \sigma_{-}^i + {e}^{-{{\rm{i}}}\pi/4} \sigma_{+}^i$. One has ESD for all initial states save for ${| \psi (0) \rangle} = ({| \!\! \uparrow \uparrow \rangle}\pm {{\rm{i}}}{|\!\! \downarrow \downarrow \rangle})/\sqrt{2}$. Since $\kappa_{\rm{QJ}}=0$, (\[eq-main\_result\]) implies $\overline{C (t)} = C_0$. If the two qubits are maximally entangled at $t=0$, then $C_{\psi (t)}=\overline{C (t)} = C_0 = 1$ for [*all*]{} quantum trajectories at any time $t \geq 0$. Therefore, for pure dephasing one can protect perfectly the qubits by measuring continuously and locally the two independent baths.
We can now give the optimal measurement scheme to protect the entanglement of two qubits coupled to independent baths at positive temperatures. Let us replace the photon-counting jump operators $J_\pm^i = \sigma_\pm^i$ by $J_\mu^i = \sum_{m=\pm} (\gamma^i_{m}/\gamma_\mu^i)^{\frac{1}{2}} u_{\mu m}^i \sigma_m^i$ where $U_i=(u_{\mu m}^i)_{\mu=1,\cdots, N}^{m=\pm}$ are unitary $2 \times N$ matrices. This corresponds to a rotation of the measurement basis and gives another unraveling of the master equation (\[eq-Lindblad\]). Let us stress that the new jump operators $J_\mu^i$ still act locally on each qubit. By (\[eq-kappa\]), the new rate is $\kappa
= \sum_{\mu,i} ( \sqrt{\gamma_{-}^i} | u_{\mu -}^i | - \sqrt{\gamma_+^i} | u_{\mu+}^i | )^2/2$. By using $\sum_\mu |u_{\mu\,\pm}^i |^2 = 1$ and optimizing over all unitaries $U_i$, one finds that the smallest disentanglement rate arises when, for example, $u_{1\,\pm}^i=\pm u_{2\,\pm}^i=1/\sqrt{2}$ ($N=2$) and is given by $$\label{eq-optimal_kappa_T>0}
\kappa_{\rm{QJ}}^{\rm{opt}} =
\frac{1}{2} \sum_{i=A,B} \Bigl( \sqrt{\gamma_{-}^i} - \sqrt{\gamma_{+}^i} \Bigr)^2\,.$$ Note that ${\kappa}_{\rm{QJ}}^{\rm{opt}}={\kappa}_{\rm{QJ}}$ at zero temperature and ${\kappa}_{\rm{QJ}}^{\rm{opt}}=0$ (perfect protection) at infinite temperature. The decay of $\overline{C(t)}$ for this optimal measurement is shown in Fig. \[fig-concurrence\_2\_baths\_T>0\] (green dashed-dotted line).
![\[fig-concurrence\_2\_baths\_deph\] (Color online) Same as in Fig.\[fig-concurrence\_2\_baths\_T>0\] for pure dephasing and the initial state ${| \psi(0) \rangle}=\frac{1}{\sqrt{2}}({| \!\! \uparrow \uparrow \rangle}+ e^{-{{\rm{i}}}\varphi} {|\!\! \downarrow \downarrow \rangle})$: (1a) $C_{\rho(t)}$ for $\varphi=\frac{\pi}{2}$ (blue dashed line); (1a’) $C_{\rho(t)}$ for $\varphi=0$ (blue line showing ESD); (1b,1c) $C_{\psi (t)}=\overline{C_{\psi (t)}}$ (red solid line). ](fig11.eps){width="1\columnwidth"}
Homodyne and heterodyne detection {#sec-QSD}
=================================
Let us come back to our example of two atoms coupled to the electromagnetic field initially in the vacuum. If homodyne photo-detection is used instead of photon counting, the jump operators become $J^{i}_{\pm \alpha}= \sigma_{-}^i \pm \alpha_i$, $\alpha_i$ being the amplitude of a classical laser field (there are now four jump operators since each homodyne detector involves two photon counters) [@Wiseman93]. Assuming that the two photon beams emitted by the atoms are combined with the two laser fields via 50% beam splitters, the jump rates associated with $J^{i}_{\pm \alpha}$ are equal, $\gamma^{i}_{\pm \alpha} =\gamma_i/2$. Thanks to (\[eq-kappa\]), one easily finds that the disentanglement rate for the new QJ dynamics, $\kappa_{\rm{QJ}} (\alpha)=(\gamma_A + \gamma_B)/2$, is the same as for photon counting.
In contrast, $\kappa_{\rm{QJ}} (\alpha)$ depends on the laser amplitudes for pure dephasing (jump operators $J^i_{\pm \alpha}={{\bf{v}}}_i \cdot {{\mathbf{\sigma}}}\pm \alpha_i$ with ${{\bf{v}}}_i \in \real^3$, $\| {{\bf{v}}}_i \|=1$, and ${{\mathbf{\sigma}}}$ the vector formed by the Pauli matrices $\sigma_x$, $\sigma_y$, and $\sigma_z$): then $\kappa_{\rm{QJ}} (\alpha)=2\sum_i \gamma_i \min \{ \alpha_i^2,1\}$ for real $\alpha_i$’s. One reaches perfect entanglement protection ($\overline{C(t)}=C_0$) only for vanishing laser intensities $\alpha_i^2$. In the case of two qubits coupled to two baths at positive temperatures, a general choice of jump operators such that the density matrix (\[eq-def\_rho\]) satisfies the master equation (\[eq-Lindblad\]) with the four Lindblad operators $\sigma_{\pm}^i$, $i=A,B$, is $J_{\mu,\pm \alpha}^i = J_\mu^i \pm \alpha_\mu^i$ with the jump rates $\gamma_{\mu,\pm \alpha}^i= \gamma_\mu^i/2$, laser amplitudes $\alpha_\mu^i \in \complex$, and $J_\mu^i=
\sum_{m=\pm} (\gamma^i_{m}/\gamma_\mu^i)^{\frac{1}{2}} u_{\mu m}^i \sigma_m^i$ for an arbitrary unitary matrix $(u_{\mu m}^i)_{\mu=1,\cdots, N}^{m=\pm}$ (see the discussion in the preceding section). The corresponding disentanglement rate, $\kappa_{\rm{QJ}} (\alpha)= \sum_{\mu,i} \gamma_\mu^i
[ {\operatorname{tr}}_{\complex^2} (J_\mu^{i \dagger} J_\mu^i ) + 2 | \alpha_\mu^i |^2
- 2 | \det_{\complex^2} (J_\mu^i )
+ (\alpha_\mu^i )^2 |]/2$, is equal to $\kappa_{\rm{QJ}} (0)$ if $\det ( J_\mu^i)=0$ or for complex laser amplitudes $\alpha_\mu^i = |\alpha_\mu^i| e^{{{\rm{i}}}\theta_\mu^i}$ satisfying $2\theta_\mu^i = \arg (\det ( J_\mu^i))$; otherwise, $\kappa_{\rm{QJ}} (\alpha)$ is larger then $\kappa_{\rm{QJ}} (0)$. We can conclude that the smallest disentanglement rate is given by (\[eq-optimal\_kappa\_T>0\]) and the best unravelings to protect the entanglement of the qubits are either the QJ model with jump operators $J_1^i \propto (\gamma_+^i)^{\frac{1}{2}} \sigma_+^i + (\gamma_-^i)^{\frac{1}{2}} \sigma_-^i$ and $J_2^i \propto (\gamma_+^i)^{\frac{1}{2}} \sigma_+^i - (\gamma_-^i)^{\frac{1}{2}} \sigma_-^i$ or the corresponding homodyne unraveling with laser phases $\theta_1^i=\pi/2$ and $\theta_2^i = 0$.
Let us now consider a general QJ model with jump operators $J_m^i$. A new unraveling of (\[eq-Lindblad\]) is obtained from the QJ model with jump operators $J_{m,\pm \alpha}^{i} = J_{m}^i \pm \alpha^i_m$ and rates $\gamma_{m,\pm \alpha}^{i}
= \gamma_m^i/2$. For large positive laser amplitudes $\alpha_m^i \gg 1$, this dynamics converges after an appropriate coarse graining in time to the QSD model described by the stochastic Schrödinger equation [@Wiseman93; @moi+Orszag] $$\begin{aligned}
\label{eq-QSD}
{\nonumber}& &
\displaystyle {| {{\rm{d}}}\psi \rangle}
=
\Bigl[ ( -{{\rm{i}}}H_0 - K){{\rm{d}}}t
+
\sum_{m,i} \Bigr( \sqrt{\gamma^i_m } \bigl( J^{i}_m - \Re \langle J^{i}_m \rangle_{\psi}
\bigr)
\\
& &
\times {{\rm{d}}}w_{m}^i
+ \gamma^i_m \Bigl( \Re \langle J_{m}^{i} \rangle_{\psi}\, J_{m}^{i}
- \frac{(\Re \langle J^{i}_m \rangle_{\psi})^2}{2} \Bigr){{\rm{d}}}t
\Bigr)
\Bigr] {| \psi \rangle}\end{aligned}$$ where ${{\rm{d}}}w_{m}^i$ are the Itô differentials for independent real Wiener processes satisfying the Itô rules ${{\rm{d}}}w_{m}^i {{\rm{d}}}w_{n}^j = \delta_{ij} \delta_{mn} {{\rm{d}}}t$. One can determine the mean concurrence for the QSD model (\[eq-QSD\]) by taking the limit of the mean concurrence for the QJ dynamics with jump operators $J_{m ,\pm \alpha}^i$. This gives again the exponential decay (\[eq-main\_result\]) but with a new rate $$\label{eq-kappaQDS}
\kappa_{\rm{ho}}
=
\frac{ {\operatorname{tr}}_{\complex^4} ( K )}{2} -
\sum_{m,i} \! \gamma_{m}^{i}
\Bigl(
\Re {\det}_{\complex^2} (J_m^i)
+ \frac{1}{2} \bigl( \Im {{\operatornamewithlimits{{\operatorname{tr}}}}}_{\complex^2} (J_m^i) \bigr)^2
\Bigr).$$ In fact, if $2 \theta_{m,\pm \alpha}^i$ is the argument of $\det ( J_{m}^i\pm \alpha_m^i ) = (\alpha_m^i)^2\pm \alpha_m^i {\operatorname{tr}}(J_m^i) +
{\mathcal{O}} (1)$ then for $\alpha_m^i \gg 1$, $\alpha_m^i >0$, one has $e^{2 {{\rm{i}}}\theta_{m,\pm \alpha}^i} \sim 1 \pm {{\rm{i}}}\Im {\operatorname{tr}}( J_m^i ) /\alpha_m^i$. Using (\[eq-kappa\_m\]), a short calculation gives (\[eq-kappaQDS\]).
Unlike $\kappa_{\rm{QJ}}$, $\kappa_{\rm{ho}}$ changes when the operators $J_m^i$ in (\[eq-QSD\]) acquire a phase factor, $J_m^i \rightarrow {e}^{-{{\rm{i}}}\theta_m^i}J_m^i$. This arises for homodyne detection with complex laser amplitudes $\alpha_m^i=|\alpha_m^i| {e}^{{{\rm{i}}}\theta_m^i}$, $|\alpha_m^i|\gg 1$. Minimizing over the laser phases $\theta_m^i$ yields $$\begin{aligned}
\label{eq-kappa_opt_hom}
\nonumber
\kappa_{\rm{ho}}^{\rm{opt}}
& = &
\frac{1}{2} {\operatorname{tr}}_{\complex^4} (K) - \sum_{m,i} \gamma_m^i
\Bigl( \bigl|\det_{\complex^2} (J_m^i) - \frac{1}{4} \bigl( {\operatorname{tr}}_{\complex^2}
(J_m^i) \bigr)^2 \bigr|
\\
& &
+\frac{1}{4} \bigl| {\operatorname{tr}}_{\complex^2} (J_m^i) \bigr|^2
\Bigr)\,.\end{aligned}$$ It is easy to show that $\kappa_{\rm{ho}}^{\rm{opt}} \leq \kappa_{\rm{QJ}}$, this inequality being strict excepted if the two eigenvalues of $J_m^i$ have the same modulus for all $(m,i)$. Thus optimal homodyne detection protects entanglement better than - or, if the aforementioned condition is fulfilled, as well as - photon counting. Let us stress that the optimal measurements (in particular, the laser phases $\theta_m^i$ minimizing the rate $\kappa_{\rm{ho}}$) only depend on the Lindblad operators $J_m^i$ in the master equation (\[eq-Lindblad\]) and are thus the same for all initial states of the qubits.
Let us now discuss the case of heterodyne detection. The corresponding jump operators $J^i_{m,\pm \alpha} (t_q) = J^i_m \pm \alpha_m^i e^{{{\rm{i}}}\Omega_m^i t_q }$ depend on the time $t_q$ of the $q$-th jump due to the oscillations of the laser amplitudes [@Knight98]. The associated rates are $\gamma_{m,\pm \alpha}^i = \gamma_m^i /2$ as for homodyne detection. We assume here that $\alpha_m^i >0$. In the limit $(\alpha_m^i)^2 \gg \Omega_m^i/\gamma_m^i \gg 1$ of large laser intensities and rapidly oscillating laser amplitudes, the QJ dynamics with jump operators $J^i_{m,\pm \alpha} (t_q)$ converges to the QSD model given by the stochastic Schrödinger equation [@Breuer] $$\begin{aligned}
\label{eq-heter}
{\nonumber}& &
\displaystyle {| {{\rm{d}}}\psi \rangle}
=
\Bigl[( -{{\rm{i}}}H_0 - K){{\rm{d}}}t
+
\frac{1}{2} \sum_{m,i}
\gamma_m^{i} \Bigl( \langle J_{m}^{i} \rangle_{\psi}^\ast \, J_{m}^{i}
\\
{\nonumber}& &
- \frac{1}{2} \bigl| \langle J^{i}_m \rangle_{\psi} \bigr|^2 \Bigr){{\rm{d}}}t
+ \sum_{m,i} \sqrt{\gamma^i_m }
\Bigl(
\bigl( J^{i}_m - \frac{1}{2} \langle J^{i}_m \rangle_{\psi} \bigr) {{\rm{d}}}\xi_m^i
\\
& &
- \frac{1}{2} \langle J^{i}_m \rangle_{\psi}^\ast ( {{\rm{d}}}\xi_m^i)^\ast
\Bigr)
\Bigr] {| \psi \rangle}\end{aligned}$$ where ${{\rm{d}}}\xi_m^i$ are the Itô differential of independent complex Wiener processes satisfying the Itô rules ${{\rm{d}}}\xi_m^i {{\rm{d}}}\xi_n^j =0$ and ${{\rm{d}}}\xi_m^i ( {{\rm{d}}}\xi_n^j)^\ast =\delta_{ij} \delta_{mn} {{\rm{d}}}t$. Eq. (\[eq-heter\]) describes the coarse-grained evolution of the normalized wavefunction ${| \psi (t) \rangle}$ on a time scale $\Delta t$ such that (i) many jumps and many laser amplitude oscillations occur in a time interval of length $\Delta t$ and (ii) ${| \psi (t) \rangle}$ does not change significantly on such a time interval. These conditions are satisfied when $(\alpha_m^i )^2 \gamma_m^i \Delta t \gg \Omega_m^i \Delta t \gg 1$ and $\gamma_m^i \Delta t \ll 1$. We now show that the mean concurrence for the QSD model (\[eq-heter\]) is given by (\[eq-main\_result\]) and determine the rate $\kappa$ of its exponential decay. This can be done by calculating the derivative ${{\rm{d}}}\overline{C (t)}/{{\rm{d}}}t$ in a similar way as in Sec. \[sec-proof\_main\_result\], using (\[eq-heter\]) and the Itô rules. It turns out to be simpler to estimate directly the average concurrence of the QJ model for heterodyne detection in the aforementioned limits, in analogy with our previous analysis for homodyne detection. Let us first remark that the results of Sec. \[sec-proof\_main\_result\] remain valid if the jump operators $J_m^i(t)$ vary slowly in time, on a time scale $(\Omega_m^i)^{-1}$ much larger than the mean time $(\alpha_m^i )^{-2}/\gamma_m^i$ between consecutive jumps. Hence ${{\rm{d}}}\overline{C} /{{\rm{d}}}t = - \kappa_{\rm{het}} (t)\, \overline{C}(t)$ and thus $\overline{C(t)} = C_0 \, e^{- \int_0^t {{\rm{d}}}t'\, \kappa_{\rm{het}} (t')}$ with a time-dependent rate $\kappa_{\rm{het}} (t)$ given by (\[eq-kappa\_m\]). To simplify notations, we temporarily omit the sum in (\[eq-kappa\_m\]) and do not write explicitly the lower and upper indices $m$ and $i$. Let us set $\tau = {\operatorname{tr}}( J )/2 = |\tau| e^{{{\rm{i}}}\varphi}$ and $\delta = \det (J) = e^{2{{\rm{i}}}\theta} | \delta|$. Let $2 \theta_{\pm \alpha} (t)$ denote the argument of $\det ( J \pm \alpha e^{{{\rm{i}}}\Omega t})$. Generalizing the calculation outlined above for homodyne detection, one gets $e^{2 {{\rm{i}}}\theta_{\pm \alpha} (t)} \sim
e^{2 {{\rm{i}}}\Omega t} ( 1 \pm 2 i \Im \{ \tau\, e^{-{{\rm{i}}}\Omega t} \}/\alpha)$ as $\alpha \gg 1$. By (\[eq-kappa\_m\]) this yields $$\begin{aligned}
& & \kappa_{\rm{het}} (t)
=
\frac{\gamma}{2}
\Bigl(
\bigl|
{\langle \uparrow \!\! |}{J} {| \!\! \uparrow \rangle}- e^{2 {{\rm{i}}}\Omega t} {\langle\downarrow \!\! |}J^{\dagger} {|\!\! \downarrow \rangle}\\
& &
\hspace*{1.2cm} - 2 {{\rm{i}}}e^{{{\rm{i}}}\Omega t}\, \Im \{ \tau e^{-{{\rm{i}}}\Omega t} \}
\bigr|^2 +
\bigl|
{\langle \uparrow \!\! |}\bigl( {J} + e^{2 {{\rm{i}}}\Omega t } J^{\dagger} \bigr) {|\!\! \downarrow \rangle}\bigr|^2
\Bigr)
\\
& & =
\frac{{\operatorname{tr}}_{\complex^4} (K)}{2} - \gamma
\bigl(
| \delta | \cos ( 2 \theta - 2 \Omega t ) + 2 | \tau |^2 \sin^2 (\varphi - \Omega t )
\bigr)
\end{aligned}$$ up to terms of order $\alpha^{-1}$. By neglecting the oscillatory integral $\int_{0}^{t} {{\rm{d}}}t'\,\cos (2 \theta - 2 \Omega t')$ (which is of order $\Omega^{-1} \ll \Delta t \leq t$) and approximating $\int_{0}^{t} {{\rm{d}}}t'\,\sin^2 ( \varphi - \Omega t')$ by $t/2$, one obtains $
\int_{0}^{t} {{\rm{d}}}t' \, \kappa_{\rm{het}} (t') \simeq
t
({\operatorname{tr}}_{\complex^4} ( K) /2 - \gamma |\tau |^2 )
$. Putting together the previous results, this shows that $\overline{C(t)} \rightarrow C_0 e^{-\kappa_{\rm{het}} t}$ in the limit $\alpha^2 \gg \Omega/\gamma \gg 1$ and $\Omega \gg (\Delta t)^{-1} \gg \gamma$, with $$\label{eq-kappa_het}
\kappa_{\rm{het}}
=
\frac{ {\operatorname{tr}}_{\complex^4} ( K )}{2} -
\frac{1}{4} \sum_{m,i} \gamma_{m}^{i}
\bigl| {\operatorname{tr}}(J_m^i) \bigr|^2
\, .$$ We note that $\kappa_{\rm{het}} \geq \kappa_{\rm{ho}}^{\rm{opt}}$. For given jump operators $J_m^i$, the measurement scheme which better protects the qubits against disentanglement is thus given by homodyne detections with optimally chosen laser phases. In this scheme, the average concurrence decays exponentially with the rate (\[eq-kappa\_opt\_hom\]).
Although (\[eq-heter\]) is different from the QSD equation for the normalized wavefunction introduced by Gisin and Percival [@Gisin92], the quantum trajectories $t \mapsto {| \psi (t) \rangle}$ for the two dynamics are the same up to a random fluctuating phase [@Breuer] which does not affect the concurrence $C_{\psi(t)}$. More generally, one can show that the mean concurrence for the QSD model with correlated complex noises satisfying the Itô rules ${{\rm{d}}}\xi_m^i {{\rm{d}}}\xi_n^j =u_{mn}^{ij} {{\rm{d}}}t$ and ${{\rm{d}}}\xi_m^i ( {{\rm{d}}}\xi_n^j)^\ast =\delta_{ij} \delta_{mn} {{\rm{d}}}t$ [@Wiseman01], which gives back the model of Gisin and Percival when $u_{mn}^{ij}=0$, decays exponentially as in (\[eq-main\_result\]) if the two baths are independent, [i.e.]{}, if $u_{mn}^{AB}=0$ for any $m,n$.
![(Color online) \[fig\_common\_bath\] Concurrence of two qubits coupled to a common bath versus $\gamma t$ for ${| \psi(0) \rangle}=\frac{2}{\sqrt{5}} {| \!\! \uparrow \downarrow \rangle}+ \frac{1}{\sqrt{5}} {| \!\! \downarrow \uparrow \rangle}$: (1a) $C_{\rho(t)}$ (blue dashed line); (1b) $C_{\psi (t)}$ for a single trajectory (black dotted line); (1c) $\overline{C_{\psi (t)}}$ given by (\[eq-result\_1\_bath\]) (red line superimposed on the blue line). Inset (2) is the same for ${| \psi(0) \rangle}=\frac{7{{\rm{i}}}}{\sqrt{53}} {| \!\! \uparrow \uparrow \rangle}+ \frac{2{{\rm{i}}}}{\sqrt{53}}
{|\!\! \downarrow \downarrow \rangle}$. ](fig3.eps){width="0.99\columnwidth"}
Qubits coupled to a common bath {#sec-common_bath}
===============================
We focus here on a specific model of two qubits with equal frequencies coupled resonantly to the same modes of the electromagnetic field initially in the vacuum. A photon counter $D$ makes a click when a photon is emitted by qubit $A$ or $B$. The jump operator in the rotating wave approximation, $J= \sigma_{-} \otimes 1_B+1_A \otimes \sigma_{-}$, is now non-local. We take $H_0=0$. Proceeding as for independent baths, the contribution to the mean concurrence of quantum trajectories without jump between $0$ and $t$ is $p_{\rm nj} (0,t) C_{\rm nj} (t)=
|\langle \sigma_y \otimes \sigma_y T \rangle_{{e}^{-t K} {| \psi(0) \rangle}}|$ and can be determined with the help of (\[eq-D(t)\]). By calculating the exponential of $K=\gamma J^\dagger J/2$, one finds $e^{- (t-t_0) K} {| \psi (t_0) \rangle} = \sum_{s,s'} c_{s s'} (t) {| s,s' \rangle}$ with $c_{{\uparrow}{\uparrow}} (t) = e^{-\gamma (t-t_0)} c_{{\uparrow}{\uparrow}} (t_0)$, $2 c_{s s'} (t) = (e^{-\gamma (t-t_0)} + 1 ) c_{s s'} (t_0)
+ (e^{-\gamma (t-t_0)} - 1 ) c_{s's } (t_0)$ for $s s'={\uparrow}{\downarrow}$ or ${\downarrow}{\uparrow}$, and $c_{{\downarrow}{\downarrow}}(t) = c_{{\downarrow}{\downarrow}}(t_0)$. Quantum trajectories having one jump in $[0,t]$ give a nonzero contribution. The probability density that the jump occurs at time $t_j \in [0,t]$ is given by $\gamma p_{\rm{nj}} (t_j,t) \| J {| \psi (t_j-) \rangle}\|^2 p_{\rm{nj}} (0,t_j)
=\gamma\, {{\cal N}}_{\text{1j}, t_j} (t)^{2}$ with ${{\cal N}}_{\text{1j}, t_j} (t) =\| {e}^{-(t-t_j)K} J {e}^{-t_j K} {| \psi (0) \rangle}\|$ (this follows from the formula $p_{\rm{nj}} (t_0,t)= \| e^{-(t-t_0) K} {| \psi (t_0) \rangle}\|^2$, see Sec. \[sec-QJM\]). The contribution of trajectories having one jump in $[0,t]$ is then obtained by multiplying this density by $C_{\text{1j}, t_j} (t) = 2 {{\cal N}}_{\text{1j}, t_j} (t)^{-2} {e}^{-2 \gamma t} | c_{{\uparrow}{\uparrow}}|^2$ and integrating over $t_j$. After two clicks, ${| \psi (t) \rangle} = {|\!\! \downarrow \downarrow \rangle}$ is in an invariant separable state. Therefore, trajectories with more than one jump do not contribute to the mean concurrence. Setting $c_\pm = c_{{\uparrow}{\downarrow}}\pm c_{{\downarrow}{\uparrow}}$, one gets $$\label{eq-result_1_bath}
\overline{C(t)} \!=\!
\frac{1}{2}\bigl| c_{-}^2
\!- c_{+}^2 e^{-2 \gamma t}
\!+ 4 c_{\uparrow \uparrow} c_{\downarrow \downarrow}e^{-\gamma t}
\bigr| +2| c_{\uparrow \uparrow}|^2 \gamma te^{-2\gamma t} .$$ The time behavior of the concurrence (\[eq-result\_1\_bath\]) depends strongly on the initial state. Unlike in the case of independent baths, $\overline{C(t)}$ may vanish at nonzero finite discrete times $t_0$. A necessary and sufficient condition for this loss of entanglement (immediately followed by a revival) is $c_{{\uparrow}{\uparrow}}=0$ and $\arg (c_{{\uparrow}{\downarrow}}) = \arg (c_{{\downarrow}{\uparrow}}) $ ([i.e.]{}, $c_{+}/c_{-} \in ]-\infty,-1[ \cup ]1,\infty[$). If this condition is fulfilled, $\overline{C(t)}$ vanishes at time $t_0=\gamma^{-1} \ln ( |c_+/c_-|)$, see Fig. \[fig\_common\_bath\]. It is not difficult to show by solving the master equation (\[eq-Lindblad\]) with $J= \sigma_{-} \otimes 1_B+1_A \otimes \sigma_{-}$ that, for any initial state containing at most one excitation ([i.e.]{}, such that $c_{{\uparrow}{\uparrow}}=0$), $\overline{C(t)}=| c_-^2 - c_+^2 e^{-2 \gamma t} |/2$ coincides at all times with the concurrence $C_{\rho(t)}$ for the density matrix. In contrast, if $c_{{\uparrow}{\uparrow}}\not = 0$ then $\overline{C(t)}$ increases at small times whereas $C_{\rho(t)}$ decreases, as shown in the inset of Fig. \[fig\_common\_bath\]. For any initial state, $\overline{C(t)}$ converges at large times $t \gg \gamma^{-1}$ to the same asymptotic value $C_\infty = | c_-|^2/2$ as the concurrence $C_{\rho(t)}$ [@Maniscalco08; @Orszag10].
A non-local measurement scheme depending on the initial state ${| \psi (0) \rangle}$ and such that $\overline{C (t)} = C_{\rho(t)}$ at all times $t \in [0,t_{\rm EDS}]$ has been found recently [@Viviescas-arXiv] for two qubits coupled to two baths at zero temperature in the rotating-wave approximation. If one neglects the Hamiltonian of the qubits, this scheme is time-independent. The corresponding quantum trajectories are given by a QSD equation [@Wiseman01] for homodyne detection with two jump operators $J_1$ and $J_2$ similar to the jump operator $J$ introduced in this section, combined with intense laser fields via 50% beam splitters, as described in Sec. \[sec-QSD\] (the main difference between $J_{1,2}$ and $J$ comes from the presence of appropriately chosen phase factors in front of $\sigma_{-}$ and $\sigma_+$ making $J_{1,2}$ non-symmetric under the exchange of the two qubits). It is striking that we also find in our model that $\overline{C (t)} = C_{\rho(t)}$ for specific initial states even though the dynamics in the absence of measurements - and thus the density matrix concurrence $C_{\rho(t)}$ - are not the same in the two models (here the two qubits are coupled to a common bath, whereas they are coupled to distinct baths in Ref. [@Viviescas-arXiv]).
Conclusion {#sec-conclusion}
==========
We have found explicit formulas for the mean concurrence $\overline{C(t)}$ of quantum trajectories and have shown that the measurements on the baths may be used to protect the entanglement of two qubits. These results shed new light on the phenomenon of entanglement sudden death. For independent baths, $\overline{C(t)}$ is either constant in time or vanishes exponentially with a rate depending on the measurement scheme only, whereas for a common bath $\overline{C(t)}$ depends strongly on the initial state and may coincide with the concurrence $C_{\rho (t)}$ of the density matrix for some initial states. A constant $\overline{C(t)}$ implies a perfect protection of maximally entangled states for all quantum trajectories. In the case of pure dephasing and for Jaynes-Cumming couplings at infinite temperature, we have found measurement schemes independent of the initial state of the qubits which lead to such a perfect entanglement protection. Despite obvious analogies, this way to protect entanglement differs from the strategy based on the quantum Zeno effect proposed in Ref. [@Maniscalco08]. In fact, in the QJ and QSD models considered here the time interval between consecutive measurements is not arbitrarily small with respect to the damping constant $\gamma^{-1}$. In the QJ model this time interval ${{\rm{d}}}t$ must be chosen such that the jump probability ${{\rm{d}}}p (t) \propto \gamma\, {{\rm{d}}}t$ is very small but one cannot let $\gamma \, {{\rm{d}}}t$ go to zero since this would amount to replacing ${{\rm{d}}}p (t)$ by $0$ and $e^{-{{\rm{i}}}H_{\rm{eff}} {{\rm{d}}}t}$ by $e^{-{{\rm{i}}}H_0 {{\rm{d}}}t}$ in (\[eq-no\_jump\]). In contrast, a perfect entanglement protection is reached in [@Maniscalco08] in the idealized limit $\gamma\, {{\rm{d}}}t \rightarrow 0$ ([i.e.]{}, when the measurements completely prevent the decay of the superradiant state [@Fisher01]).
For independent baths, $\overline{C(t)}$ is strictly greater than $C_{\rho (t)}$ if the latter concurrence vanishes after a finite time. Therefore, if there exists a measurement scheme such that the mean entanglement of formation $\overline{E (t)}$ is equal to the entanglement of formation of the density matrix (which would imply $\overline{C(t)} \leq C_{\rho (t)}$), this scheme must necessarily involve measurements of non-local (joint) observables of the two baths. Let us finally note that it should be possible to check our findings experimentally by using similar optical devices as in Ref. [@Almeida07].
[**[ACKNOWLEDGMENTS]{}**]{}
We thank P. Degiovanni, A. Joye, and C. Viviescas for interesting discussions. We acknowledge financial support from the Agence Nationale de la Recherche (Grant No. ANR-09-BLAN-0098-01).
[**Note added:**]{} after the completion of this work we learned that related results have been obtained in [@Santos-arXiv].
[33]{}
L. Diósi, in [*Irreversible Quantum Dynamics*]{},
[Lecture Notes in Physics]{} [**622**]{}, 157, edited by F. Benatti and R. Floreanini (Springer, Berlin, 2003)
P.J. Dodd and J.J. Halliwell, Phys. Rev. A [**69**]{}, 052105 (2004)
T. Yu and J.H. Eberly, Phys. Rev. Lett. [**93**]{}, 140404 (2004)
M.P. Almeida [*et al.*]{}, Science [**316**]{}, 579 (2007)
M.O. Terra Cunha, New J. Phys. [**9**]{}, 237 (2007)
D. Braun, [Phys. Rev. Lett.]{} [**89**]{}, 277901 (2002)
Z. Ficek and R. Tanás
, [Phys. Rev. A]{} [**74**]{}, 024304 (2006)
L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, and B.M. Garraway, [Phys. Rev. A]{} [**79**]{}, 042302 (2009)
J. Dalibard, Y. Castin, and K. M[ø]{}lmer, Phys. Rev. Lett. **68**, 580 (1992)
H.J. Carmichael, *An Open System Approach to Quantum Optics*, Lecture Notes in Physics, Series [**[M18]{}**]{} (Springer, Berlin, 1993)
H.M. Wiseman and G.J. Milburn, [Phys. Rev. A]{} [**47**]{}, 642 (1993)
N. Gisin and I.C. Percival, J. Phys A: Math. Gen. **25**, 5677 (1992)
H. Nha and H.J. Carmichael, [Phys. Rev. Lett.]{} [**93**]{}, 120408 (2004)
A.R.R. Carvalho, M. Busse, O. Brodier, C. Viviescas, and A. Buchleitner, [Phys. Rev. Lett.]{} [**98**]{}, 190501 (2007)
S. Maniscalco, F. Francica, R.L. Zaffino, N. Lo Gullo, and F. Plastina, Phys. Rev. Lett. [**100**]{}, 090503 (2008)
D.M. Mundarain and M. Orszag, [Phys. Rev. A]{} [**79**]{}, 052333 (2009)
A.R.R. Carvalho and J.J. Hope, [Phys. Rev. A]{} [**76**]{}, 010301(R) (2007)
E. Mascarenhas, B. Marques, D. Cavalcanti, M.O. Terra Cunha, and M. França Santos, Phys. Rev. A [**81**]{}, 032310 (2010)
C. Viviescas, I. Guevara, A.R.R. Carvalho, M. Busse, and A. Buchleitner, arXiv:1006.1452v1 \[quant-ph\]
C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters, [Phys. Rev. A]{} [**54**]{}, 3824 (1996)
W.K. Wootters, [Phys. Rev. Lett.]{} [**80**]{}, 2245 (1998)
H.-P. Breuer and F. Petruccione, [*The Theory of Open Quantum Systems*]{} (Oxford University Press, 2002)
M.B. Plenio and P.L. Knight, [Rev. Mod. Phys.]{} [**[70]{}**]{}, 101 (1998)
S. Haroche and J.-M. Raimond, [*Exploring the quantum: atoms, cavities and photons*]{} (Oxford Univ. Press, 2006)
D. Spehner and M. Orszag, J. Math. Phys. [**43**]{}, 3511 (2002)
H.M. Wiseman and L. Diósi, Chem. Phys. [**268**]{}, 91 (2001)
M. Orszag and M. Hernandez, Adv. in Optics and Photonics [**2**]{}, 229 (2010)
M.C. Fischer, B. Gutierrez-Medina, and M.G. Raizen, [Phys. Rev. Lett]{} [**87**]{}, 040402 (2001); P.E. Toscheck and C. Wunderlich, [Eur. Phys. J.]{} D [**14**]{}, 387 (2001)
E. Mascarenhas, D. Cavalcanti, V. Vedral, and M. França Santos, arXiv:1006.1233 \[quant-ph\]
|
---
abstract: 'We prove Pólya’s conjecture of 1943: For a real entire function of order greater than $2$ with finitely many non-real zeros, the number of non-real zeros of the $n$-th derivative tends to infinity as $n\to\infty$. We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.'
author:
- 'Walter Bergweiler[^1] and Alex Eremenko[^2]'
title: Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions
---
[**1. Introduction**]{}
The theory of entire functions begins as a field of research in the work of Laguerre [@Laguerre], soon after the Weierstrass product representation became available. Laguerre introduced the first important classification of entire functions, according to their genera. We recall this notion. Let $f$ be an entire function and $(z_k)$ the sequence of its zeros in $\C\backslash\{0\}$ repeated according to their multiplicities. If $s$ is the smallest integer such that the series $$\sum_{k=1}^\infty |z_k|^{-s-1}$$ converges, then $f$ has the Weierstrass representation $$\displaystyle f(z)=z^me^{P(z)}\prod_{k=1}^\infty \left(1-\frac{z}{z_k}\right)
\exp\left(\frac{z}{z_k}+\ldots+\frac{1}{s}\left(\frac{z}{z_k}\right)^s
\right).$$ If $P$ in this representation is a polynomial, then the genus is defined as $g:=\max\{ s,\deg P\}$. If $P$ is a transcendental entire function, or if the integer $s$ does not exist, then $f$ is said to have infinite genus. A finer classification of entire functions by their order $$\rho=\limsup_{r\to\infty}\frac{\log\log M(r,f)}{\log r},\quad
\mbox{\rm where}\quad M(r,f)= \max_{|z|\leq r}|f(z)|,$$ was later introduced by Borel, based on the work of Poincaré and Hadamard. For functions of non-integral order we have $g=[\rho]$, and if $\rho$ is a positive integer, then $g=\rho-1$ or $g=\rho$. So the order and the genus are simultaneously finite or infinite.
The state of the theory of entire functions in 1900 is described in the survey of Borel [@Bo]. One of the first main problems of the theory was finding relations between the zeros of a real entire function and the zeros of its derivatives. (An entire function is called real if it maps the real axis into itself.) If $f$ is a real polynomial with all zeros real, then all derivatives $f^{(n)}$ have the same property. The proof based on Rolle’s theorem uses the finiteness of the set of zeros of $f$ in an essential way. Laguerre discovered that this result still holds for entire functions of genus $0$ or $1$, but in general fails for functions of genus $2$ or higher [@Laguerre]. Pólya [@P1] refined this result as follows. Consider the class of all entire functions which can be approximated uniformly on compact subsets of the plane by real polynomials with all zeros real. Pólya proved that this class coincides with the set of functions of the form $$f(z)=e^{az^2+bz+c}w(z),$$ where $a\leq 0$, $b$ and $c$ are real, and $w$ is a canonical product of genus at most one, with all zeros real. This class of functions is called the [*Laguerre–Pólya class*]{} and denoted by $\LP$. Evidently $\LP$ is closed with respect to differentiation, so all derivatives of a function in $\LP$ have only real zeros. For a function of class $\LP$, the order and the genus do not exceed $2$.
The class $\LP$ has several other interesting characterizations, and it plays an important role in many parts of analysis [@HW; @K].
The results of Laguerre and Pólya on zeros of successive derivatives inspired much research in the 20th century. In his survey article [@P4], Pólya writes: [*“The real axis seems to exert an influence on the non-real zeros of $f^{(n)}$; it seems to attract these zeros when the order is less than $2$, and it seems to repel them when the order is greater than $2$”.*]{} In the original text Pólya wrote “complex” instead of “non-real”. We replaced this in accordance with the modern usage to avoid confusion. Pólya then put this in a precise form by making the following two conjectures.
[**Conjecture A.**]{} [*If the order of the real entire function $f$ is less than $2$, and $f$ has only a finite number of non-real zeros, then its derivatives, from a certain one onwards, will have no non-real zeros at all.* ]{}
[**Conjecture B.**]{} [*If the order of the real entire function $f$ is greater than $2$, and $f$ has only a finite number of non-real zeros, then the number of non-real zeros of $f^{(n)}$ tends to infinity as $n\to\infty$.*]{}
Conjecture A was proved by Craven, Csordas and Smith [@CCS]. Later they proved in [@CCS1] that the conclusion holds if $\log M(r,f)=o(r^2)$. The result was refined by Ki and Kim [@K1; @KK] who proved that the conclusion holds if $f=Ph$ where $P$ is a real polynomial and $h\in\LP$.
In this paper we establish Conjecture B. Actually we prove that its conclusion holds if $f=Ph$, with a real polynomial $P$ and a real entire function $h$ with real zeros that does not belong to $\LP$.
[**Theorem 1.**]{} [*Let $f$ be a real entire function of finite order with finitely many non-real zeros. Suppose that $f$ is not a product of a real polynomial and a function of the class $\LP$. Then the number $N(f^{(n)})$ of non-real zeros of $f^{(n)}$ satisfies $$\label{card1}
\liminf_{n\to\infty}\frac{N(f^{(n)})}{n}>0.$$* ]{}
For real entire functions of infinite order with finitely many non-real zeros we have the recent result of Langley [@L] that $N(f^{(n)})=\infty$ for $n\geq 2$. Theorem 1 and Langley’s result together prove Conjecture B. Combining Theorem 1 with the results of Langley and Ki and Kim we conclude the following:
[*For every real entire function $f$, either $f^{(n)}$ has only real zeros for all sufficiently large $n$, or the number of non-real zeros of $f^{(n)}$ tends to infinity with $n$.*]{}
We give a short survey of the previous results concerning Conjecture B. These results can be divided into two groups: asymptotic results on the zeros of $f^{(n)}$ as $n\to\infty$, and non-asymptotic results for fixed $n$.
The asymptotic results begin with the papers of Pólya [@P2; @P3]. In the first paper, Pólya proved that if $$\label{polya}
f=Se^T,$$ where $S$ and $T$ are real polynomials, and $f^{(n)}$ has only real zeros for all $n\geq 0$, then $f\in \LP$. In the second paper, he introduced the [*final set*]{}, that is the set of limit points of zeros of successive derivatives, and he found this limit set for all meromorphic functions with at least two poles, as well as for entire functions of the form (\[polya\]). It turns out that for a function of the form (\[polya\]), the final set consists of equally spaced rays in the complex plane. Namely, if $T(z)=az^d+bz^{d-1}+\ldots$, then these rays emanate from the point $-b/(da)$ and have directions $\arg z=(\arg a+(2k+1)\pi)/d,$ with $k=0,\ldots,d-1$. It follows that the number of non-real zeros of $f^{(n)}$ for such a function $f$ tends to infinity as $n\to\infty$, unless $d=2$ and $a<0$, or $d\leq 1$, that is, unless $\exp(T)\in\LP$. Notice that the final set of a function (\[polya\]) is independent of the polynomial $S$.
This result of Pólya was generalized by McLeod [@Mc] to the case that $S$ in (\[polya\]) is an entire function satisfying $\log M(r,S)=o(r^{d-1}),\; r\to\infty.$ This growth restriction seems natural: a function $S$ of faster growth will influence the final set. This is seen from Pólya’s result where the final set depends on $b$.
All papers on final sets use some form of the saddle point method to obtain an asymptotic expression for $f^{(n)}$ when $n$ is large. This leads to conclusions about the zeros of $f^{(n)}$. McLeod used a very general and powerful version of the saddle point method which is due to Hayman [@H]. We do not survey here many other interesting results on the final sets of entire functions, as these results have no direct bearing on Conjecture B, and we refer the interested reader to [@B; @Ge; @Shen2] and references therein.
Passing to the non-asymptotic results, we need the following definition. For a non-negative integer $p$ we define $V_{2p}$ as the class of entire functions of the form $$e^{az^{2p+2}}w(z),$$ where $a\leq 0$, and $w$ is a real entire function of genus at most $2p+1$ with all zeros real. Then we define $U_0:=V_0$ and $U_{2p}:=V_{2p}\backslash V_{2p-2}$. Thus $\LP=U_0$.
Many of the non-asymptotic results were motivated by an old conjecture, attributed to Wiman (1911) by his student [Å]{}lander [@Alan1; @Alan], that every real entire function $f$ such that $ff^{\prime\prime}$ has only real zeros belongs to the class $\LP$. For functions of finite order, Wiman made the more precise conjecture that $f^{\prime\prime}$ has at least $2p$ non-real zeros if $f\in U_{2p}$.
Levin and Ostrovskii [@LO] proved that if $f$ is a real entire function which satisfies $$\label{fast}
\limsup_{r\to\infty}\frac{\log\log M(r,f)}{r\log r}=\infty,$$ then $ff''$ has infinitely many non-real zeros. Hellerstein and Yang [@HY] proved that under the same assumptions $ff^{(n)}$ has infinitely many non-real zeros if $n\geq 2$. Thus Conjecture B was established for functions satisfying (\[fast\]).
A major step towards Conjecture B was then made by Hellerstein and Williamson [@HW1; @HW2]. They proved that for a real entire function $f$, the condition that $ff'f^{\prime\prime}$ has only real zeros implies that $f\in\LP$. It follows that for a real entire function $f$ with only real zeros, such that $f\notin\LP$, the number $N(f^{(n)})$ of non-real zeros of $f^{(n)}$ satisfies $$\limsup_{n\to\infty}N(f^{(n)})>0.$$ The next breakthrough was made by Sheil-Small [@SS] by proving Wiman’s conjecture for functions of finite order: $f^{\prime\prime}$ has at least $2p$ non-real zeros if $f\in U_{2p}$. This was a new and deep result even for the case $f=e^T$ with a polynomial $T$. By refining the arguments of Sheil-Small, Edwards and Hellerstein [@EH] were able to prove that $$N(f^{(n)})\geq 2p,\quad\mbox{for}\quad n\geq 2\quad\mbox{and}\quad f=Ph,$$ where $P$ is a real polynomial and $h\in U_{2p}$.
In the paper [@BEL], the result of Sheil-Small was extended to functions of infinite order, thus establishing Wiman’s conjecture in full generality. As we already mentioned, Langley [@L] extended the result of [@BEL] to higher derivatives. He proved the following: Let $f$ be a real entire function of infinite order with finitely many non-real zeros. Then $N(f^{(n)})=\infty$ for all $n\geq 2$. Thus Conjecture B was established for functions of infinite order.
To summarize the previous results, we can say that all asymptotic results were proved under strong a priori assumptions on the asymptotic behavior of $f$, while the non-asymptotic results estimated $N(f^{(n)})$ for $f\in U_{2p}$ from below in terms of $p$ rather than $n$.
In order to state a refined version of our result, we denote by $N_{\gamma,\delta}(f^{(n)})$ the number of zeros of $f^{(n)}$ in $$\displaystyle\{ z: |\Ima z|> \gamma |z|, |z|>n^{1/\rho - \delta}\},$$ where $\rho$ is the order of $f$, and $\gamma$ and $\delta$ are arbitrary positive numbers.
[**Theorem 2**]{}. [*Suppose that $f=Ph$ where $P$ is a real polynomial and $h\in U_{2p}$ with $p\geq 1$. Then there exist positive numbers $\alpha$ and $\gamma$ depending only on $p$ such that $$\label{card2}
\liminf_{n\to\infty}\frac{N_{\gamma,\delta}(f^{(n)})}{n}\geq \alpha>0,$$ for every $\delta>0$.*]{}
Estimates (\[card1\]) and (\[card2\]) seem to be new even for functions of the form (\[polya\]) with polynomials $S$ and $T$. We recall that for a real entire function $f$ of genus $g$ with only finitely many non-real zeros we have $N(f')\leq N(f)+g$ by a theorem of Laguerre and Borel [@Bo; @G]. Thus (\[card1\]) and (\[card2\]) give the right order of magnitude. The question remains how $\alpha$ and $\gamma$ in (\[card2\]) depend on $p$.
We conclude this Introduction with a sketch of the proof of Theorem 1, which combines the saddle point method used in the asymptotic results with potential theory and the theory of analytic functions with positive imaginary part in the upper half-plane. The latter theory was a common tool in all non-asymptotic results since the discovery of Levin and Ostrovskii [@LO] that for a real entire function $f$ with real zeros, the logarithmic derivative is a product of a real entire function and a function which maps the upper half-plane into itself (Lemma 4 below).
Our proof consists of three steps.
1\. [*Rescaling.*]{} We assume for simplicity that all zeros of $f$ are real. The Levin–Ostrovskii representation gives $L:=f'/f=P_0\psi_0$, where $P_0$ is a real polynomial of degree at least $2$, and $\psi_0$ has non-negative imaginary part in the upper half-plane (Lemma 4). So we have a good control of the behavior of $f'/f$ in the upper half-plane (Lemma 1). For any sequence $\sigma$ of positive integers we can find a subsequence $\sigma'$ and positive numbers $a_k$ such that for $k\to\infty,\; k\in\sigma'$, the following limits exist in the upper half-plane (Section 3): $$q(z)=\lim_{k\to\infty} a_k L(a_kz)/k,\quad\mbox{and}\quad
u(z)=\lim_{k\to\infty} (\log|f^{(k)}(a_kz)|-c_k)/k,$$ with an appropriate choice of real constants $c_k$. The second limit makes sense in $L^1_{\mathrm{loc}}$, and $u$ is a subharmonic function in the upper half-plane (Lemma 5). If the condition (\[card1\]) is not satisfied, that is the $f^{(k)}$ have few zeros in the upper half-plane, then $u$ will be a harmonic function. Our goal is to show that this is impossible.
2\. [*Application of the saddle point method to find a functional equation for $u$.*]{} We express the $f^{(k)}$ as Cauchy integrals over appropriately chosen circles in the upper half-plane, and apply the saddle point method to find asymptotics of these integrals as $k\to\infty$ (Section 4). The Levin–Ostrovskii representation and properties of analytic functions with positive imaginary part in the the upper half-plane give enough information for the estimates needed in the saddle point method. As a result, we obtain an expression of $u$ in terms of $q$ in a [*Stolz angle*]{} at infinity, by which we mean a region of the form $\{ z:|z|>R,\, \varepsilon<\arg z<\pi-\varepsilon\}$ with $R>0$ and $\varepsilon>0$. The expression we obtain for $u$ is $$\label{1}
u(z-1/q(z))=\Rea \int_i^z q(\tau)d\tau+\log |q(z)|.$$ This equation, which plays a fundamental role in our proof, can be derived heuristically as follows: When applying Cauchy’s formula to obtain an expression for $f^{(k)}(a_kw)$, it seems reasonable to write the radius of the circle in the form $a_kr_k$, so that $$\begin{aligned}
f^{(k)}(a_kw)
&=&
\frac{k!}{2\pi i}
\int\limits_{|\xi|=a_kr_k}\frac{f(a_kw+\xi)}{\xi^k}\frac{d\xi}{\xi}\\ \\
&=&
\frac{k!}{2\pi i}\int\limits_{|\zeta|=r_k}
\exp\left( \log f(a_k(w+\zeta))-k\log a_k\zeta\right)
\frac{d\zeta}{\zeta}.\end{aligned}$$ The saddle point method of finding the asymptotic behavior of such an integral as $k\to\infty$ involves a stationary point of the function in the exponent, that is a solution of the equation $$0=\frac{d}{d\zeta}\left(\frac{\log f(a_k(w+\zeta))}{k}-\log a_k\zeta\right)
=
\frac{a_kL(a_k(w+\zeta))}{k}-\frac{1}{\zeta}
=: q_k(w+\zeta)-\frac{1}{\zeta}.$$ This suggests to take $r_k=|\zeta|$ where $\zeta$ is a solution of the last equation. Setting $z=w+\zeta,$ we obtain $w=z-1/q_k(z),$ and the saddle point method gives $$\frac{1}{k}\left(\log f^{(k)}\left(a_k
(z-1/q_k(z))\right)-c_k\right)\sim
\int_i^z q_k(\tau)d\tau+\log q_k(z),$$ with some constants $c_k$. From this we derive (\[1\]).
3\. [*Study of the functional equation $(\ref{1})$.*]{} If the $f^{(k)}$ have few zeros in the upper half-plane, the function $u$ should be harmonic in the upper half-plane. On the other hand, we show that a harmonic function $u$ satisfying (\[1\]) in a Stolz angle cannot have a harmonic continuation into the whole upper half-plane (Section 5). Here again we use the properties of analytic functions with positive imaginary part in the upper half-plane.
The authors thank David Drasin, Victor Katsnelson, Jim Langley, Iosif Ostrovskii and the referees for useful comments on this work.
[**2. Preliminaries**]{}
In this section we collect for the reader’s convenience all necessary facts on potential theory and on functions with positive imaginary part in the upper half-plane $H:=\{ z:\Ima z>0\}.$
[**Lemma 1.**]{}
*Let $\psi\not\equiv 0$ be an analytic function in $H$ with non-negative imaginary part. Then $$\label{21}
|\psi(i)|\frac{\Im z}{(1+|z|)^2}
\leq|\psi(z)|\leq |\psi(i)| \frac{(1+|z|)^2}{\Im z},$$*
$$\label{22}
\left|\frac{\psi'(z)}{\psi(z)}\right|\leq\frac{1}{\Ima z},$$
and $$\label{23}
\left|\log\frac{\psi(z+\zeta)}{\psi(z)}\right|\leq 1
\quad\mbox{for}\quad |\zeta|<\frac{1}{2}\Ima z.$$
Inequality (\[21\]) is a well-known estimate due to Carathéodory, see for example [@V §26]. Inequality (\[22\]) is the Schwarz Lemma: it says that the derivative of $\psi$ with respect to the hyperbolic metric in $H$ is at most $1$. Finally (\[23\]) follows from (\[22\]) by integration.
[**Lemma 2.**]{} [*A holomorphic function $\psi$ in $H$ has non-negative imaginary part if and only if it has the form $$\psi(z)=a+\lambda z+\int_{-\infty}^{\infty}
\left(\frac{1}{t-z}-\frac{t}{1+t^2}\right)d\nu(t),$$ where $\lambda\geq 0$, $a$ is real, and $\nu$ is a non-decreasing function of finite variation on the real line.*]{}
In the case that $\psi$ is a meromorphic function in the whole plane, $\nu$ is piecewise constant with jumps at the poles of $\psi$. Then the representation in Lemma 2 is similar to the familiar Mittag–Leffler representation. Lemma 2 can be found in [@KacK], where it is derived from the similar Riesz–Herglotz representation of functions with positive imaginary part in the unit disc.
[**Lemma 3.**]{} [*Let $G$ be a holomorphic function in $H$ with non-negative imaginary part. Then there exists $\lambda\geq 0$ such that $$G(z)=\lambda z+G_1(z),$$ where $\Im G_1(z)\geq 0,\; z\in H$, and $G_1(z)=o(z)$ as $z\to\infty$ in every Stolz angle.*]{}
This is a theorem of Wolff; see, for example, [@V §26], where it is derived from the Schwarz Lemma. Another proof can be easily obtained from Lemma 2. The number $\lambda$ is called the [*angular derivative*]{} of $G$ at infinity.
[**Lemma 4.**]{} [*Let $h$ be a function of class $U_{2p}$. Then the logarithmic derivative of $h$ has a representation $$\label{lo}
h'/h=P_0\psi_0,$$ where $P_0$ is a real polynomial, $\deg P_0= 2p$, the leading coefficient of $P_0$ is negative, and $\psi_0\not\equiv 0$ is a function with non-negative imaginary part in $H$.*]{}
Equation (\[lo\]) is the Levin–Ostrovskii representation already mentioned, which was used in many papers on the subject [@BEL; @EH; @HW1; @HY; @L; @LO; @SS]. Levin and Ostrovskii [@LO] did not assume that $h$ has finite order and showed that (\[lo\]) holds with some real entire function $P_0$. Hellerstein and Williamson [@HW1 Lemma 2] showed that $P_0$ is a polynomial if $h$ has finite order, and they gave estimates of the degree of $P_0$ in [@HW1 Lemma 8]. As they noted, an upper bound for the degree of $P_0$ follows from an old result of Laguerre [@Laguerre p. 172].
The factorization (\[lo\]) is not unique, and the degree of $P_0$ is not uniquely determined by $h$. As our version of this factorization is different from [@HW1 Lemma 8], we include a proof.
[*Proof of Lemma 4.*]{} First we consider the simple case that $h=e^T$.
If $\deg T=2p+2$, then the leading coefficient of $T$ is negative by definition of the class $U_{2p}$. As the degree of $T'$ is odd, $T'$ has a real root $c$. So we can put $P_0(z):=T'(z)/(z-c)$ and $\psi_0(z):=(z-c)$.
If $\deg T=2p+1$, then we set $P_0:=\pm T'$ and $\psi_0:=\pm 1$, where the sign is chosen to ensure that the leading coefficient of $T'$ is negative.
Finally, if $\deg T=2p$, then the condition $h\in U_{2p}$ implies that the leading coefficient of $T$ is positive, and we set $P_0(z):=-zT'(z)$ and $\psi_0(z):=-1/z$.
From now on we assume that $h$ has at least one real zero $a_0$. If $h$ has only finitely many negative zeros and $h(x)\to 0$ as $x\to
-\infty$, then we also consider $-\infty$ as a zero of $h$. Similarly we consider $+\infty$ as a zero of $h$ if $h$ has only finitely many positive zeros and $h(x)\to 0$ as $x\to +\infty$. We arrange the zeros of $h$ into an increasing sequence $(a_j)$, where each zero occurs once, disregarding multiplicity. The range of the subscript $j$ will be $M<j<N$, where $-\infty\leq M< 0\leq N\leq \infty$, with $a_{M+1}=-\infty$ and $a_{N-1}=+\infty$ in the cases described above.
By Rolle’s theorem, each open interval $(a_j,a_{j+1})$ contains a zero $b_j$ of $h'$. To make a definite choice, we take for $b_j$ the largest or the smallest zero in this interval. Each $b_j$ occurs in this sequence only once, and we disregard multiplicity. We define $$\psi(z):=\frac{1}{z-a_{N-1}}\prod_{M<j<N-1}\frac{1-z/b_j}{1-z/a_j}$$ where the factor $z-a_{N-1}$ is omitted if $a_{N-1}=+\infty$ or $N=\infty$, and the factor $1-z/a_{M+1}$ is omitted if $a_{M+1}=-\infty$. If for some $j\in (M,N-1)$ we have $a_j=0$ or $b_j=0$ then the $j$-th factor has to be replaced by $(z-b_j)/(z-a_j)$. As in [@LO; @Levin] we see that the product converges and is real only on the real axis. We define $P:=h'/(h\psi)$. Then standard estimates using the Lemma on the Logarithmic Derivative and Lemma 1 imply that $P$ is a polynomial [@HW1; @LO]. In particular, $P$ has only finitely many zeros. The zeros of $P$ are precisely the zeros that $h'$ has in addition to the $b_j$ and possible multiple zeros $a_j$ of $h$. These additional zeros were called [*extraordinary*]{} zeros by [Å]{}lander [@Alan1]. An extraordinary zero may be real or not, it can be real but different from all $b_j$, or may be one of the $b_j$: if $b_j$ is a zero of $h'$ of multiplicity $n\geq 2$, then $b_j$ is considered an extraordinary zero of multiplicity $n-1$. Since the number of zeros of $h'$ in every interval $(a_j,a_{j+1})$ is odd (counted with multiplicity), the number of extraordinary zeros of $h'$ in this interval is even. As the non-real extraordinary zeros come in complex conjugate pairs, their number is also even. Overall, $h'$ has an even number of extraordinary zeros, counted with multiplicity. Thus the degree of $P$ is even.
We choose $\varepsilon=\pm 1$ so that $P_0:=\varepsilon P$ has negative leading coefficient and define $\psi_0:=\varepsilon\psi$. Then (\[lo\]) holds. Since the number of extraordinary zeros in each interval $(a_j,a_{j+1})$ is even, $P_0$ has an even number of zeros to the left of $a_0$, and, as $P_0(x)\to-\infty$ as $x\to-\infty$, we conclude that $P_0(a_0)<0$. Since $h'/h$ has a simple pole with positive residue at $a_0$, and $P_0(a_0)<0$, we obtain that $\psi_0$ has a pole with negative residue at $a_0$. Since $\psi_0$ is real only on the real axis this implies that $\psi_0(H)\subset H$. We note that $$\psi_1:=-1/\psi_0$$ also maps $H$ into itself.
Next we show that $\deg P_0\leq 2p$. We recall that $h$ has the form $$\label{xxx}
h(z)=e^{az^{2p+2}}w(z),$$ where $a\leq 0$, and $w$ is a real entire function of genus at most $2p+1$. Then $\log M(r,w)=o(r^{2p+2})$ as $r\to\infty$; see, for example, [@Levin I.4]. Schwarz’s formula [@Levin I.6] shows that $|w'(z)/w(z)|=o(z^{2p+1})$ as $z\to\infty$ in every Stolz angle. We thus have $h'(z)/h(z)= b z^{2p+1}+o(z^{2p+1})$ as $z\to\infty$ in every Stolz angle, with $b:=(2p+2)a\leq 0$. Denote by $\lambda_1$ the angular derivative of $\psi_1$ (see Lemma 3). Then $\psi_1(z)=\lambda_1 z +o(z)$ as $z\to\infty$ in every Stolz angle. Altogether we find that $$P_0(z)=-\frac{h'(z)}{h(z)}\psi_1(z)=
-b \lambda_1 z^{2p+2}+o(z^{2p+2}),$$ as $z\to\infty$ in every Stolz angle. If $b<0$ and $\lambda_1>0$, then $P_0$ has the positive leading coefficient $-b\lambda_1$, a contradiction. Thus $b=0$ or $\lambda_1=0$ so that $|P_0(z)|=o(z^{2p+2})$ and hence $\deg P_0< 2p+2$. Since $\deg P_0$ is even we thus obtain $\deg P_0\leq 2p$.
Now we show that $\deg P_0\geq 2p$. We have $$P_0(z)=cz^d+\ldots,\quad\mbox{where}\quad c<0.$$ Let $\lambda_0\geq 0$ be the angular derivative of $\psi_0$. Then in every Stolz angle $$\frac{h'(z)}{h(z)}=c\lambda_0z^{d+1}+o(z^{d+1}),\quad z\to\infty.$$ Integrating this along straight lines, we conclude that $$\log h(z)=\frac{c\lambda_0}{d+2}z^{d+2}+o(z^{d+2}),\quad z\to\infty.$$ If $c\lambda_0<0$, we compare this with (\[xxx\]) and obtain that $d=2p$. If $\lambda_0=0$ we obtain that $a=0$ in (\[xxx\]), so the genus $g$ of $h$ is at most $2p+1$.
Now we follow [@HW1 Lemma 8]. The logarithmic derivative of $h$ has the form $$\label{raz}
\frac{h'(z)}{h(z)}=Q(z)+z^g\sum_j\frac{m_j}{a_j^g(z-a_j)},$$ where $g\leq 2p+1$ is the genus of $h$, $m_j$ the multiplicity of the zero $a_j$ and $Q$ a polynomial. On the other hand, (\[lo\]) combined with Lemma 2 gives $$\label{dwa}
\frac{h'(z)}{h(z)}=P_0(z)\left\{ \lambda_0z+c_0+\sum_j A_j\left(\frac{1}{a_j-z}-
\frac{a_j}{1+a_j^2}\right)\right\},$$ where $\lambda_0\geq 0$, $A_j\geq 0$ and $c_0$ is real. We also have $$\label{3.54}
\sum_j \frac{A_j}{a_j^2}<\infty.$$ Equating the residues at each pole in the expressions (\[raz\]) and (\[dwa\]) we obtain $$\label{3.57}
P_0(a_j)=-m_j/A_j<0.$$ Now we choose $C>0$ such that $0<-P_0(a_j)\leq C|a_j|^d$. Then (\[3.54\]) and (\[3.57\]) imply $$\frac{1}{C}\sum_j\frac{m_j}{|a_j|^{d+2}}\leq\sum_j
\frac{-m_j}{a_j^2P_0(a_j)}
=\sum_j\frac{A_j}{a_j^2}<\infty,$$ which shows $d+2\geq g+1$, that is $d\geq g-1$. If $g=2p+1$, we have $d\geq 2p$. If $g=2p$, then also $d\geq 2p$ because $d$ is even. It remains to notice that one cannot have $g\leq 2p-1$, because $h$ is of genus $g$ and belongs to $U_{2p}$. This completes the proof of Lemma 4.
[**Lemma 5.**]{} [*Let $(u_k)$ be a sequence of subharmonic functions in a region $D$, and suppose that the $u_k$ are uniformly bounded from above on every compact subset of $D$. Then one can choose a subsequence of $(u_k)$, which either converges to $-\infty$ uniformly on compact subsets of $D$ or converges in $L^1_{\mathrm{loc}}$ (with respect to the Lebesgue measure in the plane) to a subharmonic function $u$. In the latter case, the Riesz measures of the $u_k$ converge weakly to the Riesz measure of $u$.*]{}
This result can be found in [@Hor Theorem 4.1.9]. We recall that the Riesz measure of a subharmonic function $u$ is $(2\pi)^{-1}\Delta u$ in the sense of distributions.
[**3. Beginning of the proof: rescaling**]{}
We write $$f=Ph,$$ where $P$ is a real polynomial and $h$ is a real entire function of finite order, with all zeros real, and $h\notin\LP$. By Lemma 4, we have $$h'/h=P_0\psi_0,$$ where $\psi_0$ is as in Lemma 4, and $P_0$ is a polynomial of degree $2p$ whose leading coefficient is negative. Note that $p\geq 1$ since $h\notin \LP$. We have $$\label{3a}
f'/f=P_0\psi_0+P'/P.$$ Using (\[21\]) we obtain $$\label{3b}
\frac{rf'(ir)}{f(ir)}\to\infty\quad\mbox{as}\quad r\to\infty,\; r>0.$$
Fix $\gamma,\delta>0$, and let $\sigma$ be a sequence of positive integers along which the lower limit in (\[card2\]) is attained; that is, $$\beta:=\liminf_{k\to\infty}\frac{N_{\gamma,\delta}(f^{(k)})}{k}=
\lim_{k\to\infty,\; k\in\sigma}\frac{N_{\gamma,\delta}(f^{(k)})}{k}.$$ In the course of the proof we will choose subsequences of $\sigma$, and will continue to denote them by the same letter $\sigma$.
By (\[3b\]), for every large $k\in \sigma$, there exist $a_k>0,\; a_k\to\infty,$ such that $$\left|\frac{a_kf'(ia_k)}{f(ia_k)}\right|=k.$$ An estimate of the logarithmic derivative using Schwarz’s formula implies that $$\left|\frac{a_kf'(ia_k)}{f(ia_k)}\right|\leq \left|a_k\right|^{\rho+o(1)}\quad
\mbox{as}\quad k\to\infty,$$ where $\rho$ is the order of $f$. We may thus assume that $$\label{a_k}
|a_k|\geq k^{1/\rho-\delta/2}$$ for $k\in\sigma$. We define $$\label{qk}
q_k(z):=\frac{a_kf'(a_kz)}{kf(a_kz)}.$$ Then $|q_k(i)|=1$. From (\[3a\]) and (\[21\]) we deduce that the $q_k$ are uniformly bounded on compact subsets of $H$, and thus the $q_k$ form a normal family in $H$. Passing to a subsequence, we may assume that $$\label{3c}
q_k\to q,$$ as $k\to\infty$, $k\in\sigma$, uniformly on compact subsets in $H$. We choose a branch of the logarithm in a neighborhood of $f(ia_k)$, put $b_k:=\log f(ia_k)$, and define $$\label{defQ}
Q(z):=\int_i^zq(\zeta)d\zeta$$ and $$Q_k(z):=\int_i^zq_k(\zeta)d\zeta.$$ It follows from (\[3c\]) that $$Q_k(z)=\frac{1}{k}\left(\log f(a_kz)-b_k\right)\to Q(z),\quad
k\to\infty,\; k\in \sigma,$$ uniformly on compact subsets of $H$, and $Q_k(i)=0$. The chosen branches of the $\log f$ are well defined on every compact subset of $H$, if $k$ is large enough, because $f$ has only finitely many zeros in the upper half-plane, and $a_k\to\infty$.
Let $z$ be a point in $H$, and $0<t<\Ima z$. Then the disc $\{\zeta:|\zeta-a_kz|<ta_k\}$ is contained in $H$ and does not contain any zeros of $f$ if $k$ is large enough. Thus, by Cauchy’s formula, $$\begin{aligned}
\displaystyle
\nonumber
f^{(k)}(a_kz)&=&\frac{k!}{2\pi i}\int_{|\zeta|=a_kt}
\frac{f(a_kz+\zeta)}{\zeta^k}\frac{d\zeta}{\zeta}\nonumber\\
\label{3d}\\
&=&\frac{k!}{2\pi i}\int_{|\zeta|=a_kt}\frac{\exp\left(kQ_k(z+\zeta/a_k)+b_k\right)}{
\zeta^k}\frac{d\zeta}{\zeta}.
\nonumber\end{aligned}$$ So $$|f^{(k)}(a_kz)|\leq\frac{k!}{(a_kt)^k}\exp\left(\Rea b_k+k\max_{|\zeta|=t}\Rea Q_k(z+\zeta)\right),$$ and with $$u_k(z):=\frac{1}{k}\left(\log|f^{(k)}(a_kz)|-\Rea b_k-\log k!\right)+
\log a_k
\label{defuk}$$ we obtain $$\label{bounduk}
u_k(z)\leq\max_{|\zeta|=t}\Rea Q_k(z+\zeta)-\log t.$$ Since $Q_k\to Q$, we deduce that the $u_k$ are uniformly bounded from above on compact subsets of $H$. By Lemma 5, after choosing a subsequence, we obtain $$\label{converg}
u_k\to u,$$ where $u$ is a subharmonic function in $H$ or $u\equiv-\infty$. The convergence in (\[converg\]) holds in $H$ in the sense described in Lemma 5. We will later see that $u\not\equiv-\infty$. We will then show that $u$ cannot be harmonic in $H$. This will prove Theorem 1.
Moreover, we will see that there exist positive constants $\gamma$ and $\alpha$ depending only on $p$, such that the total Riesz measure of $u$ in the region $\{ z: \Ima z> \gamma |z|\}$ is at least $\alpha/2.$
On the other hand, it follows from the definition of $u_k$ and $N_{\gamma,\delta}(f^{(k)})$ that $N_{\gamma,\delta}(f^{(k)})/(2k)$ is the total Riesz measure of $u_k$ in the region $\{ z: \Ima z> \gamma |z|, |z|>k^{1/\rho - \delta}/a_k \}.$ Note that $k^{1/\rho - \delta}/a_k \to 0$ by (\[a\_k\]). Passing to the limit as $k\to\infty$, $k\in\sigma$, we obtain $\beta\geq \alpha>0$, which will complete the proof of Theorem 2.
[**4. Application of the saddle point method**]{}
It follows from (\[3a\]) and the definitions of $q_k$ in (\[qk\]) and $q$ in (\[3c\]) that $$\label{4a}
q(z)=-z^{2p}\psi(z),$$ where $\psi:H\to\overline{H}\backslash\{0\}$ is defined by $$\psi(z):=\lim_{k\to\infty}\frac{\psi_0(a_kz)}{|\psi_0(i a_k)|}.$$ Here $\psi_0$ is the real meromorphic function in $\C$ from (\[3a\]). Note that $$\label{i}
|q(i)|=|\psi(i)|=1.$$
Now we define $$F(z):=z-1/q(z).$$ From (\[4a\]) and (\[21\]), it follows that $1/q(z)\to 0$ as $z\to\infty$ in every Stolz angle. So $$F(z)\sim z,\quad z\to\infty$$ in every Stolz angle. We fix $\delta_0\in (0,\pi/2)$, for example $\delta_0=\pi/4$. It follows that there exists $R>0$ such that $\Ima F(z)>0$ in the region $$\label{defSR}
S_{R}:=\{ z:|z|>R,\; \delta_0<\arg z<\pi -\delta_0\}.$$ Moreover, if $\delta_1\in(0,\delta_0)$, then $$F(z)\in S':=\{ w:\delta_1<\arg z<\pi-\delta_1\},\quad\mbox{when}
\quad z\in S_{R},$$ provided that $R$ is large enough, say $R>R_0$. We notice that $R_0$ is independent of $f$; this follows from (\[21\]), (\[4a\]) and (\[i\]). We will enlarge $R_0$ in the course of the proof but it will be always a constant independent of $f$. We will obtain for $R>R_0$ the identity $$\label{identity}
u(F(z))=\Rea Q(z)+\log|q(z)|,\quad z\in S_R.$$
In order to do this, we use Cauchy’s formula (\[3d\]) with $z$ replaced by $F_k(z):=z-r_k(z)$, and $t=|r_k(z)|,$ where we have set $r_k:=1/q_k$ to simplify our formulas.
We obtain $$\begin{aligned}
\ds
f^{(k)}(a_kF_k(z))&=&\frac{k!}{2\pi}\int_{-\pi}^\pi
\frac{\exp\left(kQ_k(F_k(z)+r_k(z)e^{i\theta})+b_k\right)}{
\left(a_kr_k(z)e^{i\theta}\right)^k}d\theta\nonumber\\
\label{4a1}\\
&=&\frac{k!q_k^k(z)}{2\pi a_k^k}e^{b_k}\int_{-\pi}^\pi
\exp\left(kQ_k(F_k(z)+r_k(z)e^{i\theta})-ik\theta\right)d\theta.
\nonumber\end{aligned}$$
To determine the asymptotic behavior of this integral as $k\to\infty$, we expand $$L_k(\theta):=Q_k(F_k(z)+r_k(z)e^{i\theta})$$ into a Taylor series: $$L_k(\theta)=L_k(0)+L^\prime_k(0)\theta+\frac{1}{2}L_k^{\prime\prime}(0)
\theta^2+\frac{1}{6}E_k(\theta)\theta^3,$$ where $$|E_k(\theta)|\leq\max_{t\in[0,\theta]}\left|L_k^{\prime\prime\prime}(t)
\right|.$$ We notice that $$L_k(0)=Q_k(F_k(z)+r_k(z))=Q_k(z),$$ and $$L_k^\prime(\theta)=iQ_k^\prime(F_k(z)+r_k(z)e^{i\theta})r_k(z)e^{i\theta}
= iq_k(F_k(z)+r_k(z)e^{i\theta})r_k(z)e^{i\theta},$$ so that $L_k^\prime(0)=i$. Moreover, $$L_k^{\prime\prime}(\theta)=
-q_k^\prime(F_k(z)+r_k(z)e^{i\theta})r_k^2(z)e^{2i\theta}
-q_k(F_k(z)+r_k(z)e^{i\theta})r_k(z)e^{i\theta},$$ so that $$\label{Lpp}
L_k^{\prime\prime}(0)=-\frac{q_k^\prime(z)}{q^2_k(z)}-1.$$ Finally, we have $$\begin{aligned}
L_k^{\prime\prime\prime}(\theta)&=&-iq_k^{\prime\prime}(F_k(z)+r_k(z)e^{i\theta})
r_k^3(z)e^{3i\theta}\\ \\
&&-3iq_k^{\prime}(F_k(z)+r_k(z)e^{i\theta})r_k^2(z)e^{2i\theta}\\ \\
&&-iq_k(F_k(z)+r_k(z)e^{i\theta})r_k(z)e^{i\theta}.\end{aligned}$$ To estimate $L_k^{\prime\prime}(0)$ and $L_k^{\prime\prime\prime}(\theta)$ we notice that $$\frac{q'(\zeta)}{q(\zeta)}=\frac{2p}{\zeta}+\frac{\psi^\prime(\zeta)}{\psi
(\zeta)}.$$ It follows from (\[22\]) that $$\left|\frac{q'(\zeta)}{q(\zeta)}\right|\leq\frac{2p+1}{\Ima\zeta}.$$ From this we deduce that $$\begin{aligned}
\left|\frac{d}{d\zeta}
\left(\frac{q^{\prime}(\zeta)}{q(\zeta)}\right)\right|&=&
\frac{1}{2\pi}\left|\int_{|z-\zeta|=\frac12 \Ima\zeta}
\frac{q'(z)}{q(z)(z-\zeta)^2}dz \right|\\
&\leq& \frac{2}{\Ima\zeta}\max_{|z-\zeta|=\frac12 \Ima\zeta}
\left|\frac{q'(z)}{q(z)}\right|\\
&\leq&
\frac{4(2p+1)}{(\Ima\zeta)^2}.\end{aligned}$$ so that $$\left|\frac{q^{\prime\prime}(\zeta)}{q(\zeta)}\right|
=
\left|\frac{d}{d\zeta}
\left(\frac{q^{\prime}(\zeta)}{q(\zeta)}\right)+
\left(\frac{q^{\prime}(\zeta)}{q(\zeta)}\right)^2\right|
\leq
\frac{(2p+5)(2p+1)}{(\Ima\zeta)^2}.$$ Since $q_k\to q$ we deduce from the above estimates that $$\label{4b2}
\left|\frac{q_k'(\zeta)}{q_k(\zeta)}\right|\leq\frac{2p+2}{\Ima\zeta}
\ \ \ \mbox{\rm and}\ \ \
\left|\frac{q_k^{\prime\prime}(\zeta)}{q_k(\zeta)}\right|\leq
\frac{(2p+5)^2}{(\Ima\zeta)^2}$$ on any compact subset of $H$, provided $k$ is sufficiently large, $k\in\sigma$.
Fix $\eta>0$. It follows from (\[Lpp\]) and (\[4b2\]) that if $z\in S_{R}$, where $R>R_0$ and $k$ is large enough, then $$|L_k^{\prime\prime}(0)+1|<\eta.$$ In particular, $$\label{4c}
\Rea L_k^{\prime\prime}(0)\leq -1+\eta.$$ Moreover, if $z\in S_{R}$ with $R>R_0$, then $$\label{zeta}
\zeta:=F_k(z)+r_k(z)e^{i\theta}\in S',\quad\mbox{and}\quad\zeta\to\infty\quad
\mbox{as}\quad z\to\infty.$$ For large $|z|$ we have $$|L_k^{\prime\prime\prime}(\theta)|\leq\left|\frac{q_k(\zeta)}{q_k(z)}
\right|\left(\frac{|q_k^{\prime\prime}(\zeta)|}{|q_k(\zeta)||q_k(z)|^2}
+
3\left|\frac{q_k^\prime(\zeta)}{q_k(\zeta)q_k(z)}\right|+1\right).$$ By (\[23\]) and (\[4a\]) we have $$q_k(\zeta)/q_k(z)\to1,$$ where $\zeta$ is defined in (\[zeta\]), uniformly with respect to $z\in S_{R}$ and $\theta\in[-\pi,\pi]$ as $R\to\infty$. Combining these estimates with (\[4b2\]) we find that $|L_k^{\prime\prime\prime}(\theta)|\leq 1+\eta$ and hence $$\label{4d}
|E_k(\theta)|\leq 1+\eta,$$ for $z\in S_{R}$ and $|\theta|\leq \pi$, provided $R>R_0$. Altogether we have the Taylor expansion $$\label{4e}
L_k(\theta)=Q_k(z)+i\theta+\frac{1}{2}L_k^{\prime\prime}(0)\theta^2+
\frac{1}{6}E_k(\theta)\theta^3,$$ with $L_k^{\prime\prime}(0)$ and $E_k(\theta)$ satisfying (\[4c\]) and (\[4d\]). We define $C_k(z):=-\frac12 L_k^{\prime\prime}(0)$ and notice that in view of (\[Lpp\]) $$C_k(z)\to C(z):=\frac{1}{2}\left(1+\frac{q^\prime(z)}{q^2(z)}\right),\quad k\to\infty,k\in\sigma,$$ uniformly on any compact subset in $H$.
We note that if $$|\theta|\leq\theta_0:=\frac{3(1-3\eta)}{1+\eta},$$ then $$\begin{aligned}
& & \Rea\left(-C_k(z)
\theta^2+\frac{1}{6}E_k(\theta) \theta^3\right)\nonumber\\
&=&
\Rea\left(\frac{1}{2}L_k^{\prime\prime}(0)
\theta^2+\frac{1}{6}E_k(\theta) \theta^3\right)\nonumber\\
&\leq&\theta^2\left(\frac{1}{2}(-1+\eta)+\frac{1}{6}(1+\eta)|\theta|\right)
\label{dominated}\\
&=&\theta^2\left(\frac{1}{2}(-1+\eta)+\frac{1}{2}(1-3\eta)\right)\nonumber\\
&\leq&-\eta\theta^2.\nonumber\end{aligned}$$ Using (\[4e\]) we obtain $$\begin{aligned}
\ds
&{}&\int_{-\theta_0}^{\theta_0}\exp\left(
kQ_k(F_k(z)+r_k(z)e^{i\theta})-ik\theta\right)d\theta\\
&=&e^{kQ_k(z)}\int_{-\theta_0}^{\theta_0}\exp\left(
k\left(-C_k(z)\theta^2+\frac{1}{6}E_k(\theta)\theta^3\right)\right)d\theta\\
&=&\frac{e^{kQ_k(z)}}{\sqrt{k}}\int_{-\theta_0\sqrt{k}}^{\theta_0\sqrt{k}}\exp\left(
-C_k(z)t^2+\frac{1}{6}E_k(t/\sqrt{k})\frac{t^3}{\sqrt{k}}\right)dt\end{aligned}$$ Combining this with (\[dominated\]) and the theorem on dominated convergence we obtain $$\label{4e1}
\int_{-\theta_0}^{\theta_0}\exp\left(
kQ_k(F_k(z)+r_k(z)e^{i\theta})-ik\theta\right)d\theta
\sim\frac{e^{kQ_k(z)}}{\sqrt{k}}\sqrt{\frac{\pi}{C(z)}}$$ as $k\to\infty$, $k\in\sigma$, uniformly on compact subsets of $S_{R}$, for $R>R_0$.
In order to estimate the rest of the integral (\[4a1\]) we will show that $$\label{4s}
\Rea Q_k(F_k(z)+r_k(z)e^{i\theta})\leq\Rea Q_k(z)-\frac{1}{2}(1-\cos\theta_0),$$ for $\theta_0\leq\theta\leq\pi$. We have $$\begin{aligned}
\ds
&{}&Q_k(z)-Q_k(F_k(z)+r_k(z)e^{i\theta})\\ \\
&=&Q_k(F_k(z)+r_k(z))-Q_k(F_k(z)+r_k(z)e^{i\theta})\\ \\
&=&\int_{r_k(z)e^{i\theta}}^{r_k(z)} q_k(F_k(z)+\zeta)d\zeta\\ \\
&=&\int_0^1 q_k(F_k(z)+r_k(z)e^{i\theta}+tr_k(z)(1-e^{i\theta}))
(1-e^{i\theta})r_k(z)dt\\ \\
&=&(1-e^{i\theta})\int_0^1\frac{q_k(\zeta_t)}{q_k(z)}dt,\end{aligned}$$ where $$\zeta_t:=F_k(z)+r_k(z)e^{i\theta}+tr_k(z)(1-e^{i\theta}).$$ Now $\zeta_t/z\to 1$ as $z\to\infty,\; z\in S_{R}$, uniformly with respect to $t\in [0,1]$. Using (\[23\]) we see that $q_k(\zeta_t)/q_k(z)\to 1$ for $z\in S_{R}$ as $R\to\infty.$ In particular, we have $$\Rea\left((1-e^{i\theta})\frac{q_k(\zeta_t)}{q_k(z)}\right)\geq
\frac{1-\cos\theta}{2}\geq\frac{1-\cos\theta_0}{2},$$ for $\theta_0\leq\theta\leq \pi,\; z\in S_{R}$ and $R>R_0$, and this yields (\[4s\]).
It follows from (\[4s\]) that $$\begin{aligned}
\ds
&{}&\left|\int_{\theta_0\leq|\theta|\leq\pi}
\exp\left(kQ_k(F_k(z)+r_k(z)e^{i\theta})-ik\theta\right)d\theta\right|\\ \\
&\leq&\int_{\theta_0\leq|\theta|\leq\pi}
\exp\left\{\Re\left(kQ_k(F_k(z)+r_k(z)e^{i\theta})\right)\right\}d\theta\\ \\
&\leq& 2(\pi-\theta_0)\exp\left\{ k\left(\Rea Q_k(z)-\frac12(1-\cos\theta_0)\right)\right\}
\\ \\
&=&o\left(\frac{e^{kQ_k(z)}}{\sqrt{k}}\sqrt{\frac{\pi}{C(z)}}\right).\end{aligned}$$ Combining this with (\[4e1\]) we see that $$\ds
\int_{|\theta|\leq\pi}\exp\left(kQ_k(F_k(z)+r_k(z)e^{i\theta})
-ik\theta\right)d\theta\sim\frac{e^{kQ_k(z)}}{\sqrt{k}}\sqrt{\frac{\pi}{C(z)}}$$ as $k\to\infty$, uniformly on compact subsets in $S_R$, where $R>R_0$. Together with (\[4a1\]) this gives $$\ds
f^{(k)}(a_kF_k(z))\sim\frac{k!q_k^k(z)e^{b_k}e^{kQ_k(z)}}{
2a_k^k\sqrt{\pi k C(z)}}.$$ Taking logarithms, dividing by $k$ and passing to the limit as $k\to\infty$ we obtain using (\[defuk\]) $$u(F(z))=\Rea Q(z)+\log|q(z)|,\quad z\in S_{R},$$ for $R>R_0$. This is the same as (\[identity\]). One consequence of (\[identity\]) is that $u\not\equiv-\infty$.
[**5. Analytic continuation of $u$ and conclusion of the proof**]{}
As pointed out at the end of Section 3, in order to prove Theorem 1 it suffices to show that the function $u$ obtained is not harmonic.
[**Lemma 6.**]{} [*Let $$\label{26}
q(z)=-z^{2p}\psi(z),$$ where $p\geq 1$ and $\psi$ is an analytic function mapping $H$ to $\overline{H}\setminus \{0\}$. Define $$Q(z):=\int_i^zq(\zeta)d\zeta$$ and $$\label{29}
F(z):=z-1/q(z).$$ Let $u$ be a subharmonic function in $H$ satisfying $(\ref{identity})$, in a region $S_R$ defined in $(\ref{defSR})$. Then $u$ is not harmonic in $H$.*]{}
[*Proof.*]{} Suppose that $u$ is harmonic. Then there exists a holomorphic function $h$ in $H$ such that $u=\Rea h$, and $$\label{dur}
h(F(z))=Q(z)+\log q(z),\quad z\in S_{R}.$$ Differentiating (\[dur\]) and using $q=Q'$ and (\[29\]) we obtain $$h'(F(z))=q(z)=\frac{1}{z-F(z)}.$$ From (\[26\]), (\[29\]) and (\[21\]), it follows that there is a branch $G$ of the inverse $F^{-1}(w)$ which is defined in a Stolz angle $S$ and satisfies $$\label{kkk}
G(w)\sim w\quad\mbox{as}\quad w\to\infty,\; w\in S.$$ In particular, $G(w)\in H$ for $w\in S$ and $|w|$ large enough. We have $$\label{fa}
h'(w)=q(G(w))=\frac{1}{G(w)-w},$$ for $w\in S$ and $|w|$ large enough. Since $h$ is holomorphic in $H$ we see that $G$ has a meromorphic continuation to $H$. Using (\[26\]), the second equation in (\[fa\]) can be rewritten as $$\label{5b}
\psi(G(w))
=\frac{1}{G^{2p}(w)(w-G(w))}.$$ We will derive from (\[5b\]) that $G$ maps $H$ into itself. To show this, we establish first that $G$ never takes a real value in $H$. In view of (\[kkk\]), there exists a point $w_0\in H$ such that $G(w_0)\in H$. If $G$ takes a real value in $H$, then there exists a curve $\phi:[0,1]\to H$ beginning at $w_0$ and ending at some point $w_1\in H$, such that $G(\phi(t))\in H$ for $0\leq t<1$ but $G(w_1)=G(\phi(1))\in \R$. We may assume that $G(w_1)\neq 0$; this can be achieved by a small perturbation of the curve $\phi$ and the point $w_1$. Using (\[5b\]) we obtain an analytic continuation of $\psi$ to the real point $G(w_1)$ along the curve $G(\phi)$. We have $$\lim_{t\to 1}\Ima\psi(G(\phi(t)))\geq 0,$$ because the imaginary part of $\psi$ is non-negative in $H$.
It follows that for $w\to w_1$, the right hand side of (\[5b\]) has negative imaginary part, while the left hand side has non-negative imaginary part, which is a contradiction. Thus we have proved that $G$ never takes real values in $H$.
Since $G(w_0)\in H$ we see that $G$ maps $H$ into itself. Then Lemma 3 and (\[kkk\]) imply that $\Ima (G(w)-w)>0\quad\mbox{for}\quad w\in H.$ Combining this with the second equation of (\[fa\]) we obtain $\Ima q(G(w))<0$ for $w\in H$. Using (\[kkk\]) we find that in particular $\Ima q(e^{i\pi/(2p)}y)<0$ for large $y>0$. On the other hand, we have $$\Ima q(e^{i\pi/(2p)}y)=y^{2p}\Ima \psi(e^{i\pi/(2p)}y)\geq 0$$ by (\[26\]). This contradiction proves Lemma 6, and it also completes the proof of Theorem 1.
[*Proof of Theorem 2.*]{} It remains to show that there exist positive constants $\gamma$ and $\alpha$ depending only on $p$, such that the total Riesz measure of $u$ in the region $\{ z: \Ima z> \gamma |z|\}$ is bounded below by $\alpha/2$, with $u$ and $p$ as in Sections 3 and 4.
First we note that $$\label{boundu}
u(z)\leq\max_{|\zeta|=t}\Rea Q(z+\zeta)-\log t, \quad t=\frac12 \Ima z,$$ for $z\in H$ by (\[bounduk\]). Using (\[defQ\]), (\[4a\]), (\[i\]) and (\[21\]) we see that for every compact subset $K$ of $H$, the right hand side of (\[boundu\]) is bounded from above by a constant that depends only on $p$ and $K$.
Thus for fixed $p$, the functions $u$, $Q$ and $q$ under consideration belong to normal families. Arguing by contradiction, we assume that $\alpha$ and $\gamma$ as above do not exist. Then there exists a sequence $(u_k)$ of subharmonic functions in $H$, satisfying equations of the form $$u_k(F_k)=\Rea Q_k+\log|q_k|$$ similar to (\[identity\]), such that the Riesz measure of $u_k$ in $\{ z: \Ima z> |z|/k\}$ tends to $0$ as $k\to\infty$. (The functions $u_k, F_k, q_k, Q_k$ are not the functions introduced in Sections 3 and 4.) It is important that all these equations hold in the same region $S_R$. Using normality we can take convergent subsequences, and we obtain a limit equation of the same form, satisfied by a function $u$ harmonic in $H$. But this contradicts Lemma 6, and thus the proof of Theorem 2 is completed.
[**Remark.**]{} In [@SS] and in most subsequent work on the subject the auxiliary function $F_0(z)=z-f(z)/f'(z)$ plays an important role. Note that our function $F$ of (\[29\]) is of a similar nature, except that $f'/f$ is replaced by $q$ which is obtained from $f'/f$ by rescaling.
[11]{} lander, M., Sur les zéros extraordinaires des dérivées des fonctions entières réelles. Ark. för Mat., Astron. och Fys., 11 (1916/17), No. 15, 1–18. lander, M., Sur les zéros complexes des dérivées des fonctions entières réelles. Ark. för Mat., Astron. och Fys., 16 (1922), No. 16, 1–19. Bergweiler, W., Eremenko, A., and Langley, L., Real entire functions with real zeros and a conjecture of Wiman. Geom. Funct. Anal., 13 (2003), 975–991 Berry, M., Universal oscillations of high derivatives. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 1735–1751. Borel, É., Leçons sur les fonctions entières. Gauthier-Villars, Paris, 1900. Craven, T., Csordas, G. and Smith, W., The zeros of derivatives of entire functions and the Pólya-Wiman conjecture. Ann. of Math. (2), 125 (1987), 405–431. Craven, T., Csordas, G. and Smith, W., Zeros of derivatives of entire functions. Proc. Amer. Math. Soc., 101 (1987), 323–326. Edwards, S. and Hellerstein, S., Non-real zeros of derivatives of real entire functions and the Pólya–Wiman conjectures. Complex Var. Theory Appl., 47 (2002), 25–57. Gallagher, P., Non-real zeros of real entire functions. Math. Z., 90 (1965), 124–125. Gethner, R., On the zeros of the derivatives of some entire functions of finite order. Proc. Edinb. Math. Soc., 28 (1985), 381–407. Hayman, W. K., A generalization of Stirling’s formula. J. Reine Angew. Math., 196 (1956), 67–95. Hellerstein, S. and Williamson, J., Derivatives of entire functions and a question of Pólya. Trans. Amer. Math. Soc., 227 (1977), 227–249. Hellerstein, S. and Williamson, J., Derivatives of entire functions and a question of Pólya. II. Trans. Amer. Math. Soc., 234 (1977), 497–503. Hellerstein, S. and Yang, C. C., Half-plane Tumura-Clunie theorems and the real zeros of successive derivatives. J. London Math. Soc. (2), 4 (1971/72), 469–481. Hirschman, I. and Widder, D., The convolution transform. Princeton University Press, Princeton, NJ, 1955. Hörmander, L., Analysis of partial differential operators. I. Distribution theory and Fourier analysis. Springer, Berlin, 1990. Kac, I. and Krein, M., $R$-functions – analytic functions mapping the upper halfplane into itself. Amer. Math. Soc. Transl. (2), Vol. 103 (1974), 1-18. Karlin, S., Total positivity. Vol. 1. Stanford University Press, Stanford, CA, 1968. Ki, H. and Kim, Y.-O., On the number of nonreal zeros of real entire functions and the Fourier–Pólya conjecture. Duke Math. J., 104 (2000), 45–73. Kim, Y.-O., A proof of the Pólya–Wiman conjecture. Proc. Amer. Math. Soc., 109 (1990), 1045–1052. Laguerre, E., Oeuvres de Laguerre. Tome I, pp. 167–180. Chelsea, Bronx, NY, 1972. Levin, B., Distribution of zeros of entire functions. Amer. Math. Soc., Providence, RI, 1980. Langley, J., Non-real zeros of higher order derivatives of real entire functions of infinite order. J. Anal. Math., 97 (2006), 357–396. Levin, B. and Ostrovskii, I., The dependence of the growth of an entire function on the distribution of zeros of its derivatives. Sibirsk. Mat. Z., 1 (1960) 427–455 (Russian); English transl., Amer. Math. Soc. Transl. (2), 32 (1963), 323–357. McLeod, R., On the zeros of the derivatives of some entire functions. Trans. Amer. Math. Soc., 91 (1959), 354–367. Pólya, G., Über Annäherung durch Polynome mit lauter reellen Wurzeln. Rend. Circ. Mat. Palermo, 36 (1913) 1–17. Pólya, G., Sur une question concernant les fonctions entières. Comptes Rendus, 158 (1914), 330–333. Pólya, G., Über die Nullstellen sukzessiver Derivierten. Math. Z., 12 (1922), 36–60. Pólya, G., On the zeros of the derivatives of a function and its analytic character. Bull. Amer. Math. Soc., 49 (1943) 178–191. Sheil-Small, T., On the zeros of the derivatives of real entire functions and Wiman’s conjecture. Ann. of Math. (2), 129 (1989), 179–193. Shen, L.-C., Zeros of successive derivatives of a class of real entire functions of exponential type. Proc. Amer. Math. Soc., 100 (1987), 627–634. Valiron, G., Fonctions analytiques, Presses Universitaires de France, Paris, 1954.
*W. B.: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany*
bergweiler@math.uni-kiel.de
A. E.: Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
eremenko@math.purdue.edu
[^1]: Supported by the Alexander von Humboldt Foundation, by the NSF, and by the G.I.F., the German–Israeli Foundation for Scientific Research and Development, Grant G -809-234.6/2003.
[^2]: Supported by the NSF grants DMS-0555279 and DMS-0244547.
|
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abstract: |
We investigate the process of random sequential adsorption of polydisperse particles whose size distribution exhibits a power-law dependence in the small size limit, $P(R)\sim R^{\alpha-1}$. We reveal a relation between pattern formation kinetics and structural properties of arising patterns. We propose a mean-field theory which provides a fair description for sufficiently small $\alpha$. When $\alpha \to \infty$, highly ordered structures locally identical to the Apollonian packing are formed. We introduce a quantitative criterion of the regularity of the pattern formation process. When $\alpha \gg 1$, a sharp transition from irregular to regular pattern formation regime is found to occur near the jamming coverage of standard random sequential adsorption with monodisperse size distribution.
0.1cm [PACS numbers: 81.10.Aj, 02.50.-r, 05.40.+j, 61.43.-j]{}
address:
- '$^{(1)}$Moscow State University, Physics Department, Moscow 119899, Russia'
- '$^{(2)}$Department of Chemistry, University of Toronto, Toronto, Canada M5S 1A1'
- '$^{(3)}$University of Potsdam, Physics Department, Am Neuen Palais, D–14415 Potsdam, Germany'
- '$^{(4)}$ Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215'
author:
- 'N. V. Brilliantov$^{1,2}$, Yu. A. Andrienko$^{1,3}$, P. L. Krapivsky$^{4}$, and J. Kurths$^{3}$'
title: 'Polydisperse Adsorption: Pattern Formation Kinetics, Fractal Properties, and Transition to Order'
---
[2]{}
Introduction
============
Random sequential adsorption (RSA) is an irreversible process in which particles are adsorbed sequentially and without overlaps and deposited particles cannot diffuse or desorb from the substrate. The RSA model has been initially applied to reactions along polymer chains[@flory]. More recently, RSA processes have found a variety of other applications from adhesion of colloidal particles and proteins onto substrates[@fed] to chemisorption[@chem] and epitaxial growth[@weeks]; for reviews see[@rev].
In this paper, we study a simple generalization of RSA that creates a rich dynamic behavior and results in complex spatial patterns. Namely, we consider polydisperse random sequential adsorption (PRSA) processes. Adsorption of mixtures has been addressed in a very few studies[@baker; @talbot; @meakin; @pk]. If a mixture contains a small number of different sizes, geometric and kinetic characteristics are primarily determined by the smallest size. In many applications, however, the size distribution is continuous and spreads over several decades[@powder]. Therefore, before the smallest size will finally win, an interesting intermediate asymptotics arises. To address this intermediate regime, we consider PRSA with the power-law distribution in the small size limit, $P(R)\sim R^{\alpha-1}$. We show[@bakk] that this PRSA gives rise to fractal patterns of dimension $D_f$ depending on the exponent $\alpha$. Additionally, we measure the degree of order of the patterns formed by PRSA, and identify the local structure of the patterns arising in the limit $\alpha\to\infty$ with Apollonian packing[@mand]. The significance of Apollonian packing in surface deposition problems was also recognized in Ref.[@other].
This paper is organized as follows. In Sec. II, we introduce PRSA and present numerical results. In Sec. III, we develop scaling, exact, and mean-field approaches to PRSA. Exact results are available for the one-dimensional (1D) PRSA, and they are used to check mean-field and scaling approaches. A relation between kinetics and geometry of arising patterns is also discussed. In Sec. IV, we introduce a quantitative criterion of regularity of the pattern formation process and analyze the ordering in PRSA processes. The last Sec. V contains a summary.
Polydisperse RSA
================
In applying RSA to real processes, one should take into account that adsorbed particles are typically polydisperse. The relevant example is adsorption of colloidal particles, that have a broad radii distribution. It is usually described by the Schulz distribution[@shulz], which has a power-law dependence on the radius $R$ for small $R$ and an exponential tail for large $R$: $$P_{Sz}(R)=\left[ \frac{\alpha}{\langle R\rangle}\right]^{\alpha}
\frac {{}R^{\alpha-1}}{\Gamma(\alpha)}
\exp \left[-\frac{\alpha}{\langle R\rangle}R\right].
\label{disshulz}$$ Here $\langle R\rangle$ is the average radius and $\Gamma(x)$ the gamma function. Note that the exponent $\alpha$ should be positive to obey the normalization requirement, $\int dR\,P(R)=1$.
Only the small-size behavior of $P(R)$ affects the most interesting long time characteristics since in this regime only small particles can be adsorbed. Thus instead of (\[disshulz\]) we shall use a power-law size distribution with an upper cutoff (taken as the unit of length): $$\begin{aligned}
\label{distrib}
P(R)=\cases{\alpha \, R^{\alpha -1} & $R\leq 1$;\cr
0 & $R>1$.}\end{aligned}$$
The patterns formed by PRSA are drastically different from traditional RSA patterns, since the coverage is complete for PRSA. The pore space of the patterns is a nontrivial fractal set. This is physically evident, and in one dimension it proves possible to determine the fractal dimension $D_f(\alpha)$ analytically . In higher dimensions, we have to resort to numerical treatment.
Monte Carlo simulations of the PRSA model have been performed by implementing the following algorithm. A center of the new disc is chosen at random with a uniform probability density. The radius of the disc is generated according to the size-distribution of Eq. (\[distrib\]). If this disc does not overlap any other discs already in placed, it is deposited. Otherwise, the attempt is discarded. The maximal coverage studied in simulations was $\sim
0.9$ and about $100$ runs were performed for each $\alpha$. Some of the generated patterns are shown in Fig. 1. Clearly, the character of patterns changes when the exponent $\alpha$ increases from $0$ to $\infty$. For the small $\alpha$, the patterns look like a random set of little discs distributed uniformly over the plane, with larger discs randomly scattered in the “sea” of smaller ones. For large $\alpha$, one recognizes a structure initially formed by large discs and then reproduced by smaller discs in the holes between the large ones. These properties of patterns follow from the small size behavior of radii distribution function $P(R)$. Indeed when $\alpha\to 0$ the smallest particles primarily participate in the adsorption, and hence apparently random patterns emerge. When $\alpha \to\infty$, the particles of the largest size are deposited until the system reaches the jamming limit of monodisperse RSA. Then the next particle to arrive will be the one that fits the biggest hole. This continues, so in this second stage the process is deterministic and apparently ordered patterns emerge.
On the length scales much smaller than the upper cutoff, patterns are self-similar (Fig. 2). This suggests the fractal nature of the arising patterns. We have measured the fractal dimension $D_f$ of the pore-space of the patterns using the standard approach (see e.g. [@manna; @ab]) based on the analysis of the radii distribution function for the adsorbed particles. Namely, denote by $n(R)dR$ the number density of adsorbed discs with radii from the interval $(R,R+dR)$. Let $\epsilon$ is an arbitrary lower cutoff radius. Then $N(\epsilon) = \int_{\epsilon} ^{\infty} dR\,n(R)$ gives the number density of discs with radii greater than $\epsilon$. The power law behavior of $N(\epsilon)$ at the $\epsilon \to 0$ limit, $$N(\epsilon) \sim \epsilon ^{-D_f},
\label{frdim}$$ is a signature that the pore space is a set of fractal dimension $D_f$. Numerically, we indeed observed this power law behavior. We also found that $D_f$ monotonously decreases when $\alpha$ increases (see Fig. 3).
Theoretical approaches to pattern formation
===========================================
In the present section we employ scaling, mean-field and exact approaches to the process of pattern formation in PRSA. The first approach is based on the scaling hypothesis and gives relations between structural and kinetic characteristics of the system.
Scaling Framework
-----------------
Let $\Phi(t)$ is the fraction of uncovered area and $\Psi(R,t)$ the probability that a disc of radius $R$ can be placed onto a substrate[@bakk]. Clearly,
$$\Psi(0,t)=\Phi(t).
\label{psiphi}$$
$\Phi(t)$ evolves according to the [*exact*]{} rate equation
$$\frac{d\Phi }{dt}=-\int_0^\infty dR P(R)\Psi (R,t)V_d R^d,
\label{phi1}$$
where $V_d$ denotes the volume of the $d$-dimensional unit sphere, $V_d=\pi^{d/2}/\Gamma(1+d/2)$. Assuming a scaling behavior of $\Psi(R,t)$, we write
$$\Psi (R,t)=S^{\theta}(t)\,F\left(\frac R{S(t)}\right).
\label{psi2}$$
Here $S(t)\sim t^{-\nu }$ is a typical gap between neighboring adsorbed particles and $F(x)$ is a scaling function. The scaling description applies when $t\to\infty$ and $R\to 0$ with $R/S(t)$ finite. The existence of scaling is an assumption, which is supported by numerical evidence in 2D and by analytical results in 1D[@pk].
Eqs. (\[psiphi\]) and (\[psi2\]) imply $\Phi (t)\sim t^{-z}$ with $z=\theta\nu$. Substituting then (\[psi2\]) into (\[phi1\]) gives $$\nu=\frac{1}{\alpha+d}, \quad
z \simeq \alpha V_d\int_0^\infty dx\, x^{\alpha+d-1} F(x).
\label{nuz}$$
Scaling suggests that self-similar fractal structures should arise. Indeed, computing the number density of the absorbed particles we get $$n(R)=\int_0^\infty dt\,P(R)\Psi (R,t) \sim R^{\alpha-1+(z-1)/\nu }
\label{nr}$$ Hence, the power-law dependence of Eq. (\[frdim\]) is recovered when we identify the fractal dimension with $$D_f=d-z(d+\alpha).
\label{df}$$ Eq. (\[df\]) provides a relation between the fractal dimension of the arising patterns and the kinetic exponent $z$. Similar qualitative behaviors were observed in other pattern formation models[@ab].
We should stress that scaling provides just a framework; for instance, it gives scaling relations among the exponents but it does not allow to compute the exponents. So one should use other approaches to get a complete exact or approximate description of PRSA. In one dimension, an exact description is indeed possible. In higher dimensions, even in the extreme case of $\alpha=\infty$ the fractal dimension of the pore space remains unknown, so analytical description of PRSA is hardly possible.
[[**Fig. 1**]{} Typical PRSA patterns for $\alpha=0.1, 1, 5, 10, 50$ and Apollonian packing.]{} \[fig:1\]
[[**Fig. 2**]{} Self-similarity of the PRSA patterns.]{} \[fig:2\]
[2]{}
1D PRSA: Exact Results
----------------------
A detailed analysis of 1D PRSA is given in Ref. [@pk]. Here we sketch basic results which are necessary to determine the fractal dimension. Let $C(x,t)$ be the concentration of holes of length $x$ at time $t$. All holes have the same “shape” in 1D that significantly simplifies the problem. By definition, $C(x,t)$ is related to $\Psi(x,t)$ via
$$\Psi (x,t)=\int_x^{\infty} dy (y-x)C(y,t).
\label{chipsi}$$
The kinetic equation for $C(x,t)$ reads
$$\begin{aligned}
\frac{\partial C(x,t)}{\partial t}=
-C(x,t) \left[ \int_0^x dz(x-z)P(z)\right] \nonumber\\
+2 \int_x^{\infty}dy\,C(y,t) \int_0^{y-x} dz\,P(z).
\label{1dgen}\end{aligned}$$
The first term on the right-hand side of Eq. (\[1dgen\]) gives the loss of holes of length $x$ due to deposition of intervals of length $z$ (with $z<x$); the second term describes the gain of holes of length $x$ from larger holes. Substituting $P(z) \equiv P(R)$ given by Eq. (\[distrib\]) into Eq. (\[1dgen\]) yields
$$\left(\frac{\partial}{\partial t}+\frac{x^{\alpha +1}}{\alpha+1}\right)
C(x,t)=2\int_x^{\infty}dy\,C(y,t) (y-x)^{\alpha}.
\label{1dkin}$$
Multiplying both sides of Eq. (\[1dkin\]) by $x^{\beta}$ and integrating over $x$ gives the kinetic equation $${dM_{\beta}\over dt}=\left[
2 \frac{\Gamma(\alpha +1)\Gamma(\beta +1)}{\Gamma(\alpha +\beta +2)}-
\frac{1}{\alpha +1} \right] M_{\alpha + \beta +1}
\label{momeq}$$ for the moments of the hole-size distribution $$M_{\beta}(t)=\int_0^{\infty} dx\,x^{\beta} C(x,t).$$ Eq. (\[chipsi\]) implies $C(x,t)={\partial^2\over \partial x^2}\,\Psi(x,t)$. Combining this with the scaling ansatz of Eq. (\[psi2\]) we obtain $$\label{csf}
C(x,t)=S^{\theta-2}(t)\,F''\left(x/S(t)\right),$$ where $F''(\xi)=d^2F/d\xi^2$. Eq. (\[csf\]) allows us to express the moments $M_{\beta}(t)$ via the time-independent moments, $$M_{\beta}(t)=S^{\theta+\beta-1}(t) m_\beta, \quad
m_{\beta}=\int_0^{\infty} d \xi\,\xi^{\beta} F''(\xi).
\label{moms}$$ Choose now the exponent $\beta$ so that the numerical factor on the right-hand side of Eq. (\[momeq\]) vanishes, i.e., $$2 \frac{\Gamma(\alpha +1)\Gamma(\beta +1)}{\Gamma(\alpha +\beta +2)}=
\frac{1}{\alpha+1}.
\label{eqbeta}$$ For such $\beta=\beta(\alpha)$, Eq. (\[momeq\]) implies that $M_{\beta}(t)$ does not depend on $t$. Eq. (\[moms\]) therefore gives $\theta=1-\beta$, and then other exponents and the fractal dimension are found: $$\nu=\frac{1}{1+\alpha}, \qquad
z=\frac{1-\beta}{1+\alpha}, \qquad
D_f=\beta.
\label{zbeta}$$ A simple analysis shows that (i) Eq. (\[eqbeta\]) has only one positive solution $\beta=\beta(\alpha)$, (ii) $\beta<1$ for all $\alpha>0$, and (iii) $\beta$ decreases when $\alpha$ increases. One can determine $\beta=D_f$ explicitly in some specific cases, e.g., $D_f=(\sqrt{17}-3)/2$ for the uniform size distribution ($\alpha=1$).
Mean-Field Theory of PRSA
-------------------------
In higher dimensions we employ an approximate mean-field treatment. We shall use a mean-field description of PRSA close to the one developed in[@ab] for nucleation-and-growth processes. As usual, we shall ignore many-particles spatial correlations and account only two-particle ones. To find $\Psi(R,t)$, consider a circle of radius $R$ centered at the origin. Then one can write the following estimate to this function in the mean-field spirit: $$\begin{aligned}
\label{psimft}
&&\Psi(R,t)= \nonumber\\
&&\exp \left\{-\int_0^t d\tau \int_0^R dr\,\Omega_d
r^{d-1}\int_0^\infty d\rho\,P(\rho){\Psi(\rho,\tau)\over
\Phi(\tau)}\right\}\\
&&\times \exp \left\{-\int_0^td\tau \int_R^\infty dr\,\Omega_d r^{d-1}
\int_{r-R}^\infty d\rho\,P(\rho )\frac{\Psi(\rho,\tau )}
{\Phi(\tau)}\right\}. \nonumber \end{aligned}$$ The former exponential factor in (\[psimft\]) estimates the probability that our circle is not covered up to time $t$ by discs with centers fall inside this circle. The latter exponential factor guarantees that the free space inside our circle is not covered by the other discs which were adsorbed outside the circle. We also denote by $\Omega_d$ the surface area of unit sphere in $d$ dimensions, $\Omega_d=dV_d=2\pi^{d/2}/\Gamma(d/2)$. The exponential factors in Eq. (\[psimft\]) were derived by noting that $$d\tau\,dr\,d\rho\,\Omega _d r^{d-1}P(\rho)\Psi(\rho,\tau)$$ gives the probability that a disc of radius belonging to the interval $(\rho,\rho+d\rho)$ was adsorbed within the time interval $(\tau ,\tau+d\tau)$ in the spherical shell centered at the origin and confined by radii $r$ and $r+dr$. The factor $\Psi(\rho,\tau)$ guarantees that the disc of radius $\rho$ may be placed into the system. Such event prevents the adsorption of disc of radius $R$ at the origin. The probability that such event has not happened, $$1-d\tau\,dr\,d\rho\,\Omega_dr^{d-1} P(\rho )\Psi (\rho,\tau),$$ may be re-written as $$\exp\left\{-d\tau\,dr\,d\rho\,\Omega_dr^{d-1} P(\rho )\Psi (\rho,\tau)
\right\}.$$ The probability that none of these events has happened up to time $t$ is obtained by multiplying all these factors with $0\leq \tau \leq t$, $0\leq r\leq \infty$, and $0\leq \rho \leq \infty$. However one should take into account that above expression gives a correct estimate only for factors with $\tau=0$. For subsequent factors with $\tau>0$, one should use $\Psi(\rho ,\tau )/\Phi(\tau)$ instead of $\Psi(\rho,\tau)$ since the preceding factors guarantee that the discs are placed on the uncovered space. Using $\Phi(0)=1$ and treating separately $r\leq R$ and $r\geq R$, one arrives at Eq. (\[psimft\]). Again we assume that in the scaling regime we can use Eq. (\[psi2\]) for the function $\Psi(R,t)$. Taking into account that $\Psi(0,t)=\Phi(t)$, we get $$\begin{aligned}
\label{eqf}
&&F(Rt^{\nu})= \\
&&\exp \left\{ -\int_0^td\tau \int_0^\infty d\rho\,
\Omega_dP(\rho )\left((R+\rho )^d
-\rho ^d\right) F(\rho\tau^{\nu})\right\}. \nonumber\end{aligned}$$ Remarkably, the ansatz $$F(x)=\exp(-A_1x-\ldots -A_dx^d)
\label{fanz}
\label{fx}$$ reduces the nonlinear integral equation (\[eqf\]) to a system of $d$ algebraic equations for coefficients $A_j$. In particular, in 1D we get $\nu=1/(\alpha+1)$ and a closed equation for $A_1$: $$\label{A1}
A_1=\frac{2\alpha}{1-\nu \alpha}
\int_0^{\infty}dx\,x^{\alpha-1}e^{-A_1x}.$$ Eq. (\[A1\]) is solved to find $A_1=\left[2\Gamma(\alpha+2)
\right]^{\frac{1}{\alpha+1}}$. Thus on the mean-field level, the scaling function for 1D PRSA is pure exponential, i.e. it is clearly different from the analytical solution. To obtain a quantitative difference we substitute $F=e^{-A_1x}$ with the above value of $A_1$ into Eq. (\[nuz\]) to find $z=\frac{\alpha}{\alpha+1}$. Substituting this into Eq. (\[df\]) we obtain the fractal dimension: $D_f^{\rm
MF}=1-\alpha$ for $\alpha<1$ and $D_f^{\rm MF}=0$ for $\alpha\geq 1$. Thus the mean-field theory is clearly wrong for $\alpha\geq 1$, though in the limit $\alpha\to 0$ it becomes exact[@rem102]. For instance, $D_f^{\rm exact}-D_f^{\rm MF} =(\frac{2\pi}{3}+4\gamma^2-2-6\gamma)
\alpha^2 + {\cal O}(\alpha^3)$ (here $\gamma\cong 0.57721566$ is the Euler constant).
For 2D PRSA, the ansatz of Eq. (\[fanz\]) yields $$\begin{aligned}
A_{1}&=&2 \pi\alpha(\alpha +2) \int_0^{\infty}dx\,
x^{\alpha} e^{-A_1x-A_2x^2}, \\
A_{2}&=&\frac{\pi}{2} \alpha(\alpha +2)\int_0^{\infty} dx\,
x^{\alpha-1} e^{-A_1x-A_2x^2}.\end{aligned}$$ Solving these equations numerically, and then inserting $F(x)=e^{-A_1x-A_2x^2}$ into Eq. (\[nuz\]), one finds $z$ and $D_f$.
In the small $\alpha$ limit, we perform a perturbation analysis to find[@rem5] $$\begin{aligned}
A_{1}&=&2\pi\alpha+a_1\alpha^2
=2\pi\alpha-17.3557\alpha^2,\\
A_{2}&=&\pi+a_2\alpha + a_3\alpha^2=
\pi-1.8339\alpha +12.6546\alpha^2,\end{aligned}$$ where we have omitted terms of order ${\cal O}(\alpha^3)$. Using these expressions for $A_{1}$ and $A_{2}$, we get the kinetic exponent $$\label{zmf}
z=\alpha/2+a_4\alpha^2=\alpha/2-1.5428\alpha^2+\cdots$$ and for the fractal dimension $$\label{dmft}
D_f=2-\alpha+a_5\alpha^2=2-\alpha+2.5856\alpha^2+\cdots.$$
The mean-field predictions for $D_f(\alpha)$ and $z(\alpha)$ are shown on Fig. 3 and Fig. 4. We see that the mean-field approach provides a fair description for small $\alpha$. For $\alpha \simeq 1$, however, the spatial correlations become more and more important and the mean-field theory fails.
=2.7in=2.0in 0.3in 0.15in [[**Fig.3**]{} The fractal dimension $D_f$ vs. $\alpha$ in two dimensions. $D_f$ of the scaling theory is obtained from Eq. (\[df\]) with the “experimental” value of the kinetic exponent $z$.]{} \[fig:3\]
=2.8in=2.35in 0.3in 0.15in [[**Fig.4**]{} The kinetic exponent $z$ vs. $\alpha$ in one and two dimensions.]{} \[fig:4\]
The Limit $\alpha\to\infty$
---------------------------
When $\alpha=\infty$, the PRSA process develops through two stages. The initial stage is just RSA of monodisperse particles of radii $R=R_{\rm
max}=1$. It continues until the jamming coverage, $\Phi_{\infty}\cong
0.542$, is reached. Then the late stage begins, where the next disc to place would be the one that fits the biggest hole. The dynamics in this late stage is thus deterministic and extremal. This deterministic procedure has been applied by Apollonius of Perga 200BC to fill the space between three “kissing” discs[@mand]. In the present case, the procedure fills uncovered space obtained during the initial RSA stage. As the process develops, new discs will be placed more and more often into the curvilinear triangles confined by three kissing discs. Such curvilinear triangles are filled independently, so locally our packing should be identical to Apollonian packing. The fractal dimension quantifies local characteristics of the pattern, so we conclude that $D_f(\infty)=D_A$.
The fractal dimension of Apollonian packing is $D_A\cong 1.3057$ in two dimensions. Note that the fractal dimension of the Apollonian parking, arguably the oldest known fractal, has not been computed analytically in higher dimensions, so the exact value of $D_f(\infty)=D_A$ remains unknown. The only trivial exception is the one-dimensional case where the holes remaining after the initial stage are filled up completely during the deterministic stage. The number density $n(x)\equiv
C(x,t=\infty)$ of adsorbed intervals of length $x$ is[@mudak2] $$\label{nx}
n(x)=2\int_0^\infty dt\,t\exp
\left[-xt-2\int_0^t d\tau{1-e^{-\tau}\over \tau}\right].$$ $n(x)$ exhibits a weak integrable singularity in the small size limit, $n(x)\sim \ln(1/x)$, implying that $N(\epsilon)$ is regular and therefore $D_f(\infty)=0$[@mudak]. This agrees with previous exact results, Eqs.(\[eqbeta\]) and (\[zbeta\]), which provide the asymptotic behavior $D_f(\alpha)\simeq {\ln 2\over \ln(\alpha+2)}$ when $\alpha\to\infty$.
Ordering in the RSA
===================
For a quantitative description of the ordering processes we introduce an entropy, $S_N$, characterizing the degree of order of $N$-particle patterns. Let $C$ denotes a particular $N$-particle pattern and $p_N(C)$ denotes the probability of that pattern. We can define the Shannon entropy [@mackey; @bla]: $$S_N = -\sum_{C} p_N(C) \log_2 p_N(C)
\label{entdef}$$ As it follows from Eq. (\[entdef\]), $S_N=0$ for a regular pattern, since only one configuration (which occurs with the probability $p_N=1$) contributes to the entropy. A closely related quantity, $dS_N/dN \simeq
S_{N+1}-S_N$, gives the entropy production rate per particle and characterizes the regularity of the pattern formation process. We also introduce the conditional entropy, $ S_{N+1}^{*}(C)$, characterizing patterns built by adding a disc to a given $N$-particle pattern $C$: $$\label{entcond}
S_{N+1}^*(C)=-\sum_{R, \vec{r}}
p(R,\vec{r}\, |C)\log_2 p(R,\vec{r}\, |C).$$ Here $p\left(R,\vec{r}\, |C\right)$ is the conditional probability to add a disk of radius $R$ at point $\vec{r}$ to the particular pattern $C(N)$ of $N$ disks. The total probability of the $N+1$-particle configuration, obtained from the pattern $C$ by placing an additional disc, reads: $$\label{condprob}
p_{N+1}(R,\vec{r}\,,C)=p(R,\vec{r}\, |C) p_N( C).$$ $S_{N+1}$ can be written as $$\label{sn1}
S_{N+1}=
-\sum_{R, \vec{r},C}p_{N+1}(R,\vec{r}\,,C)
\log_2 p_{N+1}(R,\vec{r}\, , C).$$ Using Eqs. (\[entdef\])–(\[sn1\]) and the normalization condition, $\sum_{R, \vec{r}}p(R,\vec{r}\,|C)=1$, we finally arrive at the entropy production rate: $${dS_N\over dN} \simeq
S_{N+1}-S_N=\sum_{C}p_N(C) S_{N+1}^*(C).
\label{entprav}$$ Thus the entropy production rate is obtained by averaging the conditional entropy over all possible configurations.
=2.8in=2.4in 0.3in 0.15in [[**Fig.5**]{} The entropy production rate vs. the uncovered area for different values of $\alpha$. A sharp decay at $\Phi \approx 0.55$ is clearly seen for large $\alpha$.]{} \[fig:5\]
In spite of apparent simplicity of the definition of the entropy production rate, its analytical evaluation is a very challenging problem even in one dimension. Indeed, it requires knowledge of the multiparticle probability distribution function. In the simplest case of the monodisperse RSA we do know the first two moments of $p(C)$, the density[@rev] and the pair correlation function[@bon], but much more detail information is needed for determination of the entropy production rate. Thus we investigated this quantity numerically for 2D system by means of Monte Carlo simulations. We implemented the following algorithm. A disc which is added to the pattern $C(N)$ of $N$ disks was treated as a point in a the “configurational” space $(R, \vec{r})=(R,x,y)$. We divided the continuous configuration space into sufficiently small discrete cells and enumerated these cells. For computations we used cells of linear size $0.01$ and $5 \times 5$ fragment of the surface (with $R_{max}=1$, see (\[distrib\])); thus, the total number of cells was $M=2.5 \times 10^7$. Then the conditional probabilities, $p\left(R,\vec{r}\, |C(N)\right)=p_i$, that $(N+1)$-th disk comes to the $i^{\rm th}$ cell, corresponding to $(R,\vec{r})$ in the given configuration, $C(N)$, of $N$ discs was calculated numerically from $$p_i=\frac{A_iR_i^{\alpha-1}}{\sum_{j=1}^M A_j R_j^{\alpha-1}}
\label{numcondp}$$ where $A_i=0$ if the disc corresponding to the $i^{\rm th}$ cell overlaps with some disc in the pattern $C(N)$; otherwise, $A_i=1$. Eq. (\[numcondp\]) follows from the rate of adsorption, (\[distrib\]), and the normalization condition for the probability. The discrete conditional probabilities (\[numcondp\]) are used to calculate the conditional entropy $ S_{N+1}^*(C)$ via Eq. (\[entcond\]). Finally, the entropy production rate was obtained by averaging $S_{N+1}^*(C)$ over $C$ according to Eq. (\[entprav\]). The averaging was performed over $10^2$ Monte Carlo runs, and the accuracy of the method was controlled through run-to-run deviations.
To compare the entropy production rate for different values of $\alpha$, we plot $dS_N/dN$ versus $\Phi$ (see Fig. 5). The striking behavior of the entropy production rate is clearly seen in the large $\alpha$ limit: At the beginning of the process of pattern formation (i.e., when $\Phi
\approx 1$), the entropy production rate decreases slowly similar to the small $\alpha$ case. Around $\Phi \approx 0.55$, however, a sharp decay occurs. The threshold value, $\Phi \approx 0.55$, is close to the jamming density, $\Phi_{\infty}\cong 0.542$, of the RSA of identical discs. The transition demarcates the initial random stage when discs of maximal radii are deposited and the final deterministic stage when discs of maximal possible radius are inserted into maximal holes. Given the deterministic nature of the final stage, the entropy production rate should be equal to zero when $\alpha=\infty$ and $\Phi \ge
\Phi_{\infty}$. This is in agreement with our numerical findings for large $\alpha$.
Summary
=======
In summary, we investigated the adsorption kinetics and spatial properties of the arising patterns in RSA of polydisperse particles whose size distribution has a power-law form in the small size limit. We developed a scaling approach and verified that it indeed applies by comparing with exact results in one dimension and numerical results in two dimensions. We found that arising patterns are self-similar fractals that appear to be completely random when deposited particles are predominantly small; in the opposite limit highly ordered structures, locally isomorphic to Apollonian parking, are formed. The fractal dimension $D_f$ of the pore space is determined by the power-law exponent $\alpha$ of the particles size distribution. When $\alpha$ increases from $0$ to $\infty$, $D_f$ decreases from 2 to $D_A\cong
1.305$, of the Apollonian packing. We introduced the entropy production rate as a quantitative measure of the regularity of arising patterns. We observed that for sufficiently small $\alpha$, the entropy production rate smoothly decays as the coverage increases. In the complimentary case of $\alpha \gg
1$, the entropy production rate displays a similar behavior for sufficiently small coverage, followed by a sharp decay to a very low entropy production rate. Physically, it reflects a two-stage nature of the pattern formation process in the large $\alpha$ limit: The ordinary RSA, that goes until the jammed state is reached, leads to a random structure which is starting point for the second stage. During this late stage the deposition process is deterministic and extremal – at a given step, the largest hole is filled.
One of us (PLK) acknowledges NSF grant DMR-9632059 and ARO grant DAAH04-96-1-0114 for financial support.
[99]{}
P. J. Flory, [*J. Amer. Chem. Soc.*]{} [**61**]{}, 1518 (1939). J. Feder and I. Giaever, [*J. Colloid Interface Sci.*]{} [**78**]{}, 144 (1980); A. Yokoyama, K. R. Srinivasan, and H. S. Fogler, [*J. Colloid Interface Sci.*]{} [**126**]{}, 141 (1988). E. S. Hood, B. H. Toby, and W. H. Weinberg, [*Phys. Rev. Lett.*]{} [**55**]{}, 2437 (1985). J. D. Weeks and G. H. Gilmer, [*Adv. Chem. Phys.*]{} [**40**]{}, 157 (1979). M. C. Bartelt and V. Privman, [*Int. J. Mod. Phys.*]{} [**5**]{}, 2883 (1991); J. W. Evans, [*Rev. Mod. Phys.*]{} [**65**]{}, 1281 (1993); J. J. Ramsden, [*J. Stat. Phys.*]{} [**79**]{}, 491 (1995). G. S. Baker and M. J. Grimson, [*Mol. Phys.*]{} [**63**]{}, 145 (1987). J. Talbot and P. Schaaf, [*Phys. Rev. A*]{} [**40**]{}, 422 (1989). P. Meakin and R. Jullien, [*Physica A*]{} [**187**]{}, 475 (1992). P. L. Krapivsky, [*J. Stat. Phys.*]{} [**69**]{}, 135 (1992). W. Soppe, [*Powder Technology*]{} [**62**]{}, 189 (1990); N. Standish, A. B. Yu, and R. P. Zou, [*Powder Technology*]{} [**68**]{}, 175 (1991). A preliminary account of this work has been reported in N. V. Brilliantov, Yu. A. Andrienko, P. L. Krapivsky, and J. Kurths, [*Phys. Rev. Lett.*]{} [**76**]{}, 4058 (1996). B. B. Mandelbrot, [*The Fractal Geometry of Nature*]{} (San Francisco, Freeman, 1982). J. F. Douglas, H. M. Schneider, P. Frantz, R. Lipman, and S. Granick, [*J. Phys. Condens. Matter*]{} [**9**]{}, 7699 (1997). G. V. Schulz, [*Z. Phys. Chem. Abt.*]{} [**43**]{}, 25 (1939). S. S. Manna and H. J. Herrmann, [ *J. Phys. A*]{} [**24**]{}, L481 (1991). Yu. A. Andrienko, N. V. Brilliantov, and P. L. Krapivsky, [ *J. Stat. Phys.*]{} [**75**]{}, 507 (1994); N. V. Brilliantov, P. L. Krapivsky, and Yu. A. Andrienko, [ *J. Phys. A*]{} [**27**]{}, L381 (1994). It was rigorously shown in [@pk] that for $\alpha \to 0$, the concentration of holes in the scaling limit, $y \to 0$, $t \to \infty$, behaves as $C(y,t) \sim t^2 \exp(-yt)$, i.e. the scaling function is exponential as in the mean-field theory. $a_1=\frac{\pi^2}{6}-\frac{37\pi}{12} +\frac{\pi^2b_1}{3}-4\pi
b_0$;$a_2=\frac{\pi}{12}-\frac{\pi^2}{6}\left( 1+2b_1
\right)$;\
$a_3=\frac{19\pi^2}{12}-\frac{5a_2^2}{6\pi}
-\frac{5a_2}{12}-2\pi b_0 -b_1\left(2a_2+2\pi a_2 b_3
-\frac{7\pi}{2} \right)$;\
$a_4=2\pi b_1+\pi
+\frac{a_2}{2}-\frac12$;$a_5=\pi -\frac12 +\frac{a_2}{2} + 2
\pi b_1$.\
The constants $b_k$ are given by $b_k=\int_0^{\infty}
dx\,x^k \ln x\, e^{-\pi x^2}$ and may be expressed via gamma and digamma functions: $b_k={1\over 4}\,\Gamma({k+1\over
2})\,\pi^{-(k+1)/2} \left[\psi({k+1\over 2})-\ln\pi\right]$. $C(x,t)$ for $\alpha \to \infty$ is given, e.g., in [@pk]. In 1D, the appropriate version of Apollonian construction is trivial and has $D_A=0$. One can define artificial versions with finite fractal dimension, see e.g. M. K. Hassan, [*Phys. Rev. E*]{} [**55**]{}, 5302 (1997). M. C. Mackey, [*Rev. Mod. Phys.*]{} [**61**]{}, 981 (1989). The summation over a continuous set of configurations implicitly assumes that we consider a finite subset of configurations; in a numerical implementation, this is clearly the case. B. Bonnier, D. Boyer, and P. Viot, [ *J. Phys. A*]{} [**27**]{}, 3671 (1994).
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abstract: 'The perturbative all-loop derivation of the NSVZ $\beta$-function for ${\cal N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives is finalized by calculating the sum of singularities produced by quantum superfields. These singularities originate from integrals of double total derivatives and determine all contributions to the $\beta$-function starting from the two-loop approximation. Their sum is expressed in terms of the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. This allows obtaining the NSVZ equation in the form of a relation between the $\beta$-function and these anomalous dimensions for the renormalization group functions defined in terms of the bare couplings. It holds for an arbitrary renormalization prescription supplementing the higher covariant derivative regularization. For the renormalization group functions defined in terms of the renormalized couplings we prove that in all loops one of the NSVZ schemes is given by the HD+MSL prescription.'
author:
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K.V.Stepanyantz\
\
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title: 'The all-loop perturbative derivation of the NSVZ $\beta$-function and the NSVZ scheme in the non-Abelian case by summing singular contributions'
---
Introduction
============
Ultraviolet divergences in supersymmetric theories are resticted by some non-renormalization theorems. In particular, in ${\cal N}=1$ supersymmetric theories the superpotential does not receive divergent quantum corrections [@Grisaru:1979wc]. However, even in the case of ${\cal N}=1$ supersymmetry there are some other similar statements. For example, the triple gauge-ghost vertices (which have two external lines of the Faddeev–Popov ghosts and one external line of the quantum gauge superfield) are also finite in all orders [@Stepanyantz:2016gtk]. Moreover, according to Refs. [@Novikov:1983uc; @Jones:1983ip; @Novikov:1985rd; @Shifman:1986zi], the $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories is related to the anomalous dimension of the matter superfields $(\gamma_\phi)_j{}^i$ by an equation which is usually called “the exact NSVZ $\beta$-function”. For the theory with a simple gauge group $G$ and chiral matter superfields in its representation $R$ this equation is written as
$$\label{NSVZ_Exact_Beta_Function}
\frac{\beta(\alpha,\lambda)}{\alpha^2} = - \frac{3 C_2 - T(R) + C(R)_i{}^j \big(\gamma_\phi\big)_j{}^i(\alpha,\lambda)/r}{2\pi(1- C_2\alpha/2\pi)}.$$
Note that we do not so far specify the definitions of the renormalization group functions (RGFs) and use the notations $r\equiv \dim\, G$,
$$\mbox{tr}(T^A T^B) \equiv T(R)\delta^{AB};\qquad (T^A T^A)_i{}^j \equiv C(R)_i{}^j; \qquad f^{ACD} f^{BCD} \equiv C_2\delta^{AB},$$
where $T^A$ are the generators of the group $G$ in the representation $R$ such that $[T^A, T^B] = i f^{ABC} T^C$. Also we assume that the generators of the fundamental representation $t^A$ are normalized by the condition $\mbox{tr}(t^A t^B) = \delta^{AB}/2$.
Eq. (\[NSVZ\_Exact\_Beta\_Function\]) implies that the all-order $\beta$-function of the ${\cal N}=1$ supersymmetric Yang–Mills (SYM) theory without matter superfields is given by the geometric series. Moreover, if ${\cal N}=2$ supersymmetric gauge theories are considered as a special case of the ${\cal N}=1$ supersymmetric theories, then Eq. (\[NSVZ\_Exact\_Beta\_Function\]) leads to the finiteness beyond the one-loop approximation provided the quantization is made in ${\cal N}=2$ supersymmetric way [@Shifman:1999mv; @Buchbinder:2014wra]. This implies that the NSVZ $\beta$-function is closely related to the ${\cal N}=2$ non-renormalization theorem derived in [@Grisaru:1982zh; @Howe:1983sr; @Buchbinder:1997ib]. However, for its derivation ${\cal N}=2$ supersymmetry should be manifest at all steps of calculating quantum corrections. This can be achieved with the help of the harmonic superspace [@Galperin:1984av; @Galperin:2001uw] and the invariant regularization [@Buchbinder:2015eva]. The finiteness of ${\cal N}=4$ SYM theory [@Grisaru:1982zh; @Howe:1983sr; @Mandelstam:1982cb; @Brink:1982pd] follows from the ${\cal N}=2$ non-renormalization theorem and, therefore, from the NSVZ $\beta$-function.
Equations analogous to NSVZ are also known for the Adler $D$-function in ${\cal N}=1$ SQCD [@Shifman:2014cya; @Shifman:2015doa] and for the renormalization of the gaugino mass in gauge theories with softly broken supersymmetry [@Hisano:1997ua; @Jack:1997pa; @Avdeev:1997vx]. Recently an NSVZ-like equation was constructed for the renormalization of the Fayet–Iliopoulos term in $D=2$, ${\cal N}=(2,0)$ supersymmetric theories [@Chen:2019eta].
It is important that the NSVZ and NSVZ-like equations are valid only for certain renormalization prescriptions. In particular, for theories regularized by dimensional reduction [@Siegel:1979wq] supplemented by modified minimal subtraction [@Bardeen:1978yd] (this scheme is usually called $\overline{\mbox{DR}}$) Eq. (\[NSVZ\_Exact\_Beta\_Function\]) is not valid starting from the order $O(\alpha^2,\alpha\lambda^2,\lambda^4)$, which corresponds to the three-loop $\beta$-function and the two-loop anomalous dimension [@Jack:1996vg; @Jack:1996cn; @Jack:1998uj; @Harlander:2006xq; @Mihaila:2013wma].[^1] However, the detailed analysis made in these papers demonstrated that by a specially tuned finite renormalization of the gauge coupling constant one can restore the NSVZ equation (\[NSVZ\_Exact\_Beta\_Function\]), at least, in the three- and four-loop approximations. It should be noted that the possibility of this tuning is non-trivial, because of the presence of various group invariants (like $C_2$, $C(R)_i{}^j$, etc.). If one considers only finite renormalizations polynomial in these invariants, then the NSVZ equation leads to some scheme-independent consequences [@Kataev:2013csa; @Kataev:2014gxa]. This implies that, although the calculations made in the $\overline{\mbox{DR}}$-scheme do not produce the NSVZ equation, they nevertheless confirm that it is valid in a certain (NSVZ) subtraction scheme. Using the general equations describing how the NSVZ equation changes under finite renormalizations [@Kataev:2014gxa; @Kutasov:2004xu], it is possible to construct an infinite set of the NSVZ schemes [@Goriachuk:2018cac].
For a long time it was unknown, how to construct an all-order renormalization prescription giving the NSVZ scheme. However, recently it was understood that the solution can be found in the case of using the higher covariant derivative method [@Slavnov:1971aw; @Slavnov:1972sq] for regularizing supersymmetric theories. Unlike dimensional reduction [@Siegel:1980qs], this regularization is mathematically consistent and can be formulated in a manifestly supersymmetric way in ${\cal N}=1$ superspace [@Krivoshchekov:1978xg; @West:1985jx]. Although the calculations in higher derivative theories are rather complicated, some of them have been done in the last decades. For instance, a number of calculations of the one-loop effective potential for ${\cal N}=1$ higher derivative supersymmetric theories were made in [@Gomes:2009ev; @Gama:2011ws; @Gama:2013rsa; @Gama:2014fca; @BezerradeMello:2016bjn; @Gama:2017ets; @Gama:2020pte]. In theories [*regularized*]{} by higher derivatives quantum corrections are obtained in a similar way. Some higher order calculations made with this regularization (see, e.g., [@Stepanyantz:2012zz; @Shakhmanov:2017soc; @Kazantsev:2018nbl; @Kuzmichev:2019ywn]) demonstrated that the NSVZ equation is valid for RGFs defined in terms of the bare couplings. (In the Abelian case this has been proved in all orders [@Stepanyantz:2011jy; @Stepanyantz:2014ima]. Similar proofs of the NSVZ-like equations have also been constructed for the Adler $D$-function in ${\cal N}=1$ SQCD [@Shifman:2014cya; @Shifman:2015doa] and for the renormalization of the photino mass in ${\cal N}=1$ SQED with softly broken supersymmetry [@Nartsev:2016nym].) RGFs defined in terms of the bare couplings are scheme-independent for a fixed regularization [@Kataev:2013eta], but depend on a regularization, so that Eq. (\[NSVZ\_Exact\_Beta\_Function\]) is valid for them for an arbitrary renormalization prescription supplementing the higher derivative regularization. The above mentioned calculations confirmed this in such an approximation where the dependence on a regularization is essential. Note that, according to [@Aleshin:2019yqj; @Aleshin:2016rrr], RGFs defined in terms of the bare couplings do not satisfy the NSVZ and NSVZ-like equations in the case of using dimensional reduction, again, for an arbitrary renormalization prescription supplementing it.
The important statement which allows constructing the NSVZ scheme is that RGFs defined in terms of the bare couplings and RGFs standardly defined in terms of the renormalized couplings up to the renaming of arguments coincide in the HD+MSL scheme [@Kataev:2013eta], when the divergences of the theory regularized by Higher covariant Derivatives are removed by Minimal Subtractions of Logarithms [@Shakhmanov:2017wji; @Stepanyantz:2017sqg]. This implies that the renormalization constants are constructed in such a way that they include only powers of $\ln\Lambda/\mu$, where $\Lambda$ is a regularization parameter, analogous to the ultraviolet cut-off, and $\mu$ is a renormalization point. All finite constants in this case, by definition, are set to 0. The coincidence of various definitions of RGFs implies that the HD+MSL scheme appears to be one of the NSVZ schemes. In the Abelian case this has been proved in all loops [@Kataev:2013eta]. Note that another all-loop NSVZ scheme in the Abelian case is the on-shell scheme [@Kataev:2019olb]. For the renormalization of the photino mass in softly broken ${\cal N}=1$ SQED and for the Adler $D$-function in ${\cal N}=1$ SQCD the NSVZ-like schemes are also given by the HD+MSL prescription, see Refs. [@Nartsev:2016mvn] and [@Kataev:2017qvk], respectively. In the non-Abelian case there are strong indications [@Stepanyantz:2016gtk] that HD+MSL = NSVZ. Also there are some nontrivial explicit multiloop calculations confirming this fact [@Shakhmanov:2017soc; @Kazantsev:2018nbl; @Kuzmichev:2019ywn], but, nevertheless, this statement has not yet been proved in all orders. This proof (started in Refs. [@Stepanyantz:2016gtk; @Stepanyantz:2019ihw; @Stepanyantz:2019lfm]) will be finalized in this paper.
The main observation used for the derivation of the NSVZ and NSVZ-like equations for RGFs defined in terms of the bare couplings in theories regularized by higher derivatives is that the loop integrals giving the $\beta$-function are integrals of double total derivatives with respect to loop momenta.[^2] Certainly, at least one of the loop integrals can be calculated analytically using equations like
$$\label{Integral_Toy}
\int \frac{d^4Q}{(2\pi)^4} \frac{\partial^2}{\partial Q_\mu \partial Q^\mu}\Big(\frac{f(Q^2)}{Q^2}\Big) = \frac{1}{4\pi^2} f(0),$$
where $f(Q^2)$ is a nonsingular function of the Euclidean momentum $Q_\mu$ which rapidly tends to 0 at infinity. Note that the result does not vanish due to the nontrivial surface integration over the small sphere $S^3_\varepsilon$ surrounding the point $Q_\mu =0$. If we consider an $L$-loop contribution to the $\beta$-function, then Eq. (\[Integral\_Toy\]) allows calculating one of the loop integrals, so that the resulting expression will contain only $(L-1)$ loop integrations. Therefore, it is possible to suggest that the result is a certain $(L-1)$-loop quantum correction. According to [@Stepanyantz:2011jy; @Stepanyantz:2014ima], in the Abelian case it is really proportional to the $(L-1)$-loop contribution to anomalous dimension of the matter superfields, so that the Abelian NSVZ equation [@Vainshtein:1986ja; @Shifman:1985fi]
$$\frac{\beta(\alpha_0)}{\alpha_0^2} = \frac{N_f}{\pi}\Big(1-\gamma(\alpha_0)\Big),$$
where $N_f$ is a number of flavors, is naturally produced for RGFs defined in terms of the bare couplings.
In the non-Abelian case the situation is more complicated, because Eq. (\[NSVZ\_Exact\_Beta\_Function\]) relates the $\beta$-function to the anomalous dimension of the matter superfields [*in all previous orders*]{} due to the gauge coupling constant dependent denominator in the right hand side. However, according to Ref. [@Stepanyantz:2016gtk], using the all-loop finiteness of triple gauge-ghost vertices the NSVZ equation (\[NSVZ\_Exact\_Beta\_Function\]) can be presented in an equivalent form
$$\label{NSVZ_Equivalent_Form}
\frac{\beta(\alpha,\lambda)}{\alpha^2} = - \frac{1}{2\pi}\Big(3 C_2 - T(R) - 2C_2 \gamma_c(\alpha,\lambda) - 2C_2 \gamma_V(\alpha,\lambda) + \frac{1}{r} C(R)_i{}^j \big(\gamma_\phi\big)_j{}^i(\alpha,\lambda)\Big),$$
which does not contain the coupling constant dependent denominator in the right hand side similarly to the Abelian case. (Note that in Eq. (\[NSVZ\_Equivalent\_Form\]) we again do not specify the definitions of RGFs and omit some other possible arguments.) Similarly to the Abelian case, Eq. (\[NSVZ\_Equivalent\_Form\]) relates an $L$-loop contribution [*only*]{} to $(L-1)$-loop contributions to the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields, denoted by $\gamma_V$, $\gamma_c$, and $(\gamma_\phi)_j{}^i$, respectively. This fact has a simple graphical interpretation analogous to the ${\cal N}=1$ SQED case considered in [@Smilga:2004zr; @Kazantsev:2014yna]. Namely, given a supergraph without external lines, the corresponding superdiagrams contributing to the $\beta$-function are constructed by attaching two external lines of the background gauge superfield in all possible ways, while the superdiagrams contributing to the anomalous dimensions are obtained by all possible cuts of internal lines. This qualitative picture is related to the structure of loop integrals giving the $\beta$-function defined in terms of the bare couplings. According to [@Stepanyantz:2019ihw] they are integrals of double total derivatives in all orders in agreement with numerous calculations made with the higher covariant derivative regularization, see, e.g., [@Shakhmanov:2017soc; @Kazantsev:2018nbl; @Pimenov:2009hv; @Stepanyantz:2011bz; @Stepanyantz:2019lyo]. Each cut of a certain propagator corresponds to taking an integral of a double total derivative analogous to (\[Integral\_Toy\]). The sums of singularities generated by the cuts of various propagators produce the corresponding anomalous dimensions in Eq. (\[NSVZ\_Equivalent\_Form\]) even at the level of loop integrals.[^3] In the lowest orders this was explicitly demonstrated in [@Shakhmanov:2017soc; @Kazantsev:2018nbl; @Kuzmichev:2019ywn; @Shakhmanov:2017wji; @Stepanyantz:2019lyo]. In all loops the sums of singularities obtained by cutting the matter and Faddeev–Popov ghost propagators have been found in [@Stepanyantz:2019lfm] using the method based on the Schwinger–Dyson effective superdiagrams proposed in [@Stepanyantz:2004sg]. These sums coincide with the terms containing $(\gamma_\phi)_j{}^i$ and $\gamma_c$ defined in terms of the bare couplings, respectively. Therefore, to complete the derivation of the NSVZ equation, one should find a sum of all singularities produced by cuts of the gauge propagators and demonstrate that it gives the term containing $\gamma_V$ in Eq. (\[NSVZ\_Equivalent\_Form\]).[^4] This is a purpose of the present paper.
The paper is organized as follows. Section \[Section\_N=1\_Gauge\_Theories\] describes the quantization of ${\cal N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives. In section \[Section\_Perturbative\_Derivation\] we rewrite the NSVZ $\beta$-function in the form of a relation between some two-point Green functions, which will be proved below. The proof is based on the method for constructing the integrals of double total derivatives giving the function $\beta/\alpha_0^2$ which was proposed in [@Stepanyantz:2019ihw], see also [@Kuzmichev:2019ywn]. This method is described in section \[Subsection\_Idea\]. It is slightly modified in section \[Subsection\_Graphs\]. Using this modification the sums of singularities produced by the matter superfields, by the Faddeev–Popov ghosts, and by the quantum gauge superfield are calculated exactly in all loops in section \[Section\_Singularities\]. In particular, in subsection \[Subsection\_Gauge\_Singularities\] we find the sum of singularities produced by the quantum gauge superfield propagators, which is needed for finalizing the proof of the NSVZ $\beta$-function. Combining the results we derive Eq. (\[NSVZ\_Equivalent\_Form\]) for RGFs defined in terms of the bare couplings. In the next section \[Section\_NSVZ\_Scheme\] we demonstrate that the HD+MSL prescription really gives one of the NSVZ schemes in all orders for RGFs defined in terms of the renormalized couplings. The last section \[Section\_Explicit\_Calculation\] is devoted to the explicit perturbative verification of the results in the lowest orders of the perturbation theory. Doing the corresponding calculations we pay especial attention to checking the modification of the method for constructing integrals giving the function $\beta/\alpha_0^2$ proposed in section \[Subsection\_Graphs\].
${\cal N}=1$ renormalizable supersymmetric gauge theories regularized by higher covariant derivatives
=====================================================================================================
\[Section\_N=1\_Gauge\_Theories\]
We will consider a general renormalizable ${\cal N}=1$ SYM theory with a simple gauge group $G$ and chiral matter superfields $\phi_i$ in a representation $R$. In the massless limit in terms of ${\cal N}=1$ superfields [@Gates:1983nr; @West:1990tg; @Buchbinder:1998qv] this theory is described by the action
$$\begin{aligned}
\label{Action_Under_Consideration}
S = \frac{1}{2e_0^2}\, \mbox{Re}\,\mbox{tr} \int d^6x\, W^a W_a + \frac{1}{4} \int d^8x\, \phi^{*i} (e^{2{\cal F}(V)} e^{2\bm{V}})_i{}^j \phi_j
+ \Big(\frac{1}{6} \lambda_0^{ijk} \int d^6x\, \phi_i \phi_j \phi_k + \mbox{c.c.}\Big),\end{aligned}$$
where for the superspace integration measures we use the brief notations
$$d^8x \equiv d^4x\, d^4\theta_x;\qquad d^6x \equiv d^4x\, d^2\theta_x;\qquad d^6\bar x \equiv d^4x\, d^2\bar\theta_x.$$
In Eq. (\[Action\_Under\_Consideration\]) $\bm{V}$ is the Hermitian background gauge superfield and $V$ is the quantum gauge superfield, which satisfies the constraint $V^+ = e^{-2\bm{V}} V e^{2\bm{V}}$. Note that in the first term of Eq. (\[Action\_Under\_Consideration\]) the quantum and background gauge superfields are expanded in the generators of the fundamental representation as $V = e_0 V^A t^A$ and $\bm{V} = e_0 \bm{V}^A t^A$, while in the second term $V = e_0 V^A T^A$ and $\bm{V} = e_0 \bm{V}^A T^A$, where $T^A$ are the generators of the gauge group in the representation $R$.
The gauge and Yukawa couplings are denoted by $e_0$ and $\lambda_0^{ijk}$, respectively, where the subscript $0$ points the bare values. The function ${\cal F}(V) = e_0 {\cal F}(V)^A t^A$ is needed, because the quantum gauge superfield is renormalized in the non-linear way [@Piguet:1981fb; @Piguet:1981hh; @Tyutin:1983rg]. In the lowest approximation this function was found in Refs. [@Juer:1982fb; @Juer:1982mp] and is written as
$$\label{Nonlinear_Function_F}
{\cal F}(V)^A = V^A + y_0\, e_0^2\, G^{ABCD}\, V^B V^C V^D + \ldots,$$
where $y_0$ is the first bare constant in an infinite set of parameters describing the nonlinearity, and $G^{ABCD} \equiv \big(f^{AKL} f^{BLM} f^{CMN} f^{DNK} + \mbox{permutations of $B$, $C$, and $D$}\big)/6$. The necessity of introducing the function ${\cal F}(V)$ was also confirmed by the calculation of Ref. [@Kazantsev:2018kjx], where it was demonstrated that the renormalization group equations are satisfied only if the renormalization of the parameter $y$ is taken into account.
The gauge superfield strength is described by the chiral superfield
$$\label{W_a_Definition}
W_a \equiv \frac{1}{8} \bar D^2 \Big[ e^{-2\bm{V}} e^{-2{\cal F}(V)}\, D_a \Big(e^{2{\cal F}(V)}e^{2\bm{V}}\Big)\Big],$$
which is a right spinor with respect to the Lorentz group.
If the Yukawa couplings satisfy the equation
$$\label{Yukawa_Constraint}
\lambda_0^{ijm} (T^A)_m{}^k + \lambda_0^{imk} (T^A)_m{}^j + \lambda_0^{mjk} (T^A)_m{}^i = 0,$$
the theory (\[Action\_Under\_Consideration\]) is invariant under the background gauge transformations
$$\label{Background_Gauge_Invariance_Original}
e^{2\bm{V}} \to e^{-A^+} e^{2\bm{V}} e^{-A};\qquad V \to e^{-A^+} V e^{A^+};\qquad \phi_i \to (e^A)_i{}^j \phi_j$$
parameterized by the chiral superfield $A$ which takes values in the Lie algebra of the gauge group $G$. The action (\[Action\_Under\_Consideration\]) is also invariant under the quantum gauge transformations, but this invariance is broken by the gauge fixing procedure.
The regularization is implemented by inserting into the classical action the higher derivative regulator functions $R$ and $F$. They should rapidly increase at infinity and satisfy the condition $R(0)=F(0)=1$. Then the action of the regularized theory can be written as
$$\begin{aligned}
\label{Action_Regularized_Without_G}
&& S_{\mbox{\scriptsize reg}} = \frac{1}{2 e_0^2}\,\mbox{Re}\, \mbox{tr} \int d^6x\, W^a \Big[e^{-2\bm{V}} e^{-2{\cal F}(V)}\, R\Big(-\frac{\bar\nabla^2 \nabla^2}{16\Lambda^2}\Big)\, e^{2{\cal F}(V)}e^{2\bm{V}}\Big]_{Adj} W_a \qquad\nonumber\\
&& + \frac{1}{4} \int d^8x\, \phi^{*i} \Big[\, F\Big(-\frac{\bar\nabla^2 \nabla^2}{16\Lambda^2}\Big) e^{2{\cal F}(V)}e^{2\bm{V}}\Big]_i{}^j \phi_j
+ \Big(\frac{1}{6} \lambda_0^{ijk} \int d^6x\, \phi_i \phi_j \phi_k + \mbox{c.c.} \Big),\qquad\end{aligned}$$
where for $f(x) = 1 + f_1 x + f_2 x^2 +\ldots$
$$f(X)_{Adj} Y \equiv Y + f_1 [X, Y] + f_2 [X,[X,Y]] + \ldots,$$
and the explicit expressions for the covariant derivatives have the form
$$\label{Covariant_Derivative_Definition}
\nabla_a = D_a;\qquad \bar\nabla_{\dot a} = e^{2{\cal F}(V)} e^{2\bm{V}} \bar D_{\dot a} e^{-2\bm{V}} e^{-2{\cal F}(V)}.$$
The modification of the action $S \to S_{\mbox{\scriptsize reg}}$ allows regularizing all divergences beyond the one-loop approximation [@Faddeev:1980be].
The generating functional for the regularized theory should also include a gauge fixing term, ghost actions, sources, and Pauli–Villars determinants needed for removing the remaining one-loop divergences [@Slavnov:1977zf],
$$\begin{aligned}
\label{Z_Functional_Generating}
Z = \int D\mu\, \mbox{Det}(PV,M_\varphi)^{-1}\mbox{Det}(PV,M)^c \exp\Big\{i\Big(S_{\mbox{\scriptsize reg}}
+ S_{\mbox{\scriptsize gf}} + S_{\mbox{\scriptsize FP}} + S_{\mbox{\scriptsize NK}} + S_{\mbox{\scriptsize sources}}\Big)\Big\}.\end{aligned}$$
The source term is given by the expression
$$S_{\mbox{\scriptsize sources}} = \int d^8x\, J^A V^A + \Big(\int d^6x\, \Big(j^i \phi_i + j_c^A c^A + \bar j_c^A \bar c^A\Big) + \mbox{c.c.}\Big),$$
where $J^A$ and $j^i$ are the real and chiral superfields, respectively. The anticommuting chiral superfields $j_c^A$, and $\bar j_c^A$ are the sources for the anticommuting chiral Faddeev–Popov ghost and antighost superfields denoted by $c^A$ and $\bar c^A$, respectively.
We will use the gauge fixing term
$$\label{Term_For_Fixing_Gauge}
S_{\mbox{\scriptsize gf}} = -\frac{1}{16\xi_0 e_0^2}\, \mbox{tr} \int d^8x\, \bm{\nabla}^2 V K\Big(-\frac{\bm{\bar\nabla}^2 \bm{\nabla}^2}{16\Lambda^2}\Big)_{Adj} \bm{\bar\nabla}^2 V$$
invariant under the background gauge transformations (\[Background\_Gauge\_Invariance\_Original\]) due to the presence of the background covariant derivatives
$$\label{Background_Values}
\bm{\nabla}_a \equiv D_a;\qquad\quad \bm{\bar\nabla}_{\dot a} \equiv e^{2\bm{V}} \bar D_{\dot a} e^{-2\bm{V}}.$$
The corresponding actions for the Faddeev–Popov and Nielsen–Kallosh ghosts ($S_{\mbox{\scriptsize FP}}$ and $S_{\mbox{\scriptsize NK}}$, respectively) can be found in Refs. [@Stepanyantz:2019ihw; @Stepanyantz:2019lfm]. In this paper we will not use the explicit expressions for them. The bare gauge parameter $\xi_0$ and the parameters present in the function ${\cal F}(V)$ (i.e., $y_0$, etc.) can conveniently be included into a single infinite set $Y_0 \equiv (\xi_0,\,y_0,\ldots)$.
Following Refs. [@Aleshin:2016yvj; @Kazantsev:2017fdc], for regularizing the one-loop divergences we use two sets of the Pauli–Villars superfields. The first one consists of three commuting chiral superfields $\varphi_a$ in the adjoint representation of the gauge group with the mass $M_\varphi = a_\varphi\Lambda$. They cancel the divergences coming from the gauge and ghost loops. The corresponding determinant present in Eq. (\[Z\_Functional\_Generating\]) is denoted by $ \mbox{Det}(PV,M_\varphi)$. The second set giving the determinant $\mbox{Det}(PV,M)$ consists of the commuting chiral superfields $\Phi_i$ in a certain representation $R_{\mbox{\scriptsize PV}}$ that admits the gauge invariant mass term such that $M^{ij} M_{jk} = M^2 \delta^i_k$ with $M = a\Lambda$.[^5] These superfields cancel the one-loop divergences produced by the matter superfields $\phi_i$ if in Eq. (\[Z\_Functional\_Generating\]) the degree of the corresponding determinant is $c=T(R)/T(R_{\mbox{\scriptsize PV}})$. Again we will not use explicit expressions of the Pauli–Villars determinants which can also be found in Refs. [@Stepanyantz:2019ihw; @Stepanyantz:2019lfm]. It should be only mentioned that the constants $a_\varphi$ and $a$ must be independent of couplings.
Starting from the generating functional for the connected Green functions $W \equiv -i\ln Z$, one can construct the effective action
$$\Gamma[\bm{V}, V, \phi_i,\ldots] \equiv W - S_{\mbox{\scriptsize sources}}\Big|_{\mbox{\scriptsize sources}\, \to\, \mbox{\scriptsize fields}},$$
where it is necessary to express the sources $J^A, j^i,\ldots$ in terms of quantum superfields $V^A,\phi_i,\ldots$ using the equations
$$\label{Field_Definitions}
V^A \equiv \frac{\delta W}{\delta J^A},\qquad \phi_i \equiv \frac{\delta W}{\delta j_i},\qquad \mbox{etc}.$$
The NSVZ equation as a relation between two-point Green functions
=================================================================
\[Section\_Perturbative\_Derivation\]
The NSVZ equation (\[NSVZ\_Equivalent\_Form\]) for RGFs defined in terms of the bare couplings in the case of using the higher covariant derivative regularization can be written as a certain equation relating two-point Green functions of the theory. The terms in the effective action corresponding to these Green functions can be presented in the form
$$\begin{aligned}
\label{Gamma2_Background_V}
&&\hspace*{-12mm} \Gamma^{(2)}_{\bm{V}} = - \frac{1}{8\pi} \mbox{tr} \int \frac{d^4p}{(2\pi)^4}\, d^4\theta\, \bm{V}(-p,\theta) \partial^2 \Pi_{1/2} \bm{V}(p,\theta)\, d^{-1}(\alpha_0, \lambda_0, Y_0, \Lambda/p);\\
\label{Gamma2_Quantum_V}
&&\hspace*{-12mm} \Gamma^{(2)}_V - S_{\mbox{\scriptsize gf}}^{(2)} = -\frac{1}{4} \int \frac{d^4q}{(2\pi)^4}\, d^4\theta\, V^A(-q,\theta) \partial^2\Pi_{1/2} V^A(q,\theta)\, G_V(\alpha_0, \lambda_0, Y_0, \Lambda/q);\\
\label{Gamma2_Phi}
&&\hspace*{-12mm} \Gamma^{(2)}_\phi = \frac{1}{4}\int \frac{d^4q}{(2\pi)^4}\, d^4\theta\, \phi^{*i}(-q,\theta) \phi_j(q,\theta) \big(G_\phi\big)_i{}^j(\alpha_0, \lambda_0, Y_0, \Lambda/q);\\
\label{Gamma2_C}
&&\hspace*{-12mm} \Gamma^{(2)}_c = \frac{1}{4}\int \frac{d^4q}{(2\pi)^4}\, d^4\theta\, \Big(\bar c^{+A}(-q,\theta) c^{A}(q,\theta) - \bar c^A(-q,\theta) c^{+A}(q,\theta)\Big) G_c(\alpha_0, \lambda_0, Y_0, \Lambda/q).\end{aligned}$$
where $\partial^2\Pi_{1/2} \equiv - D^a \bar D^2 D_a/8$ is the supersymmetric transversal projection operator.
Writing Eq. (\[Gamma2\_Background\_V\]) we took into account that the two-point Green function of the background gauge superfield is transversal due to the manifest background gauge invariance of the effective action. Similarly, in Eq. (\[Gamma2\_Quantum\_V\]) we used the fact that quantum corrections to the two-point Green function of the quantum gauge superfield are also transversal due to the Slavnov–Taylor identity [@Taylor:1971ff; @Slavnov:1972fg].
In our notation the renormalization constants are introduced with the help of the equations[^6]
$$\begin{aligned}
&& Z_\alpha \equiv \frac{\alpha}{\alpha_0};\qquad\quad V^A \equiv Z_V (V_R)^A;\qquad\quad Z_y \equiv \frac{y_0}{y};\qquad\quad \ldots \nonumber\\
&& Z_\xi \equiv \frac{\xi}{\xi_0};\qquad \bar c^A c^B = Z_c (\bar c_R)^A (c_R)^B;\qquad \phi_i = \big(\sqrt{Z_\phi}\big)_i{}^j \big(\phi_R\big)_j,\qquad\end{aligned}$$
where the subscript $R$ stands for the renormalized superfields. The renormalization constants are constructed from the requirement that the functions $d^{-1}$, $Z_V^2 G_V$, $(Z_\phi G_\phi)_i{}^j$, and $Z_c G_c$ expressed in terms of the renormalized couplings $\alpha$, $\lambda^{ijk}$, and $Y$ should be finite in the limit $\Lambda\to\infty$. Note that due to the non-renormalization of the superpotential [@Grisaru:1979wc] the renormalized Yukawa couplings are given by
$$\label{Lambda_Renormalization}
\lambda^{ijk} = \big(\sqrt{Z_\phi}\big)_m{}^i \big(\sqrt{Z_\phi}\big)_n{}^j \big(\sqrt{Z_\phi}\big)_p{}^k \lambda_0^{mnp}.$$
We will always assume that no finite constants corresponding to finite renormalizations appear in this equation, so that Eq. (\[Lambda\_Renormalization\]) partially fixes the subtraction scheme. Similarly, due to the all-loop finiteness of the triple gauge-ghost vertices [@Stepanyantz:2016gtk][^7] it is possible to choose a subtraction scheme in which the renormalization constants satisfy the condition
$$\label{ZZZ_Constraint}
Z_\alpha^{-1/2} Z_c Z_V =1.$$
Note that this equation is compatible with minimal subtractions of logarithms, because in the HD+MSL scheme all renormalization constants $Z_\alpha$, $Z_c$, and $Z_V$ are equal to 1 for $\mu=\Lambda$.
The equation (\[ZZZ\_Constraint\]) allows to present the NSVZ equation in an equivalent form relating the $\beta$-function to the anomalous dimensions of the quantum gauge superfield, the Faddeev–Popov ghosts, and the matter superfields,
$$\begin{aligned}
\label{NSVZ_Equivalent_Form_Bare}
&& \frac{\beta(\alpha_0,\lambda_0,Y_0)}{\alpha_0^2} = - \frac{1}{2\pi}\Big(3 C_2 - T(R) - 2C_2 \gamma_c(\alpha_0,\lambda_0,Y_0)\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\ \ - 2C_2 \gamma_V(\alpha_0,\lambda_0,Y_0) + \frac{1}{r} C(R)_i{}^j \big(\gamma_\phi\big)_j{}^i(\alpha_0,\lambda_0,Y_0)\Big),\qquad\end{aligned}$$
where we take into account that RGFs (at least, $\gamma_V$ and $\gamma_c$) may in general depend on $Y_0$. RGFs defined in terms of the bare couplings entering this equation can be related to the corresponding two-point Green functions by the equations
$$\begin{aligned}
\label{Beta_Bare_Definition}
&& \frac{\beta(\alpha_0,\lambda_0,Y_0)}{\alpha_0^2} = -\frac{d}{d\ln\Lambda}\Big(\frac{1}{\alpha_0}\Big)\bigg|_{\alpha,\lambda,Y=\mbox{\scriptsize const}}
= \frac{d}{d\ln\Lambda} \Big(d^{-1} - \alpha_0^{-1}\Big)\bigg|_{\alpha,\lambda,Y=\mbox{\scriptsize const};\, p\to 0};\qquad\\
\label{Gamma_V_Bare_Definition}
&& \gamma_V(\alpha_0,\lambda_0,Y_0) = - \frac{d\ln Z_V}{d\ln\Lambda}\bigg|_{\alpha,\lambda,Y = \mbox{\scriptsize const}} = \frac{1}{2}\,\frac{d\ln G_V}{d\ln\Lambda}\bigg|_{\alpha,\lambda,Y = \mbox{\scriptsize const};\ q\to 0};\\
\label{Gamma_C_Bare_Definition}
&& \gamma_c(\alpha_0,\lambda_0,Y_0) = - \frac{d\ln Z_c}{d\ln\Lambda}\bigg|_{\alpha,\lambda,Y = \mbox{\scriptsize const}} = \frac{d\ln G_c}{d\ln\Lambda}\bigg|_{\alpha,\lambda,Y = \mbox{\scriptsize const};\ q\to 0};\\
\label{Gamma_Phi_Bare_Definition}
&& \big(\gamma_\phi\big)_i{}^j(\alpha_0,\lambda_0,Y_0) = - \frac{d\big(\ln Z_\phi\big)_i{}^j}{d\ln\Lambda}\bigg|_{\alpha,\lambda,Y = \mbox{\scriptsize const}} = \frac{d\big(\ln G_\phi\big)_i{}^j}{d\ln\Lambda}\bigg|_{\alpha,\lambda,Y = \mbox{\scriptsize const};\ q\to 0}.\end{aligned}$$
That is why Eq. (\[NSVZ\_Equivalent\_Form\_Bare\]) can also be rewritten as an equation relating these Green functions
$$\begin{aligned}
\label{NSVZ_For_Green_Functions_With_1Loop}
&& \left.\frac{d}{d\ln\Lambda} \Big(d^{-1} -\alpha_0^{-1}\Big)\right|_{\alpha,\lambda,Y = \mbox{\scriptsize const};\ p\to 0} = -\frac{1}{2\pi} \Big(3C_2-T(R)\Big) + \frac{1}{2\pi}\, \frac{d}{d\ln\Lambda} \Big( 2C_2 \ln G_c \qquad\nonumber\\
&& + C_2 \ln G_V - \frac{1}{r} C(R)_i{}^j \big(\ln G_\phi\big)_j{}^i\Big)\bigg|_{\alpha,\lambda,Y = \mbox{\scriptsize const};\ q\to 0},\qquad\end{aligned}$$
which was first proposed in Ref. [@Stepanyantz:2016gtk]. In this paper we will prove it in all orders of the perturbation theory.
Integrals of total derivatives {#Section_Integrals}
==============================
How to construct and calculate integrals of total derivatives
-------------------------------------------------------------
\[Subsection\_Idea\]
The key observation needed for deriving the NSVZ $\beta$-function is the factorization of loop integrals which give the function $\beta(\alpha_0,\lambda_0,Y_0)/\alpha_0^2$ into integrals of double total derivatives in the case of using the higher covariant derivative regularization. The all-loop proof of this fact was done in Ref. [@Stepanyantz:2019ihw]. The ideas used in this proof allowed constructing a method for obtaining these integrals in each order of the perturbation theory. For this purpose one should calculate only specially modified vacuum supergraphs[^8] instead of a much larger number of superdiagrams with two external legs of the background gauge superfield. Some higher-order calculations made with the help of this method in Refs. [@Kuzmichev:2019ywn; @Stepanyantz:2019lyo; @Aleshin:2020gec] demonstrated that it works correctly and reproduces all known results. Moreover, the structure of quantum corrections which were first obtained by this method is in excellent agreement with some general theorems. Say, the $\beta$-function in the considered theories appeared to be gauge independent and satisfies the NSVZ equation in the HD+MSL scheme. Here using this method we will find an exact all-order expression for the function
$$\label{Delta_Beta}
\frac{\beta(\alpha_0,\lambda_0,Y_0)}{\alpha_0^2} - \frac{\beta_{\mbox{\scriptsize 1-loop}}(\alpha_0)}{\alpha_0^2},$$
where
$$\label{Beta_1Loop}
\beta_{\mbox{\scriptsize 1-loop}}(\alpha_0) = -\frac{\alpha_0^2}{2\pi}\left(3C_2-T(R)\right)$$
is the one-loop $\beta$-function defined in terms of the bare couplings. (For the higher covariant derivative regularization considered in this paper it was calculated in Ref. [@Aleshin:2016yvj].)
Now, let us formulate the algorithm for constructing contributions to the function (\[Delta\_Beta\]) following Refs. [@Kuzmichev:2019ywn; @Stepanyantz:2019ihw].
1\. As a starting point we consider a vacuum supergraph with $L$ loops and construct an expression for it using the superspace Feynman rules.
2\. Next, one needs to find a point with the integration over $d^4\theta$ (or convert the integration over $d^2\theta$ into the integration over $d^4\theta$) and insert the expression $\theta^4 (v^B)^2$ to the corresponding point of this supergraph. Here $v^B$ denotes a function slowly decreasing at a very large scale $R\to \infty$. (Note that without this insertion any vacuum supergraph vanishes due to the integration over the anticommuting variables $\theta$.)
3\. We calculate the supergraph modified by the above insertion and omit terms suppressed by powers of $1/(R\Lambda)$. As a result we obtain an expression proportional to
$${\cal V}_4 \equiv \int d^4x\, \big(v^B\big)^2 \to \infty.$$
4\. At the next step it is necessary to choose $L$ propagators with independent momenta denoted by $Q_i^\mu$, where $i=1,\ldots,L$. (In our notation the capital letters denote Euclidean momenta which appear after the Wick rotation.) Let the gauge group indices corresponding to the beginnings and endings of these propagators be $a_i$ and $b_i$, respectively. Then the product of the marked propagators contains the factor $\prod_{i=1}^L \delta_{a_i}^{b_i}$.
5\. The product $\prod_{i=1}^L \delta_{a_i}^{b_i}$ coming from the marked propagators in the integrand of the (Euclidean) loop integral should formally be replaced by the operator
$$\label{Replacement}
\sum_{k,l=1}^L \Big(\prod\limits_{i\ne k,l} \delta_{a_i}^{b_i}\Big)\, (T^A)_{a_k}{}^{b_k} (T^A)_{a_l}{}^{b_l} \frac{\partial^2}{\partial Q^\mu_k \partial Q^\mu_l}.$$
6\. At the last step, it is necessary to apply the operator
$$\label{Operator}
- \frac{2\pi}{r{\cal V}_4}\cdot \frac{d}{d\ln\Lambda}$$
to the resulting expression, where the derivative with respect to $\ln\Lambda$ should be calculated at fixed vales of the renormalized couplings prior to the integration over loop momenta.
After the above described procedure we obtain a contribution to the function (\[Delta\_Beta\]) corresponding to the considered supergraph. It is produced by the sum of all two-point superdiagrams which are constructed from this supergraph by attaching two external lines of the background gauge superfield $\bm{V}$ in all possible ways. By construction, the result is given by a certain integral of double total derivatives with respect to the loop momenta.
The integrals of total derivatives do not vanish due to the singularities of the integrands, which appear when two momentum derivatives act on an inverse squared momentum,
$$\label{Delta_Singularity}
\frac{\partial^2}{\partial Q_\mu \partial Q_\mu} \Big(\frac{1}{Q^2}\Big) = -4\pi^2 \delta^4(Q).$$
We will also need a modification of this identity which is obtained when the derivatives are taken with respect to the different momenta,
$$\label{Delta_Singularity_General}
\frac{\partial^2}{\partial Q_{\mu,1} \partial Q_{\mu,2}} \Big(\frac{1}{(a_1 Q_{\mu,1}+ a_2 Q_{\mu,2}+ Q_{\mu,3})^2}\Big) = - 4\pi^2 a_1 a_2\, \delta^4\Big(a_1 Q_{\mu,1}+ a_2 Q_{\mu,2}+ Q_{\mu,3}\Big),$$
where $a_1$ and $a_2$ are some constants.
Note that in calculating the integrals of double total derivatives we should take these singular contributions with the opposite sign. Really, if $f(Q^2)$ is a non-singular function which rapidly tends to 0 at infinity, then
$$\label{Integral_And_Singularity}
\int \frac{d^4Q}{(2\pi)^4}\, \frac{\partial^2}{\partial Q_\mu \partial Q_\mu}\Big(\frac{f(Q^2)}{Q^2}\Big) = \frac{1}{4\pi^2} f(0) = \int \frac{d^4Q}{(2\pi)^4}\, f(Q^2)\cdot 4\pi^2 \delta^4(Q).$$
Note that terms in which double total derivatives act on $Q^{-4}$ are not well-defined and cannot appear in the final expression for the function (\[Delta\_Beta\]), although they can be present in expressions for separate supergraphs. This statement has been confirmed by some explicit two- and three-loop calculations in Refs. [@Kazantsev:2018nbl; @Stepanyantz:2019lyo].
Graphs and total derivatives
----------------------------
\[Subsection\_Graphs\]
Let consider a vacuum supergraph with $L$ loops, $V$ vertices, and $P$ internal lines and set directions of all internal lines in an arbitrary way. These directions will be pointed by arrows. Also we denote momenta of all internal lines by certain letters. In our conventions an incoming momentum has the sign “minus”, while an outcoming one has the sign “plus”.
Let us construct a $(V-1)\times P$ matrix $M$ corresponding to the supergraph under consideration. For this purpose we numerate the vertices in an arbitrary order and write the energy-momentum conservation laws in all vertices, except for the last one. (Evidently, the last equation is a linear combination of the others.) The resulting system of equations can be written in the form
$$\left(
\begin{array}{cccc}
M_{1,1} & M_{1,2} & \ldots & M_{1,P}\\
M_{2,1} & M_{2,2} & \ldots & M_{2,P}\\
\vdots & \vdots & \ddots & \vdots\\
M_{V-1,1} & M_{V-1,2} & \ldots & M_{V-1,P}
\end{array}
\right)
\left(
\begin{array}{c}
Q_{\mu,1}\\
Q_{\mu,2}\\
\vdots\\
Q_{\mu,P}
\end{array}
\right) =0.$$
The matrix with the elements $M_{i,J}$, where $i=1,\ldots,V-1$ and $J=1,\ldots,P$, is the required matrix matched to the considered vacuum supergraph. From the above equations, which can briefly be written as
$$\label{Conservation_Laws}
\sum\limits_{J=1}^P M_{i,J} Q_{\mu,J} = 0,$$
we conclude that only $P-V+1$ momenta are independent. According to the topological identity $L=P-V+1$, this implies that (as well known) there are $L$ independent momenta $Q_{\mu,\alpha}$, $\alpha=1,\ldots, L$ in the considered supergraph, and the others can be expressed in terms of them,
$$\label{Dependent_Momenta}
Q_{\mu,I} = \sum\limits_{\alpha=1}^L N_{I,\alpha} Q_{\mu,\alpha}.$$
(0,3) (4.05,0.2)[![Examples of vacuum supergraphs, for which we construct the matrices $M$. The bold lines denote propagators the momenta of which are considered as independent.[]{data-label="Figure_Examples"}](example_graph1.eps "fig:")]{} (5.0,2.6)[2]{} (5.05,-0.2)[1]{} (3.3,1.1)[$Q_{\mu,1}$]{} (4.3,1.3)[$Q_{\mu,2}$]{} (6.3,1.1)[$Q_{\mu,3}$]{} (10.05,0.2)[![Examples of vacuum supergraphs, for which we construct the matrices $M$. The bold lines denote propagators the momenta of which are considered as independent.[]{data-label="Figure_Examples"}](example_graph2.eps "fig:")]{} (11.02,2.6)[1]{} (11.05,-0.2)[3]{} (9.7,1.2)[2]{} (12.35,1.2)[4]{} (12.0,2.1)[$Q_{\mu,3}$]{} (11.25,0.8)[$Q_{\mu,2}$]{} (10.3,1.5)[$Q_{\mu,1}$]{} (9.6,2.1)[$Q_{\mu,4}$]{} (9.7,0.3)[$Q_{\mu,5}$]{} (12.0,0.3)[$Q_{\mu,6}$]{}
As a simple example we can consider a supergraph presented in Fig. \[Figure\_Examples\] on the left. In this case the equation (\[Conservation\_Laws\]), the matrix $M$, and the equation (\[Dependent\_Momenta\]) read as
$$\Big(1\ 1 \ 1\Big) \left(
\begin{array}{c}
Q_{\mu,1}\\
Q_{\mu,2}\\
Q_{\mu,3}
\end{array}
\right) = 0;\qquad M = \Big(1\ 1 \ 1\Big);\qquad \left(
\begin{array}{c}
Q_{\mu,1}\\
Q_{\mu,2}\\
Q_{\mu,3}
\end{array}
\right) = \left(
\begin{array}{c}
1\\
0\\
-1
\end{array}
\right) Q_{\mu,1} + \left(
\begin{array}{c}
0\\
1\\
-1
\end{array}
\right) Q_{\mu,2}.$$
For a more complicated three-loop graph presented in Fig. \[Figure\_Examples\] on the right as another example, the matrix $M$ takes the form
$$M = \left(
\begin{array}{cccccc}
0 & -1 & -1 & 1 & 0 & 0\\
1 & 0 & 0 & -1 & 1 & 0\\
0 & 1 & 0 & 0 & -1 & 1
\end{array}
\right).$$
Evidently, in this case there are three independent momenta. For example, it is possible to choose $Q_{\mu,1}$, $Q_{\mu,2}$, and $Q_{\mu,3}$ as independent variables. The corresponding propagators are denoted in Fig. \[Figure\_Examples\] by the bold lines. Then the equation (\[Dependent\_Momenta\]) takes the form
$$\left(
\begin{array}{c}
Q_{\mu,1} \\ Q_{\mu,2} \\ Q_{\mu,3} \\ Q_{\mu,4} \\ Q_{\mu,5} \\ Q_{\mu,6}
\end{array}
\right) = \left(
\begin{array}{c}
1 \\ 0 \\ 0 \\ 0 \\ -1 \\ -1
\end{array}
\right) Q_{\mu,1} + \left(
\begin{array}{c}
0 \\ 1 \\ 0 \\ 1 \\ 1 \\ 0
\end{array}
\right) Q_{\mu,2} + \left(
\begin{array}{c}
0 \\ 0 \\ 1 \\ 1 \\ 1 \\ 1
\end{array}
\right) Q_{\mu,3}.$$
An important observation made in Ref. [@Stepanyantz:2019ihw] is that the equations which reflect the gauge invariance of vertices are very similar to Eq. (\[Conservation\_Laws\]). Really, let us consider a vertex with $n$ outcoming lines corresponding to the term
$$\int d^8x\, \hat V^{I_1 I_2 \ldots I_n}\, \varphi_{I_1} \varphi_{I_2} \ldots \varphi_{I_n},$$
where $\hat V$ is an operator acting on the product of various superfields of the theory denoted by $\varphi_I$. Then the energy-momentum conservation in the considered vertex is expressed by the equation
$$\label{Conservation_In_Vertex}
Q_{\mu,1} + Q_{\mu,2} + \ldots + Q_{\mu,n} = 0,$$
which is very similar to the equation which follows from the gauge invariance of theory
$$\label{Global_Gauge_Invariance_In_Vertex}
(T^A)_{K}{}^{I_1}\, \hat V^{K I_2 \ldots I_n} + (T^A)_{K}{}^{I_2}\, \hat V^{I_1 K \ldots I_n} + \ldots + (T^A)_{K}{}^{I_n}\, \hat V^{I_1 I_2 \ldots K} = 0,$$
where $(T^A)_I{}^J$ are generators of the gauge group in a relevant representation.
For the incoming lines the signs of the corresponding momenta in equations like (\[Conservation\_In\_Vertex\]) should be changed. Similarly, in equations like (\[Global\_Gauge\_Invariance\_In\_Vertex\]) one should replace the generators by the transposed generators of the conjugated representation taken with an opposite sign, although this replacement does not change the equation, because
$$\big(T^A_{R}\big)_K{}^I \ \to\ - \big(T^A_{\bar R}\big){}^I{}\vphantom{\big)}_K = \big(T^A_R\big)_K{}^I.$$
Let us consider a vacuum supergraph and replace a $\delta$-symbol $\delta_{a_J}^{b_J}$ coming from the propagator with a number $J$ by a certain matrix $A_{a_J}{}^{b_J}$. We will denote the resulting modified supergraph by $[A]_J$. If we replace the $i$-th vertex operator $\hat V^{I_1 I_2\ldots I_n}$ in a certain vacuum supergraph by the left hand side of Eq. (\[Global\_Gauge\_Invariance\_In\_Vertex\]), then, using this notation, we obtain the equation
$$\sum\limits_{J=1}^P M_{i,J} \big[\,T^A\big]_J = 0.$$
The system of these equations is analogous to Eq. (\[Conservation\_Laws\]) and contains the same matrix $M_{i,J}$. Evidently, the solution of these equations can be written in a form similar to Eq. (\[Dependent\_Momenta\]),
$$\label{TA_Solution}
\big[\,T^A\big]_I = \sum\limits_{\alpha=1}^L N_{I,\alpha} \big[\,T^A\big]_\alpha,$$
where $N_{I,\alpha}$ are exactly the same coefficients as in Eq. (\[Dependent\_Momenta\]).
According to the algorithm described in section \[Subsection\_Idea\] we need to insert $\theta^4 (v^B)^2$ into the vacuum supergraph and make the replacement described in the item 5. We will denote supergraphs obtained in this way as
$$\sum\limits_{\alpha,\beta=1}^L \Big[\,\theta^4 (v^B)^2; \frac{\partial^2}{\partial Q_{\mu,\alpha} \partial Q_{\mu,\beta}} \big[\, T^A\big]_\alpha \big[\, T^A\big]_\beta \Big].$$
This means that $\theta^4 (v^B)^2$ is inserted into an arbitrary point of the supergraph, two $\delta$-symbols are replaced by the generators, and the double total derivatives are applied to the integrand of the loop integral.
Then the sum of all perturbative contributions to the function (\[Delta\_Beta\]) can be presented in the form
$$\label{Supergraph_Sum}
\frac{\beta(\alpha_0,\lambda_0,Y_0)}{\alpha_0^2} - \frac{\beta_{\mbox{\scriptsize 1-loop}}(\alpha_0)}{\alpha_0^2} = -\frac{2\pi}{r{\cal V}_4} \frac{d}{d\ln\Lambda} \hspace*{-2mm} \sum\limits_{\stackrel{\mbox{\scriptsize vacuum}}{\mbox{\scriptsize supergraphs}}} \sum\limits_{\alpha,\beta=1}^L \Big[\,\theta^4 (v^B)^2;\, \frac{\partial^2}{\partial Q_{\mu,\alpha} \partial Q_{\mu,\beta}} \big[\, T^A\big]_\alpha \big[\, T^A\big]_\beta \Big].$$
By construction, this expression is an integral of double total derivatives. However, it does not vanish because the double total derivatives produce singular contributions acting on $1/Q_I^2$. It is important to note that expressions for separate supergraphs (presented in the form of scalar integrals) can contain not only $1/Q_I^2$, but also $1/(Q_I^2)^n \equiv 1/Q_I^{2n}$ with $n\ge 2$ if $Q_I$ is a momentum of a quantum gauge superfield propagator. Acting on $1/Q_I^{2n}$ with $n\ge 2$ the double total derivatives produce an expression which is not well-defined due to infrared divergences. However, we know that the left hand side of Eq. (\[Supergraph\_Sum\]) is well-defined. Therefore, all bad terms coming from the $1/Q_I^{2n}$ singularities should cancel each other. The calculations in the lowest orders [@Kazantsev:2018nbl; @Stepanyantz:2019lyo] exactly confirm this statement.
Because all bad terms should cancel each other, it is necessary to consider only the $1/Q_I^2$ singularities. Then in the expression for a supergraph we need to consider only the product of all [*different*]{} inverse squared momenta multiplied by a nonsingular function $f$,
$$\label{Product_Of_Momenta}
\frac{f(Q_1,Q_2,\ldots, Q_{P'})}{Q_1^2 Q_2^2 \ldots Q_{P'}^2} \equiv f(Q_1,Q_2,\ldots, Q_{P'}) \prod\limits_{I=1}^{P'}\vphantom{\prod}' \frac{1}{Q_I^2},$$
where $Q_{\mu,I} \ne Q_{\mu,J}$ for $I\ne J$. The prime indicates that the product includes only different momenta, a number of them being denoted by $P'$ (which is evidently less or equal to $P$). This is essential for supergraphs which contain coinciding momenta. The structure of such supergraphs is illustrated in Fig. \[Figure\_Graph\_With\_Coinciding\_Momenta\]. In this figure the gray circles denote 1PI subdiagrams, which are connected by propagators with coinciding momenta. By construction, the expression (\[Product\_Of\_Momenta\]) should include only one such momentum.
(0,3) (6.8,0)[![The structure of vacuum supergraphs with coinciding momenta (which are equal to $k_\mu$). The gray circles denote 1PI subdiagrams.[]{data-label="Figure_Graph_With_Coinciding_Momenta"}](graph_example1.eps "fig:")]{} (6.5,1.3)[$k_\mu$]{}
$\delta$-singularities appear when double total derivatives act on various factors in the product (\[Product\_Of\_Momenta\]). Note that, according to Eq. (\[Delta\_Singularity\_General\]), such singularities can appear even if derivatives with respect to different momenta act on the same inverse squared momentum,
$$\frac{\partial^2}{\partial Q_{\mu,\alpha} \partial Q_{\mu,\beta}} \Big(\frac{1}{Q_I^2}\Big) = - 4\pi^2 N_{I,\alpha} N_{I,\beta}\, \delta^4(Q_I),$$
where we also took Eq. (\[Dependent\_Momenta\]) into account. This implies that acting on the product (\[Product\_Of\_Momenta\]) the double total derivatives give the singular contribution
$$\label{Singularity}
\frac{\partial^2}{\partial Q_{\mu,\alpha} \partial Q_{\mu,\beta}} \prod\limits_{I=1}^{P'}\vphantom{\prod}' \frac{1}{Q_I^2}\ \to\ - 4\pi^2 \sum\limits_{I=1}^{P'}\vphantom{\sum}' N_{I,\alpha} N_{I,\beta}\, \delta^4(Q_I) \prod\limits_{J\ne I}\vphantom{\prod}' \frac{1}{Q_J^2}.$$
Taking into account that the bad terms containing $1/Q_I^{2n}$ with $n\ge 2$ should cancel each other, we can rewrite the expression (\[Supergraph\_Sum\]) as a sum of supergraphs in which they are omitted. According to Eq. (\[Singularity\]), in the remaining terms we replace one of $1/Q_I^2$ by $4\pi^2\delta^4(Q_I)$ and sum up all expressions obtained after these substitutions. (The sign is “plus”, because the singular contribution of the form (\[Delta\_Singularity\]) should be taken with the minus sign, see Eq. (\[Integral\_And\_Singularity\]).) The result can be presented as
$$\begin{aligned}
&& -\frac{2\pi}{r{\cal V}_4} \frac{d}{d\ln\Lambda} \hspace*{-2mm} \sum\limits_{\stackrel{\mbox{\scriptsize vacuum}}{\mbox{\scriptsize supergraphs}}} \sum\limits_{I=1}^{P'} \sum\limits_{\alpha,\beta=1}^L N_{I,\alpha} N_{I,\beta}\nonumber\\
&&\qquad\qquad\qquad\qquad \times \Big[\,\theta^4 (v^B)^2;\, \big[\, T^A\big]_\alpha \big[\, T^A\big]_\beta;\, \frac{1}{Q^{2n}}\bigg|_{n\ge 2} \to 0;\, \frac{1}{Q_I^2} \to 4\pi^2\delta^4(Q_I) \Big].\qquad\end{aligned}$$
Next, using Eq. (\[TA\_Solution\]) we calculate the sums over the indices $\alpha$ and $\beta$,
$$\label{New_Expression_For_Delta_Beta}
-\frac{2\pi}{r{\cal V}_4} \frac{d}{d\ln\Lambda} \hspace*{-2mm} \sum\limits_{\stackrel{\mbox{\scriptsize vacuum}}{\mbox{\scriptsize supergraphs}}} \sum\limits_{I=1}^{P'} \Big[\,\theta^4 (v^B)^2;\, \big[\, T^A T^A\big]_I;\, \frac{1}{Q^{2n}}\bigg|_{n\ge 2} \to 0;\, \frac{1}{Q_I^2} \to 4\pi^2\delta^4(Q_I) \Big].$$
Note that for the gauge and ghost propagators $(T^A T^A)_{BC} = C_2\delta_{BC}$, while for the matter propagators $(T^A T^A)_i{}^j = C(R)_i{}^j$.
Basing on Eq. (\[New\_Expression\_For\_Delta\_Beta\]) it is possible to modify the step 5 of the algorithm described in section \[Subsection\_Idea\]. Note that in what follows we formulate this step before the Wick rotation.
5\. We construct a set of supergraphs in which one of the propagators is marked. If the original supergraph contains some propagators with the same momentum, then one can mark only one propagator with this momentum. After calculating the $D$-algebra, one should omit all terms proportional to $1/q_I^{2n}$ with $n\ge 2$ in scalar integrals and for any marked propagator with the momentum $q_\mu$ in the remaining terms make the replacement
$$\begin{aligned}
\label{Replacement_Modified}
&& \frac{1}{q^2}\quad\ \to\ \ C_2\, \frac{\partial^2}{\partial q_\mu \partial q^\mu} \Big(\frac{1}{q^2}\Big)\qquad\qquad \mbox{for gauge and ghost propagators};\nonumber\\
&& \delta_i^j\, \frac{1}{q^2}\ \to\ \ C(R)_i{}^j\, \frac{\partial^2}{\partial q_\mu \partial q^\mu} \Big(\frac{1}{q^2}\Big)\qquad \mbox{for matter propagators}.\end{aligned}$$
After these replacements it is necessary to sum up the expressions for all supergraphs in the constructed set.
Note that this prescription is the same for all vacuum supergraphs, unlike the one described in section \[Subsection\_Idea\]. Really, the form of the operator (\[Replacement\]) depends on the topology of the supergraph, while the replacement (\[Replacement\_Modified\]) does not. That is why it is possible, first, to find a sum of all vacuum supergraphs and, only after this, make these replacements. Below we will demonstrate that the sum of vacuum supergraphs really does not contain any bad terms which include $1/Q_I^{2n}$ with $n\ge 2$. This confirms the above argumentation, although the explicit mechanism of cancelling the bad contributions to the function (\[Delta\_Beta\]) should be analyzed separately.
In the next section using the modification of the algorithm described above we will find the sum of all contributions to the function (\[Delta\_Beta\]) which come from $\delta$-singularities of the form (\[Delta\_Singularity\_General\]). Various perturbative verifications of the above described modification will be made in section \[Section\_Explicit\_Calculation\].
The all-loop sum of singular contributions {#Section_Singularities}
==========================================
The starting point and main idea
--------------------------------
\[Subsection\_Main\_Idea\]
In this section we will calculate the sum of all singular contributions. They appear when the double total derivatives act on matter, ghost and gauge propagators and effectively cut the corresponding internal line. In [@Stepanyantz:2019lfm] the sums of singularities produced by the cuts of matter and ghost propagators have been calculated in all orders. However, the method which is used in this paper is different. That is why it is expedient to recalculate the sums of matter and ghost singularities and verify that both approaches give the same result. Moreover, in this section we will find the all-loop expression for a sum of singularities produced by the cuts of quantum gauge superfield propagators. The method of summation generalizes the one proposed in Ref. [@Stepanyantz:2011jy] for the Abelian case.
First, we introduce the auxiliary action
$$S_g \equiv S_{\mbox{\scriptsize total}} +\Delta S_g$$
depending on a real constant $g$, where $S_{\mbox{\scriptsize total}} \equiv S_{\mbox{\scriptsize reg}} + S_{\mbox{\scriptsize gf}} + S_{\mbox{\scriptsize FP}} + S_{\mbox{\scriptsize NK}}$ and
$$\begin{aligned}
&& \Delta S_g \equiv \frac{1}{4} \Big(\frac{1}{g}-1\Big) \int d^8x\,\Big[ - V^A R(\partial^2/\Lambda^2)\, \partial^2 \Pi_{1/2} V^A \nonumber\\
&&\qquad\qquad\qquad -\frac{1}{8\xi_0} D^2 V^A K(\partial^2/\Lambda^2) \bar D^2 V^A + \bar c^{+A} c^A -\bar c^A c^{+A} + \phi^{*i} F(\partial^2/\Lambda^2)\phi_i\Big].\qquad\end{aligned}$$
For $g=1$ the action $S_g$ evidently coincides with $S_{\mbox{\scriptsize total}}$. Moreover, all vertices generated by $S_g$ coincide with the ones produced by $S_{\mbox{\scriptsize total}}$. However, the parts of $S_g$ quadratic in the quantum gauge superfield, the Faddeev–Popov ghosts, and the matter superfields differ from the corresponding quadratic parts of $S_{\mbox{\scriptsize total}}$ by the factor $1/g$. This implies that, in comparison with the original theory, all propagators of the above mentioned quantum superfields obtained from the generating functional
$$Z_g \equiv \int D\mu\, \mbox{Det}(PV, M_\varphi)^{-1}\, \mbox{Det}(PV, M)^c \exp\Big(i S_g + i S_{\mbox{\scriptsize sources}}\Big),$$
with $c=T(R)/T(R_{\mbox{\scriptsize PV}})$, acquire the factor $g$. Note that the propagators of the massive Pauli–Villars superfields and of the Nielsen–Kallosh ghosts remain the same. We need not modify them, because the Pauli–Villars propagators do not produce singularities, and the Nielsen–Kallosh ghosts are essential only in the one-loop approximation which is considered separately. Certainly, the functional $Z_g$ is treated formally, and we do not care about regularizing divergences in the corresponding supergraphs.
Now, let us find the derivative of the functional $\Gamma_g$ (defined as a Legendre transform of $W_g = - i\ln Z_g$) with respect to the parameter $g$ at vanishing superfields and $g=1$. It is convenient to present the result in the form
$$\begin{aligned}
\label{Gamma_Derivative}
&&\hspace*{-3mm} \frac{\partial \Gamma_g}{\partial g}\bigg|_{g=1;\, \mbox{\scriptsize fields=0}} = \Big\langle\frac{\partial S_g}{\partial g}\Big\rangle\Big|_{g=1,\, \mbox{\scriptsize fields=0}}
= \Big\langle \frac{1}{4} \int d^8x\, \bigg\{ V^A\Big[ \partial^2\Pi_{1/2} R(\partial^2/\Lambda^2) + \frac{1}{16\xi_0}\Big(\bar D^2 D^2 \qquad\nonumber\\
&&\hspace*{-3mm} + D^2 \bar D^2\Big) K(\partial^2/\Lambda^2) \Big]V^A - \phi^{*i} F(\partial^2/\Lambda^2) \phi_i - \bar c^{+A} c^A + \bar c^A c^{+A} \bigg\} \Big\rangle\bigg|_{g=1;\, \mbox{\scriptsize fields=0}}.\qquad\end{aligned}$$
The angular brackets entering this equation are defined in the standard way
$$\langle A \rangle \equiv \frac{1}{Z}\int D\mu\, A\, \mbox{Det}(PV, M_\varphi)^{-1}\, \mbox{Det}(PV, M)^c \exp\Big(i S_{\mbox{\scriptsize total}} + i S_{\mbox{\scriptsize sources}}\Big),$$
where the sources should be expressed in terms of fields using Eq. (\[Field\_Definitions\]).
Evidently, the expression (\[Gamma\_Derivative\]) can be rewritten in terms of the inverse two-point Green functions of the quantum superfields,
$$\begin{aligned}
\label{Gamma_Derivative_In_Terms_Of_Inverse_Functions}
&& \frac{\partial \Gamma_g}{\partial g}\bigg|_{g=1;\, \mbox{\scriptsize fields=0}}
= \frac{i}{4}\int d^8x\,\bigg\{\Big[ \partial^2\Pi_{1/2} R(\partial^2/\Lambda^2) + \frac{1}{16\xi_0}\Big(\bar D^2 D^2 + D^2 \bar D^2\Big) K(\partial^2/\Lambda^2) \Big]_x
\nonumber\\
&& \times \Big(\frac{\delta^2\Gamma}{\delta V^A_{x}\delta V^A_y}\Big)^{-1} - F(\partial^2/\Lambda^2)_x\Big(\frac{\delta^2\Gamma}{\delta\phi^{*i}_{\,,x}\delta\phi_{i,y}}\Big)^{-1}
- \Big(\frac{\delta^2\Gamma}{\delta c^A_{x}\,\delta \bar c^{+A}_{y}}\Big)^{-1}
- \Big(\frac{\delta^2\Gamma}{\delta \bar c^A_{x}\,\delta c^{+A}_{y}}\Big)^{-1}
\bigg\}\bigg|_{y=x}.\qquad\end{aligned}$$
(Certainly, in the last expression we also assume that $g=1$ and the (super)fields are set to 0.)
The original effective action $\Gamma$ at vanishing fields is given by the sum of vacuum supergraphs. So far we do not calculate these supergraphs and work only with the formal expressions constructed with the help of supersymmetric Feynman rules. The derivative (\[Gamma\_Derivative\]) is contributed by the same vacuum supergraphs, but each of them is multiplied by the number of propagators. Really, as we have already mentioned, each propagator of a quantum superfield is proportional to the parameter $g$, while the vertices do not contain $g$. Evidently, the first and second terms in Eq. (\[Gamma\_Derivative\_In\_Terms\_Of\_Inverse\_Functions\]) are contributed by vacuum supergraphs multiplied by the number of gauge and matter propagators, respectively. The last two terms give a sum of vacuum supergraphs multiplied by the number of ghost propagators.
Note that if a supergraph has some coinciding momenta as in Fig. \[Figure\_Graph\_With\_Coinciding\_Momenta\], all propagators with the coinciding momenta should be taken into account when calculating the number of propagators. The contribution of these graphs to $\partial\Gamma_g/\partial g$ is given by a relevant term in Eq. (\[Gamma\_Derivative\_In\_Terms\_Of\_Inverse\_Functions\]) and is proportional to a certain inverse Green function. If a usual two-point Green function is proportional to the function
$$G \equiv 1+ \Delta G$$
depending on the momentum, then the inverse Green function will include
$$\label{Inverse_G}
G^{-1} = 1 + \sum\limits_{n=1}^\infty (-1)^n \big(\Delta G\big)^n.$$
Evidently, a term containing $(\Delta G)^n$ corresponds to supergraphs of the structure presented in Fig. \[Figure\_Graph\_With\_Coinciding\_Momenta\] with $n$ coinciding momenta $k_\mu$. The gray circles in this supergraph certainly give various $\Delta G$.
As we discussed in the previous section, for constructing the $\beta$-function one should cut only one propagator with the momentum $k_\mu$ in a supergraph which have the structure presented in Fig. \[Figure\_Graph\_With\_Coinciding\_Momenta\]. (This cut is made by $\delta^4(k)$ which appears after the replacement (\[Replacement\_Modified\]).) From the other side, the corresponding contribution to the expression (\[Gamma\_Derivative\]) obtained by counting the propagators with the momentum $k_\mu$ is proportional to $n$. Therefore, for calculating the contribution to the $\beta$-function it is necessary to replace $G^{-1}$ by
$$\sum\limits_{n=1}^\infty \frac{(-1)^{n}}{n} \big(\Delta G\big)^n = -\ln G.$$
(The first term in Eq. (\[Inverse\_G\]) corresponds to the one-loop approximation, which is considered separately.) After this we proceed according to the algorithm described in section \[Section\_Integrals\]. Details of this calculation will be considered below, separately for the cuts of gauge, ghost, and matter propagators.
The all-loop sum of matter singularities
----------------------------------------
\[Subsection\_Matter\_Singularities\]
Let us start with calculating a contribution to the function (\[Delta\_Beta\]) coming from the cuts of matter propagators. According to Eq. (\[Gamma\_Derivative\_In\_Terms\_Of\_Inverse\_Functions\]), the corresponding contribution to $\partial\Gamma_g/\partial g$ is given by the expression
$$\label{Matter_Part}
-\frac{i}{4}\int d^8x\, F(\partial^2/\Lambda^2)_x \Big(\frac{\delta^2\Gamma}{\delta\phi^{*i}_{\,,x}\delta\phi_{i,y}}\Big)^{-1}\bigg|_{y=x;\ \mbox{\scriptsize fields}=0},$$
which encodes the sum of vacuum supergraphs with a marked matter propagator. (By definition, the marking of a propagator does not change the expression for a supergraph, but supergraphs in which marked propagators are different are considered as different.) The two-point Green function of the matter superfields obtained by differentiating Eq. (\[Gamma2\_Phi\]) with respect to $\phi$ and $\phi^*$ reads as
$$\label{Matter_Green_Function}
\frac{\delta^2\Gamma}{\delta\phi^{*i}_{\,,x}\, \delta\phi_{j,y}}\bigg|_{\mbox{\scriptsize fields}=0} = \frac{1}{16} D_x^2 \bar D_y^2 \big(G_\phi\big)_i{}^j \delta^8_{xy},$$
where
$$\delta^8_{xy} \equiv \delta^4(x-y)\, \delta^4(\theta_x-\theta_y).$$
The function $(G_\phi)_i{}^j$ present in Eq. (\[Matter\_Green\_Function\]) is normalized in such a way that in the tree approximation it is equal to $\delta_i^j F(\partial^2/\Lambda^2)$. (In the limit $\Lambda\to\infty$ this expression gives $\delta_i^j$.) Therefore, it can be presented as
$$\big(G_\phi\big)_i{}^j = \delta_i^j F + \big(\Delta G_\phi\big)_i{}^j,$$
where the sum of quantum corrections is denoted by $(\Delta G_\phi)_i{}^j$. The function inverse to (\[Matter\_Green\_Function\]) entering Eq. (\[Matter\_Part\]) is given by the expression
$$\begin{aligned}
\label{Inverse_Matter_Green_Function}
&& \Big(\frac{\delta^2\Gamma}{\delta\phi^{*i}_{\,,x}\, \delta\phi_{j,y}}\Big)^{-1} = -\frac{D_x^2 \bar D_y^2}{4\partial^2} \big(G_\phi^{-1}\big)_j{}^i \delta^8_{xy} = -4 \Big(\frac{D_x^2 \bar D_x^2}{16\partial^2 F} \delta_j^i + \frac{D_x^2 \bar D_x^2}{16\partial^2 F}\, \big(\Delta G_\phi\big)_j{}^i\, \frac{D_x^2 \bar D_x^2}{16\partial^2 F}\qquad\vphantom{\Bigg(}\nonumber\\
&& + \frac{D_x^2 \bar D_x^2}{16\partial^2 F}\, \big(\Delta G_\phi\big)_j{}^k\, \frac{D_x^2 \bar D_x^2}{16\partial^2 F}\, \big(\Delta G_\phi\big)_k{}^i\, \frac{D_x^2 \bar D_x^2}{16\partial^2 F} +\ldots \Big) \delta^8_{xy}. \vphantom{\Bigg(}\end{aligned}$$
This can easily be verified using the first of the identities
$$\label{Chiral_Identities}
\bar D^2 D^2 \bar D^2 = -16 \bar D^2\partial^2;\qquad D^2 \bar D^2 D^2 = -16 D^2\partial^2.$$
To obtain an expression which encodes the sum of vacuum supergraphs in which only one of propagators with coinciding momenta is marked, one should multiply terms with $n$ insertions of $\Delta G_\phi$ by $1/n$. (As we have already mentioned, the first term, which corresponds to the one-loop approximation, should be omitted.) After this we obtain the function
$$\begin{aligned}
&& -4 \Big(\frac{D_x^2 \bar D_x^2}{16\partial^2 F} \big(\Delta G_\phi\big)_j{}^i \frac{D_x^2 \bar D_x^2}{16\partial^2 F} + \frac{1}{2}\cdot \frac{D_x^2 \bar D_x^2}{16\partial^2 F} \big(\Delta G_\phi\big)_j{}^k \frac{D_x^2 \bar D_x^2}{16\partial^2 F} \big(\Delta G_\phi\big)_k{}^i \frac{D_x^2 \bar D_x^2}{16\partial^2 F} + \ldots \Big) \delta^8_{xy}\qquad\vphantom{\Bigg(}\nonumber\\
&& = \frac{D_x^2 \bar D_x^2}{4\partial^2 F}\Big(\frac{\Delta G_\phi}{F} - \frac{(\Delta G_\phi)^2}{2 F^2} + \frac{(\Delta G_\phi)^3}{3 F^3} + \ldots\Big)_j{}^i \delta^8_{xy} = \frac{D_x^2 \bar D_x^2}{4\partial^2 F} \Big(\ln \frac{G_\phi}{F}\Big)_j{}^i \delta^8_{xy}.\vphantom{\Bigg(}\end{aligned}$$
Therefore, the first step for calculating the matter contribution to the function (\[Delta\_Beta\]) is to make the replacement
$$\Big(\frac{\delta^2\Gamma}{\delta\phi^{*i}_{\,,x}\, \delta\phi_{i,y}}\Big)^{-1}\bigg|_{\mbox{\scriptsize fields}=0}\ \to \ \frac{D_x^2 \bar D_x^2}{4\partial^2 F} \Big(\ln \frac{G_\phi}{F}\Big)_i{}^i \delta^8_{xy}$$
in Eq. (\[Matter\_Part\]). We see that no bad terms proportional to $1/q^{2k}$ with $k\ge 2$ appear. Next, following the algorithm described in section \[Section\_Integrals\], we should insert $\theta^4 (v^B)^2$ to an arbitrary point of the supergraph. Evidently, in this case it is expedient to insert this expression to the point $x$. Moreover, we need to make a replacement
$$\Big(\ln \frac{G_\phi}{F}\Big)_i{}^i = \Big(\ln \frac{G_\phi}{F}\Big)_j{}^i \delta_i^j \ \to \ \Big(\ln \frac{G_\phi}{F}\Big)_j{}^i C(R)_i{}^j \frac{\partial^2}{\partial q_\mu \partial q^\mu},$$
where $q_\mu$ is the Minkowski momentum of the matter line which is cut. Note that (as we discussed in section \[Subsection\_Graphs\]) the operator $\partial^2/\partial q_\mu \partial q^\mu$ should act only on $1/q^2$. (After the Wick rotation it gives $-\partial^2/\partial Q_\mu^2$, where $Q_\mu$ is the corresponding Euclidean momentum.) The matter line is cut in the point $x$, so that $q_\mu$ is the momentum of the propagator coming out of this point. Finally the result should be multiplied by $-2\pi/(r{\cal V}_4)$ and differentiated with respect to $\ln\Lambda$ at fixed values of renormalized couplings. This implies that the matter contribution to the function (\[Delta\_Beta\]) can be written in the form
$$\begin{aligned}
\label{Delta_Matter_Original}
&& \Delta_{\mbox{\scriptsize matter}}\Big(\frac{\beta}{\alpha_0^2}\Big) = - \frac{i\pi}{8 r{\cal V}_4} C(R)_i{}^j \frac{d}{d\ln\Lambda} \int d^8x\,d^8y\, \big(\theta^4\big)_x \big(v^B\big)^2_x\nonumber\\
&& \qquad\qquad\qquad\qquad\quad \times \int \frac{d^4q}{(2\pi)^4}\, \delta^8_{xy}(q)\, \Big(\ln \frac{G_\phi}{F}\Big)_j{}^i \frac{\partial^2}{\partial q_\mu \partial q^\mu} \Big(\frac{1}{q^2}\Big) D_x^2 \bar D_x^2 \delta^8_{xy},\qquad\end{aligned}$$
where the function $(\ln G_\phi)_i{}^j$ depends on the momentum $q_\mu$ and
$$\delta^8_{xy}(q) \equiv \delta^4(\theta_x-\theta_y) e^{iq_\mu (x^\mu - y^\mu)}.$$
The momentum integral is calculated in the Euclidean space after the Wick rotation,
$$\begin{aligned}
&& \frac{d}{d\ln\Lambda} \int \frac{d^4q}{(2\pi)^4}\, \delta^8_{xy}(q)\, \Big(\ln \frac{G_\phi}{F}\Big)_j{}^i \frac{\partial^2}{\partial q_\mu \partial q^\mu} \Big(\frac{1}{q^2}\Big) \nonumber\\
&&\ \to\ - 4i\pi^2 \delta^4(\theta_x-\theta_y) \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4}\, \Big(\ln \frac{G_\phi}{F}\Big)_j{}^i\, \delta^4(Q)\nonumber\\
&& = - \frac{i}{4\pi^2} \delta^4(\theta_x-\theta_y) \frac{d}{d\ln\Lambda}\big(\ln G_\phi\big)_j{}^i\Big|_{Q=0} = - \frac{i}{4\pi^2} \delta^4(\theta_x-\theta_y) \big(\gamma_\phi\big)_j{}^i(\alpha_0,\lambda_0,Y_0),\qquad\end{aligned}$$
where we took into account that $F(0)=1$ and wrote the result in terms of the anomalous dimension $(\gamma_\phi)_j{}^i$ using Eq. (\[Gamma\_Phi\_Bare\_Definition\]). Therefore, the expression (\[Delta\_Matter\_Original\]) takes the form
$$\Delta_{\mbox{\scriptsize matter}}\Big(\frac{\beta}{\alpha_0^2}\Big) = - \frac{1}{32\pi r{\cal V}_4} C(R)_i{}^j \big(\gamma_\phi\big)_j{}^i(\alpha_0,\lambda_0,Y_0) \int d^8x\,d^8y\,\big(\theta^4\big)_x \big(v^B\big)^2_x\, \delta^4(\theta_x-\theta_y) D_x^2 \bar D_x^2 \delta^8_{xy}.$$
Next, it is necessary to use the identity
$$\label{Theta_Identity}
\delta^4(\theta_x-\theta_y) D_x^2 \bar D_x^2 \delta^8_{xy} = 4\, \delta^8_{xy}$$
and calculate the integral over $d^8y$ with the help of the $\delta$-function,
$$\Delta_{\mbox{\scriptsize matter}}\Big(\frac{\beta}{\alpha_0^2}\Big) = - \frac{1}{8\pi r{\cal V}_4} C(R)_i{}^j \big(\gamma_\phi\big)_j{}^i(\alpha_0,\lambda_0,Y_0) \int d^8x\,\big(\theta^4\big)_x \big(v^B\big)^2_x.$$
Taking into account that
$$\label{Coordinate integral}
\int d^8x\, \big(\theta^4\big)_x \big(v^B\big)^2_x = 4{\cal V}_4,$$
we obtain the final expression for the sum of matter singularities
$$\label{Delta_Matter_Final}
\Delta_{\mbox{\scriptsize matter}}\Big(\frac{\beta}{\alpha_0^2}\Big) = - \frac{1}{2\pi r} C(R)_i{}^j \big(\gamma_\phi\big)_j{}^i(\alpha_0,\lambda_0,Y_0).$$
It exactly coincides with the term containing the anomalous dimension of the matter superfields in the NSVZ equation written in the form (\[NSVZ\_Equivalent\_Form\_Bare\]) and agrees with another all-loop calculation made in [@Stepanyantz:2019lfm] by a different method. This can be considered as a correctness test of the method proposed in this paper.
The all-loop sum of ghost singularities
---------------------------------------
\[Subsection\_Ghost\_Singularities\]
A contribution of ghost singularities to the function (\[Delta\_Beta\]) is calculated similarly to the matter contribution, but it is necessary to take into account that the ghost superfields are anticommuting. As a starting point we consider a part of the expression (\[Gamma\_Derivative\_In\_Terms\_Of\_Inverse\_Functions\]) containing the ghost Green functions,
$$\label{Ghost_Part}
-\frac{i}{4}\int d^8x\,\bigg\{
\Big(\frac{\delta^2\Gamma}{\delta c^A_{x}\,\delta \bar c^{+A}_{y}}\Big)^{-1}
+ \Big(\frac{\delta^2\Gamma}{\delta \bar c^A_{x}\,\delta c^{+A}_{y}}\Big)^{-1}
\bigg\}\bigg|_{y=x;\ \mbox{\scriptsize fields}=0}.\qquad$$
The two-point Green functions of the Faddeev–Popov ghosts are obtained by differentiating Eq. (\[Gamma2\_C\]),
$$\label{Ghost_Green_Functions}
\frac{\delta^2\Gamma}{\delta \bar c^{A}_{x}\, \delta c^{+B}_{y}}\bigg|_{\mbox{\scriptsize fields}=0} = \frac{\delta^2\Gamma}{\delta c^{A}_{x}\, \delta \bar c^{+B}_{y}}\bigg|_{\mbox{\scriptsize fields}=0} = \delta_{AB} G_c \frac{\bar D_x^2 D^2_y}{16} \delta^8_{xy}.$$
In the tree approximation the function $G_c$ is equal to 1, so that it is convenient to present it in the form $G_c = 1 + \Delta G_c$, where $\Delta G_c$ is a sum of quantum corrections coming from 1PI supergraphs with two external ghost legs.
By definition, the functions inverse to (\[Ghost\_Green\_Functions\]) satisfy the equations
$$\begin{aligned}
&& \int d^6z\, \Big(\frac{\delta^2\Gamma}{\delta c^{+A}_{x}\, \delta \bar c^{D}_{z}}\Big)^{-1} \frac{\delta^2\Gamma}{\delta \bar c^{D}_{z}\, \delta c^{+B}_{y}}\bigg|_{\mbox{\scriptsize fields}=0} = -\frac{1}{2}\delta_{AB} D^2_x \delta^8_{xy};\nonumber\\
&& \int d^6z\, \Big(\frac{\delta^2\Gamma}{\delta \bar c^{+A}_{x}\, \delta c^{D}_{z}}\Big)^{-1} \frac{\delta^2\Gamma}{\delta c^{D}_{z}\, \delta \bar c^{+B}_{y}}\bigg|_{\mbox{\scriptsize fields}=0} = -\frac{1}{2}\delta_{AB} D^2_x \delta^8_{xy}\end{aligned}$$
and are explicitly written as
$$\Big(\frac{\delta^2\Gamma}{\delta \bar c^{A}_{x}\, \delta c^{+B}_{y}}\Big)^{-1}\bigg|_{\mbox{\scriptsize fields}=0} = \Big(\frac{\delta^2\Gamma}{\delta c^{A}_{x}\, \delta \bar c^{+B}_{y}}\Big)^{-1}\bigg|_{\mbox{\scriptsize fields}=0} = \delta_{AB}\,\frac{\bar D^2_x D^2_y}{4\partial^2 G_c} \delta^8_{xy}.$$
Repeating the transformations similar to the ones made in the previous section for the matter singularities, we conclude that for constructing a ghost contribution to the function (\[Delta\_Beta\]), first, it is necessary to perform a substitution
$$\begin{aligned}
&& \Big(\frac{\delta^2\Gamma}{\delta c^A_{x}\, \delta \bar c^{+A}_{y}}\Big)^{-1}\bigg|_{\mbox{\scriptsize fields}=0}\ \to \ - \delta_{AA} \frac{\bar D_x^2 D_y^2}{4\partial^2} \ln G_c\, \delta^8_{xy};\qquad\nonumber\\
&& \Big(\frac{\delta^2\Gamma}{\delta \bar c^A_{x}\,\delta c^{+A}_{y}}\Big)^{-1}\bigg|_{\mbox{\scriptsize fields}=0}\ \to \ - \delta_{AA} \frac{\bar D_x^2 D_y^2}{4\partial^2} \ln G_c\, \delta^8_{xy}\end{aligned}$$
in the expression (\[Ghost\_Part\]). Next, according to the algorithm described in section \[Section\_Integrals\], $\theta^4 (v^B)^2$ should be inserted to the point $x$. After this, it is necessary to make the replacement[^9]
$$\ln G_c\cdot \delta_{AA}\ \to\ \ln G_c\cdot r C_2\, \frac{\partial^2}{\partial q_\mu \partial q^\mu}$$
and apply the operator (\[Operator\]) to the resulting expression. Note that the differentiation with respect to $\ln\Lambda$ should be made at fixed values of the renormalized couplings. As a result of this algorithm, we obtain the ghost contribution to the function (\[Delta\_Beta\])
$$\begin{aligned}
\label{Delta_Ghost_Original}
&& \Delta_{\mbox{\scriptsize ghost}}\Big(\frac{\beta}{\alpha_0^2}\Big) = \frac{i\pi C_2}{4{\cal V}_4} \frac{d}{d\ln\Lambda} \int d^8x\,d^8y\, \big(\theta^4\big)_x \big(v^B\big)^2_x\nonumber\\
&& \qquad\qquad\qquad\qquad\qquad \times \int \frac{d^4q}{(2\pi)^4}\, \delta^8_{xy}(q) \ln G_c\, \frac{\partial^2}{\partial q_\mu \partial q^\mu} \Big(\frac{1}{q^2}\Big) \bar D_x^2 D_x^2 \delta^8_{xy}.\qquad\end{aligned}$$
(Note that again no bad terms appear in this expression.) Similarly to the case of matter singularities, the momentum integral is calculated after the Wick rotation in the Euclidean space,
$$\frac{d}{d\ln\Lambda} \int \frac{d^4q}{(2\pi)^4}\, \delta^8_{xy}(q) \ln G_c\, \frac{\partial^2}{\partial q_\mu \partial q^\mu} \Big(\frac{1}{q^2}\Big)\ \to\ - \frac{i}{4\pi^2} \delta^4(\theta_x-\theta_y) \gamma_c(\alpha_0,\lambda_0,Y_0).$$
With the help of this equation the considered contribution to the function (\[Delta\_Beta\]) can be brought to the form
$$\Delta_{\mbox{\scriptsize ghost}}\Big(\frac{\beta}{\alpha_0^2}\Big) = \frac{C_2}{16\pi {\cal V}_4} \gamma_c(\alpha_0,\lambda_0,Y_0) \int d^8x\,d^8y\,\big(\theta^4\big)_x \big(v^B\big)^2_x\, \delta^4(\theta_x-\theta_y) \bar D_x^2 D_x^2 \delta^8_{xy}.$$
Then we use the identity (\[Theta\_Identity\]) and calculate the integral over $d^8y$ with the help of resulting $\delta^8_{xy}$,
$$\Delta_{\mbox{\scriptsize ghost}}\Big(\frac{\beta}{\alpha_0^2}\Big) = \frac{C_2}{4\pi {\cal V}_4} \gamma_c(\alpha_0,\lambda_0,Y_0) \int d^8x\,\big(\theta^4\big)_x \big(v^B\big)^2_x.$$
According to Eq. (\[Coordinate integral\]), the remaining integral present in this equation is equal to $4{\cal V}_4$, so that the sum of singular contributions produced by cuts of the Faddeev–Popov ghost propagators takes the form
$$\label{Delta_Ghost_Final}
\Delta_{\mbox{\scriptsize ghost}}\Big(\frac{\beta}{\alpha_0^2}\Big) = \frac{C_2}{\pi} \gamma_c(\alpha_0,\lambda_0,Y_0).$$
This result agrees with the one found in Ref. [@Stepanyantz:2019lfm] by a different method and coincides with the term containing the anomalous dimension $\gamma_c$ in Eq. (\[NSVZ\_Equivalent\_Form\_Bare\]). Note that the sign of the expression (\[Delta\_Ghost\_Final\]) is different from the sign of the matter contribution (\[Delta\_Matter\_Final\]) due to the anticommutation of the ghost superfields. The factor 2 in the ghost contribution appears because there are two sets of the chiral ghost superfields (the ghost $c$ and the antighost $\bar c$) in the adjoint representation of the gauge group.
Effective propagator of the quantum gauge superfield
----------------------------------------------------
\[Subsection\_Effective\_Propagator\]
Taking into account that (for $g=1$) due to the Slavnov–Taylor identity quantum corrections to the two-point Green function of the [*quantum*]{} gauge superfield are transversal, it is possible to write the corresponding part of the effective action in the form (\[Gamma2\_Quantum\_V\]). Substituting the explicit expression for the gauge fixing action, we equivalently present this equation as
$$\Gamma^{(2)}_V = -\frac{1}{4} \int d^8x\, V^A G_V \partial^2\Pi_{1/2} V^A - \frac{1}{32 \xi_0} \int d^8x\, D^2 V^A K \bar D^2 V^A,$$
where the regulator function $K$ depends on $\partial^2/\Lambda^2$, and the argument $q^2$ of function $G_V$ should be replaced by $-\partial^2$. Differentiating this expression with respect to the quantum gauge superfield, we obtain the corresponding two-point Green function
$$\frac{\delta^2\Gamma}{\delta V_x^A \delta V_y^B}\bigg|_{\mbox{\scriptsize fields}=0} = \delta_{AB} \Big[ - \frac{1}{2} G_V \partial^2\Pi_{1/2} - \frac{1}{32 \xi_0} K \Big(D^2\bar D^2 + \bar D^2 D^2\Big)\Big] \delta^8_{xy}.$$
By definition, the inverse function satisfies the equation
$$\int d^8z\,\Big(\frac{\delta^2\Gamma}{\delta V_x^A \delta V_z^C}\Big)^{-1} \frac{\delta^2\Gamma}{\delta V_z^C \delta V_y^B}\bigg|_{\mbox{\scriptsize fields}=0} = \delta_{AB} \delta^8_{xy}.$$
Using the identity (see, e.g., [@West:1990tg])
$$0 = \partial^2 + \partial^2\Pi_{1/2} + \frac{1}{16}\Big(D^2 \bar D^2 + \bar D^2 D^2\Big),$$
which can be proved by (anti)commuting the covariant derivatives, it is possible to demonstrate that the explicit expression for the inverse two-point Green function of the quantum gauge superfield is written as
$$\label{Gauge_Propagator}
\Big(\frac{\delta^2\Gamma}{\delta V_x^A \delta V_y^B}\Big)^{-1}\bigg|_{\mbox{\scriptsize fields}=0} = - \delta_{AB}\, \Big[\,\frac{2}{G_V \partial^4} \partial^2\Pi_{1/2} + \frac{\xi_0}{8 \partial^4 K} \Big(D^2\bar D^2 + \bar D^2 D^2\Big)\Big] \delta^8_{xy}.$$
Thus, we see that the effective propagator is proportional to $Q^{-4}$, where $Q_\mu$ is the corresponding Euclidean momentum.
The all-loop sum of singularities produced by the quantum gauge superfield
--------------------------------------------------------------------------
\[Subsection\_Gauge\_Singularities\]
A part of the expression (\[Gamma\_Derivative\_In\_Terms\_Of\_Inverse\_Functions\]) contributed by supergraphs in which one gauge propagator is marked is written as
$$\label{Gauge_Part}
\frac{i}{4}\int d^8x\, \Big[ \partial^2\Pi_{1/2} R(\partial^2/\Lambda^2) + \frac{1}{16\xi_0}\Big(\bar D^2 D^2 + D^2 \bar D^2\Big) K(\partial^2/\Lambda^2) \Big]_x
\Big(\frac{\delta^2\Gamma}{\delta V^A_{x}\delta V^A_y}\Big)^{-1} \bigg|_{y=x;\ \mbox{\scriptsize fields}=0},$$
where the exact propagator of the quantum gauge superfield is given by Eq. (\[Gauge\_Propagator\]). All quantum corrections inside this exact propagator are encoded in the function $G_V$, which is equal to $R(\partial^2/\Lambda^2)$ in the tree approximation. That is why it is convenient to present it in the form
$$G_V = R + \Delta G_V,$$
where $\Delta G_V$ is a sum of relevant quantum corrections. Exactly as for the matter and ghost contributions, we rewrite the inverse two-point Green function in Eq. (\[Gauge\_Part\]) in the form of a series using the identities
$$\label{Transversal_Identity}
\big(\Pi_{1/2}\big)^2 = - \Pi_{1/2};\qquad \Pi_{1/2} D^2 = 0;\qquad \Pi_{1/2} \bar D^2 = 0,$$
and (\[Chiral\_Identities\]). To simplify the resulting expression, we introduce the notation
$$P \equiv \frac{1}{\partial^4 R} \partial^2\Pi_{1/2} + \frac{\xi_0}{16 \partial^4 K} \Big(D^2\bar D^2 + \bar D^2 D^2\Big).$$
This expression is proportional to the usual (tree) propagator of the quantum gauge superfield. Then the inverse Green function present in Eq. (\[Gauge\_Part\]) can equivalently be rewritten as
$$\begin{aligned}
\label{Gauge_Inverse_Function_Expansion}
&& \Big(\frac{\delta^2\Gamma}{\delta V_x^A \delta V_y^A}\Big)^{-1}\bigg|_{\mbox{\scriptsize fields}=0} = - r\, \Big[\,\frac{2}{G_V \partial^4} \partial^2\Pi_{1/2} + \frac{\xi_0}{8 \partial^4 K} \Big(D^2\bar D^2 + \bar D^2 D^2\Big)\Big] \delta^8_{xy}\qquad\nonumber\\
&& = - 2 r\,P\Big(1 - \partial^2\Pi_{1/2} \Delta G_V P + \partial^2\Pi_{1/2} \Delta G_V P\cdot \partial^2\Pi_{1/2} \Delta G_V P - \ldots \Big)\delta^8_{xy}.\vphantom{\frac{1}{2}}\end{aligned}$$
The factors $\partial^2\Pi_{1/2} G_V$ in this expression correspond to the 1PI subdiagrams (denoted in Fig. \[Figure\_Graph\_With\_Coinciding\_Momenta\] by the gray circles), which are evidently transversal. As we discussed in section \[Subsection\_Main\_Idea\], for constructing a contribution to the function (\[Delta\_Beta\]), one should first divide a term with the $n$-th power of $\Delta G_V$ by $n$. (Certainly, the first term corresponding to the one-loop approximation should be omitted.) Then the function (\[Gauge\_Inverse\_Function\_Expansion\]) will be replaced by the expression
$$\begin{aligned}
&& - 2 r\, P\Big( - \partial^2\Pi_{1/2} \Delta G_V P + \frac{1}{2}\partial^2\Pi_{1/2} \Delta G_V P \cdot \partial^2\Pi_{1/2} \Delta G_V P - \ldots \Big)\delta^8_{xy}\nonumber\\
&& = - 2 r\, P \sum\limits_{n=1}^\infty \frac{1}{n} \big(-\partial^2\Pi_{1/2} \Delta G_V P\big)^n \delta^8_{xy} = - 2 r\, \frac{1}{\partial^4 R} \partial^2\Pi_{1/2}\sum\limits_{n=1}^\infty \frac{1}{n} \Big(
-\frac{\Delta G_V}{R}\Big)^n \delta^8_{xy}.\qquad\end{aligned}$$
Calculating the sum of this series, we obtain that, at the first step, it is necessary to make in Eq. (\[Gauge\_Part\]) the formal substitution
$$\Big(\frac{\delta^2\Gamma}{\delta V_x^A \delta V_y^A}\Big)^{-1}\bigg|_{\mbox{\scriptsize fields}=0}\ \to \ \frac{2 r}{\partial^4 R}\, \partial^2\Pi_{1/2} \ln \frac{G_V}{R} \delta^8_{xy}.$$
After this, with the help of Eq. (\[Transversal\_Identity\]) we obtain
$$\begin{aligned}
\label{Gauge_Part_Modified}
&& (\ref{Gauge_Part})\ \to\ -\frac{i r}{2}\int d^8x\, \partial^2\Pi_{1/2} \frac{1}{\partial^2}\, \ln \frac{G_V}{R} \delta^8_{xy}\bigg|_{y=x}
\nonumber\\
&&\qquad\qquad\qquad\qquad
= \frac{i r}{2}\int d^8x\, d^8y\, \int \frac{d^4q}{(2\pi)^4} \delta^8_{xy}(q)\, \partial^2\Pi_{1/2} \frac{1}{q^2}\, \ln \frac{G_V}{R} \delta^8_{xy}.\qquad\end{aligned}$$
It should be noted that all possible bad terms proportional to $1/q^{2k}$ with $k\ge 2$ vanish, and the above expression really contains only the $1/q^2$ singularity. To construct a contribution to the function (\[Delta\_Beta\]) coming from the gauge singularities, we need to modify the expression (\[Gauge\_Part\_Modified\]) in a special way. Namely, it is necessary to insert $\theta^4 (v^B)^2$ to the point $x$, apply the operator $C_2 \partial^2/\partial q_\mu \partial q^\mu$ to $1/q^2$ coming from the gauge propagator which is cut in the point $x$, multiply the result by $-2\pi/(r{\cal V}_4)$, and differentiate it with respect to $\ln\Lambda$. The sum of the gauge singularities constructed according to this procedure is given by
$$\begin{aligned}
\label{Delta_Gauge_Original}
&& \Delta_{\mbox{\scriptsize gauge}}\Big(\frac{\beta}{\alpha_0^2}\Big) = -\frac{i\pi C_2}{{\cal V}_4} \frac{d}{d\ln\Lambda} \int d^8x\,d^8y\, \big(\theta^4\big)_x \big(v^B\big)^2_x\nonumber\\
&& \qquad\qquad\qquad\qquad\qquad \times \int \frac{d^4q}{(2\pi)^4}\, \delta^8_{xy}(q)\, \ln \frac{G_V}{R}\, \frac{\partial^2}{\partial q_\mu \partial q^\mu} \Big(\frac{1}{q^2}\Big) \big(\partial^2\Pi_{1/2}\big)_x \delta^8_{xy}.\qquad\end{aligned}$$
As earlier, we calculate the momentum integral in the Euclidean space after the Wick rotation taking into account that $R(0)=1$,
$$\begin{aligned}
&& \frac{d}{d\ln\Lambda} \int \frac{d^4q}{(2\pi)^4}\, \delta^8_{xy}(q)\, \ln \frac{G_V}{R}\, \frac{\partial^2}{\partial q_\mu \partial q^\mu} \Big(\frac{1}{q^2}\Big)\nonumber\\
&&\qquad\quad\ \to\ -\frac{i}{4\pi^2}\, \delta^4(\theta_x-\theta_y)\, \frac{d}{d\ln\Lambda} \ln G_V\Big|_{Q=0} = -\frac{i}{2\pi^2}\, \delta^4(\theta_x-\theta_y)\, \gamma_V(\alpha_0,\lambda_0,Y_0).\qquad\end{aligned}$$
This implies that
$$\label{Delta_Gauge_Intermediate}
\Delta_{\mbox{\scriptsize gauge}}\Big(\frac{\beta}{\alpha_0^2}\Big) = - \frac{C_2}{2\pi {\cal V}_4} \gamma_V(\alpha_0,\lambda_0,Y_0) \int d^8x\,d^8y\, \big(\theta^4\big)_x \big(v^B\big)^2_x\, \delta^4(\theta_x-\theta_y) \big(\partial^2\Pi_{1/2}\big)_x \delta^8_{xy}.$$
For calculating the remaining integral we use the identity
$$\delta^4(\theta_x-\theta_y) \big(\partial^2\Pi_{1/2}\big)_x \delta^8_{xy} = -\frac{1}{8}\, \delta^4(\theta_x-\theta_y) \big(D^a \bar D^2 D_a\big)_x \delta^8_{xy} = - \frac{1}{2}\,\delta^8_{xy}.$$
Therefore,
$$\int d^8x\,d^8y\, \big(\theta^4\big)_x \big(v^B\big)^2_x\, \delta^4(\theta_x-\theta_y) \big(\partial^2\Pi_{1/2}\big)_x \delta^8_{xy}
= - \frac{1}{2} \int d^8x\, d^8y\, \big(\theta^4\big)_x \big(v^B\big)^2_x\, \delta^8_{xy} = - 2 {\cal V}_4.$$
Substituting this expression into Eq. (\[Delta\_Gauge\_Intermediate\]) we obtain the final result for the sum of singular contributions to the function (\[Delta\_Beta\]) produced by the quantum gauge superfield,
$$\label{Delta_Gauge_Final}
\Delta_{\mbox{\scriptsize gauge}}\Big(\frac{\beta}{\alpha_0^2}\Big) = \frac{C_2}{\pi} \gamma_V(\alpha_0,\lambda_0,Y_0).$$
We see that it coincides with the corresponding term in Eq. (\[NSVZ\_Equivalent\_Form\_Bare\]) in exact agreement with the guess made in Ref. [@Stepanyantz:2016gtk].
The $\beta$-function defined in terms of the bare couplings
-----------------------------------------------------------
\[Subsection\_Result\]
The overall result for the function (\[Delta\_Beta\]) is obtained by summing the contributions produced by cuts of matter, ghost, and gauge propagators, which are given by Eqs. (\[Delta\_Matter\_Final\]), (\[Delta\_Ghost\_Final\]), and (\[Delta\_Gauge\_Final\]), respectively,
$$\begin{aligned}
&& \frac{\beta(\alpha_0,\lambda_0,Y_0)}{\alpha_0^2} - \frac{\beta_{\mbox{\scriptsize 1-loop}}(\alpha_0)}{\alpha_0^2} = \Delta_{\mbox{\scriptsize matter}}\Big(\frac{\beta}{\alpha_0^2}\Big) + \Delta_{\mbox{\scriptsize ghost}}\Big(\frac{\beta}{\alpha_0^2}\Big) + \Delta_{\mbox{\scriptsize gauge}}\Big(\frac{\beta}{\alpha_0^2}\Big)\qquad\nonumber\\
&& = - \frac{1}{2\pi r} C(R)_i{}^j \big(\gamma_\phi\big)_j{}^i(\alpha_0,\lambda_0,Y_0) + \frac{1}{\pi} C_2 \gamma_c(\alpha_0,\lambda_0,Y_0) + \frac{1}{\pi} C_2 \gamma_V(\alpha_0,\lambda_0,Y_0).\end{aligned}$$
Taking into account that the one-loop contribution to the $\beta$-function is given by Eq. (\[Beta\_1Loop\]), we see that the resulting expression for the $\beta$-function defined in terms of the bare couplings coincides with the NSVZ equation written in the form (\[NSVZ\_Equivalent\_Form\_Bare\]). Certainly, it is highly important that the theory is regularized by higher covariant derivatives. As we have already mentioned, for RGFs defined in terms of the bare couplings the NSVZ equation does not hold in the case of using the regularization by dimensional reduction [@Aleshin:2016rrr] due to a different structure of loop integrals [@Aleshin:2015qqc].
Note that, in fact, in the previous sections we have also proved the equation (\[NSVZ\_For\_Green\_Functions\_With\_1Loop\]) relating the two-point Green function of the considered theory in the limit of the vanishing external momenta.
To obtain the NSVZ relation in the usual form, one needs to involve the non-renormalization theorem for the triple gauge-ghost vertices. Namely, following Ref. [@Stepanyantz:2016gtk], we differentiate Eq. (\[ZZZ\_Constraint\]) (which is a consequence of this theorem) with respect to $\ln\Lambda$ at fixed values of renormalized couplings and obtain the equation
$$\label{Beta_Triple_Relation}
\beta(\alpha_0,\lambda_0,Y_0) = 2\alpha_0\Big(\gamma_c(\alpha_0,\lambda_0,Y_0) + \gamma_V(\alpha_0,\lambda_0,Y_0)\Big).$$
Excluding the sum $\gamma_c+\gamma_V$ from Eqs. (\[NSVZ\_Equivalent\_Form\_Bare\]) and (\[Beta\_Triple\_Relation\]) we obtain the NSVZ relation in the original form of the relation between the $\beta$-function and the anomalous dimension of the matter superfields,
$$\label{NSVZ_Exact_Beta_Function_Bare}
\frac{\beta(\alpha_0,\lambda_0,Y_0)}{\alpha_0^2} = - \frac{3 C_2 - T(R) + C(R)_i{}^j \big(\gamma_\phi\big)_j{}^i(\alpha_0,\lambda_0,Y_0)/r}{2\pi(1- C_2\alpha_0/2\pi)}.$$
Thus, we have proved that it is valid in all loops for RGFs defined in terms of the bare couplings in the case of using the regularization by higher covariant derivatives. The subtraction scheme in which the NSVZ equations hold for RGFs defined in terms of the renormalized couplings will be constructed in the next section.
NSVZ scheme for RGFs defined in terms of the renormalized couplings
===================================================================
\[Section\_NSVZ\_Scheme\]
We have proved that the NSVZ relations (\[NSVZ\_Exact\_Beta\_Function\]) and (\[NSVZ\_Equivalent\_Form\]) are satisfied by RGFs defined in terms of the bare couplings in the case of using the higher covariant derivative regularization described in section \[Section\_N=1\_Gauge\_Theories\]. Note that these RGFs are scheme-independent [@Kataev:2013eta], so that the NSVZ relations for them are valid independently of a renormalization prescription. However, the standard RGFs are defined in terms of the renormalized couplings and depend on both a regularization and a subtraction scheme. It is known that the NSVZ relation for RGFs defined in terms of the renormalized couplings is satisfied only in certain subtraction schemes, which are called the NSVZ schemes. In particular, the $\overline{\mbox{DR}}$-scheme is not the NSVZ scheme [@Jack:1996vg; @Jack:1996cn; @Jack:1998uj]. However, using the results described above it is possible to demonstrate that an NSVZ scheme can be obtained in all orders with the help of the HD+MSL prescription [@Stepanyantz:2016gtk]. For completeness, here we briefly explain, how this statement can be obtained.
In terms of the renormalized couplings RGFs are defined by the equations
$$\begin{aligned}
\label{RGFs_Renormalized}
&& \widetilde\beta(\alpha,\lambda,Y) \equiv \left.\frac{d \alpha}{d\ln\mu}\right|_{\alpha_0,\lambda_0,Y_0 = \mbox{\scriptsize const}};\qquad\quad\ \ \widetilde\gamma_V(\alpha,\lambda,Y) \equiv \left. \frac{d\ln Z_V}{d\ln\mu}\right|_{\alpha_0,\lambda_0,Y_0 = \mbox{\scriptsize const}};\nonumber\\
&& \widetilde\gamma_c(\alpha,\lambda,Y) \equiv \left. \frac{d\ln Z_c}{d\ln\mu}\right|_{\alpha_0,\lambda_0,Y_0 = \mbox{\scriptsize const}};\qquad\ \ (\widetilde\gamma_\phi)_i{}^j(\alpha,\lambda,Y) \equiv \left. \frac{d(\ln Z_\phi)_i{}^j}{d\ln\mu}\right|_{\alpha_0,\lambda_0,Y_0 = \mbox{\scriptsize const}}.\qquad\end{aligned}$$
Unlike Eqs. (\[Beta\_Bare\_Definition\]) — (\[Gamma\_Phi\_Bare\_Definition\]), the derivatives in Eq. (\[RGFs\_Renormalized\]) are taken with respect to $\ln \mu$ (instead of $\ln\Lambda$) at fixed values of the bare couplings (instead of the renormalized ones).
The key observation made in [@Kataev:2013eta] is that both definitions of RGFs give the same functions (of different arguments) if certain boundary conditions are imposed on the renormalization constants. In the non-Abelian case considered here these boundary conditions (in the point $x_0$ which is a fixed value of $\ln\Lambda/\mu$) take the form
$$\begin{aligned}
\label{Boundary_Conditions}
&& \alpha(\alpha_0,\lambda_0,Y_0,\ln\Lambda/\mu \to x_0) = \alpha_0;\qquad\ Z_V(\alpha_0,\lambda_0,Y_0,\ln\Lambda/\mu \to x_0) = 1;\qquad\vphantom{\Big(}\nonumber\\
&& Z_c(\alpha_0,\lambda_0,Y_0,\ln\Lambda/\mu\to x_0) = 1;\qquad\ (Z_\phi)_i{}^j(\alpha_0,\lambda_0,Y_0,\ln\Lambda/\mu \to x_0) = \delta_i^j;\vphantom{\Big(}\nonumber\\
&&\qquad\qquad\qquad\qquad Y(\alpha_0,\lambda_0,Y_0,\ln\Lambda/\mu \to x_0) = Y_0.\vphantom{\Big(}\end{aligned}$$
We also assume that the renormalization of the Yukawa couplings is related to the renormalization of the matter superfields by Eq. (\[Lambda\_Renormalization\]). Note that, due to the boundary conditions (\[Boundary\_Conditions\]) and the nonrenormalization of the triple ghost-gauge vertices, Eq. (\[ZZZ\_Constraint\]) is satisfied automatically, because in this case the equation
$$\frac{d}{d\ln\Lambda} (Z_\alpha^{-1/2} Z_c Z_V) = 0$$
has the only solution $Z_\alpha^{-1/2} Z_c Z_V=1$. Similarly, it is easy to see that the condition $Z_\xi Z_V^2=1$ is also satisfied automatically.
The conditions (\[Boundary\_Conditions\]) define a class of the HD+MSL-like schemes. Here HD means that the higher covariant derivative method is used for regularizing the theory under consideration. This is always assumed in this paper. The HD+MSL scheme is obtained for $x_0 = 0$. In this case all renormalization constants include only powers of $\ln\Lambda/\mu$ and all finite constants which define a subtraction scheme vanish. The schemes corresponding to $x_0\ne 0$ are related to the HD+MSL scheme by the redefinition of the renormalization point $\mu = \exp(-x_0)\mu_{\mbox{\scriptsize HD+MSL}}$ exactly as in the Abelian case considered in [@Goriachuk:2018cac].
Under the boundary conditions (\[Boundary\_Conditions\]) RGFs (\[RGFs\_Renormalized\]) can be related to RGFs (\[Beta\_Bare\_Definition\]) — (\[Gamma\_Phi\_Bare\_Definition\]) by the equations
$$\begin{aligned}
\label{RGFs_Relations}
&& \widetilde\beta(\alpha,\lambda,Y) = \beta(\alpha_0\to \alpha, \lambda_0\to\lambda,Y_0\to Y);\vphantom{\Big(}\nonumber\\
&& \widetilde\gamma_V(\alpha,\lambda,Y) = \gamma_V(\alpha_0\to \alpha, \lambda_0\to\lambda,Y_0\to Y);\vphantom{\Big(}\nonumber\\
&& \widetilde\gamma_c(\alpha,\lambda,Y) = \gamma_c(\alpha_0\to \alpha, \lambda_0\to\lambda,Y_0\to Y);\vphantom{\Big(}\nonumber\\
&& (\widetilde\gamma_\phi)_i{}^j(\alpha,\lambda,Y) = (\gamma_\phi)_i{}^j(\alpha_0\to \alpha, \lambda_0\to\lambda,Y_0\to Y),\qquad\vphantom{\Big(}\end{aligned}$$
where the arrows point that it is necessary to make a formal change of the argument, say, to write $\alpha$ instead of $\alpha_0$, etc.
All these equations can be proved repeating the argumentation of [@Kataev:2013eta] (in which similar equations were derived in the Abelian case). For example, taking into account that $\alpha = \alpha_0 Z_\alpha$, we obtain
$$\begin{aligned}
\label{Beta_Relation}
&&\hspace*{-4mm} \widetilde\beta(\alpha,\lambda,Y) = \frac{d\alpha}{d\ln\mu} \bigg|_{\alpha_0,\lambda_0,Y_0 = \mbox{\scriptsize const}} = \frac{d}{d\ln\mu} \Big(\alpha_0 Z_\alpha(\alpha,\lambda,Y,\ln\Lambda/\mu)\Big)\bigg|_{\alpha_0,\lambda_0,Y_0 = \mbox{\scriptsize const}}\quad\nonumber\\
&&\hspace*{-4mm} = \alpha_0 Z_\alpha \bigg(\frac{\partial \ln Z_\alpha}{\partial\ln\mu} + \frac{\partial \ln Z_\alpha}{\partial \alpha}\, \frac{d\alpha}{d\ln\mu} + \frac{\partial \ln Z_\alpha}{\partial \lambda^{ijk}}\, \frac{d\lambda^{ijk}}{d\ln\mu} + \frac{\partial \ln Z_\alpha}{\partial \lambda^*_{ijk}}\, \frac{d\lambda^*_{ijk}}{d\ln\mu} + \frac{\partial \ln Z_\alpha}{\partial Y}\, \frac{dY}{d\ln\mu}\bigg).\qquad\end{aligned}$$
Here the derivative $d/d\ln\mu$ acts on both $\ln\Lambda/\mu$ inside $\alpha$, $\lambda$, $Y$ and the explicitly written $\ln\Lambda/\mu$. The partial derivative $\partial/\partial\ln\mu$, by contrast, acts only on the explicitly written $\ln\Lambda/\mu$ and does not act on $\ln\Lambda/\mu$ inside $\alpha$, $\lambda$, and $Y$. Therefore,
$$\label{Z_Derivatives}
\frac{\partial \ln Z_\alpha}{\partial\ln\mu} = -\frac{\partial \ln Z_\alpha}{\partial\ln\Lambda} \equiv -\frac{d\ln Z_\alpha}{d\ln\Lambda}\bigg|_{\alpha,\lambda,Y=\mbox{\scriptsize const}} = \frac{d\ln (Z_\alpha)^{-1}}{d\ln\Lambda}\bigg|_{\alpha,\lambda,Y=\mbox{\scriptsize const}}.$$
Next, we consider Eq. (\[Beta\_Relation\]) in the point $\ln\Lambda/\mu = x_0$. Due to the boundary conditions (\[Boundary\_Conditions\]) $\ln Z_\alpha(\alpha,\lambda,Y,\ln\Lambda/\mu\to x_0) = 0$ for all values of $\alpha$, $\lambda$, and $Y$. This implies that
$$\begin{aligned}
\label{Z_Derivatives_In_X0}
&& \frac{\partial \ln Z_\alpha}{\partial \alpha}\bigg|_{\ln\Lambda/\mu=x_0} = 0;\qquad\qquad \frac{\partial \ln Z_\alpha}{\partial \lambda^{ijk}}\bigg|_{\ln\Lambda/\mu=x_0} = 0;\qquad\nonumber\\
&& \frac{\partial \ln Z_\alpha}{\partial \lambda^*_{ijk}}\bigg|_{\ln\Lambda/\mu=x_0} = 0;\qquad\qquad \frac{\partial \ln Z_\alpha}{\partial Y}\bigg|_{\ln\Lambda/\mu=x_0} = 0.\end{aligned}$$
Taking into account that the left hand side of Eq. (\[Beta\_Relation\]) does not depend on the value of $\ln\Lambda/\mu$ at fixed values of the renormalized couplings due to the renormalization group equation, from Eqs. (\[Beta\_Relation\]), (\[Z\_Derivatives\]), and (\[Z\_Derivatives\_In\_X0\]) we obtain
$$\widetilde\beta(\alpha,\lambda,Y) = \alpha \left.\frac{d\ln (Z_\alpha)^{-1}}{d\ln\Lambda}\right|_{\alpha,\lambda,Y=\mbox{\scriptsize const}} = \left.\frac{d\alpha_0}{d\ln\Lambda}\right|_{\alpha,\lambda,Y=\mbox{\scriptsize const}} = \beta(\alpha_0,\lambda_0,Y_0)\Big|_{\ln\Lambda/\mu = x_0}.$$
However, according to Eq. (\[Boundary\_Conditions\]), in the point $\ln\Lambda/\mu= x_0$ the values of the bare and renormalized couplings coincide,
$$\label{Couplings_Relation}
\alpha_0\Big|_{\ln\Lambda/\mu = x_0} = \alpha;\qquad \lambda_0^{ijk}\Big|_{\ln\Lambda/\mu = x_0} = \lambda^{ijk};\qquad Y_0\Big|_{\ln\Lambda/\mu = x_0} = Y.$$
(For the Yukawa couplings it is also necessary to use Eq. (\[Lambda\_Renormalization\]), which relates the renormalization of the Yukawa couplings to the renormalization of the chiral matter superfields.)
Taking into account that the left hand side of Eq. (\[Beta\_Relation\]) is considered as a function of the renormalized couplings, the right hand side should also be expressed in terms of them using Eq. (\[Couplings\_Relation\]). This implies that we should formally replace the bare couplings by the renormalized ones, so that
$$\widetilde\beta(\alpha,\lambda,Y) = \beta(\alpha_0\to\alpha,\lambda_0\to\lambda,Y_0\to Y).$$
The other equations in (\[RGFs\_Relations\]) can be proved in a similar way. For example, with the help of the chain rule for the derivative with respect to $\ln\mu$ we can present the anomalous dimension of the quantum gauge superfield in the form
$$\widetilde\gamma_V(\alpha,\lambda,Y) = \frac{\partial \ln Z_V}{\partial\ln\mu} + \frac{\partial \ln Z_V}{\partial \alpha}\, \frac{d\alpha}{d\ln\mu} + \frac{\partial \ln Z_V}{\partial \lambda^{ijk}}\, \frac{d\lambda^{ijk}}{d\ln\mu} + \frac{\partial \ln Z_V}{\partial \lambda^*_{ijk}}\, \frac{d\lambda^*_{ijk}}{d\ln\mu} + \frac{\partial \ln Z_V}{\partial Y}\, \frac{dY}{d\ln\mu}.$$
Again we consider this equation in the point $\ln\Lambda/\mu = x_0$ and take into account that due to Eq. (\[Boundary\_Conditions\]) $\ln Z_V(\alpha,\lambda,Y,\ln\Lambda/\mu\to x_0) = 0$. Then, repeating the above argumentation, we obtain
$$\widetilde\gamma_V(\alpha,\lambda,Y) = \left. - \frac{d\ln Z_V}{d\ln\Lambda}\right|_{\alpha,\lambda,Y=\mbox{\scriptsize const}} = \gamma_V(\alpha_0\to\alpha,\lambda_0\to\lambda,Y_0\to Y).$$
Earlier we have demonstrated that RGFs defined in terms of the bare couplings satisfy the NSVZ relations (\[NSVZ\_Equivalent\_Form\_Bare\]) and (\[NSVZ\_Exact\_Beta\_Function\_Bare\]) for theories regularized by higher covariant derivatives independently of a renormalization prescription. (Let us recall that these RGFs are scheme-independent for a fixed regularization.) After the formal change of arguments $\alpha_0\to \alpha$, $\lambda_0 \to \lambda$, $Y_0\to Y$ these equalities certainly remain valid. Therefore, from Eq. (\[RGFs\_Relations\]) we conclude that in the case of using the higher covariant derivative regularization supplemented by the renormalization prescription (\[Boundary\_Conditions\]) RGFs defined in terms of the renormalized couplings also satisfy the equations
$$\begin{aligned}
&&\hspace*{-8mm} \frac{\widetilde\beta(\alpha,\lambda,Y)}{\alpha^2} = - \frac{1}{2\pi}\Big(3 C_2 - T(R) - 2C_2 \widetilde\gamma_c(\alpha,\lambda,Y) - 2C_2 \widetilde\gamma_V(\alpha,\lambda,Y) + \frac{1}{r} C(R)_i{}^j \big(\widetilde\gamma_\phi\big)_j{}^i(\alpha,\lambda,Y)\Big);\nonumber\\
\label{NSVZ_Renormalized}
&&\hspace*{-8mm} \widetilde\beta(\alpha,\lambda,Y) = - \frac{\alpha^2\Big(3 C_2 - T(R) + C(R)_i{}^j
\big(\widetilde\gamma_\phi\big)_j{}^i(\alpha,\lambda,Y)/r\Big)}{2\pi(1- C_2\alpha/2\pi)}.\end{aligned}$$
This implies that in the non-Abelian case the prescription (\[Boundary\_Conditions\]) also provides the NSVZ scheme in all loops. The HD+MSL prescription is obtained by imposing the boundary conditions (\[Boundary\_Conditions\]) with $x_0=0$. For other values of $x_0$ we obtain a family of schemes which differ from HD+MSL by redefinitions of the normalization point $\mu$. Certainly, in these schemes the NSVZ relation is also valid in all loops. These statements agree with the explicit calculations (of some scheme dependent terms in RGFs) made in Refs. [@Shakhmanov:2017soc; @Kazantsev:2018nbl].
Verifications in the lowest orders
==================================
\[Section\_Explicit\_Calculation\]
In this section we verify the general argumentation discussed above by explicit calculations in the lowest orders of the perturbation theory made with the help of the higher covariant derivative regularization. For this purpose we will use the results for various groups of supergraphs obtained earlier. Namely, using the method described in section \[Subsection\_Idea\] the two-loop $\beta$-function for a general ${\cal N}=1$ supersymmetric gauge theory with a simple gauge group has been calculated in Ref. [@Stepanyantz:2019lyo]. Also using this method the parts of the three-loop $\beta$-function containing the Yukawa couplings and ghost loops have been found in Refs. [@Stepanyantz:2019ihw] and [@Kuzmichev:2019ywn], respectively. Note that before this the part of the three-loop $\beta$-function containing the Yukawa couplings has also been calculated with the help of the standard technique in Refs. [@Shakhmanov:2017soc; @Kazantsev:2018nbl]. For ${\cal N}=1$ SQED with $N_f$ flavors the complete three-loop $\beta$-function in a general $\xi$-gauge has been calculated by the algorithm of section \[Subsection\_Idea\] in Ref. [@Aleshin:2020gec]. Here all these calculations are used for checking the exact results described in the previous sections. Note that we will not verify that the equations (\[NSVZ\_Equivalent\_Form\_Bare\]), (\[NSVZ\_Exact\_Beta\_Function\_Bare\]), and (\[NSVZ\_Renormalized\]) really hold, because this has already been done in Refs. [@Shakhmanov:2017soc; @Kazantsev:2018nbl; @Kuzmichev:2019ywn; @Stepanyantz:2019lyo; @Aleshin:2020gec]. The main purpose of this section is to test the argumentation which was used for the all-loop derivation of the NSVZ equations at intermediate steps.
The two-loop approximation
--------------------------
\[Subsection\_Two\_Loop\]
First, we reanalyse the two-loop calculation made in Ref. [@Stepanyantz:2019lyo]. The corresponding contribution to the $\beta$-function is generated by the vacuum supergraphs presented in Fig. \[Figure\_Two\_Loop\]. The standard technique requires calculating all superdiagrams contributing to the two-point Green function of the background gauge superfield. They are obtained from the ones presented in Fig. \[Figure\_Two\_Loop\] by attaching two external $\bm{V}$-legs in all possible ways. The method of Ref. [@Stepanyantz:2019ihw] considerably simplifies the calculation, because it deals only with vacuum supergraphs. In this paper we have made some modifications, which will be verified here. This allows checking the general argumentation used for deriving the NSVZ equation and illustrating it by explicit calculations.
(0,4) (1.5,2.2)[![Vacuum supergraphs generating the two-loop contribution to the $\beta$-function.[]{data-label="Figure_Two_Loop"}](beta_yukawa.eps "fig:")]{} (1.3,3.6)[B1]{} (4.5,2.2)[![Vacuum supergraphs generating the two-loop contribution to the $\beta$-function.[]{data-label="Figure_Two_Loop"}](beta_matter2.eps "fig:")]{} (4.1,3.6)[B2]{} (8.8,2.2)[![Vacuum supergraphs generating the two-loop contribution to the $\beta$-function.[]{data-label="Figure_Two_Loop"}](beta_matter1.eps "fig:")]{} (8.4,3.6)[B3]{} (11.9,2.2)[![Vacuum supergraphs generating the two-loop contribution to the $\beta$-function.[]{data-label="Figure_Two_Loop"}](beta_ghost2.eps "fig:")]{} (11.5,3.6)[B4]{} (3.5,0)[![Vacuum supergraphs generating the two-loop contribution to the $\beta$-function.[]{data-label="Figure_Two_Loop"}](beta_ghost1.eps "fig:")]{} (3.1,1.4)[B5]{} (6.5,0)[![Vacuum supergraphs generating the two-loop contribution to the $\beta$-function.[]{data-label="Figure_Two_Loop"}](beta_gauge2.eps "fig:")]{} (6.1,1.4)[B6]{} (11.0,0)[![Vacuum supergraphs generating the two-loop contribution to the $\beta$-function.[]{data-label="Figure_Two_Loop"}](beta_gauge1.eps "fig:")]{} (10.6,1.4)[B7]{}
Let us start with the supergraph B1. The expression for it has been calculated in Ref. [@Stepanyantz:2019ihw]. If we include the factor $-2\pi/(r{\cal V}_4)\cdot d/d\ln\Lambda$, the result can be written as
$$\mbox{B1}\ \to\ -\frac{2\pi}{r{\cal V}_4} \frac{d}{d\ln\Lambda} \cdot \frac{2}{3} {\cal V}_4 \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4} \frac{\lambda_0^{ijk} \lambda^*_{0ijk}}{Q^2 F_Q K^2 F_K (Q+K)^2 F_{Q+K}},$$
where $F_K\equiv F(K^2/\Lambda^2)$. The integrand here contains three inverse squared momenta. Therefore, we should sum up three (equal) expressions obtained by replacing one of these inverse squared momenta $1/P^2$ (together with the corresponding $\delta$-symbol $\delta_m^n$) by $4\pi^2 C(R)_m{}^n \delta^4(P)$. This gives the contribution to the function (\[Delta\_Beta\]) of the form
$$\begin{aligned}
&& \Delta_{\mbox{\scriptsize Yukawa}}\Big(\frac{\beta}{\alpha_0^2}\Big) = -\frac{4\pi}{r} C(R)_i{}^m \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4}\, 4\pi^2 \delta^4(Q)\frac{\lambda_0^{ijk} \lambda^*_{0mjk}}{F_Q K^2 F_K (Q+K)^2 F_{Q+K}}\qquad\nonumber\\
&& = -\frac{1}{\pi r} C(R)_i{}^j \frac{d}{d\ln\Lambda} \int \frac{d^4K}{(2\pi)^4} \frac{\lambda_0^{imn} \lambda^*_{0jmn}}{K^4 F_K^2}.\end{aligned}$$
This result exactly agrees with the one obtained in Ref. [@Shakhmanov:2017soc] by the direct calculation of two-point superdiagrams with two external $\bm{V}$-legs.
The supergraphs B2 and B3 containing a matter loop produce two different contributions. If a matter loop corresponds to the superfields $\phi_i$ and to the Pauli–Villars superfields $\Phi_i$, then (for an arbitrary value of $\xi_0$) the result for the (properly modified) vacuum supergraphs multiplied by the operator (\[Operator\]) is given by
$$\begin{aligned}
\label{Matter1}
&&\hspace*{-5mm} \mbox{B2} + \mbox{B3}\ \to\ \frac{4\pi}{r}\, \mbox{tr}\, C(R)\, \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4} \frac{e_0^2}{K^2 R_K} \bigg\{\frac{1}{2Q^2(Q+K)^2} + \frac{1}{((K+Q)^2-Q^2)}\nonumber\\
&&\hspace*{-5mm} \times \bigg[\frac{F_{K+Q}^2}{2((K+Q)^2 F_{K+Q}^2 + M^2)} - \frac{F_Q^2}{2(Q^2 F_Q^2 + M^2)} - \frac{M^2 F_{K+Q}'}{\Lambda^2 F_{K+Q} ((K+Q)^2 F_{K+Q}^2 + M^2)}\qquad\nonumber\\
&&\hspace*{-5mm} + \frac{M^2 F_Q'}{\Lambda^2 F_Q (Q^2 F_Q^2 + M^2)}\bigg] \bigg\},\end{aligned}$$
where $R_K\equiv R(K^2/\Lambda^2)$. Unlike the separate supergraphs B2 and B3, it does not contain bad singularities proportional to the inverse momenta to the fourth power and terms which depend on $\xi_0$. Next, we should find a sum of the expressions obtained from (\[Matter1\]) either by replacing $1/K^2$ (which comes from the gauge propagator) by $4\pi^2 C_2 \delta^4(K)$ or by replacing $\delta_i^j/Q^2$ or $\delta_i^j/(Q+K)^2$ (coming from the propagators of the superfields $\phi_i$) by $4\pi^2 C(R)_i{}^j \delta^4(Q)$ or $4\pi^2 C(R)_i{}^j \delta^4(Q+K)$, respectively. Note that no replacement should be made for the non-singular Pauli–Villars propagators. The above procedure gives the result
$$\begin{aligned}
&& \Delta_{\mbox{\scriptsize matter}}\Big(\frac{\beta}{\alpha_0^2}\Big) = \frac{64\pi^4}{r}\,\mbox{tr}\left(C(R)^2\right) \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4} \frac{\alpha_0}{K^2 R_K} \bigg\{\frac{1}{2(Q+K)^2}\,\delta^4(Q) \nonumber\\
&& + \frac{1}{2Q^2}\, \delta^4(Q+K) \bigg\}
+ \frac{64\pi^4}{r}\,C_2\, \mbox{tr}\,C(R)\, \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4}\, \delta^4(K) \frac{\alpha_0}{R_K} \nonumber\\
&& \times \bigg\{\frac{1}{2Q^2(Q+K)^2} + \frac{1}{((K+Q)^2-Q^2)} \bigg[\frac{F_{K+Q}^2}{2((K+Q)^2 F_{K+Q}^2 + M^2)} - \frac{F_Q^2}{2(Q^2 F_Q^2 + M^2)} \qquad\nonumber\\
&& - \frac{M^2 F_{K+Q}'}{\Lambda^2 F_{K+Q} ((K+Q)^2 F_{K+Q}^2 + M^2)} + \frac{M^2 F_Q'}{\Lambda^2 F_Q (Q^2 F_Q^2 + M^2)}\bigg] \bigg\}.\end{aligned}$$
After calculating the integrals of the $\delta$-functions and some transformations we obtain the expression
$$\begin{aligned}
&& \Delta_{\mbox{\scriptsize matter}}\Big(\frac{\beta}{\alpha_0^2}\Big) = \frac{4}{r}\,\mbox{tr}\left(C(R)^2\right) \frac{d}{d\ln\Lambda} \int \frac{d^4K}{(2\pi)^4} \frac{\alpha_0}{K^4 R_K} \qquad\nonumber\\
&&\qquad\qquad\qquad + \frac{1}{2r} \,C_2\, \mbox{tr}\,C(R)\,\frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{\partial^2}{\partial Q_\mu^2}\bigg[\frac{\alpha_0 }{Q^2} \ln\Big(1+\frac{M^2}{Q^2 F_Q^2}\Big)\bigg],\qquad\end{aligned}$$
which exactly coincides with the one found in Ref. [@Stepanyantz:2019lyo] with the help of a different prescription.
The solid lines in the supergraphs B2 and B3 can also stand for the propagators of the Pauli–Villars superfields $\varphi_a$. In this case the calculation of the vacuum superdiagrams with an insertion of $\theta^4 (v^B)^2$ multiplied by the operator (\[Operator\]) for an arbitrary value of $\xi_0$ gives
$$\begin{aligned}
&&\hspace*{-5mm} \mbox{B2} + \mbox{B3}\ \to\ 4\pi C_2 \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4} \frac{e_0^2}{K^2 R_K} \bigg\{\frac{1}{(Q^2+M_\varphi^2)((Q+K)^2+M_\varphi^2)}\nonumber\\
&&\hspace*{-5mm} - \frac{1}{((K+Q)^2-Q^2)} \bigg[\frac{R_{K+Q}^2}{2((K+Q)^2 R_{K+Q}^2 + M_\varphi^2)} - \frac{R_Q^2}{2(Q^2 R_Q^2 + M^2)} -\frac{1}{\Lambda^2 R_{K+Q}}\qquad\nonumber\\
&&\hspace*{-5mm} \times \frac{M_\varphi^2 R_{K+Q}'}{((K+Q)^2 R_{K+Q}^2 + M_\varphi^2)} + \frac{M_\varphi^2 R_Q'}{\Lambda^2 R_Q (Q^2 R_Q^2 + M_\varphi^2)} + \frac{R_{K+Q}'}{\Lambda^2 R_{K+Q}} - \frac{R_Q'}{\Lambda^2 R_Q} \bigg] \bigg\}.\end{aligned}$$
In this case all matter propagators are massive and do not produce singularities. Therefore, it is only necessary to make the replacement $1/K^2\ \to\ 4\pi^2 C_2 \delta^4(K)$, after which the contribution of the Pauli–Villars superfields $\varphi_a$ to the function (\[Delta\_Beta\]) takes the form
$$\begin{aligned}
&& \Delta_\varphi\Big(\frac{\beta}{\alpha_0^2}\Big) = C_2^2 \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4}\bigg\{\frac{6\alpha_0}{Q^4} - \frac{\partial^2}{\partial Q_\mu^2}\bigg[\frac{\alpha_0}{Q^2}\ln\Big(1+ \frac{M_\varphi^2}{Q^2}\Big)\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad - \frac{\alpha_0}{2Q^2} \ln \Big(1+\frac{M_\varphi^2}{Q^2 R_Q^2}\Big) - \frac{\alpha_0}{Q^2} \ln R_Q \bigg]\bigg\}.\qquad\end{aligned}$$
This expression again coincides with the one found in Ref. [@Stepanyantz:2019lyo] with the help of a different technique.[^10]
The contribution of the vacuum supergraphs B4, B5, B6, and B7 is written in the form
$$\begin{aligned}
&&\hspace*{-5mm} \mbox{B4} + \mbox{B5} + \mbox{B6} + \mbox{B7}\ \to\ 4\pi C_2\, \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4} \frac{e_0^2}{R_K R_Q} \bigg\{-\frac{R_K}{2 Q^2 K^2 (K+Q)^2}\nonumber\\
&&\hspace*{-5mm} - \frac{1}{2Q^2 K^2} \bigg(\frac{R_Q-R_K}{Q^2-K^2}\bigg)
- \frac{1}{R_{K+Q} K^2} \Big(1-\frac{Q^2}{2(K+Q)^2}\Big) \bigg(\frac{R_Q-R_K}{Q^2-K^2}\bigg)\bigg(\frac{R_{K+Q}-R_Q}{(K+Q)^2-Q^2}\bigg)\nonumber\\
&&\hspace*{-5mm} - \frac{1}{R_{K+Q} (K+Q)^2}\bigg(\frac{R_Q-R_K}{Q^2-K^2}\bigg)^2 +\frac{2}{K^2 \big((K+Q)^2-Q^2\big)^2} \bigg[ R_{K+Q} - R_Q - R_Q' \Big(\frac{(K+Q)^2}{\Lambda^2} \nonumber\\
&&\hspace*{-5mm} - \frac{Q^2}{\Lambda^2}\Big)\bigg] - \frac{Q_\mu K^\mu}{Q^2 K^2} \bigg[\frac{R_{K+Q}}{\big((K+Q)^2-K^2\big) \big((K+Q)^2-Q^2\big)} + \frac{R_{K}}{\big(K^2-(K+Q)^2\big) \big(K^2-Q^2\big)} \nonumber\\
&&\hspace*{-5mm} + \frac{R_{Q}}{\big(Q^2-(K+Q)^2\big)\big(Q^2-K^2\big)}\bigg]
\bigg\}.\end{aligned}$$
We see that all bad terms containing inverse momenta to the fourth power really cancel each other, although they are present in expressions for the separate supergraphs. This agrees with the general argumentation of section \[Section\_Singularities\]. Also all terms dependent on the gauge parameter $\xi_0$ cancel each other. To find a contribution to the function (\[Delta\_Beta\]) we should replace one of the gauge or ghost inverse squared momenta by the corresponding $\delta$-function multiplied by $4\pi^2 C_2$ and sum up all expression thus obtained. The result is
$$\begin{aligned}
&&\hspace*{-5mm} \Delta_{\mbox{\scriptsize gauge+ghost}}\Big(\frac{\beta}{\alpha_0^2}\Big) = 64\pi^4 C_2^2\, \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4} \frac{\alpha_0}{R_K R_Q} \bigg\{-\frac{R_K}{2 K^2 (K+Q)^2}\, \delta^4(Q) \nonumber\\
&&\hspace*{-5mm} -\frac{R_K}{2 Q^2 (K+Q)^2}\, \delta^4(K) -\frac{R_K}{2 Q^2 K^2}\, \delta^4(K+Q) - \bigg(\frac{1}{2 K^2}\,\delta^4(Q) + \frac{1}{2Q^2} \delta^4(K) \bigg)\bigg(\frac{R_Q-R_K}{Q^2-K^2}\bigg)\nonumber\\
&& \hspace*{-5mm} - \frac{1}{R_{K+Q}} \bigg[\,\delta^4(K) \Big(1-\frac{Q^2}{2(K+Q)^2}\Big) - \frac{Q^2}{2 K^2}\delta^4(K+Q) \bigg] \bigg(\frac{R_Q-R_K}{Q^2-K^2}\bigg)\bigg(\frac{R_{K+Q}-R_Q}{(K+Q)^2-Q^2}\bigg)\nonumber\\
&&\hspace*{-5mm} - \frac{1}{R_{K+Q}}\, \delta^4(K+Q) \bigg(\frac{R_Q-R_K}{Q^2-K^2}\bigg)^2 +\frac{2}{\big((K+Q)^2-Q^2\big)^2}\,\delta^4(K) \bigg[ - R_Q' \Big(\frac{(K+Q)^2}{\Lambda^2} - \frac{Q^2}{\Lambda^2}\Big) \nonumber\\
&&\hspace*{-5mm} +R_{K+Q} - R_Q \bigg] - \bigg(\frac{Q_\mu K^\mu}{K^2}\,\delta^4(Q) + \frac{Q_\mu K^\mu}{Q^2}\,\delta^4(K) \bigg)\bigg[\frac{R_{K+Q}}{\big((K+Q)^2-K^2\big) \big((K+Q)^2-Q^2\big)} \nonumber\\
&&\hspace*{-5mm} + \frac{R_{K}}{\big(K^2-(K+Q)^2\big) \big(K^2-Q^2\big)} + \frac{R_{Q}}{\big(Q^2-(K+Q)^2\big)\big(Q^2-K^2\big)}\bigg]
\bigg\}.\end{aligned}$$
After calculating the integrals of the $\delta$-functions this expression can be rewritten as
$$\begin{aligned}
&& \Delta_{\mbox{\scriptsize gauge+ghost}}\Big(\frac{\beta}{\alpha_0^2}\Big) = 4 C_2^2\, \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4}\, \bigg\{-\frac{3\alpha_0}{2 Q^4} - \frac{\alpha_0}{\Lambda^4 R_Q^2} (R_Q')^2 + \frac{\alpha_0}{\Lambda^4 R_Q} R_Q''\bigg\}\qquad\nonumber\\
&& = C_2^2 \frac{d}{d\ln\Lambda}\int \frac{d^4Q}{(2\pi)^4}\bigg\{-\frac{6\alpha_0}{Q^4} + \frac{\partial^2}{\partial Q_\mu^2} \Big[\frac{\alpha_0}{Q^2}\ln R_Q\Big]\bigg\}.\end{aligned}$$
Again we see that the result coincides with the one obtained by the method of Ref. [@Stepanyantz:2019ihw] in Ref. [@Stepanyantz:2019lyo]. This confirms the correctness of the argumentation used in this paper for deriving the NSVZ equation.
The overall two-loop expression for the $\beta$-function reads as
$$\begin{aligned}
&& \frac{\beta(\alpha_0,\lambda_0,Y_0)}{\alpha_0^2} - \frac{\beta_{\mbox{\scriptsize 1-loop}}(\alpha_0)}{\alpha_0^2} = \int \frac{d^4Q}{(2\pi)^4} \frac{d}{d\ln\Lambda} \bigg\{ - \alpha_0 C_2^2\, \frac{\partial^2}{\partial Q_\mu^2}\bigg[\frac{1}{2Q^2}\ln\Big(1+\frac{M_\varphi^2}{Q^2 R_Q^2}\Big) \qquad\nonumber\\
&& - \frac{1}{Q^2} \ln\Big(1+\frac{M_\varphi^2}{Q^2}\Big)\bigg] + \frac{\alpha_0}{2r}\, C_2\,\mbox{tr}\, C(R)\, \frac{\partial^2}{\partial Q_\mu^2}\bigg[\frac{1}{Q^2}\ln\Big(1+\frac{M^2}{Q^2 F_Q^2}\Big)\bigg] + \frac{4\alpha_0}{r}\, \mbox{tr}\left(C(R)^2\right) \nonumber\\
&& \times \frac{1}{Q^4 R_Q} - \frac{1}{\pi r} \lambda_0^{imn} \lambda^*_{0jmn} C(R)_i{}^j \frac{1}{Q^2 F_Q^2}\bigg\}
+ O(\alpha_0^2,\alpha_0\lambda_0^2,\lambda_0^4)\end{aligned}$$
and coincides with the one obtained in Ref. [@Stepanyantz:2019lyo]. Certainly, this expression satisfies the NSVZ equations (\[NSVZ\_Equivalent\_Form\_Bare\]) and (\[NSVZ\_Exact\_Beta\_Function\_Bare\]) and agrees with the previous calculations (first made in Ref. [@Jones:1974pg]).
The three-loop approximation: supergraphs with Yukawa vertices
--------------------------------------------------------------
\[Subsection\_Three\_Loop\_Yukawa\]
Next, we will verify the argumentation of this paper on the example of the three-loop contribution to $\beta$-function which contains the Yukawa couplings. It is generated by the supergraphs B8 — B11 presented in Fig. \[Figure\_3Loop\_Yukawa\]. The direct calculation of the two-point superdiagrams obtained from them by attaching two external lines of the background gauge superfield $\bm{V}$ has been made in Refs. [@Shakhmanov:2017soc; @Kazantsev:2018nbl] in the Feynman gauge $\xi_0=1$. Subsequently, the result has been reobtained by the method described in section \[Subsection\_Idea\] in Ref. [@Stepanyantz:2019ihw]. Now we will demonstrate how the modification proposed in this paper works in this case.
(0,3) (1.2,0.3)[![The three-loop vacuum supergraphs containing vertices with the Yukawa couplings.[]{data-label="Figure_3Loop_Yukawa"}](3loop_y_graph1.eps "fig:")]{} (0.8,2.0)[B8]{} (3,1.05)[$K_\mu$]{}(1.9,-0.1)[$L_\mu$]{} (1.9,2.2)[$Q_\mu$]{} (5.2,0.3)[![The three-loop vacuum supergraphs containing vertices with the Yukawa couplings.[]{data-label="Figure_3Loop_Yukawa"}](3loop_y_graph2.eps "fig:")]{} (4.8,2.0)[B9]{} (5.55,0.75)[$K_\mu$]{}(5.45,1.35)[$L_\mu$]{} (6.8,1.8)[$Q_\mu$]{} (9.2,0.3)[![The three-loop vacuum supergraphs containing vertices with the Yukawa couplings.[]{data-label="Figure_3Loop_Yukawa"}](3loop_y_graph3.eps "fig:")]{} (8.7,2.0)[B10]{} (9.9,1.2)[$K_\mu$]{}(10,-0.1)[$L_\mu$]{} (11,1.05)[$Q_\mu$]{} (13.2,-0.1)[![The three-loop vacuum supergraphs containing vertices with the Yukawa couplings.[]{data-label="Figure_3Loop_Yukawa"}](3loop_y_graph4.eps "fig:")]{} (12.5,2.0)[B11]{} (14.8,2.0)[$K_\mu$]{}(14.78,0)[$L_\mu$]{} (14.8,1.2)[$Q_\mu$]{}
First, we consider the supergraph B8 which is quartic in the Yukawa couplings. The result for the modified vacuum supergraph is given by the expression
$$\begin{aligned}
\label{B8_Vacuum}
&& B8\ \to\ \frac{4\pi}{r} \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4} \frac{d^4L}{(2\pi)^4} \lambda_0^{ijk} \lambda^*_{0ijl} \lambda_0^{mnl} \lambda^*_{0mnk} \nonumber\\
&& \qquad\qquad\qquad\qquad\qquad \times \frac{1}{Q^2 F_Q (K+Q)^2 F_{K+Q} L^2 F_L (K+L)^2 F_{K+L} K^2 F_K^2},\qquad\end{aligned}$$
which does not contain bad terms proportional to the inverse momenta to the fourth power. As earlier, to construct the corresponding contribution to the function (\[Delta\_Beta\]), we should sum all expressions obtained from (\[B8\_Vacuum\]) by replacing squared inverse momenta multiplied by $\delta$-symbols coming from the corresponding propagator by momentum $\delta$-functions multiplied by $4\pi^2 C(R)$. The tensor structure of the resulting factors can easily be viewed from the structure of the graph under consideration. After constructing and calculating the integrals of the $\delta$-functions we obtain the required contribution
$$\begin{aligned}
&& \Delta_{\mbox{\scriptsize B8}}\Big(\frac{\beta}{\alpha_0^2}\Big) = \frac{1}{\pi r} C(R)_p{}^l \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4L}{(2\pi)^4}\, \lambda_0^{ijk} \lambda^*_{0ijl} \lambda_0^{mnp} \lambda^*_{0mnk} \frac{1}{Q^4 F_Q^2 L^4 F_L^2}\nonumber\\
&& + \frac{4}{\pi r} C(R)_i{}^p \frac{d}{d\ln\Lambda} \int \frac{d^4K}{(2\pi)^4} \frac{d^4L}{(2\pi)^4}\, \lambda_0^{ijk} \lambda^*_{0pjl} \lambda_0^{mnl} \lambda^*_{0mnk} \frac{1}{K^4 F_K^3 L^2 F_L (K+L)^2 F_{K+L}},\qquad\end{aligned}$$
which exactly coincides with the corresponding result of Refs. [@Shakhmanov:2017soc; @Stepanyantz:2019ihw] found by different methods.
According to [@Stepanyantz:2019ihw] the properly modified vacuum supergraph B9 is given by the expression
$$\begin{aligned}
\label{B9_Vaccum}
&& B9\ \to\ -\frac{8\pi}{r} \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4} \frac{d^4L}{(2\pi)^4}\, e_0^2 \lambda_0^{ijk} \lambda^*_{0imn} (T^A)_j{}^m (T^A)_k{}^n\nonumber\\
&& \times \frac{N(Q,K,L)}{K^2 R_K L^2 F_L Q^2 F_Q (Q+K)^2 F_{Q+K} (Q-L)^2 F_{Q-L} (Q+K-L)^2 F_{Q+K-L}},\qquad\end{aligned}$$
where $K_\mu$ corresponds to the propagator of the quantum gauge superfield and, following Ref. [@Kazantsev:2018nbl], we use the notation
$$\begin{aligned}
&&\hspace*{-5mm} N(Q,K,L) \equiv L^2 F_{Q+K} F_{Q+K-L} - Q^2 \Big((Q+K)^2 - L^2\Big) F_{Q+K-L} \frac{F_{Q+K}-F_Q}{(Q+K)^2-Q^2}\nonumber\\
&&\hspace*{-5mm} - (Q-L)^2 \Big((Q+K-L)^2-L^2\Big) F_{Q+K} \frac{F_{Q+K-L}-F_{Q-L}}{(Q+K-L)^2-(Q-L)^2} + Q^2 (Q-L)^2 \nonumber\\
&&\hspace*{-5mm} \times \Big(L^2 - (Q+K)^2 - (Q+K-L)^2\Big)\bigg(\frac{F_{Q+K}-F_Q}{(Q+K)^2-Q^2}\bigg)\bigg(\frac{F_{Q+K-L}-F_{Q-L}}{(Q+K-L)^2-(Q-L)^2}\bigg).\qquad\end{aligned}$$
Again, Eq. (\[B9\_Vaccum\]) does not contain bad terms. Next, we proceed according to the algorithm of section \[Subsection\_Graphs\]. In this case the relevant replacements are $1/K^2 \to 4\pi^2 C_2 \delta^4(K)$ and $\delta_i^j/P^2 \to 4\pi^2 C(R)_i{}^j \delta^4(P)$, where $P_\mu$ stands for momenta of the matter superfields, namely, $Q_\mu$, $(Q+K)_\mu$, $(Q-L)_\mu$, and $(Q+K-L)_\mu$. After some transformations involving the identities
$$\begin{aligned}
&&\hspace*{-5mm} \lambda_0^{ijk} \lambda^*_{0imn} (T^A)_j{}^m (T^A)_k{}^n = -\frac{1}{2} \lambda_0^{imn} \lambda^*_{0jmn} C(R)_i{}^j;\\
&&\hspace*{-5mm} \lambda_0^{ijk} \lambda^*_{0imn} C(R)_j{}^l (T^A)_l{}^m (T^A)_k{}^n = -\frac{1}{2} \lambda_0^{imn} \lambda^*_{0jmn} \big(C(R)^2\big)_i{}^j;\\
&&\hspace*{-5mm} \lambda_0^{ijk} \lambda^*_{0lmn} C(R)_i{}^l (T^A)_j{}^m (T^A)_k{}^n = \frac{1}{2} \lambda_0^{imn} \lambda^*_{0jmn} \big(C(R)^2\big)_i{}^j - \lambda_0^{ijm} \lambda^*_{0klm} C(R)_i{}^k C(R)_j{}^l, \qquad\end{aligned}$$
which follow from Eq. (\[Yukawa\_Constraint\]), we get the result
$$\begin{aligned}
&&\hspace*{-5mm} \Delta_{\mbox{\scriptsize B9}}\Big(\frac{\beta}{\alpha_0^2}\Big) = \frac{1}{\pi r}\, \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4L}{(2\pi)^4}\, e_0^2 \lambda_0^{imn}\lambda^*_{0jmn} C_2 C(R)_i{}^j \frac{N(Q,0,L)}{Q^4 F_Q^2 L^2 F_L (Q-L)^4 F_{Q-L}^2}\nonumber\\
&&\hspace*{-5mm} - \frac{1}{\pi r}\, \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4}\, e_0^2 \Big(\lambda_0^{imn}\lambda^*_{0jmn} \big(C(R)^2\big)_i{}^j - 2 \lambda_0^{ijm} \lambda^*_{0klm} C(R)_i{}^k C(R)_j{}^l\Big) \frac{1}{K^2 R_K} \nonumber\\
&&\hspace*{-5mm} \times \frac{N(Q,K,0)}{Q^4 F_Q^2 (Q+K)^4 F_{Q+K}^2} + \frac{4}{\pi r}\, \frac{d}{d\ln\Lambda} \int \frac{d^4K}{(2\pi)^4} \frac{d^4L}{(2\pi)^4}\, e_0^2 \lambda_0^{imn}\lambda^*_{0jmn} \big(C(R)^2\big)_i{}^j
\frac{1}{K^4 R_K F_K L^4}
\nonumber\\
&&\hspace*{-5mm} \times \frac{N(0,K,L)}{F_L^2 (K-L)^2 F_{K-L}},\end{aligned}$$
which agrees with the one found in [@Kazantsev:2018nbl; @Stepanyantz:2019ihw] by different methods.
The supergraph B10 is given by the expression
$$\begin{aligned}
&& B10\ \to\ -\frac{8\pi}{r} \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4} \frac{d^4L}{(2\pi)^4}\, e_0^2 \lambda_0^{ijk} \lambda^*_{0ijl} (T^A)_k{}^m (T^A)_m{}^l\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad \times \frac{L(Q,Q+K)}{K^2 R_K Q^2 F_Q^2 (Q+L)^2 F_{Q+L} (Q+K)^2 F_{Q+K} L^2 F_L},\qquad\quad\end{aligned}$$
where $K_\mu$ denotes the momentum of the quantum gauge superfield propagator and, again following Ref. [@Kazantsev:2018nbl],
$$L(Q,P) \equiv F_Q F_P + \frac{F_P-F_Q}{P^2-Q^2}\, \Big(F_Q Q^2 + F_P P^2\Big) + 2 Q^2 P^2 \bigg(\frac{F_P-F_Q}{P^2-Q^2}\bigg)^2.$$
In this case the algorithm of section \[Subsection\_Graphs\] produces the expression
$$\begin{aligned}
&&\hspace*{-5mm} \Delta_{\mbox{\scriptsize B10}}\Big(\frac{\beta}{\alpha_0^2}\Big) = -\frac{2}{\pi r} \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4L}{(2\pi)^4}\, e_0^2\, \lambda_0^{imn} \lambda^*_{0jmn} C_2 C(R)_i{}^j \frac{L(Q,Q)}{Q^4 F_Q^3 L^2 F_L (Q+L)^2 F_{Q+L}} \nonumber\\
&&\hspace*{-5mm} -\frac{2}{\pi r} \frac{d}{d\ln\Lambda} \int \frac{d^4K}{(2\pi)^4} \frac{d^4L}{(2\pi)^4}\, e_0^2\, \lambda_0^{imn} \lambda^*_{0jmn} \big(C(R)^2\big)_i{}^j \bigg(\frac{L(0,K)}{K^4 R_K F_K L^4 F_L^2} + \frac{L(K,0)}{K^4 R_K F_K^2 L^2 F_L} \nonumber\\
&&\hspace*{-5mm} \times\frac{1}{(K+L)^2 F_{K+L}}\bigg) - \frac{4}{\pi r} \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4}\, e_0^2\, \lambda_0^{ijm} \lambda^*_{0klm} C(R)_i{}^k C(R)_j{}^l \frac{L(Q,Q+K)}{K^2 R_K Q^4 F_Q^3} \qquad\nonumber\\
&&\hspace*{-5mm} \times \frac{1}{(Q+K)^2 F_{Q+K}},\end{aligned}$$
which also coincides with the results of Refs. [@Kazantsev:2018nbl; @Stepanyantz:2019ihw].
The expression for the last supergraph B11 reads as
$$\begin{aligned}
&& B11\ \to\ \frac{8\pi}{r} \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4} \frac{d^4L}{(2\pi)^4}\, e_0^2 \lambda_0^{ijk} \lambda^*_{0ijl} (T^A)_k{}^m (T^A)_m{}^l\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \times \frac{K(Q,K)}{K^2 R_K Q^2 F_Q^2 L^2 F_L (Q+L)^2 F_{Q+L}},\qquad\end{aligned}$$
where the momentum of the gauge superfield propagator is denoted by $K_\mu$ and
$$K(Q,K) \equiv \frac{F_{Q+K}-F_Q-2Q^2 F_Q'/\Lambda^2}{(Q+K)^2 - Q^2} + \frac{2Q^2 (F_{Q+K}-F_Q)}{((Q+K)^2-Q^2)^2}.$$
Replacing the squared inverse momenta and the corresponding $\delta$-symbols according to the prescription given in section \[Subsection\_Graphs\] we obtain the part of the function (\[Delta\_Beta\]),
$$\begin{aligned}
&& \Delta_{\mbox{\scriptsize B11}}\Big(\frac{\beta}{\alpha_0^2}\Big) = \frac{2}{\pi r} \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4L}{(2\pi)^4}\, e_0^2\, \lambda_0^{imn} \lambda^*_{0jmn} C_2 C(R)_i{}^j \frac{K(Q,0)}{Q^2 F_Q^2 L^2 F_L (Q+L)^2 F_{Q+L}}\nonumber\\
&& + \frac{2}{\pi r} \frac{d}{d\ln\Lambda} \int \frac{d^4K}{(2\pi)^4} \frac{d^4L}{(2\pi)^4}\, e_0^2\, \lambda_0^{imn} \lambda^*_{0jmn} \big(C(R)^2\big)_i{}^j \frac{K(0,K)}{K^2 R_K L^4 F_L^2}\nonumber\\
&& + \frac{4}{\pi r} \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4}\, e_0^2\, \lambda_0^{ijm} \lambda^*_{0klm} C(R)_i{}^k C(R)_j{}^l \frac{K(Q,K)}{K^2 R_K Q^4 F_Q^3}.\end{aligned}$$
Again, it agrees with the calculations made previously.
The three-loop approximation: supergraphs with ghost loops
----------------------------------------------------------
All three-loop vacuum supergraphs containing loops of the Faddeev–Popov ghosts have been calculated in Ref. [@Kuzmichev:2019ywn]. These supergraphs are presented in Fig. \[Figure\_3Loop\_Ghosts\]. Note that in this approximation the first nonlinear term in the function ${\cal F}(V)$ (see Eq. (\[Nonlinear\_Function\_F\])) is essential. It generates the vertex
$$- \frac{3}{4} e_0^2\, y_0\, G^{ABCD} \int d^8x\, (\bar c^A + \bar c^{+A}) V^C V^D (c^B - c^{+B})$$
present in the supergraph B21, see Ref. [@Kazantsev:2018kjx] for details. This vertex is denoted by a cross.
Expressions for the supergraphs in Fig. \[Figure\_3Loop\_Ghosts\] contain $1/K^4$ singularities, where $K_\mu$ is the Euclidean momentum of the quantum gauge superfield. As in the two-loop approximation, these singularities should disappear after adding the purely gauge vacuum supergraphs. However, the sum of the three-loop diagrams containing only the gauge propagators has not yet been calculated with the higher covariant derivative regularization. Nevertheless, the sum of the supergraphs with ghost loops does not contain any singularities proportional to the inverse ghost or matter momenta to the fourth power. This implies that it is possible to compare the sums of singularities coming from the cuts of ghost and matter lines with the corresponding anomalous dimensions $\gamma_c(\alpha_0,\lambda_0,Y_0)$ and $(\gamma_\phi)_i{}^j(\alpha_0,\lambda_0,Y_0)$. With the help of the method described in section \[Subsection\_Idea\] this has been done in Ref. [@Kuzmichev:2019ywn]. Also it is possible to use these results for checking the modification of the algorithm discussed in section \[Subsection\_Graphs\]. This is made as follows:
(0,21) (2.4,-0.4)[$+$]{} (3.2,-1.0)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_matter1.eps "fig:")]{} (5.5,-0.4)[$+$]{} (6.4,-1.1)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_matter2.eps "fig:")]{} (8.8,-0.4)[$+$]{} (9.5,-0.9)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_matter4.eps "fig:")]{} (12.35,-0.4)[$+$]{} (13.2,-0.9)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_matter3.eps "fig:")]{}
(0.7,1)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_beta11.eps "fig:")]{} (0,2.3)[B22]{} (2.7,1.6)[$+$]{} (3.7,1)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_beta12.eps "fig:")]{} (3,2.3)[B23]{} (7.0,1.7)[(1,0)[1.5]{}]{} (9.5,1.2)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma14.eps "fig:")]{} (12.35,1.6)[$+$]{} (13.2,1.17)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma13.eps "fig:")]{}
(0.5,2.9)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_beta_y.eps "fig:")]{} (0,4.2)[B21]{} (3.9,3.6)[(1,0)[1.6]{}]{} (6.5,3.1)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma_y.eps "fig:")]{}
(0.5,4.8)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_beta15.eps "fig:")]{} (0,6.1)[B20]{} (3.9,5.5)[(1,0)[1.6]{}]{} (6.2,5.4)[${\displaystyle -\frac{1}{2}\ \Bigg(}$]{} (7.4,5.1)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma2.eps "fig:")]{} (9.9,5.4)[${\displaystyle\Bigg)^{\mbox{2}}}$]{}
(1,6.7)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_beta10.eps "fig:")]{} (0,8.0)[B19]{} (3.5,7.4)[(1,0)[1.5]{}]{} (6,6.9)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma4.eps "fig:")]{} (8.8,7.3)[$+$]{} (9.4,6.9)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma5.eps "fig:")]{} (12.2,7.3)[$+$]{} (12.85,6.8)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma7.eps "fig:")]{}
(1,8.6)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_beta9.eps "fig:")]{} (0,9.9)[B18]{} (3.5,9.3)[(1,0)[2]{}]{} (6.5,9.0)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma10.eps "fig:")]{}
(1,10.5)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_beta8.eps "fig:")]{} (0,11.8)[B17]{} (3.5,11.2)[(1,0)[2]{}]{} (6.5,10.9)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma11.eps "fig:")]{} (9.5,11.1)[$+$]{} (10.3,10.9)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma12.eps "fig:")]{}
(1,12.4)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_beta7.eps "fig:")]{} (0,13.7)[B16]{} (3.5,13.1)[(1,0)[2]{}]{} (6.5,12.4)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma9.eps "fig:")]{}
(1.1,14.3)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_beta6.eps "fig:")]{} (0,15.6)[B15]{} (3.5,15.0)[(1,0)[2]{}]{} (6.5,14.5)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma6.eps "fig:")]{} (9.4,14.9)[$-$]{} (10.0,14.9)[${\displaystyle \frac{1}{2}\ \Bigg(}$]{} (10.9,14.7)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma1.eps "fig:")]{} (13.4,14.9)[${\displaystyle\Bigg)^{\mbox{2}}}$]{}
(1.1,16.2)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_beta5.eps "fig:")]{} (0,17.5)[B14]{} (3.5,16.9)[(1,0)[2]{}]{} (6.5,16.2)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma8.eps "fig:")]{}
(0.7,18.1)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_beta4.eps "fig:")]{} (0,19.4)[B13]{} (3.5,18.8)[(1,0)[2]{}]{} (6.6,17.8)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma3.eps "fig:")]{}
(0.5,20)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_beta3.eps "fig:")]{} (0,21.3)[B12]{} (3.5,20.7)[(1,0)[1]{}]{} (5.1,19.9)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma15.eps "fig:")]{} (8.1,20.6)[$-$]{} (8.6,20.6)[${\displaystyle\Bigg(}$]{} (9.1,20.3)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma1.eps "fig:")]{} (12.0,20.6)[$\times$]{} (12.7,20.3)[![The three-loop vacuum supergraphs containing a ghost loop (or ghost loops) and the corresponding superdiagrams contributing to the ghost and matter anomalous dimensions. The gray circles denote insertions of the one-loop polarization operator of the quantum gauge superfield, see Ref. [@Kazantsev:2017fdc] for details.[]{data-label="Figure_3Loop_Ghosts"}](ghost_gamma2.eps "fig:")]{} (15.15,20.6)[${\displaystyle\Bigg)}$]{}
1\. First, we consider a vacuum supergraph containing a ghost loop (or loops) with an insertion of $\theta^4 (v^B)^2$ and calculate it using the $D$-algebra.
2\. Next, we replace one of inverse squared momenta (say, $1/Q^2$) of a [*ghost*]{} propagator by $4\pi^2 C_2 \delta^4(Q)$. If matter propagators are present in the considered supergraph, then $\delta_i^j/Q^2$ coming from a matter propagator is replaced by $4\pi^2 C(R)_i{}^j \delta^4(Q)$. All expressions obtained after the replacements of one propagator should be summed.
3\. After calculating the integrals of the $\delta$-functions the results are compared with the corresponding contributions to the ghost (or matter) anomalous dimensions. These contributions are given by the sums of all two-point supergraphs which are produced by all possible cuts of ghost (or matter) propagators in the considered supergraph, see Fig. \[Figure\_3Loop\_Ghosts\].
We have done this for all three-loop vacuum supergraphs presented in Fig. \[Figure\_3Loop\_Ghosts\]. As a result, we have obtained
$$\begin{aligned}
&& \Delta_{\mbox{\scriptsize B}i} \Big(\frac{\beta}{\alpha_0^2}\Big)\ \to\ \frac{C_2}{\pi} \Delta_{\mbox{\scriptsize B}i} \gamma_c,\qquad \mbox{for}\qquad i=12,\ldots,21;\\
&& \Delta_{\mbox{\scriptsize B}22} \Big(\frac{\beta}{\alpha_0^2}\Big) + \Delta_{\mbox{\scriptsize B}23} \Big(\frac{\beta}{\alpha_0^2}\Big)\ \to\ \frac{C_2}{\pi} \Delta_{\mbox{\scriptsize B}22 + \mbox{\scriptsize B}23} \gamma_c - \frac{1}{2\pi r} C(R)_i{}^j (\Delta_{\mbox{\scriptsize B}22 + \mbox{\scriptsize B}23}\gamma_\phi)_j{}^i,\qquad\end{aligned}$$
in agreement with Eqs. (\[Delta\_Matter\_Final\]) and (\[Delta\_Ghost\_Final\]). Note that this verification is different from the one made in Ref. [@Kuzmichev:2019ywn], because the algorithm used for constructing the contribution to the function (\[Delta\_Beta\]) is different.
${\cal N}=1$ SQED in the three-loop approximation
-------------------------------------------------
As one more example for checking the method proposed in section \[Subsection\_Graphs\] we can consider ${\cal N}=1$ SQED with $N_f$ flavors. In this case the gauge group is $U(1)$, $r=1$, $C_2 = 0$, and $C(R)_i{}^j \to \delta_{\alpha\beta} \cdot 1_2$, where $\alpha,\beta=1,\ldots, N_f$ and $1_2$ denotes an identity matrix of the size $2\times 2$. This theory does not contain Yukawa terms triple in chiral superfields. Therefore, the calculations in this case have been done for $F(x)=1$. Although ${\cal N}=1$ SQED is a particular case of ${\cal N}=1$ supersymmetric gauge theories considered earlier, this example is not trivial, because for this theory the [*complete*]{} expression for the three-loop $\beta$-function has explicitly been calculated with the higher derivative regularization. That is why it is possible to perform one more nontrivial test of the method proposed in this paper.
Using the results of Ref. [@Aleshin:2020gec] the sum of two- and three-loop vacuum supergraphs modified by an insertion of $\theta^4 (v^B)^2$ and multiplied by $-2\pi/{\cal V}_4\cdot d/d\ln\Lambda$ in the general $\xi_0$-gauge is written as
$$\begin{aligned}
\label{SQED_Vacuum}
&&\hspace*{-5mm} 4\pi N_f \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4} \frac{e_0^2}{K^2 R_K}\Big(\frac{1}{Q^2 (Q+K)^2} - \frac{1}{(Q^2+M^2)((Q+K)^2+M^2)}\Big)\nonumber\\
&&\hspace*{-5mm} - 4\pi N_f^2 \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4} \frac{d^4L}{(2\pi)^4} \frac{e_0^4}{K^2 R_K^2} \Big(\frac{1}{Q^2 (Q+K)^2} - \frac{1}{(Q^2+M^2)((Q+K)^2+M^2)}\Big)\nonumber\\
&&\hspace*{-5mm} \times \Big(\frac{1}{L^2 (L+K)^2} - \frac{1}{(L^2+M^2)((L+K)^2+M^2)}\Big) + 8\pi N_f \frac{d}{d\ln\Lambda} \int \frac{d^4Q}{(2\pi)^4} \frac{d^4K}{(2\pi)^4} \frac{d^4L}{(2\pi)^4}\nonumber\\
&&\hspace*{-5mm} \times \frac{e_0^4}{K^2 R_K L^2 R_L} \bigg\{\frac{1}{Q^2 (Q+K)^2 (Q+L)^2} - \frac{K^2}{Q^2 (Q+K)^2 (Q+L)^2 (Q+K+L)^2} \\
&&\hspace*{-5mm} + \frac{K^2+M^2}{(Q^2+M^2)((Q+K)^2+M^2)((Q+L)^2+M^2)((Q+K+L)^2+M^2)} - \frac{1}{(Q^2+M^2)}\nonumber\\
&&\hspace*{-5mm} \times \frac{1}{((Q+K)^2+M^2)((Q+L)^2+M^2)} + \frac{2M^2}{(Q^2+M^2)^2((Q+K)^2+M^2)((Q+L)^2+M^2)}\bigg\}, \nonumber\end{aligned}$$
and does not contain inverse momenta to the fourth power. According to the method discussed in section \[Subsection\_Graphs\], to find a contribution to the function (\[Delta\_Beta\]), we should sum the expressions obtained from Eq. (\[SQED\_Vacuum\]) by replacing one of $1/P^2$ by $4\pi^2 \delta^4(P)$, where $P_\mu$ is a momentum of the (massless) matter superfields. Note that it is not necessary to take into account the gauge superfield propagators, because now $C_2=0$, and nonsingular propagators of the massive Pauli–Villars superfields. In the first two terms of Eq. (\[SQED\_Vacuum\]) the momentum of the quantum gauge superfield is denoted by $K_\mu$, while in the last term the momenta of the quantum gauge superfield propagators are $K_\mu$ and $L_\mu$. Then the above described procedure gives
$$\begin{aligned}
&& \frac{\beta(\alpha_0)}{\alpha_0^2} - \frac{\beta_{\mbox{\scriptsize 1-loop}}(\alpha_0)}{\alpha_0^2} = \frac{2 N_f}{\pi} \frac{d}{d\ln\Lambda} \int \frac{d^4K}{(2\pi)^4} \frac{e_0^2}{K^4 R_K} - \frac{4N_f^2}{\pi} \frac{d}{d\ln\Lambda} \int \frac{d^4K}{(2\pi)^4} \frac{d^4L}{(2\pi)^4} \frac{e_0^4}{K^4 R_K^2}\nonumber\\
&& \times \Big(\frac{1}{L^2 (L+K)^2} - \frac{1}{(L^2+M^2)((L+K)^2+M^2)}\Big) - \frac{N_f}{\pi} \frac{d}{d\ln\Lambda} \int \frac{d^4K}{(2\pi)^4} \frac{d^4L}{(2\pi)^4} \frac{e_0^4}{R_K R_L}\qquad\nonumber\\
&& \times \Big(\frac{4}{K^2 L^4 (K+L)^2} - \frac{2}{K^4 L^4}\Big) + O(e_0^6).\end{aligned}$$
Again, this expression correctly reproduces the result obtained earlier by different methods, see Refs. [@Stepanyantz:2011jy; @Aleshin:2020gec].
Therefore, for all considered supergraphs the results obtained with the help of the technique discussed in section \[Subsection\_Graphs\] (which is a certain modification of the one proposed in [@Stepanyantz:2019ihw]) coincided with the expressions found earlier by different methods. This confirms the correctness of the method which was used in this paper for the all-loop perturbative derivation of the exact NSVZ $\beta$-function.
Conclusion
==========
In this paper we have finished the all-order perturbative derivation of the exact NSVZ $\beta$-function (\[NSVZ\_Exact\_Beta\_Function\]) for non-Abelian ${\cal N}=1$ supersymmetric gauge theories which was started in Refs. [@Stepanyantz:2016gtk; @Stepanyantz:2019ihw; @Stepanyantz:2019lfm]. Its main ingredient is the higher covariant derivative regularization, which allows revealing some interesting features of quantum corrections in these theories. For instance, according to Ref. [@Stepanyantz:2019lfm], with this regularization all loop integrals giving the $\beta$-function defined in terms of the bare couplings are integrals of double total derivatives with respect to loop momenta. These integrals do not vanish due to singularities, which appear when double total derivatives act on massless propagators. The sum of the singularities produces all contributions to the $\beta$-function starting from the two-loop approximation. (The one-loop quantum corrections should be considered separately. This has been done in Ref. [@Aleshin:2016yvj].) It is possible to divide singular contributions into three groups depending on a propagator on which the double total derivatives act. Namely, they can act on the matter superfield propagators, on the propagators of the Faddeev–Popov ghosts, and on the propagators of the quantum gauge superfield. Qualitatively, this can be interpreted as a cutting of the corresponding internal line in a certain vacuum supergraph. All such cuts give a set of superdiagrams contributing to the anomalous dimension of the corresponding quantum superfield. From the other side, attaching two external gauge lines of the background gauge superfield to the considered supergraph we obtain a set of superdiagrams contributing to the $\beta$-function. The NSVZ equation in the form (\[NSVZ\_Equivalent\_Form\]) relates this contribution to the $\beta$-function to the parts of the anomalous dimensions of quantum superfields coming from the superdiagrams produced by cuts of internal lines. Note that the factorization into double total derivatives allows to calculate analytically one of loop integrals, so that the $\beta$-function in a certain order is really related to the anomalous dimensions in the previous order. The proof of this fact has been done in this paper. We have demonstrated that the sums of singularities coming from the cuts of certain propagators are really equal to the corresponding terms in Eq. (\[NSVZ\_Equivalent\_Form\]). (Note that in Ref. [@Stepanyantz:2019lfm] this has been done for the matter and Faddeev–Popov ghost superfields. The results of this paper obtained by a different method are the same. However, in the present paper we have also calculated the all-loop sum of singularities produced by cuts of the gauge propagators.) Thus, Eq. (\[NSVZ\_Equivalent\_Form\]) for RGFs defined in terms of the bare couplings is proved in all orders in the case of using the higher covariant derivative regularization.[^11] The original NSVZ $\beta$-function (\[NSVZ\_Exact\_Beta\_Function\]) for these RGFs can be obtained with the help of the non-renormalization theorem for the triple gauge-ghost vertices proved in [@Stepanyantz:2016gtk]. Note that RGFs defined in terms of the bare couplings are scheme-independent for a fixed regularization, so that both these equations are valid for any renormalization prescription supplementing the higher covariant derivative regularization.
Taking into account that in the HD+MSL scheme RGFs defined in terms of the renormalized couplings coincide with the ones defined in terms of the bare couplings up to the renaming of arguments, we conclude that for standardly defined RGFs one of the NSVZ schemes is given by the HD+MSL prescription in all orders. By other words, to obtain the NSVZ equation in all loops, one should regularize a theory by higher covariant derivatives and include into renormalization constants only powers of $\ln\Lambda/\mu$ (or, equivalently, set all finite constants to 0).
As we have already mentioned, explicit calculations exactly confirm the statements discussed above even in the approximations, where the dependence on a regularization and a renormalization prescription becomes essential, see, e.g., [@Shakhmanov:2017soc; @Kazantsev:2018nbl; @Kuzmichev:2019ywn; @Aleshin:2020gec]. Also the results of this paper confirm the correctness of the expression for the three-loop $\beta$-function of a general ${\cal N}=1$ supersymmetric gauge theory with matter superfields and a simple gauge group derived in Ref. [@Kazantsev:2020kfl] from the NSVZ equation.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to express the gratitude to S. S. Aleshin, A. E. Kazantsev, M. D. Kuzmichev, N. P. Meshcheriakov, S. V. Novgorodtsev, and I. E. Shirokov for numerous useful discussions and valuable comments on the manuscript.
The research was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘BASIS’, grant No. 19-1-1-45-1.
[100]{}
M. T. Grisaru, W. Siegel and M. Rocek, Nucl. Phys. B [**159**]{} (1979) 429. K. V. Stepanyantz, Nucl. Phys. B [**909**]{} (2016) 316. V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B [**229**]{} (1983) 381. D. R. T. Jones, Phys. Lett. [**123B**]{} (1983) 45. V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Phys. Lett. [**166B**]{} (1986) 329 \[Sov. J. Nucl. Phys. [**43**]{} (1986) 294\] \[Yad. Fiz. [**43**]{} (1986) 459\]. M. A. Shifman and A. I. Vainshtein, Nucl. Phys. B [**277**]{} (1986) 456 \[Sov. Phys. JETP [**64**]{} (1986) 428\] \[Zh. Eksp. Teor. Fiz. [**91**]{} (1986) 723\]. M. A. Shifman and A. I. Vainshtein, In \*Shifman, M.A.: ITEP lectures on particle physics and field theory, vol. 2\* 485-647 \[hep-th/9902018\]. I. L. Buchbinder and K. V. Stepanyantz, Nucl. Phys. B [**883**]{} (2014) 20. M. T. Grisaru and W. Siegel, Nucl. Phys. B [**201**]{} (1982) 292 Erratum: \[Nucl. Phys. B [**206**]{} (1982) 496\]. P. S. Howe, K. S. Stelle and P. K. Townsend, Nucl. Phys. B [**236**]{} (1984) 125. I. L. Buchbinder, S. M. Kuzenko and B. A. Ovrut, Phys. Lett. B [**433**]{} (1998) 335. A. Galperin, E. Ivanov, S. Kalitzin, V. Ogievetsky and E. Sokatchev, Class. Quant. Grav. [**1**]{} (1984) 469-498 \[Corrigendum ibid. [**2**]{} (1985) 127\]. A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky and E. S. Sokatchev, “Harmonic superspace”, Cambridge, UK: Univ. Pr. (2001) 306 p. I. L. Buchbinder, N. G. Pletnev and K. V. Stepanyantz, Phys. Lett. B [**751**]{} (2015) 434. S. Mandelstam, Nucl. Phys. B [**213**]{} (1983) 149. L. Brink, O. Lindgren and B. E. W. Nilsson, Nucl. Phys. B [**212**]{} (1983) 401. M. Shifman and K. Stepanyantz, Phys. Rev. Lett. [**114**]{} (2015) no.5, 051601. M. Shifman and K. V. Stepanyantz, Phys. Rev. D [**91**]{} (2015) 105008. J. Hisano and M. A. Shifman, Phys. Rev. D [**56**]{} (1997) 5475. I. Jack and D. R. T. Jones, Phys. Lett. B [**415**]{} (1997) 383. L. V. Avdeev, D. I. Kazakov and I. N. Kondrashuk, Nucl. Phys. B [**510**]{} (1998) 289. J. Chen and M. Shifman, Phys. Rev. D [**99**]{}, no. 6, 065007 (2019). W. Siegel, Phys. Lett. [**84B**]{} (1979) 193. W. A. Bardeen, A. J. Buras, D. W. Duke and T. Muta, Phys. Rev. D [**18**]{} (1978) 3998. I. Jack, D. R. T. Jones and C. G. North, Phys. Lett. B [**386**]{} (1996) 138. I. Jack, D. R. T. Jones and C. G. North, Nucl. Phys. B [**486**]{} (1997) 479. I. Jack, D. R. T. Jones and A. Pickering, Phys. Lett. B [**435**]{} (1998) 61. R. V. Harlander, D. R. T. Jones, P. Kant, L. Mihaila and M. Steinhauser, JHEP [**0612**]{} (2006) 024. L. Mihaila, Adv. High Energy Phys. [**2013**]{} (2013) 607807. S. S. Aleshin, A. L. Kataev and K. V. Stepanyantz, JHEP [**1903**]{} (2019) 196. A. L. Kataev and K. V. Stepanyantz, Phys. Lett. B [**730**]{} (2014) 184. A. L. Kataev and K. V. Stepanyantz, Theor. Math. Phys. [**181**]{} (2014) 1531 \[Teor. Mat. Fiz. [**181**]{} (2014) 475\]. D. Kutasov and A. Schwimmer, Nucl. Phys. B [**702**]{} (2004) 369. I. O. Goriachuk, A. L. Kataev and K. V. Stepanyantz, Phys. Lett. B [**785**]{} (2018) 561. A. A. Slavnov, Nucl. Phys. B [**31**]{} (1971) 301. A. A. Slavnov, Theor.Math.Phys. [**13**]{} (1972) 1064 \[Teor. Mat. Fiz. [**13**]{} (1972) 174\]. W. Siegel, Phys. Lett. [**94B**]{} (1980) 37. V. K. Krivoshchekov, Theor. Math. Phys. [**36**]{} (1978) 745 \[Teor. Mat. Fiz. [**36**]{} (1978) 291\]. P. C. West, Nucl. Phys. B [**268**]{} (1986) 113. M. Gomes, J. R. Nascimento, A. Y. Petrov and A. J. da Silva, Phys. Lett. B [**682**]{} (2009) 229. F. S. Gama, M. Gomes, J. R. Nascimento, A. Y. Petrov and A. J. da Silva, Phys. Rev. D [**84**]{} (2011) 045001. F. S. Gama, J. R. Nascimento and A. Y. Petrov, Phys. Rev. D [**88**]{} (2013) no.6, 065029. F. S. Gama, M. Gomes, J. R. Nascimento, A. Y. Petrov and A. J. da Silva, Phys. Lett. B [**733**]{} (2014) 247. E. R. Bezerra de Mello, F. S. Gama, J. R. Nascimento and A. Y. Petrov, Phys. Rev. D [**95**]{} (2017) no.2, 025028. F. S. Gama, J. R. Nascimento, A. Y. Petrov and P. J. Porfirio, Phys. Rev. D [**96**]{} (2017) no.10, 105009. F. S. Gama, J. R. Nascimento and A. Y. Petrov, Phys. Rev. D [**101**]{} (2020) no.10, 105008. K. V. Stepanyantz, J. Phys. Conf. Ser. [**343**]{} (2012) 012115. V. Y. Shakhmanov and K. V. Stepanyantz, Nucl. Phys. B [**920**]{} (2017) 345. A. E. Kazantsev, V. Y. Shakhmanov and K. V. Stepanyantz, JHEP [**1804**]{} (2018) 130. M. D. Kuzmichev, N. P. Meshcheriakov, S. V. Novgorodtsev, I. E. Shirokov and K. V. Stepanyantz, Eur. Phys. J. C [**79**]{} (2019) no.9, 809. K. V. Stepanyantz, Nucl. Phys. B [**852**]{} (2011) 71. K. V. Stepanyantz, JHEP [**1408**]{} (2014) 096. I. V. Nartsev and K. V. Stepanyantz, JHEP [**1704**]{} (2017) 047. A. L. Kataev and K. V. Stepanyantz, Nucl. Phys. B [**875**]{} (2013) 459. S. S. Aleshin, I. O. Goriachuk, A. L. Kataev and K. V. Stepanyantz, Phys. Lett. B [**764**]{} (2017) 222. V. Y. Shakhmanov and K. V. Stepanyantz, Phys. Lett. B [**776**]{} (2018) 417. K. V. Stepanyantz, Bled Workshops Phys. [**18**]{} (2017) no.2, 197. A. L. Kataev, A. E. Kazantsev and K. V. Stepanyantz, Eur. Phys. J. C [**79**]{} (2019) no.6, 477. I. V. Nartsev and K. V. Stepanyantz, JETP Lett. [**105**]{} (2017) no.2, 69. A. L. Kataev, A. E. Kazantsev and K. V. Stepanyantz, Nucl. Phys. B [**926**]{} (2018) 295. K. V. Stepanyantz, JHEP [**1910**]{} (2019) 011. K. V. Stepanyantz, JHEP [**2001**]{} (2020) 192. A. A. Soloshenko and K. V. Stepanyantz, Theor. Math. Phys. [**140**]{} (2004) 1264 \[Teor. Mat. Fiz. [**140**]{} (2004) 437\]. A. V. Smilga and A. Vainshtein, Nucl. Phys. B [**704**]{} (2005) 445. S. S. Aleshin, A. L. Kataev and K. V. Stepanyantz, JETP Lett. [**103**]{} (2016) no.2, 77. A. I. Vainshtein, V. I. Zakharov and M. A. Shifman, JETP Lett. [**42**]{} (1985) 224 \[Pisma Zh. Eksp. Teor. Fiz. [**42**]{} (1985) 182\]. M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Phys. Lett. [**166B**]{} (1986) 334. A. E. Kazantsev and K. V. Stepanyantz, J. Exp. Theor. Phys. [**120**]{} (2015) no.4, 618 \[Zh. Eksp. Teor. Fiz. [**147**]{} (2015) no.4, 714\]. A. B. Pimenov, E. S. Shevtsova and K. V. Stepanyantz, Phys. Lett. B [**686**]{} (2010) 293. K. V. Stepanyantz, “Factorization of integrals defining the two-loop $\beta$-function for the general renormalizable N=1 SYM theory, regularized by the higher covariant derivatives, into integrals of double total derivatives,” arXiv:1108.1491 \[hep-th\]. K. Stepanyantz, “The higher covariant derivative regularization as a tool for revealing the structure of quantum corrections in supersymmetric gauge theories,” arXiv:1910.03242 \[hep-th\]. J. C. Taylor, Nucl. Phys. B [**33**]{} (1971) 436. A. A. Slavnov, Theor. Math. Phys. [**10**]{} (1972) 99 \[Teor. Mat. Fiz. [**10**]{} (1972) 153\]. K. V. Stepanyantz, Theor. Math. Phys. [**142**]{} (2005) 29 \[Teor. Mat. Fiz. [**142**]{} (2005) 37\]. S. S. Aleshin, A. E. Kazantsev, M. B. Skoptsov and K. V. Stepanyantz, JHEP [**1605**]{} (2016) 014. S. J. Gates, M. T. Grisaru, M. Rocek and W. Siegel, Front. Phys. [**58**]{} (1983) 1. P. C. West, “Introduction to supersymmetry and supergravity,” Singapore, Singapore: World Scientific (1990) 425 p.
I. L. Buchbinder and S. M. Kuzenko, “Ideas and methods of supersymmetry and supergravity: Or a walk through superspace,” Bristol, UK: IOP (1998) 656 p.
O. Piguet and K. Sibold, Nucl. Phys. B [**197**]{} (1982) 257. O. Piguet and K. Sibold, Nucl. Phys. B [**197**]{} (1982) 272. I. V. Tyutin, Yad. Fiz. [**37**]{} (1983) 761. J. W. Juer and D. Storey, Phys. Lett. [**119B**]{} (1982) 125. J. W. Juer and D. Storey, Nucl. Phys. B [**216**]{} (1983) 185. A. E. Kazantsev, M. D. Kuzmichev, N. P. Meshcheriakov, S. V. Novgorodtsev, I. E. Shirokov, M. B. Skoptsov and K. V. Stepanyantz, JHEP [**1806**]{} (2018) 020. L. D. Faddeev and A. A. Slavnov, Front. Phys. [**50**]{} (1980) 1 \[Front. Phys. (1991) 1\]. A. A. Slavnov, Theor. Math. Phys. [**33**]{} (1977) 977 \[Teor. Mat. Fiz. [**33**]{} (1977) 210\]. A. E. Kazantsev, M. B. Skoptsov and K. V. Stepanyantz, Mod. Phys. Lett. A [**32**]{} (2017) no.36, 1750194. D. Dudal, H. Verschelde and S. P. Sorella, Phys. Lett. B [**555**]{} (2003) 126. M. A. L. Capri, D. R. Granado, M. S. Guimaraes, I. F. Justo, L. Mihaila, S. P. Sorella and D. Vercauteren, Eur. Phys. J. C [**74**]{} (2014) no.4, 2844. S. S. Aleshin [*et al.*]{}, Nucl. Phys. B [**956**]{} (2020) 115020. D. R. T. Jones, Nucl. Phys. B [**87**]{} (1975) 127. A. Kazantsev and K. Stepanyantz, JHEP [**2006**]{} (2020) 108.
[^1]: For the NSVZ-like equations the situation is the same, see Refs. [@Jack:1997pa] and [@Aleshin:2019yqj] for details.
[^2]: The total and double total derivative structure of the integrands has first been noted in calculating the lowest quantum corrections in Refs. [@Soloshenko:2003nc] and [@Smilga:2004zr], respectively. For theories regularized by dimensional reduction such a structure is not valid [@Aleshin:2015qqc].
[^3]: The superdiagrams contributing to $\gamma_V$ can be compared with a certain part of the $\beta$-function only if their sum is transversal. For the sum of [*all*]{} superdiagrams this is always true due to the Slavnov–Taylor identities [@Taylor:1971ff; @Slavnov:1972fg].
[^4]: The term $-(3C_2-T(R))/2\pi$ corresponds to the one-loop contribution to the $\beta$-function, which should be calculated separately. With the higher covariant derivative regularization this has been done in [@Aleshin:2016yvj].
[^5]: For simple gauge groups it is possible to choose $R_{\mbox{\scriptsize PV}} = Adj$.
[^6]: There is an infinite set of the renormalization constants for parameters of the nonlinear renormalization. Here we explicitly write only the first one ($Z_y$).
[^7]: Similar statements were also known for some theories formulated in terms of the usual fields in the Landau gauge $\xi_0=0$ [@Dudal:2002pq; @Capri:2014jqa].
[^8]: The word “vacuum” means that they do not contain any external legs.
[^9]: The derivatives should act only to the inverse squared momentum.
[^10]: The first term is not essential in the two-loop approximation, because $d\alpha_0/d\ln\Lambda = O(\alpha_0^2)$.
[^11]: This is true only if the ratios $a=M/\Lambda$ and $a_\varphi = M_\varphi/\Lambda$ (of the Pauli–Villars masses to the dimensionful parameter in the higher derivative term) do not depend on couplings. In this paper we always assumed that this condition was satisfied.
|
---
abstract: 'We show that an Anderson Hamiltonian describing a quantum dot connected to multiple leads is integrable. A general expression for the non-linear conductance is obtained by combining the Bethe ansatz exact solution with Landauer-Büttiker theory. In the Kondo regime, a closed form expression is given for the matrix conductance at zero temperature and when all the leads are close to the symmetric point. A bias-induced splitting of the Kondo resonance is possible for three or more leads. Specifically, for $N$ leads, with each at a different chemical potential, there can be $N-1$ Kondo peaks in the conductance.'
author:
- 'Sam Young Cho$^1$, Huan-Qiang Zhou$^2$, and Ross H. McKenzie$^{1,2}$'
title: Quantum Transport and Integrability of the Anderson Model for a Quantum Dot with Multiple Leads
---
[*Introduction.*]{} Since the first prediction [@Theory88] and realization of Kondo physics in a quantum dot (QD) [@Exp98], nonequilibrium effects on the Kondo resonance due to a finite bias voltage across the dot have attracted increasing attention. In the experiments, the zero bias peak of the differential conductances has been observed as a signature of the Kondo effect on electron transport through a QD. In the unitary scattering limit, observations of perfect transmission [@Wiel00; @Ji02] provide further evidence for the Kondo effect in QDs. The nonequilibrium density of states (DOS) of the dot has been predicted [@Meir93] to exhibit a splitting of the Kondo peak due to a bias voltage applied between the source and the drain. This splitting has not been observed in transport measurements. To observe the splitting of the Kondo resonance by a finite voltage bias, an experiment with extra leads [@Sun01; @Lebanon02] has been proposed. Very recently, such a splitting was observed in an experiment [@Franceschi02] where a three-lead setup was employed.
In a conventional bulk Kondo system [@Hewson] (e.g., a magnetic impurity in a metal), there is a single chemical potential and the Kondo resonance in the DOS appears at the Fermi energy due to the formation of a singlet between the local moments of the impurity and the conduction electrons. If the impurity has available a second conduction band to form singlet states, a second Kondo resonance in the DOS might be expected to occur at the chemical potential of the second conduction band. The splitting of the Kondo resonance of a QD by the differential chemical potentials of the two leads then seems to be reasonable. However, it is not still clear why the differential conductance has only a single peak at zero bias in experiments with two leads. Thus there arises a fundamental question associated with a Kondo resonance in a system with several chemical potentials that can be fabricated in nano-scale electronic devices: why the split Kondo peaks have not been seen in two-lead systems? To help answer this question we consider a QD coupled to multiple leads. The QD is described by an Anderson model generalized to a multiple-lead one. It will be shown that the multiple-lead Anderson model is integrable and exactly solvable by a unitary transformation and the Bethe ansatz [@Kawakami; @Wiegmann; @Filyov; @Tsvelick83]. By using the exact solution, a general expression for the conductance of the $N$-lead system shows that the Kondo resonance at equilibrium is split into $N-1$ peaks by increasing the difference between the chemical potentials of the different leads. This then clearly shows why only a single peak of the conductance occurs in the two-lead system.
[*Model.*]{} We consider an Anderson model in which $N$ leads are coupled to the QD, as in Fig. 1. The leads are described under the [*unfolded*]{} formalism with fermions. Within this formalism, fermions incident on the dot $(x=0)$ from a lead $( x < 0)$ are scattered away from the dot to leads $( x > 0)$. In the continuum limit, the multiple-lead Anderson model Hamiltonian is given by $$\begin{aligned}
H \!\!&=&\!\!-i \sum^{N}_{m=1;\sigma} \int^{\infty}_{-\infty} dx
\; c^\dagger_{m\sigma}(x) \partial_x c_{m\sigma}(x)
+ \sum_{\sigma} \varepsilon_d d^\dagger_\sigma d_\sigma
\nonumber \\ && \hspace*{1.1cm}
+U n_\uparrow n_\downarrow
+\sum^{N}_{m=1;\sigma}V_m
( c^\dagger_{m\sigma}(0) d_\sigma\!\!+\!{\rm h.c.}),
\end{aligned}$$ where $n_\sigma = d^\dagger_\sigma d_\sigma$ is the number of electrons of spin $\sigma$ on the dot and $U$ is the onsite Coulomb repulsion. $c_{m\sigma}$ and $d_\sigma$ are the annihilation operators with spin $\sigma$ for electrons in the lead $m$ and the dot. $\sum_m$ is a sum over the multiple leads ($m = 1, \cdots, N$). $\varepsilon_d$ is the energy level on the dot. Here the hopping amplitudes between the dot and the lead $m$, $V_m$, are allowed to be arbitrary.
Previously, it has been shown that, for the $N=2$ case, a unitary (Bogoliubov) transformation can be used to transform the Hamiltonian to a single-lead Anderson Hamiltonian [@Konik01]. We now generalize this to the case of general $N$. To do this, one performs a unitary transformation, $ {\widetilde {\bf c}} = {\bf U}_N\ {\bf c}$, for the lead electrons, where $ {\bf c}= ( c_1, \cdots, c_{N}) $ and ${\widetilde {\bf c} } = ( {\widetilde c}_1, \cdots, {\widetilde c}_{N})$. The components of the $N \times N$ matrix ${\bf U}_N$ are a function of the hopping amplitudes, $V_m$. ${\bf U}_N$ should satisfy ${\bf U}^{\dagger}_N{\bf U}_N=I$. If (i) $\sum_{m} V_m [{\bf U}_N]_{mm'}=\sum_{m} [{\bf U}^\dagger_N]_{m'm}V_m$, and (ii) $\sum_{m} V_m [{\bf U}_N]_{mm'}=\sqrt{\Gamma}$ for $m' = 1$ and $0$ for $m' \neq 1$, one obtains the one-lead Anderson Hamiltonian and $N-1$ free fermion Hamiltonians. Then a $N\times N$ unitary matrix for the multiple leads has a form satisfying with $[{\bf U}_N]_{1m}=V_m/\Gamma$ and $\Gamma=\sum_{m} V^2_m$. For $N > 2$, actually, there are more freedoms to choose a unitary matrix. The freedoms give us different matrices for a unitary transformation acting only on $({\tilde c}_2, \cdots, {\tilde c}_N)$, but leaving ${\tilde c}_1$ invariant, which does not affect the physics.
As a consequence, the unitary transformation satisfying such conditions decomposes the multiple-lead Hamiltonian into $N$ independent sub-Hamiltonians, ${\widetilde H}_m$, as $$H = \sum_m {\widetilde H}_m,$$ where $$\begin{aligned}
{\widetilde H}_1
\!\!\! &=&\!\!\sum_{\sigma}\left[ -i \int^{\infty}_{-\infty} \; dx \;
{\widetilde c}^\dagger_{1\sigma}(x) \partial_x {\widetilde c}_{1\sigma}(x)
+ \varepsilon_d d^\dagger_\sigma d_\sigma
\right. \nonumber \\ && \left. \hspace*{1.7cm}
+ U n_\uparrow n_\downarrow
\! +\! \sqrt{\Gamma}
( {\widetilde c}^\dagger_{1\sigma}(0) d_\sigma + {\rm h.c.} ) \right]\! ,
\\
{\widetilde H}_{m}
\!\!\! &=&\!\!-i \! \sum_{\sigma}\!\! \int^{\infty}_{-\infty} \!\! dx \;
{\widetilde c}^\dagger_{m\sigma}(x)
\partial_x {\widetilde c}_{m\sigma}(x)
% \mbox{~\hspace*{5.2cm} for $ m\! \in \! [2,N]$}.
\mbox{~~for $ m\! \in \! [2,N]$}.
\end{aligned}$$ This is a generalization of the $N=2$ case treated in Ref. [@Konik01]. The transformed Hamiltonian can be solved exactly because the sub-Hamiltonian, ${\widetilde H}_1$, is the one-lead Anderson model that is exactly solvable via the Bethe ansatz [@Kawakami; @Wiegmann; @Filyov; @Tsvelick83].
[*Integrable excitations and scattering amplitudes.*]{} The scattering amplitudes of electronic excitations off the QD coupled to the $N$ leads can be calculated based on the exact solution of ${\widetilde H}_1$. In the transformed $N$ leads, the integrable excitations, $\{{\widetilde \psi}_m\}$, will scatter off the dot with some pure phase shift with spin $\sigma$, $\delta^\sigma_1(\varepsilon )$, where in particular $\delta^\sigma_m(\varepsilon)=0$ for $m \in [2,N]$. With the unfolded formalism, the scattering can be described by the relation $${\widetilde \psi}_m (x>0)= e^{i\delta^\sigma_m} {\widetilde \psi}_m(x<0).
\label{excitation}$$ Equation (\[excitation\]) leads to the scattering amplitudes $S^\sigma_{mm'}(\varepsilon)$ of electronic excitations, $\{\psi_m\}$, of energy $\varepsilon$ between leads in the multiple-lead system. Assuming the relation $\psi_m = \sum_{mm'}[{\bf U}_N]_{mm'}
{\widetilde \psi}_{m'}$, the scattering matrix is straightforwardly given by $$\begin{aligned}
S^\sigma_{mm'}(\varepsilon) = \delta_{mm'}
+ 2i \Gamma_{mm'}e^{i\frac{\delta^\sigma_1}{2}}
\sin \frac{\delta^\sigma_1}{2},
\label{scattering}
\end{aligned}$$ where $\Gamma_{mm'} =
[ {\bf U} {\bf P}_1 {\bf U}^{-1}]_{mm'}$ and ${\bf P}$ is a polarization matrix: $[{\bf P}_m]_{mm} = 1$ and other entries are zero. For $m \neq m'$, $S^\sigma_{mm'}$ is a transmission amplitude $T^\sigma_{mm'}$ from $m'$ to $m$. For $m = m'$, $S^\sigma_{mm}$ corresponds to a reflection amplitude $R^\sigma_{mm}$ from $m$ to $m$. From $\Gamma_{mm'} = \Gamma_{m'm}$, $T^\sigma_{mm'}(\varepsilon)=T^\sigma_{m'm}(\varepsilon)$ is automatically preserved.
[*Differential matrix conductance.*]{} The current and the conductance through the QD can be obtained by the Landauer-Büttiker theory [@Datta95] for quantum transport through nano-devices. To describe scattering away from the Fermi energy and calculate the differential conductance, we employ an ansatz [@Konik01] verified in Refs. [@Fendley95; @Saleur00]. The ansatz allows us to use the in-equilibrium scattering matrices to calculate the contribution to the current of any given excitation. Konik and coworkers discussed the details of the implementation of the nonequilibrium computation in Ref. [@Konik02]. With $T^\sigma_{mm'}(\varepsilon)=T^\sigma_{m'm}(\varepsilon)$, at zero temperature, the current in lead $m$ is given by $$I_m = \frac{e}{h} \sum_{m'\neq m;\sigma}
\int^{\mu_{m}}_{\mu_{m'}} d\varepsilon
\;\; \Big|T^\sigma_{mm'}(\varepsilon,\{\mu_m\})\Big|^2 ,
\label{current}$$ where $\mu_m$ is the chemical potential at the lead $m$ and $$\Big|T^\sigma_{mm'}(\varepsilon,\{\mu_m\})\Big|^2
= 4 \Gamma^2_{mm'}
\sin^2\left[\frac{1}{2} \delta^\sigma_1(\varepsilon,
\{\mu_m\})\right].
\label{transmission}$$
To determine $\delta_1$, we solve ${\widetilde H}_1$ via the Bethe ansatz for the one-lead Anderson model. The integrability of ${\widetilde H}_1$ leads to a set of quantization conditions identical to that of the one-lead Anderson model. Single particle excitations with momenta $\{k_j\}$ are identified by an appropriate basis. Scattered particle eigenstates from the dot picks up the [*bare*]{} phase $\delta(k)=-2\tan^{-1}[\Gamma/(k-\varepsilon_d)]$. Calculating two particle eigenstates makes it possible to get the scattering matrices of excitations. The scattering matrices satisfying a Yang-Baxter relationship are identical to that of the one-lead Anderson model. Then a set of $N_e$ multi-particle eigenstates carrying total spin $S_z=N_e/2-M$ should satisfy the quantization conditions [@Kawakami; @Wiegmann; @Filyov] as $$\begin{aligned}
e^{ik_j L + i\delta(k_j)} &=&
\prod^M_{\alpha=1}
\frac{g(k_j)-\lambda_\alpha+i/2}{g(k_j)-\lambda_\alpha-i/2} , \nonumber \\
%
\prod^M_{\beta=1}
\frac{\lambda_\alpha-\lambda_\beta+i}{\lambda_\alpha-\lambda_\beta-i}
&=&
-\prod^{N_e}_{j=1}
\frac{g(k_j)-\lambda_\alpha-i/2}{g(k_j)-\lambda_\alpha+i/2},
\end{aligned}$$ where $g(k)=(k-\varepsilon_d-U/2)^2/2U\Gamma$ and $M$ characterizes the spin projection of the system with the auxiliary parameters, $\{\lambda_\alpha\}$. For $\varepsilon_d > -U/2$, then, $N_e$ total momenta $k$’s form an $N_e$ particle ground state configuration. $N_e-2M$ of $N_e$ momenta $k$’s is real and $2M$ is complex via $M$ real $\lambda_\alpha$’s. The $2M$ complex momenta are given by $k^{\pm}_\alpha=x(\lambda_\alpha)\pm i y(\lambda_\alpha)$ with $x(\lambda)=U/2+\varepsilon_d-\sqrt{U\Gamma}
[\lambda+(\lambda^2+1/4)^{1/2}]^{1/2}$ and $y(\lambda)=\sqrt{U\Gamma} [-\lambda+(\lambda^2+1/4)^{1/2}]^{1/2}$.
According to Andrei’s procedure for determining the momentum, $p$, of an added electron in a periodic system of size $L$ [@Andrei82], the quantization condition of the system leads to $p = 2\pi n/L$. Contributions to the momentum come from the bulk of the system and the dot: $$p = 2\pi n /L=p_{\rm bulk}+p_d/L.$$ The dot contribution scaled by the size of the system is identified with the scattering phase of the excitation off the dot, which gives the relation between the phase and the momentum from the dot as $\delta_1 = p_d$. In adding an electron with spin $\sigma$ to the system, then, the electron scattering phase shift has two contributions from the charge, $p^{Q}$, and the spin sectors, $p^{S}$, [@Konik01] as given by $$\delta^\sigma_1 = p^\sigma_{d} = p^{Q}_{d} (k) + p^{S}_{d} (\lambda).
\label{phase}$$ The electronic scattering phase shifts are related to the density of states $\rho_d(k)$ and $\sigma_d(\lambda)$ by the equations: $$\begin{aligned}
p^Q_d(k)\! &=&\! \delta(k) \!+\! \int^{\tilde q}_q \! d\lambda
[ \theta_1(g(k)\!-\!\lambda)\! -\! 2\pi ] \sigma_d(\lambda),
\\
p^S_d(k)\! &=&\! {\tilde \delta}(k) \!+\! \int^{\tilde q}_q \! d\lambda'
[ \theta_2(g(k)\!-\!\lambda')\! -\! 2\pi ] \sigma_d(\lambda')
\nonumber \\ && \hspace*{0.6cm}
+\! \int^{B}_{-D} \! dk
[ \theta_1(\lambda\!-\!g(k))\! -\! 2\pi ]\ \rho_d(k),
\end{aligned}$$ where ${\tilde \delta}=2 {\rm Re}[\delta(x(\lambda)+iy(\lambda)]$. $q/B$ are the Fermi surfaces of the seas of $k$ and $\lambda$ excitations while ${\tilde q}$ is related to the band cutoff, $D$. Here $\theta_{1,2}$ for describing the dot momentum should be chosen to ensure that $p^Q_d(k \rightarrow -\infty)=p^S_d(\lambda \rightarrow \infty)=0$. Moreover, the dot momenta are simply related to the dot density of states: $$\begin{aligned}
\partial_k p^Q_{d}(k) = 2 \pi \rho_{d} (k),
\mbox{~and~}
\partial_\lambda p^S_{d}(\lambda) = -2 \pi \sigma_{d} (\lambda).
\end{aligned}$$ Integrating the density of states gives us the dot momenta. Consequently, the scattering phase shift is given by $$\delta^\sigma_1 = 2\pi \int^B_{-D} dk \rho_d(k)
+ 2\pi \int^{\tilde q}_{q} d\lambda' \sigma_d(\lambda').$$ This phase shift satisfies the Langreth-Friedel sum rule, $\delta^\sigma_1=2\pi n_\sigma$, relating the phase shift to the total number of electrons $n_d$ in the dot [@Langreth66].
To obtain the matrix conductance of the multiple-lead system away from the symmetric point $(\varepsilon_d-\mu_m = -U/2)$, we need to do a numerical calculation for the associated integral equations. But at the symmetric point the scattering phase shift is obtained by using an exact expression for $\rho_d(k<0)$ [@Tsvelick83] and a direct relation between the phase shifts for the electron with spin $-\sigma$ and the hole with spin $\sigma$ from a property of electron-hole transformation based on the SU(2) spin symmetry. The phase shift is given by [@Konik02] $$\delta_1(\varepsilon)\!
= \!\frac{3}{2}\pi \!
-\! \sin^{-1}\!\!\left[
\frac{4T_{K,m}^2-\pi^2(\varepsilon-\mu_m)^2}
{4T_{K,m}^2+\pi^2(\varepsilon-\mu_m)^2}
\right]\!+ C(\varepsilon),
\label{phase2}$$ where the Kondo temperature for a lead at chemical potential $\mu_m$ is $$T_{K,m} \!\!
=\!\! \sqrt{\frac{U\Gamma}{2}}
\exp\!\!\left[\frac{\pi}{2\Gamma U}
\left[(\varepsilon_d\!-\!\mu_{m})(\varepsilon_d\!-\!\mu_{m}+U)
\! - \!\Gamma^2\right]
\right].$$ Here, $C(\varepsilon)$ does not give any significant phase shift when the Kondo energy scale is much smaller than the Coulomb interaction $U$. For $|\mu_m-\mu_{m'}| \ll U$, we can assume all of the leads are at the symmetric point. This makes it possible to take into account the essence of the physics associated with the splitting of the Kondo resonance in a multiple-lead system. Then one can obtain a simple expression for the matrix conductance ($G_{mm'}=-e\partial_{\mu_{m'}}I_m$) from Eq. (\[current\]), (\[transmission\]) and (\[phase2\]). The matrix conductance in the multiple-lead Kondo-dot system is given by $$\begin{aligned}
G_{mm} \!\!\! &= &\! -\sum_{m' \neq m} G_{mm'} ,
% 4 G_0 \!\!\sum_{m'\neq m} \!\!
% \Gamma^2_{mm'} \!\!
% \left[\! 1\! +\! \frac{\pi^2}{4}\!\! \left(
% \frac{\mu_m-\mu_{m'}}{T_{K,{\rm max}[\mu_m,\mu_{m'}]}}
% \!\right)^2\!\right]^{-1} \!\!\!\!\! ,
\label{conductance1}
\\
G_{mm'\atop (m \neq m')}\!\!\!\!\! &= &\!\!\! - 4 G_0
\Gamma^2_{mm'} \!\!
\left[\! 1\! +\! \frac{\pi^2}{4}\!\! \left(
\frac{\mu_m-\mu_{m'}}{T_{K,{\rm max}[\mu_m,\mu_{m'}]}}
\!\right)^{\! 2}\!\right]^{-1} \!\!\!\!\!\!\! ,
\label{conductance2}
\end{aligned}$$ where $G_0=2e^2/h$ is the quantum of conductance, and $\Gamma_{mm'} = V_m V_{m'}/\Gamma$. This multiple-lead matrix conductance is the generalized expression of the conductance for the two-lead Kondo-dot system. It reduces to the conductance in the two-lead system [@Konik02]. For a symmetric coupling ($V_1=\cdots=V_N$) and $\mu_1 = \cdots = \mu_N$, the matrix conductance is $G_{mm}/G_0=4(N-1)/N^2$ and $G_{mm'}/G_0=-(2/N)^2$. The resultant matrix conductance agrees with that of a multi-lead quantum point-contact for free fermions [@Nayak99]. This unitary scattering limit shows the Fermi liquid nature of the multiple-lead Kondo-dot system.
Note that the multiple-lead matrix conductance in Eq. (\[conductance1\]) and (\[conductance2\]) shows clearly that a conductance peak for the transmission from $m$ to $m'$ is developed when the two chemical potentials are tuned to be equal, $\mu_m=\mu_{m'}$. As the chemical potential difference increases, the amplitude of the conductance decreases. In a $N$-lead system, if every chemical potential has a different value, the conductance $G_{mm}$ versus $\mu_m$ has a total of the $N-1$ conductance peaks, one at each of the other chemical potentials. The amplitude of the conductance $G_{mm'}$ versus $\mu_m$ has its maximum value for $\mu_m=\mu_{m'}$. The maximum values of $G_{mm'}$’s have a one-to-one correspondence to the conductance peaks of $G_{mm}$. This behavior of the conductances implies that electrons from each lead participate in screening the local moment of the dot and take part in forming a single Kondo resonance at equilibrium. Increasing the difference between the chemical potentials, the electrons from each of the $N$ leads have their own Kondo resonances with the dot. Each resonance is characterized by a Kondo temperature, $T_{K,m}$, depending on the value of the chemical potential of the lead. Since each lead creates a single lead-dot Kondo resonance, the $N$-lead system has $N$ lead-dot Kondo resonances. If the chemical potentials of two of the leads are adjusted to be equal then the two Kondo resonances corresponding to these leads merge together in $G_{mm'}$. Then this results in only a single transmission peak in the conductance $G_{mm}$. Therefore, an electron transport measurement in the two-lead system is able to capture only the single transmission peak even though there are two lead-dot Kondo resonances created by the two leads. Hence, the two-lead system is not a good probe to observe the splitting of the Kondo resonance by finite biases.
[*Three-lead and four-lead system.*]{} Before proceeding to the conclusion, we discuss the conductance for the three leads $(N=3)$ and the four leads $(N=4)$. The unitary transformation for the three-lead system is given by the unitary matrix; $${\bf U}_{3} = \frac{1}{\sqrt{\Gamma}}
\left(\begin{array}{ccc}
V_1 & V_2 & V_3 \\
V_2 & a & b \\
V_3 & b & c
\end{array}\right),$$ where $a=(-V_1V^2_2+V^2_3\sqrt{\Gamma})/\gamma$, $b=(-V_1V_2V_3-V_2V_3\sqrt{\Gamma})/\gamma$, and $c=(-V_1V^2_3+V^2_2\sqrt{\Gamma})/\gamma$ with ${\gamma}=V^2_2+V^2_3$. It can be obtained explicitly under the necessary condition we discussed above. Similarly, the unitary matrix ${\bf U}_4$ for four leads can be determined.
We plot the conductance $G_{33}$ as a function of $\mu_3$ for $N=3$ and the conductance $G_{44}$ as a function of $\mu_4$ for $N=4$ in Fig. \[fig2\] (a) and (b), respectively. When all the leads are at the same chemical potential ($\Delta\mu=0$), the amplitude of the conductance is shown to be reduced as the number of leads increases. The maximum amplitudes are $G_{33}/G_0=8/9$ and $G_{44}/G_0=3/4$. As the difference between the other chemical potentials, $\Delta\mu$, become larger than the Kondo temperature $T^0_K$ at equilibrium, the single peak at $\Delta\mu=0$ splits progressively into two and three peaks for three and four leads, respectively. Figure \[fig2\] (a) shows that for $\Delta\mu \simeq 2 T^0_K$, the amplitudes of the split peaks reduce to around half the value of that of the equilibrium Kondo peak ($\Delta\mu=0$). The suppression of the Kondo resonance is on a voltage scale $T^0_K$. This behavior agree qualitatively, but not quantitatively, with the experimental results in Ref. [@Franceschi02].
[*Summary.*]{} By using a unitary transformation and the Bethe ansatz, the multiple-lead Anderson model is shown to be integrable. A general expression for the matrix conductance from the integrability has been obtained. The conductance for the $N$-lead system shows $N-1$ split Kondo peaks located at $N-1$ different chemical potentials. This shows that a Kondo-dot system with multiple leads provides a good probe to observe the nonequilibrium effects on the Kondo resonance by a voltage bias in transport measurement.
[*Acknowledgments.*]{} This work was supported by the Australian Research Council.
L. I. Glazman and M. E. Raikh, JETP Lett. [**47**]{}, 452 (1988); T. K. Ng and P. A. Lee, Phys. Rev. Lett. [**61**]{}, 1768 (1988).
D. Goldhaber-Gordon, H. Shtrikman, D. Abush-Magder, U. Meirav, and M. A. Kastner, Nature [**391**]{}, 156 (1998); D. Goldhaber-Gordon, J. Goeres, M. A. Kastner, Hadas Shtrikman, D. Mahalu, and U. Meirav, , 5225 (1998); S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwenhoven, Science [**281**]{}, 540 (1998); J. Schmid, J. Weis, K. Eberl, and K. von Klitzing, Physica [**256B-258B**]{}, 182 (1998); F. Simmel, R. H. Blick, J. P. Kotthaus, W. Wegscheider, and M. Bichler, , 804 (1999);
W. G. van der Wiel, S. De Franceschi, T. Fujisawa, J. M. Elzerman, S. Tarucha, and L. P. Kouwenhoven, Science [**289**]{}, 2105 (2000).
Y. Ji, M. Heiblum, and H. Shtrikman, Phys. Rev. Lett. [**88**]{}, 076601 (2002).
Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett. [**70**]{}, 2601 (1993).
Q.-F. Sun and H. Guo, Phys. Rev. B [**64**]{}, 153306 (2001).
E. Lebanon and A. Schiller, Phys. Rev. B [**65**]{}, 035308 (2002).
S. Franceschi, R. Hanson, W. G. Wiel, J. M. Elzerman, J. J. Wijpkema, T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Phys. Rev. Lett. [**89**]{}, 156801 (2002).
For a review, A. C. Hewson, [*The Kondo Problem to Heavy Fermions*]{} (Cambridge University Press, Cambridge 1993).
N. Kawakami and A. Okiji, Phys. Lett. A [**86**]{}, 483 (1982).
P. Wiegmann, V. Filyov, and A. Tsvelick, JETP Lett. [**35**]{}, 77 (1982).
V. Filyov, A. Tsvelick, and P. Wiegmann, Phys. Lett. A [**89**]{}, 157 (1982).
A. Tsvelick and P. Wiegmann, Adv. Phys. [**32**]{}, 453 (1983).
R. M. Konik, H. Saleur, and A. W. W. Ludwig, Phys. Rev. Lett. [**87**]{}, 236801 (2001).
For a review, S. Datta, [*Electronic Transport in Mesoscopic Systems*]{} (Cambridge University Press, New York, 1995).
P. Fendley, A. W. W. Ludwig, and H. Saleur, Phys. Rev. B [**52**]{}, 8934 (1995).
H. Saleur and U. Weiss, Phys. Rev. B [**63**]{}, 201302 (2001).
R. M. Konik, H. Saleur, and A. W. W. Ludwig, Phys. Rev. B [**66**]{}, 125304 (2002).
N. Andrei, Phys. Lett. A [**87**]{}, 299 (1982).
D. Langreth, Phys. Rev. [**150**]{}, 216 (1966).
C. Nayak, M. P. A. Fisher, A. W. W. Ludwig, and H. H. Lin, Phys. Rev. B [**59**]{}, 15694 (1999).
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abstract: 'The relation between the quasinormal modes (QNMs) and the second order thermodynamic phase transition (SOTPT) for the Reissner-Nordström (RN) black hole is studied. It is shown that the quasinormal frequencies of the RN black hole start to get a spiral-like shape in the complex $\omega$ plane and both the real and imaginary parts become the oscillatory functions of the charge if the real part of the quasinormal frequencies arrives at its maximum at the second order phase transition point of Davies for given overtone number and angular quantum number. That is to say, we can find out the SOTPT point from the QNMs of the RN black hole. The fact shows that the quasinormal frequencies carry the thermodynamical information of the RN black hole.'
author:
- Jiliang Jing
- Qiyuan Pan
title: 'Quasinormal modes and second order thermodynamic phase transition for Reissner-Nordström black hole'
---
The QNMs of a black hole are defined as proper solutions of the perturbation equations belonging to certain complex characteristic frequencies which satisfy the boundary conditions appropriate for purely ingoing waves at the event horizon and purely outgoing waves at infinity [@Chand75]. They are entirely fixed by the structure of the background spacetime and irrelevant of the initial perturbations [@Chand75; @Andersson]. Thus, it is generally believed that QNMs carry the footprint to directly identify the existence of a black hole. Meanwhile, the study of QNMs may lead to a deeper understanding of the thermodynamical properties of black holes in loop quantum gravity [@Hod] [@Dreyer], as well as the QNMs of anti-de Sitter black holes have a direct interpretation in terms of the dual conformal field theory [@Maldacena; @Witten; @Kalyana].
On the other hand, one of important characteristics of a black hole is its thermodynamical properties. It is well known that the heat capacity of the Schwarzschild black hole is always negative and so the black hole is thermodynamical unstable. But for the RN black hole, the heat capacity is negative in some parameter region and positive in other region. Davies pointed out that the phase transition appears in black hole thermodynamics and the SOTPT takes place at the point where the heat capacity diverges [@Davies1; @Davies2; @Davies3].
Because the QNMs of a black hole are entirely fixed by the structure of the background spacetime and the SOTPT is only related to the parameters of the black hole, an interesting question is whether there are some relations between them. The aim of this paper is to study this question for the RN black hole, and we find that the quasinormal frequencies do carry the thermodynamical information of the black hole.
Assuming that the azimuthal and time dependence of the fields will be the form $e^{-i(\omega t-m\varphi)}$ and using the Newman-Penrose formalism [@Newman], we can obtain the separated equations for massless scalar, Dirac, and Rarita-Schwinger (RS) perturbations around a RN black hole [@Jing; @Khanal; @Castillo] $$\begin{aligned}
\label{w4}
&&[\Delta {\mathcal{D}}_{1-s}{\mathcal{D}}_{0}^{\dag}
+2(2s-1)i\omega r-(\lambda +2s)]\Delta^{s}R_{s} =0,~~~\nonumber \\
&&[{\mathcal{L}}_{1-s}^{\dag}{\mathcal{L}}_{s}+(\lambda
+2s)]S_{s}=0,~~\text{(s=0,~ 1/2,~ 3/2)}~~~~\\ \nonumber \\ \nonumber
\label{w5}
&&[\Delta
{\mathcal{D}}_{1+s}^{\dag}{\mathcal{D}}_{0}
+2(2s+1)i\omega r-\lambda ]R_{s} =0,~~~~~~~~~~~~~~~~\nonumber \\
&&({\mathcal{L}}_{1+s}{\mathcal{L}}_{-s}^{\dag}+\lambda )S_{s}=0,
~~~~\text{(s=0,~ -1/2,~ -3/2)}\end{aligned}$$ where $\Delta=r^2-2Mr+Q^2$, $M$ and $Q$ represent the mass and charge of the black hole, and $\lambda $ is the angular separation constant [@WS; @ET; @ET-1] $$\begin{aligned}
\label{As}
\lambda = \left\{
\begin{array}{ll} (l-s)(l+s+1),~~~~~ l=|s|,|s|+1,\cdots, \\
(j-s)(j+s+1),~~~~ j=|s|,|s|+1,\cdots,
\end{array} \right.
\end{aligned}$$ where $l$ and $j$ are the quantum number characterizing the angular distribution for the boson and fermion perturbations respectively. Introducing an usual tortoise coordinate $ dr_*=r^2/\Delta dr $ and resolving the equation in the form $ R_{s}=\Delta^{-s/2}\Psi_{s}/r,
$ we can rewrite the radial wave equations in Eqs. (\[w4\]) and (\[w5\]) as $$\begin{aligned}
\label{wave}
\frac{d^{2}\Psi_{s} }{d r_{*}^2}+[\omega ^2-V]\Psi_{s} =0,\end{aligned}$$ with $$\begin{aligned}
\label{Poten}
V= i s \omega r^2\frac{d}{d r}\frac{\Delta }{r^4}+\frac{(s+\lambda
)\Delta +\left(\frac{s}{2}\frac{d\Delta }{dr}\right)^{2}}{r^4}
+\frac{\Delta }{r^3}\frac{d}{dr} \frac{\Delta}{r^2}.\end{aligned}$$
The boundary conditions on wave function $\Psi_s$ at the horizon and infinity can be expressed as $$\begin{aligned}
\label{Bon}
\Psi_s \sim \left\{
\begin{array}{ll} (r-r_+)^{-\frac{s}{2}-\frac{i\omega}{2\kappa_{+}}} &
~~~~r\rightarrow r_+, \\
r^{-s+i\omega}e^{i\omega r} & ~~~~ r\rightarrow +\infty,
\end{array} \right.
\end{aligned}$$ where $\kappa_\pm=(r_+-r_-)/(2r^2_\pm)$ is the surface gravity on the horizons $r_\pm$. A solution to Eq. (\[wave\]) that has the desired behavior at the boundary can be written as $$\begin{aligned}
\label{expand}
\Psi_s&=&r(r-r_+)^{-\frac{s}{2}
-\frac{i\omega}{2\kappa_+}}(r-r_-)^{-1-\frac{s}{2}+2i\omega+
\frac{i\omega}{2\kappa_-}}\nonumber\\
&&\times e^{i\omega (r-r_-)}\sum_{m=0}^{\infty}
a_m\left(\frac{r-r_+}{r-r_-}\right)^m.
\end{aligned}$$ If we take $r_++r_-=1$, the sequence of the expansion coefficients $\{a_m: m=1,2,....\}$ is determined by a three-term recurrence relation staring with $a_0=1$: $$\begin{aligned}
\label{rec}
&&\alpha_0 a_1+\beta_0 a_0=0, \nonumber \\
&&\alpha_m a_{m+1}+\beta_m a_m+\gamma_m a_{m-1}=0,~~~m=1,2,....
\end{aligned}$$ The recurrence coefficient $\alpha_m$, $\beta_m$ and $\gamma_m$ are given in terms of $m$ and the black hole parameters by $$\begin{aligned}
&&\alpha_m=m^2+(C_0+1)m+C_0, \nonumber \\
&&\beta_m=-2m^2+(C_1+2)m+C_3, \nonumber \\
&&\gamma_m=m^2+(C_2-3)m+C_4-C_2+2,
\end{aligned}$$ and the intermediate constants $C_m$ are defined by $$\begin{aligned}
C_0&=&1-s-i\omega-i\omega B, \nonumber \\
C_1&=&-4+2i\omega(2+b)+2i\omega B, \nonumber \\
C_2&=&s+3-3i\omega-i\omega B, \nonumber \\
C_3&=&\omega^2(4+2b-4r_+r_-)-s-1+(2+b)i\omega\nonumber \\&&
-\lambda^2+(2\omega+i)
\omega B,
\nonumber \\
C_4&=&s+1+2\left(i \omega-s-\frac{3}{2}\right)i\omega-(2\omega+i)\omega B,
\end{aligned}$$ where $B=(r_+^2+r_-^2)/(r_+-r_-)$. The series in (\[expand\]) converges and the $r=+\infty$ boundary condition (\[Bon\]) is satisfied if, for a given $s$ and $\lambda$, the frequency $\omega$ is a root of the continued fraction equation $$\begin{aligned}
\label{ann}
&&\left[\beta_m-\frac{\alpha_{m-1}\gamma_m}{\beta_{m-1}-}
\frac{\alpha_{m-2}\gamma_{m-1}}{\beta_{m-2}-}...
\frac{\alpha_0\gamma_1}{\beta_0}\right]\nonumber \\
&&=
\left[\frac{\alpha_m\gamma_{m+1}}{\beta_{m+1}-}
\frac{\alpha_{m+1}\gamma_{m+2}}{\beta_{m+2}-}
\frac{\alpha_{m+2}\gamma_{m+3}}{\beta_{m+3}-}...\right],
~~~~~~~~~~~~~~~~~~~~~~~~(m=1,2...).
\end{aligned}$$ This leads to a simple method to find quasinormal frequencies of the RN black hole — defining a function which returns the value of the continued fraction for an initial guess at the frequency, and then use a root finding routine to find the zeros of this function in the complex $\omega$ plane. The frequency for which happens is a quasinormal frequency [@Leaver; @Leaver1].
![\[fig1\] Left four panels show trajectories in the complex $\omega$ plane of the scalar quasinormal frequencies of the RN black hole for $l=0,~ n=1,~2 $ and $l=1,~ n=3,~4 $. The other panels draw the imaginary part $Im(\omega)$ and real part $Re(\omega)$ of the quasinormal frequencies versus the charge $Q$. ](SWL0N0 "fig:")\
![\[fig1\] Left four panels show trajectories in the complex $\omega$ plane of the scalar quasinormal frequencies of the RN black hole for $l=0,~ n=1,~2 $ and $l=1,~ n=3,~4 $. The other panels draw the imaginary part $Im(\omega)$ and real part $Re(\omega)$ of the quasinormal frequencies versus the charge $Q$. ](SWL0N1 "fig:")\
![\[fig1\] Left four panels show trajectories in the complex $\omega$ plane of the scalar quasinormal frequencies of the RN black hole for $l=0,~ n=1,~2 $ and $l=1,~ n=3,~4 $. The other panels draw the imaginary part $Im(\omega)$ and real part $Re(\omega)$ of the quasinormal frequencies versus the charge $Q$. ](SWL1N3 "fig:")\
![\[fig1\] Left four panels show trajectories in the complex $\omega$ plane of the scalar quasinormal frequencies of the RN black hole for $l=0,~ n=1,~2 $ and $l=1,~ n=3,~4 $. The other panels draw the imaginary part $Im(\omega)$ and real part $Re(\omega)$ of the quasinormal frequencies versus the charge $Q$. ](SWL1N4 "fig:")\
![\[fig2\] Left four panels show trajectories in the complex $\omega$ plane of the Dirac quasinormal frequencies for $j=1/2,~ n=1,~2 $ and $j=3/2,~ n=4,~5 $. The other panels draw the imaginary part $Im(\omega)$ and real part $Re(\omega)$ of the quasinormal frequencies versus the charge $Q$.](DWL05N1 "fig:")\
![\[fig2\] Left four panels show trajectories in the complex $\omega$ plane of the Dirac quasinormal frequencies for $j=1/2,~ n=1,~2 $ and $j=3/2,~ n=4,~5 $. The other panels draw the imaginary part $Im(\omega)$ and real part $Re(\omega)$ of the quasinormal frequencies versus the charge $Q$.](DWL05N2 "fig:")\
![\[fig2\] Left four panels show trajectories in the complex $\omega$ plane of the Dirac quasinormal frequencies for $j=1/2,~ n=1,~2 $ and $j=3/2,~ n=4,~5 $. The other panels draw the imaginary part $Im(\omega)$ and real part $Re(\omega)$ of the quasinormal frequencies versus the charge $Q$.](DWL15N4 "fig:")\
![\[fig2\] Left four panels show trajectories in the complex $\omega$ plane of the Dirac quasinormal frequencies for $j=1/2,~ n=1,~2 $ and $j=3/2,~ n=4,~5 $. The other panels draw the imaginary part $Im(\omega)$ and real part $Re(\omega)$ of the quasinormal frequencies versus the charge $Q$.](DWL15N5 "fig:")\
![\[fig3\] Left four panels show trajectories in the complex $\omega$ plane of the RS quasinormal frequencies for $j=3/2,~ n=4,~5 $ and $j=5/2,~ n=8,~9 $. The other panels draw the imaginary part $Im(\omega)$ and real part $Re(\omega)$ of the quasinormal frequencies versus the charge $Q$.](RSWL1N4 "fig:")\
![\[fig3\] Left four panels show trajectories in the complex $\omega$ plane of the RS quasinormal frequencies for $j=3/2,~ n=4,~5 $ and $j=5/2,~ n=8,~9 $. The other panels draw the imaginary part $Im(\omega)$ and real part $Re(\omega)$ of the quasinormal frequencies versus the charge $Q$.](RSWL1N5 "fig:")\
![\[fig3\] Left four panels show trajectories in the complex $\omega$ plane of the RS quasinormal frequencies for $j=3/2,~ n=4,~5 $ and $j=5/2,~ n=8,~9 $. The other panels draw the imaginary part $Im(\omega)$ and real part $Re(\omega)$ of the quasinormal frequencies versus the charge $Q$.](RSWL2N8 "fig:")\
![\[fig3\] Left four panels show trajectories in the complex $\omega$ plane of the RS quasinormal frequencies for $j=3/2,~ n=4,~5 $ and $j=5/2,~ n=8,~9 $. The other panels draw the imaginary part $Im(\omega)$ and real part $Re(\omega)$ of the quasinormal frequencies versus the charge $Q$.](RSWL2N9 "fig:")\
The Figs. \[fig1\]-\[fig3\] describe the QNMs for the scalar (s=0), Dirac (s=-1/2), and RS fields (s=-3/2) obtained by the continued fraction method respectively. In Fig. \[fig1\] (or \[fig2\], \[fig3\]) left four panels show trajectories in the complex $\omega$ plane of the scalar (or Dirac, RS) quasinormal frequencies of the RN black hole with different angular quantum number and overtone number. The other panels draw the imaginary part $Im(\omega)$ and real part $Re(\omega)$ of the quasinormal frequencies versus the charge $Q$. These figures tell us that in the complex $\omega$ plane the quasinormal frequencies will get a spiral-like shape as the charge $Q$ increases to its extremal value when the overtone number equals to or exceeds a critical value $n_c$ for a fixed angular quantum number (say, $n_c=1$ with $l=0$ and $n_c=4$ with $l=1$ for the scalar field, $n_c=2$ with $j=1/2$ and $n_c=5$ with $j=3/2$ for the Dirac field, and $n_c=5$ with $j=3/2$ and $n_c=9$ with $j=5/2$ for the RS field.), and at same time both the real and imaginary parts become the oscillatory functions of the charge. The critical value of the overtone number $n_c$ increases as the angular quantum number $l$ (or $j$) increases for a given perturbation, and it also increases as the absolute value of spins $|s|$ increases for the same level of the angular quantum number (here the least level is defined as $l=0$, $j=1/2$, and $j=3/2$ for the scalar, Dirac, and RS fields respectively).
On the other hand, we know that the heat capacity $C_{Q}$ of the RN black hole is given by [@Davies1; @Davies2] $$\begin{aligned}
&&C_{Q}=T\left(\frac{\partial S}{\partial T}\right)_{Q}
=\frac{TS^{3}M}{\pi Q^{4}/4-S^{3}T^{2}},\end{aligned}$$ where $T$ is the temperature and $S$ is the entropy of the black hole. It is obvious that the singular point of the heat capacity (SPHC) occurs at $\pi Q^{4}/4-S^{3}T^{2}=0$, i.e., $Q_{sp}=\sqrt{3}
M /2 \approx 0.4330127$. Fig. \[fig5\] shows that the heat capacity is negative in the region $Q<Q_{sp}$ and positive in the region $Q>Q_{sp}$. Davies [@Davies1; @Davies2; @Davies3] showed that the SPHC is just the SOTPT point of the black hole thermodynamics.
![\[fig5\] The panel shows trajectory of $-C_Q$ versus $Q$ near the singular point of the heat capacity $Q_{sp}=0.433$.](heatP)
To find the relation between the QNMs and the SOTPT point, we present critical point $Q_{cp}$, which is the value of the charge at which the real part of the first oscillatory quasinormal frequencies arrives at its maximum, with the different critical overtone number $n_c$ for the scalar, Dirac and RS fields in Table \[table1\]. From the table we find that the SPHC $Q_{sp}$ is in good agreement with the critical point $Q_{cp}$ because the difference between $Q_{sp}$ and $Q_{cp}$ is less than $2.55\%$. Besides, for the Schwarzschild black hole there is no SPHC because its heat capacity is always negative and there is no critical point because its QNMs never get a spiral-like.
----------------------------- -------------------- -----------------
($l$, $n_c$) ( 0, 1 ) (1, 4)
$Q_{cp}$ 0.438 0.432
$\frac{|\Delta Q|}{Q_{cp}}$ 1.15% 0.23%
($j$, $n_c$) (1/2, 2 ) (3/2, 5)
$Q_{cp}$ 0.442 0.436
$\frac{|\Delta Q|}{Q_{cp}}$ 2.08% 0.69%
($j$, $n_c$) (3/2, 5) (5/2, 9)
$Q_{cp}$ 0.444 0.430
$\frac{|\Delta Q|}{Q_{cp}}$ 2.55% 0.69%
----------------------------- -------------------- -----------------
: \[table1\] The values of the charge for $Q_{cp}$.
To compare them farther more, we take $K=\frac{d\omega_I}{d\omega_R}$ as the slope of the QNMs. For $Q<Q_{cp}$, Fig. \[fig6\] shows that the QNMs have a positive slope and hence large values of $\omega_R$ correspond to large values of $\omega_I$. However, for $Q>Q_{cp}$, the slope is negative and hence large values of $\omega_R$ correspond to small values of $\omega_I$. At $Q=Q_{cp}$, the slope becomes infinity. The transition from negative to positive value of $K$ occurs via a infinite discontinuity, characteristic of a second order phase transition. The trajectories of the slope of the QNMs take the same form as that of $-C_{Q}$ versus $Q$.
![\[fig6\] The panels show trajectories of the slope $K=d
\omega_I/d\omega_R$ versus $Q$ near the critical point $Q_{cp}$ for the first oscillatory QNMs of the scalar, Dirac, and RS fields.](SPhase "fig:")\
![\[fig6\] The panels show trajectories of the slope $K=d
\omega_I/d\omega_R$ versus $Q$ near the critical point $Q_{cp}$ for the first oscillatory QNMs of the scalar, Dirac, and RS fields.](DPhase "fig:")\
![\[fig6\] The panels show trajectories of the slope $K=d
\omega_I/d\omega_R$ versus $Q$ near the critical point $Q_{cp}$ for the first oscillatory QNMs of the scalar, Dirac, and RS fields.](RSPhase "fig:")\
We all know that there are two characteristic parameters for any perturbation of a black hole background: the oscillation time scale $\tau_R=1/\omega_R$ and the damping time scale $\tau_I=1/|\omega_I|$. Near the critical point, although the increment of $\tau_I$ increases monotonously as $Q$ increases, the increment of $\tau_R$ decreases as $Q$ increases for $Q<Q_{cp}$ and it dose not change at all at $Q=Q_{cp}$, but it increases as $Q$ increases for $Q>Q_{cp}$. It is curious that the change of the increment of $\tau_R$ presented here is similar to that of the temperature of the black hole when it takes the same quality of heat. If we take a slope of the time scale as $K_\tau=d\tau_I/d\tau_R$, the trajectory of the slope takes the same form as that of $C_{Q}$ versus $Q$ because $K_\tau=-K(\omega_R/\omega_I)^2$. Above discussions show us that the critical point of the QNMs may be associated with the SOTPT point.
In summary, we study the relation between the QNMs and the SOTPT for the RN black hole and find the following results: if the real part of the quasinormal frequencies arrives at its maximum at the SOTPT point for given overtone number and angular quantum number, the QNMs will start to get a spiral-like shape in the complex $\omega$ plane, and both the real and imaginary parts will become the oscillatory functions of the charge. The QNMs will (not) take a spiral-like shape if the angular quantum number or the overtone number larger (less) than this given value. If a black hole does not possess the SOTPT point, its QNMs never take a spiral-like in the complex $\omega$ plane. Besides the fact that the critical point $Q_{cp}$ of the QNMs is in good agreement with the SOTPT point $Q_{sp}$, the transition from negative to positive value of the slope $K$ ($K_\tau$) of the QNMs (time scale) occurs via a infinite discontinuity, and the trajectories of $K$ and $K_\tau$ take the same form as that of the heat capacity. These facts show that the critical point of the QNMs may be associated with the SOTPT point and the quasinormal frequencies carry the thermodynamical information of the RN black hole.
\[[**[Acknowledgments]{}**]{}\] This work was supported by the National Natural Science Foundation of China under Grant No. 10675045; the FANEDD under Grant No. 200317; and the Hunan Provincial Natural Science Foundation of China under grant no. 07JJ3016.
[99]{}
S. Chandrasekhar and S. Detweller, Proc. R. Soc. Lond. A [**344**]{}, 441 (1975).
N. Andersson and H. Onozawa, Phys. Rev. D [**54**]{}, 7470 (1996).
S. Hod, Phys. Rev. Lett. [**81**]{}, 4293 (1998).
O. Dreyer, Phys. Rev. Lett. [**90**]{}, 081301 (2003).
J. Maldacena, Adv. Theor. Math. Phys. [**2**]{}, 231 (1998).
E. Witten, Adv. Theor. Math. Phys. [**2**]{}, 253 (1998).
S. Kalyana Rama and B. Sathiapalan, Mod. Phys. Lett. A [**14**]{}, 2635 (1999).
P. C. W. Davies, Proc. Roy. Soc. Lond. A [**353**]{}, 499 (1977).
P. C. W. Davies, Rep. Prog. Phys. [**41**]{}, 1313 (1978).
P. C. W. Davies, Class. Quant. Grav. [**6**]{}, 1909 (1989).
E. Newman and R. Penrose, J. Math. Phys. (N. Y.) [**3**]{}, 566 (1962).
Jiliang Jing, Phys. Rev. D [**71**]{}, 124006 (2005).
U. Khanal, Phys. Rev. D [**28**]{}, 1291 (1983).
G. F. Torres del Castillo and G. Silva-Ortigoza, Phys. Rev. D [**42**]{}, 4082 (1990).
W. H. Press and S. A. Teukolsky, Astrophys. J. [**185**]{}, 649 (1973).
E. T. Newman and R. Penrose, J. Math. Phys. (N. Y.) [**7**]{}, 863 (1966).
J. N. Goldberg, A. J. Macfarlane, E. T. Newman, F. Rohrlich and E. C. G. Sudarshan, J. Math. Phys. (N. Y.) [**8**]{}, 2155 (1967).
E. W. Leaver, Pro. R. Soc. Lond. A [**402**]{}, 285 (1985).
E. W. Leaver, Phys. Rev. D [**34**]{}, 384 (1986).
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---
abstract: 'We report on the erosion of flat linoleum “pebbles” under steady rotation in a slurry of abrasive grit. To quantify shape as a function of time, we develop a general method in which the pebble is photographed from multiple angles with respect to the grid of pixels in a digital camera. This reduces digitization noise, and allows the local curvature of the contour to be computed with a controllable degree of uncertainty. Several shape descriptors are then employed to follow the evolution of different initial shapes toward a circle, where abrasion halts. The results are in good quantitative agreement with a simple model, where we propose that points along the contour move radially inward in proportion to the product of the radius and the derivative of radius with respect to angle.'
author:
- 'A. E. Roth$^1$, C. M. Marques$^2$, and D. J. Durian$^1$'
bibliography:
- 'PebbleRefs.bib'
title: Abrasion of flat rotating shapes
---
Pebbles on a rocky beach or river bank are often flat, and exhibit a wide variety of smooth rounded forms. This must arise from the combined effects of the initial pebble shapes, the material properties of the pebbles, and the entire history of erosion processes. For Geology, an important issue would be to decipher this history from the observed collection of pebble shapes [@boggs]. For Physics, an important issue would be to isolate and understand the physical action of different classes of erosion processes. It is not known, for example, whether the variety of shapes in some actual set of pebbles reflects the initial conditions and the duration of an erosion process that would eventually produce perfectly circular pebbles. Another possibility is that the responsible erosion process is stochastic, giving rise to a variety of shapes for any initial conditions.
Several models for the kinetics of two-dimensional pebble erosion have recently been proposed. The simplest is a [*“polishing”*]{} model, where the normal velocity of contour points is proportional to the local curvature and is zero where the curvature is negative [@PebblePRL]. Under this action, any initial pebble shape approaches a circle in the limit of vanishing area [@grayson]. This is similar in spirit to what might be called the [*“Aristotle”*]{} model, where the velocity of contour points would be directed toward the center of mass and grow with radial distance [@aristotle]. We are aware of no actual data that are explained by either of these models. A stochastic [*“cutting”*]{} model has also been proposed, where a straight cut is made from a random contour point with a length drawn from an exponential distribution [@PebblePRL]. This model successfully captures some features of laboratory erosion of clay pebbles in a rotating tray. However it is incapable of generating concave regions of negative curvature, which exist in the laboratory experiments and which may or not be important for natural erosion processes. And more recently, an analytically tractable [*“chipping”*]{} model has been proposed, where a randomly selected corner is broken off [@RednerPRE07]. This model produces nontrivial anisotropic shapes.
Comparison of data to such models requires that shape be quantified. In Geology, shapes are often described verbally (angular, rounded, elongated, platy) or by comparison to a standardized charts [@boggs]. It is also common practice to construct dimensionless ratios from measured values of long vs intermediate vs short axes [@illenberger; @benn; @howard; @hofmann; @graham]. To better connect with the microscopic action of erosion, other shape quantifiers have been constructed in terms of the curvature [@PebblePRL] or the turning angle [@RednerPRE07] at each point along the contour. Intuitively, regions of high positive curvature are more exposed and hence subjected to faster erosion. Unfortunately, as reviewed in the Appendix of Ref. [@PebblePRE], it remains difficult to reliably measure curvature from digital images because this involves numerical computation of a second derivative.
Thus it would be useful to explore a specific erosion process with reproducible deterministic action. And it would be useful to establish reliable means for extracting curvature from digital images. Towards these ends, we conduct experiments on the abrasion of soft flat shapes by rotation in a slurry of abrasive grit. We show that the erosion is deterministic and reproducible, and gives rise to circular shapes of nonzero size. This does not correspond to either the “polishing” or “Aristotle" models, but can be described by another similarly simple evolution equation. In addition we introduce a straightforward measurement procedure in which multiple digital photographs are taken at different orientations, in order to effectively average over pixelation noise. We show that this permits the local curvature to be measured with an uncertainty that is purely statistical and of well-controlled magnitude.
Materials and methods
=====================
General
-------
![Binarizied photographs of four flat linoleum squares, prior to erosion. Note that the construction process produces some level of variability and degree of negative curvature in the contours. []{data-label="negcurv"}](negcurv){width="3.0in"}
{width="5.5in"}
In order to have a set of flat pebbles with uniform isotropic consistency, we choose linoleum tiles of thickness 1/8 inch (3.175 mm). Linoleum is a commercial floor covering made from renewable materials such as solidified linseed oil, pine rosin, ground cork dust, wood flour, and mineral fillers such as calcium carbonate. The product we chose has no backing or fibrous content. Initial shapes are formed with a standard tile cutter, and then filed down to remove surface texture. These include four squares with approximate edge lengths of 2.5 and 5 cm; a hexagon with edge length 3 cm; a triangle with edge length 6 cm; and a $2.5\times 5~{\rm cm}^2$ rectangle. Photographs of the squares are displayed in Fig. \[negcurv\], converted from grayscale to binary. Note that the tile cutter does not produce identical shapes, and that the edges all possess slight concave regions with small negative curvature. A 6 mm mounting hole is drilled through the center of each shape. An additional 3 mm fiducial marker hole is drilled 1 cm from the center, for determining the orientation of the shape in photographs.
Erosion is accomplished by rotation in a slurry of silicon carbide grit (16 mesh, McMaster-Carr product No. 4780A34) completely submerged in water. The container holding the slurry has diameter of 12.5 cm, and is filled with grit to a depth of 10 cm; water covers the slurry by a few cm. The grains have irregular shapes, an average size of $d=1.1$ mm, a polydispersity of about 50%, a density of $\rho_g=3.21$ g/cc, and a packing fraction of about 60%. The pebbles are mounted by screw and lock-washer to a vertical steel rod attached to a Barnant series 20 mixer. The pebbles are carefully lowered $H=5.5$ cm into the grit. The rotation is exclusively clockwise at a rate of 250 rpm, except for one square where the rate is 150 rpm. As erosion proceeds, linoleum debris floats to the surface of the water, where it is regularly skimmed off.
At regular intervals the pebble is removed from the grit, laid on a lightbox, and photographed from directly above with a Nikon D70 six-megapixel digital camera equipped with a Nikon AF Micro Nikkor 60mm lens. The magnification is such that pixels collect light from $0.04\times0.04~{\rm mm}^2$ regions on the pebble. Images are converted to binary, and the skeletonized contour is identified, using built-in LabVIEW commands. Example contours are shown for four shapes in Fig. \[contours\]. Note that the erosion is chiral and is faster at leading edges, in accord with the clockwise sense of rotation. Note also that the contour spacing decreases, showing that the abrasion slows down as the final circular shape is approached. One convenient feature of our choice of system is that this process comes to completion within roughly one day. Another convenient feature is that the grit is much harder than the pebbles, and does not change as the pebble erodes.
The flow of the slurry in response to pebble rotation, and its variation around the perimeter of the pebble, would be important for a first-principles model of the abrasion process. Unfortunately, however, the grit is opaque so we cannot visualize the flow very well. At the translucent wall of the container, some motion could be observed in the plane of the pebble with a height about 7 to 8 mm. The rate of this flow decreases toward zero as the pebbles become circular. The surface of the slurry always remains at rest. Several dimensionless numbers help characterize the forces at play. The first is the Reynolds number based on grain size $d$ and the speed $v$ at the perimeter, which also sets the scale for relative grain motion near the perimeter: Re=$ \rho_f v d/\eta \approx 10^3$, where $\rho_f$ and $\eta$ are the fluid density and viscosity respectively. This means that the flow of the water at the edge of the pebble and also between the surrounding grains is mildly turbulent, such that the viscosity of the water plays no major role. The corresponding Stokes number, for the ratio of grain inertial to fluid viscosity forces, is three times larger since the grains are three times denser than the water. Another important number would be the ratio of grain inertial to friction forces, which can be estimated as $\rho_g v^2 / [ \mu (\rho_g-\rho_f)gH] \approx 2$, assuming a friction coefficient $\mu$ of order one. In short, the fluid inertial, grain inertial, and friction forces are all comparable and much greater than viscous forces.
The remainder of this section concerns experimental details and is organized as follows. The following two subsections describe our multiple photograph method for eliminating systematic pixelation errors in the contour location and for calculating statistical errors. Then the final subsection reviews the shape descriptors to be employed for quantifying shape evolution by rotational abrasion and for comparing to a model in subsequent sections.
Multiple photographs
--------------------
{width="5.75in"}
![(Color online) Normalized histogram of deviation of pixel points from contour. Positive values are for points outside the contour and negative values are for points inside the contour. The blue dashed curve depicts a Gaussian with standard deviation of 0.5 pixels. []{data-label="deviationvsangle"}](deviationvsangle){width="3.00in"}
The skeletonized contour points given by analysis of a digital image is satisfactory only for computing the area and linear dimensions of the pebble. Since the points are all on a square grid, neither the number of points nor the sum of distances between adjacent points give an accurate measure of the perimeter. The difficulty is compounded for computing the unit tangent vector ${\bf T}$, and even more so for the curvature vector ${\bf K}={\rm d}{\bf T}/{\rm d}s$, where $s$ is arclength. The approach taken in Refs. [@PebblePRL; @PebblePRE] was to fit radius vs angle to a cubic polynomial, averaging over a range of acceptable fitting windows. Here we develop an alternative approach in which the pebble is photographed at multiple orientations with respect to the grid of pixels in the digital camera. For this, the lightbox on which the pebble rests is placed on a rotation stage directly under the camera. Both the camera and the stepper-motor for the stage are automated by LabVIEW to take 100 photographs at equal angle intervals over a range $0-\pi/4$. Pixelized contour points are then aligned to a common coordinate system according to the location of the mounting and fiducial holes. An example of the final cloud of raw pixel points is shown in Fig. \[cloudpoints\] at three levels of magnification, zooming from the entire contour down to the pixel scale. Note that this pebble is about 1200 pixels across, and that the alignment of the multiple images is good to the pixel scale. Also note that the pixel points cluster densely with only a little systematic structure.
The nature of the noise in the pixel points is investigated in Fig. \[deviationvsangle\] by a normalized histogram for the distance between pixel points and the estimated location of the actual contour. Note that this distribution is approximately Gaussian, and has a standard deviation close to 1/2 pixel, $\sigma_o=\ell/2$. Thus we may safely treat the pixel points as having random uncorrelated Gaussian noise. By contrast, the uncertainties in adjacent points on a single skeletonized image are highly correlated, and can lead to unknown systematic errors in the computation of the local tangent.
Polygonal contour and uncertainty
---------------------------------
![(Color online) Fractional uncertainty in curvature, Eq. (\[DK2\]), plotted vs the number $M$ of pixel points per photograph that are averaged together into a vertex point. Examples are shown for a contour of perimeter $P/\ell=5000$ pixel units photographed at $N$ different angles, as labeled. The open plus sign marks the combination of parameters used in our experiments.[]{data-label="curvuncertainty"}](curvuncertainty){width="3.00in"}
The final step in treating the data is to construct a polygonal contour with roughly equal segment lengths based on the cloud of pixel points obtained from $N$ multiple images. For this we begin by sorting all contours by angle. Then we divide one of the contours into intervals with $M$ pixel points, and average all the points in each interval to create ‘seed’ points for the vertices of the final polygon. The position of each seed point is then refined by averaging together the closest $M$ pixel points from each of the $N$ images. This process is repeated three times, which is sufficient for convergence. The final vertex points for a polygonal contour are shown in Fig. \[cloudpoints\], as solid blue circles, for the choice $M=50$. In this example, there are $N=100$ images consisting of roughly 5000 pixels; therefore, there are roughly 100 evenly-spaced vertices in the final polygonal contour, each formed by the average of $NM=5000$ pixel points. The choice $M=50$ is made so that the statistical uncertainty in the local curvature falls below the one percent level, as demonstrated next.
The statistical uncertainty of the local curvature may be estimated as follows, using the Fig. \[deviationvsangle\] result that each pixel point has a random Gaussian uncertainty of one half pixel size, $\sigma_o=\ell/2$. First note that the curvature at a vertex is $K=\theta/S$ where $\theta$ is the turning angle between adjoining straight line segments of approximate length $S=M\ell$. The uncertainty in $K$ is due entirely to turning angle uncertainty, which equals the uncertainty $\sigma_o/\sqrt{NM}$ in the vertex positions perpendicular to the contour divided by $S$. The $\sqrt{NM}$ reduction assumes that the $NM$ pixel points per vertex are all uncorrelated. Three vertices are involved in defining the bending angle, and this gives an additional factor of $\sqrt{6}$. Normalizing by the average curvature $\langle K\rangle = 2\pi/P$, where $P$ is the perimeter, gives the estimated percent uncertainty in curvature as $$\begin{aligned}
{\Delta K \over \langle K\rangle} &=& {\left(\sqrt{6} {\sigma_o\over S\sqrt{NM}} \right) \over S} {P\over 2\pi}, \label{DK1}\\
&=& {\sqrt{6}\over 4\pi} {P/\ell \over \sqrt{NM^5}}.\label{DK2}\end{aligned}$$ The term in round brackets in Eq. (\[DK1\]) is the uncertainty in turning angle, and the simplification to Eq. (\[DK2\]) was made using $\sigma_o=\ell/2$ and $S=M\ell$. Thus the curvature uncertainty scales as the number $P/\ell$ of pixel points in the skeletonized image contours divided by $\sqrt{NM^5}$ where $N$ is the number of images and $M$ is the number of pixel points per image that contribute to each vertex of the final polygonal contour.
The Eq. (\[DK2\]) result for the fractional uncertainty in curvature, $\Delta K / \langle K\rangle$, is plotted in Fig. \[curvuncertainty\] vs the number $M$ of pixel points per image that are averaged together into vertex points for the final polygonal contour. Here the value of $P/\ell$ was taken as 5000, which is the typical number of pixel points in skeletonized contours for a compact pebble that fills the field of view of a six megapixel digital camera such as ours. For only one image, $N=1$, a window size of $M\approx20$ is needed for $\Delta K$ to be smaller than $\langle K\rangle$; this explains the difficulties and pains taken to deduce the curvature from polynomial fits in Refs. [@PebblePRL; @PebblePRE]. For $N=100$ images, the curvature uncertainty falls below the one percent level for $M=50$, as denoted in Fig. \[curvuncertainty\] by a large open plus sign. This corresponds to the experiments reported here, as illustrated by the solid blue vertex points in the example of Fig. \[cloudpoints\]. While a larger choice for $M$ would reduce the uncertainty further, it would give fewer than 100 points in the final polygonal contour and the resulting straight-line segments would eventually begin to deviate from the cloud of pixel points. Further reduction in curvature uncertainty could also be obtained by increasing the number $N$ of images taken. To obtain an independent sampling of the contour against the grid of pixels, the minimum rotation increment between successive images should cause each contour point to move by at least one pixel; furthermore the maximum total rotation should be $\pi/4$. Therefore, a hard upper limit on $N$ would be one-eight $P/\ell$. Our choice of $N=100$ is large enough for good statistics, but safely below this limit. The statistical uncertainty in other quantities, such as perimeter and area, could also be estimated; these will be significantly less than $\Delta K/ \langle K\rangle$ since curvature computation involves differentiation.
Shape descriptors
-----------------
![(Color online) Dimensionless shape descriptors vs number of rotations, for the four different square pebbles pictured in Fig. . Consistent with evolution toward the largest inscribed circle, the normalized perimeter and area both approach $\pi/4$, the caliper aspect ratio $C/A$ and compactivity both approach 1, and the angularity, concavity, and width $\sigma / \langle K \rangle$ of the curvature distribution all decrease toward zero. []{data-label="squares"}](squares){width="3.00in"}
The concept of “shape” is somewhat nebulous and subjective. We choose to quantify it using several different descriptors, all of which are demonstrated in Fig. \[squares\] showing evolution vs number of rotations in the abrasive slurry of grit for the four linoleum squares pictured in Fig. \[negcurv\]. The first two shape descriptors are simply the perimeter $P$ and the area $A$, both normalized by their initial values. The second two are the caliper aspect ratio $C/A$ and the compactivity. The caliper aspect ratio is the ratio of the largest to smallest values measured by a caliper as the pebble is rotated. The compactivity is a standard measure of circularity, equal to $P^{2} / (4\pi A)$. Note in Fig. \[squares\] that all four of these measures are consistent with the pebble evolving from a square to the largest inscribed circle, for which the normalized perimeter and area both decay from 1 to $\pi/4$, the caliper aspect ratio decays from $\sqrt{2}$ to 1, and the compactivity decays from $4/\pi$ to 1.
The remaining three shape descriptors shown in Fig. \[squares\] are all based on the curvature, measured at each vertex of the polygon as the turning angle per segment length. The simplest is the “angularity”, which we define as the fraction of the perimeter with negative curvature. This quantifies a similar notion found in textbooks [@boggs]. Next is the “compactivity”, which is a standard quantity defined as the difference in area between the convex hull [@ConvexHull] and the actual shape, divided by the area of the actual shape. Both the angularity and the concavity are zero for a shape that is purely convex. And last is the width $\sigma/\langle K\rangle$ of the curvature distribution around the perimeter, divided by the average curvature. This quantity, along with the cumulative distribution function of the curvature, were used in Refs. [@PebblePRL; @PebblePRE]. For the four squares, the initial angularity is large but the compactivity is small, consistent with the small wavy imperfections and regions of slight negative curvature seen in the images of Fig. \[negcurv\]. Note that the angularity, the concavity, and the width of the curvature distribution all decrease toward zero as the the squares abrade into circles. Note also that of all the shape descriptors, $\sigma/\langle K\rangle$ is farthest from its asymptotic value when the experiment was stopped; the curvature distribution is thus most sensitive in detecting the unabraded flat regions seen by eye in the image sequences of Fig. \[contours\].
Shape Evolution
===============
{width="6.0in"}
Using the above procedures, we now analyze image data in terms of shape descriptors in order to quantify the evolution of the initial shapes toward final circular shapes. The qualitative evolution was already seen in Fig. \[contours\] for different initial shapes, and the quantitative evolution was already seen in Fig. \[squares\] for four squares. The latter includes two nominally two-inch squares rotated at 250 rpm. Comparison of their respective shape descriptors shows a fair degree of reproducibility, both in initial shape details and also in evolution. Fig. \[squares\] also includes a nominal two-inch square rotated more slowly, at 150 rpm. All its shape descriptors agree reasonably well with those for the faster abrasion, when plotted vs number of rotations rather than vs time. The same holds for a nominal one-inch square rotated at 250 rpm, though in this case the initial shape is closer to a perfect square and the perimeter, area, and caliper aspect ratio all approach their asymptotic values a bit more slowly than the other shape descriptors. Altogether, these observations show that abrasion by rotation in a slurry of grit is essentially independent of size and rate, and hence is controlled by geometry and materials properties alone.
In Fig. \[stats\] we display the evolution of all the shape descriptors for the square, hexagon, rectangle, and triangle, whose contour sequences are depicted Fig \[contours\]. For all four shapes, the top plots in Fig. \[stats\] show the normalized area and perimeter, plus the asymptotic values for the largest inscribed circle; the middle plots show the caliper aspect ratio and compactivity, which both asymptote to one; the bottom plots show the width of the curvature distribution, the angularity, and the concavity, which all asymptote to zero for the largest inscribed circle. Note that initial shapes that are closer to a circle decay more rapidly toward the final shape. In particular, the $1/e$ decay constants for the normalized areas and perimeters are approximately 30K revolutions for the hexagon, 60K revolutions for the square, 90K revolutions for the triangle, and 160K revolutions for the rectangle.
Rotational Abrasion Model
=========================
![(Color online) Schematic illustration of a geometric “cutting” model. Due to rotational motion, material is removed by cuts normal to the radial direction: $|{\rm d}r| = r(\theta)-r(\theta-{\rm d}\theta)\cos({\rm d}\theta)$. []{data-label="cutting"}](cutting){width="2.00in"}
In this final section we attempt to model the abrasion processes and compare with quantitative shape data. Since the abrasion is due to rotation, we seek the rate of change of the radial coordinates $r$ of the vertex points. For the first ingredient, in accord with Aristotle [@aristotle], we suppose that the erosion is faster for points farther from the rotation axis, in proportional to a power of the tangential speed $(\omega r)^\alpha$. For the second ingredient, we consider the extent to which a segment moves into the slurry. This is determined by the magnitude and sign of the derivative of radius vs angle, ${\rm d}r/{\rm d}\theta$. If zero or negative, there is no abrasion since the segment moves parallel to or away from the slurry. The greater the positive magnitude, the more the segment penetrates into the slurry during rotation. Altogether, we thus propose the rate of change of vertex radii to be $$\frac{{\rm d}r}{{\rm d}t} \propto \left\{ \begin{array}{ll} -r^{\alpha} (\frac{{\rm d}r}{{\rm d}\theta})^{\beta} &\quad {\rm d}r/{\rm d}\theta > 0 \\ 0 &\quad \textrm{otherwise} \end{array} \right.
\label{model}$$ For any positive values of the powers $\alpha$ and $\beta$ this model gives abrasion that halts as the shape approaches a circle, where the radius is constant independent of $\theta$ around the entire contour. Note, however, that this model becomes unphysical for pebbles where $r(\theta)$ is not single-valued. For shapes far from a circle, where $r$ varies greatly with $\theta$, higher-order derivatives as well as non-local effects could become important.
The evolution of a given set $\{ r_i \}$ of vertex radii under Eq. (\[model\]) may be found by finite differencing as follows. At each time step, points with $r_i>r_{i-1}$ are incremented by $${\rm d}r_i = -\Delta r_{max} \left( { r_i \over \Delta r_{max} } \right)^\alpha
\left( {r_i - r_{i-1} \over \Delta r_{max}} {\Delta\theta_{min}\over \theta_i-\theta_{i-1}} \right)^\beta,
\label{dr}$$ where $\Delta r_{max}$ is the largest difference $r_i-r_{i-1}$, and $\Delta\theta_{min}$ is the smallest difference $\theta_i-\theta_{i-1}$. This corresponds to a variable time step of $${\rm d}t \propto \Delta r_{max} \left( { 1 \over \Delta r_{max} } \right)^\alpha
\left( {\Delta\theta_{min}\over \Delta r_{max} } \right)^\beta,
\label{dt}$$ so that the ratio of Eq. (\[dr\]) to (\[dt\]) gives Eq. (\[model\]). This time step is sufficiently small by construction, as confirmed by repeating with even smaller time steps. While the model is not linear, analytic solution has been achieved; see companion paper [@Bryan].
The pebble evolution given by Eq. (\[model\]) for the simplest choice of $\alpha=1$ and $\beta=1$ is depicted qualitatively by contours in Figs. \[contours\] and quantitatively by the shape descriptors in Fig. \[stats\]. In these figures the agreement with actual data for the hexagon, square, and triangle is very good. For the rectangle, the agreement is satisfactory at early stages but becomes less so at later times. On the other hand, good agreement is found if the model is initiated with a later-stage contour that is more compact. For all four shapes the [*same*]{} proportionality constant was used in Eq. (\[model\]), as determined by matching the $1/e$ decay constant for the hexagon. We note that similar degrees of agreement are found by fixing $\beta=1$ and taking $\alpha$ as 1/2, 1, 2, or 3; thus the model is relatively insensitive to the value of $\alpha$, which we thus take as 1 for simplicity. By contrast, poor agreement is found by fixing $\alpha=1$ and taking $\beta$ as 1/2 or 2.
The observations $\alpha=\beta=1$ can be understood as follows in terms of a geometric cutting model. Given two consecutive angles $\theta-d\theta$, $\theta$ and the associated radii values $r(\theta-d\theta,t)$, $r(\theta,t)$ at time $t$, one can compute the time evolution of $r(\theta,t)$ under a microscopic cut of the profile. In our case of rotating pebbles, the cutting forces act normally to the radial direction, as displayed in Fig. \[cutting\]. One thus has ${\rm d}r = r(\theta,t+{\rm d}t) - r(\theta,t) = - [r(\theta,t) - r(\theta-{\rm d}\theta,t) \cos({\rm d}\theta)]$, and taking the limit of continuous variables gives ${\rm d}r/{\rm d}t= - w\ {\rm d}r/{\rm d}\theta$, where $w$ is the fraction of angle removed by unit time. Note that the dependence of ${\rm d}r/{\rm d}t$ on ${\rm d}r/{\rm d}\theta$, is a direct consequence of the assumed tangential orientation of the cuts: a value of $\beta=1$ is imposed in our experiments by the rotation geometry. The parameter $w$ carries information on the length and frequency of each successive microscopic cut, which is a function, for a given material and abrasion agent, of the tangential velocity only. One would thus expect $w$ to be proportional to the radius $r$, compatible with a value $\alpha=1$ in the model presented above.
Conclusion
==========
In summary we have developed a method for measuring the contour of flat pebbles using multiple photographs from different angles. This method produces accurate contours, and allows curvature to be deduced with a known degree of statistical uncertainty without systematic error. It is our hope that this general procedure will be broadly applicable to research involving shape quantification, in the field and in the lab. Using this advance, we have explored an erosion process where abrasion is caused by steady rotation in a slurry of grit. By comparing different size squares and different rotation speeds, we found that the sequence of shapes evolves deterministically toward the largest inscribed circle. By comparing different initial shapes, we have found that those closest to a circle approach the limiting shape more rapidly. We have successfully modeled this behavior quantitatively with a simple differential equation, where contour points move radially inward in proportion to radius and the derivative of radius with respect to angle. This model is different from both the deterministic “polishing” and “Aristotle” models, and is the only deterministic model of which we are aware that accounts for actual data. It is our hope that these models may serve as a starting point for future theories of stochastic erosion, perhaps by the addition of a noise term, in order to compare with natural erosion processes.
We thank B.G. Chen for helpful conversations. A.E.R. thanks Roy and Diana Vagelos for their Science Challenge Award undergraduate scholarship. This work was supported by the National Science Foundation through grant DMR-0704147.
|
---
abstract: 'Due to non-homogeneous mass distribution and non-uniform velocity rate inside the Sun, the solar outer shape is distorted in latitude. In this paper, we analyze the consequences of a temporal change in this figure on the luminosity. To do so, we use the Total Solar Irradiance (TSI) as an indicator of luminosity. Considering that most of the authors have explained the largest part of the TSI modulation with magnetic network (spots and faculae) but not the whole, we could set constraints on radius and effective temperature variations. Our best fit of modelled to observed irradiance gives $dT$ = 1.2 $K$ at $dR$ = 10 mas. However computations show that the amplitude of solar irradiance modulation is very sensitive to photospheric temperature variations. In order to understand discrepancies between our best fit and recent observations of Livingston et al. (2005), showing no effective surface temperature variation during the solar cycle, we investigated small effective temperature variation in irradiance modeling. We emphasized a phase-shift (correlated or anticorrelated radius and irradiance variations) in the ($dR$, $dT$)–parameter plane. We further obtained an upper limit on the amplitude of cyclic solar radius variations between 3.87 and 5.83 km, deduced from the gravitational energy variations. Our estimate is consistent with both observations of the helioseismic radius through the analysis of $f$-mode frequencies and observations of the basal photospheric temperature at Kitt Peak. Finally, we suggest a mechanism to explain faint changes in the solar shape due to variation of magnetic pressure which modifies the granules size. This mechanism is supported by an estimate of the asphericity-luminosity parameter, [**[*w* ]{}**]{} = -7.61 $10^{-3}$, which implies an effectiveness of convective heat transfer only in very outer layers of the Sun.'
address:
- 'Observatoire de la Côte d’Azur, GEMINI Dpt., and UNSA University (Fizeau Dpt.), Av. Copernic, 06130 Grasse, France, email to: nayyer.fazel@obs-azur.fr and jean-pierre.rozelot@obs-azur.fr'
- 'University of Tabriz, Faculty of Physics, Dept. of Theoretical Physics and Astrophysics, Tabriz, Iran, email to: a-adjab@tabrizu.ac.ir'
- 'Laboratoire AIM, CEA/DSM, CNRS, Université Paris Diderot, DAPNIA/SAp, 91191 Gif sur Yvette cedex, France, email to:sandrine.lefebvre@cea.fr'
- 'Previously at: Observatoire de la Côte d’Azur, ARTEMIS Dpt., Av. Copernic, 06130 Grasse, France; Now at: Observatoire Royal de Belgique, Dpt. 1, 3 Av. Circulaire, 1180 Bruxelles, Belgique, email to: sophie.pireaux@oma.be'
author:
- 'Z. Fazel'
- 'J.P. Rozelot'
- 'S. Lefebvre'
- 'A. Ajabshirizadeh'
- 'S. Pireaux'
title: Solar gravitational energy and luminosity variations
---
and
Sun: characteristic and properties, 96.60.–j; helioseismology, 96.60.Ly; radiation (irradiance), 92.60.Vb; solar magnetism, 96.60.Hv.
Introduction
============
If to first order the Sun may be considered as a perfect sphere, it is clear that due to its axial rotation, the final outer shape will be a spheroid. Moreover, the distribution of the rotation velocity being far from uniform both at the surface and in depth, this final figure will be more complex. Although the resulting asphericities are very small, some open questions which remain are: to know if the passage from a sphere to a distorted shape will affect the luminosity, and if so, to quantify this effect. The first point has been partially studied in Rozelot $\&$ Lefebvre (2003) and in Rozelot et al. (2004). The second point was first addressed in Fazel et al. (2005) or Lefebvre et al. (2005). The present paper shows how irradiance and temperature observations allow us to put strong upper limits on radius variations. We use the TSI as an indicator of solar luminosity. Indeed as luminosity changes, so does the basic level of the TSI, which is additionally modulated by surface magnetic activity (spots, faculae, and network). This is not a minor question as the TSI variation is often claimed to be of magnetic origin alone. Mechanisms which may produce changes in irradiance have been discussed since years, but we are still unable to propose a full comprehensive model. As pointed out by Kuhn (2004), two different processes are proposed. One involves surface effects (see for instance Krivova et al. 2003), and the other is due to a complex heat transport function from the tachocline to the surface, including global properties, mainly magnetic field, temperature and radius (Sofia, 2004). Models based on the assumption that the irradiance variations on time-scales longer than a day are entirely and uniquely caused by changes in surface magnetism are rather successful (Krivova and Solanki, 2005), as correlative functions between observed and modelled data show an agreement of [$_{\verb ~ } $]{} 90-94 %. However, the main observations which have not yet been reproduced by these models are brightness changes measured by limb photometry (Kuhn et al., 1988; Kuhn and Libbrecht, 1991). Furthermore, the recent SoHO/MDI experiment has proved that exceedingly small solar shape fluctuations are measurable from outside our atmosphere (Emilio et al., 2007). Accordingly, efforts should be made to use these additional observations to better constrain solar model parameters (radius, temperature) and possibly the proportion of irradiance changes produced by surface magnetism. We think that there is still room for improvements. This paper is an attempt to clarify if some of the $6-10\%$ of total solar irradiance left unmodelled by surface magnetism could be of other origin: from this point of view the variability of the global distorted shape of the Sun must be explored.
In the following section, we will show how variations of the distorted outer shape of the Sun contribute to a fraction of TSI variations, assuming the main part of TSI variations being modelled by magnetic mechanisms. We will also emphasize the key role of surface effective temperature. In Section 3, we will illustrate the lack of consensus between present observations of solar radius variations (apparent radius) from the point of view of amplitude and phase with respect to the solar cycle. Moreover, there exist discrepancies between observations and theoretical models regarding such variations. Hence new observations (especially space–dedicated missions) are needed. In Section 4, we will explain how variations of the gravitational energy in the upper layers of the convective zone may imply solar radius variations. According to the observed amplitude of irradiance variations, we set an upper bound, of a few kilometers only, on solar shape changes. This last model shows that solar radius variations are anticorrelated with irradiance variations during the solar cycle. We will then provide additional information on the localization of luminosity variations by computing the asphericity-luminosity parameter ([**[*w*]{}**]{}). In Section 5, we suggest a mechanism to describe the connection between solar radius and magnetic activity. Finally, in Section 6, we present our conclusions.
Solar radius variations and luminosity changes
================================================
The “outer shape" of the Sun must be defined: the Sun has an extended atmosphere and it is not so simple to determine the upper limit of its photosphere. One of the most simple approach is to define this shape as an equipotential surface with respect to the total potential (gravitational and rotational). But, a contrario, if this definition has a physical meaning, the method to measure the true radius of the Sun, whether from space or from the ground, is unclear. The observed solar radius, which is apparent, may be different from the theoretical radius, whatever the definition of the latter is (see section \[solarradius\]). Moreover, it is expected from the above definition that the radius, $R$, is a function of latitude ($\theta$), both from an observational point of view (Rozelot et al. 2003, Lefebvre et al. 2004) and a theoretical one (Armstrong and Kuhn, 1999, Lefebvre and Rozelot, 2004). That is, at a constant pressure $p$ : $$R(\theta){\mid}_{p} = R_{sp} \left[1 + \sum_{n,~even} c_{n} P_n(\theta) \right]
\label{rayonvecteur}$$ where $R_{sp}$ is the radius of the best sphere fitting both polar ($R_{pol}$) and equatorial ($R_{eq}$) radii $\left( = \sqrt [3] {R_{eq}^2 R_{pol}}~ \right)$, $c_{n}$ are the shape coefficients (related to “asphericities") and $P_n(\theta)$ are the Legendre polynomials of degree $n$ ($n$ being even due to axial-symmetry). We need to compute the solar surface area $A$, corresponding to Eq. \[rayonvecteur\]: $$A = 4\pi \int^{\pi/2}_{0} R(\theta) \left[1+\left(\frac{dR \,(\theta)}{d\theta}\right)^2\right]^\frac{1}{2} d \theta.
\label{surface}$$ Armstrong and Kuhn (1999) or Rozelot et al. (2004) provided estimates of the shape coefficients. The best available values are $c_2$ $\in$ \[$-2$$\times$$10^{-6}$, $-1$$\times$$10^{-5}$\] and $c_4$ $\in$ \[$6$$\times$$10^{-7}$, $1$$\times$$10^{-6}$\]. For convenience, we express these results in fractional parts of the best sphere $A_{sp}$ = 6.087 $\times$$10^{+18}$ $m^{2}$ which corresponds to the radius $R_{sp}$= 6.959892$\times$$10^{8}$ $m$. Computations were carried up to $n$ = 4, leading to $dA$($c_2$,$c_4$)/$A_{sp}$ $\in$ \[$1.82$$\times$$10^{-6}$, $6.37$$\times$$10^{-6}$\], where the minimum corresponds to the lower bound of $c_2$ and $c_4$ given above, while the maximum corresponds to their upper bound. Those values can be compared to the ones deduced from an ellipsoid[^1] of radii $R_{eq}$ = $a$ and $R_{pol}$ = $b$ with $R_{eq}$ = 6.959918$\times$$10^{8}$ $m$ and $R_{pol}$ = 6.959844$\times$$10^{8}$ $m$, when $dR$ (= $da$ = $db$) varies from 10 mas to 200 mas (the choice of these two values will be explained later; see also Rozelot and Lefebvre 2003): $dA/A_{ell}$ $\in$ \[$3.08$$\times$$10^{-6}$, $6.16$$\times$$10^{-5}$\].
Let us call $F_{r}$, the radial component of the energy flux vector **F**. In the two-dimensional case, the luminosity, $L$, depends on $\theta$ : $$L = 2 \pi \int^{0}_{\pi} r^2 F_{r}(r, \theta, t) \sin\theta d(\theta)
\label{lumi}$$
We start from the suggestion previously made by Sofia and Endal (1980), that changes in the solar luminosity (L) might be accompanied by a change in radius. In order to check the influence of (tiny) solar radius variations on the luminosity, we use the Eddington approximation in Eq. \[lumi\] which leads to $dL/ L$ = 4$dT/T$ + $dA/A$ (Li et al., 2005), where $T$ is the effective temperature and $A$ is computed through Eq. \[surface\]. We are aware that the Sun does not radiate like a black body. If this model is appropriate for the infra-red part of the spectrum or almost true for the visible, by contrast the far UV part departs from it. However, our objective is not to provide a fully comprehensive model of $L(R)$, but to illustrate the effects of observed solar radius variations on global solar parameters such as the luminosity. In this sense this preliminary approximation used is a good indicator: the results obtained may only be illustrative, but are promising. It is then straightforward to express $dL/L$, either in the case of an ellipsoidal surface (deriving Eq. \[aireellipsoide\], as a function of the parameter $dR$ assuming $dR_{eq}$ = $dR_{pol}$ = $dR$), or in the case of a distorted shape (Eq. \[rayonvecteur\] with the time-dependent shape coefficients $c_{n}(t)$). Using the Total Solar Irradiance, *I*, as an indicator of luminosity ($dI/I \propto dL/L$), the modelled irradiance, can now be directly compared to observation data, two parameters being involved: the effective temperature, $T$, and the shape variations, $dR$. We used the irradiance composite dataset updated to October 1, 2003 for which the composite method was established by Fröhlich and Lean (1998)[^2]. We investigated both the ellipsoidal and distorted shape cases. However, the distorted shape leads to results comparable to the ellipsoid ones (see also Lefebvre and Rozelot, 2003, section 3.2). Hence, we present here only the latter results. In the case of an ellipsoid, the irradiance temporal variations will be reproduced by a variation $dR$ in the range \[10, 200\] mas, $dR$ = 0 being the case of a sphere of radius[^3] $R_{\odot}$. The choice of the upper limit (200 mas) is given hereafter. Alternatively, we can adjust the observed $dI/I$ datat to an irradiance model of mean value $I_{0}$, with a temporal sinusoidal variation of period $P$, equal to the solar cycle one, and phase $\phi$: $$I_{model} = I_{0} + sin\left( 2\pi t/P + \phi\right) dI .
\label{irradiance}$$ The best fit of the data by $I_{model}$ gives $P$ = 10.09 yrs and $\phi$ = 1.026 rad. Fig. \[irradiancessa\] shows the observed irradiance together with the $I_{model}$ best fit and the first component ([*RC1*, i.e. the trend)]{} in the Singular Spectrum Analysis (SSA) [^4]. [*RC1*]{} represents the first Component in the Reconstruction of the signal. The [*RC1*]{} fit is $\chi^2$ = 0.76, better than the sinusoidal $L_{model}$ fit for which $\chi^2$ = 1.17. Four other curves are shown: the computed irradiance through Eq. \[irradiance\] for a solar ellipsoidal surface (Eq. \[aireellipsoide\]) with different ($dR$, $dT$). Computations for an irregular solar shape (Eqs. \[rayonvecteur\] and \[surface\]) lead to similar results.
![Total irradiance variations with time. This figure shows the observed composite irradiance versus time (called IR, dots), according to dataset updated to 01/10/2003 (Fröhlich and Lean, 1998); the first component [*RC1*]{} in the Singular Spectrum Analysis (trend); the best sinusoidal curve fit to the observed composite data with $P$= 10.09 yrs and $\phi$= 1.026 rad; and four sinusoidal models with different appropriate pairs of \[$T$ (in K), $R$ (in mas)\], as indicated in the right box. []{data-label="irradiancessa"}](irradiance_020605.eps){width="12.3cm" height="6.4cm"}
Computed irradiance is very sensitive to the effective surface temperature. Two main results appear: (1) Observed irradiance variations can be reproduced with $dR$ = 200 mas and $dT$ $\approx$ 2$K$, but such a large radius change is rather unlikely, leaving to no involvement of the magnetic field; (2) an effective surface temperature variation amplitude $dT$ = 5$K$, whatever $dR$ is, also matches the observed irradiance variations, but is unlikely too (for the same reason). Hence, in order to quantitatively appreciate the influence of the pair \[dR, dT\], we computed, inside the limits \[0, 200\] (mas) for the radius and \[0,5\] (K) for the temperature, the residuals obtained between the first component in the Singular Spectrum Analysis, [*RC1*]{} and our simplified model (for each data point and over nearly two solar cycles). A minimum occured when dT = 1.2 K, for dR = 10 mas, as illustrated in Fig. \[irr\_dT\], which is, among all the figures obtained, that for which the lowest minimum take place (giving thus the best fit; in other words, dR = 10 mas is the [*lowest minimum*]{} for all dT).
A variation of the effective temperature $dT$ = 1.2 $K$ over nearly two solar cycles is close to that obtained by Gray and Livingston (1997) and Caccin et al. (2002) using the ratios of spectral line depths as indicators of the stellar effective temperature. They showed that the solar effective temperature varies systematically during the activity cycle with an amplitude modulation of 1.5 $K$ $\pm$ 0.2 $K$. However, monitoring the spectrum of the quiet atmosphere at the center of the solar disk during thirty years at Kitt Peak, Livingston and Wallace (2003) and Livingston et al. (2005) have shown an immutable basal photosphere temperature within the observational accuracy.
We conclude that our fits of modelled irradiance variations (numerical integration through Eqs. \[surface\] and \[irradiance\]) to observations should be refined. Thus, we further investigated small solar surface effective temperature variations ($dT$ $\in$ \[0,1.5\] $K$) in irradiance modeling in order to understand the discrepancies between our best fit $dT$ = 1.2 $K$ at $dR$ = 10 mas, and the latest observations at Kitt Peak showing $dT$ $\approx$ 0. This yields an unexpected result. For small values, the phase of irradiance variations with respect to radius ones reverses when crossing the curve plotted in the ($dR$, $dT$)-plane given by $$dT_{critical} = 5. 10^{-8} dR^2 + 4.10^{-4} dR + 0.0005
\label{phase}$$ where $dT$ is in $K$ and $dR$ in mas. This curve distinguishes between correlated (above the $dT_{critical}$ curve) and anticorrelated (below the $dT_{critical}$ curve) solar radius variations with irradiance variations. Consequently, a precise knowledge of $dT$ over the solar cycle is crucial.
In this section, we used the interval $dR$ $\in$ \[0, 200\] mas to model variations of the irradiance. The lower bound corresponds to a spherical Sun and the upper bound to the value necessary to model all the irradiance variations with only solar radius variations. Those two bounds are unrealistic cases. With respect to the latter interval, $dT_{critical}$ belongs to \[0, 0.082\]. Hence, we understand the sensitivity of irradiance modeling to very small temperature variations. For example, if observations show that $dT$ $\approx$ 0 with sufficiently small error bars, the Sun is in a state where its radius variations are anticorrelated with irradiance variations (below the $dT_{critical}$ curve). Since, observations do show that irradiance variations are correlated with the solar activity cycle, we can conclude that solar radius variations are anticorrelated with the solar cycle within the framework of the assumption $dT$ $\approx$ 0 (or, in any case, dT is lower than 0.082 $K$).
![Computed residuals between the first component [*RC1*]{} (SSA decomposition) of the observed irradiance and the computed irradiance (see Fig. \[irradiancessa\]), according to different $dT$. This plot is obtained for $dR$ = 10 mas. The best fit occurs for $dT$ = 1.2 $K$ (other $dR$ leads to larger residuals). []{data-label="irr_dT"}](deviation_1.eps){width="7.0cm" height="5.5cm"}
Note that the solar subsurface is organized in thin layers (Godier & Rozelot, 2001) and that changes in these layers have been explored through helioseismology $f$-mode frequencies over the last 9 years. Indeed, Lefebvre & Kosovishev (2005) and Lefebvre et al. (2007) report a variability of the “helioseismic" radius in antiphase with solar activity, the strongest variations of the stratification being just below the surface (around 0.995 $R_{\odot}$, the so-called [*“leptocline”*]{} (Bedding et al. 2007)) while the radius of the deepest layers (between 0.97 and 0.99 $R_{\odot}$) change in phase with 11-year activity cycle. These results are fully compatible with ours and this leptocline layer certainly deserves further investigations since it is the seat of important effects (ionization of Hydrogen and Helium, turbulent pressure, shears, inversion of radial rotation gradient, ...).
Apparent solar radius variation measurements {#solarradius}
==============================================
So far, the apparent radius of the Sun has been measured from the Earth by different techniques and from different sites. There is an abundant literature on the subject, but authors still give conflicting results regarding solar radius variations, both in amplitude and in phase. The discrepancies may come from the determination of the absolute solar apparent radius from the outer layer of the Sun (limb and photosphere) due to solar atmospheric phenomena (absorption, emission, scattering...), interstellar environment, Earth atmospheric effects and instrumental errors. Let us illustrate the state of the art. Considering only data obtained at the 150-foot solar tower of the Mount Wilson Observatory, La Bonte and Howard (1981) found no significant variation of the solar radius with the solar cycle (which was during its ascending phase) when they analyzed magnetograms (Fe I line at 525.0 nm) obtained routinely from 1974 to 1981. In contrast, Ulrich and Bertello (1995), with the same method, found that the solar radius varied in phase with the solar cycle over the investigated period 1982–1994 (descending phase), with an amplitude of about 0.4 arcsec. This variation could be explained by a 3% change of the line wing intensities during the solar cycle, assuming an apparent faculae and plage surface coverage of about 15-35% near the limb, a rather high percentage as emphasized by Bruls and Solanki (2004). The latter authors also suggest other mechanisms such as a change in the average temperature structure of the quiet Sun (unlikely, according to Livingston and Wallace, 2003) or an increase in the intensity profile due to the presence of plage emission (faculae, prominence feet...) near the solar limb, associated with magnetic activity variations during a solar cycle. It can also be argued that the difference between solar radius measurements may come, as suggested by Kosovichev (2005), from an incorrect reduction of the apparent radius measurements made at different optical depths which are sensitive to the temperature structure. A recent re-analysis of the magnetograms over 1974–2003 (Lefebvre et al. 2004b, 2006) shows no evident correlation of solar radius variations with magnetic activity (average error bar of 0.07 arcsec). A similar result was found by Wittmann and Bianda (2000), using a drift-time method at Iza$\tilde{n}$a[^5] from 1990 to 2000: measurements do not show long-term variations in excess of about $\pm$ 0.0003 arcsec/yr and do not show a solar cycle dependency in excess of about $\pm$ 0.05 arcsec.
Regarding space measurements of the solar radius, Kuhn et al. (2004) reported an helioseismic upper bound on solar radius variations of only 7 mas ($\pm$ 4 mas) from the MDI experiment on board SOHO over 1996–2004. The same authors also deduced an absolute value of the solar radius, (6.9574 $\pm$ 0.0011)$\times$$10^8$ $m$ or 959.28 $\pm$ 0.15 arcsec, from the Mercury transit of May 7, 2003, even if the instrument was not designed to perform such an astrometric measurement. This value agrees with that deduced from helioseismology, giving confidence in the latter method.
Based upon observations, the conclusion is that the solar radius may vary with time (on yearly and decennial time scales), but with a very weak amplitude, certainly not exceeding some 10–15 mas. We need additional dedicated solar space-based observations (at least balloon flights) to constrain the phase and the amplitude of radius variations. And if such observations can be made, we still need a physical model to explain such solar radius variation observations. We address this latter point in the following section.
Solar radius and luminosity versus gravitational energy variations
====================================================================
According to the definition of gravitational energy, $E_{g}$= $-\int (G m/r) dm$ (where $r$ is the radial coordinate and $G$ the gravitation constant), and assuming hydrostatic equilibrium, a thin shell of radius $dr$ containing a mass $dm$ in equilibrium under gravitational and pressure gradient forces will be expanded or contracted if any perturbation of these forces occurs. However, energy could be stored through gravitational or magnetic fields, each of them being able to perturb the equilibrium stellar structure, yielding at the end, changes in shape. A possible mechanism could be the following: if the central energy source remains constant while the rate of energy emission from the surface varies, there must be a reservoir where energy can be stored or released, depending on the variable rate of energy transport and through several mechanisms like gravitational or magnetic fields. (Pap et al. 1998, Emilio et al. 2000).
In order to study the consequences of gravitational energy changes on solar radius variations, Callebaut et al. (2002) used a self-consistent approach, assuming either a homogeneous or a non-homogeneous sphere. They calculated $\Delta R/R$ and $\Delta
L/L$ associated with the energies responsible for the expansion of the upper layer of the convection zone. We use here the same formalism for a few percent reminder of the modelling TSI (details of the computations can be found in the above–mentioned paper), but we consider an ellipsoidal surface (Eq. \[aireellipsoide\]). Let $\alpha$ be the fractional radius ($0$ $<$ $\alpha$ $<$ $1$): if the layer above $\alpha R$ expands, the expansion is zero at $\alpha$$R$ and is $\Delta$$ R$ at $R$. The increase in height at a radial distance $r$ in the layer interval $\left(\alpha R,
R\right)$, with $R$ = $R_{sp}$, is given by $$h(r) = \frac{(r-\alpha R)^n \Delta R}{R^n (1-\alpha )^n}
\label{height}$$ where $r$ is the usual radial coordinate and $n$= 1, 2, 3... is the order of the development. The relative increase in thickness for an infinitesimally thin layer at $r=R_{sp}$ is $(dh/dr)_{R_{sp}}= \frac{n \Delta R}{(1-\alpha ) R}$ . Considering the ideal gas law, $p= \frac{\rho}{m} k T$, and polytropic law, $p= K \rho ^\Gamma$ (where $\rho$ is the density; $k$, the Boltzmann constant; $K$, the polytropic constant, and $\Gamma$, the polytropic exponent –surely an ideal state–), the relative change in temperature expressed in terms of the relative change in radius is $$\left(\frac{\Delta T}{T}\right)_{R_{sp}} = -\frac{(\gamma -1) n \Delta R }{(1-\alpha ) R}
\label{temp.}$$ where $\Gamma$ can be replaced by $\gamma$, the ratio of the specific heats. We now apply the above approach to an ellipsoid with $R_{sp} = \sqrt [3] {R_{eq}^2 R_{pol}}$, using Eq. \[aireellipsoide\], and assuming $dR_{eq}\!$ = $\!dR_{pol}\!$ = $\!dR_{sp}$. When substituting Eq. \[temp.\] in Eq. \[lumi\] (Eddington approximation), we obtain
$$\frac{\Delta L}{L}=-\left[\frac{4n(\gamma -1)}{1-\alpha} + \frac{\frac{a}{c^2}(2a^2-b^2-ab)+\frac{b}{c^3}(2a^3-b^3-ab^2)\ln(\frac{a+c}{b})}{a+\frac{b^2}{c}\ln(\frac{a+c}{b})} \right]$$ $$\left. ~~~~~~~~ \times~~
\frac{3b}{2b+a}\frac{\Delta R_{sp}}{R_{sp}}
\right.
\label{exp.}$$
We made two computations, one with $n$=1 (monotonic expansion with radius) and the other one with $n$=2 (non monotonic expansion, as shown in Lefebvre and Kosovichev, 2005), using $\gamma$= 5/3, and $\alpha$ $\approx$ 0.96.
Eq. \[exp.\] implies that a decrease of $R_{sp}$ corresponds to an increase of $L$; that is solar radius and luminosity variations are anticorrelated.
$\Delta$L/L = 0.0011 $\Delta$L/L = 0.00073
-------------------------------------- -------- -------------------------------------- -------
$\Delta$R/R = -1.70 $\times 10^{-5}$ (n=1), $\Delta$R/R = -1.13 $\times 10^{-5}$ (n=1)
(or $\Delta$R = 11.8 km) (or $ \Delta$R = 7.86 km)
$\Delta$R/R = -8.38 $\times 10^{-6}$ (n=2), $\Delta$R/R = -5.56 $\times 10^{-6}$ (n=2)
(or $\Delta$R = 5.83 km) (or $\Delta$R = 3.87 km)
: Variations of the solar radius computed in two cases: monotonic (n=1) and non monotonic (n=2) expansion, and for two mean values of $L_{\odot}$. The sign (-) indicates a shrinking. The case $n=2$ is the most likely. []{data-label="delta(r)"}
Table \[delta(r)\] gives the results for two values of $\Delta L / L$ = $\Delta I / I$: the usual adopted value, 0.0011, using TSI composite data from 1987 to 2001 (Dewitte et al. (2005); mean value $L_{\odot}$ = 1366.495 $W/m^2$); and 0.00073, determined through a re-analyzis of the composite TSI data over the period of time 1978–2004 (Fröhlich, 2005; mean value $L_{\odot}$ = 1365.993 $W/m^2$). For n=2 (the most likely case consistent with recent other results), our absolute estimate of $\Delta R_{sp}$ is smaller than the 8.9 km obtained in the case of a spherical Sun by Callebaut et al. (2002). However our $\Delta R_{sp}/R_{sp}$ agrees with that of Antia (2003), i.e. $\Delta R/R$ = 3$\times$$10^{-6}$, who used $f$-mode frequencies data sets from MDI (from May 1996 to August 2002) to estimate the solar seismic radius with an accuracy of about 0.6 km (see also among other authors, Schou et al., 1997 or Antia, 1998 for such a determination of the solar seismic radius to a high accuracy).
Three points result from the analysis of the data. The first concerns the “helioseismic radius" which does not coincide with the photospheric one, the photospheric estimate always being larger by about 300 km (Brown and Christensen-Dalsgaard, 1998).
The second point, directly related to our subject, is the shrinking of the Sun with magnetic activity as pointed out by Dziembowski et al. (2001), using $f$-mode data from the MDI instrument on board SOHO, from May 1996 to June 2000. They found a contraction of the Sun’s outer layers during the rising phase of the solar cycle and inferred a total shrinkage of no more than 18 km. Using a larger data base of 8 years and the same technique, Antia and Basu (2004) set an upper limit of about 1 km on possible radius variations (using data sets from MDI, covering the period of May 1996 to March 2004). However, they demonstrated that the use of $f$-modes frequencies for $l$ $<$ 120 seems unreliable.
Finally, the third point concerns the luminosity production mechanism, through the parameter [**[*w*]{}**]{}, called the asphericity-luminosity parameter. This parameter is defined as $$\mbox{\bf {\textit w}} = (dR/R)/(dL/L) .
\label{asph.lum.}$$ According to small observed values of $dR$, a small [**[*w*]{}**]{} means that $L$ is produced in the upper–most layers (Gough, 2001), whereas a large [**[*w*]{}**]{} would imply luminosity production in layers deeper inside the Sun. >From the above computations and Eq. \[asph.lum.\], we can estimate [**[*w*]{}**]{} as\
$$\mbox{\bf {\textit w}} = -1.55~10^{-2} ~~~~ (n=1) ~~~~ \mbox{\rm and }~~~~ \mbox{\bf {\textit w}} = -7.61~10^{-3} ~~~~ (n=2)$$ These values[^6] (the second is the more likely) can be compared to the ones computed by Sofia and Endal (1980), -7.5 $10^{-2}$; Dearborn and Blake (1980), 5.0 $10^{-3}$; Spruit (1992), 2.0 $10^{-3}$; Gough (2001), 2.0 $10^{-3}$ if the origin of luminosity variations is located in surface layers, or 1.0 $10^{-1}$ if they are more deeply seated; and finally to the lower limit given by Lefebvre and Rozelot (2004), -7.5 $10^{-2}$.
Solar radius variation versus magnetic activity
================================================
As suggested by Livingston et al. (2005), magnetic flux tubes pass between solar granules without interacting with them. Due to magnetic pressure, one could expect a change in the mean size of granules that would be shifted toward the smaller sizes as magnetic activity increases.
Such features were confirmed by observations made by Hanslmeier and Muller (2002) at the Pic du Midi Observatory, using the 50-cm refractor (images taken on August 28, 1985 and September 20, 1988).
As a consequence, if the number of granules per unit area is constant, the whole size of the Sun would decrease. This means solar radius variations are anticorrelated with solar magnetic activity.
Conclusions {#discussion}
===========
In this study, using a preliminary black-body radiation model for the Sun, we have shown that temporal radius variations must be taken into account in the present efforts to model solar irradiance (we do not claim that irradiance variability is due to radius variability alone). Distortions with respect to sphericity, albeit faint, are related to variations of solar gravitational energy, of surface effective temperature and to variations of luminosity (as solar irradiance is an indicator of solar luminosity). Even if a major simplification was made (using a preliminary black-body radiation model, neglecting magnetic fields which can influence the limb extension), we have obtained constraints on radius and temperature variations through fits to observed irradiance data. Our best fit gives $dT$ = 1.2 $K$ at $dR$ = 10 mas. This surface effective temperature variation agrees with that found by Gray and Livingston (1997) or Caccin et al. (2002). Recent results of Livingston et ! al. (2005) support a more immutable atmosphere ($dT$ $\approx$ 0). But we have shown that irradiance variation modelling is very sensitive to small surface effective temperature variation (between 0 and 0.085 K). Indeed, we underlined a phase-shift in the ($dR$, $dT$)–parameter plane between correlated or anticorrelated radius versus irradiance variations. Better observations of $dT$ might be crucial to determine the phase of radius variations (especially near the limb) with respect to solar cycle activity, noting that observed irradiance variations are in phase with the solar cycle.
We further obtained an upper limit on the amplitude of $dR$, i.e. 3.87 – 5.83 km, by applying Callebaut’s method but taking into account the ellipsoidal shape of the Sun, in a non-monotonic expansion of the radius with depth (in the sub-surface), and composite Total Solar Irradiance. Our estimate of dR is substantially smaller than the estimate obtained by Callebaut et al. (2002) for a spherical Sun, but it agrees with those derived from helioseismology.
Equating the decrease of radiated energy with the increase of gravitational energy corresponding to the expansion of the upper layer of the convection zone leads to solar radius variations anticorrelated with luminosity ones. An estimate of the asphericity-luminosity parameter (***w*** = - 7.61 $10^{-3}$) supports this upper layer mechanism as the source of luminosity variations.
Finally, assuming a constant numbers of granules per unit area, we suggest that solar radius variations might be associated with variations of magnetic pressure between the granules. A possible mechanism could be as follows: as magnetic activity increases, magnetic flux tubes which do not interact with solar granules at the near surface, force the latter to decrease in size; the whole Sun shrinks and radius variations are thus anticorrelated with solar activity.
The present study was conducted on a large time scale (two solar cycles), and the question of smaller temporal variations (minutes, hours) is not considered here. The above mentioned mechanism may act at a smaller time scale too, but it needs to be confirmed. Space–dedicated missions might be able to answer this question.
[The authors cordially thank the referees for their remarks which have been used in this version of the paper.]{}
Antia, H.M.: 1998, A&A, 330, 336 Antia, H.M.: 2003, Ap.J., 590, 567 Antia, H.M. and Basu, S.: 2004, ESA SP-559, 301 Armstrong, J. and Kuhn, J.R.: 1999, Ap.J., 525, 533 Badache-Damiani, C. and Rozelot, J.P.: 2006, Mon. Not. R. Astron. Soc. 369, 83 Bedding, T., Crouch, A., Christenssen-Dalsgaard, J. et al.: 2007; Highlights of Astronomy, Vol. 14, Proceedings of [*the XXVth IAU G.A., August 2006*]{}, K.A. Van der Hucht ed., Cambridge University Press Brown, T. and Christensen-Dalsgaard, J.: 1998, Ap.J., 500, L195 Bruls, J.H.M.J. and Solanki, S.K.: 2004, A &A, 427, 735 Caccin, B., Penza, V. and Gomez, M.T.: 2002, A &A, 386, 286 Callebaut, D.K., Makarov, V.I. and Tlatov, A.: 2002, ESA SP-477, 209 Dearborn D.S.P. and Blake, J.B.: 1980, Ap.J., 237, 616 Dewitte, S., Crommelynck, D., Mekaoui, S. and Joukoff, A.: 2005, Solar Phys., 224, 209 Dziembowski, W.A., Goode, P.R. and Schou, J.: 2001, Ap.J., 553, 897 Emilio, M., Kuhn, J.R., Bush, R.I. and Scherrer, P.: 2000, Ap.J., 543, 1007 Emilio, M., Bush, R.I., Kuhn, J.R. and Scherrer, P.: 2007, Ap.J., 660, L161 Fazel, Z., Rozelot, J.P., Pireaux, S., Ajabszirizadeh A. and Lefebvre, S.: 2005, Mem. della Società Astronomica Italiana, I. Ermolli, P. Fox, and J. Pap, eds, 76, 961, Società Astronomica Italiana Fröhlich, C. and Lean, J.: 1998, Geophys. Res. Lett., 25, 437 Fröhlich, C.: 2005, WRCPMOD, Annual report, 18 Godier, S. and Rozelot, J.P.: 2001 in “The 11th Cool Stars, Stellar Systems and the Sun”, ASP Conference Series, 223, R.J. Garcia López, R. Rebolo, M.R. Zapatero Osorio, eds, 649, Astronomical Society of the Pacific Gough, D.O.: 2001, Nature, 410, 313 Gray, D.F. and Livingston, W.C.: 1997, Ap.J., 474, 802 Hanslmeier, A. and Muller, R.: 2002, ESA SP-506, 843 Kosovichev, S.: 2005, American Geophysical Union, Fall Meeting 2005, abstract SH11A-0244 Krivova, N.A., Solanki, S.K., Fligge, M. and Unruh, Y.C.: 2003, A&A, 399, L1-L4 Krivova, N.A., and Solanki, S.K.: 2005, AdSpR, 35, 361K Kuhn, J.R.: 2004, Adv. Space. Res., 34, 302 Kuhn, J. R. and K. G. Libbrecht: 1991, Ap.J. Lett., 381, L35–L37 Kuhn, J. R., Libbrecht, K. G., and Dicke R. H.: 1988, Science 242, 908 Kuhn, J.R., Bush, R.I., Emilio, M. and Scherrer, P.H.: 2004, Ap.J., 613, 1241 La Bonte, B. and Howard, R.: 1981, Science 214, 907 Lefebvre, S. and Rozelot, J.P.: 2003, ISCS, Tatranskà Lomnica, (Slovak Republic), ESA SP-535, 53 Lefebvre, S., Bertello, L., Ulrich, R.K., Boyden, J.E. and Rozelot, J.P.: 2004, SOHO14-GONG2004 Meeting (New-Haven, USA), ESA SP-559, 606-610 Lefebvre, S. and Kosovichev, A.: 2005, ApJ, 633, L149–L152 Lefebvre, S., Rozelot, J.P., Pireaux, S., Ajabshirizadeh, A. and Fazel, Z.: 2005, “Memorie della Società Astronomica Italiana”, Ilaria Ermolli, Peter Fox and Judit Pap eds., Vol. 76, 994 Lefebvre, S. and Rozelot, J.P.: 2004, A&A, 419, 1133 Lefebvre, S., Kosovichev, A.G. and Rozelot, J.P.: 2006, in [*“SOHO-17: 10 Years of SOHO and Beyond”*]{} (Giardini-Naxos, I), ESA Proceedings, ESA SP-617, 43.1 Lefebvre, S., Kosovichev, A.G., Rozelot, J.P.: Ap.J., 658, L135 Li, L.H., Ventura, P., Basu, S., Sofia, S. and Demarque, P.: 2005, arXiv: astro-ph/0511238 Livingston, W.C. and Wallace, L.: 2003, Solar Phys., 212, 227 Livingston, W.C. Gray, D., Wallace, L. and White, O.R.: 2005, in “Large-scale Structures and their Role in Solar Activity” ASP Conference Series, Vol. 346, 353, K. Sankarasubramanian, M. Penn, and A. Pevtsov, eds, Astronomical Society of the Pacific Pap, J.M., Kuhn, J.R., Fröhlich, C., Ulrich, R., Jones, A. and Rozelot, J.P.: 1998, SOHO 14-Gong meeting, New Haven, (USA), ESA SP-417, 267 Rozelot, J.P., Lefebvre, S., Pireaux, S. and Ajabshirizadeh, A.: 2004, Solar Phys., 224, 229 Rozelot, J.P., Lefebvre, S., Desnoux, V.: 2003, Solar Phys., 217, 39 Rozelot, J.P. and Lefebvre, S.: 2003, in [*“The Sun’s surface and subsurface"*]{}, J.P. Rozelot ed., LNP, 599, 4, Springer Schou, J., Kosovichev, A.G., Goode, P.R. and Dziembowski, W.A.: 1997, Ap.J., 489, 197 Sofia, S.: 2004, American Geophysical Union, San Francisco Fall Meeting, abstract SH51E-05 Sofia, S. and Endal, A.S.: 1980, in [*“The ancient Sun"*]{}, Pepin, R.O., Eddy, J.A. and Merrill, R.B. eds, Pergamon press, 139 Spruit, H.C.: 1992, in [*“The Sun in Time”*]{}, Sonett, C.P., Giampapa, M.S. and Matthews, M.SH., eds., The University of Arizona Press Ulrich, R.K. and Bertello, L.: 1995, Nature, 377, 214 Vautard, R., Yiou, P. and Ghil, M.: 1992, Physica D, 58, 95 Wittmann, A.D. and Bianda, M.: 2000, ESA SP-463, 113
[^1]: The area of an ellipsoid of radii $R_{eq}$ = $a$, $R_{pol}$ = $b$ and $c$=$\sqrt{a^2-b^2}$ is given by: $$A_{ell} = 2 \pi \left[a^2 + (a b^2 /c) \ln \frac{a + c}{b} \right ]
\label{aireellipsoide}$$
[^2]: Thanks to Fröhlich, C., unpublished data from the VIRGO Experiment on the cooperative ESA/NASA Mission SoHO.
[^3]: Note that $R_{sp}$ is different from the semi-diameter of the Sun (or standard radius), $R_{\odot}$.
[^4]: Let us recall that the SSA is a technique which has been developed by Vautard et al. (1992). It has the advantage of working in a data adaptable filter mode instead of using fixed basis functions, as it is the case for Fourier Transform or wavelet techniques. Therefore, the SSA has the possibility to get rid of some noise characteristic of a given type of data. The SSA is a powerful fast and simple method based on the Principal Component Analysis (PCA) which allows us to filter or reconstruct signals. The basis of the SSA is the eigenvalue-eigenvector decomposition of the lag-covariance matrix which is composed of the covariances determined from the shifted time series. Projection of the time series onto the Empirical Orthogonal Functions (EOFs) yields the so-called Principal Components (PCs); these are filtered versions of the original time series. The EOFs are data adaptable to the analogs of sine and cosine functions while the PCs are the analogs of coefficients in Fourier analysis.
[^5]: Other radius data from Iza$\tilde{n}$a are availlable, such as astrolabe measurements leading to controversial results, which are discussed elsewhere (Badache-Damiani and Rozelot, 2006).
[^6]: The sign of $w$ is obviously relevant; it seems that some authors quoted here have given absolute values.
|
---
abstract: '[ We explore the influence of heavy quarks on the deconfinement phase transition in an effective model for gluons interacting with dynamical quarks in color SU(3). With decreasing quark mass, the strength of the explicit breaking of the Z(3) symmetry grows and the first-order transition ends in a critical endpoint. The nature of the critical endpoint is examined by studying the longitudinal and transverse fluctuations of the Polyakov loop, quantified by the corresponding susceptibilities. The longitudinal susceptibility is enhanced in the critical region, while the transverse susceptibility shows a monotonic behavior across the transition point. We investigate the dependence of the critical endpoint on the number of quark flavors at vanishing and finite quark density. Finally we confront the model results with lattice calculations and discuss a possible link between the hopping parameter and the quark mass. ]{}'
author:
- Pok Man Lo
- Bengt Friman
- Krzysztof Redlich
date:
-
-
title: 'Polyakov loop fluctuations and deconfinement in the limit of heavy quarks\'
---
Introduction
============
In the limit of infinitely heavy quarks, deconfinement in the SU(3) gauge theory is associated with the spontaneous breaking of the Z(3) center symmetry [@Greensite; @tHooft; @Yaffe; @McLerran; @Polyakov]. The phase transition is of first-order and is therefore stable against small explicit symmetry breaking [@Binder]. The presence of dynamical quarks breaks the center symmetry explicitly, with a strength that increases as the quark mass decreases. It is thus expected that the transition remains discrete in the heavy quark region, and becomes continuous at some critical value of quark mass [@GK]. This defines the critical endpoint (CEP) for the deconfinement phase transition.
The relevant observables to study deconfinement are the Polyakov loop and its susceptibilities. Recently these quantities were computed on the lattice within SU(3) pure gauge theory [@pmlo1]. In particular, the ratio of the transverse and longitudinal Polyakov loop susceptibility was shown to exhibit a $\theta$-like behavior at the critical temperature $T_d$, with almost no dependence on temperature on either side of the transition. This feature makes such ratios ideal probes of deconfinement.
The influence of the dynamical light quarks on these susceptibilities has been studied in 2- and (2+1)-flavor QCD [@2+1QCD; @pmlo2]. The resulting susceptibility ratios are considerably smoothened, reflecting the crossover nature of the transition. However, the theoretical understanding of these observables is still incomplete. It is therefore useful to explore the properties of the Polyakov loop susceptibilities for different number of flavors, as functions of the quark mass in the heavy quark region, thus bridging the gap between pure gauge theory and QCD.
The thermodynamics of SU(3) gauge theory with heavy quarks have long been studied in an effective theory [@GK] and on the lattice [@Karsch2001579; @DeGrand; @attig; @Langelage:2009jb; @Ejiri:2012wp; @PhysRevD.60.034504; @Aarts:2001dz; @fomm; @whot]. The nature of the deconfinement CEP was recently examined within the effective matrix model [@PhysRevD.85.114029]. One key finding of this study is that the model results for the critical point depend strongly on the structure of the phenomenological gluon potential.
In this work we reexamine the phase structure of the deconfinement transition for heavy quarks. We formulate an effective theory where the gluon potential is constructed using the pure lattice gauge theory results on the equation of state and on fluctuations of the Polyakov loop [@pmlo2]. The contribution from heavy quarks is described by the one loop thermodynamic potential of fermions coupled to a background gluon field [@meisinger].
To study the deconfinement CEP, it is essential that the phenomenological gluon potential reproduces not only the equation of state and the Polyakov loop, but also the pure gauge theory results for fluctuations of the Polyakov loop. Thus, we employ the effective Polyakov loop potential of Ref. [@pmlo2], where both the location of minimum and the curvature are constrained by lattice results. This distinguishes this work from previous studies [@PhysRevD.85.114029].
We focus on the behavior of the Polyakov loop susceptibilities near the CEP at vanishing and finite quark chemical potential. We also investigate the quark mass dependence of the critical temperature of the CEP for different number of flavors. We find that the longitudinal Polyakov loop susceptibility is strongly enhanced in the critical region, and hence, can be used to probe the location of the CEP. On the other hand, the transverse fluctuations are insensitive to the continuous phase transition, showing monotonic behavior across the critical endpoint.
For a comparison of model and lattice results, we utilize the hopping parameter expansion of the fermionic determinant. Moreover, we propose a tentative relation between the hopping parameter and the quark mass in the continuum limit.
The paper is organized as follows. In the next section, we introduce the effective model for deconfinement in the presence of heavy quarks. We calculate the properties of different susceptibilities and locate the deconfinement critical endpoint. In Sec. III we relate the effective model result and lattice data. In Sec. IV we present our conclusions.
Modeling deconfinement in the presence of quarks
=================================================
To explore the influence of heavy quarks on the deconfinement phase transition in the SU(3) gauge theory, we consider the following model for the partition function [@GK; @gross; @meisinger; @kr; @PhysRevD.69.097502],
$$\begin{aligned}
\label{main_partition}
Z = \int dL d\bar{L}
\det[\hat{Q}_F] \, e^{-\beta V \hat U_G(L, \bar{L})},
\end{aligned}$$
where $\beta=T^{-1}$ is the inverse temperature, $L$ is the Polyakov loop and $\bar{L}$ its conjugate. Furthermore, $\hat U_G$ is the Z(3) invariant Polyakov loop potential extracted from pure gauge theory, including the contribution of the SU(3) Haar measure. The quark contribution is represented by the determinant of the fermionic matrix $\hat{Q}_F$, which in the uniform background gluon field $A_4$, reads
$$\begin{aligned}
\hat{Q}_F &= (-\partial_\tau + \mu - i g A_4) \gamma^0 + i \vec{\gamma} \cdot \nabla - m.\end{aligned}$$
For $N_f$ degenerate flavors of quark mass $m$, we obtain the one loop expression [@meisinger],
$$\begin{aligned}
\label{quark1}
\ln \det [ \hat{Q}_F ] &= 2V\beta N_f \int \frac{d^3 k}{(2 \pi)^3} \, ( 3 E(k) + g^+ + g^- ),\end{aligned}$$
where the first term is the vacuum contribution, with $E(k) = \sqrt{k^2 + m^2}$, and
$$\begin{aligned}
g^+ = T\ln(1 + 3L
e^{-\beta E^+} + 3
\bar{L}
e^{-2 \beta E^+} + e^{-3 \beta E^+}) \label{quark}\end{aligned}$$
is the contribution of quarks coupled to the Polyakov loop, with $ E^+ =E(k) - \mu$. The function $g^-$ describes the anti-quarks, and is obtained from Eq. by replacing $\mu \rightarrow -\mu$ and $L \leftrightarrow \bar L$.
The thermodynamic potential density $\Omega=-(T/V)\ln Z$ is obtained from Eq. in the mean field approximation,
$$\begin{aligned}
{ {T^{-4}}\Omega} = U_G(L,\bar L) +U_Q(L,\bar L)\label{MF},\end{aligned}$$
where $U_Q$ is the fermion contribution to the effective potential and the thermal averaged Polyakov loop, $L$ and $\bar L$, satisfy the gap equations,
$$\begin{aligned}
\partial \Omega /\partial L=0, ~~~~\partial \Omega /\partial {\bar L}=0 \label{gap}.
\end{aligned}$$
For the pure gluon part $U_G$, we employ the following phenomenological Polyakov loop potential [@pmlo2], $$\begin{aligned}
\label{glue}
{U}_G = &-\frac{1}{2} A(T)\bar{L} L + B(T) \ln M_H \nonumber \\
&+ \frac{1}{2} C(T) ( L^3 + \bar{L}^3 ) + D(T) (\bar{L} L)^2, $$ where the SU(3) Haar measure $M_H$, is given by $$\begin{aligned}
M_H &= 1-6 \bar{L} L + 4 ( L^3 + \bar{L}^3 ) -3 (\bar{L} L)^2.
\end{aligned}$$
The potential (\[glue\]) was constructed so as to describe the lattice data on SU(3) thermodynamics, including fluctuations of the Polyakov loop. The gluonic potential $U_G$ yields a first-order deconfinement phase transition at the critical temperature, $T_d= 0.27 \, {\rm GeV}$.
![Temperature dependence of the thermal averaged Polyakov loop for quark masses $m=0.5, 1.48$ and $3.5$ GeV at $N_f = 3$. []{data-label="fig:zero_mu_pl"}](zero_mu_pl){width="3.275in"}
The quark contribution to the mean field potential is obtained from (\[quark1\]),
$$\begin{aligned}
\label{quarkf}
{U}_Q = - 2 N_f \beta^4\int \frac{d^3 {k}}{(2 \pi)^3} \, [ g^+ + g^- ],\end{aligned}$$
where the vacuum term has been dropped, since it is a constant, independent of the Polyakov loop.
The thermodynamic potential in (\[MF\]) is the effective model for exploring the thermodynamics of gluons in the presence of heavy quarks.
{width="3.375in"} {width="3.375in"}
Heavy quarks and deconfinement
-------------------------------
The first-order nature of the deconfinement phase transition in the SU(3) pure gauge theory is directly related to the global Z(3) center symmetry and its spontaneous breaking. This transition is eventually washed out by the explicit symmetry breaking induced by the finite quark mass. These features are clearly exhibited in the mean field model. Indeed, in the limit of large quark masses, the leading order term in $U_Q$ in Eq. is linear in the Polyakov loop. One finds, for $\mu=0$,
$$\begin{aligned}
\label{h0}
{U}_Q(L,\bar L) &\approx - h(m,N_f,T) \, L_{\rm L},
\end{aligned}$$
where $L_{\rm L}$ is the thermal average of the longitudinal Polyakov loop [^1], and
$$\begin{aligned}
\label{linearb}
h(\beta m, N_f) = \frac{6 N_f}{\pi^2T^3} \int d {k} \, {k}^2 e^{-{\beta E}},\end{aligned}$$
is a dimensionless function of $N_f$ and $\beta m$.
Equation (\[h0\]) clearly shows that finite mass quarks generate an external field for the Polyakov loop that breaks the Z(3) symmetry explicitly. Clearly, in the limit $\beta m \rightarrow \infty$, the strength of this field vanishes and the thermodynamics of pure gauge theory is recovered.
The strength of the symmetry breaking term increases with decreasing quark mass, eventually turning the first-order deconfinement transition into a crossover. Consequently, there is a critical value of the quark mass, $m_{\rm CEP}$, where the first-order transition ends at a second order critical endpoint [@GK].
Using the linear symmetry breaking term (\[h0\]), the location of the second order CEP is determined by [@PhysRevD.85.114029]
$$\begin{aligned}
\label{phaseboundary}
h((\beta m)_{\rm CEP},N_f) &= h_c,\end{aligned}$$
where $h_c$ is a dimensionless constant, to be determined. The value of $h_{c}$ depends only on the Polyakov loop potential $U_{G}$. Since $h(\beta m)$ in (\[h0\]) is a decreasing function of $\beta m$, Eqs. and imply that the ratio $m_{\rm CEP}/T_{\rm CEP}$ increases with $N_f$. Naturally, such a pattern is also observed when the full one loop quark potential (\[quarkf\]) is employed.
Details of the phase structure of the deconfinement transition are revealed by examining the Polyakov loop fluctuations. In the following, we study how the position of the CEP changes with the number of quark flavors at vanishing and at finite quark density.
The deconfinement critical endpoint at $\mu=0$
----------------------------------------------
At $\mu=0$, the thermal average of the transverse Polyakov loop $L_{\rm T}$ vanishes, due to the symmetry of the partition function (\[main\_partition\]). Consequently, only the longitudinal Polyakov loop $L_{\rm L}$ serves as an order parameter for deconfinement. On the other hand, both the longitudinal and transverse fluctuations of the Polyakov loop are non-vanishing. We introduce the longitudinal $\chi_{\rm L}$ and transverse $\chi_{\rm T}$ susceptibilities
$$\begin{aligned}
\label{long_and_trans_1}
\chi_{\rm L,T} = \frac{1}{2} V \, ( \langle L \bar{L} \rangle_c \pm \langle L L \rangle_c ),\end{aligned}$$
where $\langle \dots \rangle_c$ denotes the connected part. The two terms on the r.h.s of Eq.(\[long\_and\_trans\_1\]) are obtained by taking the appropriate field derivatives [@pmlo2; @Sasaki:2006ww] of the thermodynamic potential (\[MF\]).
The influence of heavy quarks on the deconfinement transition is clearly reflected by the properties of the Polyakov loop. In Fig. \[fig:zero\_mu\_pl\] we show the Polyakov loop as a function of temperature for three values of the quark mass. The expectation value of the Polyakov loop, $\langle L\rangle$ is determined by the position of the global minimum of the potential (\[MF\]), including the complete one loop quark contribution (\[quarkf\]) for three degenerate quark flavors.
For a sufficiently large quark mass, the first-order nature of the phase transition persists, while at smaller quark masses, the explicit symmetry breaking is stronger and the transition is of the crossover type. The endpoint of the line of first-order transitions defines the critical value of the quark mass, $m_{\rm CEP}$.
To identify the CEP, we consider the longitudinal fluctuations of the Polyakov loop. At the CEP, the longitudinal susceptibility $\chi_{\rm L}$ diverges whereas the transverse susceptibility $\chi_{\rm T}$ remains finite.
In Fig. \[fig:zero\_mu\_susL\] we show the longitudinal and transverse susceptibility for three degenerate quark flavors. While both susceptibilities depend on the value of the quark mass, only the longitudinal one shows enhancement near the CEP. The transverse susceptibility decreases monotonically with decreasing quark mass. Thus, for a given $N_f$, the CEP can be located by identifying the global maximum of $\chi_{\rm L}$. Our results for the critical quark masses obtained for different number of quark flavors at the CEP are as follows,
$$\label{mass}
m_{\rm CEP} = 1.10, ~1.35,~ 1.48 \, {\rm GeV},~~
{\rm for \,} N_f = 1, 2, 3.$$
The resulting trend, with $m_{\rm CEP}$ increasing with $N_f$, is consistent with previous findings [@PhysRevD.85.114029]. However, we obtain a lower value of the critical quark mass than that found in the matrix model, $m_{\rm CEP} \simeq 2.5 \, {\rm GeV}$ at $N_f=3$. As discussed in Ref. [@PhysRevD.85.114029], the location of the deconfinement critical endpoint is very sensitive to the form of the Polyakov loop potential. In the present calculation, $U_G$ reproduces the lattice data on the equation of state as well as on the susceptibilities of the Polyakov loop. This feature is crucial for locating the CEP, which is influenced by fluctuations of the order parameter.
At a given value of the quark mass, we identify the deconfinement transition with the location of the maximum of $\chi_{\rm L}$. In Fig. \[fig:zero\_mu\_Td\] we show the resulting phase diagram in the $(T,m)$–plane for different $N_f$. We observe that the temperature of the CEP remains almost constant at $T_{\rm CEP} \simeq 0.261 \, {\rm GeV}$ for all $N_f$. This value is lower than the critical temperature in the pure gauge theory by about $9 \, {\rm MeV}$. Here, the temperature obtained in the matrix model is essentially zero [@PhysRevD.85.114029].
The very weak dependence of $T_{\rm CEP}$ on $N_f$, seen in Fig. \[fig:zero\_mu\_Td\], can be qualitatively understood. Assuming the structure of the thermodynamic potential introduced in Eq.(\[MF\]), and keeping only the leading symmetry breaking term, $U_Q \approx -h L_{\rm L}$, the conditions for the CEP are
$$\begin{aligned}
\label{cep_model}
\frac{\partial {U}_G}{\partial L_{\rm L}} = h, ~~
\frac{\partial^2 {U}_G}{\partial L_{\rm L}^2} = 0, ~~{\rm and} ~~
\frac{\partial^3 {U}_G}{\partial L_{\rm L}^3} = 0.\end{aligned}$$
The first condition in Eq. is just the gap equation, while the second and the third conditions reflect the fact that at the CEP, three extrema of the effective potential merge. The solution of Eq. fixes the critical values of the Polyakov loop, the quark mass and the temperature, $L_{\rm CEP}, m_{\rm CEP},$ and $T_{\rm CEP}$. If ${U}_G$ is independent of the quark mass, the last two conditions in uniquely determine $T_{\rm CEP}$ and $L_{\rm CEP}$. Thus, in the heavy quark limit, where the leading symmetry breaking term (\[h0\]) dominates, the critical temperature of the CEP, $T_{\rm CEP}$, is independent of $N_f$.
![Quark mass dependence of the deconfinement temperature, defined by a peak of $\chi_{\rm L}$, for different $N_f$. Full lines correspond to the first-order phase transitions, while dashed lines represent the pseudo-critical temperature of the crossover transition. The points indicate the locations of the CEP.[]{data-label="fig:zero_mu_Td"}](zero_mu_Tc){width="3.375in"}
A dependence of the effective Polyakov loop potential on the quark mass $m$ naturally appears when the complete one loop quark contribution in Eq.(\[quark1\]) is used. The fact, that $T_{\rm CEP}$ is almost $N_f$ independent, as shown in Fig. \[fig:zero\_mu\_Td\], confirms the expectation that the leading term in $U_{Q}$, shown in Eq., is sufficient in the mass range considered.
Phase diagram at finite chemical potential
------------------------------------------
At finite $\mu$, the expectation values of the Polyakov loop $ L$, and its conjugate $\bar{L}$ are both real, but in general, different [@Sasaki:2006ww; @PhysRevD.73.014019; @PhysRevD.72.065008]. This is because at non-zero $\mu$ the effective action is complex [@Sasaki:2006ww; @PhysRevD.72.065008].
The expectation values of $L$ and $ \bar{L} $ are determined by solving the gap equations:
$$\begin{aligned}
\label{finite_mu_gap}
\frac{\partial ~\Omega}{\partial L} = 0,~~ {\rm and }~~
\frac{\partial \Omega}{\partial \bar{L}} = 0,\end{aligned}$$
with the thermodynamic potential $\Omega$ given in Eq.(\[MF\]).
![The expectation values of the Polyakov loop $L$ (full lines) and its conjugate $\bar L$ (dashed lines) at $\mu=0.5 \, {\rm GeV}$, and for different values of the quark mass, $m = 3.5, 1.88, 0.5 \, {\rm GeV}$. (from right to left) []{data-label="fig:finite_mu_pl"}](finite_mu_pl){width="3.375in"}
{width="3.375in"} {width="3.375in"}
In Fig. \[fig:finite\_mu\_pl\] we show the temperature dependence of $L$ and $\bar L$ for finite $\mu$, and for several values of the quark mass with $N_f=3$. While $ \bar{L}$ and $ L $ differ below the deconfinement transition, they merge at high temperatures. This follows from the restriction of $(L_{\rm L},L_{\rm T})$ target region, imposed by the Haar measure. In the deconfined phase, as $L_{\rm L} \rightarrow 1$, the target region enforces $L_{\rm T} \rightarrow 0$, and thus the difference between $ \bar{L} $ and $ L $ vanishes. Such a feature is absent in the polynomial type potential, which does not comply with the constraints of the SU(3) color group structure [@Sasaki:2006ww].
The location of the CEP at finite density is again indicated by a divergence of the longitudinal susceptibility. However, at finite $\mu$, the definitions of the longitudinal and transverse susceptibilities are modified [@Sasaki:2006ww]
$$\begin{aligned}
\label{long_and_trans_2}
\chi_{\rm L,T} &= \frac{1}{2} V \, [\langle {L \bar{L}} \rangle_c \pm \frac{1}{2} \langle ({LL} + {\bar{L}\bar{L}}) \rangle_c].\end{aligned}$$
For $\mu\to 0$, $\langle{\bar{L}\bar{L}} \rangle \to \langle{LL} \rangle$, and the above definitions connect smoothly to that, introduced in Eq.(
\[long\_and\_trans\_1\]) for vanishing density.
The finite density results for the longitudinal and transverse susceptibilities are shown in Fig. \[fig:finite\_mu\_susL\]. As for $\mu = 0$, only the longitudinal susceptibility is enhanced near the CEP, whereas the transverse susceptibility is insensitive to criticality.
To construct the phase diagram, we trace the peak location of $\chi_{\rm L}$ as a function of $T$, $\mu$ and $m$. In Fig. \[fig:finite\_mu\_Td\] we show the transition temperature as a function of $\mu$ at fixed quark mass $m = 1.6 \, {\rm GeV}$. This value is larger than the critical mass found at $\mu=0$, $m_{\rm CEP}\simeq 1.48 \, {\rm GeV} $. Hence, in this case, the transition is first-order at small $\mu$.
As the quark chemical potential increases, the strength of the explicit breaking of the Z(3) symmetry increases and the transition turns into a crossover. These two regimes are connected by the deconfinement CEP, located at the critical chemical potential $\mu_{\rm CEP}$. Clearly, for $m < 1.48 \, {\rm GeV}$, the system is in the crossover regime for any value of $\mu$.
The critical temperature $T_{\rm CEP}\simeq 0.261 \, {\rm GeV}$, remains close to its $\mu = 0$ value. This follows from the assumption that $U_{G}$ is not renormalized by quark loops, which, as discussed above, is reasonably well justified in the parameter range of interest. As $m$ increases, the CEP moves to larger $\mu$, while the critical temperature remains almost constant.
In Fig. \[fig:finite\_mu\_cep\] we show the dependence of the critical quark mass $m_{\rm CEP}$ on the chemical potential $\mu$. The data points in the figure are extracted from the divergence of $\chi_{\rm L}$, while the line is determined using the leading term in the heavy quark limit, as discussed below. The increase of the critical quark mass with $\mu$, indicates that the first-order region shrinks with increasing density. This finding is consistent with the lattice results presented in Ref. [@Ejiri:2012wp; @Langelage:2009jb].
![The $\mu$ dependence of the deconfinement temperature, defined by the peak of $\chi_{\rm L}$, for $m = 1.6 \, {\rm GeV}$ and $N_f = 3$. The full line corresponds to a first-order phase transition, while the dashed line represents the pseudo-critical temperature of the crossover transition. The point indicates the location of the CEP.[]{data-label="fig:finite_mu_Td"}](finite_mu_Tc){width="3.375in"}
The relation between $m_{\rm CEP}$ and $\mu_{\rm CEP}$, shown in Fig. \[fig:finite\_mu\_cep\], can also be studied using the leading contribution to $U_Q$. At finite $\mu$, one finds
$$\begin{aligned}
\label{finite_mu_h}
{U}_Q \simeq - h(\beta\mu,\beta m,N_f) L_{\rm L},\end{aligned}$$
where $L_{\rm L}=(L+\bar L)/2$ is the longitudinal Polyakov loop [^2]. For $m/T \gg 1$, the strength of the symmetry breaking parameter is then given by
$$\begin{aligned}
\label{cosh}
h(\beta m, \beta\mu, N_f) \simeq \frac{6 N_f}{\pi^2} (\beta {m})^2 K_2(\beta {m})\cosh(\beta {\mu}),\end{aligned}$$
where $K_n$ is the modified Bessel function of the second kind.
As discussed above, the phase boundary is determined by $$\begin{aligned}
\label{critmu}
h(\beta m,\beta\mu,N_f) = h_c.\end{aligned}$$
Since $h_c$ is a universal constant, depending only on the effective Polyakov loop potential, it can be determined at $\mu=0$. Plugging in the critical quark mass $m_{\rm CEP} = 1.48 \, {\rm GeV}$, the critical temperature $T_{\rm CEP} = 0.261 \, {\rm GeV}$ for $N_f = 3$ and $\mu =0$, one finds, $h_c \approx 0.17$.
In the parameter range considered, the critical temperature at the deconfinement CEP is both $\mu$ and $N_f$ independent, and coincides with $T_{\rm CEP}$ at $\mu=0$. Thus, Eq.(\[critmu\]) defines the relation between the critical quark mass and chemical potential at the CEP
$$\begin{aligned}
\label{finite_mu_cep_formula}
\mu_{\rm CEP}= T_{\rm CEP} \cosh^{-1}( h_c/(\frac{6 N_f}{\pi^2} \left(((\beta {m}_{\rm CEP})^2 K_2(\beta {m}_{\rm CEP}))\right).\end{aligned}$$
The solution of the above equation for $N_f=3$ is shown in Fig. \[fig:finite\_mu\_cep\]. One finds a very satisfactory agreement between $m_{\rm CEP}(\mu)$ obtained from the global maximum of $\chi_{\rm L}$ and the approximate result from Eq.(\[finite\_mu\_cep\_formula\]). This indicates, that to quantify the heavy-quark phase diagram at finite density, it is sufficient to retain the leading term in the fermion thermodynamic potential.
Model predictions and lattice results
=====================================
In constructing the effective model we assume that the parameters in the gluonic potential $U_G$ are unaffected by the presence of heavy quarks, and that the coupling of quarks to the Polyakov loop is described by the one loop potential. It is important to note, however, that polarization corrections to the transverse gluons is a potential source of $m$ dependence of the parameters of $U_G$. This is because, in the Polyakov loop potential the transverse gluons are integrated out. If this effect can be neglected, the CEP transition temperature will remain approximately $N_f$ and $\mu$ independent. However, one expects that the dressing with fermion loops is important for light quarks. It is therefore interesting to validate these assumptions with first principle calculations on the lattice [@whot; @PhysRevD.60.034504].
Various lattice studies are available for the deconfinement phase transition of heavy flavors [@whot; @PhysRevD.60.034504; @Langelage:2009jb; @Karsch2001579]. However, the extrapolation to the continuum has not yet been done, and hence, a direct comparison to effective model is uncertain. Nevertheless, some valuable conclusions can be drawn from a comparison of the two approaches.
On the lattice, the fermionic determinant is usually expanded in terms of the lattice hopping parameter $\kappa$. The leading term in this expansion reads [@whot]
$$\begin{aligned}
\ln {\det [ \hat{Q}_F (\kappa) ]} &\simeq (2 N_f) (2 N_c) (2 \kappa)^{N_\tau} N_s ^3 \, L_{\rm L}.\label{lgt}\end{aligned}$$
A comparison of this equation in the mean-field approximation with the corresponding expression in the continuum model
$$\begin{aligned}
\ln {\det [ \hat{Q}_F ]}\simeq V T^3 h(\beta m,N_f) L_{\rm L},\end{aligned}$$
using $VT^3=(N_s/N_\tau)^3$, yields a relation between the symmetry breaking parameter $h$ in the continuum and the hopping parameter $\kappa$ on the lattice
$$\begin{aligned}
\label{kappa}
h(\beta m,N_f) \simeq (2 N_f) (2 N_c) (2 \kappa)^{N_\tau} N_\tau^3 .\end{aligned}$$
In particular, for $\mu = 0$, one finds
$$\begin{aligned}
\label{kappam}
(2 \kappa)^{N_\tau} N_\tau^3= \frac{(\beta {m})^2}{2\pi^2} K_2(\beta {m}).\end{aligned}$$
The right hand side of Eq.(\[kappam\]) represents a quantity in the continuum. The continuum extrapolation of the left hand side of Eq.(\[kappam\]) is however complicated, owing to the renormalization of the Polyakov loop and the bare quark masses represented by $\kappa$, in Eq.(\[lgt\]). Nevertheless, Eq.(\[kappam\]) indicates that $(2 \kappa)^{N_\tau} N_\tau^3$ is presumably better suited for the continuum extrapolation, than the frequently used $ (2\kappa)^{N_\tau}$ combination. Indeed, the lattice calculation in Ref. [@PhysRevD.60.034504], yields a very strong $N_\tau$ dependence of $(2 \kappa)^{N_\tau}$, consistent with the $N_\tau^3$ scaling implied by Eq.(\[kappam\]). Thus, Eq.(\[kappam\]) can be considered as an alternative prescription for relating the hopping parameter to the quark mass.
![The critical quark mass as a function of the quark chemical potential for $N_f = 3$. Solid points represent results obtained from the global maximum of $\chi_{\rm L}$, while the line is the solution of the Eq., with $h_c=0.17$ and $T_{\rm CEP}=0.261 \, {\rm GeV}$[]{data-label="fig:finite_mu_cep"}](finite_mu_cep){width="3.375in"}
However, the connection of $\kappa$ to the physical quark mass remains ambiguous. Various conversion formulae have been proposed to determine the value of $m_{\rm CEP}$ [@DeGrand; @PhysRevD.60.034504], showing that $1<m_{\rm CEP}< 1.5 \, {\rm GeV}$ for $N_f = 3$. The present model calculation is consistent with this mass range.
A less ambiguous extraction of the physical quark masses at the deconfinement CEP can be performed by computing the ratio of the pseudoscalar to vector meson masses, $m_{\rm PS}/m_{\rm V}$. This ratio is expected to be close to unity if the heavy quark assumption is valid. An estimate of the critical quark mass $m_{\rm CEP}$ on the lattice can then be inferred from the heavy quarkonium spectrum. This, in conjunction with the matching formula (\[kappam\]), provides a consistency check for the continuum extraction of the critical strength of the symmetry breaking term.
Conclusions
===========
The dependence of the SU(3) deconfinement transition in the presence of heavy dynamical quarks on the number of flavors and on the quark mass was studied within an effective theory. We have formulated the thermodynamic potential of interacting heavy quarks and gluons in a mean-field approach, which reproduces the equation of state and the fluctuations of the Polyakov loop in SU(3) pure gauge theory.
We have explored the phase diagram at finite temperature and density for different quark masses and number of flavors. In order to examine the deconfinement critical endpoint (CEP) and its characteristics, we have studied fluctuations of the Polyakov loop.
It was shown, that the CEP can be uniquely identified by a singularity of the longitudinal Polyakov loop susceptibility. The transverse susceptibility, on the other hand, remains finite at the CEP, and is much smaller than at the first-order deconfinement transition.
At vanishing chemical potential, the critical endpoint appears at the quark mass, $m_{\rm CEP} = 1.10, ~1.35,~ 1.48 \, {\rm GeV},$ for $N_f=1,2$ and 3 flavors, respectively. The corresponding critical temperature, $T_{\rm CEP}=0.261 \, {\rm GeV}$ is $9 \, {\rm MeV}$ below the deconfinement temperature in the pure SU(3) gauge theory.
The critical mass, $m_{\rm CEP}$, was shown to be increasing with $\mu$, whereas the critical temperature at the CEP is almost constant. Consequently, at finite density, the region of the first-order deconfinement phase transition shrinks with increasing $\mu$.
We have discussed the relation between the model and lattice results. In particular, we have argued, that in order to obtain the continuum limit for the strength of the $Z(3)$ symmetry breaking term in the hopping parameter ($\kappa$) expansion, one should consider the product, $(2 \kappa)^{N_\tau} N_\tau^3$, where $N_\tau$ is the lattice size in the temporal direction.
Finally, based on the matching criteria of the model and the lattice fermionic determinant, a formula connecting the hopping parameter and the quark mass was proposed.
Acknowledgments
===============
We acknowledge the stimulating discussions with Chihiro Sasaki, the BNL-Bielefeld lattice group, and the WHOT-QCD lattice Collaboration. P.M.L is grateful for the helpful comments from Constantia Alexandrou. K. R. acknowledges fruitful discussions with Frithjof Karsch and Rob Pisarski. P.M.L. is supported by the Frankfurt Institute for Advanced Studies (FIAS). B. F. is supported in part by the Extreme Matter Institute EMMI. K. R. acknowledges partial support of the Polish National Science Center (NCN), under Maestro grant 2013/10/A/ST2/00106. The numerical calculations have been performed on the GridEngine Cluster at GSI.
[10]{}
J. Greensite, Prog. Part. Nucl. Phys. [**51**]{}, 1 (2003). G. ’t Hooft, Nucl. Phys. B [**138**]{}, 1 (1978). L. G. Yaffe and B. Svetitsky, Phys. Rev. D [**26**]{}, 963 (1982).
L. D. McLerran and B. Svetitsky, Phys. Lett. B [**98**]{}, 195 (1981); Phys. Rev. D [**24**]{}, 450 (1981). A. M. Polyakov, Phys. Lett. B [**72**]{}, 477 (1978). K. Binder, Rep. Prog. Phys. [**50**]{}, 783 (1987).
F. Green and F. Karsch, Nucl. Phys. B [**238**]{}, 297 (1984).
P. M. Lo, B. Friman, O. Kaczmarek, K. Redlich, and C. Sasaki, Phys. Rev. D [**88**]{}, 014506 (2013).
C. McNeile, A. Bazavov, C. T. H. Davies, R. J. Dowdall, K. Hornbostel, G. P. Lepage and H. D. Trottier, Phys. Rev. D [**87**]{}, 034503 (2013). P. M. Lo, B. Friman, O. Kaczmarek, K. Redlich, and C. Sasaki, Phys. Rev. D [**88**]{}, 074502 (2013).
F. Karsch, E. Laermann, and A. Peikert, Nucl. Phys. B [**605**]{}, 579 (2001).
T. A. DeGrand and C. E. DeTar, Nucl. Phys. B [**225**]{}, 590 (1983).
N. Attig, B. Petersson, M. Wolff, R.V. Gavai Z. Phys. C - Particles and Fields [**40**]{}, 471 (1988).
J. Langelage and O. Philipsen, JHEP [**1001**]{}, 089 (2010). S. Ejiri [*et al.*]{} \[WHOT-QCD Collaboration\], PoS LATTICE [**2012**]{}, 089 (2012).
C. Alexandrou, A. Borici, A. Feo, P. de Forcrand, A. Galli, F. Jergerlehner and T. Takaishi, Phys. Rev. D [**60**]{}, 034504 (1999). G. Aarts, O. Kaczmarek, F. Karsch and I. -O. Stamatescu, Nucl. Phys. Proc. Suppl. [**106**]{}, 456 (2002).
M. Fromm, J. Langelage, S. Lottini, O. Philipsen JHEP [**01**]{}, 042 (2012).
H. Saito, S. Ejiri, S. Aoki, T. Hatsuda, K. Kanaya, Y. Maezawa, H. Ohno, and T. Umeda, Phys. Rev. D [**84**]{}, 054502 (2011) .
K. Kashiwa, R. D. Pisarski and V. V. Skokov, Phys. Rev. D [**85**]{}, 114029 (2012).
P. N. Meisinger, T. R. Miller, and M. C. Ogilvie, Phys. Rev. D [**65**]{}, 034009 (2002) . P. N. Meisinger and M. C. Ogilvie, Phys. Rev. D [**65**]{} , 056013 (2002) . P. N. Meisinger, M. C. Ogilvie and T. R. Miller, Phys. Lett. [**B**]{} 585 (2004) 149.
D. J. Gross, R. D. Pisarski and L. G. Yaffe, Rev. Mod. Phys. [**53**]{} (1981) 43.
H. T. Elze, D. E. Miller and K. Redlich, Phys. Rev. D [**35**]{}, 748 (1987).
Y. Hatta and K. Fukushima, Phys. Rev. D [**69**]{}, 097502 (2004).
C. Sasaki, B. Friman, and K. Redlich, Phys. Rev. D [**75**]{}, 074013 (2007) .
C. Ratti, M. A. Thaler, and W. Weise, Phys. Rev. D [**73**]{}, 014019 (2006).
A. Dumitru, R. D. Pisarski and D. Zschiesche, Phys. Rev. D [**72**]{}, 065008 (2005) .
[^1]: In the real sector of the Z(3) target space, the longitudinal $L_{\rm L}$ and transverse $L_{\rm T}$ parts of the Polyakov loop correspond to the real and imaginary parts respectively.
[^2]: We have neglected a subleading term in $U_Q$, which is linear in the transverse Polyakov loop, $L_{\rm T}=(\bar L-L)/2$, since $L_{\rm T}/L_{\rm L} \ll 1$ for $T > 0.2 \, {\rm GeV}$. (see Fig. \[fig:finite\_mu\_pl\])
|
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address:
- ' Department of Physics, University of California, Davis, CA 95616 '
- ' Semiconductor Physics Division, Sandia National Laboratories, Albuquerque, NM 87185-5800 '
- '(Submitted to Physical Review A)'
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-
-
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author:
- 'Antonio C. Cancio [^1] and C. Y. Fong'
- 'J. S. Nelson'
title: ' The Exchange-correlation Hole of the Si Atom, A Quantum Monte Carlo Study'
---
epsf.sty = 10000
Introduction
============
An understanding of the exchange-correlation hole and related quantities such as the pair correlation function and exchange-correlation energy is an important factor in the systematic development of accurate density functional theory (DFT) methods for quantum chemistry and solid state physics [@JG]. Much progress has been made in understanding the pair correlation function in the homogeneous electron gas or jellium, using accurate numerical techniques [@PickB; @OrtizB] and analytic modeling [@PW]. Although the closely related Coulomb hole has long been studied in atomic and molecular systems [@Coulson; @Sanders; @CHoleRev], the quantitative understanding of correlation holes in inhomogeneous systems is far from complete. In recent years there have been several efforts to fill the gaps, with a focus on characterizing properties closely tied to the development of density functional theories, such as the on-top hole [@EPB; @BPEontop] and system-averaged exchange and correlation holes [@EPB; @EBPmol].
In much of the previous work on the exchange-correlation hole in atoms and molecules, the focus has been on two electron systems and closed-shell atoms such as Be and Ne, with additional studies of more complicated molecules such as ${\rm N_2}$ and ${\rm H_2O}$ [@CHoleRev]. One class of systems that has received less attention is that of open-shell atoms. The valence shell of such an atom may consist only of a few electrons about a radially symmetric core, but with a degenerate ground state and the corresponding valence-shell structure, have interesting and quite complex features in its exchange-correlation hole. In particular, in the case of less-than-half filling, these include the exclusion-effect correlations involving three or more electrons [@Sinanoglu]. Modeling such systems requires proper attention both to orbital correlations best treated in a configuration interaction (CI) context and to dynamic correlations more amenable to a density functional description. The application of density functional theory to open-shell systems is an active area of research, with the question of the optimal treatment of degenerate ground states and of the formation of such atoms into molecules giving rise to intriguing problems [@Savin].
Recently, Variational Monte Carlo (VMC) methods combining highly accurate trial wavefunctions with Monte Carlo methods for evaluating ground state energies and wavefunctions have been developed and applied to atoms and molecules [@Umrigar; @SM; @Mitas]. The wavefunctions used in these calculations have the advantage of being simple and compact, typically recovering $85\%$ or more of the correlation energy of atoms and the dissociation energies of molecules with a few trial parameters. Unlike configuration interaction expansions they can describe with equal ease both “nondynamic" correlations and “dynamic" correlations such as the short-range cusp condition [@Kimball]. These features make this method a natural candidate for studying electron correlations, and it has been employed in recent studies of the pair correlation function in crystalline Si [@FWL; @Hood]. In this paper we study the exchange-correlation hole and pair correlation function of the valence shell of the Si atom with the VMC method. A quantitative knowledge of correlations in Si is important in improving DFT predictions for the cohesive energy, binding energies and surface characteristics for this and related technologically important materials. The Si atom is in its own right a useful laboratory for the study of electron correlations, with both local jellium-like and the nonlocal chemical properties due to the open shell structure of the atom playing important roles in determining the relevant physical and chemical properties. The paper is organized as follows: Sec. \[theory\_nxc\] provides theoretical background on the exchange-correlation hole, Sec. \[theory\] gives details of the system studied and of the calculational method. Results for the exchange-correlation hole are given in Sec. \[results\_nxc\], for the pair correlation function in Sec. \[results\_pcf\] and for correlation effects on the spin density in Sec. \[results\_spin\]. We close with a summary of the results and conclusion in Sec. \[conclusion\].
The Exchange Correlation Hole {#theory_nxc}
=============================
The exchange-correlation hole, including explicit spin dependence, is defined as: $$n_{xc}({\bf r}_0,\sigma_0; {\bf r},\sigma ) =
\frac{\displaystyle n^{(2)}({\bf r}_0,\sigma_0; {\bf r},\sigma)}
{\displaystyle n({\bf r}_0,\sigma_0)} - n({\bf r},\sigma).
\label{eqnxc}$$ Physically it describes the change in density at ${\bf r}$ and for spin component $\sigma$ from its mean value, $n({\bf r},\sigma)$, given the presence of another electron with spin $\sigma_0$ at position ${\bf r}_0$. The quantity $n^{(2)}({\bf r}_0,\sigma_0; {\bf r},\sigma)$ is the pair density, the expectation in the ground state of finding a pair of electrons at two given coordinates. It is defined in terms of an expectation of the ground state as: $$n^{(2)}({\bf r}_0,\sigma_0; {\bf r},\sigma) = \sum_{i,j\neq i}
\langle\delta({\bf r}_0 - {\bf r}_i) \delta_{\sigma_0\sigma_i}
\delta({\bf r} - {\bf r}_j) \delta_{\sigma\sigma_j}\rangle.
\label{eqnpair}$$ A closely related quantity, the pair correlation function $g({\bf r}_0,\sigma_0,{\bf r},\sigma)$, is a measure of the pair density relative to that expected for uncorrelated electrons with the same density distribution: $$g({\bf r}_0,\sigma_0; {\bf r},\sigma) =
\frac{\displaystyle n^{(2)}({\bf r}_0,\sigma_0; {\bf r},\sigma)}
{\displaystyle n({\bf r}_0,\sigma_0) n({\bf r},\sigma)}.
\label{eqpcf}$$ The importance of the exchange-correlation hole to density functional theory lies in its connection [@Exc2nxc; @Levy] to the exchange-correlation energy $E_{xc}$: $$E_{xc} = \frac{1}{2}\int\!d^3r\: n({\bf r}) \int\!d^3r^{\prime}
\int_0^1 \!d\lambda\:
\frac{\displaystyle n_{xc}({\bf r},{\bf r}^{\prime},\lambda)}
{\displaystyle |{\bf r} - {\bf r}^{\prime}|}.
\label{eqexc}$$ Here $n_{xc}({\bf r},{\bf r}^{\prime},\lambda)$ is the exchange-correlation hole, summed over spins, for the system with scaled Coulomb interaction $\lambda e^2$, with an external potential altered so that the density of the system remains unchanged. The integration over coupling constant strength $\lambda$ accounts for the kinetic energy cost of correlating electrons, weakening the strength of the integrated $n_{xc}$ with respect to its value at $\lambda\!=\!1$. Although $E_{xc}$ is in principle determined from a knowledge of the single particle density alone, this dependence is in general not easy to determine beyond the local density approximation. The exchange-correlation hole has a wealth of features which may be used to test and improve theoretical models of the exchange-correlation energy. As a result, many attempts to systematically improve density functional theory have this function as a starting point [@JG; @GJL; @Becke; @PBW-GGA].
In this paper, we will discuss the full coupling-constant ($\lambda\!=\!1$) case for $n_{xc}$. Although the coupling-constant integrated quantity is most directly connected to the density functional theory, the full coupling-constant case is interesting in itself, as its average is an experimentally measurable expectation of the ground state [@Thakkar]. It is also an essential ingredient in modern hybrid methods which combine elements of density functional theories and conventional Hartree-Fock methods [@Becke]. The quantity $n_{xc}$ is often analyzed by a decomposition into an exchange component, $n_x$, corresponding to the $\lambda=0$ or noninteracting system, which describes the correlations between particles arising from the Pauli exclusion principle, and a correlation component, $n_c$, determined by taking the difference between the fully interacting and noninteracting cases, which describes the additional correlation due to the Coulomb interaction between electrons [@Coulomb]. We study the explicit spin decomposition of $n_{xc}$. This choice is useful for understanding the correlation response of an open-shell atom for which the ground state has nontrivial differences in its spin components. Spin decomposition falls roughly along the lines of exchange and correlation: exchange affects only particles with the same spin, and Coulomb correlations, though also present in the same-spin case, are most noticeable in the opposite-spin channel. Some initial insight into the nature of the exchange-correlation hole in atoms and other finite systems can be obtained by considering limiting cases. The exchange-correlation hole about a reference electron in the homogeneous electron gas is isotropic, that is, a function solely of the distance from the electron, and localized about the electron with a radius determined by the average interelectron distance. Consequently, in a system of slowly varying density, the the local density approximation (LDA) holds, in which the hole is determined by the density (or each spin-component of the density if these are different) at the location of the reference electron and “moves" with the position of the reference electron. In atoms or molecules, important correlation effects often involve a pair excitation from the noninteracting ground-state into a finite number of lowlying, perhaps nearly degenerate, excited states. In this case, the resulting correlation hole is dependent on the shape of the ground-state orbitals vacated and the excited-state orbitals occupied and is largely unsensitive to the position of the reference electron. In real systems, the exchange-correlation hole will contain aspects of both limiting cases, with those that “move" with the position of the electron termed “dynamic" correlations and the orbital correlations insensitive to electron position termed “nondynamic". In open-shell atoms both play important roles.
Calculation Approach {#theory}
====================
Correlated description of the Si atom
-------------------------------------
We focus on correlations in the valence shell of the atom with the core electrons replaced by norm-conserving [*ab initio*]{} nonlocal pseudopotentials derived from LDA calculations [@HSC]. In this case the valence shell of the atom is described by the Hamiltonian $$\begin{aligned}
\nonumber H & = & \sum_i \left( \frac{\nabla_i^2} {2m} + \sum_{l_i, m_i}
V_{l_i}( r_i ) |l_i m_i\!><\!l_i m_i| \right) \\
& & + \sum_{i<j} \frac{e^2} {|{\bf r_i} - {\bf r_j}|}\end{aligned}$$ where the sums are over the valence electrons. $V_{l}$ is the nonlocal pseudopotential and $|l_im_i\!><\!l_im_i|$ the single particle projection operator onto the state with total angular momentum $l_i$ and $z$-axis projection $m_i$. The final term is the intravalence Coulomb interaction.
The Si atom has a nine-fold degenerate $(3s^2 3p^2)$ $^3P$ ground state. By maximizing the spin projection, this state can be represented by a single Slater determinant, consisting of a majority-spin component, here chosen to be spin up, with one $3s$ and two $3p$ orbitals, and a minority or down-spin component with one $3s$ electron. There then remain two angular momentum projections that lead to physically significant differences in $n_{xc}$. The $m_l\!=\!0$ projection has $p_x$ and $p_y$ orbitals and a “pancake" like shape, while the $m_l\!=\!\pm1$ projections have a “cigar" shape, with $p_0$ and $p_{\pm}$ orbitals. This distinction does not play a role in determining $E_{xc}$ for an atom, given the invariance of the energy to rotations of the atom. The presence of a quantization axis in the formation of a Si bond breaks this invariance, so that the projection-specific behavior of $n_{xc}$ becomes important in determining accurate molecular binding energies. It thus should be useful as a test of density functional theory, which typically overestimates binding energies by about 1 eV [@JG; @EBPmol]. Results in this paper focus on the $m_l\!=\!0$ projection which provides a clear comparison between the situation perpendicular to and parallel to the $p$ orbitals, though calculations were done on the other projection for comparison.
To calculate the exchange-correlation hole variationally we start with a Slater-Jastrow trial wavefunction $$\psi = exp(-F)\prod_{\sigma} D_{\sigma},
\label{eqhfpsi}$$ with $D_{\sigma}$ being a Slater determinant for spin component $\sigma$ and $F$ a Jastrow correlation factor. All single-particle orbitals are obtained from the same local DFT program that determined our pseudopotentials. We use a Boys and Handy form [@SM; @Boys] for $F$, which includes electron-electron, electron-nucleus and electron-electron-nucleus correlations expanded in a basis set of correlation functions: $$F = \sum_{l,m,n} c_{lmn} \sum_{i\neq j} (\bar{r}_{i}^l \bar{r}_{j}^m +
\bar{r}_{i}^m \bar{r}_{i}^l) \bar{r}_{ij}^n.
\label{eqjastrow}$$ The basis functions are $\bar{r}_{i} = br_i / (1 + br_i)$ and $\bar{r}_{ij} = dr_{ij} / (1 + dr_{ij})$, where $r_{ij}$ is the distance between a pair of electrons, $r_i$ is the distance between electron $i$ and the atom center. The terms $b$, $d$ and $c_{lmn}$ are variational parameters. The lowest order $\bar{r}_{ij}$ term is set separately for opposite and same-spin electron correlations to satisfy the short range electron-electron cusp condition [@Kimball] for each case. Higher order terms treat longer range effects and are determined without distinguishing electron spin. Electron-ion terms in the correlation function correct for the tendency of interelectron correlations to expand the volume of the atom and provide density-dependent corrections to the electron-electron correlation [@SM]. Since the valence shell of Si is less than half-filled, orbital or nondynamic correlations may be important [@Sinanoglu]. These can be incorporated with a multideterminant extension of the Slater-Jastrow wavefunction: $$\psi = exp(-F)\sum_{\alpha} \eta_{\alpha}
\prod_{\sigma} D^{\alpha}_{\sigma}.
\label{eqcipsi}$$
Method of Calculation
---------------------
The Variational Monte Carlo method [@CepK] is used to calculate the ground-state energy, derivatives with respect to variational parameters, and other expectation values. The heart of the method lies in the judicious statistical sampling of integration points to obtain an estimate of the many-body integrals involved in evaluating expectations of the Slater-Jastrow trial wavefunction. This estimate is limited in accuracy by statistical noise; however if the trial wavefunction is a good approximation to the ground-state wavefunction this noise can be very easily managed with a relatively small number of configurations [@Umrigar]. Optimized wavefunctions are obtained by minimizing the variance of the energy [@Umrigar]. With a trial wavefunction of 18 expansion terms including two set by the same- and opposite- spin cusp conditions, up to $l+m+n=6$ in the basis function expansion, we obtain a value of 3.8028(2) a.u. for the ground-state valence-shell energy. The correlation energy, measured with respect to a noninteracting ground-state energy of 3.7188 a.u., is 97$\%$ of the correlation energy obtained from Green’s function Monte Carlo and 95$\%$ of that obtained from CI using the same nonlocal pseudopotential [@Mitas]. A similar calculation starting from a two determinant reference point \[Eq. (\[eqcipsi\])\], adding the $3p_z^23p_x3p_y$ excited state to the noninteracting ground-state configuration, resulted in a modest improvement in energy to 3.8041(2) a.u.or 96.6$\%$ of the valence-shell correlation energy with respect to CI. To calculate correlation functions, we measure spin-decomposed single-particle densities $n({\bf r}, \sigma)$ and conditional densities $n({\bf r},\sigma|{\bf r}_0,\sigma_0)$. The spin-dependent conditional density is defined as the ground-state density distribution as a function of spin $\sigma$ and position ${\bf r}$ of the $N\!-\!1$ other particles given one with spin ${\sigma_0}$ fixed at ${\bf r_0}$. The difference between the conditional and “unrestricted" densities, gives the spin-dependent exchange-correlation hole, Eq. (\[eqnxc\]), $$n_{xc}({\bf r_0},\sigma_0;{\bf r},\sigma) =
n({\bf r},\sigma|{\bf r}_0,\sigma_0) - n({\bf r},\sigma).
\label{eqcdens}$$ Separate calculations are done to measure the density and the conditional density for various values of ${\bf r_0}$ and $\sigma_0$. These expectations are first calculated exactly for the Slater-determinant wavefunction (setting $F\!=\!0$ in our trial wavefunction). Then, the difference between the expectations obtained with the Slater determinant and the fully interacting wavefunctions is measured statistically using Monte Carlo sampling and the method of correlated estimates [@CepK]. This technique is an efficient means to estimate statistically the change in the expectation of an observable under a small perturbation of the Hamiltonian or of the variational parameters of the wavefunction, taking advantage of the high degree of correlation between the two expectations to reduce noise in the difference of their statistical estimates. In the present case, the correlation hole, which describes the difference between the correlated and Slater-determinant exchange-correlation holes, is fairly small compared to the exchange-correlation hole. It is thus a reasonable assumption that the correlated estimation approach should improve the sampling efficiency for this quantity. In practice, the procedure was observed to reduce statistical errors in our data by a factor of 5, and roughly $10^5$ random samples of the wavefunction sufficed to obtain expectation values.
The expectations for density and conditional density were expanded in a plane wave basis, taking the average of $\sum_i exp( -i{\bf G}\cdot {\bf r}_i )$ for a set of plane waves up to a 32 Ry cutoff, on a supercell 18 Bohr radii ($a_B$) in length. The coefficients were symmetrized and fast-Fourier transformed to obtain real-space densities. This expansion provides smooth profiles despite the statistical noise of the sampling. On the other hand the short-wavelength cutoff causes an unrealistic rounding off of the correlation hole cusp at short interparticle distances, and spurious oscillations at low densities. At present this cutoff is the largest error in our calculation. In the case of the exchange-correlation hole this error is notable mostly in the cusp region. A more complete discussion of errors is presented in Sec. \[results\_pcf\] in regard to the pair correlation function which is more sensitive to the cutoff error than the exchange-correlation hole.
Results for the Exchange-Correlation Hole {#results_nxc}
=========================================
The exchange hole in DFT is obtained by evaluating $n_{xc}$ from Eq. (\[eqnxc\]) for the Slater determinant wavefunction that minimizes the energy in the noninteracting ($\lambda\!=\!0$) limit: with the Coulomb interaction replaced by a single-particle potential that reproduces the true ground-state density [@Levy]. In practice the Slater determinant of local DFT orbitals used in our calculations produces a density that differs from our VMC density by a few percent. In terms of these orbitals, the exchange hole is: $$n_{x}({\bf r}_0,\sigma_0; {\bf r},\sigma) =
\frac{ - \left| \sum_{\alpha=1}^{N_\sigma}
\psi^*_{\alpha}({\bf r}_0,\sigma) \psi_\alpha({\bf r},\sigma)
\right|^2 \delta_{\sigma \sigma_0} }
{\displaystyle n({\bf r}_0,\sigma_0)}$$ where the sum runs over all the occupied single-particle orbitals $\psi_\alpha$ for the spin component $\sigma$. This expression, as a function of ${\bf r}$ for fixed ${\bf r}_0$ ([*i.e.*]{}, interpreting the hole as the change in density of the system at ${\bf r}$ given a particle observed at ${\bf r_0}$) has the form of minus the probability density of a hybrid atomic orbital. That is, it describes a normalized linear combination of single-particle orbitals with coefficients $c_{\alpha} = \psi^*_\alpha({\bf r}_0,\sigma_0) / \sqrt{n({\bf r}_0,\sigma_0)}$. This choice of $c_{\alpha}$ for each orbital $\psi_{\alpha}$ represents the unique linear combination of orbitals that maximizes the probability for an electron to be observed at ${\bf r_0}$ and spin $\sigma_0$ (conversely giving zero likelihood for any other electron to be observed at that point.) Finally the integral over ${\bf r}$ of the exchange hole is $-1$, as it measures the integrated difference in density between the $(N\!-\!1)$-electron system given one electron fixed at ${\bf r_0}$ and the full $N$-electron system.
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For a spin-up particle on the $z$ axis in the $L_z\!=\!0$ projection, this exchange-hole orbital is a $3s$ state since the occupied $p$ states ($p_x$,$p_y$) are in the $x$-$y$ plane. The exchange hole is completely insensitive to the electron position on this axis, given only one possible orbital from which to construct it. The situation along this axis thus corresponds to an extreme departure from the “dynamic" picture of the exchange hole derived from the homogeneous electron gas. For a particle on the $x$ axis, the hole is a combination of $s$ and $p_x$ orbitals. We find that the occupation probability $|c_p|^2$ of the $p$ orbital to be between $60\%$ and $75\%$ in the region $(0.5~a_B\! <\! x\! < \!2.5~a_B)$ of peak density along the $x$ axis; this is roughly equivalent to that of a $sp^2$ hybrid orbital (which is two-thirds $p$) oriented in the direction of the fixed particle. The slight variation in the $c_p$ acts to keep the hole more or less centered on the reference electron in the region of peak density on the $x$-$y$ plane. The exchange hole as a result is more sensitive to the exact location of particles in this plane, and therefore more efficient in screening them, than along the $z$ axis, leading to significant differences in the correlation holes in the two cases as well. This transition from “nondynamic" screening along the axis perpendicular to the occupied $p$ orbitals to a more “dynamic" screening in the plane occupied by them, particularly as it occurs at peak density in the valence shell, constitutes a major difference between the $n_{xc}$ of Si and that of closed-shell atoms.
In Figs. \[atomfig1\] and \[atomfig2\] we plot the exchange-correlation hole about a spin-up electron fixed at the point of peak density parallel to and perpendicular to the two $3p$ orbitals of the $m_l=0$ (pancake) projection of the Si atom. The single-configuration Slater-Jastrow wavefunction \[Eq. (\[eqhfpsi\])\] is used. Each plot shows the response to this electron in the $x$-$z$ plane, that is, the plane cutting through the center of the atom at the plot origin, with one axis ($x$) parallel to and one ($z$) perpendicular to the occupied $p$ orbitals. The hole is split into same spin (a), opposite spin (b) and total (c) response. The comparison between these two situations shows the dramatic anisotropy in $n_{xc}$ reflecting that of the exchange hole. For a spin-up electron placed on the $x$ axis, Fig. \[atomfig1\], the exchange-correlation hole is dominated by the $sp^2$-like exchange hole. The additional effects of Coulomb correlation on the same-spin channel are hard to detect, while the opposite-spin correlation hole is small (contributing $14\%$ of the total on-top hole, or value of the hole at zero interparticle separation.) It is largely confined to a narrow region about the electron, indicating that the $sp^2$ hybrid hole screens the electron efficiently. Fig. \[atomfig2\] shows the exchange-correlation hole of a spin-up electron on the $z$ axis, perpendicular to the two $p$ orbitals. With the addition of correlation, the same-spin hole (a) loses the rotational symmetry of the $3s$ state that characterizes the exchange hole, with the polarization of the two $3p$ orbitals creating a double valley on either side of the reference electron. The opposite-spin hole (b) shows the polarization of the $3s$ spin-down orbital, with a well centered about the fixed electron and a strong dipole response at longer range. The total $z$-axis exchange-correlation hole, (c), shows a smooth interpolation of the two spin contributions leading to a large crescent-shaped area near the electron from which the other electrons are repelled. The correlation hole contributes considerably to the total exchange-correlation hole, with up to $40\%$ of the total on-top hole due to correlation. Although the $3s$-orbital exchange hole is highly nonlocal and does not efficiently screen the electron, the correlation contribution goes a long way to make the total hole more local. In addition to the obvious differences in $n_{xc}$ along each axis due to the differences in the exchange hole, Figs. \[atomfig1\] and \[atomfig2\] reveal subtle differences in the opposite-spin hole. The extent of the orientational anisotropy in the opposite-spin hole can be better visualized by plotting the hole along the $x$ axis for the up-spin electron fixed on the $x$ axis and along the $z$ axis for the electron fixed on the $z$ axis, cutting through the minima and maxima of the contour plots Figs. \[atomfig1\](b) and \[atomfig2\](b). These are represented as solid lines in Figs. \[atomfig3\](a) and (b). The most notable difference between the two cases is the height of the peak on the side of the atom opposite the reference electron, which is three times as large along the $z$ axis (b) as on the $x$ axis (a). In addition the minimum is slightly deeper for the $z$-axis case. Additionally, in Figs. \[atomfig3\](a) and (b) we show trends in the opposite-spin correlation hole as one gradually removes the reference electron from the atom. In addition to the 1.4 $a_B$ case discussed above, we place a spin-up electron on the $x$ axis (a) and the $z$ axis (b), at a distance of 2.0 and 4.0 $a_B$ from atom center, and plot the on-axis response of the spin-down electron as long- and short-dashed lines respectively. These three reference radii correspond to placing an electron at the peak valence density along either axis, at the average radius from the atom, and at a low density point outside the atom respectively. As the electron is moved to lower densities, the shape of the minimum slowly gets wider and shallower, consistent with trends in the homogeneous electron gas. The position of the hole minimum stays near the atom, and thus increasingly more off center with respect to the electron. This is expected: the correlation hole, measuring the change in density in the presence of an electron at some reference point, can have an absolute value no greater than the density itself. Along the $x$ axis, this trend to a shallow off-centered hole is correlated with a gradual increase of peak height on opposite side of atom. For the $z$-axis case the peak height remains roughly constant as the electron is removed.
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It is interesting to compare these results to recent analytic studies of the asymptotic limit of the exchange-correlation hole of Be and Ne [@EPB; @EBPasymp]. In the limit of an electron far removed from the atom, $n_{xc}$ measures the collapse of the remaining $N\!-\!1$ electrons to an eigenstate of the positive ion. The correlation component of $n_{xc}$ measures the change in electron density due to the reduced Coulomb reduction between the remaining electrons. At intermediate distances, a few atomic radii from the atom center, $n_{xc}$ also shows a dipole polarization of the atom, or a reduction of electron density from the side of the atom nearer the reference electron to the farther side, more so for the more polarizable Be than for Ne. The collapse to the ion in the asymptotic limit results in a correlation hole peaked in the center with a minimum around the edge of the atom. This may possibly be seen in our data for the 4.0 $a_B$ $x$-axis case of Fig. \[atomfig3\](a), where the correlation-hole profile develops a peak on both sides of the ion core. The correlation hole along both axes shows evidence of a dipole polarization of the valence shell when the reference electron is moved outside the atom, with a larger response on the “Be-like" $z$ axis than along the “Ne-like" $x$ axis. The contrast between Si and closed-shell atoms is that this polarization of the valence shell does not die out when a reference electron on the $z$ axis is moved into the center of the valence shell, if anything becoming more pronounced, while the polarization is damped out along the $x$ axis under the same circumstances. An added complexity for Si is that the final state of the system after removing an up-spin electron in different directions corresponds to different atomic configurations: $3s3p^2$ for the $z$-axis case (removal of the up-spin $3s$ orbital) but $3s^23p$ for the $x$ axis. Thus significant differences between the correlation and exchange holes induced by electrons on different axes remain even for electrons asymptotically far from the atom.
We have also studied the local features of the exchange-correlation holes plotted, especially the opposite-spin on-top correlation hole, $n_{xc}({\bf r}, \sigma; {\bf r}, -\!\sigma)$. This measures the reduction in the opposite-spin electron density at the location of a given electron. As the correlation hole is deepest when the Coulomb repulsion is largest, the on-top hole also represents the lower bound for the correlation hole at any position in the atom [@ontop]. In previous papers, it was observed that this feature was well described in several small atoms [@EPB; @BPEontop] and in Hooke’s atom [@PSB] by a local spin-density (LSD) ansatz, using the on-top hole of the homogeneous electron gas corresponding to the density and magnetization at the location of the electron pair: $$n^{LSD}_{xc}({\bf r}, {\bf r}) = n({\bf r})
( g^{heg}( 0, n({\bf r}), \zeta({\bf r}) ) - 1 ).$$ Here $g^{heg}(u,n,\zeta)$ is the pair correlation function of the homogeneous electron gas at density $n$, magnetization $\zeta = (n_{\uparrow} - n_{\downarrow})/n$, and interparticle separation $u$. In this sense, it represents a link between the homogeneous electron gas and dynamic correlations in many inhomogeneous systems [@PEBS]. In Figs. \[atomfig3\](a) and (b) we plot the LSD on-top hole, using homogeneous electron gas values from Ref. [@Yasuhara], at each point along the $x$ and $z$ axes (crosses). These are compared to on-top holes of the Slater-Jastrow wavefunction \[Eq. (\[eqhfpsi\])\] evaluated specifically at the location of the electron for the correlation holes shown (circles at 1.4, 2.0, and 4.0 $a_B$). These values for the Slater-Jastrow on-top hole are calculated directly without recourse to a plane-wave expansion, using VMC with a constrained pair random walk [@CanCYF], and are exact within a negligible statistical error. The finite energy cutoffs employed for the rest of the data tend to lose sharp features of the correlation hole, in particular that of the cusp condition at zero interparticle separation. The resulting disagreement with the exact (for the trial wavefunction used) values is at worst $20\%$ and usually better. The LSD model agrees fairly well with the VMC data, within $10-20\%$ for most of the points calculated here, faithfully following the observed trends with respect to electron position. It provides a fairly reasonable lower limit for the correlation hole at all points in the atom. A calculation of the on-top hole about the spin-down electron was also done, with similar agreement with theory. It is interesting to consider in further detail the anisotropy in the opposite-spin contribution to $n_{xc}$ at peak density. In the cases shown in Figs. \[atomfig1\]-\[atomfig3\], it can be interpreted as the measure of the polarization of the down-spin electron density in the presence of an up-spin electron. At first glance, one might guess that this down-spin polarization should roughly be invariant with respect to the angular orientation of the up-spin electron since one is measuring the response of the down-spin $s$-orbital to the angle-independent Coulomb interaction. More precisely, one measures the correlations induced by the optimized Jastrow factor which depend only on interelectron and electron-nucleus distances. The difference in the opposite-spin holes Figs. \[atomfig1\](b) and \[atomfig2\](b), is thus a direct reflection of the influence of the determinantal structure of the three up-spin electrons on the correlation response of the down-spin one.
In the LDA, the influence of the environment on interparticle correlation is modeled by the variation with local density of the correlation hole, taken to be that of the homogeneous electron gas [@OrtizB; @PW] at the local density. Typically the largest absolute and relative effects in this model are felt at the on-top hole. At this point it seems to describe the observed behavior quite well, with the variation in density going from the maximum of the $p$ orbital on the $x$ axis to the open $z$ axis accounting for the difference in the peak density on-top holes. However, the large discrepancy between the maxima of the peak density $z$-axis and $x$-axis hole, the solid lines in Fig. \[atomfig3\]a and \[atomfig3\]b, as well as their relatively large size, is inconsistent with this model. Neither does it seem easily explained by local gradient corrections, as the two points are located at a density maximum and a saddle point respectively, where any correction would be only second order in the gradient and therefore quite small.
Within the context of a variational wavefunction calculation, a mechanism that can introduce an orientation dependence into the correlation hole of an open shell atom is the three-electron “exclusion effect" or Fermi correlation [@Sinanoglu] in configuration-interaction theory, which should be observable for second row atoms with less than half filling in the $3p$ valence shell. For Si in the $L_z\!=\!0$ projection considered here, there is a “nondynamic" contribution to the correlation between electrons along the $z$ axis due to a $3s^2$ to $3p_z^2$ excitation that is forbidden for the corresponding $x$- or $y$-axis analogs by Pauli exclusion. This contribution to the correlation hole about an up-spin electron on the $z$ or “unoccupied" axis, and the absence of the same about one on “occupied" axes leads to a difference in the holes observed with reference points on these axes. Since this particular configuration is within the $3p$ valence shell, and thus fairly close to the ground state energy, it is natural to expect it to play a significant role in the observed difference in response.
What is striking here is that the observed orientational dependence of $n_{xc}$ is obtained with a single Slater-determinant configuration modified by a Jastrow factor that depends only on interparticle distance and the distance of particles from the core. There are no explicit orbital correlations in this trial wavefunction \[Eq. (\[eqhfpsi\])\]. The Slater-Jastrow wavefunction, coupling a Jastrow factor with a determinant formed from an incomplete set of valence orbitals, in principle could have a nonzero overlap with the $3p_z^23p_x3p_y$ configuration allowed in the normal exclusion-effect model, and no overlap with the excluded excitations. However, this would be only one of an infinite number of configurations implicitly included in the Slater-Jastrow wavefunction, tied together by a “dynamic" correlation factor that gives no particular variational weight to any one configuration. Thus it is unlikely that it would fully account for the observed differences in the response along the occupied and open axes [@Node].
On the other hand, a more general argument involving screening can be invoked to explain the orientational anisotropy in our results. If the up-spin electron is placed on the $z$-axis, its exchange hole is not centered on the electron position, as the electron occupies a $3s$ orbital with an isotropic spatial extension throughout the atom’s valence shell. The net effect of the electron plus its exchange hole has the nature of a dipole field along the $z$ axis of the atom, with an imperfectly screened negative charge on the side of the atom nearest the electron and a positively charged region due to the exchange hole on the other side. The large peak in the down spin electron’s response to the up-spin electron can be viewed then as a dipole induced by the dipole field formed by the imperfect exchange screening of the up-spin electron. In the $x$-axis case, the exchange hole is centered and localized on the electron; the other two electrons are in $sp^2$ orbitals blocking out the rest of the valence shell. Therefore the up-spin electron is efficiently screened by its exchange hole, and in addition, there is no “easy" direction for redistributing the down-spin electron density. In such a picture the $3s^2 \rightarrow 3p^2$ excitation should play an important but not exclusive role in the overall dipole response along the $z$ axis. This effect is observable in a Slater-Jastrow wavefunction because the Jastrow factor includes correlations simultaneously between all particle pairs. Thus, the correlation response of the down-spin electron to a fixed up-spin electron involves not only their mutual Coulomb repulsion but that of the other up-spin electrons as well. The inhomogeneous distribution of these electrons about the fixed electron combined with the effect of the external potential in essence create a Coulomb interaction with the exchange hole.
Results for the Pair Correlation Function {#results_pcf}
=========================================
Single Configuration Wavefunction
---------------------------------
In a system such as an atom, the pair correlation function $g({\bf r_0},\sigma_0; {\bf r}, \sigma)$ is a valuable tool because, given the large variation in the density over the length-scale of the exchange-correlation hole, the shape of the hole is to a large extent determined by the variation in density. The pair correlation function can distinguish the intrinsic effects of correlation by eliminating any features that are simply proportional to the density.
In Fig. \[atomfig4\] we plot the pair correlation function (PCF) for several of the cases considered previously. Specifically, we show $g({\bf r_0},\uparrow; {\bf r}, \downarrow)$ fixing a spin-up particle at the peak in the valence density, 1.4 $a_B$ on the $x$ axis (a) and the $z$ axis (b) and plotting the variation with respect to the spin-down electron in the $x$-$z$ plane. These results are obtained by dividing $n_{xc}$ as plotted in Figs. \[atomfig1\](b) and \[atomfig2\](b) respectively by the spin-down density, with a constant shift of 1 due to the differing conventions in their definitions \[Eqs. (\[eqnxc\]) and (\[eqpcf\])\]. The value $g\!=\!1$ corresponding to $n_{xc}\!=\!0$ (shown as a thick contour in Fig. \[atomfig4\]) denotes the boundary between the region in which the spin-down density has been reduced ($g\!<\!1$) and that in which it has been enhanced ($g\!>\!1$) in the presence of the spin-up reference electron. As described in the next section the PCF can be significantly affected by statistical error at low density, so that the plot range is restricted to higher density regions where the statistical error is less than half a contour increment in either direction.
The contours of the pair correlation function in the vicinity of the spin-up electron on the $x$ axis are nearly circular and centered on the electron, showing the spin-down electron repelled from it without significant directional bias. The $g\!=\!1$ contour is slightly oblong in shape, indicating that at longer distances there is a slight bias towards shifting electron density to the high density areas on the far side of the atom ($x < 0$). A similar plot cutting through the $x$-$y$ plane (ie, cutting through both the $p$ orbitals along a plane perpendicular to the one shown) shows similar results except that the $g\!=\!1$ contour is more circular in shape. This picture is fairly consistent with that of a homogeneous electron gas PCF which depends only on the distance between particles, and indicates that the underlying physical processes are quite similar: high energy “dynamic" correlations that are quickly screened out over the radius of the exchange hole.
In comparison, the PCF in Fig. \[atomfig4\](b), showing the response to the spin-up electron on the $z$ axis shows a much greater departure from isotropy. The function has a noticeable dipole component with respect to the atom center, with a peak in $g$ directly opposite the minimum on the $z$ axis. The $g\!=\!1$ contour has a much more shallow curvature than that of Fig. \[atomfig4\](a), so that it is no longer centered on the spin-up electron, but rather coincides roughly with the $z\!=\!0$ plane in the high density region of the atom. (For the $3s^23p_x3p_y$ configuration of Si, the PCFwith a reference electron on the $z$ axis is rotationally invariant about the $z$ axis so that the contours shown in the plot can be extended to surfaces of rotation in three dimensions.) In the near vicinity of the fixed electron, the pair correlation function is no longer a function of the distance between the two coordinates ${\bf r}$ and ${\bf r_0}$ alone, but has taken on a noticeable angular dependence as well. Thus the slope of the well near 1.4 $a_B$ on the $z$ axis is steeper towards the center of the atom, shallower towards the outer edge of the atom. Although such a shape could in principle be accounted for in a region where the gradient of the density is large, using a gradient expansion of the homogeneous electron gas [@Langreth], it cannot be explained along these lines in the current situation since it occurs at local saddle point in the density (the peak of the density along the $z$ axis). These features in the PCF confirm the existence of a genuine directional anisotropy in the response to an electron located perpendicular to the occupied $p$ orbitals, and suggest that the explanation lies in a nonlocal mechanism such as the response to a poorly localized $3s$ exchange hole.
Along this line, it is instructive to consider the exchange-correlation hole about the lone down-spin electron. In this case the exchange hole is of necessity due to the $3s$ orbital regardless of the electron position, and one can expect significant departures from a compact, isotropic PCF. In the case of a down-spin electron on the $z$ axis, we observe the PCF that is very close to that about an up-spin electron, with perhaps a slightly larger dipole component. In both cases, the PCF indicates the presence of a significant dipole response that compensates for the poorly screening $3s$ exchange hole. The PCF for a down-spin electron on the $x>0$ axis has a quite complicated pattern outside the on-top hole region – neither roughly isotropic like Fig. \[atomfig4\](a) or with simple dipole anisotropy like Fig. \[atomfig4\](b). There are several unconnected regions where the electron density is enhanced: surrounding the electron for $x>0$ and focused on the $x$ axis opposite the electron for $x<0$. In this case the exchange hole should not screen the electron efficiently, but there is no lowlying $3p$ orbital with which to construct a response, so that there is a possibility that the contributions of higher-order orbitals might be observable.
Two Configuration Wavefunction
------------------------------
Since we have observed a significant change in shape of the opposite-spin PCF with respect the angular orientation of the reference electron, using a single-determinant Slater-Jastrow wavefunction, it is interesting to include explicitly the $3s^2 \rightarrow 3p_z^2$ excitation that is predicted in CI theory to play a prominent role in producing such a behavior in the Si atom. We consider the multideterminant wavefunction $\psi_{CI}$ that consists of a $3p_z^23p_x3p_y$ excited state configuration as well as the noninteracting ground state. This wavefunction can be treated alone or used as a starting point for adding further correlations via the Jastrow factor \[Eq. (\[eqcipsi\])\]; in either case the mixing amplitude $\eta_1$ for the excited state is variationally optimized, along with the Jastrow parameters for the latter case. We obtain a mixing coefficient $\eta_1$ of 0.132 and total energy 3.7263(18) a.u. in the former case and 0.056 and 3.8041(2) a.u. in the latter.
The nondynamic correlation modeled by the two-configuration wavefunction alone, $\psi_{CI}$, is to first order in $\eta_1$ a dipole-dipole correlation: $$g_{CI}({\bf r_0},\uparrow;{\bf r},\downarrow) = 1 + 2\eta_1
\frac{z_0}{r_0}\frac{z}{r}
\frac{R_p(r_0)}{R_s(r_0)}\frac{R_p(r)}{R_s(r)}
\label{eqgci}$$ where $R_p$ and $R_s$ are the radial $3s$ and $3p$ orbitals. The signature of this PCF in a contour plot with the cut through the $x$-$z$ plane and $r_0$ fixed on the $z$ axis is a series of roughly straight lines arranged antisymmetrically about the $z\!=\!0$ plane, with the $g\!=\!1$ contour at $z\!=\!0$. The shape of the function is independent of the position of the fixed particle, with the only change being in the overall amplitude. For one electron fixed on the $x$ axis, on the node of the $p_z$ orbital, the PCF is to first order zero.
In Fig. \[atomfig4\](c) we show a contour plot for the opposite-spin PCF of the combined CI plus Jastrow wavefunction, $\psi_{CI-J}$, for the same plot parameters as Fig. \[atomfig4\](b): with the up-spin electron fixed at 1.4 $a_B$ on the $z$ axis. In this case the occupation probability of the excited-state Slater-Jastrow wavefunction is very small, 0.3$\%$, and the change in the total ground-state energy from the single-determinant Slater-Jastrow wavefunction is correspondingly small. On the other hand the explicit addition of the nondynamic correlation increases the correlation energy by 1.6$\%$ and has a significant impact on the shape of the hole. The PCF has a clear dipole-dipole signature, exaggerating the spatial anisotropy already visible in the one determinant case. The $g\!=\!1$ contour is closer to the $z\!=\!0$ axis, and the short-range well about the fixed particle, which for a particle near the peak of the valence shell density could be expected to be fairly deep and isotropic, has been reduced to a shallow and open-ended dip. As expected from the form of the nondynamic correlation, little change was observed in the PCF for a particle fixed on the $x$ axis.
A quantitative comparison of $g$ for the above cases is also instructive and is shown in Figs. \[atomfig3\](c) and (d). We plot $g({\bf r_0},\uparrow;{\bf r},\downarrow)$ varying $r$ along the $x$ axis and fixing ${\bf r}_0$ at 1.4 $a_B$ on the $x$ axis (c) and the analogous situation for the $z$-axis (d). Error bars on the statistical measurements of these quantities are plotted, as well as the plane-wave cutoff dependence for the $x$-axis case. The converged result on the $x$ axis (solid line) is well localized about the reference particle’s position, with an enhancement of particle density on the opposite side of the atom.
The results for the $z$ axis, Fig. \[atomfig3\](d) include the PCF for the three different trial wavefunctions discussed above: the single Slater determinant plus Jastrow factor, $\psi_{S-J}$ (dashed line), the two-configuration wavefunction $\psi_{CI}$ (dotted line) and the same multiplied by a Jastrow factor $\psi_{CI-J}$ (solid line). In comparison to the most accurate result, $\psi_{CI-J}$, $\psi_{CI}$ predicts with some success the long-wavelength polarization response to the electron at ${\bf r_0}$, determining how much the electron density is pushed from one side of the atom to the other. The major difference is the absence of the small dip in the immediate vicinity of the electron due to the cusp condition, which naturally is not obtained from the two-configuration wavefunction. In contrast, $\psi_{S-J}$, which does not include explicit orbital-dependent correlations but can be optimized to obtain accurate results for short interparticle distances, is nearly identical to $\psi_{CI-J}$ in the vicinity of the reference electron but underestimates the long-wavelength polarization of the atom, obtaining about $70\%$ of the PCF of $\psi_{CI-J}$ on the other side of the atom. The mixing coefficient $\eta_1$ for the excited-state configuration was reduced by about $60\%$ when the Jastrow factor was added to the multiconfiguration result, indicating that part of the polarization of the atom produced by the simple two configuration wavefunction is already accounted for by the Jastrow factor. In each case the $z$-axis PCF leads to a much larger peak on the far side of the atom than the $x$-axis case.
Error Analysis
--------------
There are three sources of error in our calculation: statistical error in taking Monte Carlo estimates, the finite plane-wave cutoff of the data, and finally the variational bias due to the discrepancies of the trial from the true ground state wavefunction. The first two are closely connected and to some extent can be regulated; the third is harder to assess.
In a typical Monte Carlo calculation, sample points for evaluating the density or other single-particle expectation in a given region of space are generated with frequency proportional to the density itself. This leads to statistically precise measurements of the density at high density and large relative errors that vary roughly as $1/\sqrt{n(r)}$ at the vanishingly low densities outside the atom. Calculating the VMC expectations, $<\sum_i exp( -i{\bf G}\cdot {\bf r}_i )>$, of a set of plane-waves periodic on a supercell is equivalent to taking the Fourier transform of a histogram distribution of statistical sample points on that cell. The statistical outliers from the poorly sampled, asymptotic low-density region show up in the Fourier transform as a noise background independent of energy. This noise can to some degree be controlled by imposing a finite cutoff in reciprocal space. However, too small a cutoff leads to spurious long wavelength oscillations and is particularly poor for the short-distance region of the hole where the cusp condition results in a long range tail in the reciprocal space. We find that a good balance between controlling statistical and plane-wave error can be found by choosing a cutoff when the statistical average of the plane-wave component of the density is roughly equal to the statistical noise in its calculation. For a sampling size of around $10^5$ independent configurations, this cutoff limit proves to be about 32 Ry for a resolution of 1.0 $a_B$.
The statistical error of the Monte Carlo sampling was measured both for the individual plane-wave components of the density and conditional density, and for their real-space counterparts by measuring the standard deviation of these quantities over 10 to 20 independent runs. As shown in Fig. \[atomfig3\](c) and (d), the error bars of the PCF do in fact vary roughly as $1/\sqrt{n(r)}$ with well controlled errors in the peak density region, increasing to arbitrarily large values as one moves outside the atom. With the cutoff used, statistical errors in the PCF are limited to a value of less than 0.05 (out of a range in the PCF of the order of 1.0) for particles within 3.5 $a_B$ of the atom center, which in effect provides the limits in the plots of the PCF in Fig. \[atomfig4\].
The convergence of the plane-wave expansion was checked by plotting the PCF as a function of cutoff energy. A typical result is shown in Fig. \[atomfig3\](c) where the $x$-axis PCF for a particle fixed at 1.4 $a_B$ on the axis is plotted, for cutoff energies of 20, 28 and 32 Ry. The exact on-top hole has been measured by a direct VMC calculation and is plotted as a circle. The 20 Ry calculation shows clear deviations from the higher energy cutoff data, particularly in the region of the on-top hole, ie., in the immediate vicinity of the fixed electron, where the cusp condition contributes a high-energy tail to the exchange-correlation hole. The agreement between the two higher energy plots is well within statistical error except in the core region of the atom and in the low density tails. The 32 Ry case has not yet converged in the on-top region to the exact on-top value plotted as a circle, indicative of the slow convergence of the plane-wave expansion to the on-top hole cusp.
A final source of error is from the effect of the deviation of the variational trial wavefunctions \[Eqs. (\[eqhfpsi\]) and (\[eqcipsi\])\] from the exact ground state. The ground-state energy, being variationally optimized, is typically determined with much less error than other expectations (though one may expect that those important in the determination of the energy, such as the density and $n_{xc}$, should still be robust. An example of this problem is demonstrated in Fig. \[atomfig4\] in which two variationally optimized wavefunctions having relatively insignificant differences in total energy, give PCF’s with noticeable differences for an electron fixed on the unoccupied $z$ axis. The variational method is in general more sensitive to errors in the wavefunction at high density as this region contributes the most to the variational energy. Thus we expect worse errors in the PCFwhen one or both coordinates are at low density. In order to gauge possible errors arising from variational bias in our wavefunction, we study the change in the PCF and in $n_{xc}$ for several choices of trial wavefunction. Specifically we used trial wavefunctions with four, ten and eighteen Boys and Handy basis functions, corresponding to keeping expansion terms of order $o=l+m+n$ up to 2, 4 and 6 respectively in our Jastrow function. The lowest order function has, in addition to two terms that set the cusp condition for each spin component, only one electron-electron and one electron-nucleus term. The correlation energy of each wavefunction is 0.0640(8), 0.0812(4), and 0.0840(2) a.u. respectively, in comparison to 0.0877(2) a.u. using the multideterminant Slater-Jastrow wavefunction and 0.0883 a.u. for the CI calculation of Ref. [@Mitas].
In Fig. \[atomfig5\] we plot the opposite-spin PCF using these trial wavefunctions for two cases previously considered in Sec. \[results\_nxc\]: (a) an up-spin electron placed at 1.4 $a_B$ on the $x$ axis and (b) an up-spin electron placed at 4.0 $a_B$ on the $z$ axis. As in Fig. \[atomfig3\] we present a cut through the atom center and the reference electron, thus showing the minimum and maximum of the PCF. The first case represents a probable best-case situation, with the reference electron placed at peak density on an axis occupied by a $3p$ orbital, so that nondynamic correlations due to the open-shell structure should be neglible. In contrast the second case presents the worst case scenario: at low density along the unoccupied axis. The behavior of the PCF in the asymptotic low-density region of the atom is dominated by the statistical error of our plane-wave basis and is not shown.
In our best-case scenario, (a), the PCF near the reference electron is very well obtained with even the crudest model used ($o=2$). On the far side of the atom with respect to the reference electron position, there is a noticeable increase in the magnitude of the peak of the PCF with the increase in accuracy of the wavefunction. As expected, the addition of nondynamic correlations in form of a two-determinant Slater-Jastrow wavefunction (solid line) has no noticeable effect. An investigation of the corresponding contour plots, cutting through the $x$-$z$ plane as shown in Fig. \[atomfig4\], shows that the depth and spatial extent of the PCF is obtained with the lowest order wavefunction and varies only slightly with increase in basis set. The major difference is the gradual adjustment of the $g=1$ contour from a more isotropic shape to the elliptical one shown in Fig. \[atomfig4\](a), which accounts for the gradual increase in the peak on the far side of the $x$ axis as the $g=1$ contour moves towards the atom center.
The worst case scenario (b) shows a far greater degree of disagreement between wavefunctions. The three one-determinant Slater-Jastrow wavefunctions agree fairly well with each other, with an increasing percentage of the electron density removed from the near side of the atom and placed on the far side. The peak height on the far side is roughly the same as for the 1.4 $a_B$, high-density case shown in Fig. \[atomfig3\](d). The introduction of the nondynamic $3s^2\!\rightarrow\!3p^2$ excitation into the Slater-Jastrow wavefunction (solid line) leads to a dramatic change in the shape of the PCF. The peak of the PCF from the base value of $g=1$ representing no change due to correlations is increased by a factor of three, and only at the on-top hole (not shown) is there no significant change in the PCF. In comparison, the PCF of the optimized two-determinant wavefunction $\psi_{CI}$ discussed in Sec. \[results\_pcf\] is shown as a dot-dashed line. The two-determinant Slater-Jastrow wavefunction is an interpolation between the two limiting cases, favoring $\psi_{CI}$ at most locations.
An explanation of the “gigantic" features in the nondynamic part of the PCF in Fig. \[atomfig5\](b) comes from the observation that the ratio $zR_p(r) / R_s(r)$ between the $3p$ and $3s$ orbital, which determines the shape of the nondynamic PCF to first order \[Eq. \[eqgci\]\], increases exponentially at large distances along the $z$ axis. As a result, in the asymptotic region (that is, either the response to a reference electron outside the atom or the asymptotic tails of the response to a reference electron at high density), the nondynamic correlation introduced with the $3s^2\!\rightarrow\!3p_z^2$ substitution is no longer a small perturbation even with a small mixing parameter. The Jastrow factor basis functions however are chosen to tend to a constant at either large electron-electron or electron-nucleus distances. It is quite likely that none of the cases studied accurately represents the true asymptotic behavior of the correlation function. They should rather be considered to provide a qualitative idea of the PCF as well as a sense of the range of behavior it should reasonably lie within.
It is interesting to note that the agreement between the various PCF’s is far closer for the high density case on the $z$ axis plotted in Fig. \[atomfig3\](d). This indicates the greater weight of the exchange-correlation hole at high density in determining the total exchange-correlation energy, and the corresponding robustness of its determination even with qualitatively different wavefunctions. Given the large degree of variation in the asymptotic limit, it would be interesting to study the effect of including a larger number of configurations in a multideterminantal Slater-Jastrow wavefunction or include the effects of backflow or multielectron coordinates [@SM] into the evaluation of orbitals. At higher densities we expect the effect of such improvements in the wavefunction would be to add further refinements in the shape of the PCF along the $z$ axis, but with much less change in basic features such as its range and magnitude.
Correlation effects on the spin density {#results_spin}
=======================================
In addition to the exchange-correlation hole, the spin components of the single-particle density change with the inclusion of Coulomb correlations. In principle, the Kohn-Sham equations used to derive the density in DFT should give the exact ground-state radial density, even for a degenerate ground-state [@Savin]; however there is no such principle for the spin components of the density. Thus a change observed in the spin components of the density that does not alter the total radial density can be considered an intrinsic feature of Coulomb correlation and not simply due to the inaccuracy of the LDA density. As with $n_{xc}$, a prominent feature of the change in the spin-dependent density, $\Delta n({\bf r},\sigma)$, is anisotropy with respect to the fully occupied $x$ and open $z$ axes. This anisotropy in $\Delta n({\bf r},\sigma)$ may help to shed light on the mechanisms underlying the anisotropy in $n_{xc}$.
In Fig. \[atomfig3\], the spin-down density for the single determinant Slater-Jastrow wavefunction is plotted as a dotted line along the $x$ axis (a) and $z$ axis (b). Note that in the noninteracting wavefunction this density is that of the $3s$ orbital and thus the same along each axis. However, in the interacting case, anisotropy in the correlation response reduces the density along the $x$ axis by roughly 6% and increases it by the same amount along the $z$-axis. This change is shown in Fig. \[atomfig6\] (solid line) along with the change in the spin-up density (long-dashed line), with the right side of the plot showing a cut through the $x$ axis and the left side a cut through the $z$-axis. The corresponding density changes using the two-determinant Slater-Jastrow wavefunction ($\psi_{CI-J}$) are plotted as dotted and dashed lines. In both cases, the change in the spin-up density has a qualitative trend opposite to that of the spin-down density, showing a large enhancement in density near the peak of the $3p_x$ orbital and a drop in density along the $z$ axis. Thus, the changes in the two spin components cancel out partially in the total density, but add to the total spin density, defined as $m({\bf r}) = n({\bf r},\uparrow)-n({\bf r},\downarrow)$. In addition there are observable effects of a small reduction in the radius of the atom that occurs with the addition of the Jastrow correlation term, particularly a reduction in up-spin density at a large distance from the atom and an enhancement of the down-spin density in the core region. A convergence test along the lines of that for $n_{xc}$ in the previous section was carried out for the spin components of the density. The qualitative trends are repeatable with the less accurate Jastrow factors but with considerably slower quantitative convergence than for $n_{xc}$.
An anisotropic shift in spin density can be generated from the nondynamic Fermi correlation discussed in the previous sections: the mixing of the noninteracting ground-state with a $3s^2\! \rightarrow\! 3p_z^2$-substitution excited state and the lack thereof due to Pauli exclusion for the $3p_x$ and $3p_y$ analogs. Using the two-determinant wavefunction $\psi_{CI}$, the resulting change in each spin component of the density is $$\Delta n({\bf r},\uparrow) = \Delta n({\bf r},\downarrow) =
\eta_1^2 (|\psi_{3p_z}({\bf r})|^2 - |\psi_{3s}({\bf r})|^2)$$ where $\psi_{3s}$ and $\psi_{3p_z}$ are the wavefunctions of the $3s$ and $3p_z$ orbitals and $\eta_1$ the excited state probability amplitude. This function, shown as a dash-dotted line in Fig \[atomfig6\], corresponds to a density enhancement along the $z$ axis and reduction in the $x$-$y$ plane for [*both*]{} spin-components for a net zero change in the spin density $m({\bf r})$, in marked contrast to the mutually opposing changes of the two spin-components of the Monte Carlo data. Given the optimal value of $\eta_1^2 = 0.017$ for the occupation number of the excited-state configuration, one finds that the magnitude of the change in either spin component of the density is much smaller than observed, and qualitatively in the wrong direction for the up-spin case. When the nondynamic $s^2 \rightarrow p_z^2$ excitation is included into the Slater-Jastrow wavefunction (dotted and dashed lines), both the up-spin and down-spin densities are enhanced slightly along the $z$ axis, but the qualitative picture remains unchanged – as one might expect from the very small value of $\eta_1^2=0.003$ that was optimal for this case. For these reasons, it is unlikely that the observed anisotropic change in the spin components of the density can be explained by this type of mechanism.
On the other hand, a simple explanation of these results lies in that the observed change in the spin density reduces the spatial overlap between the two spin components. As the correlation energy is predominantly determined by the spatial correlation between opposite-spin electrons, such a change in spin density can reduce the correlation energy in a way that reduces the total energy if the total density remains unchanged. This correlation effect is similar to that of the unrestricted Hartree-Fock method [@Fulde] in which the energy of a system like ${\rm H_2}$ can be lowered from its Hartree-Fock value by breaking a symmetry of the ground state to induce the spatial separation of opposite spin-components of the density. In the Si atom, since the ground state is degenerate and lacks the symmetry of the Hamiltonian, it already has a nonzero spin density; multiplication by a Jastrow factor does not break the symmetry of the ground state in a substantial way [@HFUm]. The change in spin-density reflects the effect of the Coulomb interaction which induces the spatial separation between opposite spin electrons, in a system in which the different spin components are to some degree already spatially separated in the noninteracting state. In contrast to a filled-shell atom or other spin-unpolarized system, this spatial separation of opposite spins appears not only as a correlation hole but as a change in the mean density as well. Consistent with this picture, the direction of the change in the $L_z\!=\!0$ projection of the Si atom enhances the absolute difference in the spin density $|n({\bf r},\uparrow)-n({\bf r},\downarrow)|$ that already exists in the Hartree-Fock wavefunction, where up-spin electrons occupies $p_x$ and $p_y$ orbitals while the lone down-spin electron does not.
Summary and Conclusion {#conclusion}
======================
We have calculated the exchange-correlation hole of the valence shell of the ground state of the Si atom as a function of spin decomposition, using the Variational Monte Carlo method. This relatively simple four electron system, restricted to a single valence shell, nevertheless shows a rich variety of phenomena in exchange and correlation not present in closed-shell atoms.
The incomplete filling of the valence shell of the open-shell atom leads to dramatic anisotropy and nonlocality in the exchange hole, even at peak densities in the valence shell. This is accompanied by a significant compensating anisotropy in the correlation hole making the total exchange-correlation hole more (but not completely) local and isotropic.
Our paper has focused mostly on the exchange-correlation hole about a majority (up) spin electron. In this case, as one goes from a reference point at peak density along an “occupied" axis, along which one of the occupied $3p$ orbitals is oriented, to one on the “open" $z$ axis perpendicular to the $3p$ orbitals, the exchange hole changes from an efficiently screening $3sp^2$-like to a poorly screening $3s$ character. As a result, we observe in the response of the minority (down) spin density a significant “dipole" shift or shift of density from one side of the atom to the other, that occurs when the up-spin electron is placed in the peak density position on the open axis and not when it is placed in the center of a $p$ orbital. This difference shows up in the pair correlation function as a difference in shape, with that on the “occupied" axis being a modest distortion of the isotropic shape of a dynamic correlation, and that on the $z$ axis showing a marked dipole component along the $z$ axis. The dipole polarization observed is also notable in that it occurs not only for a reference electron outside the atom but at the peak along the $z$ axis, where the local gradient is zero; it is therefore a truly nonlocal feature not amenable to modeling by an expansion in the local density gradient.
The explicit inclusion of nondynamic correlations into our wavefunction enhances the difference between open and occupied-axis response, particularly when the reference electron is moved outside the atom. This is due to the exclusion effect in which the $3s^2 \rightarrow 3p^2$ substitution is allowed along the open axis and excluded along the occupied ones. Nevertheless, the existence of such a difference in the single-determinant Slater-Jastrow wavefunction, given the structure of the Jastrow factor, seems better explained in terms of the screening of the exchange hole, in which the Coulomb interaction of the down spin electron with the up-spin $3s$ exchange hole causes the dipole response along the open axis. In this case, it is the lowlying $3p$ orbital that best compensates for the poor screening of the exchange hole. It should thus play an important if not exclusive role in inducing the anisotropy of the opposite spin correlation hole.
A second major effect of the open-shell structure of the atom upon its response to the Coulomb interaction is the change of the spin components of the density with the addition of correlations. The down-spin density is pushed inwards and onto the unoccupied axis and the up-spin density pushed off the $z$ axis and onto the peak of the $p$ orbitals in the $x$-$y$ plane, resulting in a reduction of the spatial overlap between the two spin components in a way which leaves the radial density largely unchanged. The standard exclusion effect resulting from the $3s^2\!\rightarrow\!3p_z^2$ substitution produces a net change in density that is both quantitatively too small and qualitatively incorrect. However, both the anisotropic features of the exchange-correlation hole and the changes in spin density essentially stem from the same effect: the tendency of the Coulomb interaction to induce the spatial separation of opposite spin electrons in the context of the spatially anisotropic and spin-polarized structure of the degenerate Si ground state. We have only briefly discussed the $n_{xc}$ for a down-spin electron, where in addition to the screening of the exchange hole (always $3s$-like), the anisotropy of the determinantal structure of the up-spin electrons plays an important role. The correlation response to the down-spin electron on the occupied axis combines a compensation for poor screening by the exchange hole and the absence of a lowlying $3p$-orbital component from the correlation hole on account of Pauli exclusion. We have observed in this situation subtle structural properties in the PCF that merit further investigation.
In contrast to longer ranged features of the correlation hole we find that its on-top value – the reduction of electron density in the immediate vicinity of an electron – is reproduced by a local density ansatz within 10-20%, over a fairly wide range in density and magnetization, and is otherwise insensitive to the structure of the atom. Given the energetically reasonable shape of the LDA $n_{xc}$ and that it satisfies the short-range cusp condition and global particle-sum rule of the true $n_{xc}$ in addition to the approximate fit to the Si on-top hole, it is reasonable to expect that it will provide a good approximation for the angle- or system-averaged $n_{xc}$, that is, after averaging out many of the subtle angle or position dependent features studied here [@EPB]. Nevertheless, the complex spin-dependent phenomena observed in this paper point to the inherent difficulty of systematically improving on local or semilocal density functional theories in systems with nontrivial valence-shell structure, such as open shell atoms or multiply-bonded molecules.
Our results provide support for several recent hybrid approaches to DFT. The combination of a short ranged on-top hole that is fairly well modeled by local density functional theory and longer-ranged exchange and exclusion effects that are not indicates the usefulness of density functional schemes which include the on-top hole as a basic component [@PEBS] or involve the hybridization of short-ranged local or semilocal density functionals with a more accurate treatment of longer ranged correlations using RPA [@Langreth] or CI [@Savin]. The tendency of the correlation hole to cancel out the anisotropy in the exchange hole, and increase the sensitivity of the exchange-correlation hole to electron position, along with the quality of the on-top hole in the LSD approximation, lends support to recent hybrid approaches [@Becke] which mix the exact exchange hole with an local DFT approximation of the full-coupling constant exchange-correlation hole. These methods rely on the assumption that the full coupling constant limit of the integral in Eq. (\[eqexc\]) is more likely to be amenable to approximation by the isotropic, localized $n_{xc}$ of the homogeneous electron gas than the noninteracting limit dominated by exchange. In the case of the open-shell atom exclusion effects quite effectively cancel out the gross anisotropy in the exchange hole caused by the open-shell structure, at least within the region of peak valence density.
On the other hand, the changes in spin-density that we observe indicate that a standard Hartree-Fock or restricted CI basis set may be a less desirable starting point for implementing hybrid DFT methods in spin-polarized systems than, for example, a generalized Hartree-Fock approach that would allow for anisotropic distortions in the spin density due to Coulomb correlations. Also, it seems possible that more could be done to obtain accurate correlation energies within the local spin-density approximation with the incorporation of projection specific information.
One of us (A. C. Cancio) would like to thank Kieron Burke for helpful discussions. This work was supported by Sandia National Laboratories contract AP-7094 and the Campus Laboratories Collaboration of the University of California.
[99]{}
R. O. Jones and O. Gunnarson, Rev. Mod. Phys. [**61**]{}, 689 (1989). Warren E. Pickett and Jeremy Q. Broughton, Phys. Rev. B [**48**]{}, 14859 (1993-II). G. Ortiz and P. Ballone, Phys. Rev. B [**50**]{}, 1391 (1994). J. P. Perdew and Y. Wang, Phys. Rev. B [**46**]{}, 12947 (1992-II); J. P. Perdew, K. Burke and Wang, Phys. Rev. B [**54**]{}, 16533 (1996). C. A. Coulson and A. H. Neilson, Proc. Phys. Soc. [**78**]{}, 831 (1961). J. Sanders and K. E. Banyard, J. Chem Phys. [**96**]{}, 4536 (1992). M. A. Buijse and E. J. Baerends, in [*Density Functional Theory of Molecules, Clusters and Solids*]{} ed. D. E. Ellis, (Kluwer Academic, Dordrecht, 1995). M. Ernzerhof, J. P. Perdew and K. Burke, in [*Density Functional Theory*]{}, ed. R Nalewajski, (Springer-Verlag, Berlin, 1996). K. Burke, J. P. Perdew and M. Ernzerhof, J. Chem. Phys. [**109**]{}, 3760 (1998). M. Ernzerhof, K. Burke and J.P. Perdew, in “Recent Developments and Applications of Modern Density Functional Theory", ed. J.M. Seminario (Elsevier, 1996). O. Sinanoglu, Adv. Chem. Phys. [**6**]{}, 315 (1968). A. Savin, Int. J. Quantum Chem. Symp. [**22**]{}, 59 (1996); A. Savin, in “Recent Developments and Applications of Modern Density Functional Theory", ed. J.M. Seminario (Elsevier, 1996). C. J. Umrigar, K. G. Wilson, and J. W. Wilkins, Phys. Rev. Lett. [**60**]{}, 1719 (1988). K. E. Schmidt and J. W. Moskowitz, J. Chem. Phys. [**93**]{}, 4172 (1990). L. Mitáš, E. L. Shirley and D. M. Ceperley, J. Chem. Phys. [**95**]{} 3467, (1991); L. Mitáš in [*Computer Simulation Studies in Condensed Matter Physics V*]{}, ed. by D. P. Landau, K. K. Mon, and H. B. Schuttler (Springer Verlag, Berlin, 1993). J. C. Kimball, Phys. Rev. A [**7**]{}, 1648 (1973). S. Fahy, X. W. Wang and Steven G. Louie, Phys. Rev. Lett. [**61**]{}, 1631 (1988). Randolph Q. Hood et al., Phys. Rev. Lett., [**78**]{} 3350 (1997); Randall Q. Hood et al., Phys. Rev. B, [**57**]{} 8972 (1998). D. C. Langreth and J. P. Perdew, Solid State Commun. [**17**]{}, 1425 (1975). M. Levy, in “Recent Developments and Applications of Modern Density Functional Theory", ed. J.M. Seminario (Elsevier, 1996). O. Gunnarson, M. Jonson and B. I. Lundqvist, Phys. Rev. B [**20**]{}, 3136 (1979). A. D. Becke, J. Chem. Phys. [**98**]{}, 1372 (1993); A. D. Becke, J. Chem. Phys. [**98**]{}, 5648 (1993). J. P. Perdew, K. Burke and Y. Wang, Phys. Rev. B [**54**]{} 16533 (1996). A. J. Thakkar et al., Int. J. Quantum Chem. [**26**]{}, 157 (1984). The exchange hole of DFT is nearly identical with the Fermi hole commonly studied in quantum chemistry. However, the noninteracting ground-state in DFT and the corresponding single-particle orbitals are determined by the Kohn-Sham equation and in principle obtain the density of the exact ground state, while the Fermi hole is usually defined in terms of the Hartree-Fock ground state. Similarly, the correlation hole defined with respect to the full coupling constant exchange-correlation hole is the analog of the Coulomb hole of Ref. [@Coulson]. D. R. Hamann, M. Schlüter and C. Chiang, Phys. Rev. Lett. [**43**]{}, 1494 (1979); G. B. Bachelet, D. R. Hamann and M. Schlüter, Phys. Rev. B [**26**]{}, 4199 (1982). N. C. Handy, J. Chem. Phys. [**58**]{}, 279 (1973). D. Ceperley and M. H. Kalos in [*Monte Carlo Methods in Statistical Physics*]{}, edited by K. Binder (Springer Verlag, Berlin 1979). M. Ernzerhof, K. Burke and J.P. Perdew, J. Chem. Phys. [**105**]{}, 2798 (1996). Note that this argument refers to a perspective complementary to that held elsewhere in this paper and in the literature. One measures the correlation hole at a fixed position in the atom in response to a reference electron somewhere else, and varies the position of the [*reference*]{} electron. The corresponding hole measured at the fixed position should be largest when the reference electron is moved on top it. J. P. Perdew, A. Savin, and K. Burke, Phys. Rev. A [**51**]{}, 4531 (1995). J. P. Perdew, M. Ernzerhof, K. Burke and A. Savin, Int. J. Quantum Chem. [**61**]{}, 197 (1997). H. Yasuhara, Solid State Commun. [**11**]{}, 1481 (1972). A. C. Cancio and C. Y. Fong, to be published. In principle one should be able to measure the importance of this configuration by a calculation of the overlap between it and the Slater-Jastrow ground state; the significantly different nodal structure of the two states makes a VMC calculation of this quantity unreliable in practice. D. C. Langreth and J. P. Perdew, Phys. Rev. B [**21**]{}, 5469 (1980). see, for example, Peter Fulde, [*Electron Correlations in Molecules and Solids*]{}, (Springer-Verlag, Berlin, 1991). Because of the different treatment of opposite and same-spin cusp conditions in the Jastrow factor of Slater-Jastrow wavefunctions, the total spin quantum number $S^2$ will in general not preserved (C. J. Huang, C. Filippi and C. J. Umrigar, J. Chem. Phys. [**108**]{} 8838, (1998).) Imposing a spin-independent cusp condition was observed to change the up-spin component of the density noticeably, but preserved fairly closely the degree and anisotropy of spatial separation between spin-components induced by correlations.
[^1]: Present Address: School of Physics, Georgia Tech, Atlanta GA 30332
|
---
abstract: 'We perform direct numerical simulations of shock-wave/boundary-layer interactions (SBLI) at Mach number $M_{\infty} = 1.7$ to investigate the influence of the state of the incoming boundary layer on the interaction properties. We reproduce and extend the flow conditions of the experiments performed by @giepman16, in which a spatially evolving laminar boundary layer over a flat plate is initially tripped by an array of distributed roughness elements and impinged further downstream by an oblique shock wave. Four SBLI cases are considered, based on two different shock impingement locations along the streamwise direction, corresponding to transitional and turbulent interactions, and two different shock strengths, corresponding to flow deflection angles $\phi=3^o$ and $\phi=6^o$. We find that, for all flow cases, shock induced separation is not observed, the boundary layer remains attached at $\phi = 3^{\circ}$ and close to incipient separation at $\phi = 6^{\circ}$, independent of the state of the incoming boundary layer. We characterize the regions of instantaneous separation by computing the statistical probability ($\overline{\gamma}$) of the wall points with local flow reversal. The analysis shows that the turbulent interactions are characterized by a higher peak of $\overline{\gamma}$, although the region of separation is slightly wider in the transitional interaction cases. The extent of the interaction zone is mainly determined by the strength of the shock wave, and the state of the incoming boundary layer has little influence on the interaction length scale $L$. The scaling analysis for $L$ and the separation criterion developed by @souverein13 for turbulent interactions are found to be equally applicable for the transitional interactions. The findings of this work suggest that a transitional interaction might be the optimal solution for practical SBLI applications, as it removes the large separation bubble typical of laminar interactions and reduces the extent of the high-friction region associated with an incoming turbulent boundary layer.'
author:
- 'R. Quadros[^1] and M. Bernardini[^2]'
bibliography:
- 'Reference.bib'
title: 'Numerical investigation of supersonic shock-wave/boundary-layer interaction in transitional and turbulent regime'
---
Nomenclature {#nomenclature .unnumbered}
============
---------------------- --- -----------------------------------------------------------------------------------------
$M_{\infty}$ = freestream Mach number
$c_f$ = skin friction coefficient
$c_p$ = specific heat at constant pressure
$k_c$ = thermal conductivity
$G_3$ = scaling parameter for the interaction lengthscale
$k$ = height of roughness elements
$L$ = interaction length scale
$L_x$ = domain length in streamwise direction
$L_y$ = domain length in wall-normal direction
$L_x,L_y,L_z$ = size of the computational domain in the streamwise, wall-normal and spanwise directions
$P$ = mean pressure
$Pr$ = molecular Prandtl number
$p_{w_{rms}}$ = rms value of wall pressure fluctuation
$q_{\infty}$ = freestream dynamic pressure
$Re_\theta$ = Reynolds number based on momentum thickness
$Re_{x_{t}}$ = Reynolds number based on the inlet-trip distance
$R$ = specific gas constant
$S_e^*$ = separation criterion
$T$ = mean temperature
$U$ = mean velocity
$x,y,z$ = Cartesian coordinates in the streamwise, wall-normal and spanwise directions
$x_s$ = distance from the domain inlet to the shock impingement location
$x_t$ = distance from the domain inlet to the trip location
$\beta$ = shock angle
$\delta_v$ = viscous length scale
$\delta_{in}$ = inlet boundary-layer thickness
$\delta_{95}$ = boundary-layer thickness based on 95 % of the external velocity
$\delta^*_i$ = incompressible displacement thickness
$\theta_i$ = incompressible momentum thickness
$H_i$ = incompressible shape factor
$\gamma$ = specific heat ratio
$\mu$ = molecular viscosity
$\overline{\gamma} $ = statistical probability of reverse flow
$\phi$ = flow deflection angle
$\rho$ = density
$\tau$ = wall shear stress
---------------------- --- -----------------------------------------------------------------------------------------
\
subscripts {#subscripts .unnumbered}
----------
---------- --- -----------------------------
$\infty$ = evaluated in the freestream
$k$ = evaluated at roughness edge
$w$ = evaluated at wall
---------- --- -----------------------------
\
superscripts {#superscripts .unnumbered}
------------
----- --- -----------------------------
$+$ = normalization in wall units
----- --- -----------------------------
\
Introduction {#sec_intro}
============
Shock-wave/boundary-layer interactions (SBLI) are commonly observed in high speed engineering applications such as air intakes, turbo-machinery cascades, helicopter blades, supersonic nozzles, and launch vehicles. Shock waves can be useful for compressing the incoming flow, enhancing the turbulence mixing and increasing the internal energy of the flow. However, their interaction with an incoming boundary layer could result in boundary-layer separation, high-wall heat flux and surface pressure, and induction of large scale instabilities [@dolling01; @dupont06; @touber09; @souverein_10_b; @clemens14]. SBLI is also responsible for unsteady vortex shedding and shock/vortex interaction, which are the major reasons for broadband noise emission.
The nature of the incoming boundary layer which interacts with the shock has a significant impact on the flow topology and on the aerodynamic performance of the aerospace vehicle. A laminar boundary layer is more susceptible to separation when it encounters a shock wave because of its low resistance to the adverse pressure gradient created by the impinging shock [@adamson_messiter80; @delery85]. Such a boundary-layer separation could be a greater challenge in hypersonic intakes, where strong interactions occur leading to a reduced mass flow rate [@babinsky09]. This laminar interaction has been studied in detail through experiments [@hakkinen59; @giepman15], numerical simulations [@degrez87; @katzer89; @yao07] and theory [@gadd56]. @katzer89 showed that the separation size has a linear growth with the impinging shock strength and that, in agreement with the free interaction concept [@chapmanetal_57], at low Reynolds numbers the separation size increases with Reynolds number and decreases with Mach number.
A possible strategy to reduce the separation size is to energize the incoming boundary layer to make it turbulent. A turbulent boundary layer better sustains the adverse pressure gradient created by shock impingement, minimizing and in some cases suppressing the flow separation [@greene70; @stollery75; @delery86]. Various techniques have been used in the past to energize the incoming boundary layer, which include suction and injection [@gai77; @krogmann85], slots [@holden05; @smith04; @raghu87] and vortex generators [@mccormick93; @anderson06]. Although a turbulent boundary layer could reduce the shock-induced separation size, an early onset of turbulence has the disadvantage of increasing the skin friction and the associated drag.
In recent years, there has been a renewed interest in studying the interaction of a transitional boundary layer with a shock wave, motivated by the hope that a transitional interaction could bridge the gap between the large separation size obtained in a laminar interaction and the high friction drag associated with a turbulent interaction. Some of the recent studies that have experimentally investigated the interaction of a shock with a transitional boundary layer include those by @sandham14 [@davidson14; @davidson15; @giepman15]. @giepman15 studied the influence of the boundary-layer state (laminar, transitional and turbulent) on the properties of the interaction with an oblique impinging shock wave. The incoming laminar boundary layer transitioned to turbulence due to acoustic disturbances in the flow, and the oblique shock was impinged at varying locations based on the type of interaction desired. The shock-impingement point in the transitional interaction case was selected at a location with intermittency of about 50 %, and it resulted in a small boundary-layer separation. In a follow-up work, @giepman16 used tripping devices to promote the boundary-layer transition ahead of the interaction. They analyzed the effectiveness of three tripping devices namely stepwise trip, zigzag strip and a patch of distributed roughness. The zigzag strip and the distributed-roughness patch were found to be more effective in energizing the boundary layer and suppressing the separation region, mainly due to their three-dimensional shape.
There has been a limited number of numerical studies on transitional SBLI in the literature, and the only prominent one pertains to the hypersonic regime. @sandham14 carried out direct numerical simulations of an oblique shock impinging on a transitional boundary layer at $M_\infty=6$, and compared the numerical results with available experimental data. They triggered the boundary-layer transition by adding disturbances to the density field at the domain inlet. The shock was impinged at various locations for varying intermittencies, and they observed higher values of wall heat transfer for the transitional interactions as compared to the fully turbulent cases. The objective of the present work is to numerically investigate the flow physics of transitional supersonic shock/boundary-layer interaction. In our simulations we use hemispherical roughness elements to trip an incoming laminar boundary layer which is impinged by an oblique shock at varying locations corresponding to transitional or turbulent conditions. The flow configuration is chosen to match some of the experiments of @giepman16. We also analyze the effect of varying the shock strength on the transitional interaction by increasing the shock generator angle. The main aim is to highlight the advantages (if any) of a transitional interaction on the suppression of mean and instantaneous shock-induced separation, and also to bring out the key properties of the interaction in terms of boundary-layer growth and interaction length scales. The paper is organized as follows. The numerical set up and the methodology are described in section \[sec:methodology\]. Results pertaining to the case of the tripped boundary layer without any impinging shock are presented in section \[sec:bl\] along with a comparison with available experimental data. The effect of shock impingement on the transitional and turbulent boundary layers is described in section \[sec:sh\]. Conclusions are finally provided in section \[sec:conclusion\].
Methodology {#sec:methodology}
===========
Flow configuration {#subsec:flow}
------------------
\[\]\[\]\[1.25\][$L_x$]{} \[\]\[\]\[1.25\][$L_y$]{} \[\]\[\]\[1.25\][$L_z$]{} \[\]\[\]\[1.25\][$x_t$]{} \[\]\[\]\[1.25\][$x_s$]{} ![Schematic overview of the computational setup for the DNS cases.[]{data-label="SCHEMATIC3D"}](figs/SCHEMATIC_LATEST2 "fig:")
A schematic view of the flow configuration investigated is shown in figure \[SCHEMATIC3D\]. The overall size of the computational domain is $L_x \times L_y \times L_z = 120 \delta_{in} \times 35 \delta_{in} \times 10 \delta_{in}$ in the streamwise (x), wall-normal (y) and spanwise (z) directions, respectively, $\delta_{in}$ being the thickness (based on 99 % of the external velocity) of the laminar boundary layer imposed at the inflow station. At the domain inlet, the Reynolds number based on the momentum thickness is $Re_\theta=6424$ and the freestream Mach number is $M_{\infty} = 1.7$. A strip of roughness elements of width $4.5 \delta_{in}$ is centered around a streamwise distance $x_t=31\delta_{in}$ from the domain inlet. Ten hemispherical elements of height $k = 0.5 \, \delta_{95}$ are randomly distributed along the entire spanwise width. Here, $\delta_{95}$ is the thickness (based on 95 % of the external velocity) of the boundary layer at the trip location. The Reynolds number at the trip is $Re_{x_t}= 1.4 \times 10^6$, which coincides with the experimental value considered by @giepman16. The corresponding roughness Reynolds number, i.e. the Reynolds number associated with the element height $k$ and quantities evaluated at the roughness edge, is $Re_k = 3.1 \times 10^3$, a value sufficiently high to trigger roughness-induced transition [@tani69; @bernardini14].
We run a total of five DNS, whose parameters are listed in table \[TAB\_CASES\]. Case BL-TRIP corresponds to the simulation of a boundary layer tripped by roughness elements without any impinging shock. The other cases include shock/boundary-layer interactions for varying impingement location and shock strength. We choose two values of shock strength, corresponding to flow deviations $\phi=3^o$ and $\phi=6^o$. For a given shock strength, we choose two points of shock impingement, $x_s/\delta_{in}=48$ and $x_s/\delta_{in}=83.8$, corresponding to (see section \[sec:bl\]) a transitional- and a turbulent boundary layer, respectively. For two of the DNS cases (BL-TRIP and SH3-TU), experimental data from @giepman16 are available for comparison.
[c @[0.5cm]{} c @[0.5cm]{} c @[0.5cm]{} c @[0.5cm]{} c @[0.5cm]{} l @[0.5cm]{} c @[0.5cm]{} c ]{} Test case & $M_\infty$ & $Re_\theta$ & $\phi $ & $x_t/\delta_{in}$ & $x_s/\delta_{in}$ & Grid & Interaction\
BL-TRIP & 1.7 & 6424 & - & 31 & - & $4096\times 592 \times 384$ & No shock\
SH3-TR & 1.7 & 6424 & 3$^{\circ}$ & 31 & 48 & $4096\times 592 \times 384$ & transitional\
SH3-TU & 1.7 & 6424 & 3$^{\circ}$ & 31 & 83.8 & $4096\times 592 \times 384$ & turbulent\
SH6-TR & 1.7 & 6424 & 6$^{\circ}$ & 31 & 48 & $4096\times 592 \times 384$ & transitional\
SH6-TU & 1.7 & 6424 & 6$^{\circ}$ & 31 & 83.8 & $4096\times 592 \times 384$ & turbulent\
Numerical Method {#subsec:method}
----------------
We discretize the flow domain using a Cartesian grid and solve the three-dimensional compressible Navier-Stokes equations for a perfect gas with Fourier heat law and Newtonian viscous terms. The fluid under consideration is air with a value of specific heat ratio $\gamma=1.4$, specific gas constant $R=287$ $KJ/kg^oK$ and molecular Prandtl number ${\mbox{\textit{Pr}}}=0.72$. We assume the molecular viscosity $\mu$ to depend on the temperature $T$ through the Sutherland’s law and compute the thermal conductivity as $k_c = c_p \mu / {\mbox{\textit{Pr}}}$, where $c_p$ is the specific heat at constant pressure. We employ $4096 \times 592 \times 384$ points for discretization along the streamwise, wall-normal and spanwise directions, respectively, and generate a uniform grid spacing in the wall-parallel directions. In the wall-normal direction we cluster the grid nodes towards the wall by adopting a hyperbolic sine mapping ranging from $y = 0$ up to $y = 5 \delta_{in}$, which is succeeded by a uniform mesh spacing using a suitable smoothing in the connecting zone. We ensure sufficient grid refinement by evaluating the wall units at $x/\delta_{in}=83.8$ (turbulent regime) for case BL-TRIP. The wall units are obtained by normalizing the grid spacing in terms of the viscous length scale $\delta_v$, and take the value of $\Delta x^+ = 4.63$ and $\Delta z^+ = 4.12$ along the streamwise and spanwise directions, respectively. Along the wall-normal direction, the value ranges from $\Delta y^+ = 1.02$ at the wall to $\Delta y^+ = 21.4$ at the edge of the boundary layer. The boundary conditions are specified as follows. At the inlet, a laminar boundary layer is imposed, whose profile is determined from the solution of the generalized Blasius equations [@white74]. An oblique shock wave is introduced at the top of the domain by locally enforcing the inviscid jump relations so as to mimic the effect of the shock generator. Non-reflecting boundary conditions are enforced at the top wall and at the outlet of the domain to avoid spurious wave reflections. A characteristic wave decomposition is also applied at the adiabatic no-slip wall to ensure perfect reflection of acoustic waves.
The governing equations are solved using an in-house finite-difference flow solver, widely validated for wall-bounded flows and SBLI in the transonic and supersonic regimes [@pirozzoli10; @bernardini16h]. The solver is based on state-of-the-art numerical algorithms designed to tackle the challenging problems associated with high-speed turbulent flow solutions, allowing to accurately resolve a wide spectrum of turbulent scales and to capture steep gradients without unwanted numerical oscillations. We discretize the convective terms of the governing equations using a sixth-order central differencing scheme, and in the shock regions, identified through the Ducros sensor [@ducros99], we use a fifth-order WENO scheme. To improve the numerical stability, the convective terms are arranged in skew-symmetric form [@reiss14] and the triple split proposed by @kennedy08 is applied in a locally conservative formulation. The viscous terms are expanded to Laplacian form and discretized using a sixth-order central differencing scheme, which guarantees physical dissipation at the smallest scales resolved by the computational mesh. The solution is advanced in time using a third-order, low-storage, explicit Runge-Kutta algorithm [@bernardini09]. The presence of the roughness elements is managed by means of the immersed-boundary (IB) method, that allows to deal with embedded geometries of arbitrary shape on a Cartesian grid. In the present study, the IB method is implemented following the approach proposed by @detullio_07 for compressible flows. Additional details on the implementation can be found in @bernardini16.
We carry out the simulations using 2048 cores on the Lenovo NeXtScale platform at the Italian computing center CINECA, using a total budget of 2.25 Mio. CPU hours. The flow statistics were computed over a time period of about $327 \delta_{in}/U_{\infty}$ using around 1200 flow fields. In the present work, we normalize the streamwise distance either using the trip location i.e. $x_t^*=(x-x_t)/\delta_{95}$, or by the inviscid shock impingement point, i.e. $x_s^*=(x-x_s)/\delta_{in}$.
Boundary-layer transition {#sec:bl}
=========================
In this section, we look at the effect of the transition device on the incoming boundary layer in the absence of the impinging shock, and we provide a characterization of the boundary-layer evolution along the streamwise direction.
![Three-dimensional visualization of vortical structures past the roughness elements for the case BL-TRIP obtained as isosurfaces of the swirling strength, colored by the mean streamwise velocity using sixty contour levels ranging from $-0.125 \leq U/U_\infty \leq 1$.[]{data-label="CLEAN_3D_CONTOUR1"}](figs/CLEAN_3D_CONTOUR1)
The incoming laminar boundary layer is perturbed after encountering the roughness elements at $x_t^* = 0$, and since the roughness height is above the critical value [@bernardini12], transition to turbulence occurs further downstream. Figure \[CLEAN\_3D\_CONTOUR1\], where isosurfaces of the swirling strength criterion are reported, shows the typical pattern to transition observed in previous studies of roughness-induced transition [@acarlar87; @redford10], with the formation and shedding of hairpin vortical structures past the roughness elements, which evolve in the streamwise direction leading to the flow breakdown. This behavior is associated to the instability of the detached shear layer on the top of roughness edge and has been observed across a wide range of Mach numbers [@bernardini14].
![Distribution of the mean skin friction coefficient along the streamwise direction from DNS (solid line) for the case BL-TRIP. Also shown are the laminar $c_f$ values (dash dot dot) and the turbulent $c_f$ values from the Kármán-Schoenherr (dash dot; see Eq. (\[KARMAN\])) and Blasius (dashed;see Eq. (\[BLASIUS\])) relations.[]{data-label="CLEAN_CF"}](figs/CLEAN_CF)
We report the distribution of the time- and spanwise-averaged skin friction coefficient $c_f = \tau_w/q_\infty$ along the streamwise direction in figure \[CLEAN\_CF\], $\tau_w$ and $q_\infty$ being the wall-shear stress and free-stream dynamic pressure, respectively. Upstream of the roughness elements, the skin friction is lower than the corresponding laminar solution, as a consequence of the perturbation induced by the tripping device. Beyond the roughness strip, $c_f$ increases rapidly to about seven times the corresponding laminar value and gradually decreases further, attaining values typical of a turbulent boundary layer. For reference purpose, we also include in the figure the $c_f$ predictions obtained from two theoretical expressions for turbulent flows, the incompressible skin-friction correlation of Kármán-Schoenherr, $$c_{fi}=1/(17.08(\log_{10}Re_{\theta_i})^2 + 25.11 \log_{10}Re_{\theta_i} + 6.012),
\label{KARMAN}$$ and the Blasius correlation, $$c_{fi}=0.026/Re_{\theta_i}^{1/4},
\label{BLASIUS}$$ where the subscript ‘i’ refers to the incompressible regime. These relations are extended to compressible flows via the van Driest [ ]{} transformation (for adiabatic wall) given by $$c_{f_{i}}=F_c c_f, \hspace{1cm} Re_{\theta_{i}}=F_\theta Re_\theta,$$ where $$F_c=\frac{T_w/T_\infty-1}{\arcsin^2\alpha},\hspace{1cm} F_\theta=\frac{\mu_\infty}{\mu_w}, \hspace{1cm} \alpha=\frac{T_w/T_\infty-1}{\sqrt{T_w/T_\infty(T_w/T_\infty-1)}}.$$ We observe a good match between the $c_f$ distribution obtained from the DNS and the Kármán-Schoenherr prediction for $x_t^* > 40$, whereas the Blasius expression overpredicts the DNS result by about ten percent.
![Mean velocity profiles ($U/U_e$) along the wall-normal direction at various streamwise locations for the case BL-TRIP. Lines denote the DNS results and symbols represent the experimental data of @giepman16.[]{data-label="R_TRIP_UPROF_VARYX"}](figs/CLEAN_BL_PROFILE)
A comparison of the DNS data with the experiments of @giepman16 is reported in figure \[R\_TRIP\_UPROF\_VARYX\], where mean velocity profiles at various stations along the streamwise direction are shown. The first location ($x_t^* = -25$) corresponds to the region upstream of the tripping elements, and the velocity profile is characterized by an extended linear behavior typical of a laminar boundary layer. A strongly perturbed profile is observed at $x_t^*= 0$, with the presence of two inflection points, which indicates the onset of the instabilities due to the flow interaction with the tripping elements. As a consequence of the transition process, the mean velocity profiles at subsequent stations ($x_t^*=25$ and $x_t^*=50$) become fuller, with a steeper velocity gradient at the wall. We obtain a very good agreement with the experimental data, which demonstrates the capability of the simulation in accurately predicting the streamwise evolution of the flow and capturing the length-scale of the transition process.
a\) ![Distribution of the incompressible (a) displacement thickness, (b) momentum thickness and (c) shape factor for the case BL-TRIP. Lines denote the DNS results and symbols represent the experimental data of @giepman16. The two vertical dotted lines reported in panel (c) denote the location of shock impingment for the SBLI flow cases discussed in the later sections.[]{data-label="CLEAN_BL"}](figs/CLEAN_DSTARI "fig:") 1em b) ![Distribution of the incompressible (a) displacement thickness, (b) momentum thickness and (c) shape factor for the case BL-TRIP. Lines denote the DNS results and symbols represent the experimental data of @giepman16. The two vertical dotted lines reported in panel (c) denote the location of shock impingment for the SBLI flow cases discussed in the later sections.[]{data-label="CLEAN_BL"}](figs/CLEAN_THETAI "fig:") 1em c) ![Distribution of the incompressible (a) displacement thickness, (b) momentum thickness and (c) shape factor for the case BL-TRIP. Lines denote the DNS results and symbols represent the experimental data of @giepman16. The two vertical dotted lines reported in panel (c) denote the location of shock impingment for the SBLI flow cases discussed in the later sections.[]{data-label="CLEAN_BL"}](figs/CLEAN_HI "fig:") 1em
A further comparison between the DNS and the experiment is provided in figure \[CLEAN\_BL\], where the evolution of the boundary layer is described in terms of incompressible displacement thickness ($\delta^*_i$), momentum thickness ($\theta_i$) and shape factor ($H_i$), $$\label{thicknesses}
\delta^*_i=\int_0^{\delta_e}\left(1-\frac{U}{U_e}\right){\mathrm{d}}y, \qquad
\theta_i=\int_0^{\delta_e}\frac{U}{U_e} \left(1-\frac{U}{U_e}\right){\mathrm{d}}y, \qquad
H_i = \frac{\delta_i^*}{\theta_i}.$$ The agreement between the DNS results and the experimental data is very satisfactory, also considering the challenges of performing PIV measurements in extremely thin transitional boundary layers with non-uniform seeding distributions. Past the tripping elements the displacement thickness initially decreases as a consequence of the transition process that fills-up the velocity profile, it achieves a minimum at $x^*_t \approx 15$ and then it starts to rise. On the other hand the momentum thickness is characterized by a steady growth and its post-trip value is always larger than the value computed at the trip location. The distribution of the shape factor reflects the transition process undergone by the boundary layer. It has an initial value of about 2.7, typical of a laminar boundary layer and attains a peak of about 3.2 at the roughness location. Past the interaction, the shape factor displays a drastic drop to about 1.4, which is a value typical of a turbulent boundary layer [@smits06], achieved approximately for $x_t^* > 40$.
Effect of impinging shock {#sec:sh}
=========================
a\) 1em b) 1em c) 1em d) 1em
In this section we investigate the interaction of an impinging shock with the spatially evolving boundary layer discussed in the previous section, with the main aim of characterizing the effect of the incoming boundary-layer state (transitional or turbulent) on the properties of the interaction. To that purpose, we analyze the results of the four DNS listed in table \[TAB\_CASES\], performed for two values of shock strength ($\phi=3^o$ and $\phi=6^o$) and two shock impingement locations $x_s/\delta_{in} = 48 $ and $83.8$ ($x_t^* = 17.7$ and $55$), corresponding to transitional and turbulent SBLI, respectively. This classification is supported by the results discussed in the previous section (see in particular figure \[CLEAN\_BL\]c).
To provide a qualitative overview of the flow organization, we report in figure \[SHK\_CONTOUR\_2D\_DENSITY\] contours of the instantaneous density in a longitudinal $x-y$ plane. The density field highlights very well the wave system originated as a consequence of the interaction process, mainly consisting of the impinging and the reflected shock, as well as a series of waves (compression-expansion-compression) radiating in the freestream, arising from the perturbation of the boundary layer induced by the tripping device. We point out that the other feature typically observed in a strong SBLI involving a separation bubble, a fan of expansion waves associated with the reattachment of the boundary layer, is not observed in figure \[SHK\_CONTOUR\_2D\_DENSITY\] and both the transitional and turbulent interaction cases share the same qualitative behavior. This reveals that the present interactions do not involve a massive separation of the flow, contrary to the case of an untripped laminar oblique shock-wave reflection at $\phi = 3^\circ$, considered under the same flow conditions (Mach- and Reynolds numbers) in the experimental work by @giepman15, who highlighted the presence of a large separation bubble. A major role is played by the strength of the impinging shock, which determines the extent of the interaction zone, that significantly increases with the deflection angle $\phi$.
a\) ![Streamwise distribution of mean wall pressure for (a) $\phi=3^o$ and (b) $\phi=6^o$. Solid lines refer to transitional interactions and dash-dot lines denote turbulent interactions. $R-H$ denotes the inviscid distribution resulting from the Rankine-Hugoniot jump conditions (dashed).[]{data-label="SHK_PWALL"}](figs/SHK_3_PWALL "fig:") 1em b) ![Streamwise distribution of mean wall pressure for (a) $\phi=3^o$ and (b) $\phi=6^o$. Solid lines refer to transitional interactions and dash-dot lines denote turbulent interactions. $R-H$ denotes the inviscid distribution resulting from the Rankine-Hugoniot jump conditions (dashed).[]{data-label="SHK_PWALL"}](figs/SHK_6_PWALL "fig:") 1em
The distribution of the mean wall pressure is shown in figure \[SHK\_PWALL\], together with the inviscid pressure jumps predicted by the Rankine-Hugoniot relations. In all cases, the wall pressure exhibits a sharp rise upstream of the nominal shock impingement point, followed by a slower increase further downstream, which is more gradual in the case of the transitional SBLI. On the other hand, we observe that the beginning of the interaction is rather independent of the nature of the incoming boundary layer (transitional or turbulent), and the extent of the upstream influence region is determined by the shock strength.
![The streamwise variation of the rms wall pressure for the $\phi=3^o$ (red) and the $\phi=6^o$ (black) cases with the transitional (solid) and turbulent (dashed) interactions.[]{data-label="SHK_PRMS"}](figs/SHK_PRMS "fig:") 1em
Figure \[SHK\_PRMS\] shows the root-mean-square of wall pressure fluctuations ($p_{w_{rms}}$) for all shock-impingement cases. The $p_{w_{rms}}$ rise across the interaction region is sligthly higher for the turbulent interactions as compared to the corresponding transitional cases, for both the incident shock angles. The post-shock decay for the two types of interaction also varies, with the turbulent cases displaying a steeper decline past the interaction region compared to the corresponding transitional case. The transitional interactions have a broader post-shock decay region, with the $\phi = 6^{\circ}$ case also displaying a local minimum at the nominal shock impingement point.
a\) ![Distribution of skin friction coefficient along the streamwise direction for the shock impingement cases at (a) $\phi=3^o$ and (b) $\phi=6^o$. Solid lines refer to the transitional interactions and dashed lines denote the turbulent interactions. The skin friction coefficient for the case without shock impingement is also shown (dash-dot). The vertical dotted lines the nominal shock impingment location.[]{data-label="SHK_CF"}](figs/SHK_3_CF_NEW "fig:") 1em b) ![Distribution of skin friction coefficient along the streamwise direction for the shock impingement cases at (a) $\phi=3^o$ and (b) $\phi=6^o$. Solid lines refer to the transitional interactions and dashed lines denote the turbulent interactions. The skin friction coefficient for the case without shock impingement is also shown (dash-dot). The vertical dotted lines the nominal shock impingment location.[]{data-label="SHK_CF"}](figs/SHK_6_CF_NEW "fig:") 1em
The effectiveness of boundary-layer tripping in suppressing a shock-induced separation can be identified by the distribution of the skin friction coefficient $c_f$, displayed in Fig \[SHK\_CF\] for all the shock impingement cases. In the interaction region the wall shear stress is characterized by a remarkable drop, associated with the lift off of the boundary layer, followed by a gradual recovery. The $c_f$ curves show that in all the cases, even those at higher shock strength, mean separation is not observed, and tripping the boundary layer eliminates the large separation bubble found in a laminar interaction. This quantitatively confirms our expectations drawn from the inspection of figure \[SHK\_CONTOUR\_2D\_DENSITY\]. The skin friction levels remain well above the zero line for the cases SH3-TR and SH3-TU, whereas they are almost tangent for the cases SH6-TR and SH6-TU, which can be classified as cases with incipient separation. Quite surprisingly, the minimum value of the skin friction is lower in the turbulent interaction than in the transitional cases, although the width of the region where the $c_f$ value drops due to the shock impingement is wider in the transitional interactions.
a\) ![Distribution of statistical probability of wall points along the streamwise direction with $\partial u/\partial y < 0$ for (a) $\phi=3^o$ and (b) $\phi=6^o$, for the transitional interaction cases (solid lines) and the turbulent interaction cases (dashed lines). The horizontal lines denote the incipient detachment (ID), intermittent transitory detachment (ITD) and the transitory detachment (TD) levels.[]{data-label="SHK_STAT_PROB"}](figs/SHK_3_STAT_PROB "fig:") 1em b) ![Distribution of statistical probability of wall points along the streamwise direction with $\partial u/\partial y < 0$ for (a) $\phi=3^o$ and (b) $\phi=6^o$, for the transitional interaction cases (solid lines) and the turbulent interaction cases (dashed lines). The horizontal lines denote the incipient detachment (ID), intermittent transitory detachment (ITD) and the transitory detachment (TD) levels.[]{data-label="SHK_STAT_PROB"}](figs/SHK_6_STAT_PROB "fig:") 1em
The absence of a mean separation clearly does not preclude the possibility of instantaneous zones with locally reversed flow. We characterize the regions of instantaneous separation by plotting the statistical probability ($\overline{\gamma}$) of the the wall points with negative $\partial u/\partial y$, where $u$ is the instantaneous streamwise velocity. @simpson89 classified the boundary-layer detachment based on how frequently the backflow occurs. A statistical probability of backflow of 1% of the total sampling is denoted as incipient detachment (ID), while that amounting to 20 % is classified as intermittent transitory detachment (ITD). A 50 % probability of backflow is termed as transitory detachment (TD), which clearly indicates a separation in the mean.
Figure \[SHK\_STAT\_PROB\]a shows the statistical probability of instantaneous separation for the interactions at $\phi=3^o$. In the region of shock impingement ($x_s^*=0$), both the flow cases (SH3-TR and SH3-TU) exceed the ITD level, and the peak probability of separation for the turbulent interaction is slightly higher than that of the transitional interaction. The same plot for the flow cases at higher angle of incidence $\phi=6^o$ is shown in figure \[SHK\_STAT\_PROB\]b. As expected, we observe higher levels of instantaneous separation in comparison to the lower incidence cases, with the turbulent interaction exceeding the transitory detachment level. In the transitional interaction case, although the peak percentage of separation is not as high, the width of the separation region exceeds that of the turbulent interaction zone by about 36 % at the ITD level.
a\) ![Contours of instantaneous skin friction for the flow cases (a) SH6-TR and (b) SH6-TU. The iso-line $c_f=0$ is denoted in orange.[]{data-label="SHK_WALL_CF"}](figs/SHK_648_WALL_CF "fig:") 1em b) ![Contours of instantaneous skin friction for the flow cases (a) SH6-TR and (b) SH6-TU. The iso-line $c_f=0$ is denoted in orange.[]{data-label="SHK_WALL_CF"}](figs/SHK_683_WALL_CF "fig:") 1em
The instantaneous flow separation can be clearly visualized by plotting the wall skin friction contour, as carried out for the $\phi=6^o$ shock impingement cases in figure \[SHK\_WALL\_CF\]. The isolines correspond to zero skin friction level indicating areas of reverse flow. In agreement with the statistical probability of instantaneous separation discussed previously, the region of separation is slightly wider in the transitional interaction case as compared to the turbulent one.
a\) ![Distribution of the incompressible (a) displacement thickness, (b) momentum thickness and (c) shape factor for all the shock impingement cases. Lines denote DNS results. Symbols refer to experiments of @giepman16.[]{data-label="SHK_6_BLT"}](figs/SHK_6_DSTAR "fig:") 1em b) ![Distribution of the incompressible (a) displacement thickness, (b) momentum thickness and (c) shape factor for all the shock impingement cases. Lines denote DNS results. Symbols refer to experiments of @giepman16.[]{data-label="SHK_6_BLT"}](figs/SHK_6_THETA "fig:") 1em c) ![Distribution of the incompressible (a) displacement thickness, (b) momentum thickness and (c) shape factor for all the shock impingement cases. Lines denote DNS results. Symbols refer to experiments of @giepman16.[]{data-label="SHK_6_BLT"}](figs/SHK_6_HI "fig:") 1em
The development of the boundary layer across the interaction is described in figure \[SHK\_6\_BLT\], where we provide distributions of the incompressible displacement thickness, momentum thickness and shape factor. As a reference, we also plot in the figure the DNS results for the transitional boundary layer without any shock and the experimental data corresponding to the flow case SH3-TU. For all the interactions, the boundary-layer thicknesses show a rapid increase in the region of shock impingement, which is remarkably higher for the flow cases at $\phi=6^o$ angle of incidence. While the shock strength plays an important role, the boundary-layer growth does not seem greatly affected by the state of the incoming boundary layer, the jump of the momentum and displacement thickness being approximately the same for the corresponding interactions. For the SH3-TU case, the agreement with the experimental data is fair, especially for the prediction of the location and amplitude of the jump of $\delta_i$ and $\theta_i$. The discrepancies observed upstream of the interaction, where the experiments exhibit a spurious peak, can be attributed to the aero-optical distortion in the measurement and does not reflect the actual flow field [@giepman16]. The distributions of $H_i$ match quite well and highlight the boundary-layer distortion in the interaction region, with the velocity profile becoming emptier due to the effect of the adverse pressure gradient, leading to higher values of the shape factor.
[c @[0.2cm]{} c @[0.2cm]{} c @[0.2cm]{} c @[0.2cm]{} l @[0.3cm]{} c @[0.2cm]{} c @[0.2cm]{} c @[0.3cm]{} c @[0.3cm]{} c @[0.3cm]{} c @[0.3cm]{} c @[0.3cm]{} l]{} Test case & $M_\infty$ & $\phi$ & $\beta$ & Interaction & $\delta^*/\delta_{in}$ & $L/\delta_{in}$ & $L^*$ & $Re_\theta$ & $\overline{k}$ & $P_3/P_1$ & $S_e^*$ & Inference\
SH3-TR & 1.7 & 3 & 38.9 & transitional & 0.32 & 3.2 & 0.56 & 8150 & 3 & 1.35 & 0.51 & attached\
SH3-TU & 1.7 & 3 & 38.9 & turbulent & 0.36 & 3.5 & 0.55 & 12450 & 2.5 & 1.35 & 0.44 & attached\
SH6-TR & 1.7 & 6 & 42.1 & transitional & 0.41 & 6.1 & 1.74 & 7240 & 3 & 1.81 & 1.11 & insipient Sep.\
SH6-TU & 1.7 & 6 & 42.1 & turbulent & 0.35 & 6.1 & 2.04 & 12140 & 2.5 & 1.81 & 1.02 & insipient Sep.\
We now look at some of the characteristics of the interaction in the transitional and turbulent regime at varying shock angles. An important defining parameter for any SBLI is the characteristic interaction length $L$, which is defined as the distance between the wall-extrapolated point of the reflected shock and the nominal impinging shock location. Based on a mass-balance analysis, @souverein13 proposed a non-dimensional form of the interaction length scale applicable for turbulent SBLI with adiabatic wall boundary conditions (both, oblique shock impingement and compression corner), $$L^*=\frac{L}{\delta_{in}^*}G_3,$$ where $G_3$ is $\sin(\beta)\sin(\phi)/\sin(\beta-\phi)$, $\beta$ is the shock angle. This new dimensionless interaction length also classifies the interactions as attached, incipiently separated or fully separated, based on its value. While a value of $L^*\downarrow 1$ corresponds to an attached flow, the cases with incipient separation have $1<L^*<2$, and the separated interactions have $L^*$ values larger than two.
We provide the values of $L^*$ for our cases with shock impingement in table \[TAB\_CASES1\]. For the two cases at $\phi=3^o$ (SH3-TR and SH3-TU), the values of $L^*$ are less than unity, and correspond to attached flow, as shown by the skin-friction results presented in figure \[SHK\_CF\]a. For the cases at $\phi=6^o$, $L^*=1.74$ associated with the transitional interaction (SH6-TR) correctly classifies it as an incipient separation case. The borderline value $L^*=2.04$ associated with the flow case SH6-TU, while strictly classifies it as a separated interaction, is indicative of of an incipient separation as displayed by figure \[SHK\_CF\]b.
@souverein13 also proposed an additional parameter to characterize the SBLI in terms of the ratio in the pressures before ($P_1$) and after ($P_3$) the shock system. This non-dimensional parameter is given by $$S_e^*=\frac{2\overline{k}}{\gamma}\frac{\frac{P_3}{P_1}-1}{M_\infty^2},$$ and can be written as a function of free-stream Mach number $M_\infty$, flow deflection angle $\phi$ and specific heat ratio $\gamma$. The constant $\overline{k}$ as observed from the experimental data can either take a value of about 3 for $Re_\theta\leq 1\times 10^4$ or a value of about 2.5 for $Re_\theta>1\times 10^4$, where $Re_\theta$ is the Reynolds number based on the momentum thickness upstream of the interaction. The values of $Re_\theta$, and the associated values of $\overline{k}$ for each of the cases are listed in table \[TAB\_CASES1\].
![Scaling of the interaction length. Separation criterion $S_e^*$ plotted against the interaction length $L^*$ for the DNS flow cases (squares) compared with the available experimental data. Refer to @souverein13 for the interpretation of symbols.[]{data-label="SHK_SOUVEREIN_CURVE"}](figs/LSTAR)
According to the classification of @souverein13, a value of $S_e^*<1$ corresponds to attached interactions and $S_e^*>1$ represents interactions with boundary-layer separation. The values of $S_e^*$ for each of our cases is given in table \[TAB\_CASES1\]. @souverein13 suggested that the interaction length $L^*$ plotted in combination with the separation criterion $S_e^*$ would collapse the data along a single trend line irrespective of Reynolds number, Mach number and interaction type (oblique shock impingement and compression corner). This scaling is tested for our flow cases in figure \[SHK\_SOUVEREIN\_CURVE\], where DNS data are reported with available experiments, and colors are used to identify attached (black), incipiently separated (grey) or separated (white) interactions. We observe that, for all flow cases, our DNS data follow the experimental trend and fall within the acceptable range of scatter, suggesting that the scaling analysis proposed by @souverein13 for turbulent SBLI, could be equally applied to describe transitional interactions.
Conclusions {#sec:conclusion}
===========
We have performed a series of direct numerical simulations to investigate the effect of an oblique shock wave impinging on transitional and turbulent boundary layers at $M_{\infty} = 1.7$, with the main aim of evaluating the effectiveness of a transitional boundary layer to suppress shock-induced separation. The incoming laminar boundary layer was tripped by a strip of distributed roughness elements, which enabled a rapid transition to turbulence. A single DNS carried out without the presence of any impinging shock wave helped to characterize the boundary-layer transition region and to validate the numerical approach by means of a favorable comparison with available experimental data in terms of mean velocity profiles, boundary layer thicknesses and shape factor. Four DNS cases were considered in the present study based on varying shock impingement locations along the streamwise distance (corresponding to transitional or turbulent interactions), and also based on varying shock strength (flow deflection angles $\phi=3^o$ and $\phi=6^o$). We observed a clear suppression of shock-induced mean separation in both the transitional and turbulent interaction cases, inferred by the distribution of the skin friction coefficient. This observation applies to both the weak as well as the strong interaction cases considered in our study. A higher peak in the probability of the instantaneous separation was observed for the turbulent interactions, although the transitional cases exhibited wider regions of instantaneous separation. The scaling analysis for the interaction length proposed by @souverein13 for turbulent shock/boundary-layer interactions was tested for all the flow cases considered here and it was found to be applicable for our DNS, including the transitional interactions. Furthermore, the separation criterion $S^*_e$, only dependent on the Mach number and flow deflection angle, correctly classified the present interactions as attached or close to incipient separation.
Overall, our results provide numerical evidence that a transitional interaction retains the beneficial features of a turbulent interaction in terms of suppression of mean separation. Therefore, for practical SBLI applications, it seems to be reasonable to trip the boundary layer a short distance upstream of the impinging shock to remove the separation bubble, maximizing the region with a low skin friction coefficient. Obviously, such encouraging considerations are based on a DNS database of limited extent and further investigations are needed to completely characterize transitional SBLI. Future efforts will be devoted to include the effects of different tripping devices, as well as to expand the range of investigated Mach- and Reynolds numbers.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been supported by the SIR program 2014 (jACOBI project, grant RBSI14TKWU), funded by MIUR (Ministero dell’Istruzione dell’Università e della Ricerca). The simulations have been performed thanks to computational resources provided by the Italian Computing center CINECA under the ISCRA initiative (grant jACOBI).
References {#references .unnumbered}
==========
[^1]: Research Fellow, Dipartimento di Ingegneria Meccanica e Aerospaziale, russell.quadros@uniroma1.it
[^2]: Assistant Professor, Dipartimento di Ingegneria Meccanica e Aerospaziale, matteo.bernardini@uniroma1.it
|
---
title: Review of scientific topics for Millimetron space observatory
---
*N.S. Kardashev$^1$, I.D. Novikov$^{1,2}$, V.N. Lukash$^1$, S.V. Pilipenko$^1$, E.V. Mikheeva$^1$, D.V. Bisikalo$^3$, D.S. Wiebe$^3$, A.G. Doroshkevich$^1$, A.V. Zasov$^4$, I.I. Zinchenko$^{5,6}$, P.B. Ivanov$^1$, V.I. Kostenko$^1$, T.I. Larchenkova$^1$, S.F. Likhachev$^1$, I.F. Malov$^7$, V.M. Malofeev$^7$, A.S. Pozanenko$^8$, A.V. Smirnov$^1$, A.M. Sobolev$^9$, A.M. Cherepashchuk$^4$, Yu.A. Shchekinov$^{10}$* $^1$Lebedev Physical Institute, AstroSpace Center, Moscow, Russia\
$^2$The Niels Bohr Institute, Copenhagen, Denmark\
$^3$Institute of Astronomy of the Russian Academy of Sciences (INASAN), Moscow, Russia\
$^4$Lomonosov Moscow State University Sternberg Astronomical Institute, Moscow, Russia\
$^5$Institute of Applied Physics, Nizhnii Novgorod, Russia\
$^6$N.I. Lobachevskii Nizhnii Novgorod State University, Nizhnii Novgorod, Russia\
$^7$Lebedev Physical Institute, Pushchino RadioAstronomical Observatory, Pushchino, Russia\
$^8$Space Research Institute, Moscow, Russia\
$^9$Ural Federal University, Institute of Natural Sciences, Astronomical Observatory, Ekaterinburg, Russia\
$^{10}$South Federal University, Rostov-on-Don, Russia\
*Abstract*
This paper describes outstanding issues in astrophysics and cosmology that can be solved by astronomical observations in a broad spectral range from far infrared to millimeter wavelengths. The discussed problems related to the formation of stars and planets, galaxies and the interstellar medium, studies of black holes and the development of the cosmological model can be addressed by the planned space observatory Millimetron (the “Spectr-M” project) equipped with a cooled 10-m mirror. Millimetron can operate both as a single-dish telescope and as a part of a space-ground interferometer with very long baseline.
Introduction
============
The Millimetron space observatory is aimed at astronomical observations, which address a broad range of objects in the Universe in the 20 $\mu$m to 20 mm wavelength range. Since the beginning of 1990s, there has been a growing astrophysical and cosmological interest to objects in this range. This range is very important to different types of astronomical observations, including the continuum and spectral line studies, polarimetry and variations of different parameters.
The millimeter (1 mm $<\lambda<$ 1 cm), sub-millimeter (0.1 $<\lambda<$ 1 mm) and far infrared (FIR) (50 $<\lambda<$ 300 $\mu$m) bands are unique for astronomical observations for the following reasons:
- the cosmic microwave background (CMB) peaks at 1 mm wavelength. This is the only electromagnetic radiation that survived since the earliest stages of the Universe (the Big Bang) and fills the space almost homogeneously. A detailed study of the space structure, spectrum and polarization of CMB will help in solving fundamental issues in astrophysics and cosmology, including the development of the standard cosmological model, the origin and evolution of the first objects in the Universe, determination of dark matter and dark energy parameters, etc.;
- in FIR, there is a maximum emission from the coldest objects in the Universe, including gas and dust clouds in our and other galaxies, asteroids, comets and planets. Therefore, observations in this range allow exploring the interstellar medium evolution in the process of the gravitational contraction leading to the formation of stars and planetary systems, and ultimately to the appearance of life and civilizations;
- the sky background emission reaches minimum near 300 $\mu$m. This background is a sum of contributions from different sources along the line of sight and CMB (Fig. 1a). Therefore, observations in this range will enable probing very faint astronomical objects — galaxies, stars, black holes, exoplanets, etc., — with the highest sensitivity;
- in the submillimeter and FIR ranges there is a lot of atomic and molecular spectral lines, allowing the determination of the chemical composition and physical properties of gas in different objects ranging from protoplanetary disks to galaxies at different epochs;
- submillimeter observations can significantly increase the angular resolution using Very Long Baseline Interferometry (VLBI), which is necessary to study the most compact objects, such as surroundings of black holes, some pulsars and gamma-ray bursts;
- the medium surrounding many interesting astronomical objects is mostly transparent in these wavelength ranges compared to nearby spectral bands, both at short wavelengths (due to the interstellar dust absorption) and at long wavelengths (due to synchrotron self-absorption, thermal plasma absorption and scattering on plasma inhomogeneities).
One of the largest single ground-based telescopes specifically designed for submillimeter observations is the James Clerk Maxwell Telescope, JCMT (<http://www.jach.hawaii.edu/JCMT/>) with an aperture 15 m located near Mauna Kea (Hawaii) at an altitude of more than 4000 m from the sea level. Heterodyne detectors of JCMT cover all windows of the Earth atmosphere transparency in the 200-700 GHz frequency band, and the bolometer array detector consisting of 10240 elements operates at two wavelengths: 450 and 850 $\mu$m. JCMT is used to study the Solar System, interstellar gas and dust, as well as distant galaxies. Using JCMT, submillimeter galaxies were discovered in which submillimeter emission dominates over the optical emission.
Atacama Large Millimeter/submillimeter Array (ALMA) is the most prospective ground-based submillimeter instrument (<http://www.almaobservatory.org>), which is now at the commissioning stage. It represents a compact interferometer with a base up to 16 km consisting of 66 antennas, of which 54 have diameter of 12 meters and 12 of 7 meters. The ALMA observatory is located at the altitude of 5000 m above the sea level in the Atakama desert in Chile and is capable of conducting astronomical observations in all transparency windows from 10 mm to 0.3 mm. The fully operational ALMA observatory will provide an unprecedentedly high sensitivity of up to 50 $\mu$Jy in the continuum with an angular resolution of less than 0.1 arcsec, which allows a detailed mapping of protoplanetary disks and studies of the morphology of distant galaxies. A small field of view is a shortcoming of ALMA, which requires long observational time to carry out large surveys of point-like sources, studies of extended star formation regions in the Galaxy and mapping of large sky areas.
![(a) Extragalactic background spectrum. COB — cosmic optical background, CIB — cosmic infrared background, CMB — cosmic microwave background. Numbers show the total intensity of the background components in units nW m$^2$ sr$^{-1}$ \[1\]. (b) The possible fraction of CIB resolved into individual sources as a function of the telescope diameter for several wavelengths shown in the Figure \[2\].](14-img001.png "fig:"){width="7.5cm"} ![(a) Extragalactic background spectrum. COB — cosmic optical background, CIB — cosmic infrared background, CMB — cosmic microwave background. Numbers show the total intensity of the background components in units nW m$^2$ sr$^{-1}$ \[1\]. (b) The possible fraction of CIB resolved into individual sources as a function of the telescope diameter for several wavelengths shown in the Figure \[2\].](14-img002.png "fig:"){width="7.5cm"}
As noted above, ground-based astronomical observations at wavelengths $\lambda \lesssim 300$ $\mu$m are problematic, since they are significantly limited by properties of the terrestrial atmosphere: its proper emission and absorption by water vapour, oxygen, carbon dioxide and ozone. Therefore, observations at wavelengths shorter than 300 $\mu$m should be carried out at altitudes at least as high as 10-30 km, where partial water vapour pressure strongly decreases with altitude, and the associated absorption almost vanishes. This property has already been exploited by telescopes installed on airplanes (SOFIA (Stratospheric Observatory for Infrared Astronomy) \[3\]) or stratospheric balloons (BOOMERANG (Balloon Observations Of Millimeter Extragalactic Radiation and Geophysics) \[4\], TELIS (TEraHertz and submillimeter Limb Sounder) \[5\], Olimpo, etc.). A space telescope has clear advantages, since it is free from the negative effect of the Earth atmosphere and can be cooled down to low temperatures, thus strongly increasing its sensitivity.
The Herschel space observatory launched in 2009 and operated until the middle of 2013 is the most perfect and close predecessor of Millimetron observatory. The Herschel (<http://sci.esa.int/Herschel/> \[6\]) consists of a 3.5-m diameter telescope with passive cooling down to about 70 K. The observatory carried out observations in the wavelength range from 55 to 672 $\mu$m. The receivers of the Herschel included sensitive array photometers, an array spectrometer with moderate resolution and a high-resolution heterodyne spectrometer. The main scientific achievements of the Herschel observatory include observations of star-forming and dust regions in our and external galaxies, studies of submillimeter galaxies and Solar system bodies.
[|m[3cm]{}|m[5.8cm]{}|m[7.2cm]{}|]{} & Single dish & Space-Earth interferometer\
&
- Low spectral resolution or photometry (R=$\lambda $/$\Delta \lambda \sim$3).
- Medium spectral resolution ($R\sim10^{3}$).
- High spectral resolution ($R{\geq}10^{6}$).
- Polarimetry.
&
- Estimation of source angular size
- One dimensional source cross-section
- Maps with a priori source model adopted
\
& 20 $\mu$m – 3 mm (for R$\sim3$ and R$\sim 10^{3}$), 60 $\mu$m – 0.6 mm (for R${\geq}10^{6}$) & 0.3 – 17 mm
\
& 6 arcsec (at $\lambda $ = 300 $\mu$m) & for baseline 1.5 million km:
$\sim$2 $\mu$arcsec, ($\lambda $ = 13.5 mm),
$\sim$50 narcsec,($\lambda $ = 0.345 mm)
\
& Noise RMS $\sigma$ (for ${\lambda}$ = 300 $\mu$m, integration time t= 3600 s, effective area A=50 m$^2$, detector sensitivity NEP$^*$ ${\leq 10^{-19}}$ W Hz$^{-1/2}$):
- 20 nJy (for R$\sim$3)
- 4 $\mu$Jy (or $4\cdot 10^{-23}$ W m$^-2$ (for R$\sim 10^3$)
& $\sigma $ for bandwidth 4GHz (2 polarizations, 2 bit quantization)
[|m[2.0809999cm]{}|m[2.0809999cm]{}|m[1.8cm]{}|]{} Frequency, GHz & Coherent integration time \[9\], s & Sensitivity, mJy\
22 & 500 & 0.2\
43 & 300 & 0.3\
100 & 100 & 1.0\
240 & 70 & 1.9\
640 & 10 & 20.0\
870 & 5 & 40.0\
\
\
Millimetron space observatory represents a new step in the development of space FIR missions due to its unique characteristics: high angular resolution and unprecedentedly high sensitivity in a broad wavelength range from far infrared to millimeters. This leap forward may solve many fundamental issues of astrophysics and cosmology. The unique breakthrough scientific tasks are crucial in determining technical characteristics of the Millimetron observatory.
Millimetron will be launched in an orbit in the vicinity of the Lagrangian point L2 at a distance of 1.5 million km from Earth behind the Moon orbit, with the most favorable external conditions for the telescope cooling. To provide an unprecedentedly high sensitivity, a deep cooling of the telescope mirrors down to temperatures as low as 10 K is required. Such a regime can be realized only by joint operation of two cooling systems: passive and active. The former utilizes the solar shields, while the second one uses close-cycle cryogenic refrigerators. Being in the vicinity of the Lagrangian L2 point, Millimetron forms together with a ground-based telescope or a system of telescopes – an interferometer with maximum projection of the base on the plane perpendicular to the line of sight of a studied source of more than 1.5 million km. Such a unique instrument opens new horizons in solving astrophysical problems by enabling measurements with record high angular resolution.
The Spectr-M project started in the 1990s. A description of different concepts of the construction of the Millimetron observatory since the beginning of the project and a list of its main science tasks can be found in papers \[2, 7, 8\]. The main parameters of the Millimetron are listed in Table 1.
It should be noted that Table 1 presents the desirable parameters, which can be realized at the present level of technology. The characteristics of the real observatory will be recommended according to a careful compromise between the desirable parameters, the priority of scientific tasks and the project cost.
[|l|c|c|c|c|]{} & [**Herschel**]{} & [**ALMA**]{} & [**SPICA**]{} & [**Millimetron**]{}\
Range, $\mu$m & 50-670 & 315-9680 & 5-210 & 20-3000\
resolution, arcsec & 3.5-40 & 0.01-5 & 0.3-14 & 5-60\
field of view & up to 4$^\prime\times8^\prime$ & up to 25$^{\prime\prime}$ & $5^\prime\times5^\prime$ & $6^\prime\times6^\prime$\
\
photometry & 1 mJy & [>]{}10 $\mu$Jy & 4 $\mu$Jy & 20 nJy\
spectroscopy (R$\sim$1000) & 20 mJy & 60 $\mu$Jy & 200 $\mu$Jy & 4 $\mu$Jy\
spectroscopy (R${\geq}10^6$) & 2 Jy & 50 mJy & - & 200 mJy\
\
The comparison of the expected parameters of the Millimetron observatory with those of existing and planned instruments for observations in nearby or similar wavelength ranges (Table 2), such as the ground-based ALMA observatory, the Herschel and SPICA (Space Infrared Telescope for Cosmology and Astrophysics) \[10\] space telescopes, suggests the list of the highest-priority scientific tasks and allows the formulation of the priority of observations.
The expected sensitivity of Millimetron is at least two orders of magnitude higher than that of the Herschel telescope. Currently the best angular resolution in the 20-300 $\mu$m range is much worse than in other ranges (radio and near-IR). This is due to the fact that the 20-300$\mu$m range is virtually inaccessible for ground-based observations, and all space telescopes launched so far had small diameters of order of 1 m. Presently, the Herschel space telescope has the largest mirror (diameter of 3.5 m) optimized for observations in far infrared. For deeper studies of different astronomical objects a better angular resolution is needed, and the Millimetron observatory is planned to be the next step by providing three times higher angular resolution in the far-IR range.
The angular resolution of the telescope is also related to the well-known astronomical confusion problem: at a low angular resolution distant sources merge into a homogeneous background, which hampers measurements of individual fainter objects. In the far-IR range this background is usually called Cosmic Infrared Background (CIB). CIB is thought to be mainly due to emission of distant galaxies. Preliminary estimates show that Millimetron with the aperture 10-m will be capable of resolving more than 90% of CIB into individual sources — distant galaxies (Fig. 1b).
At wavelengths longer than 300 $\mu$m ALMA has a better angular resolution than Millimetron, and the ALMA sensitivity at longer wavelengths ($\lambda >$ 1 mm) is also higher than that of Millimetron due to a huge collective area. However, at wavelengths shorter than 300 $\mu$m, which are inaccessible for ALMA, Millimetron will have no competitors in sensitivity. A wide field of view of Millimetron is another advantage. This field of view is provided by several thousands of detectors and enables Millimetron to map large sky areas. A grating spectrometer will offer broad-band spectral measurements, facilitating measurement of redshifts of distant galaxies, which is a difficult task for ALMA.
Clearly, some tasks observations by ALMA and Millimetron can and should complement each other. In addition, another ambitious project — James Webb Space Telescope (JWST) (http://www.jwst.nasa.gov) operating at shorter wavelengths, can be an interesting completion. For example, in studies of high-redshift galaxies, Millimetron can analyze a relatively cold molecular and atomic gas, JWST can study properties of a hotter atomic gas, and ALMA will provide a detailed imaging of these galaxies.
In the interferometer mode, the Millimetron observatory can cooperate with the presently developing Event Horizon Telescope (EHT) \[11\] (http://www.eventhorizontelescope.org), which in the near future will join all largest ground-based submillimeter telescopes and observatories in a single VLBI network.
The present paper is prepared as a result of discussions of scientific objectives of Millimetron at scientific seminars and several symposia.[^1]
The Sections of the paper correspond to different astrophysical topics, where Millimetron can significantly contribute. The interferometric regime of observations is assumed to be used in tasks formulated in Sections 2.4, 4.2, 4.5, 5.1-5.5, 7.2; to solve tasks described in other sections of this paper, single-mirror regime of observations can be used. It should be noted that the list of scientific tasks presented in this paper is preliminary, and from the broad scientific community we are waiting for both new unique scientific tasks and more detailed elaboration of the proposed ones. In June 2014, principal scientific tasks of the Millimetron observatory were discussed at an international symposium in Paris [^2]. Using the results of this symposium and based on the preliminary list of scientific problems presented in this paper, the scientific program of the Millimetron observatory will be prepared.
Interstellar medium and star formation regions in the Galaxy
============================================================
Structure and kinematics of the interstellar medium
---------------------------------------------------
Currently, the problem of star formation \[12\], as well as its relation to the general evolution of the interstellar medium (ISM), is an important astrophysical issue. Observational data suggest that the birthrate of new stellar and planetary systems and their parameters are determined by the basic properties of ISM: its structure, kinematics, pressure, temperature, magnetic field, and matter returned back to ISM from evolved stars. The galaxy environment effects (accretion and ram pressure of intergalactic matter, interaction with other galaxies) can also play an important role.
Global processes of star formation imply that it is necessary to investigate it in a general context of structure, kinematics and evolution of the interstellar medium using as broad sample of objects as possible. The densest ISM regions, where star formation occurs have a low temperature and, therefore, require observations in the FIR and submillimeter ranges, which can be done only from space. For a deeper study, a high sensitivity is also necessary, resulting in the requirement of large-size telescope mirrors, cooling, as well as increased demands for the detector parameters. The Millimetron project satisfies these requirements.
The most promising targets to be observed with Millimetron include cold ($10-20$ K) gas and dust clumps, which are difficult to detect with less sensitive instruments, “hot cores” and high-speed bipolar outflows, diffuse clouds, submillimeter masers, ISM in other galaxies.
Millimetron will study general characteristics of the interstellar medium in various galaxies, statistical properties of dense condensations, structure and kinematics of interstellar clouds, the earliest stages of star formation, mechanisms of massive star formation, structure and properties of circumstellar shells and planetary nebulae, synthesis and proliferation of different molecules in ISM including complex organic molecules.
High sensitivity of Millimetron in the single-dish mode will allow observing individual clouds with a characteristic temperature of about 20 K and a mass of order of one solar mass ($M_\odot$) at a distance up to 1 Mpc. The molecular cloud complex Sgr B2, where many molecules were found playing an important role in cooling and condensation of clouds, including the molecular ion H$_3$O$^+$, can provide an example \[13\]. This ion, which decays into water or hydroxyl due to dissociative recombination, is very important to an overall understanding of the chemistry of oxygen in the interstellar medium.
The submillimeter radiation from molecules and atoms arises at much lower temperatures than in the visible and infrared ranges. This means that by analyzing the submillimeter data, one can examine the cold ISM, in particular probe the content of [*hidden hydrogen*]{} \[14\]. The most important spectral lines of these observations include HD transitions at a wavelength of 112 microns and \[CII\] at a wavelength of 158 microns. A very interesting target for ISM research is a molecule HeH$^+$ (transition $J=1-0$ at a wavelength of 149 microns), which has not yet been detected in space. Conditions of formation and excitation for this molecule are substantially different from those for the majority of interstellar molecules. Models (e.g., \[15, 16\]) indicate that HeH$^+$ should be especially abundant near the sources of extreme ultraviolet (UV) and X-ray radiation.
An important area of research is a study of the ISM diffuse component by absorption lines of different molecules at submillimeter wavelengths. The Herschel space telescope has already demonstrated great possibilities of such researches. In particular, they allow determining the rate of ionization by cosmic rays in various regions (OH$^+$, H$_2$O$^+$ and H$_3$O$^+$ lines), turbulence dissipation rate (CH$^+$ and SH$^+$ lines), and total distribution of molecular hydrogen (HD line). Such measurements require a sufficiently bright background source. For Millimetron with its much greater collecting area, the number of such sources will be larger than for the Herschel space telescope, which opens up the possibility of a much more complete coverage of the galactic plane.
Another source of information about the structure and kinematics of the interstellar medium are polarization measurements, which help to explore structure of magnetic fields in star-forming regions.
Ground-based observations of polarization in the FIR and submillimeter ranges can be carried out only in certain transparency windows. The possibility of constructing of the so-called polarization curve, i.e. the dependence of the polarization efficiency on wavelength, would allow us to explain not only the structure of magnetic field in the star-forming regions, but also a mechanism of dust particles orientation (Fig. 2).
![The polarization efficiency curve for Perfect Davies Greenstein (PDG) (the upper curve) and Imperfect Davies Greenstein (IDG) (the bottom curve) orientation of interstellar dust grains \[17\]; $P$ is the polarization degree at the wavelength $\lambda$ in per cents, $A(\lambda)$ is the interstellar absorption at the wavelength $\lambda$.](14-img003.png){width="7.5cm"}
Observations of dust emission in the submillimeter range are an important source of information about the star formation. The modern data obtained by the Herschel space telescope show that star formation occurs in thin (<0.1 pc) gas-dust filaments. Parameters of these filaments are not fully determined, and in particular a role of magnetic field in their formation is not clarified. To answer this question, we need higher angular resolution and sensitivity (better than that of the Herschel telescope), and availability of polarization measurements. The parameters of the space observatory Millimetron satisfy these requirements. In addition, contribution of Millimetron to star formation and evolution studies is enhanced by the possibility to observe more distant star forming regions, including extragalactic ones. In recent years, a new paradigm (where the primary role is given to large-scale star formation complexes \[18-22\]) is developing based on observational data obtained by the Herschel telescope and new theoretical researches. The heuristic role of this new paradigm is extremely important because it relates a large variety of phenomena: from the spiral arm and interarm flows several kiloparsecs in size to cores of molecular clouds and protostellar condensations as small as several astronomical units (AU). However, many aspects of the star formation process remain unclear and poorly investigated. Observations of IR lines at a wavelength of 158 microns show that the galactic disk contains a large amount of CO-dark molecular gas \[23\]. Theoretical studies of the formation of gas clouds in the Galaxy also show that temperature and density of molecular clouds vary strongly (see, e.g., the phase diagram in paper \[22\]), which may indicate existence of large masses of gas mostly consisting of molecular hydrogen with rather small abundance of CO molecules. Distribution and parameters of this gas have been studied only in a narrow band of the Galactic plane, and only in selected directions. Therefore, the measurements of the relative velocities and positions of gas clouds emitting in \[CII\] lines at a wavelength of 158 microns, as well as of submillimeter and millimeter radiation of dust, are needed to determine mechanisms of formation of star formation complexes, since such measurements form the basis of studies of morphology, kinematics and evolution of these large-scale objects.
Collective outflows of matter from the forming star clusters are another virtually unexplored important type of motion in star-forming regions. It is known that all star clusters eventually get rid of their parent gas. Clumps of molecular gas in the direction of forming star clusters in star-forming region S235 are likely to represent such collective outflows \[24, 25\]. A deeper understanding of the collective outflows from forming clusters requires observations of CO-dark molecular gas in the \[CII\] lines at wavelength of 158 microns, which can be effectively carried out by Millimetron.
Studies of objects with large angular dimensions should be preferably carried out step by step. During the initial stage it is necessary to determine an overall distribution of the radiation in the test line with a matrix spectrometer in the broadband spectroscopy mode with an average spectral resolution. To study a detailed kinematics, a high-resolution spectrometer is proposed to be used in a number of key areas identified in the first phase of the Millimetron observations or using data obtained previously by other instruments.
Star formation regions in the Galaxy
------------------------------------
Spectroscopic observations with high resolution will allow a detailed study of the molecular structure of protostellar objects. Here of particular importance are studies in the short-wavelength range of Millimetron at wavelengths less than 300 microns. The number of lines in this spectral range is smaller than in the longer wavelength region \[26\], which facilitates both the identification of the lines and their analysis (Fig. 3). The number of lines is still very large, including both simple compounds lines, and lines from complex organic molecules which are interesting from the astrobiology viewpoint. Some line transitions, available for Millimetron observations, are presented in Table 3.
![The predicted number $N_{0,1K}$ of some molecular lines with maximum emission at a temperature of above 0.1 K as a function of frequency \[26\].](14-img008.png){width="7.5cm"}
The high spectral resolution ($\sim10^6$) and high sensitivity achievable in the Millimetron project in observations of molecular lines make it possible to characterize the physical parameters and motion of gas in the protostellar objects.
[|m[3.166cm]{}|m[8.637cm]{}|]{} Species & Frequencies (GHz)\
C I & 492, 809\
O I & 2060, 4745\
HD & 2675, 5332\
OH & 1835, 2510, 3789, ...\
CH & 537, 1477, 1657, 2007, 2011, 4056, 4071, ...\
HF & 1232, 2463\
H$_2$O & 557, 988, 1113, 1670, 2774, 2969, ...\
HDO & 465, 894\
C II & 1901\
N II & 1461, 2459\
O III & 3393, 5787\
N III & 5230\
HeH$^+$ & 2010, 4009\
OH$^+$ & 972, 1033, 1960, ...\
CH$^+$ & 835, 1669, ...\
SH$^+$ & 526, 683, 893, 1050, ...\
H$_2$O$^+$ & 1115, 1140, ...\
H$_3$O$^+$ & 985, 1656, ...\
H$_3^+$ & 3150\
H$_2$D$^+$ & 1370, 2577\
D$_2$H$^+$ & 1477\
One of the most interesting lines shown in Table 3 belongs to atomic oxygen. The OI line at a wavelength of 63 microns significantly contributes to cooling of the warm ISM and photodissociation regions with high density and intense UV radiation. Observations of this line are needed to study the energy balance in ISM, but none of the existing instruments can observe this line. Observations of the 63 $\mu$m line by Millimetron will provide information about the content of oxygen in the interstellar medium, as well as will help to solve the problem of the intensity ratio of the oxygen lines at wavelengths 63 and 145 microns detected in observations by ISO (Infrared Space Observatory) \[27\]. Joint observations of oxygen, water, molecular oxygen and hydroxyl lines will help to explain the chemical evolution of oxygen compounds.
The rate of accretion onto protostellar objects is one of the key questions in protostellar evolution studies. As direct measurements of the accretion rate are very difficult, it is usually inferred from molecular outflow parameters \[28\], for example, using CO observations. However, to understand the transition from the outflow rate to the accretion rate more information is required: the velocity and extent of the outflow, the inclination of the system, parameters of the surrounding material. Observations of the 63 $\mu$m\[OI\] line will measure the accretion rate in a more direct way, but this will also require a high angular and high spectral resolution.
Another way of probing the accreting matter is to observe absorption lines, which would guarantee that the absorbing material is in front of a growing protostar. Recently it was shown \[29\] that a promising line in this respect could be an ammonia line having the wavelength of 166 microns. FIR spectral observations by Millimetron will provide an opportunity to study accretion onto protostars with a spatial resolution four times higher than the SOFIA telescope. This gain in spatial resolution can be critical for star formation studies.
Observations of neutral oxygen and ionized carbon lines with high angular and high spatial resolution are valuable to probe the evolution of ionized hydrogen regions. In particular, a numerical simulation shows that the width of the ionized carbon region around a young massive star significantly depends on the parameters of the star and the surrounding gas density \[30\].
One of the tasks of the Millimetron operations will be high-resolution spectral surveys covering frequency bands of a few tens or hundreds GHz. The Millimetron high sensitivity and lack of atmospheric absorption will enable surveying faint and hence poorly studied sources (for example, “hot corinos”, that is, hot regions near low-mass protostars). As a result, in addition to determining the main physical parameters and molecular composition of these sources, new molecules can be detected, including those important from the viewpoint of astrobiology.
One of the tasks for Millimetron space observatory will be studying the cosmic masers in the millimeter and submillimeter wavelengths in the single-dish mode. Bright masers occur in water vapour line $6_{1,6}-5_{2,3}$ at a frequency of 22 GHz. However, there are other known maser lines of H$_2$O in the submillimeter wavelength range. Submillimeter lines of water vapour are difficult or even impossible to observe from Earth due to strong absorption in the atmosphere, so observations of H$_2$O masers are carried out almost exclusively at frequency 22 GHz. As a result, even very crude model of these objects cannot be built. Some progress in observations of submillimeter maser lines have been made at high-altitude astronomical observatories and using space-based observatories SWAS (Submillimeter Wave Astronomy Satellite), Odin, and Herschel. Further progress will be possible after the launch of Millimetron, which will be the best submillimeter maser observatory among space observatories to be launched within the next 10-15 years.
Protoplanetary disks and protostellar objects
---------------------------------------------
One of the key issues in the physics of ProtoPlanetary disks (PPD) is their mass. Presently, to determine the PPD masses, mainly millimeter dust emission observations are used, under the assumption that the dust is well mixed with gas. However, the sensitivity of modern telescopes is insufficient for detection of low-mass and distant PPD. There are many disks, where only upper limits of radiation fluxes (<10 mJy) are obtained. Meanwhile, only the accurate disk mass evaluation can clarify whether the disk is able to form a planetary system. It would be highly desirable to find a direct indicator of the gas mass. Presumably, HD molecule radiation (Fig. 4) at 112 and 56 microns \[31\] could be such an indicator.
One of the biggest challenges for Millimetron may be observations of water in PPD. In these objects, both a “warm” (close to the star) and “cold” (more distant) water reservoirs are possible. At present, water is commonly viewed as being the main factor determining the structure of planetary systems (the “snow line”). Observations of water lines on the Herschel space telescope were carried out only for a few PPD, and observations of the “cold” water in them yielded conflicting results \[32, 33\]. Obviously, to clarify the role of water in the formation of planetary systems a more significant sample is needed, which requires instruments with greater sensitivity and better angular resolution. The combination of water lines in various parts of the Millimetron range allows to make conclusions about the spatial distribution of water in the disk, in particular, about actual location of the “snow line”.
In PPDs, observations of molecular oxygen and ro-vibrational lines of complex organic and simple compounds in the planetary formation zones as well as of less common isomers of previously detected molecules (or discovered by ALMA) will also be feasible. High temperatures (over 100 K) in planetary formation zones at distances less than 5–20 AU from the star result in populating high transitions, especially in complex molecules. An example is provided by the detection of organic molecules in the inner parts of the disks with the Spitzer telescope (see, for example, \[34\]). Calculations of the PPD chemical structure show \[35\] that in the planetary formation zones, column densities of such molecules as methyl cyanide, formic acid, etc., with lines in the Millimetron range, reach large values.
In the submillimeter and far-infrared (100 $\mu$m) range, observations of the large particles forming in PPD are possible. It is very important to calibrate receivers properly, with a wide and continuous spectral coverage. Figure 5a shows theoretical spectra of a PPD in the globule CB26 in comparison with the Herschel telescope observations. Error bars indicate the flux calibration uncertainty. The figure shows that in order to find the mass distribution and size of dust particles, high-precision observations of disks at wavelengths of about 100 microns are needed. To solve this problem the single-mirror mode is sufficient.
FIR spectroscopy with a moderate spectral resolution allows the bright CO and water to be detected, as well as the parameters of the PPD inner regions to be determined.
{width="7.5cm"}
In addition to the PPD studies, searching and studying of cold gas and dust clouds in the Galaxy is an important task. One of the main sources of information about the physical conditions in the prestellar and protostellar objects is their broadband spectral energy distribution (SED). It was SED that became the basis for the currently accepted classification system of these objects. Moreover, for objects of class -1 (prestellar core) and 0 (the earliest stage of evolution in the presence of a central IR source), the spectral maximum falls in the FIR and submillimeter range. A space telescope will allow building SED of prestellar cores without breaks caused by atmospheric transparency windows. The detailed shape of the spectrum will clarify the evolutionary status of a specific core. Until now, the identification of a core as being the pre- or protostellar one has been based on the absence or presence of a compact source in the core. The relative number of pre-stellar and protostellar objects forms the basis for estimation of the relative duration of the corresponding evolutionary stages. However, for example, observations of the low-mass core L1014 by the Spitzer space infrared telescope revealed the presence of a weak compact internal source by excess radiation at wavelengths of less than 70 microns \[37\].
Pavlyuchenkov et al. \[38\] show the existence of such a problem also for massive cores. Paper \[37\] analyzed observations of two massive cores of infrared dark clouds (IRDC). Near-IR and millimeter studies classified these cores as starless. However, analysis of the spectrum at a wavelength of 70 micrometers revealed that in both cases the cores already hide embedded compact sources of radiation — protostars.
The reasons for the importance of the 50-150 micron range are illustrated in Fig. 5b, which shows the result of fitting of the near-IR spectrum of a typical protostellar object with a model from \[39, 40\]. The formal best-fit model is shown by the solid black curve. However, similar fits can be obtained by other models with the object mass ranging from $1M_\odot$ to $10M_\odot$. Figure 5 clearly shows that the maximum emission of a typical protostellar object falls in the FIR range. Inclusion in the model of the same object the data on the radiation at a wavelength of 70 microns reduces the mass uncertainty by three times.
![(a) Spectral energy distribution (SED) of the protoplanetary disk in globule CB26 observed by the Herschel telescope. The red curve is a model of the dust with typical ISM parameters (the maximum grain size $a_{max}=0.25$ $\mu$m ), the blue curve is a model for $a_{max}$ 50 times as large as the typical value (calculations according to the model presented in paper \[36\]). (b) The fitting of the observed near IR spectrum (filled circles) from the typical protostellar object. The formal best-fit solution is shown by the thick black curve. The grey curves show spectra of objects with masses from $1M_\odot$ to 10 $M_\odot$, which equally well fit the observed near IR data but significantly deviate in FIR (http://caravan.astro.wisc.edu/protostars/). The dashed curve shows the best-fit solution for the photosphere emission from the central source.](14-img005.png "fig:"){width="7.5cm"} ![(a) Spectral energy distribution (SED) of the protoplanetary disk in globule CB26 observed by the Herschel telescope. The red curve is a model of the dust with typical ISM parameters (the maximum grain size $a_{max}=0.25$ $\mu$m ), the blue curve is a model for $a_{max}$ 50 times as large as the typical value (calculations according to the model presented in paper \[36\]). (b) The fitting of the observed near IR spectrum (filled circles) from the typical protostellar object. The formal best-fit solution is shown by the thick black curve. The grey curves show spectra of objects with masses from $1M_\odot$ to 10 $M_\odot$, which equally well fit the observed near IR data but significantly deviate in FIR (http://caravan.astro.wisc.edu/protostars/). The dashed curve shows the best-fit solution for the photosphere emission from the central source.](14-img006.png "fig:"){width="7.5cm"}
The 50-150 micron range was available for observations by the Herschel space telescope. However, the lack of high angular resolution of the instrument did not allow a detailed investigation of the protostellar objects at large distances, which significantly limits their sample. The higher angular resolution of Millimetron enables observations at longer wavelengths. In particular, using the PACS (Photodetector Array Camera and Spectrometer) detector of the Herschel telescope (70, 100 and 160 microns) many point-like objects inside many IRDC were discovered \[42\]. However, at longer wavelengths (250-350 microns) such point-like objects were not resolved. Meanwhile, it is necessary to determine temperature, mass and luminosity of these sources more accurately. The high sensitivity of Millimetron will help to discover more compact protostellar sources by including into the statistics distant and (or) denser objects.
In addition to cold gas and dust clouds in star-forming regions, hotter objects have been detected. They show a dust continuum and numerous spectral lines of gas-phase molecules. The spectral energy distribution in such objects is still poorly explored, but is a very important task.
Another important field of research is the study of high-speed bipolar outflows formed by accretion disks around protostars and young stars (for example, the object S255IR that emits 1.1 mm continuum) (Fig. 6). By the appearance, a bipolar outflow is similar to a black hole jet, which suggests a possible similarity in physical mechanisms of both phenomena. The origin of bipolar outflows is uncertain yet, so their study is an actual task.
![Example of q high-velocity bipolar outflow. Blue and red contours show the emission in the blue and red wing of the CO(3-2) line, respectively. Sky coordinates $\alpha,\; \delta$ (epoch J2000) \[41\].](14-img007.png){width="7.5cm"}
As a related problem, we can also mention submillimeter observations of asteroids and comets. Observations of asteroids in the reflected (scattered) light could not give reliable information about their size, because their albedo should be assumed. Observations of the intrinsic radiation of asteroids in this respect are more reliable. In addition, the emission spectra of asteroids allow us not only to evaluate their size, but also to obtain information on the chemical composition and surface structure \[43\]. However, the temperature of asteroids (especially away from the Sun) is low, and so their emission fall in the infrared and submillimeter range, which requires space observations. Spectral observations of comets make it possible to clarify their molecular composition and obtain information about the evolution of matter in the early Solar System.
Maser sources
-------------
Masers quantum transitions in molecules are a powerful tool to study different astrophysical sources, such as accretion disks around supermassive black holes in galactic cores, proto-stellar / proto-planetary disks and outflows from young stars in star-forming regions, regions of interaction between expanding HII regions and supernova remnants and dense surrounding gas clumps, expanding shells and jets associated with evolved star \[44, 45\]. Observations of masers are widely used to detect sources with unique physical and evolutionary status, to measure their distances, to study their parameters, kinematics and structure of accretion disks \[46-48\]. Modern scientific projects RadioAstron, BeSSeL (Bar and Spiral Structure Legacy Survey), MCP (Megamaser Cosmology Progect), and MMB (Methanol Multibeam Survey) demonstrate high possibilities of maser observations as a research tool. The value of this instrument is determined by a precise measurement of positions of the objects. Maser sources have a small angular size, so observations in interferometry mode are most important.
In the ground-Space VLBI (SVLBI) mode the receiving equipment of Millimetron allows observations of water masers at the transition frequency 22235.08 MHz. The possibility of maser observations in the SVLBI mode was shown by the RadioAstron space mission.
The scientific program to study cosmic masers using ground-space interferometer RadioAstron included observations of 19 water molecule maser sources. These masers are very compact (often not even resolved using the largest ground-based bases) objects that have the highest brightness temperature, sometimes exceeding $10^{17}$ K \[49\]. Because of these properties, masers can be used to high-precision studies of the kinematics and physical parameters of objects in our and other galaxies. Observations by the ground-space interferometer allow to resolve the most compact components and to evaluate the brightness temperature of the maser source and its size, which is necessary to clarify the pumping mechanism and to model the emitting region. During observations of the water line at the frequency of 22 GHz, radiation from extremely compact maser components in the directions to the four star formation was detected: Orion A, W31RS5, W51M/S and Cepheus A \[50\].
Some sources (Cepheus A) show the presence of very compact substructure maser details with a size of the order 10 micro arcseconds (which corresponds to 0.007 a.u.). Moreover, the objects are moving with a high relative velocity. This means that the maser source has a complex spatial and kinematic hyperfine structure on scales comparable to the size of the Sun. This picture is probably an indication that in this case the maser emission arises from the proto-stellar / proto-planetary disks or the smallest turbulence cells corresponding to the dissipation scale. For disks of all types the most important and still outstanding issue is the mechanism of the angular momentum transfer \[51\]. The turbulent viscosity is conventionally considered to be such a mechanism, but as yet there is no consensus on the mechanism of the turbulence excitation \[51, 52\].
The Millimetron equipment allows observing masers in the submillimeter range, as well as masers formed in other galaxies and in evolved stars, which requires better sensitivity than that of the RadioAstron. At the moment, the possibility of observing in the SVLBI mode is confirmed only for of masers in star-forming regions of the Galaxy at hydroxyl and water molecule centimeter transitions.
Stars and planets
=================
Direct observations of exoplanets
---------------------------------
One of the promising methods for studying extrasolar planets is their direct observations.
In the single-dish mode Millimetron is able to observe massive planets far remote from central stars. The more distant the system, the larger should be the orbital radius of the planet to be resolved by a telescope. Unfortunately, Earth-like planets at such distances have very low radiation intensities, so only gas giants can be observable in this mode. Table 4 lists extrasolar planets (as of the end of 2013), which will be available for direct Millimetron observation. Only three planets from this list could be observed by the Herschel telescope, and in addition planet Fomalhaut b was observed at the limit of sensitivity. Two planets shown in Table 4 were discovered already after the Herschel telescope operation had been completed. Note that in the range of Millimetron measurements, the star and a remote gas giant have the lowest contrast, which facilitates their observation \[53\].
Observations of known transiting planets are of particular interest. Observations of planetary transits and antitransits allow the orbital parameters of the planet, its mass and radius to be determined, as well as the absorption spectra of the upper layers of its atmosphere to be obtained. Since 2003, there has been a series of detailed spectroscopic observations of transits and antitransits of different types of exoplanets \[54-58\]. The high accuracy of Millimetron enables us to carry out spectroscopic observations of transits and antitransits and thus to obtain unprecedentedly comprehensive information about these systems.
[|l|l|m[1.5cm]{}|m[1.5cm]{}|l|l|m[2cm]{}|l|]{} Name & Mass, $M_\odot$ & semiaxis, au & distance, pc & stellar type & age & effective temperature & <<Herschel>>\
Fomalhaut b & 3 & 115 & 7.704 & A3 V & 0.44 & 8590 & limit\
HN Peg b & 16 & 795 & 18.4 & G0 V & 0.2 & - & yes\
WD 0806-661B & 8 & 2500 & 19.2 & DQ D & 1.5 & - & yes\
AB Pic b & 13.5 & 275 & 47.3 & K2 V & 0.03 & 4875 & no\
Ross 458 (AB) & 8.5 & 1168 & 114 & M2 V & 0.475 & - & no\
SR 12 AB c & 13 & 1083 & 125 & K4-M2 & 0.001 & - & no\
FU Tau b & 15 & 800 & 140 & M7.25 & 0.001 & 2838 & no\
U Sco CTIO 108 & 16 & 670 & 145 & M7 & 0.011 & 2600 & no\
HIP 78530 b & 23 & 710 & 156.7 & B9 V & 0.011 & 10500 & no\
GU Psc b & 11 & 2000 & 48 & M3 & 0.1 & - & -\
HD 106906b & 11 & 654 & 92 & F5V & 0.013 & 6516 & -\
Mass loss at late stages of stellar evolution
---------------------------------------------
At the late evolutionary stages, low- and intermediate mass stars (which include the Sun) experience an intensive mass loss. This is manifested in an expansion of the stellar envelope with the subsequent transformation into a planetary nebula. Studies of circumstellar medium during and after the asymptotic giant branch (AGB) are also closely related to the ISM enrichment by heavy elements.
High abilities of space investigation of low- and intermediate mass stars in the infrared range were demonstrated by such key programs of the Herschel telescope as HIFISTARS, “The circumstellar environment in post-main-sequence objects”, and others [^3]. Higher angular resolution, sensitivity, and a wide range of Millimetron not only can significantly improve the quality of research evolved stars, but also can formulate a number of new challenging tasks in the study of these objects.
![Image of the star R Aqr in the continuum at wavelength 70 $\mu$m (a) and 160 $\mu$m (b). Arrows show the proper motion of the object \[59\].](14-img012.jpg){width="15cm"}
Observations from the high-resolution spectrometer will allow, in particular, the following studies.
- Kinematics of outflows from AGB stars. Millimetron allows observations of the most highly excited molecular lines of CO, H$_2$O, and other lines formed close to the stars. Such observations are impossible with other instruments, including the Herschel telescope, and will help to answer a number of important issues related to the outflow mechanism. For example, according to modern concepts the radiation pressure on the dust in oxygen stars is not large enough to explain the observed mass loss rates.
- The cause of asymmetry of the stellar shells, symmetrical on the AGB stage and asymmetric at the stage of proto-planetary and planetary nebulae. Currently, there are several hypotheses. One of them is related to the binarity of the star \[59\]. To confirm or refute this hypothesis observations with high angular resolution are required, which are possible by Millimetron.
- The composition and physical parameters of the interacting regions of shells of moving evolved stars with an environment. Observations made by the Herschel telescope revealed the presence of complex structures and motions in such objects which allows us to investigate the stellar envelope, circumstellar medium, and motion of the stars \[59, 60\] (see Figure 7).
- The formation of molecules and dust in the evolved stars shells \[61, 62\]. The use of the short-wavelength matrix spectrometer will allow, in particular, to carry out the following studies:
- Observations of extended planetary nebulae in the \[NIII\], \[OIII\], \[OII\], \[CII\] lines, etc., to study the dependence of the chemical composition and physical parameters of the gas on the distance to the star. The methodology that was developed in paper \[63\], will be further improved to analyze the Millimetron data with a better sensitivity. These studies are related, in particular, to the problem of the origin of poor hydrogen stars;
- Observations of clumps and other irregularities in the structure of extended planetary nebulae to study variations of temperature and density, which largely determine the chemical composition and evolution of these objects;
- Studies of the structure of stellar shells interacting with the environment. These studies also require observations using matrix instruments.
Searches for extraterrestrial life
----------------------------------
Due to the absence of atmosphere, the Millimetron observatory will be capable of studying many spectral lines of water molecules and more complex organic compounds which are difficult or impossible to observe from the Earth surface. The study of these lines in the solar system and PPD will help to determine the origin and primary molecular composition of the Earth oceans, as well as to draw conclusions about the abundance of planets containing liquid water on the surface, and therefore having suitable conditions for life.
For years, the search for manifestations of extraterrestrial civilizations is one of the most ambitious projects of the mankind. Major efforts are now focused on interception of messages from extraterrestrial civilizations and the millimeter range is promising for these purposes. Paper \[64\] justifies the benefits of this range for the directed transmission in the midsection of the cosmic microwave background. A characteristic marker of this region of the spectrum can be the line of positronium hyperfine splitting at 203 GHz ($\lambda=1.5$ mm), an analogue of the 21 cm line of hydrogen atom. Preliminary observations have already begun \[65\]. Search for positronium using Millimetron is an independent important task.
Along with the search for signals from extraterrestrial intelligence, traces of astro-engineering activities are being searched for. In particular, a well-developed civilization is able to surround a star by a system of structures, intercepting and using a significant portion of stellar energy (the so-called Dyson sphere \[66\]) which should re-emit the whole or part of the energy at lower frequencies than a radiation frequency of the star itself. For the Sun and the Dyson sphere with radius of 1 AU the temperature of the sphere will be about 300 K. It can be expected that the use of more advanced technologies will be associated with the use of low temperatures and the position of the emission maximum will shift from 20 microns towards longer wavelengths. Therefore, such objects should be searched for most effectively in the infrared range up to the wavelength corresponding to the maximum of space radiation (1.5 mm). The first sky FIR survey aimed at the detection and spectral measurements of astronomical objects was carried out on the IRAS satellite (InfraRed Astronomical Satellite). 250 thousand point-like sources were found. Results of the search for objects similar to the Dyson sphere, are reported in papers \[67, 68\] (Fig. 8); several objects were found, where a natural origin has not been still reliably proven.
![Sky location of 16 possible Dyson sphere candidates (red symbols). Three of them (red squares) show the least deviation from a black body spectrum. The blue symbols show 2240 candidates selected from the IRAS catalogue. The green curves limit the sky region accessible for the Arecibo radio telescope participating in the SETI program \[68\].](14-img013.png){width="10.5cm"}
Important criteria are the spectral parameters and their comparison with the black body spectra. The spectral maximum determines the temperature. The spectral index in the long-wavelength part of a power-law spectrum is -2 for the black body and -3 and -4 for amorphous and metal dust particles, respectively, whose size is much smaller than a wavelength. Temperature, flux, form of the red part of the spectrum and a distance from the source can be used to estimate the size of the source, and to distinguish it from natural clouds of dust or stones emitting in the infrared (protostellar objects, old stars). To develop a reliable criterion of the search for the Dyson spheres, it is necessary to investigate in detail properties of natural sources, which can be accomplished by Millimetron.
Supernovae and supernova remnants
=================================
White dwarfs
------------
FIR photometric observations of nearby cold white dwarfs allow determination of their atmospheric composition, which can be used to obtain precise ages of the objects and of the Galaxy itself.
White dwarfs are, apparently, the most numerous sky objects. They can be separated into two large groups: hot and cold white dwarfs. A cold white dwarf represents an observable end stage of the white dwarf evolution, therefore an age estimate of cold white dwarf can be used to determine the age of the Galactic disk and halo, as well as of the nearby globular clusters. White dwarf cooling, which plays the crucial role in the white dwarf evolution, is not yet fully understood. It depends on the white dwarf atmosphere composition. Therefore, to determine the temperature, luminosity and age of cold white dwarfs their atmospheric composition should be known.
Analysis of mid-IR observations of nearby cold white dwarfs with an effective temperature of less than 6000 K revealed that the maximum of emission in this range is somewhat lower than predicted by models that well reproduced the observed luminosity function at wavelengths shorter than 1 $\mu$m \[69\]. This situation is illustrated in Fig. 9, where the observed spectral energy distribution in the wavelength range from 0.1 to 100 $\mu$m is shown for cold white dwarf LHS 1126 together with model atmospheres of a different chemical composition. Clearly, to solve the issue of atmospheric composition of cold white dwarfs, high-precision FIR measurements are required.
The expected Rayleigh-Jeans fluxes from white dwarfs at distances closer than 100 pc are about a few mJy.
An interesting problem is to search for cold dust debris disks around white dwarfs, which can be formed due to comet and asteroid collisions, by performing single-dish FIR observations. For example, white dwarf G2938 shows a FIR excess. In addition, in the atmosphere of this object, as well as in other hot white dwarfs heavy elements were discovered, such as calcium and iron, which should have plunged deeper into the star due to strong surface gravity \[70\]. A presence of a debris disk irradiated by the white dwarf could be an explanation. More than 20 white dwarfs surrounded by dust disks are currently known from FIR observations. Presently, studies of white dwarf atmospheres “polluted” by heavy elements are carried out in the middle IR range: for example, photometric data was obtained by WISE (Wide-Field Infrared Survey Explorer) \[71\]. Several white dwarfs demonstrate an excess in this range \[72\].
![Spectral energy distribution of the cold white dwarf LHS 1125 (model) \[69\]. The red and blue curves show models with different hydrogen to helium ratio in the atmosphere. Circles with error bard show the results of observations. The dashed curve, which best fits the observations, corresponds to the spectral slope –1.99. The inset zooms the spectrum at 5—8 $\mu$m.](14-img014.png){width="7.5cm"}
In the last time, a presence of exoplanets around white dwarfs is discussed. In principle, white dwarfs are sufficiently bright objects to sustain water in the liquid state on surfaces of such planets. The first planet was discovered around the star GI86, which belongs to a binary system with a white dwarf. Both photometric and spectroscopic studies of white dwarf atmospheres with heavy elements and of white dwarfs surrounded by dust disks aimed at exploring properties of the dust, atmospheric composition and discovering exoplanetary systems around them, are important and interesting tasks.
Pulsar radio emission
---------------------
Spectra of most radio pulsars rapidly decrease with frequency \[73\]. However, some objects are observed in the Gigahertz range \[74-76\]. Here, there can be several new and interesting tasks.
In four radio pulsars (B 0329+54, 0355+54, 1929+10 and 2021+51) and in two anomalous X-ray pulsars (AXP) (XTE J 1810-197 and 1E1547-5408) spectral flattening is seen at several dozens of GHz (Fig. 10 a,b) or even an intensity increases when the frequency increases up to 87 GHz \[75\]. In radio pulsars and AXPx, the 43 GHz flux density lies in the range from 0.15 to 0.50 mJy \[77\] and from 1 to 5 mJy \[78, 79\], respectively. The sensitivity of the bolometer for broad-band ($\gtrsim$ 10 GHz) observations in the 50-200 GHz frequency range with an exposure of several ten minutes is estimated to be around 0.1-0.5 mJy, which likely makes it possible to detect such pulsars.
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Moreover, estimates show that in these objects magnetic field and spin rotation axes misalignment can be small (less than 30$^\circ$). In this case, the magnetosphere can extend far beyond the light cylinder forming appreciable pitch-angles of relativistic electrons (more than 0.01), and synchrotron radiation is generated \[80, 81. For typical parameters of these pulsars the intensity of the radiation must increase with frequency starting from a few tens of GHz. When the usual pulsar parameters are assumed the maximum frequency is about $\nu_{max}\sim3\times 10^{11}$ Hz. A qualitative estimate of the millimeter intensity increase is model-dependent. For a monoenergetic energy distribution the intensity increases as $\nu^{1/3}$, suggesting twofold intensity increase at 275 GHz compared to 34 GHz. For a power-law electron energy distribution and assuming a large optical depth of the emitting region, the intensity increases as $\nu^{2.5}$, and at 275 GHz the intensity increase can be as high as 200 times (Fig. 10c). Testing these predictions is important to get deeper insight into the models of magnetospheres of both radio pulsars and AXPs.
Polarization measurements of pulsars with increased flux density at several tens of GHz play an important role. The degree of polarization of ordinary pulsars, as a rule, decreases with frequency. The switch-on of the synchrotron mechanism should increase the degree of polarization. Confirmation of this property would provide an additional argument supporting the synchrotron radiation hypothesis for pulsars. In addition, the pulse profile generated by the synchrotron mechanism differs from profiles expected in other models. Therefore, in the millimeter range the pulse profiles can be different from those at lower frequencies.
In parallel with spectral observations, it is necessary to measure the angular size of the emitting region in pulsars. Non-thermal pulsar radio emission is assumed to be generated inside the light cylinder with radius
$$r_{\text{LC}}\left(\text{cm}\right)=\frac{\text{cP}}{\text{2$\pi $}}=4.8\cdot
10^9P\left(\text{s}\right),$$
where $P$ is the pulsar spin period.
Millimeter radio fluxes are low, smaller than 1 mJy. However, some pulsars demonstrate outbursts with higher fluxes. Therefore, the detection of a sufficiently bright outburst in the regime of Earth-space interferometer would enable to resolve the light cylinder region for the first time (up to pulsar distances of several ten kiloparsecs) and would provide an invaluable information for understanding the nature of these objects and localization of the emission region of the observed electromagnetic radiation.
The angular size of the light cylinder is
$$\theta \left(\text{arcsec}\right)=\frac{r_{\text{LC}}} d=3.2\cdot
10^{-7}\frac{P\left(\text{s}\right)}{d\left(\text{kpc}\right)},$$
where $d$ is the pulsar distance. With an angular resolution of 40 nano arcsec, the light cylinder of a pulsar with period 1 s can be resolved from a distance of several kiloparsecs. The size of emission region comparable with the light cylinder may additionally support the synchrotron model.
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At magnetosphere periphery magnetic field lines are bended due to plasma rotation, and it is possible to have a situation, where the emission generated at moderate altitudes above the neutron star surface cannot reach the observer, while the synchrotron radiation at millimeter wavelengths can be seen by him. In that case the appearance of a pulsar seen only at the millimeter wavelengths is possible (Fig. 11).
The search for and detection of pulsars generated only microwave pulses is a totally new field in pulsar studies, which can significantly increase the observed neutron star populations.
Studies of pulsars with ground-space interferometer RadioAstron suggest that there are interferometric responses exceeding the diffraction spot size, i.e. when the scattering circle of pulsar radio emission on the interstellar plasma inhomogeneities is resolved. Here, the amplitude of the interferometer signal does not decrease with increasing the interferometer base up to 240 000 km, the maximum length obtained by the RadioAstron for radio pulsar B0329+54 \[82\]. The structure of the interferometer signal with such long bases is related to spectral parameters of the interstellar plasma inhomogeneities. For distant pulsars, diffraction scintillations can be observed at cm radio wavelengths. Observations of such pulsars by Millimetron in the interferometric mode together with the largest ground-based radio telescopes with 1 million km baselines will probe both the structure of the interstellar plasma inhomogeneities and that of the emission generation region in the neutron star magnetosphere.
Relativistic objects in centers of globular clusters
----------------------------------------------------
Observations of central parts of the most massive globular clusters, in particular, with large core radii, in the single-mirror mode can be used to detect millimeter emission from stellar mass black holes in binary systems. Should the signal be detected, observations in the Earth-space interferometer mode with high angular resolution can be performed.
Black holes with masses (5-20) $M_\odot$ are end products of stars with masses in the main sequence $/geq 25 M_\odot$ \[83\]. Recent theoretical studies suggest that several hundreds of stellar-mass black holes can be present in old globular clusters \[84\]. Almost all of them should be single black holes, which is in agreement with a small number of X-ray sources with black holes in globular clusters. The presence of black holes can heat up the central parts of the clusters, leading to a significant increase in the cluster core radius \[85\]. Therefore, old globular clusters with large core radii are the most suitable to searches for stellar-mass black holes.
Recent observations with VLA (Very Large Array) discovered two black hole candidates in two globular clusters M22 \[86\] and M62 \[87\]. Both these clusters are massive but have different structure. For example, cluster M22 has an extended core, in line with theoretical predictions, while cluster M62 has a rather compact dense core.
VLA observations of M22 with a maximum possible angular resolution of $\sim 1 ''$ revealed that the observed faint source can be in a binary system (Fig. 12a) with properties similar to those hosting stellar-mass black holes (Fig. 12b).
![(a) The VLA image of the globular cluster M22 in the continuum emission from the core. Two bright oval objects are sources identified as M22-VLA1 and M22-VLA2. The red cross shows the photometric center of the cluster \[86\]. (b) The X-ray to radio flux relation from stellar-mass black holes. $L_{8.4 GHz}^R$ is the radio luminosity at 8.4 GHz, $L_{3-9 keV}^X$ is the 3-9 keV X-ray luminosity. The sources in M22 and M62 have properties rather similar to black holes and not to white dwarfs or neutron stars \[87\].](14-img018.jpg "fig:"){width="7.5cm"} ![(a) The VLA image of the globular cluster M22 in the continuum emission from the core. Two bright oval objects are sources identified as M22-VLA1 and M22-VLA2. The red cross shows the photometric center of the cluster \[86\]. (b) The X-ray to radio flux relation from stellar-mass black holes. $L_{8.4 GHz}^R$ is the radio luminosity at 8.4 GHz, $L_{3-9 keV}^X$ is the 3-9 keV X-ray luminosity. The sources in M22 and M62 have properties rather similar to black holes and not to white dwarfs or neutron stars \[87\].](14-img019.png "fig:"){width="7.5cm"}
VLA observations of the globular cluster M62 with a maximum time exposure revealed the presence of a very faint central source M62-VLA1 with a flat radio spectrum and radio flux 18.7 $\pm$ 1.9 $\mu$Jy at the frequency 6.2 GHz \[87\]. Observed properties of this source in the radio, X-rays and optical are similar to those of the well-known transient X-ray source V404 Cyg, which is believed to be a (presently quiescent) stellar-mass black hole in close binary system.
High-sensitivity mm observations with a moderate angular resolution of the black hole candidates discovered in globular clusters can be extremely important to confirm their nature. Assuming a flat radio spectrum, the expected mm fluxes from these sources can be a few $\mu$Jy, while the fluxes from other stars should be much smaller. A caveat here will be separation of the sources from the confusion background due to distant galaxies. Possible globular clusters to be observed include M22, M14, M53, M62 and NGC2419.
Origin of ultraluminous X-ray sources
-------------------------------------
Search for intermediate-mass black holes can be another exciting task. Observations of possible mm and submillimeter emission from ultraluminous X-ray sources (ULXs) and hyper-luminous X-ray sources (HXSs) in other galaxies in the single-mirror mode with possible subsequent high-resolution observations in the Earth-space interferometer mode should constrain the models and can help to unveil the nature of these sources.
Ultraluminous X-ray sources are off-center point-like objects with an observed bolometric luminosity $L$ exceeding the Eddington limit for galactic stellar-mass black holes ($20 M_\odot$), $3\times10^{39}$ erg s$^{-1}$, in the 0.3–10 keV energy range. ULXs were discovered in nearby galaxies by the Einstein X-ray observatory \[88\]. For a spherical accretion of fully ionized hydrogen, the Eddington limit can be expressed as \[89\]: $$L_\mathrm{Edd}={4\pi c GMm_p \over \sigma_T \approx 1.3\cdot 10^{38} \left( {M\over M_\odot}\right) \mathrm{ergs s}^{-1}},$$ where $\sigma_T$ is the Thomson scattering cross-section, $c$ is the speed of light, $G$ is the gravitational constant, $m_p$ is the proton mass, $M$ is the black hole mass. This implies that ULXs can harbor intermediate-mass black holes, $M = 10^2-10^4 M_{\odot}$ (see, for example, \[90\]). They also can be close binary systems at an evolutionary stage that is not observed in the Milky Way. The number of ULXs is quite high: 230 \[91\]; in addition, more than 500 ULX candidates are discovered \[92\]. These objects are found in almost quarter of galaxies \[93\] of all types: in star-forming galaxies (which contain about 60% of all ULXs) \[94\]; in dwarf galaxies (Holmberg II); in elliptical galaxies with low star formation rate, as well as away from star formation regions (NGC 1313 X-2 and NGC 4595 X-10) \[95\].
Immediately after the discovery of ULXs, different models of their origin were proposed. These models can be conventionally separated in three main classes: subcritical accretion on to an intermediate-mass black hole \[96\], supercritical accretion on to a stellar-mass black hole \[97\] and collimated emission from a stellar-mass black hole accreting at about the Eddington limit \[98\]. Since ULXs with different properties are observed, it is quite possible that the ULX population includes different types of objects. Stellar-mass black holes are the most likely candidates, and intermediate-mass black holes can explain the observed properties in several exclusive cases \[99\].
It is important to note that the existence of intermediate-mass black holes is an unsolved issue in astrophysics. Until recently, there has been no direct observational evidence of their existence, as well as indirect indications of their reality, despite predictions of the modern theory of structure formation in the Universe. Thus, intermediate-mass black holes are a missing link between stellar-mass black holes and supermassive black holes located in galactic centers. According to current models, intermediate-mass black holes could be formed both during collapses of the primordial stars in the Universe and during the collapse of dense cores of young stellar clusters \[100\], as well as a result of accretion or stellar-mass black hole merging. In the last decades, indications of the presence of intermediate-mass black holes in the centers of globular clusters and in star-forming regions have been emerged.
Unresolved central parts of massive globular clusters and HLSs are the most probable intermediate-mass black hole candidates. In one of the most probable intermediate-mass black hole, ESO 243-49 HLX-1, discovered by XMM-Newton (X-ray Multi-Mirror Mission) space observatory in 2004 in the spiral galaxy ESO 243-49 situated at 8 arcseconds from its nucleus \[101\], radio observations by ATCA (Australia telescope Compact Array) discovered variable radio emission at frequencies 5 and 9 GHz \[102\].
From some nebulae around ULXs radio emission was discovered. The most well-studied examples are Holmberg II X-1 \[103\] and NGC 5408 X-1 \[104\]. Both nebulae show optically thin synchrotron radio emission, similar to radio emission from supernova remnants. Different radio sources show different morphology, but in some cases they look like binary sources \[95\].
Millimeter observations of the nearby ULXs with a sensitivity of $\sim 1$ $\mu$Jy can provide more detailed information that is needed to estimate the black hole mass.
An unexpected result was recently reported by the NuSTAR (Nuclear Spectroscopic Telescope Array) observatory \[105\]. One of the ULXs in galaxy M82 turned out to be an X-ray pulsar ($P_{pulsar} = 1.37$ s) in a binary system with orbital period 2.4 days. The mass function of the X-ray pulsar is $f(m)\sim 2M$. Most likely, this is a neutron star with a high magnetic field and strongly non-spherical magnetically collimated accretion onto accretion columns near the neutron star surface.
Supernovae
----------
One of the most plausible mechanisms of supernova explosions is the magneto-rotational mechanism \[106-108\]. In this mechanism, the explosion asymmetry arises due to the magnetic field mirror symmetry breaking \[109\]. In first days after a supernova explosion its remnant is still very compact. Early observations of the supernova remnant can be used to measure the initial explosion asymmetry before an interaction of the remnant with the surrounding medium that affects significantly particle trajectories of the expanding remnant. The Millimetron space observatory in the interferometer mode will be capable of resolving the supernova remnant up to a distance of 10 Mpc. The expected number of events is several tens per year for type II supernovae and one type Ib/c supernova per year.
Type IIn supernovae could also be interesting for Millimetron observations. Such supernovae can be related to a common envelope stage in the massive close binary evolution \[110 – 112\]. The common envelope leads to asymmetric outflow of the external parts of a red supergiant. The shock wave propagation along the expanding red giant atmosphere can be observed in the millimeter wavelength range. In this model, the outflow asymmetry can be detected starting from the second day after the explosion for a supernova located at distances above 30 Mpc.
Black holes and jets
====================
Black holes are one of the most intriguing predictions of General Relativity, and the problem of proving or refuting their existence is a major task in astronomy. Black holes are extremely compact objects, and to observe them a very high angular resolution is needed. For all currently known objects the angular size of the black hole horizon is less than 20 micro arcsec. The progress in this field, hopefully, will bring the answer in the near future. For example, the ground-space interferometer RadioAstron with a record angular resolution of 7.5 micro arcsec is already carrying out observations of the supermassive black hole in the center of galaxy M87. The ground-based EHT, operating in the millimeter diapason, will possibly resolve the central black hole in the Galaxy.
The space Millimetron observatory, operating jointly with ground-based telescopes, will enable measurements with an ultrahigh angular resolution and hence will be capable of probing smaller physical scales and a larger number of objects beyond these two nearby black holes. In addition to supermassive black holes, stellar-mass black holes in binary systems will be observed. These observations will help to probe the conditions of ultra-high energy particles generation and of the jet formation in the vicinity of black holes.
The VLBI specificity has been taken into account when planning the Millimetron science. We assume that such an interferometer can be used to estimate angular sizes of sources with ultrahigh angular resolution at submillimeter wavelengths, and with corresponding sensitivity (see Table 1) — to construct maps in the frame of a priori model assumptions of the source structure. The interferometer can also be used to construct one-dimensional intensity maps.
Critically important information on the magnetic field and non-relativistic particle number density in the vicinity of the nearby black holes can be obtained not only in the VLBI mode, but also by polarization spectral measurements in the single-dish mode. The presence of a magnetic field in plasma along the line of sight results in the Faraday effect of rotation of the polarization plane of a linearly polarized light. This effect is qualitatively characterized by the Rotation Measure (RM), which is $\phi/\lambda^2$ where $\phi$ is the angle of the polarization plane turn, $\lambda$ is the wavelength.
According to theoretical studies \[113, 114\], the magnetic field on the scale of a black hole accretion disk can be $\geq 10^4$ G. Experimental papers \[115, 116\] showed that the magnetic field indeed increases towards the source nucleus. Therefore, if there are thermal electrons in the near nuclear region, extremely large vale of RM can be expected.
Under quite reasonable assumptions on the magnetic field value and the thermal electron density, it is easy to estimate that RM$>10^4-10^8$ rad m$^{-2}$. The latest results confirm theoretical predictions of the extreme values of RM both from the Galactic center \[117, 118\[ and from distant active galaxies at millimeter wavelengths \[119\]. For linearly polarized synchrotron radiation from the nuclear region, narrow-band frequency channels of the Millimetron detectors will observe “sine-like” variations of the radiation flux density as a function of wavelength.
Detection of the extreme Faraday rotation by ground-based telescopes or by Millimetron in the single-dish mode will provide a list of candidate sources with definitely small angular sizes, which subsequently can be observed by Millimetron in the interferometer mode.
The origin of supermassive black holes in galactic nuclei is discussed in Sections 7.3 and 7.6.
Nearby black holes
------------------
Almost all nearby supermassive black holes (at distances less than 50 Mpc) have a luminosity below the Eddington limit, which is related to a comparatively weak accretion. These black holes include, in particular, the black hole in the Milky Way center (Sagittarius A\*), M87, Centaurus A.
When modeling such sources the accretion rate is assumed to be many orders of magnitude smaller than the Eddington value. For such small accretion rates the standard disk accretion theory is inapplicable and the so-called advection-dominated and radiative-inefficient models are studied. In these models, the accretion disk near the black hole is assumed to be geometrically thick with the characteristic thickness of the order of the radial distance to the black hole. As rarefied plasma of the disk radiates ineffectively, the matter heats up and expands in the vertical direction. The gas temperature can be as high as $10^{11}-10^{12}$ K, and most of the released accretion energy is advected by the radial inflow into the black hole. Therefore, the efficiency of the released energy conversion into radiation is very small, typically of order of $10^{-6}-10^{-3}$.
Since the accreting plasma is very rarefied, the optical depth for free-free processes and Thomson scattering is believed to be small. Moreover, in the submillimeter range it is possible to neglect the synchrotron self-absorption as well. This means that the medium near the black hole in the submillimeter range is optically thin, and the black hole can be directly observed. Therefore, it becomes possible, in principle, to determine the black hole mass, angular momentum, as well as parameters of the accreting flow and its geometrical structure. With sufficiently high angular resolution and high sensitivity, in such flows it can be possible to directly observe jet formation and matter outflows and, thus, to shed light into so far unsolved issue whether the jet results from magneto-hydrodynamic processes in the disk or arises due to the so-called Blandford-Znajek effect related to the black hole rotation. Provided that scales smaller than the gravitational radius can be probed, a principal possibility of studying turbulence of the accretion flow and understanding of the related phenomenon of quasi-periodic luminosity oscillations arises, which can be due to, for example, the presence of hot spots in the disk and/or the excitation of different oscillation modes in the accretion flow by turbulence.
The best studied object of type is Sgr A\* (Fig. 13) in the Galaxy. The distance to this black hole is $R=8$ kpc, its mass is $4\times10^6 M_\odot$, the gravitational radius is $r_g\sim10^{12}$ cm, the bolometric luminosity is $3\times10^{36}$ erg s$^{-1}$, the angular size of the horizon is $r_g/R\sim 10$ micro arcsec. The variability time-scale varies from several minutes to several hours in the near infrared, X-ray, submillimeter and mm ranges.
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There are several breakthrough tasks for Millimetron in the astrophysics of supermassive black holes. The primary task is to resolve the gravitational radius almost for all supermassive black holes within 50 Mpc. The Millimetron space observatory will be capable of resolving, in principle, a structure of accreting flows around black holes at scales of the order of gravitational radius for about 40 supermassive black holes. For such objects as the supermassive black holes in the Galactic center or in M87, the angular resolution of Millimetron will be sufficient to study details a few hundred times smaller than the gravitational radius (Fig. 14). This enables us to probe turbulence in the surrounding gas flows, which is a major issue in the theory of accretion.
![The ratio of the minimum angular size that can be resolved by Millimetron to the gravitational radius for 38 supermassive black holes at distances within 50 Mpc. The minimum angular size is assumed to be $2\times10^{-3}$ rad. The filled circles correspond to the sensitivity threshold in the interferometer mode $10^{-4}$ Jy, filled squares correspond to $10^{-2}$ Jy, diamonds correspond to $10^{-1}$ Jy. The square in the left bottom corresponds to the central black hole in the Galaxy, the bottom right diamond shows the black hole in M87 \[120\].](14-img022.png){width="7.5cm"}
Black holes shadows
-------------------
As noted in Section 5.1, if gas near a black hole is optically thin, the black hole can be seen directly. Indeed, some radiation of this gas is captured by the black hole, and so the region around the black hole at large distances should appear as a dark spot corresponding to the captured radiation. The brightness distribution around this spot will be very strongly changed due to a strong distortion of the light trajectories near the black hole, gravitational redshift, etc. The brightness distribution around the dark spot can be used to study main characteristics of the emitting gas, for example, to determine whether disk or jet mainly contributes to the object luminosity at a given wavelength. A form of the brightness distribution around the dark spot is also strongly dependent on the black hole spin, and a study of this region is apparently the most direct tool of determination of this principal parameter. However, the most significant provided result is that the discovery of the black hole shadow will be the direct evidence that superdense and supermassive objects in the centers of the galaxies are indeed black holes. The angular resolution of ground-based interferometric systems is limited by the Earth diameter, and generally is sufficient to search only for shadows around the closest two-three supermassive black holes, whereas observations from Millimetron in the interferometer mode offer the principal possibility to discover shadows around several dozens of such objects (see Fig. 14).
Fig. 15 shows the expected shapes of shadows from the central black holes in our Galaxy and M87, which were calculated numerically. In Fig. 15a, the black hole illumination is due to a radiatively ineffective disk, and in Fig. 15b — due to a jet. Calculations were carried out taking into account parameters of the EHT telescope. It is very important to perform similar calculations for parameters of the Millimetron space observatory.
![(a) Model image of a radiation-ineffective accretion disk and shadow from the central Galactic black hole. Shown are the intensity distribution (1st and 3d rows) and the visibility function amplitudes (2d and 4th rows) for different models of the accretion flow (in different columns). The models differ by the black hole spin parameter (from left to right): 0.50, 0.90, 0.92, 0.94. The upper and bottom two rows are calculated for wavelength 1.3 mm and 0.87 mm, respectively \[121\]. (b) The brightness distribution of the central region of galaxy M87. The black hole is assumed to be “illuminated” by jet. Calculations for the frequency 345 GHz. The left column shows the initial model distributions, the central and right columns show the model interferometric images as obtained by different methods. $M$ is the black hole mass. Geometrical units with $G=c=1$ are used. The physical units are obtained by multiplying the mass by $G/c^2$. The jet formation radius from the black hole decreases from bottom to up \[122\].](14-img023.png "fig:"){width="8.5cm"} ![(a) Model image of a radiation-ineffective accretion disk and shadow from the central Galactic black hole. Shown are the intensity distribution (1st and 3d rows) and the visibility function amplitudes (2d and 4th rows) for different models of the accretion flow (in different columns). The models differ by the black hole spin parameter (from left to right): 0.50, 0.90, 0.92, 0.94. The upper and bottom two rows are calculated for wavelength 1.3 mm and 0.87 mm, respectively \[121\]. (b) The brightness distribution of the central region of galaxy M87. The black hole is assumed to be “illuminated” by jet. Calculations for the frequency 345 GHz. The left column shows the initial model distributions, the central and right columns show the model interferometric images as obtained by different methods. $M$ is the black hole mass. Geometrical units with $G=c=1$ are used. The physical units are obtained by multiplying the mass by $G/c^2$. The jet formation radius from the black hole decreases from bottom to up \[122\].](14-img024.png "fig:"){width="6.5cm"}
Distant black holes
-------------------
Millimetron will offer the possibility to resolve gravitational radius of a black hole with mass of 100 million solar masses located at a distance of 100 Mpc, and of a black hole with mass of 1 billion solar masses from a distance of 1000 Mpc. A lot of bright active galactic nuclei and quasars can be found at such distances, which presumably contain black holes in this mass range, where accretion proceeds through the standard geometrically “thin” accretion disk. The presence of such a disk can be inferred from a “Big Blue Bump” in the UV spectrum. Although these disks are optically thick, the black hole shadows of can also be searched for (Fig. 16), since the size of such disks in the direction perpendicular to the disk plane is quite small compared to the horizon angular size.
At large distances it is also possible to observe very interesting non-stationary phenomena in active galactic nuclei and quasars, which are not observed in relatively close objects due to their low probability/ Let us briefly discuss only two such phenomena: supermassive binary back holes (SBBHs) and tidal disruption events of stars by a supermassive black hole.
SBBHs are not only interesting objects by themselves, but also can generate the most powerful bursts of gravitational radiation during their coalescence, which can in principle be detected by future space based gravitational wave interferometers from very large distances (up to cosmological scales). Apparently, the most known SBBHs candidate is the BL Lac object OJ 287 located at a distance of 1 Gpc. OJ 287 is thought to be a SBBH with component masses 10 bln and 100 million Solar masses and an orbital period of 12 years (Fig. 17). In the case of this object Millimetron can easily resolve scales of the order of the angular size of the horizon of the more massive component.
![The bolometric flux distribution from accretion disk around a Schwarzschild black hole. The disk inclination angle to the line of sight is 84.5$^\circ$ \[123\].](14-img025.png){width="7.5cm"}
Tidal disruptions of stars by q black hole are usually manifested as X-ray outbursts in inactive galactic nuclei. A characteristic decay time of the outbursts is about a few years.
If the submillimeter luminosity of these objects is sufficient to be observed by Millimetron in the interferometer mode, estimates show that Millimetron will be capable of resolving scales of order of the gravitational radius for at least several objects, including, for example, NGC5905 \[124\]. A very interesting source Swift J1644+57, discovered in 2011 \[125\], is a very powerful source in all wavebands, including the millimeter range, locating at a distance of about 1 Gpc. This source is interpreted as a jet formed after the tidal disruption of a star by a black hole. Millimetron will be able to test the origin of such events.
{width="7.5cm"}
Physics of jets
---------------
Variety of accretion theories leads to variety of theories of jets. AS in the case of disks, the Millimetron space observatory, thanks to its record parameters, will provide missing observational data needed to test different models.
In the case of accretion disks around black holes, the main issues can be listed as follows (see, for example, \[126\]). At first, it is necessary to understand the jet formation mechanism — is it either due to the black hole rotation or due to processes in the accretion disk? Secondly, to understand origin of jet, a magnetic field structure near the black hole should be known. Thirdly, it is yet unclear what kind of plasma moves in the jet (is it electron-positron or electron-ion plasma?), what is its characteristic velocity as a function of distance to the black hole and to the jet axis, what is a distribution of non-thermal particles in the jet, etc. Fourthly, it should be reliably determined whether “typical” jets are two-side or one-side (i.e. whether the formation of a jet directed only to one side is possible) \[127\]. Fifthly, it is important to obtain observational confirmation of a possible change of the jet magnetic field polarity \[128, 129\]. All these problems are interrelated.
In different numerical models, the very possibility of the jet formation depends on the structure of accretion disk and of the magnetic field (Fig. 18). Clearly, to unveil the nature of jets observations with high angular resolution are required, which can probe matter outflows and the magnetic field structure in the vicinity of black holes. Note that when polarization is taken into account the VLBI-mapping can be hampered by extreme Faraday rotation. Possible methods of the mappings are proposed in papers \[131-134\].
Jet formation studies are closely related to accretion disk studies, and observations of these objects in different types of astrophysical objects from stellar-mass black holes to quasars are crucial. With record high angular resolution in the interferometer mode, Millimetron possibly can solve these problems. We stress again that in order to study the magnetic field structure, polarization measurements are crucial.
![Velocity and magnetic field distribution (shown in grey and red, respectively) according to the relativistic jet formation model near a black hole with spin parameter $a=0.9375$ \[130\]. On axes: coordinates in the black hole Schwarzschild radii.](14-img027.png){width="7.5cm"}
Finally, millimeter observations can be decisive to understand the nature of GeV and TeV-flares from active galactic nuclei \[135\]. Such flares are observed from galaxies M87 and 3C454.3 each 2-3 years and last from several days to several weeks. Ultrahigh angular resolution observations can both localize the flare in the jet and to follow the evolution of the emitting region. To clarify the origin of the flares, it is also important to observe simultaneously with such flares central regions of active galactic nuclei \[127, 136-138\].
Jets from cosmic gamma-ray bursts
---------------------------------
An unprecedentedly high angular resolution of Millimetron in the interferometer mode offers the possibility to directly observe jets from cosmic gamma-ray bursts during first several days after the burst, when the jet expansion is relativistic. Indeed, for nearby GRB 030329, which is located at the redshift $z=0.1685$, the angular resolution of 0.16 micro arcsec corresponds to 0.0005 pc, which is equivalent to a light-time of 1 day. Thus, it becomes possible to observe and to investigate the early phase of jet expansion, and may be even before the so-called jet-break time when the GRB afterglow power-low light curve does exhibit a break.
These observations will substantially complete the ground-based VLBI observations \[139\] and will allow the determination of several important parameters, such as the jet-break time, the time of the jet deceleration, the time of the transition from relativistic to the Newtonian expansion, as well as to test the possibility of occurrence of two successive jets with different parameters \[140-142\]. Despite nearby gamma-ray bursts are quite rare, once per 5-7 years, it might be possible to detect one such an event during the Millimetron mission.
Galaxies
========
Evolution of galaxies
---------------------
Star formation leads to dust production, therefore star-forming galaxies are bright sources in the submillimeter range. On the other hand, even a warm dust ($T_d=30$ K) in early galaxies at redshifts $z\sim10$ will be seen in the submillimeter range. [The spectral line 158 $\mu$m of CII ion, which provides cooling of interstellar gas with temperatures 30 K to several thousands K will also fall into the millimeter wavelengths.]{} This [makes]{} Millimetron with its high sensitivity a [powerful instrument for detection of]{} galaxies at high redshifts up to $z\sim 6-7$ and to [draw]{} a sufficiently complete picture of their evolution. To achieve this aim, both continuum and spectral line (CO, CII, OI, etc.) observations are required. [Angular]{} resolution plays a key role [as well]{}.
Galaxies with active star formation are so numerous that under insufficient angular resolution their images can merge. For the Millimetron 10-m primary mirror this effect is much weaker than for the [Herschel]{} telescope and the SPICA project with 3.5-m mirrors (Fig. 1b). Preliminary estimates show that Millimetron will be able to measure [in total]{} spectra of at least 10000 galaxies and to make continuum observations for several [tens of millions of]{} galaxies. Thus, it will [be able to]{} obtain three orders of magnitude more information than the [Herschel]{} telescope. Millimetron will be able to [detect light from]{} galaxies at redhsifts up to about $z=6-7$ (Fig. 19).
![(a) Spectral energy distribution from typical submillimeter galaxies (in the rest frame) peaked at a wavelength of $\lambda \approx$ 100 $\mu$m \[143\]. The red rectangle shows the Millimetron operation range in the single-dish mode. (b) The distribution of submillimeter galaxies number density as a function of redshift \[144\]. $S_{500}$ and $S_{850}$ are the fluxes at 500 and 850 $\mu$m , respectively.](14-img028.png "fig:"){width="7.5cm"} ![(a) Spectral energy distribution from typical submillimeter galaxies (in the rest frame) peaked at a wavelength of $\lambda \approx$ 100 $\mu$m \[143\]. The red rectangle shows the Millimetron operation range in the single-dish mode. (b) The distribution of submillimeter galaxies number density as a function of redshift \[144\]. $S_{500}$ and $S_{850}$ are the fluxes at 500 and 850 $\mu$m , respectively.](14-img029.jpg "fig:"){width="7.5cm"}
Lyman-alpha emitting galaxies represent another [class of]{} very interesting [objects representing violent phases of]{} galactic evolution. The source of emission in these galaxies and their place in general scheme of galaxy evolution are still unknown. The issues to be solved include:
- what is the dust content in this galaxies?
- what evolutionary [stages]{} do they [re]{}present?
- what are the energy sources[: accretion on to a black hole, stellar emission, or gravitational energy?]{}
To answer these questions [millimeter and sub-millimeter]{} photometric and spectral observations of [such]{} objects, [identified primarily in optics are needed.]{} Due to high sensitivity Millimetron [detect and]{} study typical objects with fluxes of the order of several tens of $\mu$Jy \[145\].
Low star-formation regions
--------------------------
[An outstandingly peak]{} sensitivity of Millimetron enables measurements of temperature and mass of dust along the line of sight in [dilute galactic and intergalactic environment]{}, where dust is found under specific conditions of a low-density [diffuse]{} medium ([one to two orders]{} of magnitude [less dense]{} than near the Sun). It is important to understand mechanisms responsible for presence of dust [in such environment]{} and its heating. Studies of [dust in ]{} galactic disk [outskirts]{}, in elliptical galaxies and in galaxy clusters, where the dust formation process is [inhibited]{}, will allow to clarify processes of dust and gas transport into circumgalactic and intergalactic space, as well as to understand [particular]{} features of gas molecularization and star formation [in a]{} low-density interstellar matter.
Recent studies have revealed two types of [spatial]{} dust distribution at galactic peripheries. In one case, the dust-to-gas mass ratio decreases with radius proportionally to the metallicity, as in galaxies M99/M100 \[146\] or M31 \[147, 148\]. In [the]{} other case, the dust-to-gas mass ratio remains [nearly]{} constant at distances up to one and a half optical radius [even though the underlying metallicity goes down outwards]{} \[149, 150\]. The first case corresponds to [*in situ*]{}, [i.e., along with metals]{}, dust production. The second [though]{} corresponds [either]{} to a selective dust transport [in the radial direction along the disk]{}, or an overestimate of its relative [mass]{} content. [If confirmed this]{} dichotomy in the spatial dust distribution at the galactic periphery [seems to]{} reflect [particular]{} features of dynamical processes in galactic disks, and can be of primary importance for [their]{} evolution.
![Radial variation of the dust temperature in four galaxies: NGC 4254 (red curve), NGC 4303 (green curve), NGC 4321 (blue curve) and NGC 4501 (black curve) \[151\]; $R_{25}$ is [radius to the 25$^{\rm th}$ magnitude isophote]{}.](14-img031.png){width="7.5cm"}
IR-radiation fluxes from dust at the periphery suggest dust temperature of $\lesssim 20$ K ( Fig. 20). [IR fluxes from dust]{} beyond the galactic disk radius, [commonly assumed]{} to be its photometric radius, are [as a rule]{} below the [Herschel]{} sensitivity, however it [seems to]{} be [quite]{} sufficient to be detected by Millimetron. Addressing this task by Millimetron will allow:
- to find the source of dust heating in the ISM as a whole, [and]{} at the periphery beyond stellar disks;
- to study optical properties of dust at the galactic periphery, which will help to understand the dust transport ([and/]{}or production) mechanisms far from the main dust production sources;
- to study the [gas-to-dust]{} [ratio]{} as a function of the local star formation rate, the surface gas density and its [total amount]{}. This in turn will give a new [approach]{} to estimate the total mass of molecular gas and to understand its role in the star formation process;
- to evaluate the mass of [CO-dark]{} molecular gas [which is not traced by CO emission]{}, [though manifested by gamma-ray and ionized carbon emission, and by presence of dust in there]{} \[23, 152\]. [Such]{} low-density [CO-dark]{} regions can contain most of the molecular gas. The example of the Galaxy shows that fraction of CO-dark molecular gas increases with decreasing gas density, reaching 80% at a distance of 10 kpc from the center \[23\];
- [with making use the dust-to-gas ratio]{} to determine density and spatial distribution of [the]{} ISM and to connect [them]{} to the observed star formation rate in low-density gas regions: at [outlying]{} periphery of galactic disks \[153\], in tidal structures (tails and bars of interacting galaxies) and in the vicinity of interacting galaxies in the intergalactic space where star-forming regions [are]{} also found \[154\]. Of a special interest are studies of faint dust emission in very low-surface brightness spiral galaxies ([such as]{} Malin-1, Malin-2) and with very low column density of HI in disk which, nevertheless, have spiral structure and at least in some cases contain molecular gas detectable in CO-lines \[155\]. Faint FIR emission has been registered only from a few such objects \[156\]. The dust mass estimate will [make possible to determine CO-dark]{} molecular gas content and its [spatial]{} distribution in these objects, which is important to explain spiral structure and low star formation rate in [their]{} disks, as well as details of their evolution \[157\];
- to carry out a detailed investigation of FIR emission from star formation regions [nearby]{} interacting galaxies (in bridges and connecting bars as observed, for example, around the Antennae galaxies or in the M81/M82 group), where [much]{} dust can be found. [Such study seems to be of principal importance as the dust ]{} can affect gas thermodynamics [and stimulate]{} a star formation burst. In [the tidal bridge of M81/M82 group]{} [dust has been observed in extinction]{} with the dust-to-gas ratio six times [of]{} the standard value in the Galaxy \[158\]. Unfortunately, this region of the sky [might be contaminated by effects from ]{} Galactic [cirrus clouds]{} (clouds of [relatively]{} low surface density above the Galactic plane), which complicates a correct interpretation of the extinction measurements. Since gas and dust temperature in [this tidal bridge seems to be]{} low, the Millimetron observations could give a definite answer on the amount of dust in the M81/M82 region.
[Study of general properties of dust: its mass, optical characteristics, spatial distribution — for low-surface brightness galaxies are of great importance for understanding physics of galaxies in general. This not only will provide estimates of the amount of CO-dark molecular gas, which is essential for correct dynamical modeling of such galaxies and their dark matter halos, but bring also understanding star formation mechanisms under the conditions when standard criteria are not met. IR-fluxes expected from low-surface brightness galaxies are below the Herschel sensitivity limits, but seem to be measurable by Millimetron.]{}
![Comparison of fluxes $F$ at 24, 70 and 160 $\mu$m (filled circles) for three elliptical galaxies with model spectral energy distribution \[159\]. Dust components with different temperatures are clearly visible.](14-img032.png){width="7.5cm"}
Elliptical galaxies with low dust content and almost without cold gas provide another interesting [field]{} for studies [with Millimetron]{}. Current data suggest that dust is present there [in the form of]{} [and includes several populations]{} with different temperatures (Fig. 21). Due to a low gas density, dust in elliptical galaxies immediately [enters]{} into the hot gas phase and turns out to be [unshielded of]{} destructive effects [from]{} the hot gas. From this point of view, elliptical galaxies represent a unique laboratory, where the dust destruction mechanisms must [be]{} clearly [revealed]{}. On the other hand, in a low-density [environment]{} [of]{} ellipticals, where collisional friction is weak dust transport by radiation pressure is clearly manifested, [study of spatial]{} dust structure in [the]{} galaxies will [allow]{} to understand dust redistribution and its total “budget” in the Universe.
[Study]{} of dust in [the Universe]{} have another very important aspect: dust is responsible for optical absorption, and for correct interpretation of many optical observations dust distribution should be well known. For example, this is of principal importance for cosmological supernovae projects aimed to determine the [dark energy parameters]{} \[160, 161\]. Different estimates [of dust content]{} are controversial [by factor of several]{} \[162, 163\]. For example, [the]{} analysis of about $10^4$ galaxy clusters from SDSS (Sloan Digital Sky Survey) at low redshifts (0.1-0.2) with all background quasars within 1 Mpc around the cluster center \[164\] suggests the mean extinction $A_v=0.003\pm0.01$. Close result is obtained from the analysis of 90000 [SDSS background]{} galaxies [and 458 foreground]{} galaxy clusters at redshifts up to $z\sim0.5$ \[165\]. The corresponding mean mass dust fraction is $\rho$(dust)/$\rho$(baryons) $\sim10^{-5}-10^{-4}$, i.e. 0.1-1% of the Milky Way value. In this connection, an interesting (and intriguing) [circumstance]{} is the measured dust extinction in the intergalactic space: $0.03<A_v<0.1$ \[166\], i.e. an order of magnitude as high as in the [intracluster gas]{}. This [result]{}, if true, suggests that either dust is supplied into the intergalactic space by field galaxies (i.e. [those]{} that do not belong to clusters or groups), or a significant fraction of dust in galaxy clusters is unobservable (being apparently bound in dense cold clouds with small geometrical cross-section). Both possibilities are worth of a careful investigation.
On the other hand, from the analysis of about 7000 clusters and groups from SDSS at low redshifts ($z=$0—0.2)[ \[167\]]{}, the dust mass fraction has been derived: $\rho$(dust)/$\rho$(baryons) from 5% to 55%, [of]{} the local (Galactic) value [ for clusters and groups, respectively]{}. [Even though]{} for galaxy groups this high value [looks quite reasonable]{}, since the gas temperature in [them]{} is below the critical limit of [efficient]{} dust destruction, the higher dust concentration in clusters measured in \[164,165\] [might]{} suggest that dust in clusters is shielded from destruction by dense cold [gaseous envelopes]{} of cloud fragments [along with which dust has been expelled from galaxies]{}.
![(a) Correlation of the intensity of PAH emission with metallicity according to observations of extragalactic star formation regions. The PAH emission is represented by the parameter $P_{8,0}$, the flux ratio at 8 $\mu$m to the total flux above 70 $\mu$m . To determine $P_{8,0}$, FIR observations with the same angular resolution as in near IR are needed \[174\]. (b) Image of galaxy M82 in the line of ionized carbon at 158 mm in celestial coordinates $\alpha,\;\delta$ (epoch J2000) \[175\].](14-img033.png "fig:"){width="9.5cm"}\
![(a) Correlation of the intensity of PAH emission with metallicity according to observations of extragalactic star formation regions. The PAH emission is represented by the parameter $P_{8,0}$, the flux ratio at 8 $\mu$m to the total flux above 70 $\mu$m . To determine $P_{8,0}$, FIR observations with the same angular resolution as in near IR are needed \[174\]. (b) Image of galaxy M82 in the line of ionized carbon at 158 mm in celestial coordinates $\alpha,\;\delta$ (epoch J2000) \[175\].](14-img034.png "fig:"){width="9.5cm"}
The situation [does not become]{} completely clear even when we [turn]{} to IR-observations in emission, although some [indications]{} to a possible [solution do appear]{}: Spitzer space telescope data at wavelengths $\lambda=$24 and 160 $\mu$m from the Coma cluster direction do not [unexpectedly]{} show [signal]{} above the noise level \[168\]. However, later observations by [Herschel]{} observatory at wavelengths $\lambda=$ 100, 160, 250, 350, 500 $\mu$m revealed dust traces in the vicinity of several dwarf galaxies with extended halos in the FIR band [through clear demonstration]{} of [flattening]{} the gradient at long wavelengths. The extended FIR emission with “colder” spectrum at the periphery of elliptical galaxy M87 interpreted [initially]{} in terms of cold dust emission, later was identified with synchrotron radiation \[169\]. The example of the M87 galaxy shows that [FIR]{} and millimeter spectra of extended disks of spiral and elliptical galaxies can [stem from]{} a superposition of cold thermal dust emission and non-thermal emission [of]{} relativistic electrons [spread diffusively in disk outskirts.]{} From this point of view, submillimeter and millimeter observations could be crucial, since the difference between [contributions of]{} Jeans thermal dust spectrum and power-law spectrum of relativistic electrons with negative slope is most pronounced [ in these bands]{}. In this connection, the latest results of cold dust observations in coronae and extended disks of nearby isolated galaxies M31 \[147, 148\], M99/M100 (Fig. 22a) \[146\], in dwarf galaxies from the Virgo cluster \[169, 170\] should be mentioned. Similar dust coronae around galaxies are well observed in optics \[167\].
[Recently,]{} numerous evidences of the [presence]{} of extended (up to 300 pc in [radius]{}) [low-redshift (0.1-0.4)]{} circumgalactic [gaseous]{} coronae enriched with heavy elements up to almost the solar metallicity [have been found \[171, 172\]]{}. The dust should be expected to be present in [such]{} coronae in the same proportion as metallicity. Taking into account that the gas temperature in the coronae is not very high, typically around $10^6$ K \[173\], a significant dust fraction can be [survived against destruction]{}. Therefore, dust observations in such coronae could provide valuable information on the dust evolution during its transport into the intergalactic medium.
Extragalactic star formation regions
------------------------------------
Observations of star formation regions in dwarf galaxies similar to Holmberg II and DDO 053 require a higher angular resolution than available with the [Herschel]{} space telescope — at least as high as 10-20 arcsec at wavelengths longer than 200 microns. To observe low-metallicity star formation regions, the sensitivity level should be better than 1 $\mu$Jy per bin. Such Millimetron observations will be unprecedented and can bring breakthrough results.
A high angular resolution in the near-IR range [has been]{} attained by the Spitzer telescope. However, in order to fully describe dust [properties]{} similar angular resolution [in FIR]{} is needed [as well]{}. Observations of polycyclic aromatic hydrocarbons (PAH), in particular, study of their abundance dependence on the ISM parameters in galaxies can be a perspective task. [Such]{} observations will clarify reliability of the PAH emission as star formation rate indicators, and address the PAH survival and organic species evolution in ISM [as well]{}.
In individual galaxies and star formation regions the PAH abundance correlates with metallicity (Fig. 23a), however the nature of this correlation is unclear. The dependence of the PAH formation processes on metallicity in stars or molecular clouds, and processes of their destruction in ISM and star formation regions are considered as possible explanations. The problem is significantly complicated [though]{} due to the lack of high-resolution observations (at least 10 arcsec) at wavelengths above 200 microns. The analysis of the PAH content, i.e. their contribution to the total mass of the dust, is impossible without these data.
![(a) Correlation of the intensity of PAH emission with metallicity according to observations of extragalactic star formation regions. The PAH emission is represented by the parameter $P_{8,0}$, the flux ratio at 8 $\mu$m to the total flux above 70 $\mu$m . To determine $P_{8,0}$, FIR observations with the same angular resolution as in near IR are needed \[174\]. (b) Image of galaxy M82 in the line of ionized carbon at 158 mm in celestial coordinates $\alpha,\;\delta$ (epoch J2000) \[175\].](14-img009.png "fig:"){width="7.5cm"} ![(a) Correlation of the intensity of PAH emission with metallicity according to observations of extragalactic star formation regions. The PAH emission is represented by the parameter $P_{8,0}$, the flux ratio at 8 $\mu$m to the total flux above 70 $\mu$m . To determine $P_{8,0}$, FIR observations with the same angular resolution as in near IR are needed \[174\]. (b) Image of galaxy M82 in the line of ionized carbon at 158 mm in celestial coordinates $\alpha,\;\delta$ (epoch J2000) \[175\].](14-img010.png "fig:"){width="7.5cm"}
For extragalactic star formation regions the most important lines are \[CII\] (158 $\mu$m), HD (112 $\mu$m), HeH$^+$ (149 $\mu$m, 74 $\mu$m) , H$_3^+$ (95 $\mu$m ), H$_2$D$^+$ (219 $\mu$m, 207 $\mu$m). For example, Fig. 23b shows an image of the star-burst galaxy M82 in the ionized carbon line at 158 $\mu$m obtained by the [Herschel]{} space telescope. It is seen that low angular resolution enables studies [only]{} of a general distribution of emission [over the]{} galaxy but not of individual star-forming regions.
Observations by [Herschel]{} telescope and by other millimeter observatories have revealed the [existence of]{} dust even there where its presence has not been [expected]{} before: in elliptical galaxies, where dust must have been destroyed and at the periphery of disk galaxies far from star formation regions. New [highly sensitive]{} measurements of weak dust emission at galactic periphery, as well as observations of protostellar and protoplanetary objects are required to clarify these issues. The main method of research will be the spectral energy distribution measurements in these objects, with a high calibration accuracy being the key requirement that decreases systematic errors.
Dynamics of interstellar medium and chemical evolution of the Universe
----------------------------------------------------------------------
Chemical evolution of the Universe is of the fundamental natural and philosophical [significance]{}, since the Earth and every living organism consists of heavy elements formed in the stellar interior. Of principal importance are both a degree of homogeneity of chemically enriched matter \[176, 177\], and a homogeneity of the abundance [pattern as well]{}. For example, both from the point of view of the conversion of the CO [line emission flux]{} into the column density of H$_2$ molecules, and from the origin of life in the Universe the key question is the precise ratio of carbon to oxygen abundance, [whose spatial variations can reach an order of magnitude in the process of chemical enrichment]{} depending on initial mass function of supernova progenitors \[178\]. The [basic set of]{} observations [in framework of the study of]{} chemical evolution of the Universe includes measurements of the abundance [pattern]{} (the relative abundance of heavy elements), the [mass fraction of dust]{}, and variations of chemical composition [in their relations]{} to dynamical phenomena, such as star formation, galactic wind, etc.
[Herschel]{} observations [have]{} demonstrated a high capability of FIR observations to study heavy element migration in the Universe. These observations include measurements of high-scale galactic matter outflows, such as galactic wind in M82 \[175\], observations of bright infrared galaxies \[179\] and dust migration in galactic disks and [their outskirts]{} (see the discussion in Section 6.2). Thus, [observations by Herschel]{} telescope provided new insights into the nature of driving mechanisms of [such outflows]{}, and [posed]{} several new questions about their role in the dynamical and chemical evolution of galaxies and the Universe [as a whole]{} (for a discussion see \[175, 180\]).
Thanks to the high angular resolution, Millimetron will be able to measure the difference in chemical composition and the abundance [pattern]{} of interstellar gas [starting from scales where stars and supernovae inject heavy elements out]{}. [Optical observations of relative elemental abundances in the Crab nebula suggest that different elements show in general qualitatively similar but quantitatively strongly distinct spatial distributions \[181\].]{} The angular resolution of scales of the spatial variations is relatively low (about 10-15 arcsec), therefore observations of the Crab nebula in the IR-lines \[CII\] at 145 $\mu$m and O\[I\] at 145 $\mu$m with an angular resolution of 3-4 arcsec are of a major importance to estimate the degree of [chemical]{} inhomogeneity at the injection scales of heavy elements into ISM. Similar observations with the same angular resolution are possible and interesting for other supernova remnants, for example Cas A, as well as for nearby Wolf-Rayet stars. High angular resolution observations of the 158 $\mu$m and 145 $\mu$m lines are essential for studies of possible spatial separation of carbon and oxygen.
Millimetron observations in the CO lines with high principal quantum numbers, as those of the \[CII\] 158 $\mu$m and \[OI\] 145 $\mu$m lines and lines of other atoms and ions, [can provide detailed investigation of the enrichment of the interstellar and intergalactic medium with heavy elements.]{} Observations of these lines in the direction of assumed galactic fountains in [the]{} Milky Way — local vertical outflows driven by collective supernova explosions in sufficiently massive OB-associations \[182\], are of a high importance for studies of chemical and dynamical evolution of the Galaxy, since they can provide [a principally]{} new information on [how]{} heavy element are transported out of the galactic disk into the medium and [form]{} their radial redistribution \[183\]. [Recently,]{} the radial distribution [of heavy elements]{} in the Galaxy is has come into the focus of studies of [its]{} dynamical history \[184\]. [Among recent important results a noticeable]{} (the change by a factor three on scales from six to 12 kpc) negative gradient of the C/O ratio \[185\], which can reflect peculiarities in the chemico-dynamical evolution of the Galaxy \[186\], [looks worth mentioning]{}.
A significant fraction (half an order of magnitude) of heavy elements is confined in solid dust particles. Therefore, [observations of dust]{} are of a principal importance not only from the point of view of dust formation, its optical properties, [but]{} for studies the chemical evolution of the Galaxy [as well]{}. The detection of dust in spectra of distant quasars at redshifts $z>6$ \[187\] revealed that the dust can be formed in type II supernova shells. Details of this process, as well as the amount of dust that can be produced by an individual supernova, remain obscure. Recent observations of the Crab nebula, as well as some historical type Ia supernova remnants by the [Herschel]{} space telescope revealed the presence of a significant amount of dust in these remnants, which was produced and expelled during the supernova explosions \[188, 189\]. However, reliable observational confirmations of the possibility of the dust production by massive supernovae are absent yet (see discussion in \[190, 191\]). The possibility of observing emission from dust particles and their seeds in the submillimeter and millimeter range from massive supernovae in the local Universe, in particular in the Galaxy, would be of fundamental interest.
Gravitational lensing at high redshifts
---------------------------------------
Observations at high redshifts of dusty star-forming galaxies (DSFGs) will be carried out to study their evolution (the redshift and luminosity determination) using homogeneous samples in the single-dish mode (photometric and spectroscopic millimeter observations).
Observations of the last decade significantly changed our understanding of the galactic evolution by demonstrating that bright dusty star-forming galaxies at high redhifts are about 1000 times as numerous as in the present-day Universe (see, for example, \[192\]). Spectroscopic observations of 47 sources from a high-redhsift galaxy catalog, obtained by the South Pole Telescope (SPT) at 1.4 and 2 mm \[193\], carried out by the ALMA telescope in the frequency range 84.2-114.9 GHz showed that the brightest DSFGs are gravitationally lensed sources \[194\]. The remote galaxies are lensed by foreground galaxies in the strong gravitational lensing regime leading to multiple images of a lensed galaxy. These results fully confirmed the hypothesis of the gravitational lensing nature of DSFGs \[195, 196\].
Since the studies of properties and evolution of DSFGs are based on observations of bright lensed sources, to determine the proper luminosity of the source the amplification coefficient should be known. To estimate the amplification coefficient, a full geometry of the gravitationally lensed system should be known. To achieve this aim, the following parameters should be measured: redshifts of the source and the lens, relative positions of the lens end the source, a flux ratio from the observed images, as well as galaxy-lens properties. The latter are required to choose an adequate model of the density distribution in the galaxy-lens.
By the launch of Millimetron, large millimeter and submillimeter sky surveys by the Herschel telescope and SPT (220 GHz, 150 GHz) will be completed. These surveys will be used to select lensed DSFG candidates. Selected sample sources should meet the following main criteria: 1) sources must have a thermal spectrum, since the dust radiation in the infrared and millimeter range is produced due to re-emission of absorbed short-wavelength photons emitted by stars; 2) sources with negative K-corrections, which are the first-order redshift corrections to a wavelength, frequency bands and intensities, should be selected from the photometric data; 3) nearby IR sources with $z\leq0.03$ (according to the IRAS data) and radio-loud quasars with flat spectrum, which also emit in the millimeter range, should be excluded.
The carbon monoxide (CO) lines are the most general indicators of the presence of molecular gas at high redshifts. The main observable CO transitions include $J=$ 1–0, 2–1, 3–2, 4–3, 5–4, 6–5. For example, the rest-frame emission frequency of the transition CO $J=1-0$ is 115.27 GHz, of the transition CO $J=5-4$ is 576.3 GHz, and of the transition CO $J=6-5$ is 691.5 GHz. The transition CO – produces the brightest line indicating the presence of dense star-formation nuclei with compact morphology. For example, from a source at $z\sim 5$, the CO $J=6-5$ transition will be observed at a frequency of 111 GHz. For sources from such redshifts the \[CII\] emission line will be shifted from 158 $\mu$m to 948 $\mu$m .
![Broadband spectrum of gravitational lenses \[199\].](14-img035.png){width="7.5cm"}
Undoubtedly, the ALMA telescope will solve most of the tasks by constructing a redshift distribution of DSFGs. However, a redhsift range $z=1.74-2.0$ is unobservable for ALMA \[193\]. In addition, ALMA can operate only within the atmosphere transparency windows, which complicates construction of broad-band spectra. In this respect, Millimetron has advantages in constructing broad-band millimeter and submillimeter spectra.
In addition, it can be possible to solve the satellite problem in the $\Lambda$CDM-model (Lambda Cold Dark Matter) by performing photometric observations of gravitationally lensed sources with anomalous image flux ratios and high-resolution spectroscopic measurements of high-redshift DSFGs in a wide frequency band.
The $\Lambda$CDM model predicts that large dark matter clumps (halos), which host large galaxies like Milky Way should be surrounded by several hundreds of small dark matter clumps (subhalos) where, seemingly, dwarf satellite galaxies should be located. However, only two-three dozen satellites have been actually found around the Milky Way. Moreover, the Milky Way satellite galaxies are distributed not spherically symmetric and are rather confined within an elongated pancake tilted to the Galactic plane.
One of possible solutions of the satellite problem can be observations of gravitationally lensed systems with anomalous image flux ratio \[197\], such as MG0414+0534, MG2016+112, H1413+117 \[198\]. Importantly, the source H1413+117 shows a significant flux ($\sim0.1$ Jy) at a frequency of 1 THz (Fig. 24). In addition, a pixel lensing modeling of the gravitational lensJVAS (Jodrell/VLA Astrometric Survey) B1938+666 in the FIR revealed the presence of satellite with mass $10^8 M_\odot$ \[200\]. In this system, the source is a bright galaxy located at $z=2.059$, the lens is a massive elliptic galaxy at $z=0.881$, and an almost full Einstein-Khvolson ring with diameter $\sim0.9'$ is observed. The presence of a low-mass substructure, i.e. a luminous or dark satellite in the galaxy-lens, could locally perturb the observed brightness distribution of the extended Einstein-Khvolson arcs. Since these arcs are formed by multiple images of the gravitational lens system, the “surface brightness anomalies” can be found and analyzed using the pixel modeling technique and then used for the gravitational detection, mass and location measurement of a substructure with a mass as small as 0.1% of the lens mass inside the Einstein-Khvolson ring \[200\].
Another way to address the “satellite problem” can be high-resolution broadband spectral observations of high-redshift DSFG lensed galaxies. Large redshifts of the observed lensed sources enable a wide range of possible lens redshifts to be obtained, which in principle can constrain the evolution of subhalo population with redshift \[201\]. Here of most interest are observations of the CO molecule transition $J=6-5$, which is the brightest emission line and indicates the presence of dense star-forming cores with compact morphology.
A breakthrough task for the space interferometer mode can be observations of gravitational lens candidates at high redshifts (up to $z\sim5$) selected by submillimeter/millimeter observations in order to confirm image multiplicity without mapping (an advantage of Millimetron) using only the visibility function for sources with sufficiently high brightness temperatures.
Cosmology
=========
Infrared background and galaxy surveys
--------------------------------------
Galaxies at redshifts $z>1$ have a maximum flux density at wavelengths $\lambda>200 \mu$m and produce the cosmological IR-background (Fig. 1a). Here, the capability of Millimetron to resolve more than 90% of the IR-background into individual galaxies (Fig. 1b) can bring breakthrough results. The expected surface density of resolved objects for photometric surveys is $\sim10^5$ per square degree.
Massive spectral observations of a large number of galaxies will enable to construct three-dimensional catalogues of submillimeter galaxies at redshifts $z>2$ and to study the evolution of large-scale dark matter distribution at an age of the Universe of less than three billion years. Millimetron can complete 3D-catalogues mentioned above by partial filling the gap between the recombination era (the age of Universe 300 thousand years) and later epochs, up to the present time (the age of Universe 3-13 billion years).
Studies of the 3D galactic space distribution, first of all, give invaluable information on the properties of galaxies themselves: by comparing cosmological numerical simulations with observations it is possible to study the relation of the dark halo mass with the observed properties of galaxies \[202\]. This information is important for planning future cosmological tests from baryonic acoustic oscillations, gravitational lensing, etc.
Cosmological angular distances
------------------------------
There are two main methods of measuring the cosmological model parameters: geometrical and structural \[203, 204\]. The first method measures the Universe expansion law, from which its geometry and equation of state of its constituents (matter, radiation, dark energy) can be inferred. In this method, independent measurements of distances and redshifts of remote astronomical sources are required.
Long-term observations of water megamasers at a frequency of 22 GHz can be used to estimate the physical sizes of accretion disks by measuring a motion of individual maser spots. Then distances to these objects can be derived from their angular sizes with quite a high accuracy \[205\]. For the ground-based interferometers with the maximum base this distance is limited to 300 Mpc.
Interferometry with superlong (cosmic) base can achieve an angular resolution better than 10 micro arcsec, therefore sufficiently bright sources can be observed from any redshift. Such distance measurements as a function of redshift can allow to probe the expansion law of the Universe and to study dark energy equation of state with an unprecedentedly high accuracy.
However, this task requires multi-year very high-precision observations, and therefore can be difficult to fulfill from the Earth even for nearby objects. Presently, little is known about distant megamasers to reliably judge whether is task is doable for Millimetron.
The same relates to another similar problem. The size of a black hole can be derived from indirect measurements of its mass. Then the direct measurement of the black hole size can be used to determine the cosmological distance. It is not clear as yet whether supermassive black holes in high redshift active galactic nuclei have parameters sufficient to be observed by Millimetron. This issue can be resolved after observations by the EHT or RadioAstron telescopes of black holes in the nearby galaxies and in the center of Galaxy.
Distant galaxies and reionization of the Universe
-------------------------------------------------
First stars (Population III stars) and galaxies must have been formed from the primordial matter which had not been enriched with heavy elements. Observations of such sources are important, firstly, to check hypotheses of formation of first stars which reionize the Universe, enrich the Universe by heavy chemical elements, secondly, to understand details of the first galaxy formation and peculiarities of the star formation in the medium with primordial chemical composition, thirdly, to solve the supermassive black hole formation problem.
A galaxy enriched with heavy elements should emit in spectral lines of atomic and ionized carbon, carbon monoxide, oxygen and other elements, and also in continuum. Correspondingly, the primordial matter should not radiate in the submillimeter diapason, excluding several spectral lines of simplest molecules: HD at $112\left(1+z\right)\mu$m , $H_2$ at $28\left(1+z\right)\mu$m , HeH${}^+$ at $149\left(1+z\right)\mu$m, where $z$ is the galaxy redshift \[15, 206, 207\]. At the same time, an atomic hydrogen emission should be observed in the near and middle IR owing to high redshift.
Search for first galaxies is one of tasks of JWST. The search includes the study of the dependence of the number of galaxies on redshift $z$: the vanishing of the number of galaxies at some $z$ would indicate the galaxy formation era. However, submillimeter and FIR observations are required to confirm the nature of such distant objects. The lack of the detection of dust and heavy elements atoms in the submillimeter range would suggest the discovery of a possible primordial galaxy. The final confirmation can be obtained through observations of the molecular lines HD (56 and 112 $\mu$m ) and H$_2$ (28, 17, 12 and 9.7 $\mu$m ), which would indicate that the gas cooling occurs without heavy elements.
The rotational lines of molecular hydrogen, as well as of its isotopic analog HD, provide the main cooling of the energy released during gravitational contraction of the first protostellar clouds. The fluxes in these lines can much exceed the sensitivity limit of Millimetron: $\sim0.1$ mJy in the HD line at 112$(1+z)$ $\mu$m and $\sim 1$ mJy in the H$_2$ line at 28$(1+z)$ $\mu$m , depending on the formation scenario of the first protostars. The detection of these lines is of primary importance at least to pin down the time of the star formation beginning in the Universe. In addition, measurements of these lines will help to determine or improve the redshift of first distant galaxies and quasars, which will help to construct the model of their evolution. As the simplest hydrogen molecule is well studied, it is a good indicator of physical conditions in the primordial ISM. Excitation conditions of the hydrogen molecule levels are also well studied, which enables physical conditions in the primordial gas to be probed.
The high sensitivity and availability of observations of molecular emission lines redshifted to FIR is the main advantage of Millimetron in probing the physical conditions in the primordial gas. Thus, in the near future only Millimetron will be able to study the relatively cold ($T<500$ K) ISM, including the primordial gas cooling.
### Spectral-space CMB perturbations
The “dark age” epoch where there was no stars and galaxies ends by $z=10-25$, when first ionization sources appear: the first stars and galaxies, as well as black holes. The ionization of the Universe in this epoch can be both directly observed and inferred from the CMB polarization studies \[209, 210\]. However, it is unclear until now which potential ionization sources dominates. Observational studies of this problem will shed light on the formation mechanism of the first stars and supermassive black holes, whose mass can increase up to $10^9 M_\odot$ already by $z=6$, i.e. during the first billion years of the age of the Universe.
The reionization sources leads to the emerging of ionized volumes that can be probed by the kinematic Sunyaev-Zeldovich effect (the thermal Sunyaev-Zeldovich effect for these objects is much smaller). The expected level of spectral-space CMB fluctuations is $\Delta T/T=10^{-7}-10^{-6}$, which corresponds to a flux of 1-10 $\mu$Jy in the Millimetron band. The size of the ionization region is $\sim 10$ Mpc, corresponding to an angular scale of $\sim$ 1 arcmin \[211\].
In addition, the emission from the HeH$^+$ molecule at redshifts $z=20-30$ falls within the CMB range, which may cause the temperature fluctuations $\Delta T/T\sim 10^{-5}$ within the spectral bands $\Delta \nu/\nu\sim 0.01$ \[212\].
![Spectrum of a remote quasar at $z=6.42$ \[214\]. The signal-to-noise ratio $\sim 3.5$ is insufficient to reliably detect the HeH$^+$ line.](14-img036.png){width="7.5cm"}
Spectral observations with low and moderate resolution in the 100-500 GHz frequency band can be used to analyze the form and evolution of the ionized clouds and to formulate constraints on the reionization scenarios: whether the Population III stars or supermassive black holes were the primary reionizaion sources.
### Search for emission from HeH$^+$ molecule
Search for the HeH$^+$ molecule will help to understand details of interstellar and intergalactic medium responsible for the formation of the first sources of ionization in the early Universe.
HeH$^+$ should be one of the mostly abundant molecules in the reionization epoch \[213\]. This molecule is formed in the primordial gas near the powerful ionization sources. The rest-frame wavelength of emission is 149.1 and 74.6 $\mu$m . An intriguing opportunity for the HeH$^+$ molecule searches can be provided by observations of a quasar with $z=6.42$ (the signal-to-noise ratio is 3.5) (Fig. 25).
Galaxy clusters
---------------
CMB photons traveling through volumes filled with sufficiently hot plasma will experience spectral changes (Fig. 26). This is the essence of the Sunyaev-Zeldovich effect (SZ) \[216, 217\]. Galaxy clusters are the most appropriate for the SZ-effect observations. Ground-based observations of the SZ-effect are restricted by the atmosphere to frequencies below 300 GHz. The use of the space telescope may lead to a breakthrough related to the separation of the thermal SZ-effect from other spectral distortions with sufficient accuracy (see Fig. 26), which can be used to measure, in particular, peculiar velocities of galaxy clusters (relative to CMB). Measurements of these velocities will bring important information to test and improve the cosmological model \[203, 204\].
![(a) The Earth atmosphere transmission coefficient for dry atmosphere. (b) The Sunyaev-Zeldovich effect spectrum (deviations from the blackbody CMB emission). The solid, dashed, dotted and dash-dotted curves show the thermal effect, kinematic effect, effect on non-thermal electrons and dust emission, respectively \[215\].](14-img037.png){width="10.5cm"}
Studies of galaxy clusters using the thermal and kinematic SZ-effect will enable the measurement of the primordial cosmological power spectrum amplitude, of the amount of dark energy in the Universe, and will help to study the growth of small perturbations and to obtain new constraints on the cosmological model, including the dark energy content. Using the SZ-effect polarization, the CMB quadrupole amplitude can be measured from the point of view of the observer co-moving with the cluster, i.e. form different regions in the Universe.
Gamma-ray burst afterglow and host galaxies
-------------------------------------------
Despite that since the discovery of the GRB afterglow in 1997 more than 500 events have been detected in the optic and more than 800 in the X-rays, the most interesting spectral part of the afterglow remains poorly investigated. Indeed, maximum $\nu_m$ in the initial energy spectrum of the afterglow (Fig. 27) falls in the millimeter range. Afterglows that have been registered in this range \[219\] are shown in Fig. 28a. Observations of bright afterglows by Millimetron can constrain, and in some cases determine the characteristic frequency of the synchrotron emission $\nu_m$ and thus put the stringent limits on the energy, parameters of the emission region and particularly the bulk relativistic gamma-factor of an ejecta.
![Theoretical spectral energy distribution of a gamma-ray burst afterglow \[218\]. The red curve shows the Ly-$\alpha$ frequency in observer’s frame as a function of redshift (right Y-axis). To the right of this curve, the optical emission is suppressed due to absorption in neutral hydrogen along the line of sight.](14-img038.png){width="10.5cm"}
Currently, about 10% of the registered gamma-ray bursts have redshifts $z>5$ \[222\]. However, the sample of high-redshift gamma-ray bursts is incomplete because of selection effects due to difficulties to observe sources at $z\gtrsim 5$ (the maximum redshift $z=9.4$ determined for GRB 090429B \[223\]), where the optical emission is effectively absorbed by the Ly-$\alpha$ forest (Fig. 28a). Therefore, the 10% is a lower limit of the total amount of distant GRBs.
An intriguing issue is the discrepancy between the star formation rate as inferred from distant galaxies and gamma-ray bursts (Fig. 28b). The submillimeter observations of GRB afterglows will increase the statistics of high-redshift GRBs. On the other hand, possibly, some of already observed sources are related to explosions of the Population III stars \[224, 225\]. The redhsift estimate $z\geq 15$ for at least one gamma-ray burst obtained from the submillimeter observations could confirm the existence of the Population III stars and their collapses producing GRB.
Optically dark GRB, when the optic to X-ray flux ratio in the afterglow phase is extremely small are burst of special interest \[226\]. The redshift determination of dark gamma-ray bursts is difficult due to low (or even undetectable) optical flux. Submillimeter observations of such gamma-ray bursts will help clarifying their nature. Indeed, the absence of the host galaxy of a gamma-ray burst in the optical (the most distant host galaxy presently known, for GRB 100219A, has $z=4.667$ \[227\]), and at the same time the detection of the submillimeter afterglow uniquely suggests a high redshift of the source. For close gamma-ray bursts, when redhsift can be determined by spectroscopic or photometric observations of the host galaxy, the detection of the submillimeter afterglow can be used to investigate parameters of the absorbing circumburst medium \[228-231\].
![(a) Submillimeter gamma-ray burst afterglows \[219\]. Data for GRB 130427A are taken from \[220\]. The dark and light symbols correspond to detection and upper limits, respectively. The dashed curves show the fluxes for sources with equal luminosity located at different redshifts. (b) The star-formation rate (SFR) as a function of redshift $z$ \[221\]. (GRB — gamma-ray burst, LBG — Lyman-Break Galaxy).](14-img039.png "fig:"){width="7.5cm"} ![(a) Submillimeter gamma-ray burst afterglows \[219\]. Data for GRB 130427A are taken from \[220\]. The dark and light symbols correspond to detection and upper limits, respectively. The dashed curves show the fluxes for sources with equal luminosity located at different redshifts. (b) The star-formation rate (SFR) as a function of redshift $z$ \[221\]. (GRB — gamma-ray burst, LBG — Lyman-Break Galaxy).](14-img040.png "fig:"){width="7.5cm"}
Primordial black and white holes, wormholes and Multiverse
----------------------------------------------------------
The most exciting issues of the modern cosmology include: how the Universe was born? Are there other universes and is it possible to obtain information about them? Today, this is a purely hypothetical field based on the analysis of GR equations \[203, 232-238\].
One of the least hypothetical assumptions is the formation of black holes in the early universe. These black holes have a non-astrophysical origin and are referred to as primordial black holes. To form them, strong density inhomogeneities at early stages of the Universe are needed \[239\].
The primordial black holes can have a very wide mass spectrum, and depending on mass various methods are used to their searches \[240\]. Millimetron will be able, as minimum, to study black holes with a mass of higher than $10^4M_\odot$ both at the present time (see Section 4.5) and in the remote past, during the reionization of the Universe (see section 7.3).
Features of the Universe passing through the singularity at the very beginning of the Big Bang can lead to the formation of interesting hypothetical objects — wormholes, the relativistic objects similar to black holes but connecting different regions of space-time or even different universes (the Multiverse). Possible observational distinctions between black holes and wormholes include the presence in wormholes of a radial monopole magnetic field and radiation that brings information (the image, physical parameters, etc.) from the different region of our and even from another universe \[241-246\]. Observational test of this hypothesis will be carried out jointly with all black hole studies (see Section 5).
General relativity also admits solutions in the form of white holes \[247, 248\]. Such objects can be probed, for example, by explosions like anomalous gamma-ray bursts without host galaxies \[249\]. Searches for host galaxies of gamma-ray bursts by Millimetron (see Section 7.5) can shed light on this possibility.
Conclusions
===========
In conclusion, we formulate three groups of unique tasks, where Millimetron can play a decisive role and can strongly contribute to solve outstanding astrophysical, astrochemical and cosmological problems.
1\. Studies of the vicinity of black holes and testing of general relativity. Studies of accreting flows and jets near the black hole horizons.
2\. The analysis of interstellar medium and star formation. Studies of protostars, protoplanets and protoplanetary disks, as well as of exoplanets and the Solar system. Studies of star formation conditions and of the interstellar medium enrichment by heavy elements.
3\. Formation end evolution of galaxies, studies of cosmological objects and the development of the standard cosmological model.
**Acknowledgements.** The authors are deeply indebted to all whose comments were used in the preparation of this paper: V.V. Akimkin, A.V. Alakoz, A.A. Andrianov, N.A. Arkhipova, V.S. Beskin, M.V. Barkov, O.V. Verkhodanov, A.A. Volnova, Yu.N. Gnedin, T. de Graauw, V.K. Dubrovich, A.A. Ermash, D.E. Ionov, N.R. Ikhsanov, P.V. Kaygorodov, S.V. Kalensky, A.V. Kasparova, M.S. Kirsanova, Yu.Yu. Kovalev, S.G. Moiseenko, Y.N. Pavlyuchenko, K.A. Postnov, M.V .Popov, O.K. Sil’chenko, V.N. Rudenko, A.V. Stepanov, S.A. Tyulbashev, M.S. Hramtsova, N.N. Shakhvorostova, A.A. Shatsky, B.M. Shustov, S.V. Chernov, as well as the staff of the P.N. Lebedev Physical Institute RAS (LPI), the P.K. Sternberg State Astronomical Institute (SAI MSU), the Institute of Astronomy RAS, the Main Astronomical Observatory RAS, the Pushchino Radio Astronomy Observatory and the Astro Space Center (ASC) LPI, who assisted in the preparation of the scientific program. The ASC LPI staff (N.S. Kardashev, I.D. Novikov, V.N. Lukash, S.V. Pilipenko, E.V. Mikheeva, A.G. Doroshkevich, P.B. Ivanov, V.I. Kostenko, T.I. Larchenkova, S.F. Likhachev, A.V. Smirnov) thanks L.N. Likhachova for support. A.G. Doroshkevich, P.B. Ivanov, T.I. Larchenkova, V.N. Lukash, E.V. Miheeva, I.D. Novikov, S.V. Pilipenko acknowledge the support from grant of the President of the Russian Federation for State Support leading Scientific Schools NSH-4235.2014.2, the program DPS RAS OFN-17 “Active processes in galactic and extragalactic objects” and the program of the Presidium of the Russian Academy of Sciences P-21 “Non-stationary phenomena in objects of the Universe”. D. V. Bisikalo and D. Z. Vibe are supported by a grant of the President of the Russian Federation for State Support of Leading Scientific Schools NSH-3620.2014.2. Yu. A. Schekinov is supported by the RFBR grant 12-02-00917-a. The work of I.F. Malov and V.M. Malofeev is supported by the RFBR grant 12-02-00661 and by the Presidium of Russian Academy of Sciences (program “The origin, structure and evolution of objects of the Universe”). A.S. Pozanenko is supported by the RFBR grants 12-02-01336, 13-01-92204, 14-02-10015. The work of I.D. Novikov is supported by the RFBR grant 12-02-00276a. The work of I.I. Zinchenko is partially supported by a grant under the agreement between the Ministry of Education and Science of the Russian Federation and Nizhny Novgorod State University 02.V.49.21.0003 of August 27, 2013, as well as by the RFBR grant 13-02-12220-ofi-m. A.M. Sobolev”s work was performed as part of the job of the State Ministry of Education and Science of the Russian Federation (project 3.1781.2014 / K). A.M. Cherepashchuk acknowledges the grant of the President of the Russian Federation for State Support of Leading Scientific Schools NSH-1675.2014.2 and RFBR grant 14-02-00825.
[299]{} Dole H et al. *A&A* **451** 417 (2006) Smirnov A V, Baryshev A M, Pilipenko S V, Myshonkova N V et al. *Proc. SPIE 8442 Space Telescopes and Instrumentation 2012: optical, infrared and millimeter wave* eds M C Clampin et al. (2012) Krabbe A, R?ser H-P *Rev. of Mod. Astronomy* **12** 1052 (1999) Crill B P, et al. *ApJ*S **148** 527 (2003) Hoogeveen, R W M, et al. *Proc. of SPIE, Infrared Spaceborne Remote Sensing XI* **5152** 347 (2004) Pilbratt G L, in *Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series* ed. Mather J C **4850** 586 (2003) Kardashev N S et al. Trudy FIAN 228 112 (2000) Wild W, Kardashev N S et al. *Exp Astron* **23** 221 (2009) Kiuchi H Coherence estimation on the measured phase noise in Allan standard deviation ALMA Memo **530** (2005) Nakagawa T, in *Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series* 7731 (2010) Doeleman S S et al. *Nature* **455** 78 (2008) McKee and Ostriker *ARAA* **45** 565 (2007) Goicoechea J R and Cernicharo J *ApJ* **554** L213 (2001) Planck Collaboration *A&A* **536** 19 (2011) Roberge W, Dalgarno A *ApJ* **255** 489 (1982) Cecchi-Pestellini C., Dalgarno A., *ApJ* **413** 611 (1993) Voshchinnikov N V, private communication Efremov Yu N *MNRAS* **405** 1531 (2010) Fukui Y, Kawamura A *ARA&A* **48** 547 (2010) Fukui Y et al. *ApJ* **780** 36 (2014) Inutsuka S-I, Koyama H *Ap&SS* **281** 67 Khoperskov S A, Vasiliev E O, Sobolev A M and Khoperskov A V *MNRAS* **428** 2311 (2013) Pineda J L, Langer W D, Velusamy T, и Goldsmith P F *A&A* **554** 103 (2013) Kirsanova M S, Sobolev A M, Thomasson M, Wiebe D S, Johansson L E B, Seleznev A F *MNRAS* **388** 729 (2008) Kirsanova M S, Wiebe D S, Sobolev A M, Henkel C, Tsivilev A P *MNRAS* **437** 1593 (2014) Crockett N R et al. *A&A* **521** L21 (2010) http://www.sron.rug.nl/millimetron/OxygenPuzzle Walmsley M, van der Tak F in procs. “Dusty and molecular universe: a prelude to Herschel and ALMA” ESA **SP-577** 55 (2005) Wyrowski F. et al. *A&A* **542** L15 (2012) Kirsanova M S, Wiebe D S, Sobolev A M Astron. Rep. **53** 611 (2009) Bergin E A et al. *Nature* **493** 644 (2013) Bergin E A, Hogerheijde M R, Brinch C, Fogel J et al. *Astron. Astrophys* **521** L33 (2010) Hogerheijde M R, Bergin E A, Brinch Ch, Cleeves L I *Science* 334 338 (2011) Lahuis F, van Dishoeck E F, Boogert A C A, Pontoppidan K M *Astrophys J* **636** L145 (2006) Akimkin V et al. *ApJ* **766** 8 (2013) Akimkin V et al. *Astrophys. Space Sci.* **335** 33 (2011) Young Ch H et al. *Astrophys. J. Suppl.* **154** 396 (2004) Pavlyuchenkov Ya N, Vibe D S, Fateeva A M, Vsyunina T S *Astron. Zurn.* **88** 3 (2011) Robitaille Th O et al. *Astrophys. J. Suppl.* **167** 256 (2006) Robitaille Th P et al. *Astrophys. J. Suppl.* **169** 328 (2007) Zinchenko I I, in preparation (2014) Henning Th et al. *Astron. Astrophys.* **518** L95 (2010) Muller T G et al. *Astron. Astrophys.* **566** A22 (2014); arXiv:1404.5847 Gray M Maser Sources in Astrophysics (Cambridge: Cambridge Univ. Press, 2012) Sobolev AM et al. *Proc. Int. Astron. Union* **242** 81 (2007) Moran J Met al. *Proc. Int. Astron. Union* **242** 391 (2007) Parfenov S Yu, Sobolev A M *Mon. Not. R. Astron. Soc.* **444** 620 (2014) Zhilkin AG, BisikaloDV, Boyarchuk A A *Phys. Usp.* **55** 115 (2012) Matveyenko L I et al. *Astron. Lett.* **30** 100 (2004) Sobolev AM et al. *Science* (2015), in print Bisikalo D V *Phys. Usp.* **51** 551 (2008) Kurbatov E P, Bisikalo D V, Kaygorodov P V *Phys. Usp.* **57** 787 (2014) Seager S Exoplanet Atmospheres: Physical Processes (Princeton, N.J.: Princeton Univ. Press, 2010) Vidal-Madjar A et al. *Nature* **422** 143 (2003) Linsky J et al. *Astrophys. J.* **717** 1291 (2010) France K et al. *Astrophys. J.* **712** 1277 (2010) Fossati L et al. *Astrophys. J.* **714** L222 (2010) Narita N et al. *Astrophys. J.* **773** 144 (2013) Mayer A et al. *Astron. Astrophys.* **549** A69 (2013) Decin L *Adv. Space Res.* **50** 843 (2012) Justtanont K et al. *Astron. Astrophys.* **537** 144 (2012) Cernicharo J et al. *ApJ* **778** L25 (2013) Ueta T et al. *Astron. Astrophys.* **565** A36 (2014) Kardashev N S *Nature* **278** 28 (1979) Mauersberger R et al. *Astron. Astrophys.* **306** 141 (1996) Dyson F *Science* **131** 1667 (1960) Slysh V I, in The Search for Extraterrestrial Life — Recent Developments: Proc. of the 112th Symp. of the Intern. Astronomical Union, Boston, Mass., USA, June 18 – 21, 1984 (Ed M D Papagiannis)(Dordrecht: D. Reidel, 1985) p. 315 Carrigan R *Astrophys. J.* **698** 2075 (2009) Kilic Met al. *Astrophys. J.* **678** 1298 (2008) Farihi J et al. *Mon. Not. R. Astron. Soc.* **432** 1955 (2013) CutriRMet al. “WISE All-sky Data Release”, 2012yCat.2311....0C (2012) Kawka A, Vennes S *Mon. Not. R. Astron. Soc. Lett.* **439** L90 (2014) Malofeev V et al. *Astron. Astrophys.* **285** 201 (1994) Xilouris KMet al. *Astron. Astrophys.* **288** L17 (1994) Morris D et al. *Astron. Astrophys.* **322** L17 (1997) Lohmer O et al. *Astron. Astrophys.* **480** 623 (2008) Kramer Met al. *Astrophys. J.* **488** 364 (1997) Camilo F et al. *Astrophys. J.* **669** 561 (2007) Camilo F et al. *Astrophys. J.* **679** 681 (2008) Malov I F *Astron. Rep.* **41** 617 (1997) Malov I F *Astron. Rep.* **58** 139 (2014) Popov M V et al. *Astrophys. J. Lett.* (2015), in print Heger A et al. *Astrophys. J.* **591** 288 (2003) Marscher A P et al. *Astrophys. J.* **763** L15 (2013) Mackey A D et al. *Mon. Not. R. Astron. Soc.* **386** 65 (2008) Strader J et al. *Nature* **490** 71 (2012) Chomiuk L et al. *Astrophys. J.* **777** 69 (2013); arXiv:1306.6624 Fabbiano G *Annu. Rev. Astron. Astrophys.* **27** 87 (1989) Frank J, King A, Raine D J Accretion Power in Astrophysics (Cambridge: Cambridge Univ. Press, 2002) Colbert E J M, Mushotzky R F *Astrophys. J.* **519** 89 (1999) Liu J-F, Bregman J N *Astrophys. J. Suppl.* **157** 59 (2005) Liu Q Z, Mirabel I F *Astron. Astrophys.* **429** 1125 (2005) Colbert E J M, Ptak A F *Astrophys. J. Suppl.* **143** 25 (2002) Swartz D A *ASP Conf. Ser.* **423** 277 (2010) Mushotzky R *Prog. Theor. Phys. Suppl.* **155** 27 (2004) Miller JMet al. *Astrophys. J. Lett.* **585** L37 (2003) Poutanen J et al. *Mon. Not. R. Astron. Soc.* **377** 1187 (2007) Kording E, Falcke H, Markoff S *Astron. Astrophys.* **382** L13 (2002) Feng H, Soria R *New Astron. Rev.* **55** 166 (2011) Miller M, Coleman H, Douglas P *Astrophys. J.* **576** 894 (2002) Farrell S A et al. *Nature* **460** 73 (2009) Webb N et al. *Science* **337** 554 (2012) MillerNA, Mushotzky R F, Neff SG *Astrophys. J.* **623** L109 (2005) Kaaret P et al. *Science* **299** 365 (2003) Harrison F, in Intern. Conf. ZELDOVICH-100, Moscow, Russia, 16 – 20 June 2014 Bisnovatyi-Kogan G S *Sov. Astron.* **14** 652 (1971) Moiseenko S G, Bisnovatyi-KoganGS, ArdeljanN V *Mon. Not. R. Astron. Soc.* **370** 501 (2006) Bisnovatyi-Kogan G S, Moiseenko S G, Ardelyan N V *Astron. Rep.* **52** 997 (2008) Bisnovatyi-Kogan G S, Moiseenko S G, Ardelyan N V *Astron. Rep.* **36** 285 (1992) Barkov M V, Komissarov S S *Mon. Not. R. Astron. Soc.* **415** 944 (2011) Chevalier R A *Astrophys. J. Lett.* **752** 2 (2012) Taam R E, Sandquist E L *Astron. Astrophys.* **38** 113 (2000) Field G B, Rogers R D *Astrophys. J.* **403** 94 (1993) Kardashev N S *Mon. Not. R. Astron. Soc.* **276** 515 (1995) Lobanov A P *Astron. Astrophys.* **330** 79 (1998) Tyul’bashev S A *Astron. Astrophys.* **387** 818 (2002) Eatough et al. *Nature* **501** 391 (2013) Marrone et al. *Astrophys. J. Lett.* **654** 57 (2007) Trippe et al. *Astron. Astrophys.* **540** A74 (2012) Ivanov P B, private communication Dexter J et al. *Astrophys. J.* **717** 1092 (2010) Lu Y et al. *Astrophys. J.* (2014), in print Muller T, Frauendiener J *Eur. J. Phys.* **33** 955 (2012) Komossa S, Bade N *Astron. Astrophys.* **343** 775 (1999) Bloom J S et al. *GCN Circ.* (11847) 1 (2011) Beskin V S *Phys. Usp.* **53** 1199 (2010) Barkov M V, Komissarov S S *Mon. Not. R. Astron. Soc.* **401** 1644 (2010) Barkov MV, Baushev A N *New Astron.* **16** 45 (2011) McKinney J C, Uzdensky D A *Mon. Not. R. Astron. Soc.* **419** 573 (2012) McKinney J C, Tchekhovskoy A, Blandford R D *Mon. Not. R. Astron. Soc.* **423** 3083 (2012) De Bruyn A G NFRA Note 655 1 (1996) BrentjensMA, de BruynAG, in Proc. of the Riddle of Cooling Flows in Galaxies and Clusters of Galaxies, Charlottesville, VA, USA, May 31 – June 4, 2003 (Eds T H Reiprich, J C Kempner, N Soker) (Charlottesville, VA: Univ. of Virginia, 2004) Vogt C, Dolag K, Enslin T A *Mon. Not. R. Astron. Soc.* **358** 726 (2005) Brentjens MA, de Bruyn A G *Astron. Astrophys.* **441** 1217 (2005) Aharonian F et al. *Astrophys. J.* **664** L71 (2007) Begelman M C, Fabian A C, Rees M J *Mon. Not. R. Astron. Soc.* **384** L19 (2008) Giannios D, Uzdensky D A, Begelman M C *Mon. Not. R. Astron. Soc.* **395** L29 (2009) Barkov MV et al. *Mon. Not. R. Astron. Soc.* **749** 119 (2012) Mesler R A, Pihlstrom YM *Astrophys. J.* **774** 77 (2013) Berger E et al. *Nature* **426** 154 (2003) Frail D A et al. *Astrophys. J.* **619** 994 (2005) Barkov M V, Pozanenko A S *Mon. Not. R. Astron. Soc.* **417** 2161 (2011) Michalowski M, Hjorth J, Watson D *Astron. Astrophys.* **514** A67 (2010) Pope A, Chary R R *Astrophys. J.* **715** L171 (2010) Dayal P, Hirashita H, Ferrara A *Mon. Not. R. Astron. Soc.* **403** 620 (2010) Pohlen Met al. *Astron. Astrophys.* **518** L72 (2010) Fritz Met al. *Astron. Astrophys.* **516** 34 (2012) Smith MWL et al. *Astrophys. J.* **756** 40 (2012) Holwerda BWet al. *Astron. Astrophys.* **444** 101 (2005) Holwerda BWet al. *Astron. Nachr.* **334** 268 (2013) Magrini L et al. *Astron. Astrophys.* **535** 13 (2011) Grenier I A, Casandjian J M, Terrier R *Science* **307** 1292 (2005) Abramova O V, Zasov A V *Astron. Lett.* **38** 755 (2012) Karachentsev I et al. *Mon. Not. R. Astron. Soc.* **415** L31 (2011) Das M, Boone F, Viallefond F *Astron. Astrophys.* **523** 63 (2010) Hinz J L et al. *Astrophys. J.* **663** 895 (2007) Kasparova A V et al. *Mon. Not. R. Astron. Soc.* **437** 3072 (2014) Xilouris E et al. *Astrophys. J.* **651** L107 (2006) Temi P, Brighenti F, Mathews WG *Astrophys. J.* **660** 1215 (2007) Agguirre A, Haiman Z *Astrophys. J.* **532** 28 (2000) Johansson J, Morstell E *Mon. Not. R. Astron. Soc.* **426** 3360 (2012) Nath B, Shchekinov Yu *Rep. Prog. Phys.* (2014), submitted Polikarpova O L, Shchekinov Yu A *Astron. Rep.* In press (2014) Chelouche D, Koester B P, BowenDV *Astrophys. J.* **671** L97 (2007) Muller S et al. *Astrophys. J.* **680** 975 (2008) Menard B, Kilbinger M, Scranton R *Mon. Not. R. Astron. Soc.* **406** 1815 (2010) McGee S L, BaloghML *Mon. Not. R. Astron. Soc.* **405** 2069 (2010) Kitayama T et al. *Astrophys. J.* **695** 1191 (2009) Grossi Met al. *Astron. Astrophys.* **518** L52 (2010) BaesM et al. *Astron. Astrophys.* **518** L53 (2010) Simcoe R A et al. *Astrophys. J.* **637** 648 (2006) Tumlinson J et al. *Science* **334** 948 (2011) Vasiliev E O, Nath B B, Shchekinov Yu *Mon. Not. R. Astron. Soc.* (2014), submitted; arXiv:1401.5070 Khramtsova MS et al. *Mon. Not. R. Astron. Soc.* **431** 2006 (2013) Contursi A et al. *Astron. Astrophys.* **549** 118 (2013) Dedikov S Yu, Shchekinov Yu A *Astron. Rep.* **48** 9 (2004) Vasiliev E O, Dedikov S Yu, Shchekinov Yu A *Astrophys. Bull.* **64** 317 (2009) Shchekinov Yu A, Vasiliev E O *Astrophys. Space Sci.* (2014), submitted Diaz-Santos T et al. *Astrophys. J. Lett.* **788** L17 (2014) Kreckel K et al. *Astrophys. J.* **790** 26 (2014) Satteréeld T J et al. *Astron. J.* **144** 27 (2012) Heiles C, ReachW T, Koo B-C *Astrophys. J.* **466** 191 (1996) Tenorio-Tagle G *Astron. J.* **111** 1641 (1996) Lemasle B et al. *Astron. Astrophys.* **558** 31 (2013) Esteban C et al. *Mon. Not. R. Astron. Soc.* **433** 382 (2013) Esteban C et al. *Mon. Not. R. Astron. Soc.* **443** 624 (2014) Maiolino R et al. *Nature* **431** 533 (2004) Gomez H L et al. *Astrophys. J.* **760** 96 (2012) Gomez H L et al. *Mon. Not. R. Astron. Soc.* **420** 3557 (2012) Krause O et al. *Nature* **432** 596 (2004) Barlow MJ et al. *Astron. Astrophys.* **518** L138 (2010) Lagache G, Puget J-L, Dole H Annu. *Rev. Astron. Astrophys.* **43** 727 (2005) Vieira J D et al. *Astrophys. J.* **719** 763 (2010) Vieira J D et al. *Nature* **495** 344 (2013); arXiv:1303.2723 Blain A W *Mon. Not. R. Astron. Soc.* **283** 1340 (1996) Negrello Met al. *Mon. Not. R. Astron. Soc.* **377** 1557 (2007) Mao S D, Kochanek C S *Mon. Not. R. Astron. Soc.* **268** 569 (1994) McLeod K K, Bechtold J *Astrophys. J.* **704** 415 (2009) Irwin MJ et al. *Astrophys. J.* **505** 529 (1998) Vegetti S et al. *Nature* **481** 341 (2012) Hezaveh Y D et al. *Astrophys. J.* **761** 20 (2012) Viero MP et al. *Astrophys. J.* **772** 77 (2013) Lukash V N, Mikheeva E V Physical cosmology (M.: Fizmatlit, 2010) Lukash V N, Mikheeva E V, Malinovsky A M *Phys. Usp.* **54** 983 (2011) Braatz R et al. *Astrophys. J.* **767** 154 (2013) Galli D, Palla F *Annu. Rev. Astron. Astrophys.* **51** 163 (2013) Shchekinov Yu A *Sov. Astron. Lett.* **12** 211 (1986) Kamaya H, Silk J *Mon. Not. R. Astron. Soc.* **332** 251 (2009) Becker R H et al. *Astron. J.* **122** 2850 (2001) Dunkley J et al. *Astrophys. J. Suppl.* **180** 306 (2009) Doroshkevich A G, Pilipenko S V *Astron. Rep.* **55** 567 (2011) Dubrovich V K, Bajkova A, Khaikin V B *New Astron.* **13** 28 (2008) Dubrovich V K, Lipovka A A *Astron. Astrophys.* **296** 307 (1995) Zinchenko I, Dubrovich V, Henkel C *Mon. Not. R. Astron. Soc.* **415** L78 (2011) de Bernardis P et al. *Astron. Astrophys.* **538** 86 (2012) Zeldovich Ya B, Sunyaev R A *Astrophys. Space Sci.* **288** 4 (1969) Zeldovich Ya B, Sunyaev R A *Astrophys. Space Sci.* **20** 9 (1969) Sari R, Piran T, Narayan R *Astrophys. J.* **497** L17 (1998) De Ugarte Postigo et al. *Astron. Astrophys.* **538** 44 (2012) Perley D A et al. *Astrophys. J.* **781** 37 (2014) Trenti Met al. *Astrophys. J. Lett.* **749** L38 (2012) Tanvir N R et al. *Astrophys. J.* **754** 46 (2012) Cucchiara A et al. *Astrophys. J.* **736** 7 (2011) Bromm V, Coppi P S, Larson R B *Astron. J.* **564** 23 (2002) Komissarov S S, Barkov M V *Mon. Not. R. Astron. Soc.* **402** L25 (2010) Jakobsson P et al. *Astrophys. J.* **617** L21 (2004) Thoene C C et al. *Mon. Not. R. Astron. Soc.* **428** 3590 (2013) Castro-Tirado A J et al. *Astron. Astrophys.* **475** 101 (2007) Perley D A et al. *Astron. J.* **138** 1690 (2009) Zauderer B A et al. *Astrophys. J.* **767** 161 (2013) Volnova A A et al. *Mon. Not. R. Astron. Soc.* **442** 2586 (2014) Linde A Particle Physics and Inflationary Cosmology (Chur: Harwood Acad. Publ., 1990) Linde A, in Universe or Multiverse? (Ed. B Carr) (Cambridge: Cambridge Univ. Press, 2007) p. 127 Mukhanov V F Physical Foundations of Cosmology (Cambridge: Cambridge Univ. Press, 2007) Smolin L The Life of the Cosmos (Oxford: Oxford Univ. Press, 1999) Lukash V N, Mikheeva E V, Strokov V N *Phys. Usp.* **55** 831 (2012) Lukash V N, Mikheeva E V, Strokov V N *Phys. Usp.* **55** 204 (2012) Garay I, Robles-Perez S *Int. J. Mod. Phys. D* **23** 1450043 (2014) Nadezhin D K, Novikov I D, Polnarev A G *Sov. Astron.* **22** 129 (1978) Carr B J et al. *Phys. Rev. D* **81** 104019 (2010) Kardashev N S, Novikov I D, Shatskii A A *Astron. Rep.* **50** 601 (2006) Doroshkevich A G, Kardashev N S, Novikov D I, Novikov I D *Astron. Rep.* **52** 616 (2008) Pozanenko A, Shatskiy A *Gravit. Cosmol.* **16** 259 (2010); arXiv:1007.3620 Shatskii A A, Novikov I D, Kardashev N S *Phys. Usp.* **51** 457 (2008) Novikov I D, KardashevNS, Shatskii A A *Phys. Usp.* **50** 965 (2007) Shatskii A A *Phys. Usp.* **52** 811 (2009) Novikov I D *Sov. Astron.* **8** 857 (1965) Novikov I D, Frolov V P Physics of Black Holes (Dordrecht: Kluwer Acad., 1989) Retter A, Heller S *New Astron.* **17** 73 (2012)
[^1]: See the report by V.N. Lukash and I.D. Novikov at Groningen (2013) ([https://streaming1.service.rug.nl/p2player/player.aspx?path=
cit\_mobiel/2013/04/12/3/video\_post.wmv&mediatyoe=recordings](https://streaming1.service.rug.nl/p2player/player.aspx?path=
cit_mobiel/2013/04/12/3/video_post.wmv&mediatyoe=recordings).
[^2]: http://workshop.asc.rssi.ru
[^3]: <http://Herschel.cf.ac.uk/key-programmes/stars>.
|
---
abstract: 'It is shown that the Lorentz group is the natural language for two-beam interferometers if there are no decoherence effects. This aspect of the interferometer can be translated into six-parameter representations of the Lorentz group, as in the case of polarization optics where there are two orthogonal components of one light beam. It is shown that there are groups of transformations which leave the coherency or density matrix invariant, and this symmetry property is formulated within the framework of Wigner’s little groups. An additional mathematical apparatus is needed for the transition from a pure state to an impure state. Decoherence matrices are constructed for this process, and their properties are studied in detail. Experimental tests of this symmetry property are possible.'
address:
- 'National Aeronautics and Space Administration, Goddard Space Flight Center, Code 935, Greenbelt, Maryland 20771'
- 'Department of Physics, University of Maryland, College Park, Maryland 20742'
- 'Department of Radiology, New York University, New York, New York 10016'
author:
- 'D. Han[^1]'
- 'Y. S. Kim[^2]'
- 'Marilyn E. Noz [^3]'
title: '**Interferometers and Decoherence Matrices**'
---
Introduction {#intro}
============
In our earlier papers [@hkn97josa; @hkn97; @hkn99; @barakat63], we have formulated polarization optics in terms of the two-by-two and four-by-four representations of the six-parameter Lorentz group. It was noted that the two-component Jones vector and the four-component Stokes parameters are like the relativistic spinor and the Minkowskian four-vector respectively. We were able to identify the attenuator, rotator, and phase shifter with appropriate transformation matrices of the Lorentz group. It was noted that the two-element Jones vector is like the two-component Pauli spinor and that the four Stokes parameters act like the elements of a Minkowskian four-vector.
The purpose of this paper is to show that the mathematics of polarization optics is applicable also to interferometers. Our reasoning is that polarization optics is basically the physics of two plane waves. The same is true for two-beam interferometers. We need mathematical devices which will perform phase shifts between the waves and which will take care of attenuations at different rates. In the case of interferometers, it is possible to achieve the beam split and synthesis by rotation matrices. We can use the matrices of the above-mentioned Lorentz group in order to achieve these basic physical operations.
In addition, in this paper, we discuss the mathematical device which will describe the decoherence effect due to random phases. For this purpose, we need density matrices. However, the coherency matrix serves as the density matrix, and its four elements constitute the four components of the Stokes vector [@born80; @fey72]. It was noted in our previous paper that it is possible to construct a four-by-four decoherence matrix which will transform a pure-state Stokes vector into a mixed-state Stokes vector. Unlike the case of attenuations, rotations, or beam splits and syntheses, the decoherence matrix does not belong to the Lorentz group.
In order to study the decoherence process more carefully, we borrow the concept of Wigner’s little group originally developed for studying internal space-time symmetries of elementary particles [@wig39; @knp86]. Wigner’s little group is the maximal subgroup of the Lorentz group whose transformations leave the four-momentum of a given particle invariant. In the present case, the little group consists of transformations on a given density matrix which will leave that matrix invariant. It is shown in this paper that the little group for pure states is like that for massless particles, while the little group for impure states is like that for massive particles. The transition of the little group from a pure to impure state is discussed in detail.
In Sec. \[formul\], we show how each element in the two-beam interferometer system corresponds to a transformation matrix in the Lorentz group. The combined effect is the two-by-two representation of the six-parameter Lorentz group. In Sec. \[denma\], it is pointed out that the coherency matrix can also be defined for the interferometer system and that this matrix serves as the density matrix. The transformation property of the density matrix is discussed in detail. In Sec. \[little\], we introduce the little group which will leave a given density matrix invariant. It is noted that the little group for pure states has a symmetry property quite different from that for impure states. In Sec. \[decohm\], the decoherence matrices are discussed in detail. Although the augmentation of this matrix to the Lorentz group leads to a large group, there exist subgroups exhibiting symmetry properties familiar to us. Possible experiments with the decoherence matrix are suggested.
Formulation of the Problem {#formul}
==========================
Typically, one beam is divided into two by a beam splitter. We can write the incoming beam as $$\label{expo1}
\Psi = \pmatrix{\psi_{1} \cr \psi_{2}} =
\pmatrix{ \exp{\left\{i(kz - \omega t)\right\}} \cr 0} .$$ Then, the beam splitter can be written in the form of a rotation matrix [@sand99]: $$\label{rot22}
R(\theta) = \pmatrix{\cos(\theta/2) & -\sin(\theta/2) \cr
\sin(\theta/2) & \cos(\theta/2) } ,$$ which transforms the column vector of Eq.(\[expo1\]) into $$\label{expo2}
\pmatrix{\psi_{1} \cr \psi_{2}} =
\pmatrix{[\cos(\theta/2)]\exp{\left\{i(kz - \omega t)\right\}} \cr
-[\sin(\theta/2)\exp{\left\{i(kz - \omega t)\right\}} } .$$ The first beam $\psi_{1}$ of Eq.(\[expo1\]) is now split into $\psi_{1}$ and $\psi_{2}$ of Eq.(\[expo2\]). The intensity is conserved. If the rotation angle $\theta$ is -$\pi/4$, the initial beam is divided into two beams of the same intensity and the same phase [@campos89].
These two beams go through two different optical path lengths, resulting in a phase difference. If the phase difference is $\phi$, the phase shift matrix is $$\label{shif22}
P(\phi) = \pmatrix{e^{-i\phi/2} & 0 \cr 0 & e^{i\phi/2}} .$$
When reflected from mirrors, or while going through beam splitters, there are intensity losses for both beams. The rate of loss is not the same for the beams. This results in the attenuation matrix of the form $$\label{atten}
\pmatrix{e^{-\eta_{1}} & 0 \cr 0 & e^{-\eta_{2}}} =
e^{-(\eta_{1} + \eta_{2})/2} \pmatrix{e^{\eta/2} & 0 \cr 0 &
e^{-\eta/2}}$$ with $\eta = \eta_{2} - \eta_{1}$ . This attenuator matrix tells us that the electric fields are attenuated at two different rates. The exponential factor $e^{-(\eta_{1} + \eta_{2})/2}$ reduces both components at the same rate and does not affect the degree of polarization. The effect of polarization is solely determined by the squeeze matrix $$\label{sq22}
S(\eta) = \pmatrix{e^{\eta/2} & 0 \cr 0 & e^{-\eta/2}} .$$
In the detector or the beam synthesizer, the two beams undergo a superposition. This can be achieved by the rotation matrix like the one given in Eq.(\[rot22\]) [@sand99]. For instance, if the angle $\theta$ is $90^{o}$, the rotation matrix takes the form $${1 \over \sqrt{2}}\pmatrix{1 & -1 \cr 1 & 1} .$$ If this matrix is applied to the column vector of Eq.(\[expo2\]), the result is $${1 \over \sqrt{2}} \pmatrix{\psi_{1} - \psi_{2} \cr
\psi_{1} + \psi_{2}} .$$ The upper and lower components show the interferences with negative and positive signs respectively.
We have shown in our previous papers that repeated applications of the rotation matrices of the form of Eq.(\[rot22\]), shift matrices of the form of Eq.(\[shif22\]) and squeeze matrices of the form of Eq.(\[sq22\]) lead to a two-by-two representation of the six-parameter Lorentz group. The transformation matrix in general takes the form $$\label{lt22}
G = \pmatrix{\alpha & \beta \cr \gamma & \delta} ,$$ applicable to the column vector of Eq.(\[expo1\]), where all four elements are complex numbers with the condition that the determinant of the matrix be one.
Although we can borrow all the elegant mathematical identities of the two-by-two representations of the Lorentz group, this formalism does not allow us to describe the loss of coherence within the interferometer system. In order to study this effect, we have to construct the coherency matrix: $$\label{cocy}
C = \pmatrix{S_{11} & S_{12} \cr S_{21} & S_{22}},$$ with $$\begin{aligned}
\label{sii}
&{}& S_{11} = <\psi_{1}^{*}\psi_{1}> , \qquad
S_{22} = <\psi_{2}^{*}\psi_{2}> , \nonumber \\[2ex]
&{}& S_{12} = <\psi_{1}^{*}\psi_{2}> , \qquad
S_{21} = <\psi_{2}^{*}\psi_{1}> .\end{aligned}$$ It is sometimes more convenient to use the following combinations of parameters. $$\begin{aligned}
\label{stokes}
&{}& S_{0} = S_{11} + S_{22}, \cr
&{}& S_{1} = S_{11} - S_{22}, \cr
&{}& S_{2} = S_{12} + S_{21}, \cr
&{}& S_{3} = -i\left(S_{12} - S_{21}\right).\end{aligned}$$ These four parameters are called the Stokes parameters in the literature [@shur62; @hecht70], usually in connection with polarized light waves. In the present paper, we are applying these parameters to two separate beams in a given interferometer system.
The Stokes parameters, originally developed for polarization optics, are becoming applicable to other branches of physics dealing with two orthogonal states. In this paper, we are using these parameters for interferometers.
We have shown previously [@hkn97] that the four-by-four transformation matrices applicable to the Stokes parameters are like Lorentz-transformation matrices applicable to the space-time Minkowskian vector $(t, z, x, y)$. This allows us to study space-time symmetries in terms of the Stokes parameters which are applicable to interferometers. Let us first see how the rotation matrix of Eq.(\[rot22\]) is translated into the four-by-four formalism. In this case, $$\alpha = \delta = \cos(\theta/2), \qquad
\gamma = -\beta = \sin(\theta/2) .$$ The corresponding four-by-four matrix takes the form [@hkn99] $$\label{rot44}
R(\theta) = \pmatrix{1 & 0 & 0 & 0 \cr
0 & \cos\theta & -\sin\theta & 0 \cr
0 & \sin\theta & \cos\theta & 0 \cr
0 & 0 & 0 & 1} .$$
Let us next see how the phase-shift matrix of Eq.(\[shif22\]) is translated into this four-dimensional space. For this two-by-two matrix, $$\alpha = e^{-i\phi/2} , \qquad \beta = \gamma = 0 , \qquad
\delta = e^{i\phi/2} .$$ For these values, the four-by-four transformation matrix takes the form [@hkn99] $$\label{shif44}
P(\phi) = \pmatrix{1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 \cr
0 & 0 & \cos\phi & -\sin\phi \cr 0 & 0 & \sin\phi & \cos\phi} .$$ For the squeeze matrix of Eq.(\[sq22\]), $$\alpha = e^{\eta/2}, \qquad \beta = \gamma = 0 , \qquad
\delta = e^{-\eta/2} .$$ As a consequence, its four-by-four equivalent is $$\label{sq44}
S(\eta) = \pmatrix{\cosh\eta & \sinh\eta & 0 & 0 \cr
\sinh\eta & \cosh\eta & 0 & 0 \cr
0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1} .$$ If the above matrices are applied to the four-dimensional Minkowskian space of $(t, z, x, y)$, the above squeeze matrix will perform a Lorentz boost along the $z$ or $S_{1}$ axis with $S_{0}$ as the time variable. The rotation matrix of Eq.(\[rot44\]) will perform a rotation around the $y$ or $S_{3}$ axis, while the phase shifter of Eq.(\[shif44\]) performs a rotation around the $z$ or the $S_{1}$ axis. Matrix multiplications with $R(\theta)$ and $P(\phi)$ lead to the three-parameter group of rotation matrices applicable to the three-dimensional space of $(S_{1}, S_{2}, S_{3})$.
The phase shifter $P(\phi)$ of Eq.(\[shif44\]) commutes with the squeeze matrix of Eq.(\[sq44\]), but the rotation matrix $R(\theta)$ does not. This aspect of matrix algebra leads to many interesting mathematical identifies which can be tested in laboratories. One of the interesting cases is that we can produce a rotation by performing three squeezes [@hkn99]. Another interesting case is a combination of squeeze and rotation will produce a matrix which will convert numerical multiplication into addition. This aspect known as the Iwasawa decomposition is discussed in detail in Ref. [@hkn99].
Density Matrices and Their Little Groups {#denma}
========================================
According to the definition of the density matrix [@fey72], the coherency matrix of Eq.(\[cocy\]) is also the density matrix. Since we discussed transformation properties of coherency matrices in our earlier papers [@hkn97josa; @hkn97], we can start here with those results on this subject.
The most effective way of formulating the symmetry property of a given physical system is to construct a group of transformations which leave the system invariant. This concept was originally developed by Wigner [@wig39] for internal space-time symmetries of relativistic particles. Wigner’s little group is the maximal subgroup of the Lorentz group whose transformations leave the four-momentum of a given particle invariant. For instance, for a particle at rest, the little group is the three-parameter rotation group. The rotations do not change the four-momentum of the particle, even though they change the direction of the spin. There are also massless particles which cannot be brought to rest. This is the reason why the little group for a massive particle is different from that of the massless particle. The little group for massless particles is like (or locally isomorphic to) the two-dimensional Euclidean group [@wig39; @knp86].
Indeed, in Ref.[@hkn99], we discussed Wigner rotations and Iwasawa decompositions rotations applicable to massive and massless particles respectively and how these little-group transformations can be applied to the Stokes four-vectors. In this section, we shall see that the Stokes vectors for pure and impure states are like the four-momentum of the massless and massive particles respectively.
In the following discussion, we will need transformations of the Stokes four-vectors and the corresponding transformations of the two-by-two density matrices. We are quite familiar with four-by-four matrices applicable to the Stokes vectors. For the two-by-two density matrices, the transformation takes the form
Under the influence of the $G$ transformation given in Eq.(\[lt22\]), this coherency matrix is transformed as $$\begin{aligned}
\label{trans22}
&{}& C' = G\,C\,G^{\dagger} =
\pmatrix{S'_{11} & S'_{12} \cr S'_{21} & S'_{22}} \nonumber \\[2ex]
&{}&\hspace{5ex} = \pmatrix{\alpha & \beta \cr \gamma & \delta}
\pmatrix{S_{11} & S_{12} \cr S_{21} & S_{22}}
\pmatrix{\alpha^{*} & \gamma^{*} \cr \beta^{*} & \delta^{*}} ,\end{aligned}$$ where $C$ and $G$ are the density matrix and the transformation matrix given in Eq.(\[cocy\]) and Eq.(\[lt22\]) respectively. According to the basic property of the Lorentz group, these transformations do not change the determinant of the density matrix $C$. Transformations which do not change the determinant are called unimodular transformations.
As we shall see in this section, the determinant for pure states is zero, while for that for mixed states does not vanish. Is there then a transformation matrix which will change this determinant within the Lorentz group. The answer is No. This is the basic issue we would like to address in this section.
If the phase difference between the two waves remains intact, the the system is said to in a pure state, and the density matrix can be brought to the form $$\label{pure22}
\pmatrix{1 & 0 \cr 0 & 0} ,$$ through the transformation of Eq.(\[trans22\]) with a suitable choice of the $G$ matrix. For the pure state, the Stokes four-vector takes the form $$\label{pure4}
\pmatrix{1 \cr 1 \cr 0 \cr 0} .$$
In order to study the symmetry properties of the density matrix, let us ask the following question. Is there a group of transformation matrices which will leave the above density matrix invariant? In answering this question, it is more convenient to use the Stokes four-vector. The column vector of Eq.(\[pure4\]) is invariant under the operation of the phase shifter $P(\phi)$ of Eq.(\[shif44\]). In addition, it is invariant under the following two matrices: $$\begin{aligned}
\label{d1d2}
&{}& F_{1}(u) = \pmatrix{ 1 + u^{2}/2 & - u^{2}/2 & u & 0 \cr
u^{2}/2 & 1 - u^{2}/2 & u & 0 \cr
u & -u & 1 & 0 \cr
0 & 0 & 0 & 1 } , \nonumber \\[2ex]
&{}& F_{2}(v) = \pmatrix{ 1 + v^{2}/2 & - v^{2}/2 & 0 & v \cr
v^{2}/2 & 1 - v^{2}/2 & 0 & v \cr
0 & 0 & 1 & 0 \cr
u & -v & 0 & 1 } .\end{aligned}$$ These mathematical expressions were first discovered by Wigner in 1939 [@wig39] in connection with the internal space-time symmetries of relativistic particles. They went through a stormy history, but it is gratifying to note that they serve a useful purpose for studying interferometers where each matrix corresponds to an operation which can be performed in laboratories.
The $F_{1}$ and $F_{2}$ matrices commute with each other, and the multiplication of these leads to the form $$\label{d44}
F_{2}(u)F_{2}(v)
= \pmatrix{1 + (u^{2} + v^{2})/2 & - (u^{2} + v^{2})/2 & u & u \cr
(u^{2} + v^{2})/2 & 1 - (u^{2} + v^{2})/2 & u & v \cr
u & -u & 1 & 0 \cr
v & -v & 0 & 1 } .$$ This matrix contains two parameters.
Let us go back to the phase-shift matrix of Eq.(\[shif44\]). This matrix also leaves the Stokes vector of Eq.(\[pure4\]) invariant. If we define the “little group” as the maximal subgroup of the Lorentz group which leaves a Stokes vector invariant, the little group for the Stokes vector of Eq.(\[pure4\]) consists of the transformation matrices given in Eq.(\[shif44\]) and Eq.(\[d44\]).
Next, if the phase relation is completely random, and the first and second components have the same amplitude, the density matrix becomes $$\label{imp22}
\pmatrix{1/2 & 0 \cr 0 & 1/2} .$$ Here is the question: Is there a two-by-two matrix which will transform the pure-state density matrix of Eq.(\[pure22\]) into the impure-state matrix of Eq.(\[imp22\])? The answer within the system of matrices of the form given in Eq.(\[lt22\]) is No, because the determinant of the pure-state density matrix is zero while that for the impure-state matrix is $1/4$. Is there a way to deal with this problem? We shall return to this problem in Sec. \[decohm\]. In this section, we restrict ourselves to the unimodular transformation of Eq.(\[trans22\]) which preserves the value of the determinant of the density matrix. The Stokes four-vector corresponding to the above density matrix is $$\label{imp4}
\pmatrix{1 \cr 0 \cr 0 \cr 0} .$$ This vector is invariant under both the rotation matrix of Eq.(\[rot44\]) and the phase shift matrix of Eq.(\[shif44\]). Repeated applications of these matrices lead to a three-parameter group of rotations applicable to the three-dimensional space of $(S_{1}, S_{2}, S_{3})$.
Not all the impure-state density matrices take the form of Eq.(\[imp22\]). In general, if they are brought to a diagonal form, the matrix takes the form $$\label{impp22}
{1 \over 2}\pmatrix{1 + \cos\chi & 0 \cr 0 & 1 - \cos\chi} ,$$ and the corresponding Stokes four-vector is $$\label{impp4}
e^{-\eta} \pmatrix{\cosh\eta \cr \sinh\eta \cr 0 \cr 0} ,$$ with $$\eta = {1 \over 2}\ln{1 + \cos\chi \over 1 - \cos\chi} .$$ The matrix which transforms Eq.(\[imp4\]) to Eq.(\[impp4\]) is the squeeze matrix of Eq.(\[sq44\]). The question then is whether it is possible to transform the pure state of Eq.(\[pure4\]) to the impure state of Eq.(\[impp4\]) or to Eq.(\[imp4\]).
In order to see the problem in terms of the two-by-two density matrix, let us go back to the pure-state density matrix of Eq.(\[pure22\]). Under the rotation of Eq.(\[rot22\]), $$\begin{aligned}
&{}& \pmatrix{\cos(\chi/2) & -\sin(\chi/2) \cr \sin(\chi/2) &
\cos(\chi/2) } \pmatrix{1 & 0 \cr 0 & 0} \nonumber\\[2ex]
&{}& \times \pmatrix{\cos(\chi/2) & \sin(\chi/2) \cr
-\sin(\chi/2) & \cos(\chi/2) } ,\end{aligned}$$ the pure-state density matrix becomes $${1 \over 2} \pmatrix{1 + \cos\chi & \sin\chi \cr
\sin\chi & 1 - \cos\chi} .$$
For the present case of two-by-two density matrices, the trace of the matrix is one for both pure and impure cases. The trace of the $(matrix)^{2}$ is one for the pure state, while it is less than one for impure states.
The next question is whether there is a two-by-two matrix which will eliminate the off-diagonal elements of the above expression that will also lead to the expression of Eq.(\[impp22\]). In order to answer this question, let us note that the determinant of the density matrix vanishes for the pure state, while it is non-zero for impure states. The Lorentz-like transformations of Eq.(\[trans22\]) leave the determinant invariant. Thus, it is not possible to transform a pure state into an impure state by means of the transformations from the six-parameter Lorentz group. Then is it possible to achieve this purpose using two-by-two matrices not belonging to this group. We do not know the answer to this question. We are thus forced to resort to four-by-four matrices applicable to the Stokes four-vector.
Decoherence Effects on the Little Groups {#little}
========================================
We are interested in a transformation which will change the density matrix of Eq.(\[pure22\]) to Eq.(\[imp22\]). For this purpose, we can use the Stokes four-vector consisting of the four elements of the density matrix. The question then is whether it is possible to find a transformation matrix which will transform the pure-state four-vector of Eq.(\[pure4\]) to the impure-state four-vector of Eq.(\[imp4\]).
Mathematically, it is more convenient to ask whether the inverse of this process is possible: whether it is possible to transform the four-vector of Eq.(\[imp4\]) to that of Eq.(\[pure4\]). This is known in mathematics as the contraction of the three-dimensional rotation group into the two-dimensional Euclidean group [@knp86]. Let us apply the squeeze matrix of Eq.(\[sq44\]) to the four-vector of Eq.(\[imp4\]). This can be written as $$\label{sqimp}
\pmatrix{\cosh\eta & \sinh\eta & 0 & 0 \cr
\sinh\eta & \cosh\eta & 0 & 0 \cr
0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1}
\pmatrix{1 \cr 0 \cr 0 \cr 0} =
\pmatrix{\cosh\eta \cr \sinh\eta \cr 0 \cr 0} .$$ After an appropriate normalization, the right-hand side of the above equation becomes like the pure-state vector of Eq.(\[pure4\]) in the limit of large $\eta$, as $\cosh\eta$ becomes equal to $\sinh\eta$ in the infinite-$\eta$ limit. This transformation is from a mixed state to a pure or almost-pure state. Since we are interested in the transformation from the pure state of Eq.(\[pure4\]) to the impure state of Eq.(\[imp4\]), we have to consider an inverse of the above equation: $$\label{inver1}
\pmatrix{\cosh\eta & -\sinh\eta & 0 & 0 \cr
-\sinh\eta & \cosh\eta & 0 & 0 \cr 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1}
\pmatrix{\cosh\eta \cr \sinh\eta \cr 0 \cr 0} =
\pmatrix{1 \cr 0 \cr 0 \cr 0} .$$ However, the above equation does not start with the pure-state four-vector. If we apply the same matrix to the pure state matrix, the result is $$\pmatrix{\cosh\eta & -\sinh\eta & 0 & 0 \cr
-\sinh\eta & \cosh\eta & 0 & 0 \cr 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1}
\pmatrix{1 \cr 1 \cr 0 \cr 0}
= e^{-\eta} \pmatrix{1 \cr 1 \cr 0 \cr 0} .$$ The resulting four-vector is proportional to the pure-state four-vector and is definitely not an impure-state four-vector.
The inverse of the transformation of Eq.(\[sqimp\]) is not capable of bringing the pure-state vector into an impure-state vector. Let us go back to Eq.(\[sqimp\]), it is possible to bring a impure-state into a pure state only in the limit of infinite $\eta$. Otherwise, it is not possible. It is definitely not possible if we take into account experimental considerations.
The story is different for the little groups. Let us start with the rotation matrix of Eq.(\[rot44\]), and apply to this matrix the transformation matrix of Eq.(\[sqimp\]). Then $$\label{3mats}
\pmatrix{\cosh\eta & \sinh\eta & 0 & 0 \cr
\sinh\eta & \cosh\eta & 0 & 0 \cr
0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1}
\pmatrix{1 & 0 & 0 & 0 \cr 0 & \cos\theta & -\sin\theta & 0 \cr
0 & \sin\theta & \cos\theta & 0 \cr 0 & 0 & 0 & 1}
\pmatrix{\cosh\eta & -\sinh\eta & 0 & 0 \cr
-\sinh\eta & \cosh\eta & 0 & 0 \cr
0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1} .$$ If $\eta$ is zero, the above expression becomes the rotation matrix of Eq.(\[rot44\]). If $\eta$ becomes infinite, it becomes the little-group matrix $F_{1}(u)$ of Eq.(\[d1d2\]) applicable to the pure state of Eq.(\[pure4\]). The details of this calculation for the case of Lorentz transformations are given in the 1986 paper by Han [*et al.*]{} [@hks86jm]. We are then led to the question of whether one little-group transformation matrix can be transformed from the other.
If we carry out the matrix algebra of Eq.(\[3mats\]), the result is $$\pmatrix{1 + \alpha u^{2} w/2 & -\alpha u^{2} w/2 & \alpha uw & 0 \cr
\alpha u^{2} w/2 & 1 - u^{2} w/2 & uw & 0 \cr
\alpha uw & -uw & 1 - (1 - \alpha^{2}) u^{2} w/2 & 0 \cr
0 & 0 & 0 & 1} ,$$ where $$\label{anal}
\alpha = \tanh\eta , \qquad
u = - 2\,\tan\left({\theta \over 2}\right), \qquad
w = { 1 \over 1 + (1 - \alpha^{2})\tan^{2}(\theta/2)}.$$ If $\alpha = 0$, the above expression becomes the rotation matrix of Eq.(\[rot44\]). If $\alpha = 1$, it becomes the $F_{1}$ matrix of Eq.(\[d1d2\]). Here we used the parameter $\alpha$ instead of $\eta$. In terms of this parameter, it is possible to make an analytic continuation from the pure state with $\alpha = 1$ to an impure state with $\alpha < 1$ including $\alpha = 0$.
On the other hand, we should keep in mind that the determinant of the density matrix is zero for the pure state, while it is non-zero for all impure states. For $\alpha = 1$, the determinant vanishes, but it is nonzero and stays the same for all non-zero values of $\alpha$ less than one and greater than or equal to zero. The analytic expression of Eq.(\[anal\]) hides this singular nature of the little group [@hks86jm].
Decoherence Matrices {#decohm}
====================
We are interested in the decoherence effect on the density matrix. We are particularly interested in the mechanism where the off-diagonal elements $S_{12}$ and $S_{21}$ become smaller due to time average or phase-randomizing process [@raymer97]. If this happens, we can apply to the Stokes four-vector the following decoherence matrix. $$\pmatrix{1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 \cr
0 & 0 & e^{-2\lambda} & 0 \cr 0 & 0 & 0 & e^{-2\lambda}} ,$$ which can also be written as $$\label{decoh1}
e^{-\lambda} \pmatrix{e^{\lambda} & 0 & 0 & 0 \cr
0 & e^{\lambda} & 0 & 0 \cr 0 & 0 & e^{-\lambda} & 0 \cr
0 & 0 & 0 & e^{-\lambda}} ,$$ where $e^{-\lambda}$ is the overall decoherence factor. For convenience, we define the decoherence matrix as $$\label{dlam}
D(\lambda) = \pmatrix{e^{\lambda} & 0 & 0 & 0 \cr
0 & e^{\lambda} & 0 & 0 \cr 0 & 0 & e^{-\lambda} & 0 \cr
0 & 0 & 0 & e^{-\lambda}} .$$ This matrix cannot be constructed from the six-parameter Lorentz group applicable to the Stokes four-vectors.
If we combine this decoherence matrix with the Lorentz group, the result will be a fifteen-parameter group of four-by-four matrices isomorphic to $O(3,3)$ which is beyond the scope of the present paper [@hkn95jm]. In order to extract the symmetry of physical interest, let us go back to the four-by-four matrices $R(\theta), P(\phi)$, and $S(\eta)$ of Eq.(\[rot44\]), Eq.(\[shif44\]), and Eq.(\[sq44\]) respectively. The phase-shift matrix of Eq.(\[shif44\]) commutes with the decoherence matrix.
As we discussed in our earlier paper on polarization optics [@hkn97], the decoherence matrix and the rotation matrix will lead to two-dimensional squeeze transformations applicable to the two-component vector $$\label{va}
V_{A} = \pmatrix{S_{1} \cr S_{2}} .$$ The four-by-four $D(\lambda)$ matrix of Eq.(\[dlam\]) and the rotation matrix of Eq.(\[rot44\]) become reduced to $$\begin{aligned}
\label{dara}
&{}& D_{A}(\lambda) = \pmatrix{e^{\lambda} & 0 \cr
0 & e^{-\lambda}} , \nonumber \\[2ex]
&{}& R_{A}(\theta) = \pmatrix{\cos\theta & -\sin\theta \cr
\sin\theta & \cos\theta} .\end{aligned}$$ As for the remaining components of the Stokes parameters, we can define another two-component vector as $$V_{B} = \pmatrix{S_{0} \cr S_{3}} .$$ The decoherence matrix applicable to this two-component vector is $$D_{B}(\lambda) = \pmatrix{e^{\lambda} & 0 \cr 0 & e^{-\lambda}} ,$$ but the rotation matrix does not change the two-component vector $V_{B}$.
Let us go back to the two-dimensional space of $V_{A}$, and its two-by-two transformation matrices. The matrices $D_{A}(\lambda)$ and $R_{A}(\theta)$ of Eq.(\[dara\]) applicable are strikingly similar to the two-by-two matrices given in Eq.(\[sq22\]) and Eq.(\[rot22\]) respectively. If we replace the parameters $\eta$ in $S(\eta)$ and $\theta$ in $R(\theta)$ by $2\lambda$ and $2\theta$ respectively, they become $D_{A}$ and $R_{B}$ of Eq.(\[dara\]).
With these two matrices, we can repeat the calculations for the Wigner rotations and Iwasawa decompostions discussed in our earlier paper [@hkn99]. It is possible to perform experiments to test these mathematical relations.
Concluding Remarks {#concluding-remarks .unnumbered}
==================
In this paper, we have discussed two-beam interferometers within the framework of the six-parameter Lorentz group. It has been shown that beam splitters and beam synthesizers can be represented by two-by-two rotation matrices. The phase shift can also be represented by two-by-two rotation matrices applicable to spinor systems. As for attenuation, we introduced two-by-two squeeze matrices. The combined effect of these transformations leads to a two-by-two representation of the six-parameter Lorentz group.
We have found that the mathematical formalism given in this paper is identical to the formalism we presented in our earlier papers for polarization optics. In this series of papers, our purpose has been to minimize the group theoretical language and write down formulas close to what we observe in the real world. In this paper, we were able to by-pass completely the group theoretical formality known as the Lie algebra of the Lorentz group consisting of generators and their closed commutation relations.
With this improved mathematical technique, we discussed two-beam physics in terms of the little groups using only matrices which are realizable in laboratories. It has been shown that the little groups for pure and impure states are different. It was noted that analytic continuation from a pure state to an impure state is possible for the little groups. On the other hand, this transformation does not exist within the six-parameter Lorentz group, but requires an extra four-by-four matrix applicable to the Stokes four-vector, called the decoherence matrix.
The augmentation of this decoherence matrix into the Lorentz group will lead to a bigger group which is beyond the scope of this paper [@hkn95jm]. However, this bigger group has $O(2,1)$-like or $SU(1,1)$-like subgroups which are quite familiar to us from the squeezed states of light, and the Lorentz group-formulation of the polarization optics [@hkn97]. We are fortunate to observe, within the framework of this decoherence matrix, mathematical consequence which will lead to experiments on Wigner rotations and Iwasawa decompositions which are possible in both polarization optics and interferometers.
It will be a challenging problem to translate what we did in this paper to the language of quantum optics. The rotation operations corresponding to phase shifts and rotations around the direction of the propagation can be formulated in terms of the two-mode squeezed states [@ymk86]. However, the squeeze transformations discussed in this paper correspond to the loss of intensity, which cannot be translated into quantum optics. On the other hand, the decoherence matrix can be accommodated into the density-matrix formalism. Indeed, they all are challenging problems.
Furthermore, unlike the case of polarization optics, there can be more than two beams for interferometers. For instance, three-beam interferometer are quite common. This will open up a new research line for studying symmetry properties in optics. The power of group theoretical approaches is that we can establish the symmetry properties in one branches of physics to those in a different field using the isomorphism and/or homomorphism of group theory. As for the three-beam case, we are happy to note a recent paper by Rowe [*et al*]{}. [@rowe99].
[99]{}
D. Han, Y. S. Kim, and M. E. Noz, J. Opt. Soc. Am. A [**14**]{}, 2290 (1997).
D. Han, Y. S. Kim, and M. E. Noz, Phys. Rev. E [**56**]{}, 6065 (1997);
For earlier and later papers on this subject, see R. Barakat, J. Opt. Soc. Am. [**53**]{}(3) 317 (1963); C. S. Brown and A. E. Bak, Opt. Engineering [**34**]{}, 1625 (1995); J. J. Monzon and L. L. Sánchez-Soto, Phys. Lett. A [**262**]{} 18 (1999).
D. Han, Y. S. Kim, and M. E. Noz, Phys. Rev. E [**60**]{}, 1036 (1999).
M. Born and E. Wolf, [*Principles of Optics, 6th Ed.*]{} (Pergamon, Oxford, 1980). The first edition of this book was published in 1959.
R. P. Feynman, [*Statistical Mechanics*]{} (Benjamin/Cummings, Reading, MA, 1972).
E. Wigner, Ann. Math. [**40**]{}, 149 (1939).
Y. S. Kim and M. E. Noz, [*Theory and Applications of the Poincaré Group*]{} (Reidel, Dordrecht, 1986); Y. S. Kim and M. E. Noz, [*Phase Space Picture of Quantum Mechanics*]{} (World Scientific, Singapore, 1991).
B. C. Sanders and A. Mann, Group 22, Proceedings of the 22nd International Colloquium on Group Theoretical Methods in Physics, S. P. Cornel [*et al*]{}. eds. (International Press, Boston, 1999). See pp 474-478.
For earlier papers on beam splitters based on the $SU(2)$ and $Sp(2)$ transformations, see R. A. Campos, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A, [**40**]{}, 1371 (1989) and A. Luis and L. L. Sánchez-Soto, Quantum Semniclass. Opt. [**7**]{}, 153 (1995), respectively.
W. A. Shurcliff, [*Polarized Light*]{} (Harvard Univ. Press, Cambridge, MA, 1962).
E. Hecht, Am. J. Phys. [**38**]{}, 1156 (1970).
D. Han, Y. S. Kim, and D. Son, J. Math. Phys. [**27**]{}, 2228 (1986).
D. F. McAlister and M. G. Raymer, Phys. Rev. A [**55**]{}, R1609 (1997).
D. Han, Y. S. Kim, and M. E. Noz, J. Math. Phys. [**36**]{}, 3940 (1995).
B. Yurke, S. McCall, and J. R. Klauder, Phys. Rev. A [**33**]{}, 4033 (1986).
D. Han, Y. S. Kim, and D. Son, 1987, Class. Quantum Grav. [**4**]{}, 1777 (1987).
D. J. Rowe, B. C. Sanders, and H. de Guise, J. Math. Phys. [**40**]{}, 3604 (1999).
[^1]: electronic mail: han@trmm.gsfc.nasa.gov
[^2]: electronic mail: yskim@physics.umd.edu
[^3]: electronic mail: noz@nucmed.med.nyu.edu
|
---
abstract: 'With VLT/X-shooter, we obtain optical and near-infrared spectra of six [Ly$\alpha$]{}blobs at $z$ $\sim$ 2.3. For a total sample of eight [Ly$\alpha$]{}blobs (including two that we have previously studied), the majority (6/8) have broadened [Ly$\alpha$]{}profiles with shapes ranging from a single peak to symmetric or asymmetric double-peaked. The remaining two systems, in which the [Ly$\alpha$]{}profile is not significantly broader than the [\[\]]{}or [H$\alpha$]{}emission lines, have the most spatially compact [Ly$\alpha$]{}emission, the smallest offset between the [Ly$\alpha$]{}and the [\[\]]{}or [H$\alpha$]{}line velocities, and the only detected and lines in the sample, implying that a hard ionizing source, possibly an AGN, is responsible for their lower optical depth. Using three measures — the velocity offset between the [Ly$\alpha$]{}line and the non-resonant [\[\]]{}or [H$\alpha$]{}line ([v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}), the offset of stacked interstellar metal absorption lines, and a new indicator, the spectrally-resolved [\[\]]{}line profile — we study the kinematics of gas along the line of sight to galaxies within each blob center. These three indicators generally agree in velocity and direction, and are consistent with a simple picture in which the gas is stationary or slowly outflowing at a few hundred [kms$^{-1}$]{}from the embedded galaxies. The absence of stronger outflows is not a projection effect: the covering fraction for our sample is limited to $<$1/8 (13%). The outflow velocities exclude models in which star formation or AGN produce “super” or “hyper” winds of up to $\sim$1000[kms$^{-1}$]{}. The [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}offsets here are smaller than typical of Lyman break galaxies (LBGs), but similar to those of compact [Ly$\alpha$]{}emitters. The latter suggests a connection between blob galaxies and [Ly$\alpha$]{}emitters and that outflow speed cannot be a dominant factor in driving extended [Ly$\alpha$]{}emission. For one [Ly$\alpha$]{}blob (CDFS-LAB14), whose [Ly$\alpha$]{}profile and metal absorption line offsets suggest no significant bulk motion, we use a simple radiative transfer model to make the first column density measurement of gas in an embedded galaxy, finding it consistent with a damped [Ly$\alpha$]{}absorption system. Overall, the absence of clear inflow signatures suggests that the channeling of gravitational cooling radiation into [Ly$\alpha$]{}is not significant over the radii probed here. However, one peculiar system (CDFS-LAB10) has a blueshifted [Ly$\alpha$]{}component that is not obviously associated with any galaxy, suggesting either displaced gas arising from tidal interactions among blob galaxies or gas flowing into the blob center. The former is expected in these overdense regions, where [*HST*]{} images resolve many galaxies. The latter might signify the predicted but elusive cold gas accretion along filaments.'
author:
- 'Yujin Yang, Ann Zabludoff, Knud Jahnke, Romeel Davé'
title: 'The Properties of [Ly$\alpha$]{}Nebulae: Gas Kinematics from Non-resonant Lines'
---
Introduction {#sec:intro}
============
Giant [Ly$\alpha$]{}nebulae, or “blobs,” are extended sources at $z$ $\sim$ 2–6 with typical [Ly$\alpha$]{}sizes of $\gtrsim$5 ($\gtrsim$50kpc) and line luminosities of $L_{\rm{Ly\alpha}}\gtrsim10^{43}$[ergs$^{-1}$]{}. The low number counts, strong clustering, multiple embedded sources, and location in over-dense environments of the largest [Ly$\alpha$]{}blobs indicate that they lie in massive ($M_{\rm halo}$ $\sim$ $10^{13}$[$M_{\sun}$]{}) dark matter halos, which will evolve into those typical of rich galaxy groups or clusters today [@Yang09; @Yang10; @Prescott08; @Prescott12b]. Therefore, [Ly$\alpha$]{}blobs are unique tracers of the formation of the most massive galaxies and their early interaction with the surrounding intergalactic medium (IGM).
This interaction is probably tied on some scale to the source of the blobs’ extended [Ly$\alpha$]{}emission, but that mechanism is poorly understood. Emission from [Ly$\alpha$]{}blobs could arise from several phenomena, which may even operate together, including shock-heating by galactic superwinds or gas photoionized by active galactic nuclei . Another possibility is smooth gas accretion, which is likely to play an important role in the formation of galaxies [e.g., @Keres05; @Keres09] and which should channel some of its gravitational cooling radiation into atomic emission lines such as [Ly$\alpha$]{}. Another scenario is the resonant scattering of [Ly$\alpha$]{}photons produced by star formation or active galactic nuclei (AGN) [@Steidel10; @Hayes11] in the embedded galaxies.
To resolve the debate about the nature of [Ly$\alpha$]{}blobs requires — at the very least — that we discriminate between between outflowing and inflowing models. Because [Ly$\alpha$]{}is a resonant line and typically optically thick in the surrounding intergalactic medium, studies even of the same [Ly$\alpha$]{}blob’s kinematics can disagree. On one hand, @Wilman05 argue that their integral field unit (IFU) spectra of a [Ly$\alpha$]{}blob are consistent with a simple model where the [Ly$\alpha$]{}emission is absorbed by a foreground slab of neutral gas swept out by a galactic scale outflow. On the other, @Dijkstra06b explain the same data as arising from the infall of the surrounding intergalactic medium. Worse, @Verhamme06 claim that the same symmetric [Ly$\alpha$]{}profiles are most consistent with static surrounding gas.
To distinguish among such possibilities requires a comparison of the center of the [Ly$\alpha$]{}line profile with that of a non-resonant line like [H$\alpha$ $\lambda$6563]{}or [\[\] $\lambda$5007]{}. These rest-frame optical nebular lines are better measure of the [Ly$\alpha$]{}blob’s systemic velocity, i.e., of the precise redshift, because it is not seriously altered by radiative transfer effects and is more concentrated about the galaxies in the [Ly$\alpha$]{}blob’s core. We illustrate this line offset technique in Figure \[fig:cartoon\]. While gas accretion models predict different line profile shapes depending on various assumptions, e.g., the location of ionizing sources, the detailed geometry, and the velocity field, they all predict that the overall [Ly$\alpha$]{}line profile, originating from the central source or the surrounding gas, will be blue-shifted with respect to the center of a non-resonant line such as H$\alpha$ that is optically thin to the surrounding gas [@Verhamme06; @Dijkstra06a]. This is because the red-side of the [Ly$\alpha$]{}profile will see higher optical depth due to the infalling (approaching) gas. In other words, the H$\alpha$ kinematics represent the true underlying velocity field if the [Ly$\alpha$]{}blob is accreting gas from the intergalactic medium. The same is true if the gas is outflowing, except that the [Ly$\alpha$]{}line will be redshifted with respect to H$\alpha$. Thus, if we measure the direction of the [Ly$\alpha$]{}–H$\alpha$ and/or [Ly$\alpha$]{}–[\[\]]{}line offset (hereafter defined as [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}), we can distinguish an inflow from an outflow.
The first such analysis for two [Ly$\alpha$]{}blobs shows that [Ly$\alpha$]{}is coincident with or redshifted by $\sim$200[kms$^{-1}$]{} from the H$\alpha$ line center [@Yang11 see also McLinden et al. 2013]. These offsets are much smaller than the $\sim$1000[kms$^{-1}$]{}expected from superwind models and even smaller than those typical of Lyman break galaxies (LBGs), which are widely believed to have galactic outflows [@Steidel04; @Steidel10]. Thus, if [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} is a proxy for outflow velocities ([$v_{\rm exp}$]{}), our initial results suggest that star formation- or AGN-produced winds may not be required for powering [Ly$\alpha$]{}blobs, making other interpretations of their emission more likely.
However, we do not yet know if these results are representative of [*all*]{} [Ly$\alpha$]{}blobs or if we have failed to detect strong flows due to the projected orientations of these two sources. For example, the gas flow may not be isotropic. As in bipolar outflows in M82 , a galactic-scale outflow may occur in the direction of minimum pressure in the surrounding interstellar medium (ISM), often perpendicular to the stellar disks. Or, if gas accretion is taking place in [Ly$\alpha$]{}blobs, numerical simulations suggest that the gas infall may occur preferentially along filamentary streams [@Keres05; @Keres09; @Dekel09]. Thus, if the bulk motion of gas (either infalling or outflowing) happens to be mis-aligned with our line of sight (LOS), then we may underestimate or even fail to detect the relative velocity shifts. Therefore, it is critical to measure [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}for a larger sample to average over any geometric effects and obtain better constraints on the incidence, direction, speed, and isotropy of bulk gas motions in blobs.
In this paper, in order to overcome this geometry effects, we present new X-shooter optical and near-infrared (NIR) spectroscopy of the six more [Ly$\alpha$]{}blobs at $z\approx2.3$. Note that the survey redshift of this [Ly$\alpha$]{}sample has been carefully selected to allow all important rest-frame optical diagnostic lines (e.g., \[\]$\lambda$3727, \[\]$\lambda$5007, H$\beta$$\lambda$4868, H$\alpha$$\lambda$6563) to fall in NIR windows and to avoid bright OH sky lines and atmospheric absorption [@Yang09; @Yang10]. With the resulting large sample of [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}measurements (a total of eight), we determine the relative frequency of gas infall versus outflow.
Benefiting from X-shooter’s high spectral resolution and wide spectral coverage (3000Å – 2.5), we constrain the gas kinematics in [Ly$\alpha$]{}blobs using three different tracers: (1) the offset of the [Ly$\alpha$]{}profile with respect to a non-resonant nebular line, (2) the offset of an interstellar metal absorption line in the rest-frame UV with respect to the nebular emission line, and (3) a new indicator, the profile of the spectrally-resolved [\[\]]{}emission line.
This paper is organized as follows. In Section \[sec:observation\], we review our sample selection and describe the X-shooter observations and data reduction. In Section \[sec:result\], we present the results from the X-shooter spectroscopy, confirming the [Ly$\alpha$]{}blobs’ redshift (Section \[sec:spec1d\]). We present 1–D and 2–D spectra in Section \[sec:spec1d\] and Section \[sec:spec2d\], respectively. We briefly summarize the properties of individual systems in Section \[sec:individual\]. In Section \[sec:kinematics\], we constrain the gas kinematics using the three different techniques noted above. In Section \[sec:shift\], we compare the [Ly$\alpha$]{}profiles with the H$\alpha$ or [\[\]]{}line centers to discriminate between simple infall and outflow scenarios and present the [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} statistics for the sample. In Section \[sec:absorption\], we describe the interstellar absorption lines detected in three galaxies. In Section \[sec:O3profile\], we inspect the [\[\]]{}emission line profiles in detail to look for possible signatures of warm outflows. In Section \[sec:column\_density\], we constrain the column density of a [Ly$\alpha$]{}blob by comparing its [Ly$\alpha$]{}profile with a simple radiative transfer (RT) model. In Section \[sec:cdfs-lab10\], we focus on a [Ly$\alpha$]{}blob with a blue-shifted [Ly$\alpha$]{}component not directly associated with any detected galaxy, a possible marker of gas inflow. Section \[sec:conclusion\] summarizes our conclusions. Throughout this paper, we adopt the cosmological parameters: $H_0$ = 70[kms$^{-1}$]{}Mpc$^{-1}$, $\Omega_{\rm M}=0.3$, and $\Omega_{\Lambda}=0.7$.
Observations and Data Reduction {#sec:observation}
===============================
Sample
------
We observe six [Ly$\alpha$]{}blobs from the @Yang10 sample. These targets were chosen such that they are not X-ray detected \[$L({\rm 2-32keV})$ $<$ (0.3 – 4.2)[$\times$$10^{43}$]{} [ergs$^{-1}$]{}; [@Lehmer05; @Luo08]\], and thus are not obvious AGN, as our primary goal is to cleanly detect gas infall or outflow. These six blobs lie in the Extended Chandra Deep Field South (ECDFS) and were discovered via deep narrowband imaging with the CTIO-4m MOSAIC–II camera and a custom narrowband filter ([*NB*]{}403). This filter has a central wavelength of $\lambda_c \approx 4030$Å, designed for selecting [Ly$\alpha$]{}-emitting sources at $z\approx2.3$. In Figure \[fig:image\], we show the images of these six [Ly$\alpha$]{}blobs (CDFS-LAB06, 07, 10, 11, 12, 13, 14) at various wavelengths ([*UBK*]{}, [Ly$\alpha$]{}, [*Spitzer*]{} IRAC 3.6and [*HST*]{} F606W; @Gawiser06a, @Yang10, @Damen11, @Rix04).
With X-shooter, we are targeting intermediate luminosity ([$L_{\rm Ly\alpha}$]{}$\sim$ 10$^{43}$ [ergs$^{-1}$]{}) [Ly$\alpha$]{}blobs: the higher [Ly$\alpha$]{}blob ID indicates the lower [Ly$\alpha$]{}luminosity in our sample. Our sample was obtained from a blind survey, so combined with the two brightest [Ly$\alpha$]{}blobs presented in @Yang11, the full sample (a total of eight) spans a wide and more representative range of [Ly$\alpha$]{}luminosity and size. Furthermore, the transition from compact [Ly$\alpha$]{}emitters (LAEs; isophotal area of a few arcsec$^2$) to extended [Ly$\alpha$]{}blobs ($>$10arcsec$^2$) is continuous [@Matsuda04; @Yang10], thus the gas kinematics for our faintest [Ly$\alpha$]{}blobs might share the properties with those of bright LAEs. We refer readers to @Yang10 for details of the sample selection and to @Yang11 for the first results of our spectroscopic campaign.
UV-to-NIR Spectroscopy
----------------------
We obtained high resolution optical–NIR (3000Å– 2.5) spectra of the six [Ly$\alpha$]{}blobs using X-shooter, a single object echelle spectrograph [@Vernet11], on the VLT UT2 telescope in service mode between 2010 November 6 and 2011 January 28. In Table \[tab:obslog\], we summarize the X-shooter observations. In Figure \[fig:image\], we show the location of the spectrograph slit on the sky, which was placed on UV-brightest galaxy or galaxies embedded at or near the [Ly$\alpha$]{}blob center. These galaxies or galaxy fragments also lie in the region of brightest [Ly$\alpha$]{}emission. Later, we assume that their redshifts mark the systemic redshift of the [Ly$\alpha$]{}blob.
X-shooter consists of three arms (UVB, VIS, NIR), which covers the spectral ranges of 3000Å–5500Å, 5500Å–1, and 1–2.5, respectively. This enormous spectral coverage allows us to obtain both [Ly$\alpha$]{}and [H$\alpha$]{}lines with a [*single*]{} exposure in contrast to our previous approach [@Yang11] involving both optical and NIR spectrograph. Furthermore, X-shooter can detect at least one of the nebular lines (\[\], [H$\beta$]{}, \[\], H$\alpha$), which will provide the systemic velocity of the embedded galaxies. Because at the redshift of our targets ($z \simeq 2.3$) most of the emission lines of interest ([Ly$\alpha$]{}, , , \[\], \[\], H$\alpha$) are located in the UVB or NIR arms, we focus only on the UVB ($\lambda_{\rm rest}$ = 900Å–1660Å) and NIR ($\lambda_{\rm rest}$ = 3020Å–7500Å) part of spectra in this paper.
The observations were carried out over eight nights and 14 observing blocks (OBs) of one hour duration each. The sky condition was either clear or photometric, and the guide camera seeing ranged from 06 to 12 with a median of 08 depending on the OB. We adopted 16 and 12-wide slits for UVB and NIR, yielding a spectral resolution of $R$ $\sim$ 3300 and $R\sim3900$, respectively. The slit length is rather small (12) compared to typical long-slits. In each OB, we placed the slit on a target using a blind offset from a nearby star. Using acquisition images taken after the blind-offset, we estimate that the telescope pointing and position angle of the slit are accurate within 0.2 and 0.5 on average, respectively. The individual exposure times were 680s and 240s for UVB and NIR, respectively, and the telescope was nodded along the slit by $\pm$ 2 while keeping the science targets always on the slit but at different detector positions. Total exposure times were 0.8 – 3.2 hours depending on the targets. In general, we were always able to detect both [Ly$\alpha$]{}and at least one of the optical nebular lines within one OB.
For accurate wavelength calibration, we took ThAr lamp frames through pinhole mask right before the science exposures and at the same telescope pointing to compensate for the effect of instrument flexure. For the UVB arm, we obtained another ThAr arc frame at the end of each OB in order to verify the wavelength solution where no bright sky lines are available. Telluric standard stars (B–type) were taken after or before the science targets with similar airmass to correct for atmospheric absorption in the NIR. Spectrophotometric standard stars were observed with 5-wide slits once during the night as a part of the observatory’s baseline calibration plan.
Data Reduction {#sec:reduction}
--------------
We reduce the data using the ESO X-shooter pipeline (version 1.3.7). In the UVB arm, the frames are overscan-corrected, bias-subtracted, and flat-fielded with halogen and deuterium lamps. The sky background is then subtracted in the “stare”-mode of the pipeline by modeling the sky in 2–D as described in @Kelson03. In the NIR arm, dark current and sky background are removed from each science frame by subtracting the dithered “sky” frame (“nodding”-mode of the pipeline). Then, we flat-field the data and correct for cosmic-ray hits and bad pixels. In both arms, these flat-fielded, sky-subtracted frames were corrected for the spatial distortions using multi-pinhole arc frames.
Because we will compare the velocity centers of the [Ly$\alpha$]{}and H$\alpha$ lines, we carefully verify the wavelength calibration. In the NIR, we compare the wavelength solutions obtained from the OH sky lines in the science frames to those from the daytime arc lamps and flexure-compensation frames, i.e., the pipeline solutions. In the UVB, we also compare the wavelength solutions obtained from the attached ThAr arc frames with the pipeline solutions. Our wavelength calibration is accurate within $\sim$4[kms$^{-1}$]{}in both arms. Furthermore, in the cases where we visited sources multiple times, all spectra agree each other. These frames are then rectified (resampled) and combined to create 2–D spectra. We collapse the 2–D spectra in the wavelength direction to measure the spatial extent of each emission line. Then we extract 1–D spectra from \[$-2\sigma$, $+2\sigma$\] apertures, where the $\sigma$ is the Gaussian width of the spatial profile. The aperture sizes are 2– 35 in the UVB and 15 – 25 in the NIR depending on the target. Finally, the 1–D spectra are corrected to heliocentric velocities and transformed to the vacuum wavelength.
Results {#sec:result}
=======
Systemic Redshift from [[\[\]]{}]{} and [[H$\alpha$]{}]{} {#sec:redshift}
---------------------------------------------------------
Various emission lines in the UVB and NIR arms confirm that [Ly$\alpha$]{}blobs lie at the survey redshift, $z\sim2.3$. In addition to [Ly$\alpha$]{}and [H$\alpha$]{}, we cover other UV emission lines and the non-resonant [\[\] $\lambda\lambda$3727,3729]{}, [\[\] $\lambda\lambda$4959,5007]{}, and [H$\beta$ $\lambda$4861]{}lines. In Figure \[fig:spec1d\], we show 1–D and 2–D spectra of the six [Ly$\alpha$]{}blobs. The first three columns show the rest-frame UV emission lines ([Ly$\alpha$]{}, , ) from the X-shooter UVB arm, and the remaining columns show the rest-frame optical nebular emission lines ([\[\]]{}, [H$\beta$]{}, [\[\]]{}, [H$\alpha$]{}) from the NIR arm.
Among the rest-frame optical nebular lines, the [\[\]]{} line is the brightest and detected with highest signal-to-noise (S/N) ratio in all cases, partly due to the low sky background and thermal instrument background in [*H*]{}-band. We determine the systemic redshift using [\[\]]{} doublets. In one case (CDFS-LAB14), the brighter [\[\]]{} line ($\lambda5007$) falls on top of an OH sky line, thus making it impossible to determine the line center. In this case we used the fainter [\[\]]{} line ($\lambda4959$). The vertical dashed lines in Figure \[fig:spec1d\] indicate the line centers determined by [\[\]]{} lines, which are then overlayed on other emission line profiles. As expected, we find that the line centers of all non-resonant emission lines agree well each other, to within $\sim$10[kms$^{-1}$]{}, showing that all of these lines are good indicators of systemic velocity. Thus, the brightest [\[\]]{} line can serve as best emission line to target for this survey redshift and instrument.
1–D Ly$\alpha$ Profiles {#sec:spec1d}
-----------------------
The Ly$\alpha$ profiles are significantly broad compared to the non-resonant lines ([H$\alpha$]{}and/or [\[\]]{}) in six cases out of the sample of eight [Ly$\alpha$]{}blobs, including the two from our previous work [@Yang11]. These integrated 1–D profile shapes range from an asymmetric single-peaked profile (CDFS-LAB06, 07, 10), to a double-peaked profile with a stronger red peak (CDFS-LAB02, 13), to a double-peaked profile with two similar intensity peaks (CDFS-LAB14). The [Ly$\alpha$]{}profiles of even this small sample show extremely diverse morphologies consistent with simple radiative model predictions [@Verhamme06; @Verhamme08; @Dijkstra06a] with varying geometry and outflow velocities [see also @Matsuda06; @Saito08; @Weijmans10].
The remaining two [Ly$\alpha$]{}profiles are narrower [*relative to*]{} the [\[\]]{}lines and show slightly extended wings (CDFS-LAB01 and 11). Note that the [Ly$\alpha$]{}line width of CDFS-LAB01 is one of the largest among our sample, but the blue side of its [Ly$\alpha$]{}profile agrees well with its [H$\alpha$]{}profile [@Yang11]. While there is an underlying broad component in CDFS-LAB11, the width of the dominant narrow component is small, comparable to that of the [\[\]]{}line (see §\[sec:type2\]). These are the two [Ly$\alpha$]{}blobs where and emission lines are also detected indicating that they contain a hard ionizing source, possibly an AGN. If photo-ionization by an AGN is indeed responsible for the and emission, and possibly the extended [Ly$\alpha$]{}emission as well, the discovery of narrow [Ly$\alpha$]{}profiles suggests that the [Ly$\alpha$]{}blob gas is highly-ionized, i.e., that the resonant scattering of [Ly$\alpha$]{}is not effective enough to alter the profile significantly. We will further investigate the details of these , emission lines and the implications for AGN in a future paper (Y.Yang in preparation).
The fraction of double-peaked profiles is significant: $\sim$38% (3/8), which is roughly consistent with the findings for LBGs and LAEs at $z$ = 2–3. Among LBGs, @Kulas12 find that $\sim$30% of LBGs with [Ly$\alpha$]{}emission show multiple-peaked profiles. @Yamada12 also find that $\sim$50% of LAE’s profiles have multiple peaks.
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2–D Ly$\alpha$ and [\[\]]{} Profiles {#sec:spec2d}
------------------------------------
Using 2–D spectra, we detect the extended [Ly$\alpha$]{}lines and identify the exact locations from which the [Ly$\alpha$]{}or [\[\]]{}line originates. In Figure \[fig:spec2d\], we show close-ups of the 2–D [\[\]]{}and [Ly$\alpha$]{}profiles in the first and second columns, respectively. To aid the comparison between the [\[\]]{}and [Ly$\alpha$]{}profiles, we overlay the rough boundary of the [\[\]]{}spectrum on each [Ly$\alpha$]{}panel with ellipses. The third column shows the spatial profiles along the slit, i.e., collapsed in the wavelength direction. The last column shows the 1–D [Ly$\alpha$]{}and [\[\]]{}profiles that are extracted from the different parts of the slit, as indicated with vertical arrows in the third column.
In four cases (CDFS-LAB06, 10, 13, 14), the Ly$\alpha$ spectrum is spatially extended relative to the [\[\]]{}line, confirming the narrowband imaging result that the [Ly$\alpha$]{}-emitting gas extends beyond the embedded galaxies that are probably responsible for the [\[\]]{}emission. Note that the X-shooter spectroscopy reaches much shallower [Ly$\alpha$]{}surface brightness limit ($\sim$1.5–3[$\times$$10^{-17}$]{}[ergs$^{-1}$cm$^{-2}$arcsec$^{-2}$]{}) than the narrowband imaging ($\sim$5[$\times$$10^{-18}$]{}[ergs$^{-1}$cm$^{-2}$arcsec$^{-2}$]{}; 3[$\sigma$]{}limit).
In two [Ly$\alpha$]{}blobs (CDFS-LAB06 and 10), the [\[\]]{}lines are spatially resolved. However, there is no evidence that the [\[\]]{}or [H$\alpha$]{}lines are extended beyond the UV continuum emission arising from stars in the embedded galaxies seen in Figure 2. Deeper NIR spectroscopic, preferentially IFU, observations are required to better define the spatial extent of these lines and to measure the spatially-resolved gas kinematics. Throughout the paper, we assume that the [\[\]]{}and [H$\alpha$]{}lines originate from the central embedded galaxies, not from the extended [Ly$\alpha$]{}-emitting gas, and thus that their line centers represent the systemic velocity of the [Ly$\alpha$]{}blob. In the next section, we briefly describe individual systems in detail.
Notes for Individual Objects {#sec:individual}
----------------------------
### CDFS-LAB06
CDFS-LAB06 has two rest-frame UV sources with small separation (08; 6.5kpc) in the [*HST*]{} image (Fig. \[fig:image\]). Both components (or clumps) were placed in the X-shooter slit and are detected in [\[\]]{}and [Ly$\alpha$]{}(Fig. \[fig:spec2d\]). These two sources are separated by only $\sim$50[kms$^{-1}$]{}in velocity space, thus it is not clear whether they belong to one galaxy or are interacting with each other. We adopt the redshift of the brighter UV source as the systemic velocity of CDFS-LAB06. The [Ly$\alpha$]{}emission detected in the spectrum (between $\Delta\theta$ = $-1$ and $+2$) is spatially extended due to the other galaxy, so it does not represent the IGM or circum-galactic medium (CGM).
### CDFS-LAB07
The galaxy within CDFS-LAB07 has a bar-like morphology in the [*HST*]{} image with which we align the slit. This galaxy or galaxy fragments were marginally resolved in the [\[\]]{}emission line. All optical nebular emission lines ([\[\]]{}, [H$\beta$]{}, [\[\]]{}, and [H$\alpha$]{}) are detected. Faint UV continuum emission is marginally detected, allowing us to study the gas kinematics with metal absorption lines (§\[sec:absorption\]). The [\[\]]{}and [H$\alpha$]{}lines show asymmetric profiles extending toward the blue. This profile can be fitted with a narrow Gaussian component at the velocity center superposed on the blueshifted broad component (§\[sec:O3profile\]).
### CDFS-LAB10
CDFS-LAB10 is the most puzzling and complex source in the X-shooter sample. The [Ly$\alpha$]{}emission in the narrow-band image is elongated over 10 ($\sim$80kpc). In the NIR spectrum, three sources are detected: two with [\[\]]{}emission lines (galaxies B and C), and the other (galaxy A) with very faint continuum (Fig. \[fig:spec2d\]). This NIR continuum source (galaxy A) is located at the slit center, while the strongest [\[\]]{}-emitting source (galaxy C) is offset by $\sim$1toward north-east from the center and barely detected in the [*HST*]{} image. The 2–D [\[\]]{}spectrum of galaxy C shows a velocity shear indicative of rotating disk. In the 2–D [Ly$\alpha$]{}spectrum, there are also three distinct components. Although the 1–D [Ly$\alpha$]{}spectrum of the entire blob looks single-peaked with a broad linewidth, it is in fact composed of these three components. We will investigate these various emission line and continuum sources in §\[sec:cdfs-lab10\].
### CDFS-LAB11
There are one compact UV source at the slit center and a diffuse emission toward northeast in the [*HST*]{} image. It appears that the [\[\]]{}emission originates from the central compact source. Unlike commonly observed broad [Ly$\alpha$]{}profiles found in [Ly$\alpha$]{}blobs and emitters, CDFS-LAB11 has a peculiar [Ly$\alpha$]{}profile that is almost symmetric and narrow. As will be discussed in §\[sec:type2\], both narrow and underlying broad components are required to explain the [Ly$\alpha$]{}profile. The [Ly$\alpha$]{}is redshifted against [H$\alpha$]{}by small amount: [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}= 84$\pm$6[kms$^{-1}$]{}. Both [\[\]]{}and [Ly$\alpha$]{}are spatially compact (Fig. \[fig:spec2d\]). Both [ $\lambda$1546]{}and [ $\lambda$1640]{} emission lines are also detected (Fig. \[fig:spec1d\]), implying the presence of hard ionizing source. A total of two [Ly$\alpha$]{}blobs from our ECDFS sample (CDFS-LAB01 and 11; 2/8) show these emission lines. Note that and [*narrow*]{} line emission are commonly detected in bright [Ly$\alpha$]{}blobs [@Dey05; @Scarlata09; @Prescott09; @Yang11], while the nature of the hard-ionizing source is unknown.
### CDFS-LAB13
CDFS-LAB13 has a double-peaked profile with a stronger red peak, and asymmetric [\[\]]{}profile (§\[sec:O3profile\]). [Ly$\alpha$]{}in CDFS-LAB13 is likely more extended, but the low S/N of the [\[\]]{}line makes the comparison difficult. There are multiple galaxy fragments in the [*HST*]{} UV continuum image, but they were not spatially resolved in the X-shooter observations.
### CDFS-LAB14
In the [*HST*]{} and narrowband images, a UV source is located at the upper boundary of the [Ly$\alpha$]{}emission contours. In the X-shooter 2–D spectra (Figure \[fig:spec2d\]), the [\[\]]{}line is centered on this UV continuum source and [Ly$\alpha$]{}is more extended toward the south in agreement with narrowband imaging. CDFS-LAB14 is one of two cases where extended [Ly$\alpha$]{}emission is well detected in the spectroscopy. The [Ly$\alpha$]{}profile has two peaks with similar intensities. The peak separation narrows as the slit distance from the central galaxy increases. Faint UV continuum is also detected at the location of the UV source allowing us to measure outflow speed from metal absorption lines (§\[sec:absorption\]). Combining the [Ly$\alpha$]{}and the absorption profiles, we will put constraints on the gas kinematics and the neutral column density of this system (§\[sec:column\_density\]). [\[\]]{}has a broad wing on top of the narrow component (§\[sec:O3profile\])
Gas Kinematics {#sec:kinematics}
--------------
In this section, we constrain the gas kinematics in the [Ly$\alpha$]{}blobs using three different techniques and compare those results. First, for the six blobs in the X-shooter survey, we compare the optically thick [Ly$\alpha$]{}and non-resonant [\[\]]{}line (either $\lambda5007$ or $\lambda4959$ in the case of CDFS-LAB14) to measure [Ly$\alpha$]{}velocity offsets from the systemic velocity of the [Ly$\alpha$]{}blobs. For the two [Ly$\alpha$]{}blobs from our previous work (CDFS-LAB01A and 02; [@Yang11]), where an [\[\]]{}line is unavailable, we use [H$\alpha$]{}. As mentioned previously, the line centers of both [\[\]]{}lines and [H$\alpha$]{}are all consistent. Second, we constrain the outflow speed from the interstellar metal absorption lines in three galaxies where we are able to detect the rest-frame UV continuum in the spectrum. Lastly, we present a new tracer of kinematics in four [Ly$\alpha$]{}blobs: characterizing the breadth and asymmetry of the [\[\]]{}line profile. Note that while we detect an asymmetric wing in some [\[\]]{}line profiles, it does not affect the line centroid, which is essential in determining the systemic velocity in the first technique above.
### [Ly$\alpha$]{} – [[\[\]]{}]{} offset {#sec:shift}
We compare the peak of each [Ly$\alpha$]{}profile with the center of a non-resonant nebular emission line, particularly [\[\]]{}. Throughout the paper, the [Ly$\alpha$]{}–[H$\alpha$]{}and [Ly$\alpha$]{}–[\[\]]{}offsets are used interchangeably to represent the [Ly$\alpha$]{}offset from the systemic velocity: [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}. Figure \[fig:line\_comparison\] shows eight [Ly$\alpha$]{}profiles from this work and from @Yang11, plotted with increasing [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}.
Because [Ly$\alpha$]{}spectra are somewhat noisy because of the small spectral dispersion (20–30[kms$^{-1}$]{} per pixel), we measure [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} using the following two methods. First, we measure the velocity of the [Ly$\alpha$]{}peak after smoothing the spectra with a Gaussian filter with FWHM = 90[kms$^{-1}$]{}, corresponding to our velocity resolution. Second, we fit the red peaks with an asymmetric Gaussian function, which consists of two Gaussian functions with different FWHMs being joined at the center. The two measurements agree to within $\sim$50[kms$^{-1}$]{}, except for CDFS-LAB06 where the second method gives [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} = 120[kms$^{-1}$]{}compared to 320[kms$^{-1}$]{}from simple smoothing due to the very sharp edge at the blue side of [Ly$\alpha$]{}profile. We adopt the second measurements in this paper and show these fits in Figure \[fig:line\_comparison\] and Table \[tab:Lyman\]. None of the conclusions in this paper are affected by this choice.
Figure \[fig:voffset\] shows the distribution of [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} of the eight [Ly$\alpha$]{}blob galaxies in comparison with 41 LBGs that have both H$\alpha$ and [Ly$\alpha$]{}spectra [@Steidel10]. Because the spectral resolutions of our [Ly$\alpha$]{}spectra obtained from VLT/X-shooter and Magellan/MagE are higher than those of the [Ly$\alpha$]{}profiles of the LBGs (FWHM $\simeq$ 370[kms$^{-1}$]{}), we test if the different spectral resolutions affect the [Ly$\alpha$]{}blob–LBG comparison. We repeat the measurement of [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} after convolving our [Ly$\alpha$]{}profiles to the spectral resolution of LBG sample. The [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} values change by a negligible amount (only $\pm$50[kms$^{-1}$]{}) except for CDFS-LAB02, where [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} increases by +100[kms$^{-1}$]{}because of its sharp red peak. Therefore, we conclude that the [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} distributions of [Ly$\alpha$]{}blobs and LBGs can be compared directly.
The [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} distribution for galaxies within our [Ly$\alpha$]{}blobs reveals smaller velocity offsets than typical of LBGs, confirming previous claims [@Yang11 see also McLinden et al. 2013]. Galaxies within blobs have [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}= $-$60 $\rightarrow$ +400[kms$^{-1}$]{} with an average of $\langle{\ifmmode\Delta v_{\rm Ly\alpha}\else$\Delta v_{\rm Ly\alpha}$\xspace\fi}\rangle$ = 160[kms$^{-1}$]{}, while LBGs at similar redshifts have [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}= 250 – 900[kms$^{-1}$]{} with $\langle{\ifmmode\Delta v_{\rm Ly\alpha}\else$\Delta v_{\rm Ly\alpha}$\xspace\fi}\rangle$ = 445[kms$^{-1}$]{}.
The remaining question is how to interpret these small [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} values. Two possibilities are: (1) [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} is a proxy for the gas outflow velocity ([$v_{\rm exp}$]{}), suggesting that the outflows here are weaker than in other star-forming galaxies at $z$ = 2–3 [@Verhamme06], or (2) there is less neutral gas close to the systemic velocity of the embedded galaxies [@Steidel10], and [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} is independent of the outflow speed. We discussed the caveats associated with interpreting [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} in @Yang11 and revisit this issue in §\[sec:dvlya\].
In the direction of the embedded galaxies, we do not find any [Ly$\alpha$]{}profile that is blue-peak dominated, i.e., [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}$<$ 0[kms$^{-1}$]{}. Therefore, there is no evidence along these lines-of-sight for infalling gas (but see §\[sec:cdfs-lab10\]). While the statistics are still small, we place an upper limit on the covering factor of infalling gas (if any) detectable with the [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} technique. Here the covering factor of any inflowing streams like those predicted by cold-mode accretion [@Keres05; @Keres09; @Dekel09] must be less than $\sim$13% (1/8). We further discuss the covering factor of cold streams in §\[sec:covering\_factor\].
### Interstellar Metal Absorption Lines {#sec:absorption}
{height="28.00000%"} {height="28.00000%"} {height="28.00000%"}
While strong emission lines such as [Ly$\alpha$]{}and [\[\]]{} are relatively easy to detect, interstellar metal absorption lines provide a less ambiguous way to measure the outflow velocity of neutral gas lying [*in front of*]{} the galaxies targeted with our spectroscopic slit. However, due to the faint UV continuum ($B$ = 23.8–26.5 mag) of our targets, even 8-m telescope struggles to detect their continuum for absorption line studies. Therefore, we obtain higher-S/N absorption line profiles by stacking several lines in each galaxy. Our aim is to test the mild outflow interpretation of our [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} results by comparing the velocity offset of the stacked ISM absorption profile to the systemic velocity of the galaxy: [$\Delta v_{\rm IS}$]{}.
Among the six [Ly$\alpha$]{}blobs, only the UV continua of the galaxies in CDFS-LAB07 and CDFS-LAB14 have S/N higher than 1.5 per spectral pixel ($\sim$20–30[kms$^{-1}$]{}), allowing us to marginally extract the absorption profiles. Figure \[fig:spec\_abs\](top) shows the rest-frame UV spectra of these two galaxies in comparison with the LBG composite [@Shapley03]. For comparison, we also show the boxcar-smoothed spectra over 20 pixels, corresponding to a velocity width of 450–600[kms$^{-1}$]{}. While it is difficult to identify the absorption lines in the un-binned spectra, the smoothed spectra show a good match to the composite spectrum, revealing several low- and high-ionization lines.
As suggested above, it is still difficult to measure individual absorption line profiles, so we stack the five low-ionization lines: $\lambda$1260, $\lambda$1302, $\lambda$1334, $\lambda$1526, $\lambda$1670 to increase the S/N. We do not include the and lines in the stacking, as these high ionization lines are contaminated with broader absorption features arising from stellar winds from massive stars [@Shapley03], which are hard to remove in our low S/N spectra. Furthermore, it is possible that these high ionization lines trace different state of gas, while the low ionization lines show similar profiles [@Steidel10]. In Figure \[fig:spec\_abs\](bottom), we show the stacked absorption profiles for three galaxies, including the re-analyzed CDFS-LAB02 spectrum that we obtained earlier with Magellan/MagE [@Yang11].
The stacked absorption profiles here are consistent with weaker outflows than required by the super/hyperwind hypothesis for [Ly$\alpha$]{} blob emission . These profiles are also consistent with the interpretation of small [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} indicating small outflow speed, as discussed in §\[sec:shift\]. In two galaxies (within CDFS-LAB02 and 07), the absorption profiles have minima around $-$200[kms$^{-1}$]{}and might extend up to $-$500[kms$^{-1}$]{}, although the exact end of the profile is uncertain due to the low S/N. The shapes of these two absorption profiles are similar to those of typical star-forming galaxies (i.e., LBGs) at the same epoch [@Steidel10], and the implied outflow velocities are comparable to or slower than in the LBGs. By fitting a Gaussian profile, we obtain [$\Delta v_{\rm IS}$]{}= $-$177$\pm$31[kms$^{-1}$]{}and $-$236$\pm$31[kms$^{-1}$]{}for CDFS-LAB02 and CDFS-LAB07, respectively. We do not find any redshifted absorption component, which is consistent with the absence of any blueshifted [Ly$\alpha$]{}emission line associated with these galaxies.
In contrast, the profile of CDFS-LAB14’s galaxy has a minimum at $v$ $\simeq$ 0[kms$^{-1}$]{} ([$\Delta v_{\rm IS}$]{}= $-$59$\pm$32 [kms$^{-1}$]{}) without any significant blueshifted (outflowing) component. While the other two profiles terminate roughly at $v$ $\simeq$ 0[kms$^{-1}$]{}, CDFS-LAB14’s profile extends to $v$ $>$ 0[kms$^{-1}$]{}. Thus, its ISM absorption and [Ly$\alpha$]{} profiles are both consistent with the simple RT model expectation that a symmetric, double-peaked [Ly$\alpha$]{}profile should emerge from static gas or the absence of of bulk motions. Furthermore, the absorption profile is almost saturated, indicating that the column density in this [Ly$\alpha$]{}blob could be higher than for the other two systems. We will place a constraint on the column density of its [Ly$\alpha$]{}-emitting gas in §\[sec:column\_density\].
For CDFS-LAB14, the location of the dip between the blue and red [Ly$\alpha$]{}velocity peaks changes little for the two extraction apertures shown in Figure \[fig:spec2d\]: one along the line of sight toward the embedded galaxy (the [\[\]]{}source) and the other encompassing the spatially extended gas around $\Delta \theta$ = $-$0.5. Note that such a trough between [Ly$\alpha$]{}peaks is often interpreted as arising from absorption by the neutral media between the [Ly$\alpha$]{}source and observers. For example, @Wilman05 claim that their IFU spectra of a [Ly$\alpha$]{}blob (SSA22-LAB02; Steidel blob 2) suggest that the [Ly$\alpha$]{}emission is absorbed by a foreground slab of neutral gas swept out by a galactic scale outflow. More recently, @Martin14b show instead that the absorption troughs in the [Ly$\alpha$]{}emission are actually located at the systemic velocity determined by the [\[\]]{}emission line, i.e., there are negligible velocity offsets between any foreground screen and the systemic velocity. CDFS-LAB14 also demonstrates that a coherent velocity trough, at least over a $\sim$10kpc scale, can arise entirely from complicated radiative transfer effects even if there are no significant bulk motions in the [Ly$\alpha$]{}-emitting gas.
### Broadening and Asymmetry in [[\[\]]{}]{} Profile {#sec:O3profile}
The central galaxies in four [Ly$\alpha$]{}blobs (CDFS-LAB06, 07, 13, 14) show a broad and/or shifted underlying component to the [\[\]]{}profile, which provides additional constraints on the kinematics of warm ionized gas in the vicinity of the galaxies. To the authors’ knowledge, this is the first detection of broadened, asymmetric [\[\]]{}profiles from narrowband-selected [Ly$\alpha$]{}-emitting galaxies at high redshifts, demonstrating that high spectral resolution is required to fully exploit the NIR spectroscopy of [Ly$\alpha$]{}galaxies.
We emphasize again that these broad underlying components, even when shifted in velocity, do not affect our measurements of [\[\]]{}line centers, because the [\[\]]{}flux density at the core is dominated by the narrower component, which is likely to arise from nebular regions in the embedded galaxy. As discussed previously, the line centers determined from other non-resonant emission lines (e.g., [H$\alpha$]{}and [H$\beta$]{}) agree to within $\sim$10[kms$^{-1}$]{}. Therefore, our measurements of systemic velocity, critical for determining [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} and [$\Delta v_{\rm IS}$]{}, remain unchanged.
In Figure \[fig:O3profile\], we show close-ups of the [\[\]]{}profiles. In CDFS-LAB11, which has the narrowest and the most symmetric [Ly$\alpha$]{}profile, the [\[\]]{}line is also symmetric. CDFS-LAB10 is excluded from the following analysis because its [\[\]]{}line is elongated in position-velocity space, suggesting some velocity shear, and the neighboring galaxy (CDFS-LAB10A) makes it difficult to reliably extract its profile (see §\[sec:cdfs-lab10\] for more details of this system). In the remaining four [Ly$\alpha$]{}blobs, the [\[\]]{}profile either has an asymmetric wing (CDFS-LAB06, 07, 13) or a symmetric broad component or components (CDFS-LAB14). Note that the CDFS-LAB06 profile in Figure \[fig:O3profile\] also includes the light from a faint neighbor or clump to the northwest that is slightly blueshifted ($\sim$50[kms$^{-1}$]{}). Therefore, the somewhat redshifted wing of CDFS-LAB06 is not due to this contamination.
To extract the underlying broad components, we fit the line profiles with two Gaussian functions (the two dot-dashed lines in Figure \[fig:O3profile\]) and list the results in Table \[tab:O3profile\]. In the Appendix, we describe our fitting procedures in detail. We adopt a two-component fit for simplicity and for comparison with previous studies (although we cannot rule out the possibility that the velocity wings consist of multiple small narrow components). The line center of the broad component ($v_{\rm broad}$) agree with that of the narrow component in CDFS-LAB14, is blueshifted in CDFS-LAB07, and in CDFS-LAB13, and is marginally redshifted in CDFS-LAB06. The widths of the broad components, corrected for the instrumental resolution, are relatively small: $\sigma_{\rm broad}$ = 45 – 120[kms$^{-1}$]{}(FWHM = 100 – 280[kms$^{-1}$]{}), which would not be detected with lower resolution or lower S/N spectra.
To quantify the contribution of the broad components, we measure the “broad-to-narrow ratio” ($F_{\rm broad}/F_{\rm narrow}$), which is defined as the ratio of the flux in the broad [\[\]]{}emission line to the flux in the narrow component. Because the fluxes in the two Gaussian components are anti-correlated with each other, this ratio has large uncertainties. We also list the “broad flux fraction” ($f_{\rm broad}$), which is the ratio of the flux in broad emission to the total [\[\]]{}flux ($F_{\rm broad}$/$F_{\rm total}$). The broad emission component is significant with $F_{\rm broad}/F_{\rm narrow}$ = 0.4 – 0.8 constituting 30%–45% of the total [\[\]]{}line flux (although the uncertainties are fairly large). The measurements of the underlying components are highly dependent on the S/N of spectra. Due to the low S/N in the $K$-band, we are not able to reliably measure the broad line components from permitted lines such as [H$\alpha$]{}or [H$\beta$]{}, but similar broad wings are also present in the [H$\alpha$]{}profiles of at least two galaxies (CDFS-LAB07 and 14; see Figure \[fig:spec1d\]).
What is the mechanism responsible for the broad component in the [\[\]]{}profiles? A broad component with a larger line-width ($\sigma_v$) of a few hundred [kms$^{-1}$]{}is generally interpreted as a signature of starburst-driven galactic winds, and often observed in local dwarf starbursts [e.g., @Westmoquette07] and in local ultraluminous infrared galaxies (ULIRGs) [e.g., @Soto12]. While broad components in forbidden lines such as [\[\]]{}are not related to the broad line region (BLR) of Type 1 AGN, they might still arise from the shocked narrow line region (NLR) in AGN. However, the composite spectrum of all of our X-shooter sample has a line ratio [$\log$(\[\]/H$\alpha$)]{}$<$ $-0.88$, excluding any significant contribution from AGN (Y. Yang et al. in preparation). Furthermore, in CDFS-LAB11, whose and emission lines hint at the presence of an AGN, no broad component is detected.
At high redshift ($z\gtrsim2$), broadened emission lines were first reported by @Shapiro09. They found that the stacked spectrum of $z\sim2$ star-forming galaxies (SFGs) shows broad (FWHM $\sim$ 550[kms$^{-1}$]{}) emission underneath the H$\alpha$+[\[\]]{}line complex. More recently, stacked spectra of higher S/N data have revealed that the broad emission is spatially extended over a half-light radius [@Newman12b]. Broad emission lines are now detected from individual SFGs and even from giant star-forming clumps within them [@Genzel11]. @Genzel11 show that $z\sim2$ SFGs show broad wings of [H$\alpha$]{}emission with FWHM $\sim$ 300 – 1000[kms$^{-1}$]{}($\sigma_{\rm broad}$ $\sim$ 125 – 425) and employ the maximum blueshifted velocity, [$\Delta v_{\rm max}$]{}= $| \langle v_{\rm broad} \rangle$ $-$ $2\,\sigma_{\rm broad} |$ as a proxy of the outflow speed, finding that SFGs have [$\Delta v_{\rm max}$]{}= 380 – 1000[kms$^{-1}$]{}. Thus, the broad component in these galaxies is attributed to powerful galactic outflows. For compact [Ly$\alpha$]{}emitters, although there are increasing number of detections of [\[\]]{}and [H$\alpha$]{}[e.g., @McLinden11; @Finkelstein11; @Nakajima12; @Hashimoto13], broad emission in [\[\]]{}has not been reported so far, perhaps due to the lower spectral resolution or lower S/N of these studies. Therefore, at the moment, it is not clear whether the broad component and sometime line asymmetry that we observe in [Ly$\alpha$]{}blobs is a general property of [Ly$\alpha$]{}-selected galaxies.
As in local ULIRGs and in high redshift SFGs, the detection of a broad component in the [\[\]]{}profile here suggests warm ionized outflows from the galaxies within [Ly$\alpha$]{}blobs, presumably driven by supernovae and stellar winds. However, while the flux fraction of our broad emission is comparable to those of SFGs, the broad [\[\]]{}wings are narrower ($\sigma_{\rm broad}$ = 45 – 120[kms$^{-1}$]{}) and the inferred velocities much smaller [$\Delta v_{\rm max}$]{}= 150 – 260[kms$^{-1}$]{}. Therefore, at face value, the warm ionized outflows from the [Ly$\alpha$]{}blob galaxies are not as strong as those in SFGs at similar redshift. Furthermore, these estimates for outflow velocity are roughly consistent with the values obtained from our two other kinematic measures, [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}and [$\Delta v_{\rm IS}$]{}(§\[sec:shift\] and §\[sec:absorption\]), independently discounting shock-heating via super/hyperwinds as a viable powering mechanism.
Constraint on Column Density {#sec:column_density}
----------------------------
Constraining the physical state of the [Ly$\alpha$]{}-emitting gas in a [Ly$\alpha$]{}blob is a critical step to understand its emission mechanism and to directly compare the observations with the numerical simulations , which predict the [Ly$\alpha$]{}emissivity maps from the gas density, temperature, and UV radiation fields. In this section, as a first step, we estimate the column density of a [Ly$\alpha$]{}blob, CDFS-LAB14.
The [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}and [$\Delta v_{\rm IS}$]{}analyses of CDFS-LAB14 are consistent with a simple RT model in which the surrounding gas is static or without bulk motions. The third kinematic indicator, the [\[\]]{}profile, which suggests a mild outflow, either contradicts the other indicators or is sensitive to gas in a different state (e.g., warm, ionized instead of cold, neutral) or distribution (e.g., around galaxy instead of in galaxy). Here we use the [Ly$\alpha$]{}emission and ISM absorption line profiles to place a constraint on the amount of neutral hydrogen in this [Ly$\alpha$]{}blob. Ultimately, by comparing the [Ly$\alpha$]{}profile with detailed RT models, one could extract a wealth of information, including outflow speed, optical depth, and column density [e.g., @Verhamme08]. We defer such detailed analysis to the future papers and consider the most simplistic case in this section.
[Ly$\alpha$]{}line transfer in an extremely thick medium of neutral gas has been studied over many decades, and the analytic solutions for simple geometries (like a static homogeneous slab or uniform sphere) are known [@Harrington73; @Neufeld90; @Dijkstra06a]. We consider a simple geometry in which a [Ly$\alpha$]{}source is located at the center of a static homogeneous slab with optical depth of $\tau_0$ from the center to the edge. Because [Ly$\alpha$]{}photons generated at the center escape the system through random scattering in the frequency space, the emergent line spectrum is double-peaked profile with its maxima at $\Delta x_{\rm peak} \equiv (\nu-\nu_0)/\nu_D = \pm 1.173 (a \tau_0)^{1/3}$, where $x$ represents the line frequency in units of Doppler width $\nu_D$. $a$ and $\tau_0$ are the Voigt parameter and [Ly$\alpha$]{}optical depth at the line center, respectively [@Dijkstra06a][^1]. In terms of velocity, the separation between blue and red peaks with same the intensity is $$\Delta v_{\rm blue-red} = 424{\ifmmode{\rm \,km\,s^{-1}}\else\,km\,s$^{-1}$\xspace\fi}\cdot \left(\frac{b}{12.85{\ifmmode{\rm \,km\,s^{-1}}\else\,km\,s$^{-1}$\xspace\fi}}
\frac{N_{\rm HI}}{10^{20}{\rm cm^{-2}}}\right)^{1/3},$$ where $b$ and $N_{\rm HI}$ represent the Doppler parameter ($\sqrt{v^2_{\rm th} + v^2_{\rm turb}}$) and the column density of neutral hydrogen, respectively. If we adopt a uniform sphere geometry instead of a slab, the coefficient of the above equation will decrease to 336[kms$^{-1}$]{}, and the product ($b$ $N_{\rm HI}$) will increase by a factor of two for a fixed $\Delta v_{\rm blue-red}$ value. Note that the blue-to-red peak separation is degenerate between two parameters, $b$ and $N_{\rm HI}$.
From the [Ly$\alpha$]{}profile of CDFS-LAB14 (Figure \[fig:spec1d\] and \[fig:spec2d\]), we measure the separation between the blue and red peaks, $\Delta v_{\rm blue-red}$ = 790$\pm$39[kms$^{-1}$]{}. In the analytic solution, the peaks on the blue and red sides of the systemic velocity are each symmetric, whereas for CDFS-LAB14 they are slightly asymmetric. As a result, the RT in this system may require ultimately a more complicated model than the assumed simple geometry.
To constrain $N_{\rm HI}$, we consider two extreme cases for the Doppler parameter $b$: First, as a lower limit, $b$ $>$ 12.85[kms$^{-1}$]{}, obtained by assuming a temperature $T$ = 10$^4$K and ignoring the turbulence term. Second, $b$ $<$ $\sigma_{\rm abs}$, the width of the metal absorption lines. In other words, we assume that the observed velocity width of absorbing material is purely due to the turbulence or random motions inside the slab. The width of the absorption line is $\sigma_{\rm abs}$ = 152 $\pm$ 35[kms$^{-1}$]{}from a single Gaussian fit (§\[sec:absorption\] and Figure \[fig:spec\_abs\]) and after being corrected for the instrumental line width. For this range of $b$, we obtain 19.7 $<$ $\log N_{\rm HI}$ $<$ 20.8.
Because the [Ly$\alpha$]{}emission is spatially resolved in CDFS-LAB14, we further apply this technique to the extended [Ly$\alpha$]{}-emitting gas in a direction other than toward the embedded galaxy. As shown in Figure \[fig:spec2d\], the [Ly$\alpha$]{}profile remains double-peaked as we move away from the sight-line directly toward the embedded galaxy. The separation between the blue and red peaks decreases, indicating that $b$ $N_{\rm
HI}$ also decreases. For $\Delta v_{\rm blue-red}$ $\simeq$ 650 [kms$^{-1}$]{}, we obtain slightly smaller estimates for the column density: 19.5 $<$ $\log N_{\rm HI}$ $<$ 20.6.
Using the accurate measurement of systemic velocity, the metal absorption lines, and the symmetric double-peaked [Ly$\alpha$]{}profile, we are able to place constraints on the $N_{\rm HI}$ toward the galaxy and the extended [Ly$\alpha$]{}-emitting gas, albeit with large uncertainties. It is intriguing that this rough estimate of the column density is similar to those of damped [Ly$\alpha$]{}absorption systems (DLA; $N_{\rm HI}$ $>$ 2[$\times$$10^{20}$]{}cm$^{-2}$). While we have identified this [Ly$\alpha$]{}blob by searching for extended [Ly$\alpha$]{}emission, it would be interpreted as a DLA if there were a background QSO whose continuum spectrum showed absorption at the blob redshift. There are similar systems where extended [Ly$\alpha$]{}emission is identified from DLAs very close to the QSO redshift [e.g., @Moller98; @Fynbo99; @Hennawi09].
Blueshifted [Ly$\alpha$]{}without Broadband Counterpart {#sec:cdfs-lab10}
-------------------------------------------------------
\
Up to this point, we have examined only [Ly$\alpha$]{}emission along the line-of-sight to galaxies embedded in our blobs. In the 2–D spectrum of CDFS-LAB10, however, we identify a region of blueshifted [Ly$\alpha$]{}emission that does not spatially coincide with any of the blob galaxies identified with [*HST*]{}. This is the first case in our sample where we detect the [Ly$\alpha$]{}emission from the extended gas itself. Given that a blueshifted [Ly$\alpha$]{}line is a long-sought signature of gas inflows as described in §\[sec:intro\] and Figure \[fig:cartoon\], we present a detailed analysis of CDFS-LAB10 here.
In Figure \[fig:slitprof\](bottom), we show the [*HST*]{} F606W image of CDFS-LAB10 now rotated and smoothed with a Gaussian kernel to increase the contrast of faint sources. There are three broadband sources labeled CDFS-LAB10A, B, and C within the [Ly$\alpha$]{}contour, which is elongated over 10 ($\sim$82kpc). Figure \[fig:slitprof\](top) shows the spatial profiles of the stellar continuum and the emission lines ([Ly$\alpha$]{}and [\[\]]{}). The latter are the same as in Figure \[fig:spec2d\]. To obtain the spatial profiles for the continuum, we collapse the 2–D spectra for the rest-frame wavelength range, \[1260Å, 1600Å\] and \[4340Å, 5400Å\], i.e., redward of [Ly$\alpha$]{}and both sides of the [\[\]]{}lines, respectively. Because of different data reduction modes, the spatial profiles from the NIR arms ([\[\]]{}) cover only the central $\sim$6, while the profiles from the UVB arm ([Ly$\alpha$]{}) cover $\sim$14.
The brightest UV source in the [*HST*]{} image, CDFS-LAB10A at the slit center ($\Delta \theta$ = 0), is detected in both the rest-frame UV and optical continua, but not in [Ly$\alpha$]{}or any strong emission lines ([\[\]]{}, [\[\]]{}, [H$\beta$]{}, [H$\alpha$]{}). It is still possible that there is a faint [\[\] $\lambda$5007]{}line under a sky line at $\Delta v$ $\sim$ $-550$[kms$^{-1}$]{}(see Figure \[fig:spec2d\]). Therefore, it is not clear whether galaxy A is at the same redshift as the [Ly$\alpha$]{}-emitting gas or is a fore- or background galaxy. The galaxy B at $\Delta \theta$ $\sim$$-$3 is detected in [Ly$\alpha$]{}, [\[\]]{}, and UV continuum and is thus a member of the CDFS-LAB10 blob. Galaxy B’s [Ly$\alpha$]{}emission is slightly offset from its UV continuum. CDFS-LAB10C, which has a filamentary or elongated morphology in the [*HST*]{} image, is the faintest ($m_{\rm F606W}$ = $27.9\pm0.2$) of the three sources, but the brightest in [Ly$\alpha$]{}and [\[\]]{}. Thus, galaxy C is the dominant source of [Ly$\alpha$]{}emission from the known galaxies in this [Ly$\alpha$]{}blob. It appears to be a classic example of an object heavily extincted in the UV whose [Ly$\alpha$]{}photons can still escape. Its 2–D [\[\]]{}spectrum shows a velocity shear, suggesting a disk.
There is no obvious counterpart of the [Ly$\alpha$]{}emission at $\Delta \theta$ = $-1$ and $\Delta v$ $\sim$ $-$500[kms$^{-1}$]{}in either the [*HST*]{} image or in the X-shooter continuum spectrum. Therefore, this isolated [Ly$\alpha$]{}emission probably arises from the extended gas. Its line profile is broad, double-peaked, and blueshifted from galaxy B and galaxy C (Figure \[fig:spec2d\]). It is not possible to distinguish whether the bulk motion is inflowing or outflowing (at $\sim$500[kms$^{-1}$]{}) with respect to galaxy C, the brightest [\[\]]{}source and marker of systemic velocity (Figure \[fig:spec2d\]), because we do not know whether the gas lies in front of or behind galaxy C. This is the first unambiguous detection of a blueshifted [Ly$\alpha$]{}line with respect to galaxies embedded within [Ly$\alpha$]{}blobs. It is likely that the blueshift is relative to the systemic velocity of the [Ly$\alpha$]{}blob as well.
It is not clear what this blueshifted [Ly$\alpha$]{}emission represents. We consider three possibilities. First, although unlikely, this [Ly$\alpha$]{}component could be associated with a heavily-extincted galaxy that lies below the detection limit of the [*HST*]{} image ($m_{\rm F606W}$ $\gg$ 28 mag) or with galaxy A, whose redshift is unknown. Second, this gas might be tidally-stripped material arising from a galaxy-galaxy interaction: the two galaxies (B and C) are separated by $\sim$30kpc in projected distance and $\sim$200[kms$^{-1}$]{}in velocity space. Lastly, but most interestingly, this blueshifted [Ly$\alpha$]{}emission could be the long-sought, but elusive cold gas accretion along filamentary streams [@Keres05; @Keres09; @Dekel09]. Note that similarly blueshifted [Ly$\alpha$]{}emission has been reported in a faint [Ly$\alpha$]{}emitter at $z$ = 3.344, which was discovered in an extremely deep, blind spectroscopic search [@Rauch11]. The spectrum of this peculiar system also has very complex structure (e.g., diffuse fan-like blueshifted [Ly$\alpha$]{}emission and a DLA system). @Rauch11 suggest that this blueshifted [Ly$\alpha$]{}emission can be explained if the gas is inflowing along a filament behind the galaxy and emits fluorescent [Ly$\alpha$]{}photons induced by the ionizing flux escaping from the galaxy. Discriminating among the above possibilities will require us to further constrain the [Ly$\alpha$]{}line profile of this blueshifted component and to more fully survey possible member galaxies (or energy sources) within CDFS-LAB10.
Still, we were able to successfully link the [Ly$\alpha$]{}and [\[\]]{}sources with the embedded galaxies in this complex system and to find blueshifted [Ly$\alpha$]{}emission that might point to gas inflow. This example raises concerns about how to identify the sources of [Ly$\alpha$]{}emission and to interpret the gas morphologies and kinematics solely from [Ly$\alpha$]{}lines. Elongated or filamentary [Ly$\alpha$]{}morphologies may be a sign of bipolar outflows [@Matsuda04] or related to filamentary cold streams. In CDFS-LAB10, the [Ly$\alpha$]{}spectrum shows two kinematically distinct components around the brightest galaxy A: the upper ($\Delta\theta$ $>$ 0) and lower parts are blue- and redshifted, respectively. In the absence of [\[\]]{}spectroscopy, the [Ly$\alpha$]{}data appear to be consistent with either of the above outflow or infalling stream scenarios. However, our detailed analysis including the [\[\]]{}line clearly shows that one of the [Ly$\alpha$]{}components arises from the faint galaxy C, which would be difficult to detect without [*HST*]{} imaging and is likely the dominant source of [Ly$\alpha$]{}emission. Therefore, we stress that interpreting the [Ly$\alpha$]{}morphology and spectra requires deep high-resolution imaging and the determination of the systemic velocity through NIR spectroscopy.
Discussion {#sec:discussion}
==========
Covering Factor of Inflows and Outflows {#sec:covering_factor}
---------------------------------------
Under the assumption that inflowing gas streams (if any) are randomly distributed, and from the non-detection of any blueshifted [Ly$\alpha$]{}–[H$\alpha$]{}or [Ly$\alpha$]{}–[\[\]]{}offset in the direction of the eight embedded galaxies tested here (although see Section \[sec:cdfs-lab10\]), we constrain the covering fraction of inflows to be $<$1/8 (13%). Likewise, if all of our [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}offsets are different projections of the same collimated outflow from the galaxies and are a proxy for outflow speed, the covering fraction of [*strong*]{} outflows, i.e., super/hyper-winds, is less than $\sim$13%.
The covering fraction of gas flows to which we are referring here has a different meaning than often discussed in the literature . In these theoretical papers, the covering factor is defined as how many sight-lines will be detected in metal or [*absorption*]{} when bright background sources close to the galaxies are targeted. What we measure in this paper is how often the inflowing gas is aligned with observer’s sight-lines so that the column of neutral gas becomes optically thick and blue-shifts [Ly$\alpha$]{}lines against systemic velocity. Nonetheless, predict that the covering factor within the virial radius ($R_{\rm vir}$ $\simeq$ 75kpc) from their simulated galaxies with a halo mass of $M_h$ = 3[$\times$$10^{11}$]{}[$M_{\sun}$]{} at $z=2$ is relatively small: $\sim$3% and 10% for DLAs and Lyman limit systems (LLS), respectively. Within 0.5$\times$$R_{\rm vir}$, this factor increases to $\sim$10% and 30%, respectively, and presumably will be much higher directly towards the galaxies (i.e., for a pencil beam or looking down-the-barrel). Note that ignore galactic winds in order to isolate the inflowing streams, so we expect that there will be enough dense material arising from gas accretion to affect the transfer of [Ly$\alpha$]{}photons in our experiments.
What is uncertain is that how the velocity field of this dense material near the galaxy will shift the emerging [Ly$\alpha$]{}profiles and the statistics of [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}, because answering this requires full [Ly$\alpha$]{}radiative transfer treatment. Unlike RT models with simple geometry [e.g., @Dijkstra06a; @Verhamme06], cosmological simulations [@Faucher-Giguere10] predict that the gas inflow can only slightly enhance the blue [Ly$\alpha$]{}peak, implying that the overall profile will be still red-peak dominated if the effects of outflows from the galaxies and IGM absorption are fully considered. Therefore, according to these simulations, our non-detection of infall signatures does not contradict the cold stream model. The [Ly$\alpha$]{}profiles calculated from cold-stream models are in general the integrated profiles of [Ly$\alpha$]{}-emitting gas around the galaxies, which are not detectable by our study. What we need are the predictions for how [Ly$\alpha$]{}profiles are modified and/or shifted against optically-thin lines along the line-of-sight to the galaxies embedded in the extended gas. In this way, the distribution of predicted [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}can be compared directly with the observations.
It is also critical to take into account the effect of simultaneous outflows and inflows. For example, the H$\alpha$ and [\[\]]{}detections suggest that the galaxies embedded in the blobs are forming stars, which could generate mechanical feedback into the surrounding gas cloud. Thus, one might have expected that the innermost part of the gas cloud, close to the galaxies, has galactic scale outflows similar to those of other star-forming galaxies at $z=2-3$ [e.g., @Steidel10]. Gas infall (if any) may dominate at larger radii (up to $\sim$50 kpc, the typical blob size). In this case, the emerging [Ly$\alpha$]{}profile will be more sensitive to the core of the [Ly$\alpha$]{}blob, presumably the densest part of the CGM, than to the gas infall. As a result, it might be difficult to detect the infalling gas by measuring [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}. This kind of more realistic, infall+outflow model has not been considered yet in RT calculations, so its spectral signatures are unknown.
Outflow Speed vs. [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} {#sec:dvlya}
--------------------------------------------------------
In our sample of eight [Ly$\alpha$]{}blobs, the embedded galaxies have smaller velocity offsets ($\langle{\ifmmode\Delta v_{\rm Ly\alpha}\else$\Delta v_{\rm Ly\alpha}$\xspace\fi}\rangle$ = 160[kms$^{-1}$]{}) than those of LBGs (445[kms$^{-1}$]{}). The remaining question is how to interpret these small [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}values. We consider two possibilities here.
First, if we assume a simple geometry where the outflowing material forms a spherical shell that consists of continuous media, [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}originates from the resonant scattering of the [Ly$\alpha$]{}photons at the shell as discussed in @Verhamme08. Note that the emerging [Ly$\alpha$]{}profile is independent of the physical size of the shell as long as the shell have the same outflow velocity. Therefore, this model is applicable to [Ly$\alpha$]{}blobs. In this shell model, a central monochromatic point source is surrounded by an expanding shell of neutral gas with varying column density ($N_{\rm H I}$) and Doppler $b$ parameter [e.g., @Verhamme06; @Verhamme08]. A generic prediction is that the [Ly$\alpha$]{}emission is asymmetric, with the details of the line shape depending on the shell velocity, Doppler parameter $b$, and optical depth of column in the shell. In this simple geometry, the [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} values can be used as a proxy for outflow velocity of expanding shell, i.e., $v_{\rm exp}$ $\simeq$ 0.5[v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}. If this is the case, outflow velocities from blob galaxies are much smaller than the values expected from models of strong galactic winds . Furthermore, these offsets are even smaller than the typical [Ly$\alpha$]{}–H$\alpha$ offsets (250–900[kms$^{-1}$]{}) of LBGs [@Steidel04; @Steidel10]. In particular, while CDFS-LAB14 has the largest [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} for its red peak, its blue and red peaks have similar intensity, which is a characteristic of a static medium or no bulk motions [@Verhamme06; @Kollmeier10].
Second, in an “expanding bullet” or “clumpy CGM” model [@Steidel10], the outflowing material, which consists of small individual clumps, has a wide velocity range rather than a single value, and the [Ly$\alpha$]{}profiles are determined by the Doppler shift that photons acquire when they are last scattered by the clumps just before escaping the system. In this case, the [Ly$\alpha$]{}–[H$\alpha$]{} offsets are primarily modulated by the amount of gas that has $v = 0$ component (though it is not clear where this material is spatially located); thus [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} is not directly correlated with outflow velocity. If this is the case, the small [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} value of the galaxies in our [Ly$\alpha$]{}blobs simply indicates that there is less neutral gas at the galaxies’ systemic velocity (presumably near the galaxy) compared to LBGs.
Testing these two hypotheses for [Ly$\alpha$]{}blobs or LBGs first requires a detailed comparison between high resolution [Ly$\alpha$]{}profiles and RT predictions. This test is beyond the scope of this paper and will be discussed in the future. As emphasized by @Steidel10, one also needs to check the consistency of the [Ly$\alpha$]{}profiles with ISM metal absorption profiles, which depends on high S/N continuum spectra that are not available here. Nonetheless, we attempted such an analysis by stacking many low-ionization absorption lines in three galaxies (section \[sec:absorption\]).
For those three embedded galaxies with measured [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} and [$\Delta v_{\rm IS}$]{}, the outflow velocity estimates from both methods roughly agree within the uncertainties arising from low S/N and from RT complications. However, closer inspection reveals a discrepancy between the observations and the simplest RT models in that we obtain [$\Delta v_{\rm IS}$]{}$\sim$ $-$200[kms$^{-1}$]{}and also [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}$\sim$ 200[kms$^{-1}$]{}while the RT model with an expanding shell geometry predicts [$\Delta v_{\rm IS}$]{}= $-$[$v_{\rm exp}$]{}= $-0.5$ [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}. This discrepancy has also appeared in LBGs [@Steidel10], and @Kulas12 show that the stacked [Ly$\alpha$]{}profile of LBGs with double-peaked profiles cannot be reproduced accurately by the shell model. For a wide range of parameters ([$v_{\rm exp}$]{}, $b$, [$N_{\rm{H I}}$]{}), @Kulas12 cannot reproduce the location of the red peak, the width of the [Ly$\alpha$]{} profile, and the metal absorption profile at the same time.
We attribute this discrepancy to the very simple nature of the shell model. By construction or by definition of the expanding “shell”, the internal velocity dispersion of the media that constitute the shell should be much smaller than the expansion velocity of the shell itself, i.e., [$v_{\rm exp}$]{}/$b$ $\gg$ 1. Therefore, one expects that the metal absorption lines arising from this thin shell have very narrow line widths $\sigma_{\rm
abs}$ similar to $b$, typically $\sim$tens of [kms$^{-1}$]{}. However, such narrow absorption lines are not observed in either LBGs or [Ly$\alpha$]{}blobs. Clearly, RT calculations with more realistic geometries and allowing a more thorough comparison with the observed [Ly$\alpha$]{}profiles are required.
[v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}in Context of LAEs and LBGs {#sec:LAE}
-----------------------------------------------------------------
Given that the galaxies within our [Ly$\alpha$]{}blobs were selected, by definition, as [Ly$\alpha$]{}emitters with high equivalent width (EW), the comparison with LAEs suggests that small [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} values are a general characteristic of all high EW [Ly$\alpha$]{}-selected populations, be they compact or extended. In addition to this work and @Yang11, there are recent studies that measure [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} for bright compact [Ly$\alpha$]{}-emitters [@McLinden11; @Finkelstein11; @Hashimoto13; @Guaita13] and [Ly$\alpha$]{}blobs [@McLinden13], which also find small [Ly$\alpha$]{}–[\[\]]{}offsets for LAEs ranging from 35 – 340[kms$^{-1}$]{}. A Kolmogorov–Smirnov test fails to distinguish the [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}distribution of our eight [Ly$\alpha$]{}blobs from that of ten compact LAEs compiled from the four studies mentioned above. Thus, we now have a fairly large sample of high–EW [Ly$\alpha$]{}emitters (compact or extended) with [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} = $-$60 $\rightarrow$ +400[kms$^{-1}$]{}, which is clearly different from the distribution for LBGs (see Section \[sec:shift\]).
The smaller [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}of LAEs and LABs might be related to their higher [Ly$\alpha$]{}equivalent widths. An anti-correlation between EW([Ly$\alpha$]{}) and [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}in a compilation of LAE and LBG samples has been suggested [@Hashimoto13]. Even within the LBG population itself there is an indication that [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}decreases as EW([Ly$\alpha$]{}) increases. @Shapley03 find an anti-correlation between the EW([Ly$\alpha$]{}) and the velocity offsets between interstellar absorption and [Ly$\alpha$]{}emission lines, $\Delta v_{\rm
Ly\alpha-abs}$.[^2] From the stacked spectra of LBGs binned at different EW([Ly$\alpha$]{}), they show that with increasing [Ly$\alpha$]{}line strength from $-$15Åto 53Å, $\Delta v_{\rm Ly\alpha-abs}$ decreases from 800[kms$^{-1}$]{}to 480[kms$^{-1}$]{}. Because $\Delta v_{\rm Ly\alpha-abs}$ = [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}+ $|{\ifmmode{\Delta v_{\rm IS}}\else$\Delta v_{\rm IS}$\xspace\fi}|$, we can infer that [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}is likely to decrease as EW([Ly$\alpha$]{}) increases unless the observed anti-correlation is entirely due to $|{\ifmmode{\Delta v_{\rm IS}}\else$\Delta v_{\rm IS}$\xspace\fi}|$.
The origin of the apparent relationship between larger EW([Ly$\alpha$]{}) and smaller [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} is not understood. It is possible that other physical properties drive this anti-correlation: for example, LBGs tend to have brighter continuum magnitudes, thus likely higher star formation rates and stellar masses, than are of typical LAEs. @Hashimoto13 propose that low column density, and thus a small number of resonant scatterings of [Ly$\alpha$]{}photons, might be responsible for the strong [Ly$\alpha$]{}emission and small [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}of [Ly$\alpha$]{}emitters.
What we do know from our [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} measurements is that LAEs and [Ly$\alpha$]{}blobs are kinematically similar. Therefore, the mystery remains as to what powers [Ly$\alpha$]{} nebulae, or, in other words, why certain galaxies have more spatially-extended [Ly$\alpha$]{} gas (i.e., blobs) than others (i.e., compact LAEs) with similar [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} and EW([Ly$\alpha$]{}). While the answer may be related to photo-ionization from (buried) AGN (e.g., see Section \[sec:type2\] or @Yang14), an extended proto-intracluster medium that can scatter or transport the [Ly$\alpha$]{}photons out to larger distances, or less dust to destroy [Ly$\alpha$]{} photons [cf. @Hayes13], discriminating among these possibilities will require a multi-wavelength analysis of a large sample of [Ly$\alpha$]{}blobs. For now, the similarity here of LABs and LAEs suggests that differences in gas kinematics are not responsible for the extended [Ly$\alpha$]{}halos.
CDFS-LAB11: Photo-ionization by AGN? {#sec:type2}
------------------------------------
From the nearly symmetric [Ly$\alpha$]{}profile (Fig. \[fig:cdfs-lab11\]), and detection, and rest-frame optical line ratios, we hypothesize that an AGN in CDFS-LAB11 ionizes the gas surrounding the galaxies, making [Ly$\alpha$]{} [*relatively*]{} optically thin and preventing the resonant scattering of [Ly$\alpha$]{}photons from dominating the shape of the profile. While fitting the profile requires an underlying broad component (FWHM $\simeq$ 480[kms$^{-1}$]{}), the narrow component (redshifted by $\simeq$ 80[kms$^{-1}$]{}) has a small velocity width (FWHM $\simeq$ 110[kms$^{-1}$]{}), comparable to that of the [\[\]]{}line ($\simeq$80[kms$^{-1}$]{}). Therefore, if we assume that [Ly$\alpha$]{}and [\[\]]{}photons originate from the same region, resonant scattering by the IGM/CGM does not broaden the intrinsic profile significantly. In other words, at least along our LOS, the optical depth of CDFS-LAB11’s IGM/CGM is smaller than for the other [Ly$\alpha$]{}blobs in our sample and for typical [Ly$\alpha$]{}emitters at high redshifts.
A nearly symmetric and narrow [Ly$\alpha$]{}profile such as CDFS-LAB11’s is rarely observed among high-$z$ [Ly$\alpha$]{}-emitting galaxies. However, [Ly$\alpha$]{}radiative transfer calculations show that such [Ly$\alpha$]{}profiles can be generated in the presence of a central ionizing source. For example, @Dijkstra06a investigate the emerging [Ly$\alpha$]{}profiles from an collapsing/expanding sphere centered on an ionizing source (an AGN) for different luminosities (see their Figures 11 and 12). They find that if the ionizing source is strong and the [Ly$\alpha$]{}is [*relatively*]{} optically thin (line-center optical depth $\tau_0$ $<$ $10^3$), the separation of the characteristic double-peaks become smaller ($\Delta v_{\rm blue-red}$ $\ll$ 100[kms$^{-1}$]{}) as the ionizing source gets stronger. The overall profile becomes symmetric, and is blue or redshifted depending on the flow direction, but again by small amount ([v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}$\ll$ 100[kms$^{-1}$]{}). These authors were skeptical that the sign of the [Ly$\alpha$]{}shift and the small [Ly$\alpha$]{}offset ([v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}$\sim$ a few tens [kms$^{-1}$]{}) from the systemic velocity could be measured. However, with the right strategy (i.e., high spectral resolution and a careful choice of survey redshift), we are able to reliably measure the predicted small offset. Clearly, more detailed comparisons with the RT models are required.
A broad, asymmetric [Ly$\alpha$]{}profile with a sharp blue edge (e.g., CDFS-LAB06 and 07) is characteristic of high-$z$ [Ly$\alpha$]{}-emitters. In spectroscopic follow-up observations of [Ly$\alpha$]{}-emitter candidates, where spectral coverage is limited, these characteristics alone are often used to discriminate high-$z$ [Ly$\alpha$]{}emitters from possible low-$z$ interlopers. The example of CDFS-LAB11 demonstrates that caution is required because a narrow and symmetric [Ly$\alpha$]{}line can also arise when the ISM or CGM of a candidate galaxy is significantly photo-ionized, e.g., by AGN. On the bright side, our [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{} velocity offset technique could be used for studying gas infall or outflow in high-$z$ QSOs instead of relying on only the [Ly$\alpha$]{}line profile [e.g., @Weidinger04].
Conclusions {#sec:conclusion}
===========
Exploring the origin of [Ly$\alpha$]{}nebulae (“blobs") at high redshift requires measurements of their gas kinematics that are difficult with only the resonant, optically-thick [Ly$\alpha$]{}line. To define gas motions relative to the systemic velocity of the nebula, the [Ly$\alpha$]{}line must be compared with non-resonant lines, which are not much altered by radiative transfer effects. We made first comparison of non-resonant [H$\alpha$ $\lambda$6563]{}to extended [Ly$\alpha$]{}emission for two bright [Ly$\alpha$]{}blobs in @Yang11, concluding that, within the context of a simple radiative transfer model, the gas was static or mildly outflowing at $\lesssim$250[kms$^{-1}$]{}. However, it was unclear if these two [Ly$\alpha$]{}blobs, which are the brightest in the sample, are representative of the general [Ly$\alpha$]{}blob population. Furthermore, geometric effects — infall along filaments or bi-polar outflows — might hide bulk motions of the gas when only viewed from the two directions toward these two [Ly$\alpha$]{}blobs. With VLT X-shooter, we obtain optical and near-infrared spectra of six additional [Ly$\alpha$]{}blobs from the @Yang10 sample. With a total of eight [Ly$\alpha$]{}blobs, we investigate the gas kinematics within [Ly$\alpha$]{}blobs using three techniques: the [Ly$\alpha$]{}offset from the systemic velocity ([v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}), the shape and shift ([$\Delta v_{\rm IS}$]{}) of the ISM metal absorption line profiles, and the breadth of the [\[\]]{}line profile.
Our findings are:
1. Both [Ly$\alpha$]{}and non-resonant lines confirm that these blobs lie at the survey redshift ($z\sim2.3$). We also detect the [\[\] $\lambda\lambda$3727,3729]{}, [\[\] $\lambda$4959]{}, [\[\] $\lambda$5007]{}, and [H$\beta$ $\lambda$4861]{}lines. All non-resonant line velocities are consistent with each other and with arising from the galaxy or galaxies embedded in the [Ly$\alpha$]{}blob. [\[\]]{}, which is observed at high signal-to-noise in all cases and whose profile is an RT-independent constraint on the gas kinematics, is a particularly good diagnostic line for this redshift and instrument.
2. The majority of the blobs (6/8) have broadened [Ly$\alpha$]{}profiles indicating radiative transfer effects. These [Ly$\alpha$]{}profiles are consistent with being in the same family of objects as predicted by RT, with profile shapes ranging from symmetric double-peaked, to asymmetric red peak dominated, to a single red peak. The fraction of double-peaked profiles is $\sim$38% (3/8).
3. The narrow [Ly$\alpha$]{}profile systems (CDFS-LAB01A, CDFS-LAB11), whose [Ly$\alpha$]{}profile is not significantly broader than the [\[\]]{}or [H$\alpha$]{}lines, have the smallest [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}offsets, the most spatially compact [Ly$\alpha$]{}emission, and the only and lines detected, implying that a hard ionizing source, possibly an AGN, is responsible for the lower optical depth toward the central embedded galaxies.
4. With a combination of [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}, the interstellar metal absorption line profile, and a new indicator, the spectrally-resolved [\[\]]{}line profile, we detect gas moving along the line of sight to galaxies embedded in the [Ly$\alpha$]{}blob center. Although not all three indicators are available for all [Ly$\alpha$]{}blobs, the implied speeds and direction are roughly consistent for the sample, suggesting a simple picture in which the gas is stationary or slowly outflowing at a few hundred [kms$^{-1}$]{}from the embedded galaxies. These outflow speeds are similar to those of LAEs, suggesting that outflow speed is not the dominant driver of extended [Ly$\alpha$]{}emission. Furthermore, these outflow speeds exclude models in which star formation or AGN produce “super” or “hyper” winds of up to $\sim$1000[kms$^{-1}$]{}.
More specifically:
1. We compare the non-resonant emission lines [\[\]]{}and [H$\alpha$]{}to the [Ly$\alpha$]{}profile to obtain the velocity offset [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}. The galaxies embedded within our [Ly$\alpha$]{}blobs have smaller [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}than those of LBGs, confirming the previous claims [@Yang11]. The galaxies within [Ly$\alpha$]{}blobs have [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}= $-$60 $\rightarrow$ +400[kms$^{-1}$]{} with an average of $\langle{\ifmmode\Delta v_{\rm Ly\alpha}\else$\Delta v_{\rm Ly\alpha}$\xspace\fi}\rangle$ = 160[kms$^{-1}$]{}, while LBGs at similar redshifts have [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}= 250 – 900[kms$^{-1}$]{}. The small [v\_[Ly]{}$\Delta v_{\rm Ly\alpha}$]{}in the [Ly$\alpha$]{}blobs are consistent with those measured for compact LAEs.
2. By stacking low-ionization metal absorption lines, we measure the outflow velocity of neutral gas in front of the galaxies in the [Ly$\alpha$]{}blobs. Galaxies in two [Ly$\alpha$]{}blobs show an outflow speed of $\sim$250[kms$^{-1}$]{}, while another has an almost symmetric absorption line profile centered at [$\Delta v_{\rm IS}$]{}= 0[kms$^{-1}$]{}, consistent with no significant bulk motion. The ISM absorption line profiles here have low S/N, but are very roughly consistent with those of some LBGs (at the level of several hundred [kms$^{-1}$]{}outflows).
3. The high spectral resolution of our data reveals broad wings in the [\[\]]{} profiles of four [Ly$\alpha$]{}blobs. This new kinematic diagnostic suggests warm ionized outflows driven by supernovae and stellar winds. These broad line components are narrower ($\sigma_{\rm broad}$ = 45 – 120[kms$^{-1}$]{}) and have a maximum blueshifted velocity ([$\Delta v_{\rm max}$]{}= 150 – 260[kms$^{-1}$]{}) smaller than those of $z\sim2$ star-forming galaxies (SFGs), implying weaker outflows here than for LBGs and SFGs at similar redshifts.
4. If we assume that the detected outflows are different projections of the same outflow from the [Ly$\alpha$]{}blob center, we can estimate the effects of flow geometry on our measurements given that our large sample size allows averaging over many lines-of-sight. The absence of any strong ($\sim$1000[kms$^{-1}$]{}) outflows among the eight galaxies tested is not a projection effect: their covering fraction is $<$1/8 (13%). Likewise, the lack of a blue-peak dominated [Ly$\alpha$]{}profile, at least in the direction of the embedded galaxies (see point 6. below), implies that the covering factor of any cold streams [@Keres05; @Keres09; @Dekel09] is less than 13%. The channeling of gravitational cooling radiation into [Ly$\alpha$]{}may not be significant over the radii probed by our techniques here.
5. Constraining the physical state of [Ly$\alpha$]{}-emitting gas in a [Ly$\alpha$]{}blob is a critical step in understanding its emission mechanism and in comparing to simulations. For one [Ly$\alpha$]{}blob whose [Ly$\alpha$]{}profile and ISM metal absorption lines suggest no significant bulk motion (CDFS-LAB14), at least in its cool and neutral gas, we assume a simple RT model and make the first column density measurement of gas in a embedded galaxy, finding that it is consistent with a DLA.
6. For one peculiar system (CDFS-LAB10), we discover blueshifted [Ly$\alpha$]{}emission that is [*not*]{} directly associated with any embedded galaxy. This [Ly$\alpha$]{}emitting gas is blueshifted relative to two embedded galaxies, suggesting that it arose from a tidal interaction between the galaxies or is actually flowing into the blob center. The former is expected in these overdense regions, where [*HST*]{} images resolve many galaxies. The latter might signify the predicted but elusive cold gas accretion along filaments.
We thank the anonymous referee for her or his thorough reading of the manuscript and helpful comments. The authors thank Daniel Eisenstein for his contributions at the start of this project. We thank Jason X. Prochaska for helpful discussions. YY thanks the MPIA ENIGMA group for the helpful discussions. YY also thanks the Theoretical Astrophysics Center at the University of California, Berkeley for the travel support, as well as Claude-André Faucher-Gigu[è]{}re, Du[š]{}an Kere[š]{} and Daniel Kasen for the helpful discussions during that stay. YY also thanks Sangeeta Malhotra for a helpful discussion regarding the LAE and LBG connection. YY acknowledges support from the BMBF/DLR grant Nr. 50 OR 1306. A.I.Z. thanks the Max-Planck-Institut für Astronomie and the Center for Cosmology and Particle Physics at New York University for their hospitality and support during her stays there. A.I.Z. acknowledges support from the NSF Astronomy and Astrophysics Research Program through grant AST-0908280 and from the NASA Astrophysics Data Analysis Program through grant NNX10AD47G. She also thank the generosity of the John Simon Guggenheim Memorial Foundation.
Facilities:
Bland, J., & Tully, B. 1988, , 334, 43 Cen, R., & Zheng, Z. 2013, , 775, 112 Colbert, J. W., Teplitz, H., Francis, P., Palunas, P., Williger, G. M., & Woodgate, B. 2006, , 637, L89 Damen, M., Labb[é]{}, I., van Dokkum, P. G., et al. 2011, , 727, 1 Dekel, A., et al. 2009, , 457, 451 Dey, A., et al. 2005, , 629, 654 Dijkstra, M., Haiman, Z., & Spaans, M. 2006, , 649, 14 Dijkstra, M., Haiman, Z., & Spaans, M. 2006, , 649, 37 Dijkstra, M., & Loeb, A. 2009, , 400, 1109 Erb, D. K., Bogosavljevi[ć]{}, M., & Steidel, C. C. 2011, , 740, L31 Fardal, M. A., Katz, N., Gardner, J. P., Hernquist, L., Weinberg, D. H., & Dav[' e]{}, R. 2001, , 562, 605 Faucher-Gigu[è]{}re, C.-A., Kere[š]{}, D., Dijkstra, M., Hernquist, L., & Zaldarriaga, M. 2010, , 725, 633 Faucher-Gigu[è]{}re, C.-A., & Kere[š]{}, D. 2011, , 412, L118 Finkelstein, S. L., Hill, G. J., Gebhardt, K., et al. 2011, , 729, 140 Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, , 125, 306 Francis, P. J., et al. 2001, , 554, 1001 Fumagalli, M., Prochaska, J. X., Kasen, D., et al. 2011, , 418, 1796 Fynbo, J. U., M[ø]{}ller, P., & Warren, S. J. 1999, , 305, 849 Gawiser, E., et al. 2006, , 162, 1 Geach, J. E., et al. 2009, , 700, 1 Genzel, R., Newman, S., Jones, T., et al. 2011, , 733, 101 Goerdt, T., Dekel, A., Sternberg, A., Ceverino, D., Teyssier, R., & Primack, J. R. 2010, , 933 Guaita, L., Francke, H., Gawiser, E., et al. 2013, , 551, A93 Haiman, Z., & Rees, M. J. 2001, , 556, 87 Haiman, Z., Spaans, M., & Quataert, E. 2000, , 537, L5 Harrington, J. P. 1973, , 162, 43 Hashimoto, T., Ouchi, M., Shimasaku, K., et al. 2013, , 765, 70 Hayes, M., Scarlata, C., & Siana, B. 2011, , 476, 304 Hayes, M., [Ö]{}stlin, G., Schaerer, D., et al. 2013, , 765, L27 Hennawi, J. F., Prochaska, J. X., Kollmeier, J., & Zheng, Z. 2009, , 693, L49 Keel, W. C., Cohen, S. H., Windhorst, R. A., & Waddington, I. 1999, , 118, 2547 Kelson, D. D. 2003, , 115, 688 Kere[š]{}, D., Katz, N., Weinberg, D. H., & Dav[é]{}, R. 2005, , 363, 2 Kere[š]{}, D., Katz, N., Fardal, M., Dav[é]{}, R., & Weinberg, D. H. 2009, , 395, 160 Kimm, T., Slyz, A., Devriendt, J., & Pichon, C. 2011, , 413, L51 Kollmeier, J. A., Zheng, Z., Dav[é]{}, R., Gould, A., Katz, N., Miralda-Escud[é]{}, J., & Weinberg, D. H. 2010, , 708, 1048 Kulas, K. R., Shapley, A. E., Kollmeier, J. A., et al. 2012, , 745, 33 Latif, M. A., Schleicher, D. R. G., Spaans, M., & Zaroubi, S. 2011, , 413, L33 Lehmer, B. D., Brandt, W. N., Alexander, D. M., et al. 2005, , 161, 21 Luo, B., Bauer, F. E., Brandt, W. N., et al. 2008, , 179, 19 Martin, D. C., Chang, D., Matuszewski, M., et al. 2014, arXiv:1402.4809 Matsuda, Y., et al. 2004, , 128, 569 Matsuda, Y., Yamada, T., Hayashino, T., Yamauchi, R., & Nakamura, Y. 2006, , 640, L123 Matsuda, Y., Yamada, T., Hayashino, T., et al. 2011, , 410, L13 McLinden, E. M., Finkelstein, S. L., Rhoads, J. E., et al. 2011, , 730, 136 McLinden, E. M., Malhotra, S., Rhoads, J. E., et al. 2013, , 767, 48 M[ø]{}ller, P., Warren, S. J., & Fynbo, J. U. 1998, , 330, 19 Nakajima, K., Ouchi, M., Shimasaku, K., et al. 2012, , 745, 12 Neufeld, D. A. 1990, , 350, 216 Newman, S. F., Genzel, R., F[ö]{}rster-Schreiber, N. M., et al. 2012, , 761, 43 Nilsson, K. K., Fynbo, J. P. U., M[ø]{}ller, P., Sommer-Larsen, J., & Ledoux, C. 2006, , 452, L23 Ohyama, Y., et al. 2003, , 591, L9 Ouchi, M., et al. 2009, , 696, 1164 Prescott, M. K. M., Kashikawa, N., Dey, A., & Matsuda, Y. 2008, , 678, L77 Prescott, M. K. M., Dey, A., & Jannuzi, B. T. 2009, , 702, 554 Prescott, M. K. M., Dey, A., & Jannuzi, B. T. 2012, , 748, 125 Prescott, M. K. M., Dey, A., Brodwin, M., et al. 2012, , 752, 86 Rauch, M., Becker, G. D., Haehnelt, M. G., et al. 2011, , 418, 1115 Rix, H.-W., et al. 2004, , 152, 163 Rosdahl, J., & Blaizot, J. 2012, , 423, 344 Saito, T., Shimasaku, K., Okamura, S., Ouchi, M., Akiyama, M., & Yoshida, M. 2006, , 648, 54 Saito, T., Shimasaku, K., Okamura, S., Ouchi, M., Akiyama, M., Yoshida, M., & Ueda, Y. 2008, , 675, 1076 Scarlata, C., et al. 2009, , 706, 1241 Schaerer, D., & Verhamme, A. 2008, , 480, 369 Shapiro, K. L., Genzel, R., Quataert, E., et al. 2009, , 701, 955 Shapley, A. E., Steidel, C. C., Pettini, M., & Adelberger, K. L. 2003, , 588, 65 Smith, D. J. B., & Jarvis, M. J. 2007, , 378, L49 Soto, K. T., Martin, C. L., Prescott, M. K. M., & Armus, L. 2012, , 757, 86 Steidel, C. C., Adelberger, K. L., Shapley, A. E., Pettini, M., Dickinson, M., & Giavalisco, M. 2000, , 532, 170 Steidel, C. C., Shapley, A. E., Pettini, M., Adelberger, K. L., Erb, D. K., Reddy, N. A., & Hunt, M. P. 2004, , 604, 534 Steidel, C. C., Erb, D. K., Shapley, A. E., et al. 2010, , 717, 289 Taniguchi, Y., Shioya, Y., & Kakazu, Y. 2001, , 562, L15 Taniguchi, Y. & Shioya, Y. 2000, , 532, L13 Verhamme, A., Schaerer, D., & Maselli, A. 2006, , 460, 397 Verhamme, A., Schaerer, D., Atek, H., & Tapken, C. 2008, , 491, 89 Vernet, J., Dekker, H., D’Odorico, S., et al. 2011, , 536, A105 Weidinger, M., M[ø]{}ller, P., & Fynbo, J. P. U. 2004, , 430, 999 Weijmans, A.-M., Bower, R. G., Geach, J. E., et al. 2010, , 402, 2245 Westmoquette, M. S., Exter, K. M., Smith, L. J., & Gallagher, J. S. 2007, , 381, 894 Wilman, R. J., Gerssen, J., Bower, R. G., Morris, S. L., Bacon, R., de Zeeuw, P. T., & Davies, R. L. 2005, , 436, 227 Yamada, T., Matsuda, Y., Kousai, K., et al. 2012, , 751, 29 Yang, Y., Zabludoff, A., Tremonti, C., Eisenstein, D., & Dav[é]{}, R. 2009, , 693, 1579 Yang, Y., Zabludoff, A., Eisenstein, D., & Dav[é]{}, R. 2010, , 719, 1654 Yang, Y., Zabludoff, A., Jahnke, K., et al. 2011, , 735, 87 Yang, Y., Walter, F., Decarli, R., et al. 2014, , 784, 171
Decomposition of [\[\]]{}profiles
=================================
To test if two velocity components are needed to fit our spectrally-resolved [\[\]]{}profiles and to extract the line parameters, we employ the [\[\]]{}profiles with two Gaussian functions using a Markov Chain Monte Carlo (MCMC) technique. We use the [emcee]{} software [@emcee] to sample the distributions of six parameters: line center, width, and fluxes for the two Gaussian profiles. We require that (1) the line widths of both components are larger than the instrumental line width (31[kms$^{-1}$]{}; the vertical dotted lines in Figure \[fig:O3MCMC\]), (2) the peak of each component should be at least 5% of the observed peak intensity, and (3) the peak of the narrow component is higher than that of the broad component. The latter two priors are imposed to prevent the MCMC chains from getting stuck in parameter spaces with extremely broad lines but with negligible fluxes.
In Figure \[fig:O3MCMC\], we show the likelihood distributions of the line widths of the two components: $\sigma_{\rm narrow}$ and $\sigma_{\rm
broad}$. Note that we exclude CDFS-LAB10 in this analysis because its neighboring galaxy (CDFS-LAB10A) makes it difficult to reliably extract its profile (see Sections \[sec:O3profile\] and \[sec:cdfs-lab10\]). Except for CDFS-LAB11, the likelihood distributions of the two line-widths do not overlap significantly, and the peaks of the joint 2–D distributions are not located near the $\sigma_{\rm narrow}$ = $\sigma_{\rm broad}$ line (the dashed line). Therefore, we conclude that the remaining four systems (CDFS-LAB06, 07, 13, 14) are likely to consist of two components. The uncertainties of the parameters are determined from the 68.2% confidence interval of the marginalized distribution.
{width="30.00000%"} {width="30.00000%"} {width="30.00000%"}\
{width="30.00000%"} {width="30.00000%"} {width="30.00000%"}
[c cc cc ccccc]{} CDFS-LAB06 & 1.57 $\pm$ 0.07 & 56 $\pm$ 3 & 03:32:19.78 & -27:47:31.5 & 2011-01-07 & 1.07 & 0.80 & 0.76 & 0.80\
CDFS-LAB07 & 1.14 $\pm$ 0.05 & 58 $\pm$ 3 & 03:32:03.36 & -27:45:24.4 & 2011-01-06 & 1.05 & 0.77 & 0.76 & 0.80\
& & & & & 2011-01-06 & 1.18 & 0.98 & 0.76 & 0.80\
& & & & & 2011-01-28 & 1.12 & 0.81 & 0.76 & 0.80\
CDFS-LAB10 & 0.71 $\pm$ 0.04 & 318 $\pm$ 36 & 03:32:37.45 & -28:02:05.7 & 2010-11-06 & 1.00 & 0.71 & 0.76 & 0.80\
& & & & & 2010-11-07 & 1.00 & 0.68 & 0.76 & 0.80\
& & & & & 2010-11-07 & 1.08 & 1.13 & 0.76 & 0.80\
& & & & & 2010-11-08 & 1.08 & 1.23 & 0.76 & 0.80\
CDFS-LAB11 & 1.02 $\pm$ 0.04 & 120 $\pm$ 10 & 03:32:43.25 & -27:42:58.3 & 2011-01-08 & 1.07 & 0.75 & 0.76 & 0.80\
& & & & & 2011-01-08 & 1.23 & 0.85 & 0.76 & 0.80\
CDFS-LAB13 & 0.94 $\pm$ 0.04 & 184 $\pm$ 25 & 03:32:32.75 & -27:39:06.4 & 2011-01-07 & 1.22 & 0.79 & 0.76 & 0.80\
& & & & & 2011-01-08 & 1.01 & 0.70 & 0.76 & 0.80\
CDFS-LAB14 & 0.93 $\pm$ 0.05 & 67 $\pm$ 5 & 03:32:32.29 & -27:41:26.4 & 2011-01-09 & 1.07 & 0.80 & 0.76 & 0.80\
& & & & & 2011-01-09 & 1.24 & 0.73 & 0.76 & 0.80\
\
CDFS-LAB01 & 8.02 $\pm$ 0.24 & 512 $\pm$ 50 & & & & & & &\
CDFS-LAB02 & 2.88 $\pm$ 0.12 & 43 $\pm$ 2 & & & & & & & \[tab:obslog\]
[c ccc r ccc l]{} CDFS-LAB01 & $-$65 $\pm$ 20 & 228 $\pm$ 14 & 45.2 $\pm$ 1.6 && & & & (1)\
CDFS-LAB11 & 84 $\pm$ 6 & 79 $\pm$ 3 & 26.5 $\pm$ 0.7 && & & & (1)\
CDFS-LAB10 & 247 $\pm$ 147 & 425 $\pm$ 97 & 18.7 $\pm$ 2.4 && & & & (1)\
CDFS-LAB13 & 152 $\pm$ 14 & 100 $\pm$ 9& 13.7 $\pm$ 0.7 && $-$277 $\pm$ 8 & 43 $\pm$ 8 & 4.0 $\pm$ 0.5 & (2)\
CDFS-LAB06 & 123 $\pm$ 15 & 150 $\pm$ 13 & 25.7 $\pm$ 1.3 && & & & (1)\
CDFS-LAB02 & 211 $\pm$ 43 & 192 $\pm$ 35 & 8.2 $\pm$ 0.9 && $-$342 $\pm$ 117 & 273 $\pm$ 137 & 3.3 $\pm$ 1.3 & (2)\
CDFS-LAB07 & 181 $\pm$ 15 & 168 $\pm$ 10 & 22.2 $\pm$ 0.8 && & & & (1)\
CDFS-LAB14 & 371 $\pm$ 34 & 202 $\pm$ 24 & 18.0 $\pm$ 1.2 && $-$404 $\pm$ 17 & 176 $\pm$ 18 & 12.4 $\pm$ 1.1 & (3) \[tab:Lyman\]
[c cc c cc c]{} CDFS-LAB06 & 38 $\pm$ 3& $+$39 $\pm$ 27 & 55 $\pm$ 10 & 0.51$^{ +0.58}_{ -0.29}$ & 0.35 $\pm$ 0.17 & 152 $\pm$ 15\
CDFS-LAB07 & 44 $\pm$ 2& $-$97 $\pm$ 23 & 64 $\pm$ 10 & 0.44$^{ +0.31}_{ -0.18}$ & 0.32 $\pm$ 0.12 & 222 $\pm$ 11\
CDFS-LAB13 & 35 $\pm$ 5& $-$56 $\pm$ 21 & 60 $\pm$ 11 & 0.84$^{ +0.55}_{ -0.38}$ & 0.46 $\pm$ 0.13 & 174 $\pm$ 18\
CDFS-LAB14 & 67 $\pm$ 16 & $-$12 $\pm$ 71 & 125 $\pm$ 40 & 0.47$^{ +0.66}_{ -0.20}$ & 0.33 $\pm$ 0.20 & 262 $\pm$ 75\
CDFS-LAB10 & 71 $\pm$ 1& & & & &\
CDFS-LAB11 & 47 $\pm$ 1& & & & & \[tab:O3profile\]
[^1]: Note that this coefficient is slightly different from the traditional Neufeld solution (1.066).
[^2]: The notation, $\Delta v_{\rm em-abs}$ is used in @Shapley03.
|
---
abstract: 'In [@BG], Bazzoni and Glaz conjecture that the weak global dimension of a Gaussian ring is $0,1$ or $\infty$. In this paper, we prove their conjecture.'
address:
- |
G. Donadze\
Department of Algebra, University of Santiago de Compostela, 15782, Spain.
- |
V.Z. Thomas\
School of Mathematics, Tata Institute of Fundamental Research, Mumbai, Maharashtra 400005, India.
author:
- Guram Donadze
- 'Viji Z. Thomas'
title: 'Bazzoni-Glaz Conjecture'
---
Introduction
============
In her Thesis [@T], H. Tsang, a student of Kaplansky introduced Gaussian rings. Noting that the content of a polynomial $f$ over a commutative ring $R$ is the ideal $c(f)$ generated by the coefficients of $f$, we now define a Gaussian ring.
A polynomial $f\in R[x]$ is called Gaussian if $c(f)c(g)=c(fg)$ for all $g\in R[x]$. The ring $R$ is called Gaussian if each polynomial in $R[x]$ is Gaussian.
In her thesis, Tsang determined conditions under which a polynomial is Gaussian. In particular, she proved
Let $R$ be a commutative ring and let $f\in R[x]$ be a polynomial in one variable over $R$. If $c(f)$ is an invertible ideal, or more generally, locally a principal ideal, then $f$ is a Gaussian polynomial.
The converse of this statement has received considerable interest in the recent past and is related to the following conjecture of Kaplansky.
Let $R$ be a commutative ring and let $f\in R[x]$ be a Gaussian polynomial. Then $c(f)$ is an invertible or, at least, a locally principal ideal.
A number of authors contributed to the solution of the above conjecture of Kaplansky. D.D. Anderson and Kang ([@AK]) initiated the renewed interest in Kaplansky’s conjecture; Glaz and Vasconcelos ([@GV], [@GV1]) showed that the conjecture holds for integrally closed Noetherian domains; Heinzer and Huneke ([@HH]) proved that the conjecture holds for Noetherian domains; Loper and Roitman ([@LR]) solved the conjecture for all domains; Lucas ([@L], [@L1]) extended this result to a partial answer for non-domains. In general, Kaplansky’s conjecture is false (see [@GV], [@GV1]).
A Prüfer (see Definition \[pru\]) domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context, in particular, a Noetherian Prüfer domain is a Dedekind domain. Among other things, H. Tsang ([@T]) proved that an integral domain is Gaussian if and only if it is Prüfer, a result also proved independently by R. Gilmer in [@RG]. Thus Gaussian rings provide another class of rings extending the class of Prüfer domains to rings with zero divisors. A homological generalization to rings $R$ of the Prüfer property for domains, is the condition: $R$ has weak global dimension less or equal to one, $\operatorname{w.gl.dim}\leq 1$. A subfamily of this class of rings is the class of semihereditary rings. Recall that a ring is semihereditary if every finitely generated ideal of $R$ is projective. Semihereditary rings $R$ are precisely those rings $R$ with $\operatorname{w.gl.dim}\leq 1$ which are coherent. Domains of weak global dimension less or equal to 1, in particular, semihereditary domains, are Prüfer domains. A lot of work and progress has been made in investigating Prüfer like conditions in commutative rings in the last 15 years, this can be found in the survey article [@GS]. To give a homological characterization to several classes of rings has been a topic of research for several years, for example a ring $R$ is semi-simple if and only if the global dimension of $R$ is $0$ ([@SG1]). Similarly, $\operatorname{w.gl.dim}R=0$ if and only if $R$ is a Von-Nuemann regular ring (see [@H]). We note that it follows from Osofsky [@O] that arithmetical rings have weak global dimension at most one or $\infty$. In the same vein, the author of [@SG] is concerned with giving a homological characterization to Gaussian rings. She considers the question: For a Gaussian ring $R$, what possible values may $\operatorname{w.gl.dim}R$ take? In [@BG], the authors consider five possible extensions of the Prüfer domain notion to the case of commutative rings with zero divisors, two among which are Gaussian rings and rings with weak global dimension (see Definition \[weak\]) at most one. They consider the problem of determining the possible values for the weak global dimension of a Gaussian ring. At the end of their paper, they make the following conjecture.
The weak global dimension of a Gaussian ring is either $0,1$ or $\infty$.
Our aim in this article is to give a proof of the above conjecture. The above conjecture is also listed as an open question in the recent survey article [@GS]. In a recent paper [@AJK], the authors have validated the Bazzoni-Glaz conjecture for the class of rings called fqp-rings. The class of fqp-rings fall strictly between the classes of arithmetical rings and Gaussian rings. In [@SG], the author shows that the weak global dimension of a coherent Gaussian ring is either $\infty$ or at most one. She also shows that the weak global dimension of a Gaussian ring is at most one if and only if it is reduced. So to prove the conjecture it is enough to show that $\operatorname{w.gl.dim}R=\infty$ for all non-reduced Gaussian rings $R$. Since $\operatorname{w.gl.dim}R=\sup\{\operatorname{w.gl.dim}R_{\mathfrak{p}}\mid \mathfrak{p}\in \operatorname{Spec}(R)\}$, it is enough to prove the conjecture for non-reduced local Gaussian rings. For any non reduced local Gaussian ring $R$ with nilradical $\mathcal{N}$, either $(i)$ $\mathcal{N}$ is nilpotent or $(ii)$ $\mathcal{N}$ is not nilpotent. Except when $\mathcal{N}^2=0$, the authors of [@BG] prove that if $R$ satisfies $(i)$, then $\operatorname{w.gl.dim}R=\infty$. In this paper we prove that if $R$ satisfies $(ii)$, then $\operatorname{w.gl.dim}R=\infty$ (cf. Theorem \[main\]). We also give a complete proof of $(i)$. Now we briefly describe the strategy of the proof. After localization at minimal prime ideal, $\mathcal{N}$, the maximal ideal and the nilradical of $R_{\mathcal{N}}$ coincide, let us denote it by $\mathcal{N'}$. If $\mathcal{N}^2\neq 0$, then $\mathcal{N'}\neq 0$. Hence to prove the Bazzoni-Glaz conjecture in this case, it suffices to show that the weak global dimension of a local Gaussian ring with the maximal ideal coinciding with the nilradical is infinite. If $\mathcal{N}^2=0$, then $\mathcal{N'}$ can be zero (see Example \[EG\]). Hence we discuss this case separately in Section 6.
In Section 3, we consider some homological properties of local Gaussian rings. In particular we consider local Gaussian rings $(R,\mathfrak{m})$ which are not fields, with the property that each element of $\mathfrak{m}$ is a zero divisor. In this case we prove that $\operatorname{w.gl.dim}R\geq 3$.
In [@BG Section 6], the authors consider local Gaussian rings $(R,
\mathfrak{m})$ such that the maximal ideal $\mathfrak{m}$ coincides with the nilradical of $R$. With this set up in Section 4, we prove that if $\mathcal{N}$ is not nilpotent, then $\operatorname{w.gl.dim}R=\infty$.
In Section 5, we prove the conjecture except the case $\mathcal{N}^2=0$, and in the last section we prove it completely.
Throughout this paper, $R$ is a commutative ring with unit, $(R,\mathfrak{m})$ is a local ring(not necessarily Noetherian) with unique maximal ideal $\mathfrak{m}$. For $R$-modules $M$ and $N$, we write $M\otimes N$ and $\operatorname{Tor}_*(M,N)$ instead of $M\otimes_R N$, and $\operatorname{Tor}_*^R(M, N)$, respectively. We denote the set of all prime ideals of $R$ by $\operatorname{Spec}(R)$ and the set of all maximal ideals by $\operatorname{Max}(R)$.
Preliminary Results
===================
In this section we will recall some definitions and results that we will need in later sections.
Let $M$ be a module over $R$. The weak global dimension of $M$ (denoted by $\operatorname{w.gl.dim}_R(M)$) is the minimum integer (if it exists) such that there is a resolution of $M$ by flat $R$ modules $0\to F_n\to \cdots\to F_1\to F_0\to M\to 0$. If no finite resolution by flat $R$ modules exists for $M$, then we set $\operatorname{w.gl.dim}_R(M)=\infty$.
Now we define the weak global dimension of a ring $R$ denoted as $\operatorname{w.gl.dim}(R)$. It is also sometimes suggestively called as the $\operatorname{Tor}$-dimension.
\[weak\] $\operatorname{w.gl.dim}(R)$=$\sup${$\operatorname{w.gl.dim}_R(M) \mid M$ is an $R$-module}.
Recall that $\operatorname{w.gl.dim}(R)$=$\sup${$d \mid \operatorname{Tor}_d(M,N)\neq 0$ for some $R$-modules $M,N$}. The $\operatorname{w.gl.dim}(R)\leq 1$ if and only if every ideal of $R$ is flat, or equivalently, if and only if every finitely generated ideal of $R$ is flat.
We now define a Prüfer domain.
\[pru\] An integral domain $R$ is called a Prüfer domain if every finitely generated non-zero ideal of $R$ is invertible.
L. Fuchs introduced the class of arithmetical rings in [@F].
A ring $R$ is arithmetical if the lattice of the ideals of $R$ is distributive.
In [@J], the author characterized arithmetical rings by the property that in every localization at a maximal ideal, the lattice of the ideals is linearly ordered by inclusion. Hence in a local arithmetical ring, the lattice of the ideals is linearly ordered by inclusion. Thus local arithmetical rings provide another class of rings extending the class of valuation domains to rings with zero-divisors.
The next theorem appears in Tsang’s (see [@T]) unpublished thesis.
\[thm2.1\] Let $R$ be a Gaussian ring. If $R$ is local, then
- $R$ is Gaussian if and only if $R_{\mathfrak{m}}$ is Gaussian for all $\mathfrak{m}\in \operatorname{Max}(R)$;
- $R$ is Gaussian if and only if $R_{\mathfrak{p}}$ is Gaussian for all $\mathfrak{p}\in \operatorname{Spec}(R)$;
- the prime ideals of $R$ are linearly ordered under inclusion; and
- the nilradical of $R$ is the unique minimal prime ideal of $R$.
We will need several equivalent characterizations of local Gaussian rings, which we now state.
\[thm2.2\] Let $(R,\mathfrak{m})$ be a local ring with maximal ideal $\mathfrak{m}$. The following conditions are equivalent.
- $R$ is a Gaussian ring;
- If $I$ is a finitely generated ideal of $R$ and $(0:I)$ is the annihilator of $I$, then $I/{I\cap(0:I)}$ is a cyclic $R$-module;
- Condition $(ii)$ for two generated ideals;
- For any two elements $a,b\in R$, the following two properties hold:
- $(a,b)^2=(a^2)$ or $(b^2)$;
- If $(a,b)^2=(a^2)$ and $ab=0$, then $b^2=0$.
- If $I=(a_1,a_2,\ldots,a_n)$ is a finitely generated ideal of $R$, then $I^2=(a_{i}^2)$ for some $1\leq i\leq n$.
The implication $(iv)\Rightarrow (i)$ was noted by Lucas in [@L] and the rest of Theorem \[thm2.2\] was proved by Tsang in [@T]. The next two results can be found in [@BG].
\[thm2.3\] Let $(R,\mathfrak{m})$ be a local Gaussian ring and let $D=\{x\in R \mid x^2=0\}$. The following hold:
- $D$ is an ideal of $R$, $D^2=0$, and $R/D$ is an arithmetical ring;
- For every $a\in R$, $(0:a)$ and $D$ are comparable and $D\subseteq
Ra+(0:a)$;
- If $a\in \mathfrak{m}\setminus D$, then $(0:a)\subseteq D$;
- Let $\mathfrak{m}$ be the nilradical of $R$. If $\mathfrak{m}$ is not nilpotent, then $\mathfrak{m}=\mathfrak{m^2}+D$ and $\mathfrak{m^2}=\mathfrak{m^3}$.
\[prop2.1\] Let $(R,\mathfrak{m})$ be a local Gaussian ring. If $\mathfrak{m}$ is non-zero and nilpotent, then $\operatorname{w.gl.dim}_R \mathfrak{m}=\infty$.
Some results on local Gaussian rings
====================================
Throughout this section (as well as in the sequel) $D$ is supposed to be as in Theorem \[thm2.3\]. It is well known that if the $\operatorname{w.gl.dim}_R(M)=n$, then there exists a cyclic $R$-module, say $R/I$ such that $\operatorname{Tor}_n(R/I, M)\neq 0$. In the next lemma, we show that this cyclic module can be chosen with some additional properties.
\[lemma2\] Let $R$ be local Gaussian ring and $M$ be a module over $R$ with $\operatorname{w.gl.dim}_R(M)=n$. Let $I$ be an ideal of $R$. If $\operatorname{Tor}_n(R/I, M)\neq 0$, then there exists an ideal $J\subset R$ such that $\operatorname{Tor}_n(R/J, M)\neq 0$ and $J\supseteq I+D$.
Suppose the lemma is not true. Then we will prove that the natural projection $R/I \to
R/(I+x_1R+\cdots +x_mR)$ induces an inclusion $$\label{tor1}
\operatorname{Tor}_n (R/I, M)\hookrightarrow \operatorname{Tor}_n (R/(I+x_1R+\cdots +x_mR), M)$$ for any finite subset $\{x_1, \dots, x_m\}\subset D$. Towards that end, set $I_0=I$ and define $I_p$ inductively as $I_p=I_{p-1}+x_p R$ for all $1\leq p\leq m$. We have the following short exact sequence $$0\to (x_pR+I_{p-1})/I_{p-1} \to R/I_{p-1} \to R/I_p \to 0$$ for all $1\leq p\leq m$. The homomorphism $f: R \to (x_pR+I_{p-1})/I_{p-1}$ defined by $f(r)=rx_p+I_{p-1}$ for all $r\in R$ induces an isomorphism $R/\operatorname{Ker}f \cong (x_p
R+I_{p-1})/I_{p-1}$. Furthermore we have that $\operatorname{Ker}f\supset I_{p-1}+ (0:x_p)
\supset I+D$. If $\operatorname{Tor}_n ((x_pR+I_{p-1})/I_{p-1}, M)\neq 0$, then the lemma is true with $J=\operatorname{Ker}f$. So assume that $\operatorname{Tor}_n ((x_pR+I_{p-1})/I_{p-1}, M)= 0$. In this case the natural projection $R/I_{p-1} \to R/I_p$ induces an inclusion $\operatorname{Tor}_n (R/I_{p-1}, M)\hookrightarrow \operatorname{Tor}_n (R/I_p, M)$ for all $1\leq p\leq m$, proving (\[tor1\]).
Now let $\mathcal{X}$ denote the following class of ideals: $J\in \mathcal{X}$ iff $J\subset D$ and $J$ is finitely generated. Then $$\label{directlim}
\varinjlim_{J\in \mathcal{X}} \operatorname{Tor}_n(R/(I+J),M)=\operatorname{Tor}_n(R/(I+D),M)$$
Using (\[tor1\]) and (\[directlim\]), we obtain an inclusion $\operatorname{Tor}_n (R/I,
M)\hookrightarrow \operatorname{Tor}_n (R/(I+D), M)$. Thus $\operatorname{Tor}_n (R/(I+D), M)\neq 0$ and the lemma is proved.
The next lemma is an immediate consequence of the long exact sequence of $\operatorname{Tor}$ groups applied to the given short exact sequence. We note it here for the readers convenience.
\[lemma3\] Let $R$ be a commutative (not necessarily local Gaussian) ring. Let $M, M_1$ and $M_2$ be $R$-modules and $f:M_1\to M_2$ be an injective homomorphism. If the $\operatorname{w.gl.dim}_R(M)=n$, then $f_*:\operatorname{Tor}_n(M_1, M) \to
\operatorname{Tor}_n(M_2, M)$ is also injective.
The proof is a direct consequence of the long exact sequence of $\operatorname{Tor}$ groups applied to the given short exact sequence and the fact that $\operatorname{Tor}_{n+1} (M_2/M_1, M)=0$.
The next lemma is very useful for us in the sequel.
\[lemma4\] Let $(R,\mathfrak{m})$ be a local Gaussian ring and $M$ be a module over $R$. If $\operatorname{w.gl.dim}_R(M)=n\geq 1$, then $\operatorname{Tor}_n (R/D, M )=0$.
Suppose the lemma is not true. Consider a free resolution of $M$: $\cdots \xrightarrow{\partial_{n+2}} R^{X_{n+1}}\xrightarrow{\partial_{n+1}}
R^{X_n}\xrightarrow{\partial_{n}}
\cdots \xrightarrow{\partial_1} R^{X_0}\xrightarrow{\partial_0} M$, where $X_i$ are sets. By assumption $\operatorname{Tor}_n (R/D, M)=\operatorname{Ker}(\overline{\partial_n})/ \operatorname{Im}(\overline{\partial_{n+1}})\neq
0$, where $\overline{\partial_i}$ is the natural homomorphism $\overline{\partial_i}:(R/D)^{X_i}\to (R/D)^{X_{i-1}}$ obtained after tensoring the above resolution by $R/D$ for all $i\in \mathbb{N}$. Since $\operatorname{Tor}_n (R/D, M)\neq 0$, there exists a $\overline{w}\in \operatorname{Ker}(\overline{\partial_n})$ such that $\overline{w}\notin \operatorname{Im}(\overline{\partial_{n+1}})$. Let $w$ be the representative of $\overline{w}$ in $R^{X_n}$. Hence $\partial_n(w)\in D^{X_{n-1}}$. Let $\lambda_1, \dots,\lambda_m\in D$ be the finitely many non-zero entries of $\partial_n(w)$. Now we consider two cases.
[**Case 1**]{}. There exists an $a\in \mathfrak{m}\setminus D$ such that $a\lambda_j=0$ for all $1\leq j\leq m$.\
Define a homomorphism $f:R/D \to R/a D$ which is multiplication by $a$. Using Theorem \[thm2.3\]$(iii)$, it follows that $(0:a)\subset D$. This gives the injectivity of $f$. Therefore by Lemma \[lemma3\], $f_*:\operatorname{Tor}_n (R/D, M) \to \operatorname{Tor}_n
(R/(aD), M)$ is injective and hence $f_*(\overline{w})\neq 0$. It is easy to verify that $aw$ is a representative of $f_*(\overline{w})$ in $R^{X_n}$. Since $a\in (0:\lambda_j)$ for all $1\leq j\leq m$, we obtain that $\partial_n(aw)=a\partial_n(w)=0$. This would imply that $f_*(\overline{w})=0$, a contradiction.
[**Case 2**]{}. For all $a\in R\setminus D$ at least one $a\lambda_j\neq 0$.\
We have an injective homomorphism $g: R/D\to R^m$ defined by $g(1)=(\lambda_1, \dots
, \lambda_m)$. By Lemma \[lemma3\], the induced homomorphism $g_*:\operatorname{Tor}_n (R/D, M) \to \operatorname{Tor}_n (R^m,M)$ is injective. This is a contradiction as $\operatorname{Tor}_n (R^m,M)=0$.
\[llemma\] Let $(R,\mathfrak{m})$ be a local Gaussian ring and $M$ be a module over $R$. If $\operatorname{w.gl.dim}_R(M)=n\geq 1$, then there exists an $a\in \mathfrak{m}\setminus D$ such that $\operatorname{Tor}_n(R/(aR+D), M)\neq 0$.
There exists an ideal $I\subset R$ such that $\operatorname{Tor}_n(R/I, M)\neq 0$. Without loss of generality one can assume that $D\subset I$ (see Lemma \[lemma2\]). Using Lemma \[lemma4\], we obtain that $I\neq D$. Let $\mathcal{X}$ denote the following class of ideals: $J\in \mathcal{X}$ iff $J\subset I$ and $J$ is finitely generated. Since $\operatorname{Tor}_n(R/I, D)= \varinjlim_{J\in \mathcal{X}}\operatorname{Tor}_n(R/ J, M)\neq 0$, there exist $a_1,\dots, a_m\in I$ such that $\operatorname{Tor}_n (R/ (a_1R+\cdots +a_mR+D), M)\neq 0$. By Theorem \[thm2.3\]$(i)$, $R/D$ is a local arithmetical ring. Hence there exists an $i$ such that $a_1R+\cdots +a_mR+D=a_iR+D$ for some $1\leq i\leq m$.
Let $(R,\mathfrak{m})$ be a local ring such that each element of $\mathfrak{m}$ is a zero divisor. If $b\in \mathfrak{m}$, then there exists an element $r_b\in \mathfrak{m}$ such that $br_b=0$. In general, the element $r_b$ depends on $b$. In the next lemma, we show that if $(R,\mathfrak{m})$ is a local Gaussian ring, then a slightly stronger result is true. In particular, we show that for finitely many elements in $\mathfrak{m}$, there exists a single element in $\mathfrak{m}$ that annihilates all of them.
\[singleann\] Let $(R,\mathfrak{m})$ be a local Gaussian ring such that each element of $\mathfrak{m}$ is a zero divisor. If $\lambda_1, \dots, \lambda_n\in \mathfrak{m}$, then there exists a non-trivial element $a\in \mathfrak{m}$ such that $a\lambda_j=0$ for all $1\leq j\leq n$.
We divide the proof into two cases.
[**Case 1**]{}: $\lambda_1,\dots,\lambda_n\in D$.
For $1\leq i\leq n$, choose any $\lambda_i\in D\setminus 0$. Using Theorem \[thm2.3\]$(i)$, the desired result follows.
[**Case 2**]{}: There exist $j\in \{1,2,\ldots,n\}$ such that $\lambda_j\notin D$.
By Theorem \[thm2.3\]$(iii)$, it follows that $(0:\lambda_j)\subseteq D$. Set $I=(\lambda_1,\ldots, \lambda_n)$. So $(0:I)\subset (0:\lambda_j)\subseteq D$. Using Theorem \[thm2.2\]$(ii)$, we obtain that $I/{I\cap (0:I)}$ is a cyclic $R$-module, say its generator is $\lambda$. Hence we can write $\lambda_i=r_i\lambda+d_i$ for all $1\leq i\leq n$, where $d_i\in I\cap (0:I)$ and $r_i\in R$. Observe that $\lambda\in \mathfrak{m}\setminus D$. Choose any $d\in (0:\lambda)\setminus 0$. Using Theorem \[thm2.3\]$(iii)$, it follows that $d\in D$. Multiplying the equation expressing $\lambda_i$ in terms of $\lambda$ with $d$, we obtain $d\lambda_i=dr_i\lambda + dd_i$ for all $1\leq i\leq n$. Using Theorem \[thm2.3\]$(i)$, we obtain that $dd_i=0$. Thus $d\lambda_i=0$ for all $1\leq i\leq
n$.
\[lemma5\] Let $(R,\mathfrak{m})$ be a local Gaussian ring and $M$ be a module over $R$. If each element of $\mathfrak{m}$ is a zero divisor and $\operatorname{w.gl.dim}_R(M)=n\geq 1$, then $\operatorname{Tor}_n (R/\mathfrak{m}, M )=0$.
The proof of this Lemma follows by substituting $\mathfrak{m}$ for $D$ in Lemma \[lemma4\]. As a result of Lemma \[singleann\], the proof of lemma \[lemma5\] falls under Case 1 of Lemma \[lemma4\].
The next result seems to be restricted but it is useful for our further purposes.
\[prop1.1\] Let $(R,\mathfrak{m})$ be a local Gaussian ring. If $\mathfrak{m}\neq 0$ and each element of $\mathfrak{m}$ is a zero divisor, then $\operatorname{w.gl.dim}(R)\geq 3$.
If $\mathfrak{m} =D$, then Proposition \[prop2.1\] implies that $\operatorname{w.gl.dim}(R) = \infty$. If $\mathfrak{m}\neq D$, then take any $x\in
\mathfrak{m}\setminus D$ and consider the following resolution of $R/xR$: $$0\to (0:x)\to R\xrightarrow{x_m}R\xrightarrow{\pi}R/xR ,$$ where $\pi$ is the natural projection and $x_m$ is multiplication by $x$. If $\operatorname{w.gl.dim}(R)<3$, then $(0:x)$ must be flat. Thus it suffices to show that $(0:x)$ is not flat. We will use the fact that if $M$ is a flat $R$ module then $I\otimes M=IM$ for all ideals $I\subset R$. Set $I=xR$ and $M=(0:x)$ and observe that $IM=0$. Hence it suffices to show that $I\otimes
(0:x)\neq 0$. Since $x\notin D$, Theorem \[thm2.3\]$(iii)$ implies that $(0:x)\subset D$. Define a homomorphism $\theta : I\otimes (0:x)\to (0:x)$ as follows: if $a\in I$ and $b\in (0:x)$, then set $\theta(a\otimes b)=rb$, where $r\in
R$ is such that $a=xr$. If there is another $r'\in R$ such that $a=xr'$, then $(r-r')\in (0:x)$ which implies that $(r-r')b=0$. Taking into account the last remark, it is easy to check that $\theta$ is well defined. Moreover the homomorphism $\theta':(0:x)\to I\otimes (0:x)$ defined by $\theta'(c)=x\otimes c$ for all $c\in
(0:x)$ is an inverse of $\theta$. Hence we have an isomorphism $\theta:I\otimes (0:x)\cong (0:x)$ which shows that $I\otimes (0:x)\neq 0$, proving that $(0:x)$ is not flat.
Let $R$ be a local Gaussian ring which admits the following property:
\[hypo1\] For all $x\in D\setminus 0$, $(0:x)$ is not cyclic modulo $D$. In other words there is no $a\in R\setminus D$ such that $(0:x)=aR+D$.
By Theorem \[thm2.2\], we know that if $I=(a_1,a_2,\cdots,a_n)$ is a finitely generated ideal of a local Gaussian ring, then $I^2=(a_i^2)$ for some $i$, where $1\leq i\leq n$. But for this particular $i$, $a_i$ could have the property that $a_i^4=0$. In the next lemma, we show that if $(R,\mathfrak{m})$ is a local Gaussian ring with some additional hypothesis, then the square of any finitely generated proper ideal of $R$ is contained in the square of an element, say $x\in \mathfrak{m}$ with $x$ having the property that $x^4\neq 0$. We also show that $\mathfrak{m}^2$ is flat.
\[flat\] Let $(R,\mathfrak{m})$ be a local Gaussian ring such that each element of $\mathfrak{m}$ is a zero divisor. If $R$ admits Property (\[hypo1\]) and $\mathfrak{m}\neq D$ , then
- $\mathfrak{m}=\mathfrak{m}^2+D$;
- for any finitely generated ideal $J\subset \mathfrak{m}$ there exist $x\in \mathfrak{m}$ such that $J^2\subset x^2R$ and $x^2\notin D$;
- $\mathfrak{m}^2$ is flat.
(i): Let $a\in \mathfrak{m}\setminus D$. Since every element of $\mathfrak{m}$ is a zero divisor, there exists $x\in D\setminus 0$ such that $ax=0$. By Property (\[hypo1\]), $(0:x)\neq aR+D$. So there exists some $b\in
\mathfrak{m}$ such that $b\in (0:x)$ and $b\notin aR+D$. Theorem \[thm2.3\]$(i)$ implies that $R/D$ is a local arithmetical ring. So $a\in b R+D$ and hence $a=b r +
d$ for some $r \in R$ and $d\in D$. Moreover $b\notin aR+D$ which implies that $r$ is not a unit and hence $r\in \mathfrak{m}$. Thus $a\in \mathfrak{m}^2+D$.
(ii): First we will show that if $x^2\in D$ for all $x\in \mathfrak{m}$, then $\mathfrak{m^2}\subset D$. Towards that end let $z\in \mathfrak{m^2}$. Such a $z$ is of the form $z=\sum_{i=1}^n x_iy_i$, where $x_i,y_i\in m$ for all $1\leq i\leq n$. Using Theorem \[thm2.2\]$(iv)$, it follows that $(x_i,y_i)^2=(x_i^2)\;\text{or}\;(y_i^2)$ for all $1\leq i\leq n$. This shows that $x_iy_i\in D$ for all $1\leq i\leq n$. Recalling that $D$ is an ideal of $R$, it follows that $z\in D$. Hence we have proved that $\mathfrak{m^2}\subset D$. By $(i)$, this would imply that $\mathfrak{m}=D$, a contradiction. Thus there exists an $x\in
m$ such that $x^2\notin D$. By Theorem \[thm2.2\]$(v)$, for any finitely generated ideal $J$ we have $J^2=y^2R$ for some $y\in J$. If $y^2\notin D$, then we are done. If $y^2\in D$, choose any $x\in \mathfrak{m}$ with $x^2\notin D$ and observe that $y^2\in x^2R$. Thus $J^2\subset x^2 R$.
(iii): To prove that $\mathfrak{m}^2$ is flat over $R$, we show that for any ideal $I\subset R$, the natural homomorphism $f: I\otimes \mathfrak{m}^2 \to
\mathfrak{m}^2$ is injective. Assume that $w\in I\otimes \mathfrak{m}^2$ is such that $f(w)=0$. Set $w = \sum_{i=1}^k z_i\otimes x_i y_i$, where $z_i\in I$ and $x_i,y_i\in \mathfrak{m}$. By (ii), there exist $x\in \mathfrak{m}$ such that $x^2\notin D$ and $x_iy_i\in x^2 R$ for all $1\leq i\leq n$. Put $x_iy_i=x^2r_i$, where $r_i\in R$. Then $w=z\otimes x^2$, where $z =\sum_{i=1}^k z_ir_i \in I$. Hence $f(z \otimes x^2)=0 \Leftrightarrow zx^2=0$. If $z=0$, then $w=0$ and the proof is finished. So assume that $z\neq 0$. Using Theorem \[thm2.3\]$(iii)$, we obtain that $(0:a)\subseteq D$ for all $a\in \mathfrak{m}\setminus D$. Since $x^2\in
\mathfrak{m}\setminus D$, it follows that $z\in D$ and either $zx = 0$ or $zx\neq
0$. If $zx = 0$, then $z\in D\setminus 0$ and $x\in (0:z)$. It follows from Property (\[hypo1\]) that $(0:z)\neq xR+D$. So there exists $y\in
\mathfrak{m}$ such that $y\in (0:z)$ and $y\notin xR+D$. By Theorem \[thm2.3\]$(i)$, we obtain that $R/D$ is a local arithmetical ring. Hence $(y)\not\subset (x)$. So $x=cy+d'$, where $c\in \mathfrak{m}$ and $d'\in D$. Computing $w$, we obtain $$w = z\otimes x^2 = z\otimes (cy+d')^2=z\otimes (c^2y^2+2cyd'+d'^2)=zy^2\otimes
c^2+zd'\otimes 2cy+z\otimes d'^2 .$$ Noting that $d'^2,zd'\in D^2=0$ and that $zy^2=0$, we obtain $w=0$. If $zx\neq 0$, we have $zx\in D\setminus 0$ and $x\in (0:zx)$. By (\[hypo1\]), there exists $h\in \mathfrak{m}$ such that $h\in (0:zx)$ and $h\notin xR+D$. Using the same argument as above, there exists an $a \in \mathfrak{m}$ such that $x=ah+d''$. Observing that $zd'',d''^2\in D^2$ we obtain that $w = z\otimes x^2=z\otimes
(ah+d'')^2=z\otimes a^2h^2$. Furthermore, by (i) we can write $a=b+d$ where $b\in
\mathfrak{m}^2$ and $d\in D$. Therefore $$\label{tensor}
w=z\otimes (ah^2(b+d))= z\otimes (ah^2b)+z\otimes (ah^2d) =(zah^2)\otimes
b+(zd)\otimes (ah^2).$$ Substituting $0=zd\in D^2$ and $ah=x-d''$ in (\[tensor\]) and recalling that $h\in
(0:zx)$, we obtain $w=(zxh)\otimes b-zhd''\otimes b=0$.
\[lemma1.2\] Let $(R,\mathfrak{m})$ be a local Gaussian ring such that each element of $\mathfrak{m}$ is a zero divisor. If $R$ admits Property (\[hypo1\]), $\mathfrak{m}\neq D$ and $\operatorname{w.gl.dim}R<\infty$, then $\mathfrak{m}$ is flat.
If $\mathfrak{m} =\mathfrak{m^2}$, then the result follows by Lemma \[flat\]. So assume that $\mathfrak{m} \neq \mathfrak{m^2}$. By assumption, the $\operatorname{w.gl.dim}_R(R/\mathfrak{m})=n<\infty$. Hence there exists an $R$-module $M$ such that $\operatorname{Tor}_n(R/\mathfrak{m}, M)\neq 0$. If we show that $n\leq 1$, then the lemma will be proved. Suppose that $n\geq 2$. Using Lemma \[flat\]$(iii)$, we obtain that $\operatorname{Tor}_{n}(R / \mathfrak{m^2}, M)=0$. Now consider the short exact sequence $0\to \mathfrak{m} / \mathfrak{m^2} \to R / \mathfrak{m^2}\to R / \mathfrak{m} \to 0$ and the following segment of the corresponding long exact sequence of $\operatorname{Tor}$ groups $$\operatorname{Tor}_{n+1}(R / \mathfrak{m} , M)\to \operatorname{Tor}_{n}(\mathfrak{m}/ \mathfrak{m^2}, M)\to
\operatorname{Tor}_{n}(R / \mathfrak{m^2}, M) .$$ The above sequence implies that $\operatorname{Tor}_{n}(\mathfrak{m}/ \mathfrak{m^2}, M)=0$. Observing that $\mathfrak{m} / \mathfrak{m^2}$ is a vector space over $R/\mathfrak{m}$, we obtain that $\mathfrak{m} /\mathfrak{m^2} = \bigoplus R/\mathfrak{m}$. Hence $0=\operatorname{Tor}_{n}(\mathfrak{m} / \mathfrak{m^2}, M)=\operatorname{Tor}_{n}(\bigoplus
R/\mathfrak{m}, M)=\bigoplus \operatorname{Tor}_{n}(R/\mathfrak{m}, M)$, a contradiction.
Local Gaussian rings with nilradical being the maximal ideal
============================================================
The idea of the next lemma is taken from [@O], but we give a more general result and with a slightly different proof.
\[lemma6\] Let $(R,\mathfrak{m})$ be a local arithmetical ring with nilradical $\mathfrak{m}$. For any $x\in \mathfrak{m}\setminus 0$, if $(0:x)=I$ then $(0:I)=(x)$.
Clearly $(x)\subseteq (0:I)$. We want to show that $(0:I)\subseteq
(x)$. Towards that end assume that there exists a $z\in (0:I)$ such that $z\notin
(x)$. Recalling that the ideals in a local arithmetical ring are linearly ordered under inclusion, we obtain that $x=\lambda z$ where $\lambda\in \mathfrak{m}$. Hence $\lambda\notin I$ which implies that $I\subset (\lambda)$. By induction on $k$, we will show that $I\subset (\lambda^k)$ for all $k\in \mathbb{N}$. The case $k=1$ is obvious. Let $b\in I$ be arbitrary. Since $I\subset (\lambda)$ there exists a $t\in
\mathfrak{m}$ such that $b=\lambda t$. Notice that we have $0=zb=z\lambda t=xt$. Hence $t\in I=(0:x)$. By the induction hypothesis, $I\subset (\lambda^k)$. So $t=\lambda^kt_1$ where $t_1\in \mathfrak{m}$. Hence $b=\lambda^{k+1} t_1$, where $t_1\in \mathfrak{m}$. Thus $I\subset (\lambda^{k+1})$ for all $k\in \mathbb{N}$. Since $\lambda$ is nilpotent, we obtain that $I=0$, a contradiction.
In what follows let $R'=R/D$ and $\mathfrak{m'}= \mathfrak{m}/D$. Recall that if $R$ is a local Gaussian ring, then $R'$ is a local arithmetical ring by Theorem \[thm2.3\]$(i)$.
\[lemma7\] Let $(R,\mathfrak{m})$ be a local Gaussian ring with nilradical $\mathfrak{m}$ and let $M$ be an $R$-module. If $\operatorname{w.gl.dim}_R (M)=n\geq 1$, then the following conditions hold.
- there is a non trivial element $x\in \mathfrak{m'}$ such that $\operatorname{Tor}_n(R'/xR', M)\neq 0$;
- for any non trivial element $z\in \mathfrak{m'}$ and ideal $J\subset
R'$ such that $z\in J$, $zR'\neq J$, the natural projection $R'/zR'\to R'/J$ induces a trivial map $0:\operatorname{Tor}_n(R'/zR', M)\to \operatorname{Tor}_n(R'/J, M)$;
- $\operatorname{Tor}_n(R'/zR', M)\neq 0$ for any non trivial element $z\in \mathfrak{m'}$.
(i): This is a special case of Lemma \[llemma\].
(ii): Let $I=(0: z)\subset R'$. Using Lemma \[lemma6\], we obtain that $(0:I)=zR'$. This implies that $(0:I)\subset J$ and $(0:I)\neq J$. Hence there exists $y\in I$ such that $(0:y)\subset J$ and $(0:y)\neq J$. Thus we have the inclusions $zR' \subset (0:y)\subset J$ which give rise to the natural projections $R'/zR' \to R'/(0:y)\to R'/J$. Using Lemma’s \[lemma3\], \[lemma4\] and the fact that $R'/(0:y)\cong yR' \subset R'$, we obtain that $\operatorname{Tor}_n(R'/(0:y), M)=0$. Hence the composition of the following maps $\operatorname{Tor}_n(R'/zR', M) \to \operatorname{Tor}_n(R'/(0:y), M)\to \operatorname{Tor}_n(R'/J, M)$ is trivial.
(iii): By (i), we have a non trivial element $x\in \mathfrak{m'}$ such that $\operatorname{Tor}_n(R'/xR', M)\neq 0$. For any non trivial element $z\in \mathfrak{m'}$, either $z\in xR'$ or $x\in zR'$. If $z\in xR'$ and $z\neq x$, then there exists $\lambda \in
\mathfrak{m'}$ such that $z=\lambda x$. Define a map $\alpha : R'/xR'\to R'/zR'$ by $\alpha(r+xR')= \lambda r+zR'$ for all $r\in R'$. Since $x\notin (0:\lambda)$, it follows that $(0:\lambda)\subset xR'$. This shows that $\alpha$ is injective. Using Lemma \[lemma3\], we obtain that $\alpha$ induces an inclusion $\operatorname{Tor}_n(R'/xR',
M)\hookrightarrow \operatorname{Tor}_n(R'/zR', M)$. Thus $\operatorname{Tor}_n(R'/zR', M)\neq 0$.
In the case when $x\in zR'$ and $x\neq z$, there exists $\lambda'\in \mathfrak{m'}$ such that $x=\lambda' z$. Define a map $\sigma : R'/zR'\to R'/xR'$ by $\sigma(r+zR')=r +xR'$ for all $r\in R'$. Since $z\notin (0:\lambda')$, we obtain that $(0:\lambda')\subset zR'$. Thus $\sigma$ is injective. Observe that $xR'\subset \lambda' R'$ and $xR'\neq \lambda' R'$. Now consider the short exact sequence $$0 \rightarrow R'/zR'\xrightarrow{ \ \sigma \ } R'/xR'\xrightarrow{ \ \tau \ }
R'/\lambda' R' \rightarrow 0\,,$$ where $\tau $ is the natural projection. Using $(ii)$, we see that $\tau$ induces the trivial map $0:\operatorname{Tor}_n( R'/xR', M)\to \operatorname{Tor}_n( R'/\lambda'R', M)$. Therefore $\sigma$ induces an epimorphism $\operatorname{Tor}_n( R'/zR', M)\twoheadrightarrow \operatorname{Tor}_n( R'/xR', M)$, which implies that $\operatorname{Tor}_n( R'/zR', M)\neq 0$.
Let $\deg(r)$ denote the degree of nilpotency of $r\in R$. Noting that the nilpotency degree of an element $r\in R$ is the smallest $k\in \mathbb{N}$ such that $r^k=0$, we state our next lemma.
\[deg\] Let $(R, \mathfrak{m})$ be a local Gaussian ring with nilradical $\mathfrak{m}$ and let $\lambda\in \mathfrak{m}$. If $\mathfrak{m}$ is not nilpotent, then there exists $z\in \mathfrak{m}$ such that $\deg(z) > \deg(\lambda)$.
Let $\deg(\lambda)=n$. Suppose the lemma is not true, then $\deg(z)\leq
\deg(\lambda)$ for all $z\in \mathfrak{m}$. We will show that $\mathfrak{m}^n=0$. This will give us a contradiction, as $\mathfrak{m}$ is not nilpotent. Towards that end, let $z_1,\ldots,z_n\in \mathfrak{m}$ and consider $I=(z_1,\ldots,z_n)$. Using Theorem \[thm2.2\]$(ii)$, we can write $z_i=r_iz+d_i$ for some $z\in I$, $r_i\in R$ and $d_i\in I\cap (0:I)\subset (0:z_i)$ for all $1\leq
i\leq n$. So $$\label{prod}
z_1z_2\cdots z_n=\prod_{i=0}^n (zr_i+d_i)$$ After expanding the right hand side of (\[prod\]), observe that every term of the expansion except the term $d_1\cdots d_n$ contains a $z$ and some $d_i$, where $1\leq i\leq n$. By Theorem \[thm2.3\]$(i)$, it follows that $d_1\cdots d_n=0$. Since $d_iz=0$ for all $1\leq i\leq n$, every term in the expansion is zero. Thus $\mathfrak{m}^n=0$, a contradiction.
\[lemma8\] Let $(R,\mathfrak{m})$ be a local Gaussian ring with nilradical $\mathfrak{m}$ and let $M$ be a $R$-module. If $\operatorname{w.gl.dim}_R (M)=n\geq 1$ and $\mathfrak{m}$ is not nilpotent, then $\operatorname{Tor}_n(R'/a\mathfrak{m'}, M)=0$ for all non trivial $a\in
\mathfrak{m'}$.
If $a\mathfrak{m'}=0$, then Lemma \[lemma4\] gives the desired result. So assume that $a\mathfrak{m'}\neq 0$. We claim that $a\mathfrak{m'}$ is not a finitely generated ideal. Suppose $a\mathfrak{m'}$ is a finitely generated ideal. Since $R'$ is a local arithmetical ring, there exists an element $\lambda \in \mathfrak{m'}$ such that $a\mathfrak{m'} =
a\lambda R'$. Let $\deg(x)$ denote the degree of nilpotency of $x$ for all $x\in \mathfrak{m'}$. Since $\mathfrak{m}$ is not nilpotent, $\mathfrak{m'}$ is not nilpotent. By Lemma \[deg\], there exists $z\in \mathfrak{m'}$ such that $\deg(z)>\deg(\lambda)$. Observe that $\lambda \in z\mathfrak{m'}$, i.e. $\lambda =zh$ for some $h\in
\mathfrak{m'}$. Hence $az\neq 0$. Furthermore $1-hr$ is a unit for all $r\in R'$. This implies that $az-a\lambda r=az(1-hr)\neq 0$. Thus $az\notin a\lambda R'$, a contradiction.
Now let $\mathcal{X}$ be the following class of ideals: $J\in \mathcal{X}$ iff $J\subset a\mathfrak{m'}$ and $J$ is finitely generated. Then $\operatorname{Tor}_n(R'/a\mathfrak{m'}, M)= \varinjlim_{J\in \mathcal{X}} \operatorname{Tor}_n(R'/ J, M)$. Since $R'$ is a local arithmetical ring, there exists $c\in \mathfrak{m'}$ such that $I=cR'$ for all $I\in \mathcal{X}$. As $a\mathfrak{m'}$ is not finitely generated, $I\neq a\mathfrak{m'}$. Using Lemma \[lemma7\]$(ii)$, we obtain that the natural projection $R'/I\to R'/a\mathfrak{m'}$ induces a trivial homomorphism $0:\operatorname{Tor}_n(R'/
I, M)\to \operatorname{Tor}_n(R'/a\mathfrak{m'}, M)$. Thus the canonical homomorphism $\operatorname{Tor}_n(R'/ I, M)\to \varinjlim_{J\in \mathcal{X}}\operatorname{Tor}_n(R'/ J, M)$ is trivial for all $I\in \mathcal{X}$. This implies that $\varinjlim_{J\in
\mathcal{X}}\operatorname{Tor}_n(R'/ J, M)=0$.
Now we are ready to prove the main theorem of this section.
\[thm1\] Let $(R,\mathfrak{m})$ be a local Gaussian ring with nilradical $\mathfrak{m}$. If $\mathfrak{m}$ is not nilpotent, then $\operatorname{w.gl.dim}(R)=\infty$.
Suppose the theorem is not true, then the $\operatorname{w.gl.dim}(R)=n<\infty$. Using Proposition \[prop1.1\], we obtain that $n\geq 3$. Let $M$ be a $R$-module with $\operatorname{w.gl.dim}_R(M)=n$. We divide the proof into two cases.
[**Case 1.**]{} $R$ does not admit Property (\[hypo1\]).
Hence there exists $x\in D\setminus 0$ and $a\in R\setminus D$ such that $(0:x)=aR+D$. Thus we have an isomorphism $R/(aR+D) \cong xR$. Using Lemma \[lemma3\] and noting that $xR\subset R$, we obtain an inclusion $\operatorname{Tor}_n
(R/(aR+D), M)\hookrightarrow \operatorname{Tor}_n (R, M)$. Hence $\operatorname{Tor}_n (R/(aR+D), M)=0$. But using Lemma \[lemma7\]$(iii)$, we obtain that $\operatorname{Tor}_n (R/(aR+D), M)\neq 0$, a contradiction.
[**Case 2.**]{} $R$ admits Property (\[hypo1\]).
Consider the short exact sequence $0\to aR'/a\mathfrak{m'}\to R'/a\mathfrak{m'} \to
R'/aR'\to 0$ where $a\in \mathfrak{m}\setminus D$. From the corresponding long exact sequence of $\operatorname{Tor}$ groups, consider the following segment $$\operatorname{Tor}_n(R' / a\mathfrak{m'} , M) \to \operatorname{Tor}_{n}(R' / aR', M) \to \operatorname{Tor}_{n-1}(aR' /
a\mathfrak{m'},M)\,.$$ Applying Lemma \[lemma8\], we obtain that $\operatorname{Tor}_n(R' / a\mathfrak{m'} ,M)=0$. Observing that $aR' / a\mathfrak{m'}\cong R/\mathfrak{m}$ and using Lemma \[lemma1.2\] yields $\operatorname{Tor}_{n-1}(aR' / a\mathfrak{m'}, M)=0$. Hence $\operatorname{Tor}_{n}(R' / aR', M)=0$. But using Lemma \[lemma7\] $(iii)$, we obtain that $\operatorname{Tor}_{n}(R' / aR', M)\neq 0$, a contradiction.
From the proof of Theorem \[thm1\], we see that for any module $M$, if $\operatorname{w.gl.dim}_R(M)\geq 3$, then $\operatorname{w.gl.dim}_R(M)=\infty$. In particular, $\operatorname{w.gl.dim}_R(R/aR)=\infty$ for all $a\in \mathfrak{m}
\setminus D$.
Bazzoni-Glaz Conjecture
=======================
In this section, we restate [@BG Theorem 6.4] with an additional hypothesis and prove the theorem under this additional hypothesis. We also give an example to show that the proof of Theorem 6.4 as given in [@BG] is not complete. We need the next lemma to give a proof of the modification of [@BG Theorem 6.4]. We can use the same idea to give a proof of Theorem \[main\].
\[lemma5.1\] Let $R$ be a local Gaussian ring with nilradical $\mathcal{N}$. If $\mathcal{N}\neq
D$, then the maximal ideal of $R_{\mathcal{N}}$ is non-zero.
Using Theorem \[thm2.1\], it follows that the nilradical $\mathcal{N}$ is the unique minimal prime ideal of $R$. Thus the maximal ideal and the nilradical of $R_{\mathcal{N}}$ coincide and let us denote it by $\mathcal{N'}$. We want to show that $\mathcal{N'}\neq 0$. Towards that end, let $x\in \mathcal{N}\setminus D$. We will show that $0\neq \frac{x}{1}\in R_\mathcal{N}$. Suppose not, then there exists $y\in R\setminus \mathcal{N}$ such that $xy=0$. Using Theorem \[thm2.2\]$(iv)$, it follows that $x^2=0$ or $y^2=0$, a contradiction.
Noting that the nilpotency degree of an ideal $I$ of $R$ is the smallest $k\in
\mathbb{N}$ such that $I^k=0$, we now restate and prove Theorem 6.4 of [@BG].
\[thm5.1\] Let $R$ be a Gaussian ring admitting a maximal ideal $\mathfrak{m}$ such that the nilradical $\mathcal{N}$ of the localization $R_{\mathfrak{m}}$ is a non-zero nilpotent ideal. If the nilpotency degree of $\mathcal{N}\geq 3$, then $\operatorname{w.gl.dim}(R)=\infty$.
Let $\mathfrak{m}$ be a maximal ideal of $R$ such that $R_{\mathfrak{m}}$ has a non-zero nilpotent nilradical $\mathcal{N}$. Using Theorem \[thm2.1\], it follows that $\mathcal{N}$ is the unique minimal prime ideal. Recall that the Nilradical($S^{-1}R$)=$S^{-1}$(Nilradical($R$)) for any multiplicative closed set $S\subset R$. Hence $\mathcal{N}=S^{-1}\mathfrak{n}$, where $\mathfrak{n}$ is the nilradical of $R$ and $S=R\setminus \mathfrak{m}$ is a multiplicatively closed set in $R$. Furthermore $\mathfrak{n}$ is a prime ideal of $R$. It is clear that the maximal ideal of $R_{\mathfrak{n}}$ is nilpotent. It follows from Lemma \[lemma5.1\] that the maximal ideal of $R_{\mathfrak{n}}$ is non-zero. The rest of the proof follows [@BG Theorem 6.4] mutatis mutandis.
The hypotheses that the nilpotency degree of $\mathcal{N}\geq 3$ in the above Theorem ensures that $\mathcal{N}\neq D$.
We now give an example to show that the hypothesis on the nilpotency degree in Theorem \[thm5.1\] is necessary for the conclusion of Lemma \[lemma5.1\] to hold.
\[EG\] let $\mathbf{k}$ be a field and $\mathbf{k}[X,Y]$ be a polynomial ring in two variables. Consider a set $S\subset \mathbf{k}[X,Y]/(XY, Y^2)$ defined by $$S=\{a+bY+a_1X+ a_2X^2+\cdots+a_nX^n , \;a,b,a_i\in \mathbf{k}, a\neq 0, n\geq 0\}.$$ Then $S$ is multiplicative set in $\mathbf{k}[X,Y]/(XY, Y^2)$. Define $R=S^{-1}\big(
\mathbf{k}[X,Y]/(XY, Y^2)\big)$. It is easy to see that the unique maximal ideal of $R$ is given by $\mathfrak{m}=\{xf(x)+b_1y \mid f(x)\in \mathbf{k}[x]\; and\; b_1\in
\mathbf{k}\}$, where $x, y$ are the images of $X, Y$ in $R$. Any $c,d\in
\mathfrak{m}$ has the form, $c=\lambda_1 y+c_1x+c_2x^2+\cdots c_nx^n$ and $d=\lambda_2 y+d_1x+d_2x^2+\cdots +d_mx^m$, where $m,n\in \mathbb{N}$ and $c_i,
d_j\in \mathbf{k}$ for $1\leq i\leq n$ and $1\leq j\leq m$. Let $i,j\in \mathbb{N}$ be such that $c_i\ne 0$ and $d_j\neq 0$ and let $i,j$ be the least natural numbers with this property. Then one can rewrite $c,d$ as $c=\lambda_1
y+x^i(c_i+c_{i+1}'x+\cdots+c_{n}'x^{n-i})$ and $d=\lambda_2
y+x^j(d_j+d_{j+1}'x+\cdots+d_{m}'x^{m-j})$, where $c_{t}'=c_{t+1}c_{i}^{-1}$ and $d_{s}'=d_{s+1}d_{j}^{-1}$ for all $i\leq t\leq n-i$ and $j\leq s\leq n-j$. Observe that $c_i+c_{i+1}'x+\cdots+c_{n}'x^{n-i}$ and $d_j+d_{j+1}'x+\cdots+d_{m}'x^{m-j}$ are units in $R$. Furthermore $(c,d)^2=(c^2,cd,d^2)$, $c^2=x^{2i}u_1^2$, $d^2=x^{2j}u_{2}^2$ and $cd=x^{i+j}u_1u_2$, where $u_1,u_2\notin \mathfrak{m}$. Now using Theorem \[thm2.2\]$(iv)$, it can be verified that $R$ is a local Gaussian ring. Its nilradical $\mathfrak{n}=(y)\subset R$ is not trivial, while the nilradical of $R_\mathfrak{n}$ is trivial as $\frac{y}1=\frac{xy}x=0$.
\[main\] Let $R$ be a non-reduced local Gaussian ring with nilradical $\mathcal{N}$. If $\mathcal{N}\neq D$, then $\operatorname{w.gl.dim}(R)=\infty$.
By Theorem \[thm2.1\], the nilradical $\mathcal{N}$ is the unique minimal prime ideal. Thus the maximal ideal and the nilradical of $R_{\mathcal{N}}$ coincide and let us denote it by $\mathcal{N'}$. Since $\operatorname{w.gl.dim}(R)\geq
\operatorname{w.gl.dim}(R_{\mathcal{N}})$, it suffices to show that $\operatorname{w.gl.dim}(R_\mathcal{N})=\infty$. Using Lemma \[lemma5.1\], it follows that $\mathcal{N'}\neq 0$. Hence we have a local Gaussian ring $(R_\mathcal{N}, \mathcal{N'})$ with $\mathcal{N'}\neq 0$. If $\mathcal{N'}$ is nilpotent, then Proposition \[prop2.1\] implies that $\operatorname{w.gl.dim}(R_\mathcal{N})=\infty$. If $\mathcal{N'}$ is not nilpotent, then Theorem \[thm1\] implies that $\operatorname{w.gl.dim}(R_\mathcal{N})=\infty$.
We claim that to prove the Bazzoni-Glaz Conjecture, it remains to consider the case of a local Gaussian ring with nilradical $\mathcal{N}=D\neq 0$. Let $R$ be a Gaussian ring (not necessarily local) and let $\mathcal{N}_{\mathfrak{p}}$ denote the nilradical of $R_{\mathfrak{p}}$ for any $\mathfrak{p}\in \operatorname{Spec}(R)$. We have the following cases:
- $R_{\mathfrak{p}}$ is domain for all $\mathfrak{p}\in \operatorname{Spec}(R)$;
- there exists a $\mathfrak{p}\in \operatorname{Spec}(R)$ such that the $\mathcal{N}_{\mathfrak{p}}\neq 0$ and $\mathcal{N}^2_{\mathfrak{p}}\neq 0$;
- there exists a $\mathfrak{p}\in \operatorname{Spec}(R)$ such that $\mathcal{N}_{\mathfrak{p}}\neq 0$ and $\mathcal{N}^2_{\mathfrak{p}}= 0$.
We remind the reader that if $R_{\mathfrak{p}}$ is not a domain, then $\mathcal{N}_{\mathfrak{p}}\neq 0$ and hence all possible cases are listed above. In case (i) $\operatorname{w.gl.dim}(R)\leq 1$, while in case (ii) $\operatorname{w.gl.dim}(R)= \infty$. Hence to prove the Bazzoni-Glaz Conjecture it remains to show the following.
\[thm5.5\] Let $R$ be a non-reduced local Gaussian ring with nilradical $\mathcal{N}$. If $\mathcal{N}^2=0$, then $\operatorname{w.gl.dim}(R)=\infty$.
We prove this in the next section.
Local Gaussian rings with square free nilradical
================================================
Throughout this section $R$ is a local Gaussian ring with maximal ideal $\mathfrak{m}$ and nilradical $\mathcal{N}=D\neq 0$. Without loss of generality, we may assume that each element of $\mathfrak{m}$ is a zero divisor and $\mathfrak{m}\neq D$. Since the nilradical is the unique minimal prime ideal in a local Gaussian ring, one easily checks that $D_\mathfrak{p}$ is also the nilradical of $R_\mathfrak{p}$ for all $\mathfrak{p}\in Spec (R)$. We show that the assumption $\operatorname{w.gl.dim}(R)<\infty$ leads to a contradiction. By Proposition \[prop1.1\], we know that $\operatorname{w.gl.dim}(R)\geq 3$.
\[adlemma1\] Let $\operatorname{w.gl.dim}(R)=n$. If $n<\infty$, then $\operatorname{w.gl.dim}_R (R/D)=n-1$.
By Lemma \[llemma\], there exists an $a\in \mathfrak{m}\setminus D$ such that $\operatorname{w.gl.dim}_R(R/(aR+D))=n$. Since $R/D$ is domain, we have the following short exact sequence, $0\rightarrow R/D\overset{a_{m}}{\rightarrow}R/D\overset{\pi}{\rightarrow} R/(aR+D)\rightarrow 0$, where $a_{m}$ is multiplication by $a$ and $\pi$ is the natural projection. This implies $\operatorname{w.gl.dim}_R(R/D)\geq \operatorname{w.gl.dim}_R (R/(aR+D))-1=n-1$. On the other hand, using Lemma \[lemma4\], we obtain that $\operatorname{w.gl.dim}_R (R/D)\leq n-1$.
\[adlemma2\] Let $i\geq 1$. If $a\in \mathfrak{m}\setminus D$ is a zero-divisor on $\operatorname{Tor}_{i-1}(R/D, R/D)$, then $\operatorname{Tor}_{i}(R/(aR+D), R/D)\neq 0$. Furthermore, if $\operatorname{w.gl.dim}R=n<\infty$, then $\operatorname{Tor}_{n-1}(R/(aR+D), R/D)\neq 0$ if and only if $a$ is a zero-divisor on $\operatorname{Tor}_{n-2}(R/D, R/D)$.
We have a short exact sequence: $0\rightarrow R/D\overset{a_m}{\rightarrow}R/D\overset{\pi}{\rightarrow} R/(aR+D)\rightarrow 0$, where $a_m$ is multiplication by $a$ and $\pi$ is the natural projection. Consider the following segment of the corresponding long exact sequence of Tor groups $$\operatorname{Tor}_{i}(R/(aR+D), R/D)\rightarrow \operatorname{Tor}_{i-1}(R/D, R/D) \xrightarrow{a_m^{*}} \operatorname{Tor}_{i-1}(R/D, R/D),$$ where $a_m^{*}$ is multiplication by $a$. This exact sequence implies the first part of the lemma. If $\operatorname{w.gl.dim}R=n<\infty$, then first using Lemma \[adlemma1\] and then using Lemma \[lemma4\], we obtain that $\operatorname{Tor}_{n-1}(R/D, R/D)=0$. Hence the above exact sequence for $i=n-1$ implies that $\operatorname{Tor}_{n-1}(R/(aR+D), R/D)=
\operatorname{Ker}\{a_m^{*} : \operatorname{Tor}_{n-2}(R/D, R/D)\rightarrow \operatorname{Tor}_{n-2}(R/D, R/D)\}$. This completes the proof.
The following easy observation will be used several times in the sequel.
\[adlemma3\] Let $M$ be a $R$-module. If $x\in M$, then the set $I=\{a\in \mathfrak{m}| a^qx=0 \;\text{for some}\; q\in \mathbb{N} \}$ is a prime ideal of $R$.
Set $J=\{a\in \mathfrak{m} , ax=0 \}$. Observe that $J$ is an ideal of $R$. It can be easily verified that $I/J$ is the nilradical of $R/J$. Since $R/J$ is a local Gaussian ring, $I/J$ is prime ideal and so is $I$.
\[adlemma4\] Let $\operatorname{w.gl.dim}R=n$. If $n<\infty$, then there exist a $\mathfrak{p}\in Spec (R)$ such that the following conditions hold.
- $D_\mathfrak{p}\neq \mathfrak{p}_\mathfrak{p}$ and there exist $\omega\in \operatorname{Tor}^{R_\mathfrak{p}}_{n-2}(R_\mathfrak{p}/D_\mathfrak{p}, R_\mathfrak{p}/D_\mathfrak{p})$ such that $\omega\neq 0$ and for all $c\in \mathfrak{p}_\mathfrak{p}$, $c^q\omega =0$ for some $q\in \mathbb{N}$ ;
- for all $c\in \mathfrak{p}_\mathfrak{p}\setminus D_\mathfrak{p}$, $\operatorname{Tor}^{R_\mathfrak{p}}_{n-1}(R_\mathfrak{p}/(cR_\mathfrak{p}+D_\mathfrak{p}), R_\mathfrak{p}/D_\mathfrak{p})\neq 0$;
- each element of $\mathfrak{p}_\mathfrak{p}$ is a zero-divisor;
- $\operatorname{w.gl.dim}(R_\mathfrak{p})=n$.
(i): By Lemma’s \[llemma\] and \[adlemma1\], there exists an $a\in \mathfrak{m}\setminus D$ such that $\operatorname{Tor}_{n-1}(R/(aR+D), R/D)\neq 0$. Using Lemma \[adlemma2\], there exist a $\omega\in \operatorname{Tor}_{n-2}(R/D, R/D)$ such that $\omega\neq 0$ and $a\omega =0$. Set $\mathfrak{p}= \{b\in \mathfrak{m} , b^q \omega=0 \;\text{for some}\; q\in \mathbb{N} \}$. By Lemma \[adlemma3\], it follows that $\mathfrak{p}\in Spec (R)$. Moreover $a\in \mathfrak{p}$ and since $D$ is a prime ideal, $\frac{a}{1}\notin D_\mathfrak{p}$. Hence $D_\mathfrak{p}\neq \mathfrak{p}_\mathfrak{p}$. We claim that $\omega=\frac{\omega}{1}\in \operatorname{Tor}_{n-2}(R/D, R/D)_\mathfrak{p} =
\operatorname{Tor}^{R_\mathfrak{p}}_{n-2}(R_\mathfrak{p}/D_\mathfrak{p}, R_\mathfrak{p}/D_\mathfrak{p})$ is not trivial. If this is not the case, then there exist $b\in R\setminus \mathfrak{p}$ such that $b \omega =0$, a contradiction. By definition, for any $c\in \mathfrak{p}_\mathfrak{p}$, there exist $q\in \mathbb{N}$ such that $c^q\omega =0$.
(ii): By (i), $c\in \mathfrak{p}_\mathfrak{p}\setminus D_\mathfrak{p}$ is a zero-divisor on $\operatorname{Tor}^{R_\mathfrak{p}}_{n-2}(R_\mathfrak{p}/D_\mathfrak{p}, R_\mathfrak{p}/D_\mathfrak{p})$. Hence Lemma \[adlemma2\] implies (ii).
(iii): If this is not the case, then there exists $c\in \mathfrak{p}_\mathfrak{p}\setminus D_\mathfrak{p}$ which is regular in $R_\mathfrak{p}$. Using Theorems \[thm2.1\](ii) and \[thm2.3\](ii), we obtain that $D_\mathfrak{p} \subset cR_\mathfrak{p}+(0:c)=cR_\mathfrak{p}$. This implies that $cR_\mathfrak{p} =
cR_\mathfrak{p} + D_\mathfrak{p}$. By (ii), $\operatorname{Tor}^{R_\mathfrak{p}}_{n-1}(R_\mathfrak{p}/cR_\mathfrak{p}, R_\mathfrak{p}/D_\mathfrak{p})\neq 0$. On the other hand, we have the following free resolution of $R_\mathfrak{p}/cR_\mathfrak{p}$: $0\rightarrow R_\mathfrak{p}\overset{c_m}{\rightarrow}R_\mathfrak{p}\overset{\pi}{\rightarrow}
R_\mathfrak{p}/cR_\mathfrak{p}\rightarrow 0$, where $c_m$ is multiplication by $c$ and $\pi$ is the natural projection. This implies that $\operatorname{w.gl.dim}_{R_\mathfrak{p}}(R_\mathfrak{p}/cR_\mathfrak{p})\leq 1$, but this is a contradiction since $n\geq 3$.
(iv): It suffices to check that $\operatorname{w.gl.dim}(R_\mathfrak{p})\geq n$. By (ii), $\operatorname{w.gl.dim}_{R_\mathfrak{p}}(R_\mathfrak{p}/D_\mathfrak{p})\geq n-1$. Using Lemma \[adlemma1\], we obtain that $\operatorname{w.gl.dim}(R_\mathfrak{p})\geq n$.
Thus, if the Bazzoni-Glaz Conjecture is not true, then Lemma \[adlemma4\] enables us to assume the existence of a local Gaussian ring $(R, \mathfrak{m})$ with $\operatorname{w.gl.dim}(R)=n<\infty$, satisfying the following conditions:
(C1) $D\neq \mathfrak{m}$ and $\operatorname{Tor}_{n-1}(R/(aR+D), R/D)\neq 0$ for all $a\in \mathfrak{m}\setminus D$.
Moreover, since $n\geq 3$, $\operatorname{Tor}_{n-2}(R/D, R/D)=\operatorname{Tor}_{n-3}(D, R/D)$ and we also can assume that $R$ satisfies the following:
(C2) there exists an $\omega\in \operatorname{Tor}_{n-3}(D, R/D)$ such that $\omega\neq 0$ and for all $a\in \mathfrak{m}$, $a^q\omega =0$ for some $q\in \mathbb{N}$.
\[adlemma5\] Let $\operatorname{w.gl.dim}(R)=n < \infty$. If $R$ satisfies (C1), then the following conditions hold.
- $\mathfrak{m}$ is flat;
- if $I=\mathfrak{m}, D$ or $(0:c)$ for any $c\in \mathfrak{m}$, then $I=I\mathfrak{m}$. Moreover, if $x\in I$, then there exist $x'\in I$ and $a\in \mathfrak{m}$ such that $x=ax'$;
- $\mathfrak{m}$ is not finitely generated modulo $D$.
(i): As a result of Lemma \[lemma1.2\], it suffices to check that $R$ admits Property \[hypo1\]. If this is not the case, then there exists a $d\in D\setminus 0$ and $a\in \mathfrak{m}\setminus D$ such that $(0:d)=aR+D$. This implies that $R/(aR+D) \cong dR$. First using Lemma \[adlemma1\] and then using Lemma \[lemma3\], we obtain that $\operatorname{Tor}_{n-1}(dR, R/D)= 0=\operatorname{Tor}_{n-1}(R/(aR+D), R/D)$, contradicting (C1).
(ii): By (i), we have that $\operatorname{w.gl.dim}_R(R/\mathfrak{m})=1$. Using Lemma’s \[lemma5\], \[lemma4\] and \[lemma3\] respectively with $I=\mathfrak{m}, D$ and $(0:c)$, we obtain that $\operatorname{Tor}_1(R/I,R/\mathfrak{m})=0$. Thus $I=I\mathfrak{m}$ because $\operatorname{Tor}_1(R/I,R/\mathfrak{m})=I/I\mathfrak{m}$. Consider any $x\in I$. We have $x=x_1a_1+\cdots+x_sa_s$, where $x_1,\cdots, x_s\in I$ and $a_1,\cdots,a_s\in \mathfrak{m}$. Let $J$ be an ideal generated by $x_1,\cdots, x_s,a_1,\cdots,a_s$. By Theorem \[thm2.2\]$(ii)$, there exists an $a\in J$ such that $J=aR+(0:J)$. Set $a_i=ar_i+\lambda_i$, where $r_i\in R$ and $\lambda_i\in (0:J)$ for all $1\leq i\leq s$. Then $x=x_1ar_1+\cdots +x_sar_s=ax'$ where $x'=x_1r_1+\cdots +x_sr_s\in I$.
(iii): If this is not the case, then there exists an $a\in \mathfrak{m}\setminus D$ such that $\mathfrak{m}=aR+D$. Flatness of $\mathfrak{m}$ implies that $\mathfrak{m}=\mathfrak{m}^2$. So $\mathfrak{m}=a^2 R + D$. Hence $a = a^2r+d$, where $r\in R$ and $d\in D$. Thus $a(1-ar)\in D$, which implies that $a\in D$, a contradiction.
\[adlemma6\] Let $\operatorname{w.gl.dim}(R)=n<\infty$. If $R$ satisfies (C1) and (C2), then there exist $\overline{b}\in \mathfrak{m}\setminus D$ with the following property: for all $b\in \mathfrak{m}$ with $\overline{b}\in bR+D$, there exists $\omega_b\in \operatorname{Tor}_{n-3}(D, R/D)$ such that $b\omega_b\neq 0$ and $b^q\omega_b =0$ for some $q\in \mathbb{N}$.
If $n=3$, then $\operatorname{Tor}_{n-3}(D, R/D)=D\otimes (R/D)=D$. Thus we are given $\omega\in D\setminus 0$ with the following property: for all $a\in \mathfrak{m}$ there exist $q\in \mathbb{N}$ such that $a^q\omega =0$. By Lemma \[adlemma5\]$(ii)$, there exists $\overline{b}\in \mathfrak{m}$ and $\overline{\omega}\in D$ such that $\omega=\overline{b}\overline{\omega}$. Take any $b\in \mathfrak{m}$ such that $\overline{b}\in bR+D$. Set $\overline{b}=br+d$ where $r\in R$ and $d\in D$ and let $\omega_b = r\overline{\omega}$. Then $b\omega_b=br\overline{\omega}=(br+d)\overline{\omega}=
\overline{b}\overline{\omega}=\omega\neq 0$. Moreover, according to (C2), there exists $q\in \mathbb{N}$ such that $b^q\omega =0$, implying $b^{q+1}\omega_b=0$. Hence the lemma is proved for $n=3$.
Set $n\geq 4$. Consider a free resolution of $R/D$: $\cdots\xrightarrow{\partial_2}R^{X_1}\xrightarrow{\partial_1}R^{X_0}\xrightarrow{\partial_0} R/D$, where $X_i$ are sets. Set $K =\operatorname{Ker}\partial_{n-4}$. Let $\sigma : K \rightarrow R^{X_{n-4}}$ be the natural inclusion and $\overline{\sigma}:K\otimes D \rightarrow D^{X_{n-4}}$ be the natural homomorphism obtained after tensoring by $D$. Then $\operatorname{Tor}_{n-3}(D, R/D)=\operatorname{Ker}\overline{\sigma}$. Let $\omega$ be as in (C2). Set $\omega = x_1\otimes d_1+\cdots+x_s\otimes d_s\in \operatorname{Ker}\overline{\sigma}$, where $x_i\in K$ and $d_i\in D$ for all $1\leq i\leq s$. By Lemma \[adlemma5\]$(ii)$, there exists $a_i\in \mathfrak{m}\setminus D$ such that $d_i\in a_iD$ for all $1\leq i\leq s$. Since $R/D$ is a local arithmetical ring, there exists an $a\in \{a_1,\cdots, a_s\}$ such that $a_i\in aR+D$ for all $1\leq i\leq s$. Hence $d_i\in aD$ for all $1\leq i\leq s$. Set $d_i=ad_i'$, where $d_i'\in D$ for all $1\leq i\leq s$ and set $\omega'=x_1\otimes d_1'+\cdots+x_s\otimes d_s'$. Observe that $a\omega'=\omega$, which implies that $a\overline{\sigma}(\omega')=0$. We have $\overline{\sigma}(\omega')\in D^{X_{n-4}}$. Let $\lambda_1, \cdots, \lambda_t\in D$ be the finitely many non zero entries of $\overline{\sigma}(\omega')$. Then $a\in (0:\lambda_i)$ for all $1\leq i\leq t$. By Lemma \[adlemma5\]$(ii)$, there exist $c_i\in (0:\lambda_i)$ such that $a\in c_i\mathfrak{m}$ for all $1\leq i\leq t$. Since $R/D$ is a local arithmetical ring, there exist a $c\in \{c_1,\cdots, c_t\}$ such that $c\in c_iR+D$ for all $1\leq i\leq t$. Thus $c\in (0:\lambda_i)$ for all $1\leq i\leq t$ and $a\in c\mathfrak{m}$. Set $\overline{\omega}=c\omega'$ and $a=c\overline{b}$, where $\overline{b}\in \mathfrak{m}\setminus D$. Observe that $\overline{\omega}\in \operatorname{Ker}\overline{\sigma}$ and $\overline{b}\overline{\omega}=\omega\neq 0$. Choose any $b\in \mathfrak{m}$ such that $\overline{b}\in bR+D$. Set $\overline{b}=br+d$ where $r\in R$ and $d\in D$ and set $\omega_b = r\overline{\omega}$. Then $b\omega_b=br\overline{\omega}=(br+d)\overline{\omega}=\overline{b}\overline{\omega}=
\omega\neq 0$. Moreover, according to (C2), there exists $q\in \mathbb{N}$ such that $b^q\omega =0$, implying that $b^{q+1}\omega_b=0$.
\[adlemma7\] Let $\operatorname{w.gl.dim}(R)=n<\infty$. If $R$ satisfies (C1) and (C2), then there exists a $\mathfrak{p}\in Spec(R)$ such that the following conditions hold.
- $D_\mathfrak{p}\neq \mathfrak{p}_\mathfrak{p}$ and each element of $\mathfrak{p}_\mathfrak{p}$ is a zero divisor. Furthermore, there exist $d\in D$ such that $\frac{d}{1}\in D_\mathfrak{p}$ is not trivial and for any $c\in \mathfrak{p}_\mathfrak{p}$, $\frac{d}{1}c^q=0$ for some $q\in \mathbb{N}$;
- for all $c\in \mathfrak{p}_\mathfrak{p}\setminus D_\mathfrak{p}$, $\operatorname{Tor}^{R_\mathfrak{p}}_{n-1}(R_\mathfrak{p}/(cR_\mathfrak{p}+D_\mathfrak{p}), R_\mathfrak{p}/D_\mathfrak{p})\neq 0$;
- $\operatorname{w.gl.dim}(R_\mathfrak{p})=n$.
(i): Choose $\overline{b}\in \mathfrak{m}\setminus D$ as in Lemma \[adlemma6\]. Fix a non trivial element $d\in (0:\overline{b})$. Set $\mathfrak{p}= \{a\in \mathfrak{m} , a^q d=0 \;\text{for some}\; q\in \mathbb{N} \}$. Using Lemma \[adlemma3\], it follows that $\mathfrak{p}\in Spec (R)$ and note that $\overline{b}\in \mathfrak{p}$. Since $D$ is a prime ideal, $\frac{\overline{b}}{1}\notin D_\mathfrak{p}$. Hence $D_\mathfrak{p}\neq \mathfrak{p}_\mathfrak{p}$. To prove the remaining part of (i), it suffices to check that $\frac{d}{1}\neq 0$. If $\frac{d}{1}= 0$, then there exists an $a\in R\setminus \mathfrak{p}$ such that $ad=0$, a contradiction.
(ii): Using Lemma \[adlemma2\], it suffices to check that any element $c\in \mathfrak{p}_\mathfrak{p}\setminus D_\mathfrak{p}$ is a zero-divisor on $\operatorname{Tor}^{R_\mathfrak{p}}_{n-2}(R_\mathfrak{p}/D_\mathfrak{p}, R_\mathfrak{p}/D_\mathfrak{p})$. This is the same as showing that $c$ is a zero-divisor on $\operatorname{Tor}^{R_\mathfrak{p}}_{n-3}(D_\mathfrak{p}, R_\mathfrak{p}/D_\mathfrak{p})$. It suffices to consider the case $c=\frac{b}{1}$ where $b\in \mathfrak{p}\setminus D$. Since $R/D$ is a local arithmetical ring, either $\overline{b}\in bR+D$ or $b\in \overline{b}R+D$, where $\overline{b}$ is defined as in (i).
[**Case 1.**]{} $\overline{b}\in bR+D$.
By Lemma \[adlemma6\], there exists $\omega_b\in \operatorname{Tor}_{n-3}(D, R/D)$ such that $b\omega_b\neq 0$ and $b^q\omega_b =0$ for some $q\in \mathbb{N}$. We claim that $\frac{\omega_b}{1}\in \operatorname{Tor}_{n-3}(D, R/D)_\mathfrak{p} =\operatorname{Tor}^{R_\mathfrak{p}}_{n-3}(D_\mathfrak{p}, R_\mathfrak{p}/D_\mathfrak{p})$ is not trivial. If this is not the case, then there exists an $a\in R\setminus \mathfrak{p}$ such that $a\omega_b=0$. Since $b\in \mathfrak{p}$, we obtain that $b\in aR+D$. Thus $b\omega_b=0$, a contradiction. Hence we have a nontrivial element $\frac{\omega_b}{1}\in \operatorname{Tor}^{R_\mathfrak{p}}_{n-3}(D_\mathfrak{p}, R_\mathfrak{p}/D_\mathfrak{p})$ such that $\frac{\omega_b}{1}b^q =0$ for some $q\in \mathbb{N}$. This shows that $b$ is a zero-divisor on $\operatorname{Tor}^{R_\mathfrak{p}}_{n-3}(D_\mathfrak{p}, R_\mathfrak{p}/D_\mathfrak{p})$.
[**Case 2.**]{} $b\in \overline{b}R+D$.
Set $b=\overline{b}r+\lambda$, where $r\in R$ and $\lambda\in D$. As in the previous case, there exists $\overline{\omega}\in \operatorname{Tor}^{R_\mathfrak{p}}_{n-3}(D_\mathfrak{p}, R_\mathfrak{p}/D_\mathfrak{p})$ such that $\overline{\omega}\neq 0$ and $\overline{b}\overline{\omega}=0$. Notice that $b \overline{\omega} = \overline{b}r \overline{\omega}=0$, proving that $b$ is a zero-divisor on $\operatorname{Tor}^{R_\mathfrak{p}}_{n-3}(D_\mathfrak{p}, R_\mathfrak{p}/D_\mathfrak{p})$.
(iii): It suffices to check that $\operatorname{w.gl.dim}(R_\mathfrak{p})\geq n$. By (ii), $\operatorname{w.gl.dim}_{R_\mathfrak{p}}(R_\mathfrak{p}/D_\mathfrak{p})\geq n-1$. Using Lemma \[adlemma1\], we obtain that $\operatorname{w.gl.dim}(R_\mathfrak{p})\geq n$.
Thus, if the Bazzoni-Glaz conjecture is not true, then using Lemma \[adlemma7\], it is possible to assume the existence of a local Gaussian ring $(R, \mathfrak{m})$ such that $\operatorname{w.gl.dim}(R)=n<\infty$ and satisfying the following conditions:
(C1) $D\neq \mathfrak{m}$ and for all $c\in \mathfrak{m}\setminus D$, $\operatorname{Tor}_{n-1}(R/(cR+D), R/D)\neq 0$;
(C3) there exists $d\in D$ such that $d\neq 0$ and for any $c\in \mathfrak{m}$, $c^q d =0$ for some $q\in \mathbb{N}$.
We need one more Lemma.
\[adlemma8\] Let $\operatorname{w.gl.dim}(R)=n<\infty$. If $R$ satisfies (C1) and (C3), then there exists an $a\in \mathfrak{m}\setminus D$ such that $\operatorname{Tor}_{n-1}(R/(a\mathfrak{m}+D), R/D)= 0$.
Choose $d$ as in (C3). By Lemma \[adlemma5\]$(ii)$, there exists $d'\in D$ and $c\in\mathfrak{m}\setminus D$ such that $d=cd'$. Using the same lemma, there exists $a,b\in \mathfrak{m}\setminus D$ such that $c=ab$. Let $\mathcal{X}$ be the following class of ideals: $I\in \mathcal{X}$ iff $aR+bR+D\subset I\subset \mathfrak{m}$ and $I/D$ is finitely generated. Then $\operatorname{Tor}_{n-1}(R/(a\mathfrak{m}+D), R/D)=\lim_{I\in \mathcal{X}}\operatorname{Tor}_{n-1}(R/(aI+D), R/D)$. Take any $I_1\in \mathcal{X}$. To prove the lemma it is sufficient to show that the canonical map $\operatorname{Tor}_{n-1}(R/(aI_1+D), R/D)\rightarrow \lim_{I\in \mathcal{X}}\operatorname{Tor}_{n-1}(R/(aI+D), R/D)$ is trivial. Since $I_1$ is finitely generated modulo $D$, there exist $b_1\in \mathfrak{m}\setminus D$ such that $I_1=b_1R+D$. By Lemma \[adlemma5\]$(iii)$, $\mathfrak{m}$ is not finitely generated modulo $D$. Thus there exists $I_2\in \mathcal{X}$ such that $I_1\subset I_2$ and $I_1\neq I_2$. Set $I_2=b_2R+D$, where $b_2\in \mathfrak{m}\setminus D$. Using Theorem \[thm2.3\]$(ii)$, there exist $x\in \mathfrak{m}$ and $d_1\in (0:b_2)$ such that $b_1=xb_2+d_1$. Notice that $a\in b_2R+D$, implying that $(0:b_2)\subset(0:a)$. Hence $a d_1=0$ and $ab_1=xab_2$. Consider the following commutative diagram with exact rows: $$\xymatrix {
0 \ar[r] & (0:ab_2) \ar[d]^{inclusion}\ar[r] & R \ar[d]^{1_R}\ar[r]^{f_2} &ab_2R \ar[d]^{f_3}\ar[r] & 0\;\;\\
0 \ar[r] & (0:ab_1) \ar[r] & R \ar[r]^{f_1} & ab_1R \ar[r] & 0 \;,}$$ where $f_1,f_2$ and $f_3$ are defined as multiplications by $ab_1, ab_2$ and $x$, respectively.
Claim: The natural inclusion $(0:ab_2)\subset (0:ab_1)$ is not onto.
If this is not the case, then the first two vertical homomorphisms of the above diagram are isomorphisms. The short five lemma would imply that $f_3:ab_2R \rightarrow ab_1R$ is also an isomorphism. Since $b\in I_2$, $b=b_2r+d_2$, where $r\in R$ and $d_2\in D$. So $d=d'c=d'ab=d'a(b_2r+d_2)=d'ab_2r\in ab_2R$. This implies that $x^id\in ab_2R$ for all $i\geq 0$. By (C3), there exists $q\in \mathbb{N}$ such that $x^qd=0$ and $x^{q-1}d\neq 0$. Then $f_3(x^{q-1}d)=x^{q}d=0$. Hence $f_3$ is not injective, a contradiction.
Observing that $ab_1\in \mathfrak{m}\setminus D$, our claim implies that there exist a $\lambda \in D$ such that $\lambda \in (0:ab_1)$ and $\lambda \notin (0:ab_2)$. This implies that $ab_1R+D \subset (0:\lambda) \subset ab_2R+D$. Hence $aI_1+D \subset (0:\lambda) \subset aI_2+D$. Thus we have natural projections $R/(aI_1+D) \rightarrow R/(0:\lambda) \rightarrow R/(aI_2+D)$. Using Lemma’s \[lemma3\], \[adlemma1\] and the fact that $R/(0:\lambda)\cong \lambda R \subset R$, we obtain that $\operatorname{Tor}_{n-1}(R/(0:\lambda) ,R/D)=0$. Hence the natural map $\operatorname{Tor}_{n-1}(R/(aI_1+D) ,R/D)\rightarrow\operatorname{Tor}_{n-1}(R/(aI_2+D) ,R/D)$ is trivial, as it is the composition of the following natural maps $$\operatorname{Tor}_{n-1}(R/(aI_1+D) ,R/D)\rightarrow \operatorname{Tor}_{n-1}(R/(0:\lambda) ,R/D) \rightarrow\operatorname{Tor}_{n-1}(R/(aI_2+D) ,R/D).$$ This implies that $\operatorname{Tor}_{n-1}(R/(aI_1+D), R/D)\rightarrow \lim_{I\in \mathcal{X}}\operatorname{Tor}_{n-1}(R/(aI+D), R/D)$ is also trivial.
[**Proof of Theorem \[thm5.5\].**]{} If the theorem is not true, then there exists a local Gaussian ring $R$ which satisfies (C1) and (C2). Choose $a\in \mathfrak{m}\setminus D$ prescribed by Lemma \[adlemma8\]. Consider the following short exact sequence: $$0\rightarrow (aR+D)/(a\mathfrak{m}+D)\rightarrow R/(a\mathfrak{m}+D)\rightarrow R/(aR+D)\rightarrow 0.$$ From the corresponding long exact sequence of Tor groups, consider the segment: $$\operatorname{Tor}_{n-1}(R/(a\mathfrak{m}+D), R/D)\rightarrow \operatorname{Tor}_{n-1}(R/(aR+D), R/D)\rightarrow$$ $$\rightarrow\operatorname{Tor}_{n-2}((aR+D)/(a\mathfrak{m}+D), R/D).$$ There is an isomorphism $(aR+D)/(a\mathfrak{m}+D)\cong R/\mathfrak{m}$. Moreover, Lemma \[adlemma5\]$(i)$ implies that $\operatorname{w.gl.dim}_R(R/\mathfrak{m})=1$. Applying Lemma \[lemma4\] and noting that $n\geq 3$, we obtain that $$\operatorname{Tor}_{n-2}((aR+D)/(a\mathfrak{m}+D), R/D)=0.$$ Now using the above exact sequence we obtain that $\operatorname{Tor}_{n-1}(R/(aR+D), R/D)=0$, but this contradicts (C1).
[\[\*\*\*\]]{}
J. Abuhlail, M. Jarrar, S. Kabbaj, Commutative rings in which every finitely generated ideal is quasi-projective, [*Journal of Pure and Applied Algebra*]{} **215** (2011), 2504–2511.
D.D. Anderson and B.G. Kang, Content formulas for polynomials and power series and complete integral closure, [*J. of Algebra*]{} **181** (1996), 82–-94.
S. Bazzoni and S. Glaz, Gaussian Properties of total rings of quotients, [*J. of Algebra*]{} **310** (2007), 180–193.
L. Fuchs, Ber die Ideale arithmetischer ringe, [*Math. Helv.*]{} **23** (1949), 334–341.
R. Gilmer, [*Multiplicative Ideal Theory*]{}, Marcel Dekker, 1972.
S. Glaz, [*Commutative Coherent Rings*]{}, Lecture Notes in Mathematics 1371. Springer-Verlag, Berlin, 1989.
S. Glaz, Weak dimension of Gaussian rings, [*Proc. Amer. Math. Soc.*]{} **133**, (2005), 2507–2513.
S. Glaz and R. Schwarz, Prüfer Conditions in Commutative Rings, [*The Arabian Journal of Science and Engineering*]{} **36**, (2011), 967–983.
S. Glaz and W.V. Vasconcelos, Gaussian polynomials, Marcel Dekker Lecture Notes, **186**, (1997), 325–-337.
S. Glaz and W.V. Vasconcelos, The content of Gaussian polynomials, [*J. of Algebra*]{} **202**, (1998), 1–-9.
M. Harada, Note on the dimension of modules and algebras, [*J. Inst. Polytech. Osaka City Univ. Ser. A.*]{} **7**, (1956), 17–27.
W. Heinzer and C. Huneke, Gaussian polynomials and content ideals, [*Proc. Amer. Math. Soc.*]{} **125** (1997), 739–-745.
C.U. Jensen, Arithmetical rings, [*Acta Math. Hungr.*]{} **17** (1966), 115–123.
K.A. Loper and M. Roitman, The content of a Gaussian polynomial is invertible, [*Proc. Amer. Math. Soc.*]{} **133** (2005), 1267–-1271.
T.G. Lucas, Gaussian polynomials and invertibility, [*Proc. Amer. Math. Soc.*]{} **133** (2005), 1881–-1886.
T.G. Lucas, The Gaussian Property for Rings and Polynomials, [*Houston J. of Math*]{} **34** (2008), 1–18.
B. Osofsky, Global dimension of commutative rings with linearly ordered ideals, [*J. London Math. Soc.*]{} **44** (1969), 183–185.
H. Prüfer, Untersuchungen uber Teilbarkeitseigenshaften in Korpern, [*J. Reine Angew. Math.*]{} **168** (1932), 1–-36.
H. Tsang, [*Gauss’s Lemma*]{}, Ph.D. Thesis, University of Chicago, 1965.
|
---
abstract: 'We present spectroscopic identifications for a sample of 55 submillimeter(submm)-selected ‘SCUBA’ galaxies, lying at redshifts $z=0.7$ to 3.7, that were pinpointed in deep 1.4-GHz VLA radio maps. We describe their properties, especially the presence of active galactic nuclei (AGN) in the sample, and discuss the connection of the SCUBA galaxies and the formation of spheroidal components of galaxies, which requires knowledge of their masses and the timescales of their very luminous activity. For a subset of the galaxies, we show their disturbed and diverse [*Hubble Space Telescope (HST)*]{} optical morphologies.'
author:
- 'Scott C. Chapman$^1$, Andrew W. Blain$^1$, Rob J. Ivison$^2$, and Ian Smail$^3$.'
title: 'The spectroscopic redshifts of SCUBA galaxies: implications for spheroid formation'
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =
\#1 1.25in .125in .25in
Introduction
============
SCUBA galaxies (Smail, Ivison & Blain 1997; Barger et al. 1998; Hughes et al. 1998; Eales et al. 1999) are an important population (Blain et al. 1999, 2002), but frustratingly difficult to mesh with semi-analytic model frameworks (Guiderdoni et al. 1998; C.Lacey et al., in preparation). They are both numerous, with a surface density about 10% that of optically-selected Lyman-break galaxies (LBGs; Steidel et al., 1999, 2003) at the relative depths of the current surveys, and luminous (possibly because of very high SFRs), with typical bolometric luminosities about 10 times greater than LBGs, adopting plausible spectral energy distributions (SEDs). The existing SCUBA galaxies produce most of the extragalactic submm background radiation intensity, and a significant fraction of the background at far-infrared (FIR) wavelengths (Blain et al. 2002; Smail et al. 2002; Cowie et al. 2002). They represent a population which is often difficult to detect at optical wavelengths, with at least the fainter half clearly being missed in current optical cosmological surveys. Unfortunately, the SCUBA galaxy population is notoriously difficult to study! Until now, we have been able to gather almost nothing about their redshifts and morphologies as fundamental observable properties. As a consequence, deriving properties such as total mass and dust temperature, and finding both the fraction of power contributed by AGN and starbursts, and their connection to optically-selected star-forming LBGs, have been the topic of largely idle speculation over the five years since their discovery.
The principal hurdle has always been identification at other wavelengths. SCUBA/MAMBO surveys have large beam sizes (10“–15”), a situation that will remain true for upcoming single-antenna, wide-field instruments: Bolocam, SHARC-II, a bolometer camera on the APEX telescope and SCUBA2. Candidate counterparts to SCUBA galaxies cannot therefore be identified unambiguously without interferometry to pinpoint their positions. However, mm/submm interferometry is currently an arduous process, with tens of hours of integration required to detect a single object at the OVRO MMA or IRAM PdB interferometers.
As a consequence, the 20-cm radio emission from SCUBA galaxies has become an important, but [*second best*]{} surrogate with which to probe both the energy generation processes and morphology of submm galaxies (Ivison et al. 1998; Smail et al. 2000; Barger, Cowie & Richards 2000; Chapman et al. 2001, 2002a, 2003a; Ivison et al. 2002). The problem remains that in order to use a radio wavelength as a surrogate for submm/FIR emission, we would like a clear physical principle connecting the emission at the different wavelengths. We don’t yet, but we do have a strong empirical connection: the far-IR–radio correlation (e.g., Helou et al. 1985), which has an RMS dispersion of only 0.2dex over a large range in luminosity at low redshifts. Radio identification of SCUBA galaxies has allowed their optical properties to be explored in detail (Chapman et al. 2003a; Ivison et al. 2002). A large range of optical properties is observed for the radio-selected SCUBA galaxies, with 65% fainter than $I>23.5$. They have red optical-IR colors with $I-K=3$ to 6 and $<I-K>=4.3$, but they are not all extremely red objects (EROs). Deep radio observations allow SCUBA galaxies to be pre-selected with an efficiency of about 40%.
Redshifts for SCUBA galaxies
============================
We have been able to secure redshifts for 55 radio-identified submm galaxies through deep Keck/LRIS observations (Chapman et al. 2003b, S.C. Chapman, in preparation.), and the sample is growing rapidly.[^1] The sources lie in several distinct fields: CFRS03hr, Lockman-Hole, HDF, SSA13, CFRS14hr, Elais-N2, SSA22. The spectra for several new examples are shown in Fig. 1. The redshifts allow, for the first time, accurate calculation of luminosities and dust temperatures for the SCUBA galaxy population. We emphasize that obtaining redshifts is not easy, relying heavily on the superb blue sensitivity of the new LRIS-B mutliobject spectrograph (Steidel et al. 2003), and often have no detectable continuum emission with which to extract the spectrum. In addition the galaxies are hard to identify, typically being faint, messy, composite objects in optical images. Because of their small radio/optical offsets ($\sim0.5$") it is often difficult to assess the best position at which to align the slit. Often we designated several slit positions on different masks for each target. Our spectroscopic completeness is $\sim$50% over the magnitude range of the sample from $I=22$ to 27.
While the issue of correctly identifying the submm galaxy is concerning, three (out of 3) CO detections with IRAM-PdB have already been made, realizing an unequivocal confirmation of the redshifts (Neri et al. 2003), as well as dynamical mass estimates $>10^{11}$M$_\odot$. One goal of the IRAM-PdB program is to obtain CO detections for a statistical sample $\sim$30 of our submm galaxies, to understand the range in molecular gas properties. We also note that beyond the radio positional identification, we are finding the correct [*type*]{} of object through optical spectroscopy. The identifications are at high-$z$, have apparent SFRs of several 100M$_\odot$yr$^{-1}$, often show AGN features, and are very rare objects in the LBG distribution. We ask, what else can they be?
We further note that we believe we have spectroscopically identified a sample of blank-field SCUBA galaxy counterpart candidates without radio detections, effectively through trial and error by targeting faint optical sources lying within the SCUBA beam using otherwise redundant spectrograph slits. Observations of these identifications were tried based on our cumulative experience of with the properties of the radio-identified sample: often faint, distorted blue/red composite sources near the SCUBA beam centroid, are found to have Type-II AGN spectra and/or inferred star formation rates of order 100M$_\odot$yr$^{-1}$. Again, CO detection will be the final arbiter concerning the validity of the identifications. These identifications overlap significantly in redshift with our radio-submm sample, their radio [*non*]{}-detections implying colder dust temperatures.
In Fig. 2, we show the observed redshift distribution, and a toy model for the radio and submm distributions derived from an evolving far-IR luminosity function (Chapman et al. 2003c, Lewis et al. 2003). The model is very useful for understanding selection effects, and in particular the bias against the highest-redshift galaxies due to the requirement of a radio selection. This bias has a strong dependence on dust temperature ($T_{\rm d}$). The sources missed by the radio are expected to fill the region lying between the model radio and submm distributions, overlapping significantly in redshift with the radio-identified sample if ther dust temperatures are in the cold to warm regime ($T_{\rm d}<35$K). The redshift distribution of a radio-selected QSO sample (Shaver et al. 1998) (which is unlikely to be affected by dust obscuration) is overplotted, suggesting a remarkable correspondence with the submm galaxy population.
If our identifications for submm galaxies [*without*]{} radio detections are correct, then this leaves only a $\sim20$% tail of submm galaxies which can lie above $z>4$. With a surface density of $\sim200$/deg$^2$, this 20% tail is still considerably larger than the $z>4$ QSOs detected in the mm/submm (Carilli et al. 2001, Isaak et al. 2002). Continued effort to identify these sources will be a worthy pursuit of the newest instruments and techniques.
Spheroids in Formation?
=======================
We have presented a plausible redshift distribution for submm galaxies, the most important ingredient for addressing whether they could be spheroids in formation. Encouragingly, they lie at the correct redshifts to be proto-ellipticals or the forming bulges of spiral galaxies. (all the stellar population constraints point to most stars being formed at $z$=2–3). In particular, we have demonstrated that most submm galaxies do not lie at $z>4$, and thus most are not very high redshift Population-III sources. Quite surprisingly, even this small sample of spectroscopic identifications reveals a redshift clustering signal, bolstering their association with massive halos (Blain et al. 2003).
However, to assess what type of formation mechanism is at work we must study the AGN/starburst contribution to the dust heating, the timescales of their luminous phase, and compare their [*HST*]{} morphologies. The AGN versus starburst issue is always difficult to address. We have firstly from our restframe UV spectra, the possibility of diagnosing the presence of an AGN through high ionization lines. Indeed, we find signs of CIV, and other lines which cannot easily be excited in a starburst, in approximately 50% of our sources. Our single near-IR spectrum thus far (Smail et al. 2003) reveals an OIII/H$\beta$ ratio typical of Seyfert-2 galaxies as well as strong NeV emission. X-ray detections of 7 submm galaxies in the Chandra Deep Field North were presented in Alexander et al. (2003), while we have measured significant X-ray flux from a further seven of our sample, together implying that $\sim$70% of the radio-identified SCUBA galaxies have X-ray detections. However, only 1/3 of these are too X-ray bright to be generated by star formation alone (scaling from Nandra et al. 2002 for LBGs). Finally, $\sim$20% of our sources appear to be unusually bright at radio wavelengths, departing significantly from the far-IR–radio distribution for low-redshift starburst galaxies. Together, this suggests that AGN are prevalent in the sample (as expected from the likely coeval evolution of the BH and bulge required to generate the tight correlation between their masses; Magorrian et al.1998), but not necessarily dominant in the bolometric energy released by SCUBA galaxies.
The duration of the very luminous activity of the SCUBA galaxies are important: are their timescales 10Myr or 1Gyr? If the former, then several bursts would be required to form a massive elliptical galaxy. In the spectra with the highest signal-to-noise ratios, stellar and interstellar features have been observed. These can be fitted using starburst synthesis models, to yield a timescale for the starburst activity visible at ultraviolet (UV) wavelengths, and a relation between the star-formation rates at observed optical and submm wavelengths. For our brightest object, N2.4 (Smail et al. 2003) the best fit is for a $\sim$5Myr-long instantaneous burst of star formation. The implied star formation rate is about 1000M$_\odot$yr$^{-1}$, which is similar to that infered directly from its submm and radio flux densities. While not all sources may contain such a short burst of star-formation dominated energy output, this example is likely to define the paradigm for the nature of a significant fraction of SCUBA galaxies.
The morphologies of a subset of SCUBA galaxies have recently been studied with [*HST*]{}-STIS by Chapman et al. (2003d). Fig. 3 shows a montage of several representative examples, typically revealing multi-component, distorted galaxy systems that are reminiscent of mergers in progress. There are no examples of isolated, compact sources, and we conclude generally that the morphologies of SCUBA galaxies are generally consistent with hierarchical galaxy formation scenarios in which the most intense activity occurs when gas-rich, high-redshift galaxies collide and merge. Although they could be the sites of spheroid formation, most spheroids do not form in a monolithic collapse of primordial gas clouds at extreme redshifts.
References {#references .unnumbered}
==========
Alexander, D., et al., ApJ, 2003, in press (astro-ph/0211267)\
Barger, A., et al., Nature, 1998, 394, 293\
Barger, A., Cowie, L., Richards, E., 2000, AJ 119, 2092\
Blain, A.W., Smail, I., Ivison, Kneib, J.-P., 1999, MNRAS, 302, 632\
Blain, A.W., Smail, I., Ivison, Kneib, J.-P., Frayer D.T., 2002, Physics Reports, 369, 111 (astro-ph/0202228)\
Blain, A.W., Chapman, S.C., Smail, I., Ivison, R., ApJ, submitted\
Carilli, C., et al., 2001, ApJ, 555, 625\
Chapman, S.C., Richards, E., Lewis, G., Wilson, G., Barger, A., 2001, ApJ, 548, L147\
Chapman, S.C., Lewis, G.F., Scott, D., Borys, C., Richards, E., 2002a, ApJ, 570, 557\
Chapman S.C., Barger A., Cowie L., et al., 2003a, ApJ, 585, 57\
Chapman, S.C., Blain, A., Ivison, R., Smail, I., 2003b, Nature, April 17 issue\
Chapman, S.C., Helou, G., Lewis, G., Dale, D., 2003c, ApJ, in press, astro-ph/0301233\
Chapman, S.C., Windhorst, R., Odewahn, S., Haojing, Y., Muxlow, T., ApJ, 2003d, submitted\
Cowie, L., Barger, A., Kneib, J.-P., 2002, AJ, 123, 2197\
Eales, S., et al., 1999, ApJ 515, 518\
Guiderdoni, B., et al. 1998, MNRAS, 298, 708\
Helou, P., et al., 1985, ApJ 440, 35\
Hughes, D., et al., Nature, 1998, 394, 241\
Isaak, K., et al., 2002, MNRAS, 329, 149\
Ivison, R., et al., 1998, MNRAS, 298, 583\
Ivison, R., et al., 2002, MNRAS, 337, 1\
Lewis, G., Chapman, S., Helou, G. 2003, ApJ, submitted\
Magorrian J. et al., 1998, AJ, 115, 2285\
Nandra, P., et al. 2002, ApJ,\
Neri, R., et al. 2003, A&A, in preparation\
Shaver, P., et al., 1998, APS Conf series 156, Highly Redshifted Radio Lines, ed. C.Carilli, S.Radford, K. Menten, G. Langston, (San Francisco: ASP163)\
Smail, I., Ivison, R.J., Blain, A.W., 1997, ApJ 490, L5\
Smail, I., Ivison, R.J., Blain, A.W., Kneib, J.-P., 2002, MNRAS, 331, 495\
Smail, I., et al., 2003, MNRAS, in press, astro-ph/0303128\
Steidel C., Adelberger K., Giavalisco M., Dickinson M., Pettini M., 1999, ApJ 519, 1\
Steidel, C., et al., 2003, ApJS, in press\
[^1]: Data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership between Caltech, the University of California and NASA. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation.
|
---
abstract: |
There is an ongoing effort to develop tools that apply distributed computational resources to tackle large problems or reduce the time to solve them. In this context, the Alternating Direction Method of Multipliers (ADMM) arises as a method that can exploit distributed resources like the dual ascent method and has the robustness and improved convergence of the augmented Lagrangian method. Traditional approaches to accelerate the ADMM using multiple cores are problem-specific and often require multi-core programming. By contrast, we propose a problem-independent scheme of accelerating the ADMM that does not require the user to write any parallel code. We show that this scheme, an interpretation of the ADMM as a message-passing algorithm on a factor-graph, can automatically exploit fine-grained parallelism both in GPUs and shared-memory multi-core computers and achieves significant speedup in such diverse application domains as combinatorial optimization, machine learning, and optimal control. Specifically, we obtain 10-18x speedup using a GPU, and 5-9x using multiple CPU cores, over a serial, optimized C-version of the ADMM,
which is similar to the typical speedup reported for existing GPU-accelerated libraries, including cuFFT (19x), cuBLAS (17x), and cuRAND (8x).
author:
-
-
-
-
-
bibliography:
- 'IEEEabrv.bib'
- 'ref.bib'
title: 'Testing fine-grained parallelism for the ADMM on a factor-graph'
---
ADMM; Distributed Optimization; Message-passing algorithm; GPU computing; Shared-memory multi-core computing
|
---
abstract: 'Pixel detectors with cylindrical electrodes that penetrate the silicon substrate (so called 3D detectors) offer advantages over standard planar sensors in terms of radiation hardness, since the electrode distance is decoupled from the bulk thickness. In recent years significant progress has been made in the development of 3D sensors, which culminated in the sensor production for the ATLAS Insertable B-Layer (IBL) upgrade carried out at CNM (Barcelona, Spain) and FBK (Trento, Italy). Based on this success, the ATLAS Forward Physics (AFP) experiment has selected the 3D pixel sensor technology for the tracking detector. The AFP project presents a new challenge due to the need for a reduced dead area with respect to IBL, and the in-homogeneous nature of the radiation dose distribution in the sensor. Electrical characterization of the first AFP prototypes and beam test studies of 3D pixel devices irradiated non-uniformly are presented in this paper.'
address:
- 'ICREA and Institut de Física d’Altes Energies (IFAE), Barcelona, Spain'
- 'Centro Nacional de Microelectronica, CNM-IMB (CSIC), Barcelona , Spain'
- 'Fondazione Bruno Kessler, FBK-CMM, Trento, Italy'
- 'U.S. Naval Research Laboratory, Washington, USA'
- 'School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom'
- 'Universita degli Studi di Trento and INFN, Trento, Italy'
- 'INFN Sezione di Genova, Genova, Italy'
- 'Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, USA'
- 'SLAC National Accelerator Laboratory, Menlo Park, USA'
- 'University of Hawaii, c/o Lawrence Berkeley Laboratory, Berkeley, USA'
- 'University of Bonn, Bonn, Germany'
- 'Institute of Physics ASCR v.v.i., Prague, Czech Republic'
author:
- 'S. Grinstein'
- 'M. Baselga'
- 'M. Boscardin'
- 'M. Christophersen'
- 'C. Da Via'
- 'G.-F. Dalla Betta'
- 'G. Darbo'
- 'V. Fadeyev'
- 'C. Fleta'
- 'C. Gemme'
- 'P. Grenier'
- 'A. Jimenez'
- 'I. Lopez'
- 'A. Micelli'
- 'C. Nelist'
- 'S. Parker'
- 'G. Pellegrini'
- 'B. Phlips'
- 'D.-L. Pohl'
- 'H. F.-W. Sadrozinski'
- 'P. Sicho'
- 'S. Tsiskaridze'
title: |
Beam Test Studies of 3D Pixel Sensors Irradiated Non-Uniformly\
for the ATLAS Forward Physics Detector
---
Introduction {#sec:introduction}
============
ATLAS [@atlas] plans to install a Forward Physics detector (AFP) in order to identify diffracted protons at $\approx$210 m from the interaction point [@phase-1-LoI]. The current AFP design foresees a high resolution pixelated silicon tracking system combined with a timing detector for the removal of pile up protons. The AFP tracker unit will consist of an array of six pixel sensors placed at 2-3 mm from the Large Hadron Collider (LHC) proton beam. The proximity to the beam is essential for the AFP physics program as it directly increases the sensitivity of the experiment [@phase-1-LoI]. Thus, there are two critical requirements for the AFP pixel detector: first, the active area of the detector has to be as close as possible to the LHC beam, which means that the dead region of the sensor has to be minimized. Second, the device has to be able to cope with a very inhomogeneous radiation distribution. Preliminary estimations indicate that the side of the sensors close to the beam will have to sustain doses on the order of $5\times 10^{15}$ n$_{eq}$/cm$^2$, while the opposite site is expected to receive several orders of magnitude less radiation [@afp_cite].
Based on the successful performance of CNM [@cnm] and FBK [@fbk] 3D sensors productions [@3dtech] for the ATLAS Insertable B-Layer (IBL) [@ibl_module_paper], the 3D technology was chosen for the AFP tracking detector. However, the IBL sensors have a large inactive edge on the side which will be closest to the beam in the AFP configuration. To meet the AFP requirements, different dicing techniques have been investigated to reduce the dead area of the IBL 3D sensors. Section \[sec:prototypes\] of this paper presents the results of electrical tests performed on slim-edge prototypes for AFP.
The bias voltage needed to deplete a silicon sensor increases with radiation dose. Since the radiation distribution of the AFP silicon sensors will be highly non-uniform, there can be a scenario where the breakdown voltage of the non-irradiated zone is lower than the depletion voltage of the irradiated one. This will degrade the device performance: a high bias voltage will increase the leakage current leading to noise (to a point where the device may not longer be operated), while an intermediate voltage will reduce the amount of charge collected in the irradiated area, degrading the hit reconstruction efficiency. Thus sensors with high initial breakdown voltage will be ideal for AFP. Section \[sec:tb\] of this paper presents the first beam test results of 3D sensors irradiated non-uniformly up to fluencies of $9.4\times 10^{15}$ n$_{eq}$/cm$^2$.
The AFP Pixel Module
====================
The pixel readout electronics of AFP will be the FE-I4 [@fei4] chip, which was developed for IBL. Built in the 130 nm CMOS process, the FE-I4 offers an increased tolerance to radiation with respect to the current ATLAS pixel readout chip, the FE-I3 [@fei3]. The IBL chip features an array of $80\times 336$ pixels with a pixel size of $50\times250$ $\mu$m$^2$. The total size of the FE-I4 chip is $20.2\times 19.0$ mm$^2$ and the active fraction is 89%. The sensors are DC coupled to the chip with negative charge collection. Each readout channel contains an independent amplification stage with adjustable shaping, followed by a discriminator with independently adjustable threshold. The chip operates with a $40$ MHz externally supplied clock. The time over threshold (ToT) with $4$-bit resolution together with the firing time are stored for a latency interval until a trigger decision is taken.
The selected sensor for AFP is the IBL 3D sensor, since it provides high radiation tolerance at low bias voltage [@sgrinstein_nima]. The IBL 3D sensor is 230 $\mu$m thick with the n- and p-type columns etched from the opposite sides of the p-type substrate. The pixel configuration consists of two n-type readout electrodes connected at the wafer surface along the 250 $\mu$m long pixel direction, surrounded by six p-type electrodes which are shared with the neighboring pixels. The CNM 3D sensor design features $210$ $\mu$m long columns which are isolated on the n$^+$ side with p-stop implants. The edge isolation is accomplished with a combination of a n$^+$ 3D guard ring, which is grounded, and fences, which are at the bias voltage potential from the ohmic side.
The IBL sensor design has a large ($\approx$1mm) dead region in the side opposite to the wirebonds, see Fig \[fig:design\]. This dead region was added to accommodate the bias stab needed for the active edge technology which was finally not incorporated into the IBL (the side opposite to the wirebonds is not critical for the IBL, since the sensors overlap in the $r-\phi$ direction). However, for the AFP project, it is critical to substantially reduce the mentioned dead area, since it will be closest to the beam.
In order to adapt the 3D IBL sensors for AFP, different techniques have been investigated. Detectors were post-processed using the scribe-cleave-passivate (SCP) technology to reduce the dead area. Details of SCP method have been described in other publications [@scp]. Very briefly, the method relies on passivating the sidewall with low defect density. A low defect density is obtained in a two-step process. First, the surface of the sensor is scribed at the desired edge location in the direction that coincides with one of the silicon crystal planes. Then the peripheral region is mechanically cleaved off. The resulting sidewall follows the crystal plane. The passivation step uses dielectric material that depends on the bulk type of the silicon. For p-type wafers used in this study, a dielectric with negative interface charge on the border with silicon is needed. This is accomplished by depositing an atomic layer (ALD) of Al$_2$O$_3$. Standard diamond-saw cuts were also investigated [@fbk_marco_povoli_iword_2011], but are not presented here.
![Design mask of the CNM 3D sensor for IBL. The large dead area in the region opposite to the wirebonds is visible. A magnified view shows details of pixel geometry.[]{data-label="fig:design"}](cnm_design_resmdd12.png){width="7cm"}
Electrical Tests of the First 3D Sensors Prototypes for AFP {#sec:prototypes}
===========================================================
The SCP technology has been used to reduce the inactive area of the IBL sensors for AFP. Since a limited number of sensors were available at the time, the technique was first tested on 3D sensors designed for the FE-I3 readout chip. Though the pixel geometry is different, the results are expected to be relevant for the FE-I4 sensors, since the slim edge performance mostly depends on characteristics of the sensor periphery: sensor thickness, slim edge distance, and sidewall treatment. Seven CNM 3D FE-I3 devices which presented breakdown voltages above $40$ V were sent to NRL [@nrl] to reduce the dead region by applying the SCP technology, from the original $1$ mm to $50-100\mu$m. The devices were returned to Barcelona to be bump-bonded to FE-I3 readout chips and characterized. Figure \[fig:fei3s\] shows the current versus bias voltage (“IV”) measurements at room temperature. Two assemblies show a resistive behavior. This is probably due to the slim-edge being too close to the 3D guard-ring, which effectively connects the ohmic side to ground. The charge collection was verified in all devices with good breakdown voltage using a 90-Sr source and an external trigger provided by a scintillator, see Fig. \[fig:fei3s\]. Since the 230 $\mu$m thick devices were calibrated to 30 ToT for a 20000 electron signal, almost full charge collection was obtained at 10 V.
![Current versus bias voltage for CNM FE-I3 3D sensors with slim edges (top). Two devices have resistive behavior. The rest has expected low leakage currents and high breakdown voltage. The MPV of the Landau fit to the ToT spectrum obtained with a Sr-90 source is shown below for the AFPs2 device.[]{data-label="fig:fei3s"}](IV_AFP_25C.pdf "fig:"){width="7cm"} ![Current versus bias voltage for CNM FE-I3 3D sensors with slim edges (top). Two devices have resistive behavior. The rest has expected low leakage currents and high breakdown voltage. The MPV of the Landau fit to the ToT spectrum obtained with a Sr-90 source is shown below for the AFPs2 device.[]{data-label="fig:fei3s"}](sr90_fei3.png "fig:"){width="6.0cm"}
Non-uniform Irradiation of 3D Sensors {#sec:irrad}
=====================================
One critical aspect of the pixel devices for the AFP forward detector is the non-uniform nature of the radiation distribution across the sensor. AFP pixel devices must be able to operate with high efficiency when only a portion of the sensor is irradiated to $5\times 10^{15}$ n$_{eq}$/cm$^2$. This means that the less irradiated areas of the sensor have to sustain high bias voltages keeping the leakage current at moderate values. Previous experiences with FE-I4 CNM 3D devices indicated that the leakage current has to be lower than 200 $\mu$A (front-end chip not powered) in order to maintain low noise levels and ensure high hit reconstruction efficiency [@sgrinstein_nima].
To study the effect of non-uniform irradiations, two FE-I4 IBL prototypes (CNM-57 and CNM-83) were irradiated with protons non-uniformly at the IRRAD1 facility at CERN-PS [@irrad1]. The irradiation levels are listed in Table \[table:fei4\_devices\]. Figure \[fig:irrad\] shows the irradiation dose distribution on CNM-57, a similar profile was obtained for CNM-83 but with a larger maximum dose.
![Dose distribution on the CNM-57 sample [@dosimetry]. A similar profile was obtained for CNM-83, but with a maximum dose of $9.4\times 10^{15}$ n$_{eq}$/cm$^2$, see also Table \[table:fei4\_devices\].[]{data-label="fig:irrad"}](fig3_cnm57_irrad_f.png){width="8cm"}
The leakage currents measured as a function of the bias voltage for CNM-57 and CNM-83 are shown in Fig. \[iv\_fei4\_irrad\]. The measurements were taken at $-20^{\circ}$ C with the front-end electronics not powered. The device with early breakdown before irradiation shows a large leakage current at intermediate bias voltages ($\approx$150 V), an indication that its performance may not be optimal due to the noise induced by the dark current.
![Measurement of the leakage current as a function of the bias voltage for the two irradiated CNM devices at $-20^{\circ}$C with the front-end chip not powered.[]{data-label="iv_fei4_irrad"}](IV_CNM57_CNM83.pdf){width="7cm"}
-------- ----------------------------- ------------- ------------------
Sample Max. Dose Breakdown Bias Voltage
($10^{15}$ n$_{eq}$/cm$^2$) Voltage (V) at Beam Test (V)
CNM-57 4.0 75 130
CNM-83 9.4 10 130
-------- ----------------------------- ------------- ------------------
: Samples irradiated non-uniformly for AFP. The breakdown voltages shown are before irradiation. []{data-label="table:fei4_devices"}
Before the performance of the devices is studied in beam tests, it is necessary to verify that the front-end electronics was not damaged as a result of the proton irradiation. The operational threshold was set to $1700$ electrons based on previous experience with the IBL devices [@ibl_module_paper]. The threshold distributions of the two irradiated devices, shown in Fig. \[fig:thresholds\], present good uniformity, while the ENC (Equivalent Noise Charge) is below 200 electrons.
![Threshold (top) and noise (bottom) distribution of the FE-I4 CNM AFP prototype irradiated devices. []{data-label="fig:thresholds"}](resmdd12_thrs_f.png){width="8cm"}
Beam Test Studies of Non-uniformly Irradiated 3D Sensors {#sec:tb}
========================================================
Test beam studies of non-uniformly irradiated devices are essential to understand the AFP pixel detector module performance. Critical parameters, such as hit efficiency and position resolution, can only be determined at beam tests. Both CNM-83 and CNM-57, together with an un-irradiated reference device, have been characterized using 120 GeV pions at the CERN SPS H6 beam line in August 2012.
Beam particle trajectories were reconstructed using the high resolution EUDET telescope [@eudet]. The telescope consists of six Mimosa tracking planes, the readout data acquisition system and the trigger hardware, and provides a $\approx3\mu$m track pointing resolution. The devices under test were placed between the telescope planes. Data presented here were recorded at perpendicular incident angle. The devices under test were cooled to $-15^{\circ}$ C. The hit efficiency is determined from extrapolated tracks on the devices, after track quality cuts have been applied. A hit on the device under test is searched for in a $3\times3$ pixel window around the track position.
Since the active area of the FE-I4 devices is larger than the Mimosa sensors of the telescope, separate sets of data were taken to cover the irradiated and non-irradiated regions of the sensors. Fig. \[fig:cnm57\_eff\_sensor\] shows the efficiency map for sensor CNM-57. The non-irradiated region has an average efficiency of $98.9\%$, while the efficiency for the irradiated side is $92.7\%$. If the dead and noise pixel cells (due to front-end issues) are removed, the efficiency increases to $98.0\%$. In order to highlight the pixel structure, the corresponding efficiency as function of the track hit position folded into a two by two cell is also shown in Fig. \[fig:cnm57\_eff\_sensor\]. As the device is positioned perpendicular to the particle beam the effect of the pass-through electrodes is evident. The efficiency for sensor CNM-83 on the irradiated side was much lower, about $60\%$, due to the large noise induced by the leakage current. This was expected from the electrical measurements done before the beam tests. In fact, it was difficult to operate the device during the beam test because the noise affected the data readout synchronization.
![Hit efficiency for CNM-57. The top plot shows the efficiency across the sensor. The area labeled “a” (“b”) corresponds to the region which was less (more) irradiated (see Fig. \[fig:irrad\]). After dead and noise pixels are removed, the average efficiency for “a” (“b”) is $99.2\%$ ($98.0\%$). The efficiency for “a” folded into a two by two pixel area is shown below. []{data-label="fig:cnm57_eff_sensor"}](cnm57_eff_sensor.png "fig:"){width="7cm"} ![Hit efficiency for CNM-57. The top plot shows the efficiency across the sensor. The area labeled “a” (“b”) corresponds to the region which was less (more) irradiated (see Fig. \[fig:irrad\]). After dead and noise pixels are removed, the average efficiency for “a” (“b”) is $99.2\%$ ($98.0\%$). The efficiency for “a” folded into a two by two pixel area is shown below. []{data-label="fig:cnm57_eff_sensor"}](cnm57_eff_pixel.png "fig:"){width="7cm"}
Conclusions
===========
The AFP project presents a new challenge for pixel detectors: reduced dead areas are needed to maximize the physics potential of the experiment, and the sensors have to be able to sustain a highly non-uniform irradiation distribution. The scribe-cleave-passivate technology used to reduce the inactive edge of 3D sensors showed promising results with FE-I3 devices. The first FE-I4 prototypes have already been produced and will be tested shortly. The first beam test studies of non-uniformly irradiated 3D pixel sensors for AFP have been presented. For devices with low initial breakdown, the performance is poor due to the large leakage current caused by the bias voltage needed to deplete the irradiated area. Devices that show good electrical behavior before irradiation are able to sustain the voltage needed to achieve excellent efficiency ($>98\%$ at perpendicular incidence) throughout the sensor.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was partially funded by the MINECO, Spanish Government, under the grant FPA2010-22060-C02-01, and the European Commission under the FP7 Research Infrastructures project AIDA, grant agreement no. 262025. We would like to thank the Institute for Nanoscience (NSI) at the U.S. Naval Research Laboratory (NRL) and the NSI staff. The work done at NRL was sponsored by the Office of Naval Research (ONR). The work at SCIPP was supported by Department of Energy, grant DE-FG02-04ER41286.
[00]{}
The ATLAS Collaboration, “The ATLAS Experiment at the CERN Large Hadron Collider”, JINST 3 S08003 (2008).
The ATLAS Collaboration, “Letter of Intent for the Phase-I Upgrade of the ATLAS Experiment”, CERN-LHCC-2011-012, 2011.
L. Adamczyk , “AFP: A proposal to install proton detectors at 220 m around ATLAS to complement the ATLAS high luminosity physics program”, ATL-COM-LUM-2011-006, CERN (2011), AFP technical proposal.
Centro Nacional de Microelectronica (CNM), Barcelona (Spain), http://www.imb-cnm.csic.es/. Fondazione Bruno Kessler (FBK), Trento (Italy), http://www.fbk.eu/.
C. Da Via , Nucl. Instrum. and Methods in Phys. Res. A [**694**]{} (2012) 321–330
The ATLAS IBL Collaboration, “Prototype ATLAS IBL Modules using the FE-I4A Front-End Readout Chip”, submitted to JINST, http://arxiv.org/abs/1209.1906.
M. Garcia-Sciveres , Nucl. Instrum. Methods Phys. Res. A [**636**]{}, S155 (2011). I. Peric, , Nucl. Instrum. Methods in Phys. Res. A [**565**]{}, 178 (2006).
S. Grinstein, The ATLAS Collaboration, Nucl. Instrum. Methods in Phys. Res. A [**699**]{}, 61 (2013).
M. Christophersen, V. Fadeyev, B.F. Phlips, H. F.-W. Sadrozinski, C. Parker, S. Eli, J.G. Wright, Nucl. Instrum. Meth. in Phys. Res. A [**699**]{} (2013) 14.
M. Povoli , iWoRID2011 Zurich 2011 (to be fixed!).
U.S. Naval Research Laboratory, Washington, USA.
M. Glaser, M. Huhtinen, F. Lemeilleur, C. Leroy, P. Roy, M. Tavlet, “Radiation Test Facilities in the New PS East Hall at CERN”, in Proc.5th Eur. Conf. Radiation and Its Effects Components and Systems, Fontevraud, France, 2000, pp. 136-141.
M. Glaser, F. Ravotti, M. Moll, “Dosimetry Assessments in the Irradiation Facilities at the CERN-PS Accelerator”, IEEE Trans. Nucl. Sci, vol. 53 (4), pp. 2016-2022, 2006.
E. Corrin, Nuclear Science Symposium Conference Record (NSS/MIC), 2009 IEEE, 816 (2009).
|
---
abstract: |
If perturbations beyond the horizon have the velocities prescribed everywhere then the dragging of inertial frames near the origin is suppressed by an exponential factor. However if perturbations are prescribed in terms of their angular momenta there is no such suppression.
We resolve this paradox and in doing so give new explicit results on the dragging of inertial frames in closed, flat and open universe with and without a cosmological constant.
author:
- |
Jiři Bičák,$^{1,2,4}$ Donald Lynden-Bell,$^{2,1,4}$ and Joseph Katz$^{3,1,4}$\
$^1$[*Institute of Theoretical Physics, Charles University, *V Holešovick' ach 2, 180 00 Prague 8, Czech Republic**]{}\
[*$^2$Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, United Kingdom*]{}\
[*$^3$The Racah Institute of Physics, Givat Ram, 91904 Jerusalem, Israel*]{}\
[*$^4$Max-Planck Institute for Gravitational Physics (Albert Einstein Institute) *14476 Golm, Germany.**]{}
title: |
[Do Rotations Beyond the Cosmological Horizon\
Affect the Local Inertial Frame?]{}
---
Introduction
============
In a clearly written paper Schmid [@Sc] describes how rotational perturbations of a spatially flat universe influence the inertial frames. He finds that such influences are attenuated by an exponential Yukawa factor whenever the perturbation lies beyond a ‘horizon’. He expressed his results in terms of a quantity that he calls the energy current ${\vec J}_\epsilon$. What corresponds to Schmid’s energy current ${\vec J}_\epsilon$ in our calculation is a quantity ${\vec J}_s$ with the dimensions of angular momentum. However his result appears to disagree with our earlier result [@LKB95] (hereafter LKB) that in a flat universe the rotation of the inertial frame , $\omega$, due to any system of spheres with small rotations about a center is given by $$\vec\omega (\overline{r},t)=\frac {2G}{c^2}\left[
\frac{1}{\overline{r}{}^3}{\vec J}(t,<\overline{r})+\int_{\overline r}^{\infty}\frac{1}{\overline
r{'}^3}\frac{{\rm d}{\vec J}}{{\rm d}\overline
r{'}}\,{\rm d}\overline r{'}\right],
\label{1}$$ where ${\vec J}(t,<\overline{r})$ is the total angular momentum within the sphere of proper radius [@foot1] $\overline r=a(t)r$. This expression demonstrates how angular momenta at all distances contribute and shows no exponential cut-off and no influence of any horizon.
Both results agree, however, that inertial ‘influences’ may be expressed instantaneously i.e. with no light travel-time delay. This is because they follow from the constraint equations of General Relativity with an appropriate mapping onto an unperturbed universe to provide a suitable gauge.
The results are in [*apparent*]{} contradiction. However, when the details of both calculations are examined it is evident that the contradiction lies in the attribution of different causes for the effect. Schmid’s ‘energy current’ is considered by him as the source of the rotational dragging of inertial frames $\omega$. Schmid’s quantity does not obey a conservation law but for a rotating sphere it can be directly expressed in terms of its angular velocity $\Omega$, density $\rho$, pressure $p$ and proper radius $\overline r$ as $$J_\epsilon =2\pi\!\!\int\!\!\!\int(\rho+p)\Omega\sin^3\theta~\overline r^4 \,{\rm d}\theta
\,{\rm d}\overline{r},$$ $\Omega$ is the ‘coordinate’ angular velocity, not that measured relative to the local inertial frame which, as we show below, is the quantity needed in the angular momentum conservation law. The contribution to the conserved angular momentum is $$2\pi(\rho+p)(\Omega -\omega)\sin^3{\theta} \, \overline r^4 \,{\rm d}\theta \,{\rm d}\overline{r}.$$ The difference comes about mathematically because the perturbed metric is not diagonal but in spherical polar coordinates is $${\rm d}s^2={\rm d}t^2-a^2(t) \Big\{ {\rm d}r^2+r^2\Big[
{\rm d}\theta^2+\sin^2\theta \, ({\rm d}\varphi^2-2\omega \, {\rm d}\varphi \,{\rm d}t)
\Big]\Big\},
\label{metric}$$ where $\omega(t,r)$ gives the small rotation of the inertial frame due to the rotational perturbations everywhere.
For a perturbed 3-flat universe, angular momentum conservation is given by the equation $$\frac{\partial}{\partial x^\mu}\left(-\sqrt{-g}\,T^\mu_\nu\eta^\nu \right)=0,$$ where $\eta^\nu$ is the angular Killing vector of the background (flat) space corresponding to the particular component of angular momentum considered.
Thus the conserved quantity is (the minus sign comes from the signature we use in the metric) $$J=\int-T^0_\varphi
\sqrt{-g}\,{\rm d}^3\!x.$$ Since the $\varphi$ component of $T^{\mu\nu}$ is brought down in this expression, it is [*not*]{} merely the motion of the fluid that is involved in $T^0_\varphi$ via its contribution $u^\mu u^\nu$ but also the off-diagonal metric component $g_{0\varphi}$ which depends on $\omega$ at the position of the source \[see our metric (\[metric\])\]. As we see from equation (\[1\]), we regard the conserved angular momentum $J$ as the source of the dragging of inertial frames, and this was the quantity we used in LKB.
Schmid’s work for a spatially flat universe is more general than our work published so far, since he considers [*all*]{} vector perturbations, nevertheless we treated closed and open universes as well as flat ones and indeed from a Machian viewpoint it is the closed universes that are more interesting by far. We also considered all spherical but inhomogeneous Lemaître-Tolman-Bondi universes with rotational perturbations that were constant on spheres. Finally we looked into the problem of the rotation of inertial frames induced by spheres of given angular velocities, rather than given angular momentum . This is a special case of Schmid’s problem but generalized to closed and open universes. In our discussion we wrote down the equations governing $\omega(t,r)$ when $\Omega(t,r)$ was given and showed how they could be solved. We carried out the detailed solution only for the static closed Einstein universe (LKB Appendix A).
Schmid’s beautiful result that the dragging is exponentially suppressed when a sphere of given angular velocity is outside his horizon, has stimulated us to work out all our solutions in detail for all FRW universes. Barring factors of $a(t)$ that Schmid seems to have omitted in error, we fully confirm his result for a flat universe. Thus we have the fascinating paradox that [*while spheres of given angular velocity have their dragging exponentially suppressed if they are outside the horizon, nevertheless the dragging of spheres of given angular momentum suffers no such suppression!*]{} How can this be!
In the prescribed angular momentum problem one may consider (for an open or 3-flat universe) having only one spherical shell of finite thickness having angular momentum. The gravity of this source will induce a rotation of inertial frames everywhere. The fluid at all other places will respond inertially and start to rotate so that $\Omega=\omega$ everywhere except on the original shell. Thus in the prescribed angular momentum problem we give one shell angular momentum , sit back and watch. We see how the inertial frames are affected everywhere else merely by watching the rotations of all other spheres.
The prescribed angular velocity problem needs more organization in the creation of the initial state. If we start one thick shell rotating at the prescribed rate then all the other will start moving so as to keep up with the induced rotation of the inertial frames. If the prescription is to have just the one thick shell rotating and none of the others we shall have to stop them. In doing so we have to give them negative angular momentum to keep $\Omega$ zero even though the inertial frame is rotating at $\omega$. When in Schmid’s problem the perturbation in angular velocity is considered as confined beyond his horizon he shows that the rotation of inertial frames is exponentially suppressed near the origin. The prime reason is that in order to keep the motion confined, the intervening spheres have to be given backward angular momentum to stop them from following their inertial tendency of rotating at the inertial rate $\omega$. The influence of all the backward angular momentum (of non-rotating spheres!) rather effectively cancels most of the rotation of the inertial frames induced by the original shell. Hence the suppression of the effect is due to all the negative angular momentum that was supplied to keep the other spheres from rotating! The remaining suppression is due to the rotation of the inertial frame at the original shell itself: $\omega$ there is a fraction of $\Omega$ so that $\Omega-\omega$, on which the source depends, is less.
There is a long history of treating dragging effects within spheres starting with Einstein’s treatment using an early version of his gravitational theories. Within General Relativity the early works of Thirring [@Th] and Lense and Thirring [@LT] were later generalized to deep potential wells by Brill and Cohen [@BC]. This raised questions as to whether the dragging would be perfect within a black-hole’s horizon. We believe that the first paper to remark on the apparent instantaneity if inertial frames is the pioneering paper of Lindblom and Brill [@LB] on inertia in a sphere that falls through its horizon. More recently we explored observational effects seen within such a sphere [@KLB98] and gave an example of strong linear dragging in a rapidly accelerated charged sphere [@LKB98]. Strong cosmological perturbations in a weakly rotating sphere surrounding a void were treated by Klein [@Kl], and in greater detail by Doležel, Bič' ak, and Deruelle [@DBD] who also discussed how an observer within such a cosmological shell views the world outside.
We owe a debt to Schmid as his work stimulated us to work out the consequences of our solutions [@LKB95] in much greater detail and, without that, we would never have raised, let alone understood, the delightful paradox emphasized above. In particular we have now investigated thoroughly the problem when $\omega$ is to be solved for with the angular velocities given everywhere at one cosmic time. Previously we had concentrated on the problem with the angular momenta given. While both are important problems we strongly believe that it is the latter that is of dynamical importance in formulating Mach’s Principle. It can nevertheless be argued that the apparent agreement between the angular positions of quasars at different epochs and the inertial frame defined by using the solar system as a giant gyroscope stimulates Machian ideas. While it is the angular momentum that is important for the physics it is the apparent kinematical agreement between the [*angular*]{} velocity of the sphere and the inertial frame that is observed. In this sense the problem with given angular velocities may be closer to Mach’s original and it is unclear how distant observations could measure the true angular momentum of a sphere including its dragging term, while its angular velocity is more directly observable. However, see [@DBD] for the complications of light bending. Unfortunately the problem of the observed agreement of frames is not that either Schmid or we have addressed since both our treatments relate instantaneous quantities at the same cosmic time whereas observers use no such world map (except in the solar system) but a world picture in which distant objects are seen as they were long ago on the backward light cone. It seems unlikely that an [*exact*]{} causal relationship exists between proper motions of masses on our past light cone and our local inertial frame, since, [*at any cosmic time the inertial frame’s rotation has contributions from objects that were never in our past light cone*]{}. Of course such objects will no doubt have been seen by some alien and the Copernican principle would suggest that the apparent agreement of the kinematic and inertial frames here will be repeated there. What is under discussion above is the influence of distant bodies on the local inertial frame. This is quite distinct from a comparison of the dynamics of the solar system with its kinematics relative to distant quasars (as seen on a hundred years of past light cones), from which the rotation of the inertial frame is computed.
Beside the resolution of the apparent contradiction with Schmid the main contributions of this paper are the following.
[*Section 2*]{}. The derivation of the equations governing general perturbations and a brief introduction to Machian gauge conditions which allow the separation of the $(h_{0k})$ vector perturbations equations from the others. The discussion of the equations of motion that must be obeyed if the contracted Bianchi identities are to be satisfied. As a consequence when axial symmetry is imposed each ring of fluid preserves its angular momentum . This section concludes with basic equations for odd parity axially symmetrical perturbations from which the remainder of the paper is derived.
[*Section 3*]{} derives the explicit expressions for rotation of inertial frames in terms of the angular momentum distribution at any one time. This is done for all FRW universes with $k=\pm 1$ or $0$ but the simplest case is solved in this section with $\Omega$ constant on spheres at the time considered. This corresponds to odd-parity vector $l=1$ perturbations with $\Omega$ independent of $\theta$. In the following paper [@BLK2] (Paper II) we allow for general $\theta$ dependence. With the integrals evaluated at fixed cosmic time and with the constants $c$ and $G$ restored we have the following results for $\vec \omega(r)$ at fixed time (for the derivation of the vector forms below see [@LKB95]):
For $k=0$, $\overline r =a(t)r$, $$\qquad \vec \omega=\frac{2G}{c^2a^3}\left[
{\vec J}(<r)r^{-3}+\int_{\overline r}^{\infty}\frac{{\rm d}{\vec J}}{{\rm d}
r} r^{-3}\,{\rm d} r\right].$$ Notice that $\vec \omega\propto[a(t)]^{-3}$ since ${\vec J}$ is conserved.
For $k=1$, $$\vec \omega=\frac{2G}{c^2a^3}\left[
{\vec J}(<\chi)W(\chi)+\int_{\chi}^{\pi}\frac{{\rm d}{\vec J}}{{\rm d}
\chi'}W(\chi')\,{\rm d} \chi'\right] + \vec \omega_0(t),$$ here $\vec \omega_0(t)$ is undetermined, $r=\sin\chi$ and $W(\chi)=
\cot^3\chi+3\cot\chi$. The arbitrariness of $\vec \omega_0(t)$ is intimately connected with Mach’s principle. The physical ${\vec J}$ involves $(\vec \Omega-\vec \omega)$ and does not change for rotating axes as it involves a difference, see [@LKB95] and below.
For $k=-1$, $$\vec \omega=\frac{2G}{c^2a^3}\left[
{\vec J}(<\chi)\overline W(\chi)+\int_{\chi}^{\infty}\frac{{\rm d}{\vec J}}{{\rm d}
\chi'}\overline W(\chi')\,{\rm d}\chi'\right],$$ where $\overline W(\chi)=
\coth^3\chi-3\coth\chi+2$, and $\overline W$ has an extra 2 so it tends to zero at $\chi\rightarrow\infty$ thus ensuring that the boundary condition $\omega\rightarrow 0$ is obeyed. When contributions from a $\theta$ dependence of $\Omega$ are included these results are supplemented by $\theta$ dependent terms that average to zero on spheres. More general results are given in the accompanying Paper II.
[*Section 4*]{} gives explicit solutions for the rotations of inertial frames for the same special forms of perturbations as in Section 3 but now it is the angular velocities of the different spheres that are given rather than their angular momenta (this is [*closer*]{} to what might be observed but cf. earlier discussion). We define $\lambda$ by $$\lambda^2=2\kappa a^2(\rho+p)= 4(k-a^2\dot
H),
\label{la}$$ $\kappa=8\pi G/c^4$, $\kappa=8\pi$ in geometrical units used in the following, the dot denotes $\partial/\partial t$, $H=\dot a/a$ is the Hubble constant. The second relation in (\[la\]) follows from the combination of the background Einstein’s equations for any $\rho$, $p$, $k$ and also for any value of the cosmological constant $\Lambda$. The rotation of inertial frames near the origin due to an $\Omega$ distribution at large $z'=\lambda r$ is for $k=0$ $$ \omega(r)=\frac{1}{3}\left(1+\frac{1}{10}\lambda^2r^2\right)
\int_0^\infty z'^2e^{-z'}\Omega(z')\,{\rm d}z',$$ [which shows Schmid’s exponential attenuation $e^{-z'}$. At the perturbation itself, close to $z_0$, we find for $z'$ large:]{} $$\omega(r_0)=\frac{1}{2}\int_0^\infty
\left(\frac{z'}{z_0}\right)^2e^{-|z'-z_0|}\Omega(z')\,{\rm d}z'.$$
For $k=1$ we give the results near the origin and at the perturbation when $\lambda^2>4$. When $\lambda^2<4$, which can occur when a $\Lambda$-term is present, there is no exponential in the expression. It is assumed that $\exp(\sqrt{\lambda^2-4}\,\chi)$ is large at the source. With $r=\sin\chi$ $$\begin{aligned}
\omega(\chi)&&=\frac{1}{3}\left(1+\frac{\lambda^2\chi^2}{10}\right)
\times\\\nonumber
&&\int_0^\infty\lambda^2\sqrt{\lambda^2-4}~e^{-\sqrt{\lambda^2-4}\chi'}\sin^2(\chi')
\Omega(\chi')\,{\rm d}\chi'.\end{aligned}$$ We have assumed $\exp(\sqrt{\lambda^2-4}~x)\gg1$ for $x=\pi$, $\chi'$, and $\pi-\chi'$. At the ‘source’ $$\begin{aligned}
&&\omega(\chi_0)=\\\nonumber
&&\frac{1}{2} \int_0^\infty\!\!\lambda^2\frac{\lambda^2-3}{\sqrt{\lambda^2-4}}\left(
\frac{\sin\chi'}{\sin\chi_0}\right)^2e^{-\sqrt{\lambda^2-4}|\chi'-\chi_0|}
\Omega(\chi')\,{\rm d}\chi'.\end{aligned}$$ Similarly for $k=-1$: $r=\sinh\chi$, $$\begin{aligned}
&&\omega(\chi)=\frac{1}{3}\left(1+\frac{\lambda^2\chi^2}{10}\right)\times\\\nonumber
&&\int_0^\infty\lambda^2\sqrt{\lambda^2+4}~e^{-\sqrt{\lambda^2+4}\chi'}\sinh^2(\chi')
\Omega(\chi')\,{\rm d}\chi',\end{aligned}$$ and at the source $$\begin{aligned}
&&\omega(\chi_0)=\\\nonumber
&&\frac{1}{2} \int_0^\infty\frac{\lambda}{\sqrt{\lambda^2+4}}\left(
\frac{\sinh\chi'}{\sinh\chi_0}\right)^2e^{-\sqrt{\lambda^2+4}|\chi'-\chi_0|}
\Omega(\chi')\,{\rm d}\chi'.\end{aligned}$$
We emphasize that all of the above relationships are true at any given instant, but that both the angular momentum distribution and the angular velocity distribution at later instants are related to those at earlier times, so can not be given [*independently*]{} of those given at an earlier epoch. In axial symmetry the angular momentum distribution follows the motion of the perfect fluid but, as the angular momentum is first order and the movement across the background is of first order, the product can be neglected. Thus to first order the angular momentum density can be considered as painted on the background. This is not true of ${\vec J}_s$ which is not conserved and nor is it true of the angular velocity $\Omega$. In both cases to find the time evolution one must appeal to the equations of motion which, in axial symmetry, leads back to local conservation of angular momentum density. Only by use of its conservation can one find how $\Omega$ and ${\vec J}_s$ can evolve consistently with Einstein equations (i.e. with the contracted Bianchi identities). In this sense the given angular momentum problem is far more physical than either Schmid’s problem or the given $\Omega$ problem to which it is equivalent. The time evolution of $\omega$ and $\Omega$ are derived and discussed in Section 5.
In a paper that has long been in gestation we give a discussion of those gauges in which the Machian relations of the local inertial frames to the motions of distant masses can be expressed instantaneously at constant cosmic time. In that paper we derive all equations that govern all perturbations. All can be solved using harmonics in the 3-space of constant time. However harmonics are not as informative as Green’s functions so in the following paper [@BLK2] we integrate the relationships between the rotations of the inertial frames and the angular momentum density for all axially symmetrical odd-parity vector perturbations, called usually “toroidal” perturbations in astrophysical and geophysical literature. These results allow $\Omega-\omega$, which enter the angular momentum density, to be any function of $(r,\theta)$ but independent of $\varphi$. However, since the background is spherically symmetric, non-axisymmetric perturbations can be generated by re-expanding axisymmetric perturbations around a new axis, and taking the component with the new $e^{im\varphi}$ as the component with general $m$.
The equations to be solved
==========================
We write the perturbed FRW metric in the form $$\begin{aligned}
{\rm d}s^2&=&(\overline g_{\mu\nu}+h_{\mu\nu}){\rm d}x^\mu {\rm d}x^\nu
\\\nonumber
&=& {\rm d}t^2-a^2(t)
f_{ij}{\rm d}x^i {\rm d}x^j+h_{\mu\nu}{\rm d}x^\mu {\rm d}x^\nu,
\label {ds2}\end{aligned}$$ where the background metric $\overline g_{\mu\nu}$ is used to move indices and the time-independent part of the spatial background metric $f_{ij}~[i,j,k=1,2,3]$ is used to define the 3-covariant derivative $\nabla_k$ and $\nabla^k=f^{kl}\nabla_l$.
In one of the standard coordinate systems the background FRW metric reads $${\rm d}s^2={\rm d}t^2-a^2\left[ \frac{{\rm d}r^2}{1-kr^2}+r^2({\rm d}\theta^2+\sin^2\theta
{\rm d}\varphi^2) \right],
\label{ds22}$$ where in positive curvature (closed) universe ($k=+1$) $r\in\left<0,1\right>$, in flat $(k=0)$ and negative curvature $(k=-1)$ open universes $r\in\left<0,\infty\right>$, and $\theta\in (0,\pi)$, $\varphi\in(0,2\pi)$. We shall also employ hyperspherical coordinates $$ds^2={\rm d}t^2-a^2\left[ {\rm d}\chi^2+r^2({\rm d}\theta^2+\sin^2\theta
{\rm d}\varphi^2) \right],
\label{ds23}$$ with $r=\sin\chi$, $\chi$, $\sinh\chi$ for $k=1$, $0$, $-1$.
In a completely general gauge for general perturbations $h_{\mu\nu}$, the (momentum) constraint equation, $\delta G^0_k=\kappa \delta T^0_k$, turns out to be $$\begin{aligned}
\frac{1}{2}\nabla^2h_{0k}&+&kh_{0k}-\frac{1}{6}
a^2\nabla_k\nabla_j h_0^j+\frac{2}{3}a\nabla_k{\cal K} -\frac{1}{2}
a^2\dot{\cal T}_k
\nonumber
\\
&=&a^2\kappa\delta T^0_k,
\label{h0k}\end{aligned}$$ where the dot denotes $\partial/\partial t$, $${\cal K}=a\left[ \frac{3}{2} H h_{00}-\frac{1}{2}
(h^j_j)^{\,\bf\dot{}}+\nabla_jh^j_0\right]
\label{K}$$ is the perturbed mean external curvature of $t={\rm constant}$ slices, $H=\dot a/a$ is the Hubble constant, $${\cal T}_k=-\nabla_j\left(h^j_k-\frac{1}{3}\delta^j_kh^i_i\right).
\label{T}$$ Notice that equation (\[h0k\]) is independent of the choice of the cosmological constant $\Lambda $ because we perturbed “mixed" components of $G^0_k$. Other perturbed Einstein’s equations will not be needed here. Since, however, we are interested primarily in perfect fluid perturbations we shall also consider the perturbed fluid equations of motion, i.e. the perturbed Bianchi identities $$\begin{aligned}
(\delta\rho)^{\,\bf\dot{}}+3H(\delta\rho+\delta
p)&+&
\\\nonumber
(\rho+p)\nabla_k(h_0^k+V^k)&+&(\rho+p)\left(\frac{3}{2}Hh_{00}-
\frac{1}{a}{\cal K}\right)=0,
\label{delta rho}\end{aligned}$$ and $$\begin{aligned}
\frac{1}{a^3}\left[
a^3(\rho+p)(a^2f_{km}V^m-h_{0k})\right]^{\,\bf\dot{}}&+&\\\nonumber
\nabla_k\delta p+
\frac{1}{2}(\rho+p)\nabla_k h_{00}&=&0,
\label{Bianchi}\end{aligned}$$ where $V^k=\frac{{\rm d}x^k}{{\rm d}t~}\simeq
\delta U^k$ and $V_k=-a^2f_{kj}V^j$ is the fluid (small) velocity. The perturbed fluid energy-momentum tensor components entering the constraint equations (\[h0k\]) read $$\delta T^0_k=(\rho+p)(h_{0k}+V_k)=(\rho+p)(h_{0k}-a^2f_{km}V^m).
\label{deltaT}$$ There have been various choices of gauges used in the literature, in particular the synchronous gauge $(h_{00}=h_{0k}=0)$. In order to understand the effect of dragging of inertial frames, in particular its ‘instantaneous’ character, it is convenient to use gauges — we call them ‘Machian’ — in which the constraint equations, and still another (combination of) the perturbed field equations are explicitly the elliptic equations. In order to achieve this it is first useful to choose coordinates on $t={\rm constant}$ slices such that the [*spatial*]{} harmonic gauge conditions are satisfied, i.e. ${\cal T}_k=0$, where ${\cal T}_k$ is given in (\[T\]) (in numerical relativity $\dot{\cal T}_k=0$ is frequently called the ‘minimal distortion’ shift vector gauge condition). Next, it is convenient to choose the time slices so that, for example, the perturbation of their external curvature vanishes: ${\cal K}=0$, ${\cal K}$ given by (\[K\]) (so called ‘uniform Hubble expansion gauge’). Under these gauge conditions (which determine the coordinates in a substantially more restrictive way than e.g. the synchronous gauge) the constraint field equations (\[h0k\]) become the elliptic equations for just the components $h_{0k}$, no other $h_{\mu\nu}$ enter.
Until now we considered general perturbations in the chosen gauge. Hereafter, we assume the vectors $h_{0k}, V^k$ to be transverse, $$\nabla^kh_{0k}=0,\qquad \nabla_kV^k=0,
\label{transverse}$$ so that also $\nabla^k\delta T^0_k=0$. If (\[transverse\]) is not satisfied, we can apply $\nabla^k$ to equation (\[h0k\]), find the elliptic equation for the scalar $\nabla^kh_{0k}$, solve it and substitute back into (\[h0k\]) where the third term on the left hand side could be viewed as the source together with $\delta T^0_k$. Since, however, the longitudinal parts do not contribute to the dragging of inertial frames, we assume equations (\[transverse\]) to be satisfied.
The constraint field equations (\[h0k\]) with our choice of gauge ${\cal K}={\cal T}_k=0$ \[cf. equations (\[K\]) and (\[T\])\] thus become $$\nabla^2h_{0k}+2kh_{0k}=2a^2\kappa\delta T^0_k=2a^2\kappa
(\rho+p)(h_{0k}-a^2f_{km}V^m),
\label{h0k2}$$ where for the perfect fluid $\delta T^0_k$ is substituted from equation (\[deltaT\]). This is our basic equation to be solved at a given time $t={\rm constant}$, with either $\delta T^0_k$ or $V^k$ given. The Bianchi identities (fluid equations of motion) determine the time evolution of perturbations, the scalar equation (\[delta rho\]) for $\delta\rho$, whereas the vector equation (\[Bianchi\]) governs the evolution of the term $$j_k\equiv a^3(\rho+p)(a^2f_{km}V^m-h_{0k}) =-a^3\delta T^0_k.$$ In the following we shall often express the background time dependent term $a^2(\rho+p)$ by using equation (\[la\]).
Consider first the flat universe ($k=0$). In Cartesian coordinates $x^k$ used by Schmid [@Sc], the 3-metric $f_{kl}=\delta_{kl}$, and (\[h0k2\]) becomes $$\nabla^2h_{0k}=2a^2\kappa\delta T^0_k=2a^2\kappa
(\rho+p)(h_{0k}+V_k),
\label{h0k3}$$ where $\nabla^2$ is the flat-space Laplacian. Substituting from equation (\[la\]) with $k=0$ in the first term in the r.h.s. of equation (\[h0k3\]), we get $$\nabla^2h_{0k}=-4a^2\dot Hh_{0k}-2\kappa a^4(\rho+p)V_k.
\label{h0k4}$$
Now comparing our general form of the perturbed FRW metric with the perturbed metric (5) in Schmid’s work (and bewaring of the opposite signature), we see that $h_{0k}=-a\beta_k$(Schmid). Considering $(\rho+p)V_k$ (denoted by ${\vec J}_\epsilon$ in Schmid) as the source, the equation (\[h0k4\]) written for Schmid’s $\beta_k$ becomes $$-\nabla^2\beta_k-4a^2\dot H\beta_k=-2\kappa a^3(\rho+p)V_k.$$ This is Schmid’s basic equation (14), up to the factors $a^2$ and $a^3$ which in Schmid’s equation (14) are missing but this does not change significantly Schmid’s conclusions.
When $\delta T^0_k$ is given, the solution of equation (\[h0k3\]) is given as the Poisson integral over the source. If, however, the matter current is given, equation (\[h0k4\]) can be written as $$\nabla^2h_{0k}-\lambda^2(t)h_{0k}=-2\kappa a^4(\rho+p)V_k,
\label{h0k5}$$ with ($k=0$) $$\lambda^2=-4a^2\dot H.$$ Usually (e.g. in the standard Friedmann models) $\dot H<0$, so $\lambda$ is real. The three equations (\[h0k5\]) are, as emphasized by Schmid, of the Yukawa-type. The Green’s functions are given by $$G(x,x')= -\frac{1}{4\pi} \frac{e^{\mp\lambda|x-x'|}}{|x-x'|}
\label{G},$$ the well-behaved solution of equation (\[G\]) is thus $$h_{0k}=-\frac{1}{2\pi}\kappa a^4(\rho+p)\int
V_k(x')\frac{e^{-\lambda|x-x'|}}{|x-x'|}\,{\rm d}x'.$$ Clearly [*if*]{} the perturbation $V_k(x')$ is located at $|x-x'|{\stackrel{>}{_\sim}}\lambda^{-1}=1/2a\sqrt{-\dot H}$, i.e. beyond the ‘$\dot H ~$radius’ $R_{\dot H}=2(-\dot H)^{-\frac{1}{2}}$ in Schmid’s terminology, the vector $h_{0k}$ which determines the dragging of inertial frames is exponentially suppressed around the origin. Although we thus verified the interesting conclusion of Schmid, we do not resonate with his view that “ because of the exponential cut-off... there is no need to impose ‘appropriate boundary conditions of some kind’....". The Green’s function in (\[G\]) with the ‘$+$’ sign in the exponential is also the solution of equation (\[h0k5\]) with a $\delta$-function source but one discards it by demanding ‘reasonable’ boundary conditions at infinity.
From the Machian viewpoint the closed universes are of course preferable. There is, however, no vector Green’s function available for equation (\[h0k2\]) with either $\delta T^0_k$ or $V_k$ considered as a source. In order to understand how Schmid’s conclusions get modified in curved universes and to generalize our previous work [@LKB95] which analyzed perturbations corresponding to rigid rotating spheres in the FRW universes, we shall study all axisymmetric, odd-parity $l$-pole perturbations corresponding to differentially rotating ‘spheres’. We now derive the basic equations for such “toroidal” perturbations. Their solutions, in particular for $l\geq 2$ and closed universes, require special treatment. These solutions are analyzed in the following Paper II.
In spherical coordinates \[as in the FRW metrics (\[ds2\]), (\[ds22\])\], the only non-vanishing vector components are $h_{0\varphi}(t,r,\theta)$ and $V_\varphi
(t,r,\theta)$. \[For the general axisymmetric even-parity vector fields $V_\varphi=0$, whereas $V_r
(t,r,\theta)$ and $V_\theta
(t,r,\theta)$ are non-vanishing, the same being true for $h_{0r},
h_{0\theta}$\]. There is now just one non-trivial constraint equation in (\[h0k2\]) to be satisfied: $$\nabla^2h_{0\varphi}+2kh_{0\varphi}=2a^2\kappa\delta T^0_\varphi,
\label{h0k6}$$ in which $\nabla^2=f^{kl}\nabla_k\nabla_l$, with $f^{kl}$ being the inverse to $f_{kl}$ given by FRW metric (\[ds22\]) (recall – see (\[ds2\]) – that $f_{kl}$ is positive definite, without factor $a^2$). Calculating $\nabla^2h_{0\varphi}$ explicitly, we find equation (\[h0k6\]) to take the form $$\begin{aligned}
&&\left[ (1-kr^2) \frac{\partial^2}{\partial r^2}-kr\frac{\partial}{\partial r} \right] h_{0\varphi} +
\\\nonumber
&+&\frac{1}{r^2} \sin \theta \frac{\partial}{\partial\theta}\left( \frac{1}{\sin\theta}
\frac{\partial}{\partial\theta}\right) h_{0\varphi}+
4kh_{0\varphi}=2a^2\kappa\delta T^0_\varphi.
\label{constraint}\end{aligned}$$ Before solving this constraint equation it is interesting to notice what the perturbed equations of motion (Bianchi identities) say for axisymmetric odd-parity perturbations. Equation (\[delta rho\]) in our gauge choice (with ${\cal K}=0$) and transverse character of $h_{0k}, V_k$ is a simple evolution equation for $\delta\rho$. The vector equation (\[Bianchi\]) for indices $1,2 ~(x^1=r, x^2=\theta)$ turns into the well known relativistic equilibrium conditions for perfect fluids, $\nabla_k\delta p=-(\rho+p)\nabla_k (\frac{1}{2} h_{00})$ (see e.g. [@MTW]). In the following the crucial role is played by equation (\[Bianchi\]) for index $3 ~(x^3=\varphi)$. Since in axisymmetric case $\nabla_\varphi\delta p =0,
\nabla_\varphi h_{00}=0$, it becomes $$\left[ a^3(\rho+p)\left( a^2r^2\sin^2\theta\/
V^\varphi-h_{0\varphi}\right)\right]^{\,\bf\dot{}}=0,
\label{Bianchi0}$$ or $$\left[ a^3\delta T^0_\varphi\right]^{\,\bf\dot{}}=0.
\label{Bianchi2}$$ This is the conservation of angular momentum of each element of each axially symmetrical ring of fluid. The total angular momentum in a spherical layer $\left<\chi_1,\chi_2\right>$ is given by $$J(\chi_1,\chi_2)=-\int_{\chi_1}^{\chi_2}\!\!\!{\rm d}\chi\int_0^\pi
\!\!\!{\rm d}\theta\int_0^{2\pi}\!\!\!{\rm d}\varphi\sqrt{-\overline g}\,\delta T^0_\mu\eta^\mu,$$ where $\eta^\mu=(0,0,0,1)$ is the rotational Killing vector, the background metric determinant $\overline g=\overline
g^{(3)}=-a^6r^4\sin^2\theta, ~r=\sin\chi, \chi, \sinh\chi$ for respectively $k=+1,0,-1$ as in equation (\[ds23\]). Integrating over $\varphi$ we have $$\begin{aligned}
J(\chi_1,\chi_2)&=&-2\pi\int_{\chi_1}^{\chi_2}{\rm d}\chi\int_0^\pi
{\rm d}\theta ~a^3r^2\sin\theta\delta
T^0_\varphi
\nonumber\\
&=&2\pi \int_{\chi_1}^{\chi_2}{\rm d}\chi\int_0^\pi {\rm d}\theta~j(\theta,\chi,t),
\label{calJ} \end{aligned}$$ where $j(\theta,\chi,t)$ is the (coordinate) angular momentum density. Hence, the Bianchi identity (\[Bianchi2\]) can be written as $$\left[ j(\theta,\chi)\right]^{\,\bf\dot{}}=0.
\label{Bianchi3}$$ This is important for studying the time evolution of the $h_{0k}$ and $V_k$ perturbations.
Defining the fluid angular velocity $$\Omega=V^\varphi=\frac{{\rm d}\varphi}{{\rm d}t},$$ we get $$V_\varphi=-a^2f_{\varphi\varphi}V^\varphi=-a^2r^2\sin^2\theta~\Omega(t,r,\theta).$$ Writing similarly $$h_{0\varphi}=a^2r^2\sin^2\theta~\omega(t,r,\theta),$$ the only non-vanishing component of $\delta T^0_k$ becomes $$\delta T^0_\varphi=(\rho+p) a^2r^2\sin^2\theta(\omega-\Omega).
\label{230}$$ The angular momentum density conservation law (\[Bianchi2\]), resp. (\[Bianchi3\]), turns then into the simple evolution equation $$\left[ a^5(\rho+p)(\omega-\Omega)\right]^{\,\bf\dot{}}=0.
\label{Bianchi4}$$
Let us now return back to the constraint equation (\[constraint\]). The second term on its left hand side suggests the decomposition into the vector spherical harmonics. It should be emphasized that, in contrast to standard practice in the cosmological perturbation theory where perturbations are decomposed into harmonics in all three spatial dimensions (see e.g. [@KS]), we decompose in the usual coordinates $\theta,\varphi$ on spheres only, and assume axial symmetry (spherical functions $Y_{lm}$ having $m=0$). Thus, we write ($Y_{l0,\theta}\equiv\partial_\theta Y_{l0}$) $$\begin{aligned}
h_{0\varphi}&=&a^2r^2\sum\limits_{l=1}^{\infty}\omega_l(t,r)\sin\theta\/ Y_
{l0,\theta},
\\V_\varphi&=&-a^2r^2\sum\limits_{l=1}^{\infty}\Omega_l(t,r)\sin\theta\/ Y_{l0,\theta},\end{aligned}$$ and $$\begin{aligned}
\delta
T^0_\varphi&=&a^2(\rho+p)r^2\sum\limits_{l=1}^{\infty}
(\omega_l-\Omega_l)\sin\theta~ Y_{l0,\theta}
\nonumber\\
&=&
\sum\limits_{l=1}^{\infty}[\delta
T^0_\varphi(t,r)]_l\sin\theta\/ Y_{l0,\theta}.
\label{234}\end{aligned}$$ Substituting these expansions into equation (\[constraint\]) and using the orthogonality of functions $\sin\theta ~Y_{l0,\theta}$ for different $l$’s, we obtain the ‘radial’ equation for each $l$: $$\begin{aligned}
&&\left[ (1-kr^2) \frac{\partial^2}{\partial r^2}-kr\frac{\partial}{\partial r} \right] (r^2\omega_l)
-l(l+1)\omega_l+4kr^2\omega_l
\nonumber\\
&&=2a^2\kappa
(\rho+p)r^2(\omega_l-\Omega_l)=\lambda^2r^2(\omega_l-\Omega_l),
\label{constraint2}\end{aligned}$$ where we used equation (\[la\]). It is easy to convert the last equation into the form $$\begin{aligned}
-\sqrt{1-kr^2}\frac{1}{r^2}\frac{\partial}{\partial r}\left[
\sqrt{1-kr^2}\frac{\partial}{\partial r}(r^2\omega_l)
\right]+
\nonumber\\
\frac{l(l+1)}{r^2}\omega_l-4k\omega_l=\lambda^2(\Omega_l-\omega_l).
\label{oml}\end{aligned}$$ For $l=1$ (and the background pressure $p=0$) this equation coincides exactly with equation (4.32) in LKB. In the language of the present paper, in LKB we analyzed dipole $(l=1)$ axisymmetric odd-parity perturbations. With $l=1$, $Y_{10,\theta}=-\sqrt{3/4\pi}\sin \theta$, so that putting $\omega=-\sqrt{3/4\pi }~\omega_{l=1}$, $\Omega=-\sqrt{3/4\pi }~\Omega_{l=1}$, we recover $$\begin{aligned}
h_{0\varphi}&=&a^2r^2\sin^2\theta~\omega(t,r),\quad V_\varphi=-a^2r^2\sin^2\theta~
\Omega(t,r),\nonumber\\
\delta T^0_\varphi&=&a^2(\rho+p)r^2\sin^2\theta(\omega-\Omega),
\label{237}\end{aligned}$$ which corresponds to the [*rigidly*]{} rotating spheres in the FRW universes considered in Section 4.4 in LKB, and, for $\Omega(t,r)$ given, analyzed in detail in Section 4 in the following.
Consider first the case $k=0$. Equation (\[oml\]) can be written with the angular momentum density $(\delta T^0_\varphi)_l$ as a source, $$\frac{1}{r^4}\frac{\partial}{\partial r}\!\!\left(\!r^4 \frac{\partial\omega_l}{\partial
r}\!\right)-\frac{l(l+1)-2}{r^2}\omega_l=\lambda^2(\omega_l-\Omega_l)
=\frac{2\kappa}{r^2}(\delta
T^0_\varphi)_l.
\label{oml8}$$ If the fluid angular velocity is taken as a source, the equation reads $$\frac{1}{r^4}\frac{\partial}{\partial r}\left( r^4 \frac{\partial\omega_l}{\partial
r}
\right)-\left[\lambda^2+\frac{l(l+1)-2}{r^2}\right]
\omega_l=-\lambda^2\Omega_l,
\label{oml2}$$ where $\lambda^2=-4a^2\dot H=2\kappa a^2(\rho+p)$ by using equation (\[la\]) with $k=0$.
In the case of spatially curved $(k\ne 0)$ backgrounds it is advantageous to write $r^2=k(1-\mu^2)$, i.e. $\mu=\sqrt{1-kr^2}$ to obtain $$\begin{aligned}
\frac{1}{[k(1-\mu^2)]^{3/2}}\frac{\partial}{\partial\mu}\Big\{
[k(1-\mu^2)]^{5/2}\frac{\partial\omega_l}{\partial\mu}\Big\}
-\frac{l(l+1)-2}{k(1-\mu^2)}\omega_l&\!\!&\!\!\nonumber\\
=\frac{2\kappa}{k(1-\mu^2)}(\delta
T^0_\varphi)_l.\hphantom{AAAAA}&\!\!&\!\!\label{oml3}\end{aligned}$$ The substitution $$\omega_l=[k(1-\mu^2)]^{-3/4}\overline \omega_l
\label{241}$$ turns equation (\[oml3\]) into the Legendre equation for $\overline\omega_l$ with $(\delta T^0_\varphi)_l$ as the source: $$\begin{aligned}
&&\frac{\partial}{\partial\mu}\left[
k(1-\mu^2)\frac{\partial\overline\omega_l}{\partial\mu}\right]+\left[ k \frac{3}{2}
(\frac{3}{2}+1)-
\frac{(l+\frac{1}{2})^2}{k(1-\mu^2)}
\right] \overline\omega_l
\nonumber\\
&&\hphantom{AAAAAAAAAAAA}
=\frac{2\kappa}{[k(1-\mu^2)]^{1/4}}(\delta T^0_\varphi)_l.
\label{242}\end{aligned}$$ Finally, considering the fluid angular velocity as the source, we can write the last equation again as the Legendre equation with a more complicated degree: $$\begin{aligned}
&&\frac{\partial}{\partial\mu}\left[
k(1-\mu^2)\frac{\partial\overline\omega_l}{\partial\mu}\right]+\left[ k\nu(\nu+1)-
\frac{(l+\frac{1}{2})^2}{k(1-\mu^2)}
\right]\overline\omega_l
\nonumber\\
&&\hphantom{AAAA}
=-K_l\equiv-\lambda^2\Omega_l[k(1-\mu^2)]^{3/4},
\label{oml4}\end{aligned}$$ where $$\left(\nu+\frac{1}{2}\right)^2=4-2k\kappa a^2(\rho+p)=4-k\lambda^2=4ka^2\dot H.
\label{nu}$$ The degree $\nu$ of the Legendre equation does not depend on $l$. For $l=1$, equation (\[oml4\]) goes over into equation 4.35 in LKB [@foot2].
Solutions for $\omega$ with given angular momentum distribution
================================================================
We shall start by making more explicit the solutions obtained in LKB which are the $l=1$ odd-parity vector solutions of the general problem. In such modes each sphere rotates with no shear but it expands (or contracts) with the background and as it does so its angular velocity changes (see Section 5).
The equation to be solved is (\[oml8\]) with $l=1$, this is 4.33 LKB $$\frac{1}{r^4}\frac{\partial}{\partial r}\left( r^4 \frac{\partial\omega}{\partial r}
\right)=-\lambda^2(\Omega-\omega)=\frac{2\kappa}{r^2}\delta T^0_\varphi,
\label{om}$$ multiplying up by $r^4$ this takes the form $$\frac{\partial}{\partial r}\left( r^4
\frac{\partial\omega}{\partial r}\right)=-\frac{6}{a^3}\frac{{\rm d}J(<r)}{{\rm d}r},$$ so $$\frac{\partial\omega}{\partial r}=-\frac{6J}{a^3r^4},$$ the constant of integration is zero since $J(<r)$ is zero at $r=0$ where $\partial\omega/\partial r$ must vanish. Integrating again and insisting that $\omega\rightarrow 0$ at $\infty$ we find $$\begin{aligned}
\omega&=&a^{-3}\!\int_r^\infty\!\frac{6J}{r{'}^4}\,{\rm d}r'=2a^{-3}\left[
\frac{J(<r)}{r^3}+\!\int_r^\infty\!\!\frac{{\rm d}J}{{\rm d}r'}r{'}^{-3}\,{\rm d}r'
\right]\nonumber
\\&=&\frac{2}{r^3} \int_0^r\!\!\!\int_0^\pi 2\pi r{'}^2\sin\theta(-\delta
T^0_\varphi)\,{\rm d}\theta\,{\rm d}r'
\nonumber\\
&&
+ 2\int_r^\infty\!\!\!\int_0^\pi 2\pi r{'}^{-1}\sin\theta(-\delta
T^0_\varphi)\,{\rm d}\theta\,{\rm d}r',
\label{0omJ}\end{aligned}$$ where we have used (\[calJ\]) to define $J(<r)$ in terms of $\delta T^0_\varphi$.
The equation to be solved is (\[oml3\]) with $l=1$ which is 4.34 LKB [@foot3] $$\begin{aligned}
\frac{\partial}{\partial\mu}\Big\{
(1-\mu^2)^{5/2}\frac{\partial\omega}{\partial\mu}\Big\}&=&2\kappa(1-\mu^2)^\frac{1}{2}\delta
T^0_\varphi= \frac{6}{a^3}\frac{{\rm d}J}{{\rm d}\mu}
\nonumber\\
\noalign{\noindent so that}
(1-\mu^2)^\frac{1}{2}\frac{\partial\omega}{\partial\mu}&=&\frac{6J}{a^3(1-\mu^2)^2}.\end{aligned}$$ As before there is no integration constant for the same reason. We now write $\mu=\cos \chi$, then $\chi$ is the normal cosmic radial angle and $$\frac{\partial\omega}{\partial\chi}=-\frac{6J}{a^3\sin^4\chi}.
\label{36}$$ Now $$\int^\chi\frac{{\rm d}\chi}{\sin^4\chi}=-\frac{1}{3}
\left(\cot^3\chi+3\cot\chi\right)=-\frac{1}{3} W(\chi).
\label{37}$$ Hence $$\omega=2a^{-3}\left[ WJ(<\chi)+\int_\chi^\pi
W\frac{{\rm d}J}{{\rm d}\chi'}\,{\rm d}\chi'\right] +\omega_0,
\label{1omJ}$$ where $$J=\int_0^\chi\!\!\int_0^\pi 2\pi a^3[r(\chi')]^2\sin^2\theta(-\delta
T^0_\varphi)\,{\rm d}\theta\,{\rm d}\chi'.
\label{J}$$
Just as in the last case $W$ diverges at $\chi=0$ like $\chi^{-3}$, however, the angular momentum of spheres near the origin is sufficiently small to make the $WJ$ tend to a constant as $\chi$ tends to zero. It is shown in LKB that the condition of convergence of the second integral at $\chi=\pi$ is that the total angular momentum of the universe is zero. If that condition is fulfilled and $\Omega-\omega$ is regular near $\chi=\pi$ then the integral converges. If the total angular momentum is not zero then the integral for $\omega$ diverges at $\chi=\pi$. Thus for $\omega$ to be finite at $\chi=\pi$ the total angular momentum must be zero in the closed universe. There is no way of fixing $\omega_0$ because there is no standard of zero rotation, as there is for the infinite universes. Indeed, according to Mach a description of the world in rotating axes is just as good in principle as a description in non-rotating ones. Note that the source $\Omega-\omega$ does not change when the axes are rotating since $\Omega$ and $\omega$ acquire the same constant $\omega_0$. An absolute rotation can arise only from spatial boundary conditions which do not occur for closed universes.
The equation to be solved is (\[oml3\]) with $l=1$. Multiplying through by $(\mu^2-1)^{3/2}$ we obtain $$\frac{\partial}{\partial\mu}\Big\{
(\mu^2-1)^{5/2}\frac{\partial\omega}{\partial\mu}\Big\}=-2\kappa \sqrt{\mu^2-1}\delta
T^0_\varphi=-\frac{6}{a^3}\frac{{\rm d}J}{{\rm d}\mu},$$ so on integration and division $$(\mu^2-1)^{1/2}\frac{\partial\omega}{\partial\mu}=-\frac{6}{a^3}(\mu^2-1)J.$$ Writing $\mu=\cosh\chi$ to introduce the natural radial variable of hyperbolic space, this becomes $$\frac{\partial\omega}{\partial\chi}=-\frac{6}{a^3}\frac{J}{\sinh^4\chi}.$$ Integrating again and insisting that $\omega\rightarrow 0$ at infinity we use the integral $$\int^\chi\frac{{\rm d}\chi'}{\sinh^4\chi'}=-\frac{1}{3}\left(
\coth^3\chi-3\coth\chi+2\right)\equiv-\frac{1}{3}\overline W(\chi),
\label{313}$$ and on integrating by parts we obtain $$\omega=2a^{-3}\left[
\overline W J(<\chi)+\int_{\chi}^{\infty}\overline W(\chi')\frac{dJ}{{\rm d}\chi'}{\rm d} \chi'\right],
\label{-1omJ}$$ where $J$ is the same as in (\[J\]) with $r=\sinh\chi$. We have chosen the above definition of $\overline W$ so that $\overline W\rightarrow 0$ at infinity; so no constant of integration is needed to incorporate the boundary condition that $\omega\rightarrow 0$.
Solutions for $\omega$ with given $\Omega$
===========================================
The method of solution was outlined in LKB but here we work through all the details starting with the simplest case.
The relevant equation to be solved is (\[oml2\]) with $l=1$, equation 4.33 in LKB, rewritten as $$\frac{1}{r^4}\frac{\partial}{\partial r}\left( r^4 \frac{\partial\omega}{\partial r}
\right)-\lambda^2\omega=-\lambda^2\Omega.
\label{om2}$$ Here $\lambda^2=2a^2\kappa(\rho+p)>0$. $\lambda^{-1}a$ has the units of a length and we shall call it, following Schmid [@Sc], the distance to the horizon. In dimensionless comoving coordinates this corresponds to $r=\lambda^{-1}$. We write $z=\lambda r$ and $\partial\omega/\partial z=\omega'$. Then equation (\[om2\]) reduces to $$\omega''+\frac{4}{z}\omega-\omega=-\Omega.
\label{42}$$ The corresponding homogeneous equation is Bessel’s equation for $z^{-3/2}J_{3/2}(iz)$, which has real solutions $\omega=\overline
{\cal I}$ and $\omega=\overline
{\cal K}$, where $\overline{\cal I}=z^{-3/2}I_{3/2}(z)$ and $\overline{\cal K}=z^{-3/2}K_{3/2}(z)$. For small $z$, $\overline{\cal I}\rightarrow\frac{1}{3}\sqrt{2/\pi}(1+z^2/10)$; $\overline{\cal K}\rightarrow\sqrt{\pi/2}~
z^{-3}$. For large $z$, $\overline{\cal I}\rightarrow(1/\sqrt{2\pi})z^{-2}e^z$; $\overline{\cal K}\rightarrow\sqrt{\pi/2}~z^{-2}e^{-z}$.
We use the method of variation of parameters to solve the inhomogeneous equation with boundary conditions that $\omega$ tends to zero at infinity and to a constant at the origin. We thus obtain $$\begin{aligned}
\omega(z)=&&\overline{\cal K}(z)\int_0^z (z')^4\overline{\cal I}(z')\Omega(z')\,{\rm d}z'
\nonumber\\
&+&\overline{\cal I}(z)
\int_z^\infty
(z')^4\overline{\cal K}(z')\Omega(z')\,{\rm d}z'.
\label{0om}\end{aligned}$$ For the solutions near the origin with sources that are not so close, we may neglect the first term and then for small $z$, $$\omega(z)=\frac{1}{3}\sqrt{\frac{2}{\pi}}\left(1+\frac{z^2}{10}\right)\int_z^
\infty(z')^4\overline{\cal K}(z')\Omega(z')\,{\rm d}z'.
\label{44}$$ When the source $\Omega$ is beyond the horizon $z=1$, i.e. $z'\gg1$, $$\omega(z)=\frac{1}{3}\left(1+\frac{z^2}{10}\right)\int_z^
\infty(z')^2e^{-z'}\Omega(z')\,{\rm d}z';$$ so for a source localized in $r_0(1\pm\Delta)$ with $\Delta\ll1/\lambda$, $$\omega(z)=\frac{1}{3}\left(1+\frac{\lambda^2r^2}{10}\right)(\lambda r_0)^3e^{-\lambda r_0}\overline\Omega 2\Delta,
\label{46}$$ which clearly shows the exponential decline of influence remarked on by Schmid [@Sc]. When $\Omega$ is concentrated near $z_0$, in $z_0\pm\lambda\Delta$, then with $z_0\gg 1$ and $\Omega=\overline\Omega$ we get $$\omega(z_0)=\frac{1}{2}\int_0^\infty\!\left(\frac{z'}{z_0}\right)^2e^{-|z'-z_0|}\Omega(z')\,{\rm d}z'\simeq
\lambda\Delta\overline\Omega.
\label{47}$$ Thus [*at*]{} the source the inertial frame rotates at $\lambda\Delta\overline\Omega$ and $\overline\Omega-\omega=(1-\lambda\Delta)\overline\Omega$.
We now turn to the solutions for a closed universe.
The relevant equation is Legendre’s equation for $\overline\omega=(1-\mu^2)^{3/4}\omega
$ with an inhomogeneous term written below. This is LBK equation 4.35 and the same as the equation (\[oml4\]) of Section 2 of this paper specialized for $l=1$: $$\begin{aligned}
\frac{\partial}{\partial\mu}\Big\{
(1-\mu^2)\frac{\partial\overline\omega}{\partial\mu}\Big\}&+&\Big\{\nu(\nu+1)-
\frac{(\frac{3}{2})^2}{(1-\mu^2)}
\Big\}\overline\omega=-K,
\nonumber\\
&&K\equiv\lambda^2\Omega(1-\mu^2)^{3/4},
\label{48}\end{aligned}$$ where $(\nu+\frac{1}{2})^2=4-\lambda^2$ as in (\[nu\]). Since $k=+1$ the space is hyperspherical and the convention is to write $\mu=\cos\chi$ so that $\chi$ becomes the radial variable. The solutions of the homogeneous equation are the Legendre functions $P^{3/2}_\nu(\mu)$ and $Q^{3/2}_\nu(\mu)$ and a recurrence relation that generates $P^{\mu+1}_\nu$ from $P^\mu_\nu$ and $P^\mu_{\nu-1}$. (Here the order $\mu$ of the Legendre function has nothing to do with the variable $\mu=\sqrt{1-kr^2}$.) Thus $$\begin{aligned}
P^{1/2}_\nu(\cos\chi)&=&\hphantom{-}\Big(\frac{\pi}{2}\Big)^{-\frac{1}{2}}(\sin\chi)^{-\frac{1}{2}}
\cos\Big[\Big(\nu+\frac{1}{2}\Big)\chi\Big],\nonumber\\
Q^{1/2}_\nu(\cos\chi)&=&-\Big(\frac{\pi}{2}\Big)^\frac{1}{2}(\sin\chi)^{-\frac{1}{2}}
\sin\Big[\Big(\nu+\frac{1}{2}\Big)\chi\Big].
\label{49}\end{aligned}$$ To keep $P^{3/2}_\nu(\cos\chi)$ and $Q^{3/2}_\nu(\cos\chi)$ real, we use $(1-\mu^2)^{-\frac{1}{2}}=(\sin\chi)
^{-1}$ in place of $(\mu^2-1)^{-\frac{1}{2}}$ in the recurrence relation 8.5.1 of Abramowitz and Stegun [@AS] (this merely multiplies the results by $-i$). $$P^{3/2}_\nu(\cos\chi)=\frac{1}{\sin\chi}\!\left[\Big(\nu-\frac{1}{2}\Big) P^\frac{1}{2}_\nu\cos\chi
+\Big(\nu+\frac{1}{2}\Big)P^\frac{1}{2}_{\nu-1}\right],
\label{410}$$ the same relation holds for the $Q^{3/2}_\nu$. It turns out to be convenient to write $n=\nu+\frac{1}{2}$. We note that (\[nu\]) and (\[48\]) involve this quantity and that $n$ can be real but is often imaginary. Thus $$\begin{aligned}
&&P^{3/2}_{n-\frac{1}{2}}(\cos\chi)=\\\nonumber
&&-\left(\frac{\pi}{2}\right)^{-\frac{1}{2}}\frac{1}{\sin^{3/2}\chi}
\left[
\cos\chi \cos(n\chi)-n\sin\chi\sin(n\chi)\right],\end{aligned}$$ similarly writing $n$ when it is real but $n=iN$ when it is imaginary: $$\begin{aligned}
&&Q^{3/2}_{n-\frac{1}{2}}(\cos\chi)=
\\
&&\hphantom{i}\Big(\frac{\pi}{2}\Big)^{\frac{1}{2}}\frac{1}
{\sin^{3/2}\chi}
\left[
\cos\chi \sin(n\chi)-n\sin\chi\cos(n\chi)\right],
\nonumber\\
&&Q^{3/2}_{iN-\frac{1}{2}}(\cos\chi)=
\nonumber\\\nonumber
&&i\Big(\frac{\pi}{2}\Big)^{\frac{1}{2}}\frac{1}{\sin^{3/2}\chi}
\left[
\cos\chi \sinh(N\chi)-N\sin\chi\cosh(N\chi)\right].
\label{412}\end{aligned}$$ We shall be concerned to have functions which, after multiplication by another $(\sin\chi)^{-3/2}$, are nevertheless still finite at the origin $\chi=0$. A little expansion around $\chi=0$ shows that the $P$ function diverges but $Q$ function satisfies this stringent test. Our next job is to find a solution that satisfies this stringent convergence not at $\chi=0$ but at the ‘other’ $r=0$ at $\chi=\pi$. Since that is an alternative origin it is clear that $Q^{3/2}_{n-\frac{1}{2}}[\cos(\pi-\chi)]$ passes that test. A little work shows that it is indeed the linear combination $(2/\pi)\sin(n\pi)P^{3/2}_{n-\frac{1}{2}}(\chi)-\cos(n\pi)
Q^{3/2}_{n-\frac{1}{2}}(\chi)$. Finally we notice that $n=0$, which is needed in some of our solutions, gives $Q_{-1/2}^{3/2}\equiv 0$. This is not a solution at all! However $\lim\limits_{n\rightarrow 0}[(1/n)Q_{n-1/2}^{3/2}]$ gives the finite limit $$\Big(\frac{\pi}{2}\Big)^\frac{1}{2}\frac{1}{\sin^{3/2}\chi}\left[ \chi\cos\chi
-\sin\chi \right].$$ We shall therefore use the functions $$\begin{aligned}
q_n&=&\Big(\frac{\pi}{2}\Big)^{\frac{1}{2}}\frac{1}{\sin^{3/2}\chi}S_n(\chi)\;,
\\\nonumber
p_n&=&\Big(\frac{\pi}{2}\Big)^{\frac{1}{2}}\frac{1}{\sin^{3/2}\chi}S_n(\pi-\chi)
\label{414}\end{aligned}$$ as our independent solutions of the Legendre equation. These functions have the added advantage that they remain real when $n=iN$: $$\begin{aligned}
S_n(\chi)&=&-\cos\chi
\frac{\sin(n\chi)}{n}+\sin\chi\cos(n\chi),\\\nonumber
S_{iN}(\chi)&=&-\cos\chi
\frac{\sinh(N\chi)}{N}+\sin\chi\cosh(N\chi).\end{aligned}$$ The Wronskian may be shown to be $$p_n\frac{{\rm d}q_n}{{\rm d}\mu}-q_n\frac{{\rm d}p_n}{{\rm d}\mu}=\frac{\pi}{2}\frac{\sin(n\pi)}{n}
\frac{n^2-1}{1-\mu^2}=\frac{{\cal W}}{1-\mu^2}.
\label{416}$$ Having formed solutions $p$ and $q$ each of which satisfy [*one*]{} of the boundary conditions we look for solutions of the inhomogeneous equation of the form $$\overline\omega=A(\mu)p+B(\mu)q.$$ We choose $A'p+B'q=0$, and then the equation demands that $$(1-\mu^2)[A'p'+B'q']=-K,$$ where a dash denotes $\partial/\partial\mu$. Solving for $A'$ and $B'$ we have, using the Wronskian ${\cal W}/(1-\mu^2)$ defined earlier, $A'=Kq/{\cal W}$. Now $p$ does not satisfy the boundary conditions at $\chi=0$, so $A$ must be zero there; hence $$A=-\int_\mu^1\frac{Kq}{{\cal W}}\,{\rm d}\mu=-\int_0^\chi\frac{Kq}{{\cal W}}\sin\chi~
\,{\rm d}\chi.$$ Similarly $B'=-Kp/{\cal W}$ and to satisfy the boundary conditions at $\mu=-1,
\chi=\pi$, $$B=-\int_{-1}^\mu\frac{Kp}{{\cal W}}\,{\rm d}\mu=-\int_\chi^\pi\frac{Kp}{{\cal W}}
\sin\chi\,{\rm d}\chi.$$ Thus the solution by variation of the parameters is $$\overline\omega=-\left[ p(\chi)\int_0^\chi\!\!\frac{Kq}{{\cal W}}\sin\chi'\,{\rm d}\chi'+q(\chi)
\int_\chi^\pi\!\!\frac{Kp}{{\cal W}}
\sin\chi'\,{\rm d}\chi'
\right],$$ which gives our solution for $\omega(\chi)=(\sin\chi)^{-3/2}\overline\omega$: $$\begin{aligned}
\omega(\chi)=&-&\frac{\pi/2}{{\cal W}\sin^3\chi}\left[
S_n(\pi-\chi)\!\int_0^\chi\!\!\!\lambda^2\Omega S_n(\chi')\sin\chi'\,{\rm d}\chi'
\right.
\nonumber\\
&+&
\left.
S_n(\chi)\int_\chi^\pi\!\!\lambda^2\Omega S_n(\pi-\chi')\sin\chi'\,{\rm d}\chi'
\right].
\label{1om}\end{aligned}$$ For $\chi$ small, $$\begin{aligned}
S_n&\rightarrow&
\frac{(1-n^2)}{3}\chi^3\left[1-\frac{(1+n^2)\chi^2}{10}\right],~~{\rm i.e.}
\nonumber\\
\frac{1}{\sin^3\chi}S_n&\rightarrow& \frac{(1-n^2)}{3}\left[
1+\frac{(4-n^2)}{10}\chi^2\right],\end{aligned}$$ and for $n=iN$, $$\frac{1}{\sin^3\chi}S_{iN}\rightarrow
\frac{(1+N^2)}{3}\left[1+\frac{(4+N^2)}{10}\chi^2\right].$$ We note that with $k=+1~,~4+N^2=\lambda^2$, and $${\cal W}=\frac{\pi}{2}(n^2-1)\frac{\sin(n\pi)}{n}=-\frac{\pi}{2}(1+N^2)\frac{\sinh(N\pi)}{N}.$$ For $N$ large and $\chi$ small $$\frac{S_{iN}}{{\cal W}\sin^3\chi}\rightarrow
-\frac{4}{3\pi}Ne^{-N\pi}\left( 1+\frac{\lambda^2\chi^2}{10}\right).$$ For $N$ large and $\chi$ not small nor near $\pi$, $$S_{iN}(\chi)= \frac{1}{2} \sin\chi ~e^{N\chi},\qquad S_{iN}(\pi-\chi)= \frac{1}{2}
\sin\chi ~e^{N(\pi-\chi)}.$$ Hence our solution near the origin is $$\omega(\chi)=\frac{1}{3}\left( 1+\frac{\lambda^2\chi^2}{10}\right) N \int_\chi^\pi
\lambda^2\Omega(\chi')\sin^2\chi' e^{-N\chi'}\,{\rm d}\chi',
\label{428}$$ and near the perturbation $$\omega(\chi_0)=\frac{1}{2}\int_0^\pi
\frac{\lambda^2 N}{N^2+1}\left(\frac{\sin\chi'}{\sin\chi_0}\right)^2
e^{-N|\chi'-\chi_0|}\Omega(\chi')\,{\rm d}\chi',
\label{429}$$ where at the last line we consider a perturbation with a mean $\Omega$ of $\overline\Omega$ in $r_0\pm\Delta$ with $N\Delta\ll1$.
The equation to be solved is (\[oml4\]) with $k=-1$ and $l=1$. Now we write $\mu=\cosh\chi~,~(\nu+\frac{1}{2})^2=\lambda^2+4$. Space is now hyperbolic and $\mu$ runs from 1 to $\infty$. The relevant solutions of the homogeneous equation are $$\begin{aligned}
p&=&-\left( P^{3/2}_\nu +\frac
{2}{\pi}iQ^{3/2}_\nu\right)=\frac{1}{2}\Big(\frac{\pi}{2}\Big)^{-\frac{1}{2}}
\frac{1}{\sinh^{3/2}\chi}S_e(\chi),\nonumber\\
q&=& iQ^{3/2}_\nu=\frac{1}{2}\left(\frac{\pi}{2}\right)^{\frac{1}{2}}
\frac{1}{\sinh^{3/2}\chi}E(\chi),\end{aligned}$$ where $n=(\nu+\frac{1}{2})$, $$\begin{aligned}
E(\chi)&=&-(n-1)e^{-(n+1)\chi}+(n+1)e^{-(n-1)\chi},
\nonumber\\
S_e(\chi)&=&\frac{1}{2}\left[ E(\chi)-E(-\chi) \right].
\label{431}\end{aligned}$$ The Wronskian $$p\frac{{\rm d}q}{{\rm d}\mu}-q\frac{{\rm d}p}{{\rm d}\mu}=-\frac{(n^2-1)n}{\mu^2-1}.$$ The solution by variation of parameters is $$\overline\omega=-\frac{1}{n(n^2-1)}\left[
p\int_\mu^\infty qK\,{\rm d}\mu+q\int^\mu_1pK\,{\rm d}\mu\right],$$ hence, changing the integrations from $\mu$ to $\chi$ and $\overline \omega$ to $\omega$, we have $$\begin{aligned}
\omega=&&\frac{(\sinh\chi)^{-3}}{4(n^2-1)n}\left[ E(\chi)\int_0^
\chi\lambda^2\Omega(\chi') S_e(\chi')\sinh\chi'\,{\rm d}\chi'
\right.
\nonumber\\
&+& \left. S_e(\chi)\int_\chi^\infty\lambda^2\Omega(\chi') E(\chi')\sinh\chi'\,{\rm d}\chi'
\right].
\label{434}\end{aligned}$$ For small $\chi$ $$\begin{aligned}
E(\chi)&=&2-(n^2-1)\chi^2\times
\\\nonumber
&&\left[ 1-
\frac{2n}{3}\chi+\frac{3n^2+1}{12}\chi^2-\frac{n(n^2+1)}{15}
\chi^3+...\right],\end{aligned}$$ so $$\begin{aligned}
S_e(\chi)&=&\frac{2n}{3}(n^2-1)\chi^3\left[ 1-
\frac{(n^2+1)}{15}\chi^2\right] .\end{aligned}$$ At large $\chi$ $$\begin{aligned}
E(\chi)&=&(n+1)e^{-(n-1)\chi}=2(n+1)e^{-n\chi}\sinh\chi,\nonumber\\
S_e(\chi)&=& \frac{1}{2} (n-1)e^{(n+1)\chi}=(n-1)e^{n\chi}\sinh\chi.\end{aligned}$$ Near the origin $$\begin{aligned}
\omega&=&\frac{1}{3}\left[1-\frac{(4-n^2)\chi^2}{10}\right]\times
\\\nonumber
&&\int_\chi^\infty(n^2-4)(n+1)\sinh^2\chi'e^{-n\chi'}\Omega (\chi')\,{\rm d}\chi'.
\label{437}\end{aligned}$$ At the perturbation $$\omega(\chi_0)=\frac{1}{2}\frac{n^2-4}{n}\int_0^\infty
\left(\frac{\sinh \chi'}{\sinh \chi_0}\right)^2
e^{-n|\chi'-\chi_0|}\Omega(\chi')\,{\rm d}\chi'.
\label{438}$$
The time evolution of the dragging
==================================
The evolution of $\omega$ and $\Omega$ as functions of cosmic time is governed by the equations of motion (contracted Bianchi identities) (\[Bianchi\]). For axisymmetric, odd-parity perturbations these become the angular momentum density conservation law, as discussed in equations (\[Bianchi0\])–(\[Bianchi3\]) in Section 2. In terms of $\omega(t,r,\theta)$ and $\Omega(t,r,\theta)$ the conservation law simply becomes (\[Bianchi4\]), i.e. $$\left[ a^5(\rho+p)(\omega-\Omega) \right]^{\,\bf\dot{}}=0$$ or, in terms of the angular momentum density, we get $$\Omega-\omega=\frac{1}{a^5(\rho+p)}\cdot\frac{j(\chi,\theta)}{r^4\sin^3\theta}.
\label{Om-om}$$ In this formula the first factor singles out the time dependence of $\Omega-\omega$. Notice that we have already obtained $\omega(t,r,\theta)$ as a function of the angular momentum within $\chi$, $J(<\chi)$, in all three cases $k=+1,0,-1$ \[see equations (\[0omJ\]), (\[1omJ\]), (\[-1omJ\])\]. We found $\omega$ to depend on the time as $1/a^3(t)$. Equation (\[Om-om\]) then can be regarded as a solution $\Omega(t,r,\theta)$ implied by the equations of motion.
On the other hand, for $\Omega-\omega$ given at some time $t=t_0$ as a function of $\chi,\theta$, equation (\[Om-om\]) determines the density $j(\chi,\theta)$ which in turn gives $J(<\chi)$ and $\omega(t,\chi,\theta)$ is then obtained from equations (\[0omJ\]), (\[1omJ\]), (\[-1omJ\]). Angular velocity of matter, $\Omega(t,\chi,\theta)$, is then given again by equation (\[Om-om\]).
If we are interested in proper azimuthal velocities, we can write $$V=ar\sin\theta~\Omega,\qquad v=ar\sin\theta~\omega,$$ and rewrite (\[Om-om\]) as $$V-v=\frac{1}{a^4(\rho+p)}\cdot\frac{j(\chi,\theta)}{r^3\sin^2\theta}.
\label{V-v}$$ Since $|\Omega r|,|\omega r|\ll1$, we have also $|V|,|v|\ll1$. In the case of the dust universes ($p=0$) the density obeys the conservation law $\rho a^3={\rm
constant}\equiv C$. Equation (\[V-v\]) then implies $$V-v=\frac{j(\chi,\theta)}{Cr^3\sin^2\theta}\cdot\frac{1}{a}.$$ This is not valid near $t\sim 0$ when $a\rightarrow 0$ due to our approximation. For $a\rightarrow \infty$, $V-v\rightarrow 0$ — the dragging becomes perfect.
This work started during our meeting at the Institute of Theoretical Physics of the Charles University in Prague and continued during our stay at the Albert Einstein Institute in Golm. We are grateful to these Institutes for their support.
A partial support from the grant GAČR 202/02/0735 of the Czech Republic is also acknowledged.
[99]{} C. Schmid, in [*Proceedings of the International Workshop on Particle Physics and Early Universe COSMO-01*]{}, Finland 2001, gr-qc/0201095. D. Lynden-Bell, J. Katz, and J. Bičák, Mon. Not. R. Astron. Soc. [**272**]{}, 150 (1995); Errata: [**277**]{}, 1600 (1995). Here the rotations of different spherical shells may be about different axis. H. Thirring, Phys. Z. [**19**]{}, 33 (1918), Errata: [**22**]{}, 29 (1921). J. Lense and H. Thirring, Phys. Z. [**19**]{}, 156 (1918). D. R. Brill and J. M. Cohen, Phys. Rev. [**143**]{}, 1011 (1966). L. Lindblom and D. R. Brill, Phys. Rev. [**D10**]{}, 3151 (1974). J. Katz, D. Lynden-Bell, and J. Bičák, Class. Quantum Grav. [**15**]{}, 3177 (1998). D. Lynden-Bell, J. Bičák, and J. Katz, Ann. Phys. (N.Y.) [**271**]{}, 1 (1999). C. Klein, Class. Quantum Grav. [**10**]{}, 1619 (1993); [**11**]{}, 1539 (1994). T. Doležel, J. Bičák, and N. Deruelle, Class. Quantum Grav. [**17**]{}, 2719 (2000). J. Bičák, D. Lynden-Bell, and J. Katz, Phys. Rev. D, submitted — the following paper (Paper II). C. Misner, K. S. Thorne and J. A. Wheeler, [*Gravitation*]{} (Freeman and Co., San Fransisco 1973). H. Kodama and M. Sasaki, Prog. Theor. Phys. Supp. [**78**]{}, 1 (1984). There $(3/2)$ should read $(3/2)^2$ and $(1-\mu^2)$ should be replaced by $k(1-\mu^2)$ to cover both $k=\pm 1$ consistently. In the latter there are typographical errors in that the final $(1-\mu)^{3/2}$ should be $(1-\mu^2)^{3/2}$ and the minus after the final equals sign should be a + because ${\rm d}J/{\rm d}\mu$ is negative since $\mu=+1$ at the origin. Also $a^3$ is missing. M. Abramowitz and I. A. Stegun, [*Handbook of Mathematical Functions*]{} (Dover Publ., New York 1972).
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abstract: 'Using Composition–Diamond Lemma we construct presentations of groups $G = \langle x_1,\ldots,x_n \, | \, r_1,\ldots, r_m \rangle$ with the following property; for a fixed $1 \le i \le n$, and for all $1 \le j \le m$, Fox derivatives ${\partial r_j / \partial x_i}$ have common divisor. It follows that in some cases the group ring $\mathbb{Z}[G]$ has zero divisors.'
title: '**On Common Divisors of Fox Derivatives with towards to Zero Divisors of Group Rings**'
---
Viktor Lopatkin[^1]
Introduction {#introduction .unnumbered}
============
Let $\mathsf{F}$ be a field, and $G$ a group with torsion, say $g^n = 1$ with $1 < n < \infty$. Consider the group ring $\mathsf{F}[G]$. We then have $$(1-g)(1+g-g^2 + \cdots + (-1)^{n-1}g) = 0$$ in $\mathsf{F}[G]$.
Assume now that $G$ is a torsion-free group. *Kaplansky’s zero divisor conjecture* states the the group ring $\mathsf{F}[G]$ does not contain nontrivial zero divisors, that is, it is a domain.
In this paper we aim to construct groups with nontrivial zero divisors using methods of homological algebra. The main result is Theorem \[themainresult\].
Preliminaries
=============
Let $G$ be a group is generated by $x_1,\ldots, x_n$ and is defined by relations $r_1,\ldots, r_m$, *i.e.,* the $G$ is presented as follows (= a group presentation) $$G = \langle x_1,\ldots, x_n \, | \, r_1,\ldots, r_m \rangle.$$
Thus, $G \cong F/N$ where $F$ is the free group with basis $X:=\{x_1,\ldots, x_n\}$ and $N$ is the normal closure in $F$ of the set $\{r_1,\ldots, r_m\}$ of words in $X\cup X^{-1}$.
Further, as is customary, $2$-complexes will be specified by means of groups presentations. The *cellular model* of a presentation of $G$ ()is the $2$-complex $\mathcal{K}_G$ that has a single $0$-cell, one $1$-cell for each generator $x_i$ and one $2$-cell fore each relator $r_j$. An orientation of the cells in the one-skeleton $\mathcal{K}_G^{(1)}$ determines an isomorphism $\pi_1\mathcal{K}_G^{(1)} \cong F$. The $2$-cell corresponding to a relator $r_j$ is attached along a based loop in the one-skeleton that spells the word $r_j$. The inclusion $\mathcal{K}_G^{(1)} \subseteq \mathcal{K}_G$ induces a surjection $F \to \pi_1 \mathcal{K}_G$ with kernel $N$. In particular, $\pi_1\mathcal{K}_G$ is canonically isomorphic to $G$, and so $\pi_2\mathcal{K}_G$ is a left $\mathbb{Z}[G]$-module under the homotopy action of $\pi_1 \mathcal{K}_G$.
Consider the $\mathbb{Z}[G]$-modules $\bigoplus_{i=1}^m\mathbb{Z}[G]$, $\bigoplus_{i=1}^n\mathbb{Z}[G]$. Define $$\mathrm{d}_0: \bigoplus_{i=1}^n\mathbb{Z}[G] \to \mathbb{Z}[G]$$ by setting $$\mathrm{d}_0 :(\alpha_1, \ldots,\alpha_n)^T\mapsto \sum_{i=1}^n\alpha_i (x_i-1).$$
Next, define $$\mathrm{d}_1: \bigoplus_{i=1}^m\mathbb{Z}[G] \to \bigoplus_{i=1}^n\mathbb{Z}[G]$$ by setting $$\mathrm{d}_1: (\beta_r)_{r\in \{r_1,\ldots, r_m\}} \mapsto \sum_{r\in \{r_1,\ldots, r_m\}} (J_{rx}\beta_r)_{x \in X}, \qquad \beta_r \in \mathbb{Z}[G],$$ where $J_{rx}$ is the image in $\mathbb{Z}[G]$ of the (left) partial derivative $\partial r / \partial x$ (= Fox derivative, see [@MKS Sec. 5.15]).
Further, the second homotopy $\mathbb{Z}[G]$-module $\pi_2(\mathcal{K}_{G})$ can be viewed as the kernel of the map $\mathrm{d}_1$. This was first observed by K. Reidemeister [@R]. More precisely, we have the following result.
\[exact\] There is an exact sequence of $\mathbb{Z}[G]$-modules: $$0 \to \pi_2(\mathcal{K}_G) \xrightarrow{\mathfrak{p}} \bigoplus_{i=1}^m\mathbb{Z}[G] \xrightarrow{\mathrm{d}_1} \bigoplus_{i=1}^n\mathbb{Z}[G] \xrightarrow{\mathrm{d}_0} \mathbb{Z}[G] \xrightarrow{\varepsilon} \mathbb{Z} \to 0,$$ where $\varepsilon$ is the augmentation map.
Take a $\beta \in \pi_2(\mathcal{K}_G)$, we then get $\mathfrak{p}(\beta) = (\beta_1,\ldots,\beta_m)^T$, and hence $$\sum_{j=1}^m\beta_j \dfrac{\partial r_j}{\partial x_i} = 0,$$ for every $1 \le i \le n$.
Assume now that for a fixed $i$, $\dfrac{\partial r_j}{\partial x_i} = D_jf$, $1 \le j \le m$, then by the previous equality we then have $$\label{foxcommon}
\sum_{j=1}^m(\beta_j D_j) f = 0,$$ it follows that if $\pi_2(\mathcal{K}_G) \ne 0$ then $\mathbb{Z}[G]$ has zero divisors.
The aim of this paper is thus to construct such groups. We will use Composition–Diamond Lemma technique. More precisely. Consider a free algebra $\mathsf{F}\langle X\rangle$ over a field $\mathsf{F}$. Given polynomials $\varphi, f \in \mathsf{F}\langle X\rangle$. When the polynomial $\varphi$ is divided by $f$? The Composition–Diamond Lemma can help us to answer this question.
[**Acknowledgements:**]{} the author would like to express his deepest gratitude to <span style="font-variant:small-caps;">Dr. Roman Mikhailov</span> who told about the problem in the course “Groups and Homotopy Theory” (YouTube channel “Lectorium”). Special thanks are due to <span style="font-variant:small-caps;">Prof. James Howie</span> for very useful discussions and for having kindly clarified some very important details. Thanks are due to Czech Technical University in Prague for a great hospitality, where the core part of this paper was written, especially to <span style="font-variant:small-caps;">Prof. Pavel Štovíček</span> and <span style="font-variant:small-caps;">Prof. Čestmír Burdík</span>.
Composition–Diamond Lemma
=========================
Here we present the concepts of Composition–Diamond lemma and Gröbner–Shirshov basis. In the classical version of Composition–Diamond lemma, it assumed that considered algebras is over a field, here we consider the general case.
CD-Lemma for associative algebras
---------------------------------
Let $\mathsf{K}$ be an arbitrary commutative ring with unit, $\mathsf{K} \langle X \rangle$ the free associative algebra over $\mathsf{K}$ generated by $X$, and let $X^*$ be the free monoid generated by $X$, where empty word is the identity, denoted by $\mathbf{1}_{X^*}$. Assume that $X^*$ is a well-ordered set. Take $f\in \mathsf{K} \langle X \rangle$ with the leading word (term) $\mathrm{LT}({f})$ and $f= \kappa\mathrm{LT}({f}) + r_{f}$, where $0\neq \kappa\in \mathsf{K}$ and $\mathrm{LT}({r_f})<\mathrm{LT}({f})$. We call $f$ is *monic* if $\kappa = 1$. We denote by $\mathrm{deg}(f)$ the degree of $\mathrm{LT}({f})$.
A well ordering $\leqslant$ on $X^*$ is called [*monomial*]{} if for $u,v \in X^*$, we have: $$u\leqslant v \Longrightarrow \bigl.w\bigr|_u \leqslant \bigl.w\bigr|_v, \qquad \forall w \in X^*,$$ where $\bigl.w\bigr|_u : = \bigl.w\bigr|_{x \to u}$ and $x$’s are the same individuality of the letter $x \in X$ in $w$.
A standard example of monomial ordering on $X^*$ is *the deg-lex ordering* (i.e., degree and lexicographical), in which two words are compared first by the degree and then lexicographically, where $X$ is a well-ordering set.
Fix a monomial ordering $\leqslant$ on $X^*$, and let $\varphi$ and $\psi$ be two monic polynomials in $\mathsf{K}\langle X\rangle$. There are two kinds of compositions:
- If $w$ is a word (i.e, it lies in $X^*$) such that $w = \mathrm{LT}({\varphi}) b = a \mathrm{LT}({\psi})$ for some $a,b\in X^*$ with $\mathrm{deg}(\mathrm{LT}({\varphi}))+\mathrm{deg}(\mathrm{LT}({\psi})) >\mathrm{deg}(w)$, then the polynomial $(\varphi,\psi)_w:=\varphi b - a \psi$ is called the [*intersection composition*]{} of $\varphi$ and $\psi$ with respect to $w$.
- If $w = \mathrm{LT}({\varphi}) = a\mathrm{LT}({\psi}) b$ for some $a,b \in X^*$, then the polynomial $(\varphi,\psi)_w:=\varphi -a\psi b$ is called the [*inclusion composition*]{} of $\varphi$ and $\psi$ with respect to $w$.
We then note that $\mathrm{LT}{(\varphi,\psi)}_w\leq w$ and $(\varphi,\psi)_w$ lies in the ideal $(\varphi,\psi)$ of $\mathsf{K}\langle X\rangle$ generated by $\varphi$ and $\psi$.
Let $\mathbf{S}\subseteq \mathsf{K}\langle X \rangle$ be a monic set (i.e., it is a set of monic polynomials). Take $f \in \mathsf{K}\langle X\rangle$ and $w\in X^*$. We call $f$ is [*trivial modulo*]{} $(\mathbf{S},w)$, denoted by $$f \equiv 0 \bmod(\mathbf{S},w),$$ if $f = \sum_{s\in \mathbf{S}} \kappa a s b$, where $\kappa \in \mathsf{K}$, $a,b \in X^*$, and $a \mathrm{LT}({s})b \leqslant w$.
A monic set $\mathbf{S}\subseteq \mathsf{K}\langle X \rangle$ is called a *Gröbner–Shirshov basis* in $\mathsf{K}\langle X \rangle$ with respect to the monomial ordering $\leq$ if every composition of polynomials in $\mathbf{S}$ is trivial modulo $\mathbf{S}$ and the corresponding $w$.
The following Composition–Diamond lemma was first proved by Shirshov [@Sh2] for free Lie algebras over fields (with deg-lex ordering). For commutative algebras, this lemma is known as Buchberger’s theorem [@Buchberger].
\[CDA\] Let $\mathsf{K}$ be an arbitrary commutative ring with unit, $\leqslant$ a monomial ordering on $X^*$ and let $I(\mathbf{S})$ be the ideal of $\mathsf{K} \langle X \rangle$ generated by the monic set $\mathbf{S}\subseteq \mathsf{K} \langle X \rangle$. Then the following statements are equivalent:
- $\mathbf{S}$ is a Gröbner–Shirshov basis in $\mathsf{K} \langle X \rangle$.
- if $f \in I(\mathbf{S})$ then $\mathrm{LT}({f}) = a\mathrm{LT}({s})b$ for some $s\in \mathbf{S}$ and $a,b\in X^*$.
- the set of irreducible words $$\mathrm{Irr}(\mathbf{S}):=\left\{u \in X^*: u \ne a \mathrm{LT}({s})b,\,s \in \mathbf{S},\,a,b\in X^*\right\}$$ is a linear basis of the algebra $\mathsf{K} \langle \bigl.X\bigr|\mathbf{S}\rangle:=\mathsf{K} \langle X\rangle/I(\mathbf{S})$.
Let $\mathsf{K}$ be an arbitrary commutative ring and consider the following algebra $\Lambda = \mathsf{K} \langle x,y \rangle/(x^2 - y^2)$. Let us consider the polynomials $\varphi = x^2 - y^2$, $\psi = xy^2 - y^2x$, and let $y \leqslant x$. It is not hard to see that the set $\mathbf{S} = \{\varphi,\psi\}$ is a Gröbner–Shirshov basis of $\Lambda$. Indeed, $$\begin{aligned}
(\varphi, \varphi)_{w}&=&{\varphi}x - x {\varphi}\\
&=& x^3-y^2x - (x^3 - xy^2) = \psi,\end{aligned}$$ for $w =x^3,$ and $$\begin{aligned}
(\varphi,\psi)_{w} &=& \varphi y^2 - x\psi \\
&=&x^2y^2 - y^2y^2 - (x^2y^2 - xy^2x)\\
&=& \psi x + y^2 \varphi,\end{aligned}$$ for $w = x^2y^2.$ Since the set $\mathbf{S}$ is monic, then the set $$\mathrm{Irr}(\mathbf{S}) = \bigcup\limits_{n > 0}\Bigl\{1,x, xy, y^n,y^nx, (xy)^n, (yx)^n, (yxy)^n\Bigr\}$$ is the $\mathsf{K}$-basis for $\Lambda$, by Theorem \[CDA\].$\square$
CD-Lemma for Semigroups and Groups
----------------------------------
Given a set $X$ consider $S \subseteq X^*\times X^*$ the congruence $\rho(S)$ on $X^*$ generated by $S$, the quotient semigroup $$P = \mathbf{sgr}\langle X \, | \, S \rangle = X^*/\rho(S),$$ and the semigroup algebra $\mathsf{K}[P]$. Identifying the set $\{u=v\, | \, (u,v) \in S\}$ with $S$, it is easy to see that $$\tau: \mathsf{K}\langle X\, | \, S \rangle \to \mathsf{K}(X^*/ \rho(S)), \quad \sum \kappa f + I(S) \mapsto \sum \kappa \mathrm{LT}(f)$$ is an algebra isomorphism.
The Shirshov completion $S^c$ of $S$ consists of semigroup relations, $S^c:=\{f-g\}$. Then $\mathrm{Irr}(S^c)$ is a linear $\mathsf{K}$-basis of $\mathsf{K}\langle X \, | \, S \rangle$, and so $\tau(\mathrm{Irr}(S^c))$ is a linear $\mathsf{K}$-basis of $\mathsf{K}(X^*/ \rho(S))$. This shows that $\mathrm{Irr}(S^c)$ consists precisely of the normal forms of the elements of the semigroup $\mathbf{sgr}\langle X \, | \, S \rangle$.
Therefore, in order to find the normal forms of the semigroup $\mathbf{sgr}\langle X \, | \, S \rangle$, it suffices to find a Gröbner–Shirshov basis $S^c$ in $\mathsf{K}\langle X \, | \, S \rangle.$ In particular, consider a group $G = \mathbf{gr}\langle X \, | \, S \rangle$, where $S = \{(u,v) \in F(X) \times F(X)\}$ and $F(X)$ is the free group on a set $X$. Then $G$ has a [*semigroup presentation*]{} $$G = \mathbf{sgr}\langle X \cup X^{-1} \, | \, S, \, x^{\varepsilon}x^{-\varepsilon} = 1, \varepsilon = \pm, x \in X \rangle, \qquad X \cap X^{-1} = \varnothing,$$ as a semigroup.
First Examples of Groups
========================
Let $G = \mathbf{gr} \langle x, y_1,\ldots, y_\ell\, | \, r_{11}=r_{12}, \ldots, r_{n1} = r_{n2} \rangle$ be a group and let $\dfrac{\partial r_i}{\partial x} = D_i f$ for $1 \le i \le n$, here $r_i:=r_{i1}r^{-1}_{i2}$, and $D_i, f\in \mathbb{Z}[G]$. We assume that every $r_{ij}$ does not contain other term $r_{pq}$ as a subword, and all $r_i$ are not reduced words, *i.e.,* they do not contain a word (as a subword) of form $aa^{-1}$.
Consider now $G$ as a semigroup and set $x >x^{-1} > y_{j}$ for $1 \le j \le \ell$, and deg-lex order the free monoid $\mathfrak{W}$ generated by ${x,x^{-1},y_1,y_1^{-1},\ldots, y_\ell, y_\ell^{-1}}$.
Fix $1 \le i \le n$ and consider $r_i$, we have $$\dfrac{\partial r_{i}}{\partial x} = \dfrac{\partial r_{i1}}{\partial x} - \dfrac{\partial r_{i2}}{\partial x}.$$
Without loss of generality, we may put $r_{i1}>r_{i2}$ for $1 \le i \le n$. Hence $\mathrm{LT}( \partial r_{i}/ \partial x) = \mathrm{LT}(\partial r_{i1}/\partial x)$.
Set $\mathrm{LT}( \partial r_{i} /\partial x ) = u_i\bar f$, then $r_{i1} = u_i\bar f x \widetilde{u_i}$, where $\widetilde{u_i}$ does not involve $x$ and $x^{-1}$. We have $$\varphi_0(r_i):=\dfrac{\partial r_{i}}{\partial x} = \dfrac{\partial u_{i}}{\partial x} + u_i\dfrac{\partial \bar f}{\partial x} - \dfrac{\partial r_{i2}}{\partial x} + u_i\bar f.$$
Then $$\begin{aligned}
\varphi_1(r_i) &:=& \left( \dfrac{\partial r_{i}}{\partial x},f \right)_{u_i\bar f} := \dfrac{\partial r_{i}}{\partial x} - u_if \\
&=& \dfrac{\partial u_{i}}{\partial x} + u_i\dfrac{\partial \bar f}{\partial x}- \dfrac{\partial r_{i2}}{\partial x} + u_i\bar f - u_if \\
&=&\dfrac{\partial u_{i}}{\partial x} + u_i\dfrac{\partial \bar f}{\partial x} - \dfrac{\partial r_{i2}}{\partial x} - u_i f_1,\end{aligned}$$ where $f_1:=f - \bar f$.
We thus have to consider the following possibilities: (1) $\varphi_1(r_i) = 0$, (2) $\mathrm{LT}(\varphi_1(r_i)) = u_i\mathrm{LT}(\partial \bar f / \partial x),$ (3) $\mathrm{LT}(\varphi_1(r_i)) = \mathrm{LT}(\partial r_{i2}/\partial x)$, (4) $\mathrm{LT}(\varphi_1(r_i)) = u_i\mathrm{LT}(f_1)$.
To get some examples we consider cases (1) and (3). Other cases will be considered the next section.
\(1) Let $\varphi_1(r_i) = 0$. We have $$\dfrac{\partial u_{i}}{\partial x} + u_i\dfrac{\partial \bar f}{\partial x} - \dfrac{\partial r_{i2}}{\partial x} - u_i f_1 = 0.$$
If we assume that the $\partial u_i / \partial x, \partial r_{i2}/\partial x \ne 0$ have common terms it then implies that a $\mathstrut^\bullet r_{i2}$ contain $\mathstrut^\bullet u_i$ as a subword, here, for a word $w\in \mathfrak{W}$, we set $w = \mathstrut^\bullet w w^\bullet$. Similarly, one can easy see that the polynomials $u_i \partial \bar f/\partial x$ $ \partial r_{i2}/\partial x$ have no similar terms.
Thus we may put $\partial u_i/\partial x = 0$, $ \partial \bar f/\partial x = f_1$ and $ \partial r_{i2}/\partial x = 0$, [*i.e.,*]{} $r_{i2}$ does not involve the terms $x$, $x^{-1}$.
\(3) Let $\mathrm{LT}(\varphi_1(r_i)) = \mathrm{LT}\left( \dfrac{\partial r_{i2}}{\partial x} \right) = v_i\bar f$. Hence $r_{i2} = v_i\bar f x \widetilde{v_i}$, where $\widetilde{v_i}$ does not involve $x$ and $x^{-1}$, and $$\dfrac{\partial r_{i2}}{\partial x} = \dfrac{\partial v_{i}}{\partial x} + v_i\dfrac{\partial \bar f}{\partial x} + v_i \bar f.$$
We then get $$\begin{aligned}
\varphi_2(r_i) &:=& (-\varphi_1, f)_{v_i\bar f} := -\varphi_1 - v_if \\
&=& -\dfrac{\partial u_{i}}{\partial x} - u_i\dfrac{\partial \bar f}{\partial x} + \dfrac{\partial v_{i}}{\partial x} + v_i\dfrac{\partial \bar f}{\partial x} + v_i \bar f + u_i f_1 - v_if \\
&=& \dfrac{\partial v_{i}}{\partial x} - \dfrac{\partial u_{i}}{\partial x} + (v_i - u_i)\dfrac{\partial \bar f}{\partial x} + (u_i-v_i)f_1 \\
&=& \dfrac{\partial v_{i}}{\partial x} - \dfrac{\partial u_{i}}{\partial x} + (v_i-u_i)\left(\dfrac{\partial \bar f}{\partial x} - f_1 \right).\end{aligned}$$
Setting $f_1 =\partial \bar f / \partial x$ we then get $\varphi_2(r_i) = \partial v_i / \partial x - \partial u_i / \partial x$. Thus we have the same problem as for $\varphi_0(r_i):=\partial r_{i1}/ \partial x - \partial r_{i_2} / \partial x$. It follows that we then get the following set of groups $$\label{G1}
G_{\ell,n} = \mathbf{gr}\langle x, y_1,\ldots, y_\ell\, | \, r_{11} = r_{12}, \ldots, r_{n,1} = r_{n,2}\rangle,$$ where$$\begin{aligned}
r_{i1} &=& \mathsf{u}_{i,1}\mathrm{w}x\mathsf{u}_{i,2} \cdots \mathsf{u}_{i,p-1}\mathrm{w}x\mathsf{u}_{i,p_i}, \\
r_{i2} &=& \mathsf{v}_{i,1}\mathrm{w}x\mathsf{v}_{i,2} \cdots \mathsf{v}_{i,q-1}\mathrm{w}x\mathsf{v}_{i,q_i},\end{aligned}$$ here for $1 \le i \le n$, $p_i \ge 1$, $q_i \ge 0$, and if $q_i=0$ then $r_{i2} = \mathsf{v}_{i,0}$, further all $r_{i1}$, $r_{i2}$ are not reduced, all $\mathsf{u}_{i,j},\mathsf{v}_{i,k}$ do not involve $x,x^{-1}$, $\mathrm{w}\ne 1$, and every term of any relation does not contain, as a subword, a term of other relations. Therefore we get
\[themainresult\] For a group $G_{\ell,n}$ presented by (\[G1\]), with $\pi_2(\mathcal{K}_{G_{\ell,n}}) \ne 0$, the group ring $\mathbb{Z}[G_{\ell,n}]$ has nontrivial zero divisors.
Indeed, for all $1 \le i \le n$, we have $$\begin{aligned}
\dfrac{\partial r_i}{\partial x} &=& \mathsf{u}_{i,1} \left(\dfrac{\partial \mathrm{w}}{\partial x} + \mathrm{w}\right) + \cdots + \mathsf{u}_{i,1}\mathrm{w}x\cdots \mathsf{u}_{i,p-1} \left( \dfrac{\partial \mathrm{w}}{\partial x} + \mathrm{w} \right) \\
&& - \mathsf{v}_{i,1} \left(\dfrac{\partial \mathrm{w}}{\partial x} + \mathrm{w}\right) - \cdots - \mathsf{v}_{i,1}\mathrm{w}x\cdots \mathsf{v}_{i,q-1} \left( \dfrac{\partial \mathrm{w}}{\partial x} + \mathrm{w} \right),
\end{aligned}$$ hence $$\dfrac{\partial r_i}{\partial x} = \left(\sum_{k=p-1}^{1} \mathsf{u}_{i,1}\mathrm{w}x \cdots \mathsf{u}_{i,p-k} - \sum_{t=q-1}^{1} \mathsf{v}_{i,1}\mathrm{w}x \cdots \mathsf{v}_{i,q-t} \right) \left( \dfrac{\partial \mathrm{w}}{\partial x} + \mathrm{w} \right).$$
If $\pi_2(\mathcal{K}_{G_{\ell,n}}) \ne 0$, then by (\[exact\]), we obtain nontrivial zero divisors in $\mathbb{Z}[G]$, as claimed.
The Other Possibilities
=======================
In this section we consider other possibilities which appeared in the construction of $G_{\ell,n}$ and we will see that we again get the same set (\[G1\]) of groups.
\[commondiv\] Let $\mathfrak{w}, \mathfrak{p}_1, \ldots, \mathfrak{p}_\ell \in \mathfrak{F}$ and $P = \sum_{i=1}^\ell \varepsilon_i \mathfrak{p}_i \in \mathbb{Z}[\mathfrak{F}]$, where $\varepsilon = \pm 1$. If the polynomials $\partial \mathfrak{w} / \partial x$, $P$ have a common term, say $\mathfrak{p}_k$, then the words ${\mathfrak}{w}, {\mathfrak}{p}_k$ have common left divisor, i.e., there exist nonempty words ${\mathfrak}{u}, {\mathfrak}{w}',{\mathfrak}{p}_k' \in {\mathfrak}{F}$ such that ${\mathfrak}{w} = {\mathfrak}{u}{\mathfrak}{w}'$, ${\mathfrak}{p}_k = {\mathfrak}{u}{\mathfrak}{p}_k'$.
Indeed, let ${\mathfrak}{w} = {\mathfrak}{w}_1 x^{n_1} {\mathfrak}{w}_2 x^{n_2} \cdots {\mathfrak}{w}_m x^{n_m}{\mathfrak}{w}_{n_{m+1}}$, where for $1 \le j \le n_{m+1}$ every ${\mathfrak}{w}_{j}$ does not involve $x$, $x^{-1}$ and $n_j \in \mathbb{Z}$. Thus we have $$\dfrac{\partial {\mathfrak}{w}}{\partial x} = {\mathfrak}{w}_1 \dfrac{\partial x^{n_1}}{\partial x} + {\mathfrak}{w}_1 x^{n_1} {\mathfrak}{w}_2 \dfrac{\partial x^{n_2}}{\partial x} + \cdots + {\mathfrak}{w}_1 x^{n_1} {\mathfrak}{w}_2 x^{n_2} \cdots {\mathfrak}{w}_m \dfrac{\partial x^{n_m}}{\partial x},$$ where $$\dfrac{\partial x^{n_j}}{\partial x} = \begin{cases}
1 + x + \cdots + x^{n_j-1}, & n_j \ge 0, \\
-x^{-1} - x^{-2} - \cdots - x^{-|n_j|}, & n_j < 0,
\end{cases}$$ and the statement follows.
Give a group $G = \mathbf{gr} \langle x, y_1,\ldots, y_\ell \, | \, r_1,\ldots, r_m \rangle $. Take a $\bar f \in \mathfrak{F}$. Let $r_1 = \{r_{11} = r_{12}\}$ and let $\mathrm{LT}(\partial r_{11}/\partial x) = u_1 \bar f$. Hence, $r_{11} = u_1\bar f x\widetilde{u_1}$ with $\widetilde{u_1} \ne \widetilde{u_1}(x)$.
We get $$\dfrac{\partial r_1}{\partial x} = \dfrac{\partial r_{11}}{\partial x} - \dfrac{\partial r_{12}}{\partial x} = \dfrac{\partial u_1}{\partial x} + u_1 \dfrac{\partial \bar f}{\partial x} - \dfrac{\partial r_{12}}{\partial x}.$$
Then $$\varphi_1: = \left( \dfrac{\partial r_1}{\partial x}, f \right)_{u_1\bar f} = \dfrac{\partial r_1}{\partial x} - u_1f = \dfrac{\partial u_1}{\partial x} + u_1 \dfrac{\partial \bar f}{\partial x} - \dfrac{\partial r_{12}}{\partial x} - u_1f_1.$$ where $f_1:=f-\bar f.$
A monomial $\bar f$ involves $x^{\pm 1}$
----------------------------------------
Let $\bar f = w_1 x^{n_1} w_2 \cdots w_k x^{n_k} w_{k+1}$, where $n_i \ne 0$, $w_i \ne w_{i}(x)$ for $i=1,\ldots, k+1$.
\[LT(udf)\] Let $\mathrm{LT}(\varphi_1) = \mathrm{LT} (u_1 \partial \bar f/\partial x) = u_2 \bar f$. Then $\bar f = wxw\cdots wxw$ and $u_1 = u_2wx$.
Let $\bar f = w_1 x^{n_1} w_2 \cdots w_k x^{n_k} w_{k+1}$, where $n_i \ne 0$, $w_i \ne w_{i}(x)$ for $i=1,\ldots, k+1$.
\(1) Assume that $n_k >0$ then $\dfrac{\partial x^{n_k}}{\partial x} = 1 +x + \cdots + x^{n_k-1}$, and we get
$$\dfrac{\partial (f_\lambda \cdot x^{n_k} w_k)}{\partial x} = \dfrac{\partial f_\lambda}{\partial x} + f_\lambda(1+x \cdots + x^{n_k-1}),$$ where $f_\lambda := w_1x^{n_1}\cdots w_{k}$.
Then, by assumption, $$\mathrm{LT}\left(u_1 \dfrac{\partial \bar f}{\partial x}\right) = u_1 w_1x^{n_1}\cdots w_{k} x^{n_k-1} = u_2 w_1 x^{n_1} w_2 \cdots w_k x^{n_k} w_{k+1},$$ hence either $w_{k+1}=1$ or $n_{k}=1$.
Let $w_{k+1} = 1$, $n_k>1$, then $u_1 w_1x^{n_1}\cdots w_{k} = u_2 w_1 x^{n_1} w_2 \cdots w_k x$, and hence $w_k = 1$. Thus, $u_1w_1x^{n_1} \cdots w_{k-1}x^{n_{k-1}} = u_2w_1x^{n_1}w_2 \cdots w_{k-1}x^{n_{k-1}+1}$, hence we must put $w_{k-1} = \cdots = w_1 = 1$ and we then obtain $u_1x^{n_1 + \cdots n_{k-1}} = u_2 x^{n_1 + \cdots n_{k-1} +1}$. Hence $u_1 = u_2x$.
Let $w_{k+1} \ne 1$, $n_k=1$, we then have $$u_1 w_1x^{n_1}\cdots xw_{k} = u_2 w_1 x^{n_1} w_2 \cdots w_k x w_{k+1}.$$
Hence, $w_{k+1} = w_k = \cdots w_1= w$ and $u_1 = u_2wx$.
\(2) Assume that $n_k <0$ then $\dfrac{\partial x^{n_k}}{\partial x} = - (x^{-1} + x^{-2} + \cdots + x^{n_k})$, hence $\mathrm{LT} \left( u_1 \dfrac{\partial \bar f}{\partial x} \right) = - u_1 w_1 x^{n_1} \cdots w_k x^{n_k}$. By $\mathrm{LT}\left(u_1 \dfrac{\partial \bar f}{\partial x}\right) = u_2 \bar f$, $w_{k+1} = 1$, and $u_1 = -u_2$. But it follows that $r_{11}$ is not a reduced word.
If $\mathrm{LT}(\varphi_1) = \mathrm{LT}(u_1 \partial \bar f / \partial x) = u_2\bar f$ and $\varphi_k(r_1)=0$ for $k \ge 2$ we then get the set of group are described by (\[G1\]).
We have $$\varphi_2(r_1):=\varphi_1(r_1) - u_2f = \dfrac{\partial u_1} {\partial x} + u_1 \dfrac{\partial \bar f}{ \partial x} - \dfrac{\partial r_{12} }{ \partial x} - u_1 f_1 - u_2 f.$$
By Lemma \[LT(udf)\], we have $\bar f = (wx)^nw$, $u_1 = u_2wx$.
We get $$\dfrac{\partial \bar f}{\partial x} = w + wxw + \cdots + (wx)^{n-1}w, \qquad \dfrac{\partial u_1}{\partial x} = \dfrac{\partial u_2}{\partial x} + u_2w.$$
Then
$$\begin{aligned}
\varphi_2(r_1) &=& \dfrac{\partial u_2}{\partial x} + u_2w + u_2wx(w + wxw + \cdots + (wx)^{n-1}w) \\
&&- \dfrac{\partial r_{12}}{\partial x} - u_2wxf_1 -u_2\bar f - u_2 f_1 \\
&=& \dfrac{\partial u_2}{\partial x} + u_2w + u_2wx(w + wxw + \cdots + (wx)^{n-2}w) \\
&&- \dfrac{\partial r_{12}}{\partial x} - u_2wxf_1 - u_2 f_1 \\
&=& \dfrac{\partial u_2}{\partial x} + u_2 (w + wxw + \cdots + (wx)^{n-1}w) - \dfrac{\partial r_{12}}{\partial x} - (u_2wx + u_2)f_1\\
&=& \dfrac{\partial u_2}{\partial x} + u_2 \dfrac{\partial \bar f}{\partial x} - \dfrac{\partial r_{12}}{\partial x} - (u_2wx + u_2)f_1.\end{aligned}$$
Suppose that $\varphi_2(r_1) = 0$. By Lemma \[commondiv\], $\partial u_2 / \partial x, \partial r_{12}/ \partial x = 0$. Thus we have the following equation $$w + wxw + \cdots + (wx)^{n-1}w - wx f_1 - f_1 = 0,$$ which has a solution. Indeed, we may put $n=2$ and $f_1 = w$.
It follows that we get the following set of groups $G = \langle x, y_1,\ldots,y_\ell \, |\, r_{11} = r_{12}, \ldots, r_{1m} = r_{2m} \rangle$, where $r_{i1} = u_i wxwxw u_i'$, $r_{i2} = v_i$, $1\le i \le m$, here the $w$ and all the words $u_i,, u_i',v_i$ do not involve $x$, $x^{-1}$, and they each of them does not contain as a subword another word. Thus, we have got the set of groups (\[G1\]).
Assume now that $\varphi_2(r_1) \ne 0$ and $\mathrm{LT}(\varphi_2(r_1)) = \mathrm{LT}(\partial r_{12}/ \partial x) = v_1\bar f$. Then $r_{12} = v_1\bar f x v_1'$, where $v_1'$ does not involve $x$, $x^{-1}$.
We have $$\begin{aligned}
\varphi_3(r_1) &: =& -\varphi_2(r_1)-v_1f \\
&=& -\dfrac{\partial u_2}{\partial x} - u_2 \dfrac{\partial \bar f}{\partial x} + (u_2wx + u_2)f_1 \\
&& \dfrac{\partial v_1}{\partial x} + v_1 \dfrac{\partial \bar f}{\partial x} + v_1\bar f - v_1f \\
&=& \dfrac{\partial v_1}{\partial x} -\dfrac{\partial u_2}{\partial x} - u_2 \dfrac{\partial \bar f}{\partial x} + (u_2wx + u_2)f_1 + v_1 \dfrac{\partial \bar f}{\partial x} - v_1f_1.\end{aligned}$$
If we put $\varphi_3(r_1) = 0$ then by Lemma \[commondiv\], $\partial v_1 / \partial x, \partial u_2 / \partial x = 0$ because of $u_2$, $v_1$ have no common left divisors by assumption. Thus, to have a common divisor for $\partial r_i / \partial x$ we have to put $\varphi_3(r_1) \ne 0$ and $\mathrm{LT}(\varphi_3(r_1)) = \mathrm{LT}(v_1 \partial \bar f / \partial x) = v_2 \bar f$. Similarly, as for $\varphi_2(r_1)$, we then obtain $$\varphi_4(r_1) = \dfrac{\partial v_2}{\partial x} -\dfrac{\partial u_2}{\partial x} - u_2 \dfrac{\partial \bar f}{\partial x} + (u_2wx + u_2)f_1 + v_2 \dfrac{\partial \bar f}{\partial x} - (v_2wx + v_2)f_1.$$
Hence the equation has a solution. Indeed, we may put $\partial v_2/ \partial x, \partial u_2 / \partial x = 0$, $\bar f = (wx)^2w$, $f_1 = w$. Thus we have the same groups are described by (\[G1\]).
A monomial $\bar f$ does not involve $x^{\pm 1}$
------------------------------------------------
We then have $\partial r_i / \partial x = \partial u_i / \partial x - \partial r_{i2} / \partial x$, and if we put $\mathrm{LT}(\partial r_i / \partial x) = \mathrm{LT}(\partial u_{i1} / \partial x) = u_{i2}\bar f$. It follows that $u_{i1} = u_{i2}\bar f x u_{i2}'$ where $u_{i2}'$ does not involve $x$, $x^{-1}$. It is easy to see that it is impossible.
Indeed,we have $$\begin{aligned}
\varphi_1(r_i) &=& \dfrac{\partial r_{i}}{\partial x} - u_{i2} f \\
&=& \dfrac{\partial u_{i2}}{\partial x} + u_{i2}\bar f - \dfrac{\partial r_{i2}}{\partial x} - u_{i2}f \\
&=& \dfrac{\partial u_{i2}}{\partial x} - \dfrac{\partial r_{i2}}{\partial x} - u_{i2}f_1,\end{aligned}$$ hence by Lemma \[commondiv\], $\varphi_k(r_i) \ne 0$ for $k\ge 1$.
Conclusion {#conclusion .unnumbered}
==========
We have seen that homological point of view on the Kaplansky’s zero divisor conjecture is very useful and very easy for understanding. The author is going to study when this groups have no a torsion and when $\pi_2(\mathcal{K}_{G_{\ell,n}}) \ne 0$ in the future papers.
[20]{} L.A. Bokut and Y. Chen, Gröbner–Shirshov basis and their calculation, [*Bull. Math. Sci.*]{} [**4**]{}, (2014), 325–395.
B. Buchberger, An algorithmical criteria for the solvability of algebraic systems of equations, [*Aequationes Math.,*]{} [**4**]{}, 374–383 (1970).
S. Cojocaru, A. Podoplelov, and V. Ufnarovski, Non commutative Gröbner bases and Anick’s resolution, [*Computational methods for representations of groups and algebras (Essen, 1997), Birkhäuse, Basel.*]{} (1999) 139–159.
W. Magnus, A. Karras and D. Solitar, “Combinatorial Group Theory,” [*Interscience,*]{} New York, 1966.
K. Reidemeister, Complexes and homotopy chains, [*Bull. Amer. Math. Soc.,*]{} **56**, 297–307 (1950).
A. I. Shirshov, Some algorithm problems for Lie algebras, [*Sibirsk. Mat. Z.*]{} [**3**]{}(2) 1962 292–296 (Russian); English translation in *SIGSAM Bull*. [**33**]{}(2) (1999) 3–6.
[^1]: Laboratory of Modern Algebra and Applications, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, Russia, St. Petersburg Department of Steklov Mathematical Institute.\
The author is supported by the grant of the Government of the Russian Federation for the state support of scientific research carried out under the supervision of leading scientists, agreement 14.W03.31.0030 dated 15.02.2018. 1\
*Email address:* `wickktor@gmail.com`
|
---
abstract: 'We analyze the luminosity function of the globular clusters (GCs) belonging to the early-type galaxies observed in the ACS Virgo Cluster Survey. We have obtained maximum likelihood estimates for a Gaussian representation of the globular cluster luminosity function (GCLF) for 89 galaxies. We have also fit the luminosity functions with an “evolved Schechter function”, which is meant to reflect the preferential depletion of low-mass GCs, primarily by evaporation due to two-body relaxation, from an initial Schechter mass function similar to that of young massive clusters in local starbursts and mergers. We find a highly significant trend of the GCLF dispersion $\sigma$ with galaxy luminosity, in the sense that the GC systems in smaller galaxies have narrower luminosity functions. The GCLF dispersions of our Galaxy and M31 are quantitatively in keeping with this trend, and thus the correlation between $\sigma$ and galaxy luminosity would seem more fundamental than older notions that the GCLF dispersion depends on Hubble type. We show that this narrowing of the GCLF in a Gaussian description is driven by a steepening of the cluster mass function above the classic turnover mass, as one moves to lower-luminosity host galaxies. In a Schechter-function description, this is reflected by a steady decrease in the value of the exponential cut-off mass scale. We argue that this behavior at the high-mass end of the GC mass function is most likely a consequence of systematic variations of the initial cluster mass function rather than long-term dynamical evolution. The GCLF turnover mass $M_{\rm TO}$ is roughly constant, at $M_{\rm TO}\simeq(2.2\pm0.4)\times10^5\,M_\odot$ in bright galaxies, but it decreases slightly (by $\sim\!35\%$ on average, with significant scatter) in dwarf galaxies with $M_{B, {\rm gal}} \ga -18$. It could be important to allow for this effect when using the GCLF as a distance indicator. We show that part, though perhaps not all, of the variation could arise from the shorter dynamical friction timescales in less massive galaxies. We probe the variation of the GCLF to projected galactocentric radii of 20–35 kpc in the Virgo giants M49 and M87, finding that the turnover point is essentially constant over these spatial scales. Our fits of evolved Schechter functions imply average dynamical mass losses ($\Delta$) over a Hubble time that vary more than $M_{\rm TO}$, and systematically but non-monotonically as a function of galaxy luminosity. If the initial GC mass distributions rose steeply towards low masses as we assume, then these losses fall in the range $2\times10^5\,M_\odot\la \Delta < 10^6\,M_\odot$ per GC for all of our galaxies. The trends in $\Delta$ are broadly consistent with observed, small variations of the mean GC half-light radius in ACSVCS galaxies, and with rough estimates of the expected scaling of average evaporation rates (galaxy densities) versus total luminosity. We agree with previous suggestions that if the full GCLF is to be understood in more detail, especially alongside other properties of GC systems, the next generation of GCLF models will have to include self-consistent treatments of dynamical evolution inside time-dependent galaxy potentials.'
author:
- 'Andrés Jordán, Dean E. McLaughlin, Patrick Côté, Laura Ferrarese, Eric W. Peng, Simona Mei, Daniela Villegas, David Merritt, John L. Tonry, Michael J. West'
title: |
The ACS Virgo Cluster Survey. XII.\
The Luminosity Function of Globular Clusters in Early Type Galaxies
---
=cmr8
Introduction {#sec:intro}
============
One of the remarkable features of the systems of globular clusters (GCs) found around most galaxies is the shape of their luminosity function, or the relative number of GCs with any given magnitude. Historically most important has been the fact that these distributions always appear to peak, or turn over, at a GC absolute magnitude around $M_{V,{\rm TO}} \approx -7.5$ (e.g., Harris 2001), corresponding roughly to a mass of $M_{\rm TO}\sim 2\times10^5\,M_\odot$. The near universality of this magnitude/mass scale for GCs has motivated the widespread use of the globular cluster luminosity function (GCLF) as a distance indicator (see Harris 2001; also Ferrarese et al. 2000), and it has also posed one of the longest-standing challenges to theories of GC formation and evolution.
In recent years, some significant amount of attention has also been paid to the way that GCs are distributed in [*mass*]{} around the peak of the GCLF. Traditionally, the full GCLF has most often been modeled as a Gaussian distribution in magnitude, corresponding to a lognormal distribution of GC masses. However, if one focuses only on the distribution of GCs above the point where the magnitude distribution turns over, it is found that the mass function can usually be described by a power law (Harris & Pudritz 1994), or perhaps a Schechter (1976) function (Burkert & Smith 2000), which is very similar to the mass distributions of giant molecular clouds and the young massive star clusters forming in starbursts and galaxy mergers in the local universe (e.g., Zhang & Fall 1999). The main difference between ancient GCs and the present-day sites of star-cluster formation is then that the mass functions of the latter rise steeply upwards towards masses much less than $M_{\rm TO}\sim\,10^5\,M_\odot$, far exceeding the observed frequency of such low-mass GCs.
There are two main possibilities to explain this fundamental difference. The first is that the conditions of star cluster formation in the early universe when GCs were assembling may have favored the formation of objects with masses in a fairly narrow range around $\sim\! 10^5$–$10^6\,M_\odot$ (to the exclusion, in particular, of much smaller masses). These conditions would no longer prevail in the environments forming young clusters in the nearby universe. Some theoretical models along these lines invoke the $\sim\! 10^6\,M_\odot$ Jeans mass at the epoch of recombination (Peebles & Dicke 1968), the detailed properties of $\sim\! 10^6\,M_\odot$ cold clouds in a two-phase protogalactic medium (Fall & Rees 1985), and reionization-driven compression of the gas in subgalactic ($\la\! 10^7\,M_\odot$) dark-matter halos (Cen 2001).
The second possibility is that GCs were in fact born with a wide spectrum of masses, like that observed for young star clusters, extending from $\! 10^6$–$10^7\,M_\odot$ down to $\sim\! 10^3$–$10^4\,M_\odot$ or below. A subsequent transformation to the characteristic mass function of GCs today could then be effected mainly by dynamical processes (relaxation and tidal shocking) that are particularly efficient at destroying low-mass clusters over the lifetime of a GC system (e.g., Fall & Rees 1977; Ostriker & Gnedin 1997; Fall & Zhang 2001). Some observational evidence has been reported for such an evolution in the mass functions of young and intermediate-age star clusters (e.g., de Grijs, Bastian & Lamers 2003, Goudfrooij et al. 2004).
If we take the Occam’s-razor view that indeed GCs formed through substantially the same processes as star clusters today, then the picture offered by observations of old GCLFs is unavoidably one of survivors. There has been some debate as to whether it was in fact the long-term dynamical mechanisms just mentioned that were mainly responsible for destroying large numbers of low-mass globulars, or whether processes more related to cluster formation strongly depleted many low-mass protoclusters on shorter timescales (Fall & Zhang 2001; Vesperini & Zepf 2003). Even the most massive Galactic GCs have rather low binding energies $E_b \la 10^{52}$ erg (McLaughlin 2000), so that if conditions were not just right, very many protoglobular clusters could have been easily destroyed in the earliest $\sim\! 10^7$ yr of their evolution, through the catastrophic mass loss induced by massive-star winds and supernova explosions (see, e.g., Kroupa & Boily 2002; Fall, Chandar & Whitmore 2005). Furthermore, any clusters that survive this earliest mass-loss phase intact but with too low a concentration could potentially still dissolve within a relatively short time of $\sim\! 10^8$–$10^9$ yr (Chernoff & Weinberg 1990). Homogeneous observations of large samples of old GCLFs can help clarify the relative importance of such early evolution versus longer-term dynamical mass loss in the lives of star clusters generally.
The largest previous studies of GCLFs in early-type galaxies were performed with archival HST/WFPC2 data. Kundu & Whitmore (2001a, b) studied the GCLF for 28 elliptical and 29 S0 galaxies. They concluded that the turnover magnitude of the GCLF is an excellent distance indicator, and that the difference in the turnover luminosity between the $V$ and $I$ bands increases with the mean metallicity of the GCs essentially as expected if the GC systems in most galaxies have similar age and mass distributions. Larsen et al. (2001) studied the GCLF for 17 nearby early-type galaxies. They fitted Student’s $t$ distributions separately to the subpopulations of metal-rich and metal-poor GCs in each galaxy, and found that any difference in the derived turnovers was consistent with these subpopulations having similar mass and age distributions and the same GCLF turnover [*mass*]{} scale. Larsen et al. also fitted power laws to the mass distributions of GCs in the range $M\simeq 10^5$–$10^6\,M_\odot$ and found they were well described by power-law exponents similar to those that fit the mass functions of young cluster systems.
In this paper, we study the GCLFs of 89 early-type galaxies observed by HST as part of the ACS Virgo Cluster Survey (Côté et al. 2004). This represents the most comprehensive and homogeneous study of its kind to date. Some of the results in this paper are also presented in a companion paper (Jordán et al. 2006). In the next section, we briefly describe our data and present our observed GCLFs in a machine-readable table available for download from the electronic edition of the [*Astrophysical Journal*]{}. In §\[sec:models\] we discuss two different models that we fit to the GCLFs, and in § \[sec:method\] we describe our (maximum-likelihood) fitting methodology. Section \[sec:results\] presents the fits themselves, while §\[sec:trends\] discusses a number of trends for various GCLF parameters as a function of host galaxy luminosity and touches briefly on the issue of GCLF variations within galaxies. In §\[sec:disc\] we discuss some aspects of our results in the light of ideas about GC formation and dynamical evolution, focusing in particular on the relation between our data and a model of evaporation-dominated GCLF evolution. In §\[sec:conclusions\] we conclude.
Data {#sec:obs}
====
A sample of 100 early-type galaxies in the Virgo cluster was observed for the ACS Virgo Cluster Survey (ACSVCS; Côté et al. 2004). Each galaxy was imaged in the F475W ($\simeq$ Sloan $g$) and F850LP ($\simeq$ Sloan $z$) bandpasses for a total of 750 s and 1210 s respectively, and reductions were performed as described in Jordán et al. (2004a). These data have been used previously to analyze the surface-brightness profiles of the galaxies and their nuclei (Ferrarese et al. 2006ab, Côté et al. 2006), their surface brightness fluctuations (Mei et al. 2005ab; 2007), and the properties of their populations of star clusters, mainly GCs (Jordán et al. 2004b, Jordán et al. 2005, Peng et al. 2006a) but also dwarf-globular transition objects (or UCDs, Haşegan et al. 2005) and diffuse star clusters (Peng et al. 2006b).
One of the main scientific objectives of the ACSVCS is the study of the GC systems of the sample galaxies. We have developed a procedure by which we select GC candidates from the totality of observed sources around each galaxy, discarding the inevitable foreground stars and background galaxies that are contaminants for our purposes. This GC selection uses a statistical clustering method, described in detail in another paper in this series (Jordán et al. 2007, in preparation), in which each source in the field of view of each galaxy is assigned a probability $p_{\rm GC}$ that it is a GC. Our samples of GC candidates are then constructed by selecting all sources that have $p_{\rm GC} \ge 0.5$. The results of our classification method are illustrated in Figure 1 of Peng et al. (2006a). For every GC candidate we record the background surface brightness $I_b$ of the host galaxy at the position of the candidate, and we measure $z$- and $g$-band magnitudes and a half-light radius $R_h$ by fitting PSF-convolved King (1966) models to the local light distribution of the cluster (Jordán et al. 2005). Photometric zeropoints are taken from Sirianni et al. 2005 (see also Jordán et al. 2004a), and aperture corrections are applied as described by Jordán et al. (2007, in preparation).
Note that, as part of the ACSVCS we have measured the distances to most of our target galaxies using the method of surface brightness fluctuations (SBF; Tonry & Schneider 1988). The reduction procedures for SBF measurements, feasibility simulations for our observing configuration, and calibration have been presented in Mei et al. (2005ab) and the distance catalog is presented in another paper in this series (Mei et al. 2007). We use these distances in this paper[^1] to transform observed GC magnitudes into absolute ones on a per galaxy basis whenever we wish to assess GCLF properties in physical (i.e., mass-based) terms or need to compare the GCLFs of two or more galaxies. While some galaxies have larger distances, the average distance modulus that we employ is $(m-M)_0=31.09\pm0.03 \rm{(random)} \pm0.15 \rm{(systematic)}$, corresponding to $D=16.5\pm0.1 \rm{(random)} \pm 1.1 \rm{(systematic)}$ Mpc.
GCLF Histograms {#ssec:gclfhists}
---------------
There are three main ingredients we need to construct a GCLF for any galaxy. First, we have sets of magnitudes, in both the $z$ and $g$ bands, for all GC candidates. As mentioned above, we generally isolate GC candidates from a list of all detected objects by requiring that $p_{\rm GC}\ge 0.5$. Note that here and throughout, we use $g$ as shorthand to refer to the F475W filter, and $z$ denotes F850LP. Also, all GC magnitudes in this paper have already been de-reddened (see § 2.7 in Jordán et al. 2004a for details).
Second, we have the (in)completeness functions in both bandpasses. Our candidate GCs are marginally resolved with the ACS, and thus these completeness functions depend not only on the GC apparent magnitude $m$ and its position in its parent galaxy (through the local background surface brightness $I_b$), but also on the GC projected half-light radius $R_h$. Separate $z$- and $g$-band completeness functions $f(m,R_h,I_b) \le 1$ have therefore been calculated from simulations in which we first added simulated GCs with sizes $R_h=1,3,6,10$ pc and King (1966) concentration parameter $c=1.5$ to actual images from the ACSVCS (making sure to avoid sources already present), and then reduced the simulated images in an identical fashion to the survey data. We next found the fraction of artificial sources that were recovered, as a function of input magnitude and half-light radius, in each of ten separate bins of background light intensity. The final product is a three-dimensional look-up table on which we interpolate to obtain $f$ for any arbitrary values of $(m,R_h,I_b)$.
Last, we have the expected density of contaminants as a function of magnitude for each galaxy, obtained from analysis of archival ACS images (unassociated with the Virgo Cluster Survey) of 17 blank, high-latitude control fields, each observed with both $g$ and $z$ filters to depths greater than in the ACSVCS. We “customized” these data to our survey galaxies by performing object detections on every control field as if it contained each galaxy in turn. This procedure is described in more detail in Peng et al. (2006a, their §2.2). The net result is 17 separate estimates of the number of foreground and background objects, as a function of $g$ and $z$ magnitude, expected to contaminate the list of candidate GCs in every ACSVCS field.
Of the 100 galaxies in the ACSVCS, we restrict our analysis to those that have more than 5 probable GCs, as estimated by subtracting the total number of expected contaminants from the full list of GC candidates for each galaxy. We additionally eliminated two galaxies for which we could not usefully constrain the GCLF parameters. This results in a final sample of 89 galaxies. The GCLF data for these are presented in Table \[tab:gclf\_hists\].
The first column of Table \[tab:gclf\_hists\] is the galaxy ID in the Virgo Cluster Catalogue (VCC: Binggeli, Sandage & Tammann 1985; see Table 1 in Côté et al. 2004 for NGC and Messier equivalents). Column (2) contains an apparent $z$-band magnitude defining the midpoint of a bin with width $h_z$ given in column (3). This binwidth was chosen to be 0.4 for all galaxies. Columns (4)–(6) of the table then give the total number $N_{z,{\rm tot}}$ of observed sources in this bin; the number $N_{z,{\rm cont}}$ of contaminants in the bin as estimated from the average of our 17 control fields; and the average completeness fraction $f_z$ in the bin—all applying to the candidate-GC sample defined on the basis of our GC probability threshold, $p_{\rm GC}\ge 0.5$. Columns (7)–(11) repeat this information for the galaxy’s GC candidates identified in the $g$ band. Columns (12)–(21) are the corresponding $z$- and $g$-band data for an alternate GC sample defined strictly by magnitude cuts and an upper limit of $R_h<0\farcs064\simeq 5$ pc (which will include the large majority of real GCs; Jordán et al. 2005), rather than by relying on our $p_{\rm GC}$ probabilities. This provides a way of checking that selecting GC candidates by $p_{\rm GC}$ does not introduce any subtle biases into the GCLFs (see also §\[sec:method\] below).
The data in Table \[tab:gclf\_hists\] can be converted to distributions of absolute GC magnitude by applying the individual galaxy distances given in Mei et al. (2007). If they are used to fit model GCLFs, it should be by comparing the observed $N_{\rm tot}$ against a predicted $(f \times N_{\rm model} + N_{\rm cont})$ as a function of magnitude. This is essentially what we will do here, although we employ maximum-likelihood techniques rather than using the binned data. However, before describing our model-fitting methodology, we pause first to discuss in some detail the models themselves. We work with two different distributions in this paper: one completely standard, and one that is meant to elucidate the connections between observed GC mass distributions and plausible initial conditions and dynamical evolution histories.
Two GCLF Models {#sec:models}
===============
The term “globular cluster luminosity function” is customarily used to refer to a directly observed histogram of the number of GCs per unit magnitude. We follow this standard useage here, and in addition whenever we refer simply to a “luminosity function,” we in fact mean the GCLF, i.e., the distribution of magnitudes again. We denote the magnitude in any arbitrary bandpass by a lower-case $m$, and thus the GCLF is essentially the probability distribution function $dN/dm$. It is [*not*]{} equivalent to the distribution of true GC luminosities, since of course $m\equiv C-2.5\,\log\,L$ for some constant $C$, so $dN/dL = (dN/dm)|\partial m/\partial L|$ has a functional form different from that of $dN/dm$.
In this paper, when we speak of GC masses, we denote them by an upper-case $M$ and we almost always make the assumption that they are related by a multiplicative constant to GC luminosities, such that $m=C^{\prime} - 2.5\,\log\,M$, with $C^{\prime}$ another constant including the logarithm of a mass-to-light ratio (generally taken to be the same for all GCs in any one system, as is the case in the Milky Way; McLaughlin 2000). We refer to the number of GCs per unit mass, $dN/dM$, as a “mass function” or a “mass distribution.” In the literature, it is sometimes also called a “mass spectrum.” Its relation to the GCLF is $$\frac{dN}{dM} \propto \frac{1}{M} \frac{dN}{d\,\log\,M}
\propto 10^{0.4 m} \, \frac{dN}{dm} \ .
\label{eq:defs}$$
As we have already mentioned, most observed GCLFs show a “peak” or “turnover” at a cluster magnitude that is generally rather similar from galaxy to galaxy. One important consequence of equation (\[eq:defs\]) is that any such feature in the GCLF does [*not*]{} correspond to a local maximum in the GC mass distribution: if the first derivative of $dN/dm$ with respect to $m$ vanishes at some magnitude $m_{\rm TO}$, then the derivative of $dN/dM$ with respect to $M$ at the corresponding mass scale $M_{\rm TO}$ is strictly negative, i.e., the mass function still rises towards GC masses below the point where the GCLF turns over. (More specifically, the logarithmic slope of $dN/dM$ at the GCLF turnover point $M_{\rm TO}$ is always exactly $-1$; see McLaughlin 1994 for further discussion.)
The Standard Model {#ssec:gaussmod}
------------------
The function most commonly taken to describe GCLFs is a Gaussian, which is the easiest way to represent the peaked appearance of most luminosity functions in terms of number of clusters per unit magnitude. It is thus our first choice to fit to each of the observed GCLFs in this paper. Denoting the mean GC magnitude $\mu\equiv\langle m \rangle$ and the dispersion $\sigma_m = \langle (m-\mu)^2 \rangle^{1/2}$, we have the usual $$\frac{dN}{dm} = \frac{1}{\sqrt{2\pi}\,\sigma_m}
\exp\left[-\frac{(m-\mu)^2}{2\sigma_m^2}\right]\ .
\label{eq:gauss}$$ In terms of GC masses, $M$, this standard distribution corresponds to a mass function $dN/dM = (dN/dm)|\partial m/\partial M|$ or, since $m={\rm constant} - 2.5\,\log\,M$ (assuming a single mass-to-light ratio for all clusters in a sample), $$\frac{dN}{dM} =\frac{1}{(\ln 10) M}
\frac{1}{\sqrt{2\pi}\,\sigma_M}
\exp\left[-\frac{(\log\,M-\langle\log\,M\rangle)^2}{2\sigma_M^2}\right]\ ,
\label{eq:lognorm}$$ with $\sigma_M\equiv\sigma_m/2.5$.
As will be evident in what follows, the GCLFs in a large sample such as ours show a variety of detail that is unlikely to be conveyed in full by a few-parameter family of distributions. But it is also clear that a Gaussian captures some of the most basic information we are interested in investigating—the mean and the standard deviation of the GC magnitudes in a galaxy—with a minimal number of parameters. It is also the historical function of choice for GCLF fitting, and in many cases the fit is indeed remarkably good.
Nevertheless, the Gaussian does have some practical limitations. Secker (1992) showed that the tails of the GCLF in the Milky Way and M31 are heavier than a Gaussian allows, and he argued that a Student’s $t$ distribution (with shape parameter $\nu\simeq 5$) gives a better match to the data. More importantly, the observed GCLFs in our Galaxy and in M31 are [*asymmetric*]{} about their peak magnitude, a fact which has been emphasized most recently by Fall & Zhang (2001). This was implicit in the work of McLaughlin (1994), who advocated using piecewise power laws to fit the number of GCs per unit linear luminosity—or piecewise exponentials to describe the usual number of GCs per unit magnitude. Baum et al. (1997) used an asymmetric hyperbolic function to fit the strong peak and asymmetry in the combined Galactic and M31 GCLF.
However, all of these alternative fitting functions still share another shortcoming of the Gaussian, which is that there is no theoretical underpinning to it. Moreover, with the exception of the power laws in McLaughlin (1994), there is no obvious connection with the mass distributions of the young massive clusters that form in mergers and starbursts in the local universe. We therefore introduce an alternative fitting function—based on existing, more detailed studies of initial cluster mass functions and their long-term dynamical evolution—to address these issues.
An Evolved Schechter Function {#ssec:esmod}
-----------------------------
### Initial GC Mass Function {#sssec:esinit}
Observations of young star clusters indicate that the number of clusters per unit mass is well described by a power law—$dN/dM \propto M^{-\beta}$ with $\beta \approx 2$—or alternatively by a Schechter (1976) function with an index of about 2 in its power-law part and an exponential cut-off above some large mass scale that might vary from galaxy to galaxy (e.g., Gieles et al. 2006a). Perhaps the best-observed mass distribution for a young cluster system is that in the Antennae galaxies, NGC 4038/4039 (Zhang & Fall 1999). In this specific case, a pure power-law form suffices to describe the cluster $dN/dM$ as it is currently known; but a Schechter function with an appropriately high cut-off mass also fits perfectly well. Thus, assuming $dN/dM \propto M^{-\beta}\,\exp(-M/M_c)$, we find from the data plotted by Zhang & Fall (1999) that $\beta=2.00\pm0.04$ and $\log\,(M_c/M_\odot)=6.3^{+0.7}_{-0.3}$ for their sample of clusters with ages 2.5–6.3 Myr; and $\beta=1.92\pm0.14$ and $\log\,(M/M_c)=5.9^{+0.45}_{-0.25}$ for ages 25–160 Myr.
The mass functions of old globular clusters in the Milky Way and M31 can also be described by power laws with $\beta\simeq 2$ for clusters more massive than the GCLF peak (McLaughlin 1994; McLaughlin & Pudritz 1996; Elmegreen & Efremov 1997). And the GC mass distributions in large ellipticals follow power laws over restricted high-mass ranges, although here the slopes are somewhat shallower (McLaughlin 1994; Harris & Pudritz 1994) and there is clear evidence of curvature in $dN/dM$ (McLaughlin & Pudritz 1996) that is better described by the exponential cut-off at very high cluster masses in a Schechter function (e.g., Burkert & Smith 2000). Theoretical models for GC formation, which aim expressly to explain these high-mass features of GCLFs and relate them to the distributions of younger clusters and molecular clouds, have been developed by McLaughlin & Pudritz (1996) and Elmegreen & Efremov (1997).
The important difference between the mass functions of old GCs and young massive clusters, then, is not the power-law or Schechter-function form of the latter [*per se*]{}; it is the fact that the frequency of young clusters continues to rise toward the low-mass limits of observations, while the numbers of GCs fall well below any extrapolated power-law behavior for $M\la 2\times10^5\,M_\odot$, i.e., for clusters fainter than the classic peak magnitude of the GCLF. We therefore assume a Schechter function, $$\frac{dN}{dM_0} \propto M_0^{-2}\,\exp(-M_0/M_c) \ ,
\label{eq:esinit}$$ as a description of the initial mass distribution of globular clusters generally. We emphasize again that the fixed power law of $M_0^{-2}$ at low masses is chosen for compatibility with current data on systems of young massive clusters. The variable cut-off mass $M_c$ is required to match the well observed curvature present at $M\ga 10^6\,M_\odot$ in the mass distributions of old GC systems in large galaxies. This feature is certainly allowed by the young cluster data, even if it may not be explicitly required by them.
A strong possibility to explain the difference between such an initial distribution and the present-day $dN/dM$ is the preferential destruction of low-mass globular clusters by a variety of dynamical processes acting on Gyr timescales (see Fall & Zhang 2001; Vesperini 2000, 2001; and references therein).
### Evolution of the Mass Function {#sssec:esevolve}
Fall & Zhang (2001) give a particularly clear recent overview of the dynamical processes that act to destroy globular clusters on Gyr timescales as they orbit in a fixed galactic potential. The main destruction mechanisms are dynamical friction; shock-heating caused by passages through galaxy bulges and/or disks; and evaporation as a result of two-body relaxation. Only the latter two are important to the development of the low-mass end of the GCLF, since dynamical-friction timescales grow rapidly towards low $M$, as $\tau_{\rm df}\propto M^{-1}$. (Cluster disruption due to stellar-evolution mass loss does not change the shape of the GC mass function if the stellar IMF is universal, unless a primordial correlation between cluster concentration and mass is invoked; cf. Fall & Zhang 2001 and Vesperini & Zepf 2003).
Tidal shocks drive mass loss on timescales $\tau_{\rm sh}\propto \rho_h P_{\rm cr}$, where $\rho_h\propto M/R_h^3$ is the mean density of a cluster inside its half-mass radius, and $P_{\rm cr}$ is the typical time between disk or bulge crossings. Evaporation scales rather differently, roughly as $\tau_{\rm ev}\propto M/\rho_h^{1/2}$. A completely general assessment of the relative importance of the two processes can therefore be complicated. However, tidal shocks are rapidly self-limiting in most realistic situations (Gnedin, Lee, & Ostriker 1999): clusters with high enough $\rho_h$ and on orbits that expose them only to “slow” and well-separated shocks (i.e., with both the duration of individual shocks and the interval $P_{\rm cr}$ longer than an internal dynamical time, $t_{\rm dyn} \propto
\rho_h^{-1/2}$) will experience an early, sharp increase in $\rho_h$ in response to the first few shocks. Thereafter $\tau_{\rm ev}\ll \tau_{\rm sh}$, and in the long term shock-heating presents a second-order correction to the dominant mass loss caused by evaporation. Most GCs today, at least in our Galaxy, appear to be in this evaporation-driven evolutionary phase (Gnedin et al. 1999; see also Figure 1 of Fall & Zhang 2001, and Prieto & Gnedin 2006).
Fall & Zhang (2001) therefore develop a model for the evolution of the Galactic GCLF that depends largely on evaporation to erode an initially steep $dN/dM_0$ (in fact, they adopt a Schechter function, as in eq. \[\[eq:esinit\]\], for one of their fiducial cases). They assume—as is fairly standard; e.g., see Vesperini (2000, 2001)—that any cluster roughly conserves its mean half-mass density $\rho_h$ as it loses mass, at least after any rapid initial adjustments due to stellar mass loss or the first tidal shocks, and when the evolution is dominated by evaporation. The mass-loss rate is then $$\mu_{\rm ev} \equiv - dM/dt
\propto M/t_{\rm rh} \propto (M/R_h^3)^{1/2} \sim {\rm constant} \ ,
\label{eq:muev}$$ where $t_{\rm rh}\sim \rho^{-1} \langle v^2\rangle^{3/2}
\propto M^{1/2} R_h^{3/2}$ is the relaxation time at the half-mass radius $R_h$. Under this assumption, the mass of a GC at any age $t$ is just $M(t)=M_0 - \mu_{\rm ev} t$. For any collection of clusters with the same density (for example, those on similar orbits, if $\rho_h$ is set by tides at a well defined perigalacticon), $\mu_{\rm ev}$ is independent of cluster mass as well as time, and if the GCs are coeval in addition, then $\mu_{\rm ev}t$ is a strict constant. The mass function of such a cluster ensemble with any age $t$ is then related to the initial one as in equation (11) of Fall & Zhang (2001): $$\frac{dN}{dM(t)}
= \frac{dN}{dM_0}\left|\frac{\partial M_0}{\partial M}\right|
= \frac{dN}{dM_0} \ .
\label{eq:dndmtrans}$$ Thus, simply making the substitution $M_0 = M+\mu_{\rm ev}t$ in the functional form of the original GC mass function gives the evolved distribution. An initially steep $dN/dM_0$ therefore always evolves to a flat mass function, $dN/dM(t) \sim {\rm constant}$, at sufficiently low masses $M\ll
\mu_{\rm ev}t$. As Fall & Zhang show for the Milky Way, and as we shall also see for the early-type ACSVCS galaxies, this gives a good fit to observed GCLFs if the cumulative mass-loss term $\mu_{\rm ev}t > 10^5\,M_\odot$ by $t\simeq 13$ Gyr.
The key physical element of this argument, as far as the GCLF is concerned, is the linear decrease of cluster mass with time. While the quickest way to arrive at such a conclusion is to follow the logic of Fall & Zhang, as just outlined, there are some caveats to be kept in mind.
Tidal shocks can be much more important than evaporation for some globulars. In particular, clusters with low densities and low concentrations (such that shocks significantly disturb the cores as well as the halos), and/or those on eccentric orbits with very short intervals between successive bulge or disk crossings, may never recover fully from even one shock. Rather than re-adjusting quickly to a situation in which $\tau_{\rm ev}\ll \tau_{\rm sh}$, such clusters may be kept out of dynamical equilibrium for most of their lives, significantly overflowing their nominal tidal radii. Their entire evolution could then be strongly shock-driven. This appears to be the case, for example, with the well known Galactic globular, Palomar 5; see Dehnen et al. (2004). The extremely low mass and concentration of Palomar 5 make it highly unusual in comparison to the vast majority of known GCs in any galaxy currently, but many more clusters like its progenitor may well have existed in the past. This then raises the question of whether considering evaporation-dominated evolution alone gives a complete view of the dynamical re-shaping of the GC mass function. Here, however, it is important that the $N$-body simulations of Dehnen et al. (2004) show that the late-time evolution of even the most strongly shock-dominated clusters is still characterized by a closely linear decrease of mass with time: rather than $\rho_h$ being conserved in this case, the half-light radius $R_h$ is nearly constant in time, and for a given orbit the mass-loss rate is $M/\tau_{\rm sh}\propto M/\rho_h\propto R_h^3$. While the physical reasoning changes, the end result for the GCLF of clusters in this physical regime is the same as equation (\[eq:dndmtrans\]).
The importance of Fall & Zhang’s (2001) assumption of a constant $\rho_h$ for evaporation-driven cluster evolution is its implication that the mass-loss rate is constant in time. This has some direct support from N-body simulations (e.g., Vesperini & Heggie 1997; Baumgardt & Makino 2003). (Note the distinction in Baumgardt & Makino between the total cluster mass loss and that due only to evaporation; see their Figure 6 and related discussion.) But more than this, if evaporation is to be primarily responsible for the strong depletion of a GC mass function at scales $M<\mu_{\rm ev}t$, then after a Hubble time $\mu_{\rm ev}t$ must be roughly of order the current GCLF turnover mass $M_{\rm TO}\sim 2\times 10^5\,M_\odot$. Since $\mu_{\rm ev}\propto \rho_h^{1/2}$, this argument ultimately constrains the average density required of the globulars, which can of course be checked against data. In addition, satisfying observational limits on the (small) variation of GCLFs between different subsets of GCs in any one system—for example, as a function of galactocentric position—puts constraints on the allowed distribution of initial (and final) GC densities.
Fall & Zhang calculate in detail the evolution of the Milky Way GCLF over a Hubble time under the combined influence of stellar evolution (which, as mentioned above, does not change the [*shape*]{} of $dN/dM$ except in special circumstances), evaporation, and tidal shocks (which, again, contribute second-order corrections to the results of evaporation in their treatment). They relate $\mu_{\rm ev}\propto
\rho_h^{1/2}$ to GC orbits, by assuming that $\rho_h$ is set by tides at the pericenter of a cluster orbit in a logarithmic potential with a circular speed of 220 km s$^{-1}$. They then find the GC orbital distribution that allows both for an average cluster density high enough to give a good fit to the GCLF of the Milky Way as a whole, and for a narrow enough spread in $\rho_h$ to reproduce the observed weak variation in $M_{\rm TO}$ with Galactocentric radius (e.g., Harris 2001).
Ultimately, the GC distribution function found by Fall & Zhang in this way is too strongly biased towards radial orbits with small pericenters to be compatible with the observed kinematics and $\rho_h$ distributions of globulars in the Milky Way and other galaxies—as both they and others (e.g., Vesperini et al. 2003) have pointed out. However, Fall & Zhang also note that the difficulties at this level of detail do not necessarily disprove the basic idea that long-term dynamical evolution is primarily responsible for the present-day shape of the GCLF at low masses. The problem may lie instead in the specific relation adopted to link the densities, and thus the disruption rates, of GCs to their orbital pericenters. In particular, Fall & Zhang—along with almost all other studies along these lines—assume a spherical and time-independent Galactic potential. Both assumptions obviously break down in a realistic, hierarchical cosmology. Once time-variable galaxy potentials are taken properly into account in more sophisticated simulations, it could still be found that cluster disruption on Gyr timescales can both explain the low-mass side of GC mass functions and be consistent with related data on the present-day cluster orbital properties, $\rho_h$ distributions, and so on. Recent work in this vein by Prieto & Gnedin (2006) appears promising, though it is not yet decisive.
We will return to these issues in §\[ssec:evaporation\]. First, however, we describe an analytical form for $dN/dM$, which combines the main idea in Fall & Zhang (2001)—that evaporation causes cluster masses to decrease linearly with time—with a plausible, Schechter-function form for the initial $dN/dM_0$. We fit the evolved function to the GCLF of the Milky Way, to show that it provides a good approximation to the fuller, numerical models of Fall & Zhang; and then we fit it to our ACSVCS data, to produce new empirical constraints for detailed modeling of the formation and evolution of GC mass functions under conditions not specific only to our Galaxy.
### Fitting Functions for $dN/dM$ and the GCLF {#sssec:eschar}
To summarize the discussion above, we assume that the mass-loss rate of any globular cluster is constant in time. Following Fall & Zhang (2001), we expect that this will occur naturally if the disruption process most relevant to the GCLF in the long term is evaporation, which plausibly conserves the average densities $\rho_h$ of individual clusters inside their half-mass radii. Thus, we continue to denote the mass-loss rate by $\mu_{\rm ev}$. However, it should be recognized that tidal shocks can contribute second-order corrections to $\mu_{\rm ev}$ and may even, in some extreme cases, dominate evaporation (though the net result arguably could still be a constant total $dM/dt$).
For any set of clusters with similar ages $t$ and similar $\rho_h$ (and on similar orbits, if these significantly affect $\rho_h$ or add tidal-shock contributions to $\mu_{\rm ev}$), the cumulative mass loss $\Delta\equiv \mu_{\rm ev} t$ is a constant, so that each cluster has $M(t)=M_0 - \Delta$. Combining equation (\[eq:esinit\]) for the initial mass distribution with equation (\[eq:dndmtrans\]) for its evolution then yields an “evolved Schechter function” $$\frac{dN}{dM}
\propto \frac{1}{(M+\Delta)^2}\,
\exp\left(-\frac{M+\Delta}{M_c}\right) \ ,
\label{eq:esmass}$$ with $M_c$ allowed to vary between galaxies. Once again, $\Delta$ in this expression may vary between different sets of GCs, with different densities or orbits, in the same galaxy. The detailed modeling of Fall & Zhang (2001) takes this explicitly into account. But in what follows, we fit equation (\[eq:esmass\]) to GC data taken from large areas over galaxies, which effectively returns an estimate of the average mass loss per cluster over a Hubble time. Since $\mu_{\rm ev}\propto \rho_h^{1/2}$ when evaporation dominates shocks, this implicit averaging is essentially done over the distribution of GC mean half-mass densities.
To relate this evolved mass function to the standard observational definition of a GCLF—the number of GCs per unit magnitude—we write $m\equiv C-2.5\,\log\,M$, $\delta\equiv C-2.5\,\log\,\Delta$, and $m_c\equiv
C-2.5\,\log\,M_c$, where $C$ is related to the solar absolute magnitude and the typical cluster mass-to-light ratio in an appropriate bandpass. The model then reads $$\frac{dN}{dm} \propto
\frac{10^{-0.4(m-m_c)}}
{\left[10^{-0.4(m-m_c)}+10^{-0.4(\delta-m_c)}\right]^{2}} \,
\exp\left[- 10^{-0.4(m-m_c)}\right] \ .
\label{eq:esmag}$$ In both of equations (\[eq:esmass\]) and (\[eq:esmag\]), the constants of proportionality required to normalize the distributions can be evaluated numerically.
Figure \[fig:schematic\] illustrates the form of the evolved Schecter function, in terms of both the mass distribution $dN/dM$ and the GCLF $dN/dm$. (Note that mass $M$ increases to the right along the $x$-axis in the upper panel, but—as usual—larger $M$ corresponds to brighter magnitudes $m$, at the left of the axis in the lower panel.) From the equations above, it is clear that the mass $M_c$ or the magnitude $m_c$ sets the scale of the function, while the ratio $\Delta/M_c$ or the magnitude difference $(\delta-m_c)$ controls its overall shape. For very small $\Delta\ll M_c$ (faint $\delta\gg m_c$), the function approaches an unmodified Schechter (1976) function. This is drawn in Figure \[fig:schematic\] as the bold, broken curves that rise unabated toward low cluster masses or faint magnitudes. The magnitude $m_{\rm TO}$ at which the GCLF peaks in general can be found by setting to zero the derivative of equation (\[eq:esmag\]) with respect to $m$. This yields $$10^{-0.8 (m_{\rm TO}-m_c)}
+ 10^{-0.4 (m_{\rm TO}-m_c)}\left[1 + 10^{-0.4 (\delta-m_c)}\right]
- 10^{-0.4 (\delta-m_c)} = 0 \ ,
\label{eq:es_magto}$$ the solution to which corresponds to a mass of $$M_{\rm TO} =
\frac{-(M_c + \Delta) +
\sqrt{(M_c + \Delta)^2 + 4\Delta M_c}}{2} \ .
\label{eq:es_massto}$$
From either of equations (\[eq:es\_magto\]) or (\[eq:es\_massto\]), or from the sequence of curves in Figure \[fig:schematic\], it can be seen that when $\Delta\ll M_c$, the GCLF peaks at a magnitude $m_{\rm TO}\simeq \delta$, i.e., the turnover reasonably approximates the average cluster mass loss in the model (although $m_{\rm TO}$ is formally always fainter than $\delta$). As the ratio $\Delta/M_c$ increases, the GCLF turnover initially tracks $\Delta$ but eventually approaches an upper limit set by the exponential cut-off scale in the mass function: $m_{\rm TO} \rightarrow m_c$ as $(\delta-m_c) \rightarrow -\infty$ ($\Delta\gg M_c$).
For any fixed value of $\Delta/M_c$, Figure \[fig:schematic\] shows that in the limit of low masses, $M \ll \Delta$, the mass function in equation (\[eq:esmass\]) is essentially flat. As Fall & Zhang (2001) first pointed out, this is a direct consequence of the assumption of a mass-loss rate that is constant in time. It follows generically from equation (\[eq:dndmtrans\]) above, independently of the specific initial GC mass function. At the other extreme, for very high masses $M\gg \Delta$ the evolved $dN/dM$ just approaches the assumed underlying initial function with $\Delta=0$. In terms of the GCLF, this means that $dN/dm$ tends (always) to an exponential, $dN/dm \propto 10^{-0.4\, m}$, at magnitudes much fainter than the turnover; and (for initial Schechter function assumed here) to the steeper $dN/dm \propto 10^{0.4\, m} \exp\,[-10^{-0.4\, (m-m_c)}]$ for very bright magnitudes. The faint half of the GCLF in this model is therefore significantly broader than the bright half.
Finally, it is worth considering the widths of the GCLFs in the lower panel of Figure \[fig:schematic\] in more detail. For $\Delta=0$, the full width at half-maximum (FWHM) of $dN/dm$ is undefined, since there is no turnover. As the ratio $\Delta/M_c$ increases and a well-defined peak appears in the GCLF, the distribution clearly becomes narrower and narrower. As we have already discussed, even though formally $\Delta/M_c$ can increase without limit, the turnover magnitude ultimately has a maximum brightness $m_{\rm TO}\rightarrow m_c$. Similarly, the FWHM of the GCLF approaches a firm [*lower limit*]{} of ${\rm FWHM}\simeq 2.66$ mag. This includes a limiting half width at half-maximum of ${\rm HWHM}\simeq 1.59$ mag on the faint side of the GCLF, and a smaller ${\rm HWHM}\simeq 1.07$ mag on the bright side. All of these numbers can be obtained from analysis of equation (\[eq:esmag\]) by letting $(\delta-m_c)\rightarrow -\infty$, i.e., $\Delta/M_c\rightarrow + \infty$. In this limit, the GCLF approaches a fixed shape and is free only to shift left or right depending on the value of $m_c\simeq m_{\rm TO}$. This limiting shape is already essentially achieved with $\Delta/M_c=10$ or $(\delta-m_c)=-2.5$, which is plotted in Figure \[fig:schematic\] (even though the turnover is still about 0.18 mag fainter than $m_c$ in this case).
As we will see in §\[ssec:esfits\] and §\[sssec:masstrend\], the GCLFs observed in the ACSVCS are all best fit with $\Delta/M_c\ga 0.1$, or $(\delta-m_c)\la 2.5$ mag. This is the case also in the Milky Way.
Comparison with the Milky Way GCLF {#ssec:modcomp}
----------------------------------
Figure \[fig:MWGCLF\] plots the GCLF and the corresponding GC mass function in the Milky Way. The upper panel of this figure shows the GCLF $dN/dm$, in terms of clusters per unit absolute $V$ magnitude, for 143 GCs in the online catalogue of Harris (1996). (Note again that cluster luminosity and mass increase to the left in this standard magnitude distribution.) The bold, dashed line is the usual Gaussian representation (eq. \[\[eq:gauss\]\]) with parameters given by Harris (2001): $$\mu_V = -7.4 \pm 0.1\ {\rm mag} \ ; \quad \sigma_V = 1.15 \pm 0.10\ {\rm mag}
\ .
\label{eq:MWgmag}$$ The bold solid curve is our fit of the evolved Schechter function in equation (\[eq:esmag\]), with $$\delta_V = -8.0 \pm 0.3\ {\rm mag} \ ; \quad m_{c,V} = -9.3 \pm 0.3\ {\rm mag}
\ .
\label{eq:MWesmag}$$ The lighter, broken line rising steeply towards faint magnitudes is a normal Schechter function with $m_c$ as in equation (\[eq:MWesmag\]) but no mass-loss parameter, i.e., $\delta \rightarrow \infty$ in equation (\[eq:esmag\]). The shape of this curve is therefore typical of the distribution of [*logarithmic*]{} mass for young massive clusters in nearby galaxies.
The lower panel of Figure \[fig:MWGCLF\] contains a log-log representation of the Galactic GC mass function, $dN/dM$. To construct this distribution, we converted the absolute $V$ magnitude of each GC into an equivalent mass by assuming a mass-to-light ratio of $\Upsilon_V=2\ M_\odot\,L_\odot^{-1}$ for all clusters (as implied by population-synthesis models; see McLaughlin & van der Marel 2005). The curves here are the mass equivalents of those in the upper panel. Thus the bold, dashed curve traces equation (\[eq:lognorm\]) with $$\langle\log(M/M_\odot)\rangle = 5.2\pm0.04 \ ; \quad \sigma_M=0.46 \pm 0.04
\label{eq:MWgmass}$$ while the solid curve is equation (\[eq:esmass\]) with $$\log\,(\Delta/M_\odot)=5.4\pm0.1 \ ; \quad \log\,(M_c/M_\odot) = 5.9 \pm 0.1
\label{eq:MWesmass}$$ and the lighter broken curve is equation (\[eq:esmass\]) with $\log\,(M_c/M_\odot)=5.9$ and $\Delta=0$—again, representative of young cluster mass functions.
Although both model fits to the GCLF are acceptable in a statistical sense, the evolved Schechter function yields a significantly lower $\chi^2$ value. This is because of the clear asymmetry in the observed GCLF, which appears as a faintward skew in the top panel and as a failure of the mass function $dN/dM$ to decline toward low masses in the bottom panel. This behavior is described well by the evolved Schechter function but is necessarily missed by the Gaussian, which systematically underestimates the number of clusters with $M\la 3\times10^4\,M_\odot$.
As a result of this, the best-fit evolved Schechter function yields a GCLF peak which is slightly brighter than the Gaussian. From the parameters given just above and either of equations (\[eq:es\_magto\]) or (\[eq:es\_massto\]), we find a turnover magnitude of $m_{\rm TO}=-7.5\pm0.1$ in the $V$ band, some 0.1 mag brighter than the Gaussian turnover in equation (\[eq:MWgmag\]). The turnover mass implied by the evolved Schechter function is thus $M_{\rm TO}\simeq (1.75\pm0.15)\times10^5\,M_\odot$, just over 10% more massive than the Gaussian fit returns. The intrinsically symmetric Gaussian model is forced to a fainter or lower-mass turnover in order to better fit the relatively stronger low-mass tail of the observed GCLF. We find similar offsets in general between the GCLF turnovers from the two model fits to our ACSVCS data (see §\[sec:results\] and §\[sec:trends\] below).
We reiterate that the parameter $\Delta$ in the evolved Schechter function represents the average total mass loss per cluster (presumably due mostly to evaporation) that is required to transform an initial mass function like that of young clusters in the local universe, into a typical old GCLF. Both qualitatively and quantitatively, our model fits in Figure \[fig:MWGCLF\] correspond to the various similar plots in Fall & Zhang (2001). In fact, the value $\Delta\simeq
(2.5\pm0.5)\times10^5\,M_\odot$ obtained here for the Milky Way agrees well with the mass losses required by Fall & Zhang for their successful models with the second-order effects of tidal shocks included. The simple function in equation (\[eq:esmass\]) is thus a good approximation to their much fuller treatment of the GCLF.
It is also worth emphasizing just how close $\Delta$ is to the GCLF turnover mass scale. This implies that essentially [*all*]{} globulars currently found in the faint “half” of the GCLF are remnants of substantially larger initial entities. Equivalently, any clusters initially less massive than $\simeq\!2$–$3\times10^5\,M_\odot$ are inferred to have disappeared completely from the GC system.
Despite any difficulties in detail (§\[sssec:esevolve\] and §\[ssec:evaporation\]) that might remain to be resolved in this evaporation-dominated view of the GCLF, and of GC systems in general, it is important just to have at hand a fitting formula like the evolved Schechter function. In purely phenomenological terms, it fits the GCLF of the Milky Way—which is, after all, still the best defined over the largest range of cluster masses—at least as well as any other function yet tried in the literature. In particular, it captures the basic asymmetry of the distribution without sacrificing the small number of parameters and the simplicity of form that have always been the primary strengths of a Gaussian description. But at the same time, it is grounded in a detailed physical model with well specified input assumptions (Fall & Zhang 2001). Fitting it to large datasets, such as that afforded by the ACSVCS, thus offers the chance to directly, quantitatively, and economically assess the viability of these ideas, in much more general terms than has been possible to date.
Fitting Methodology and Technical Considerations {#sec:method}
================================================
Maximum-Likelihood Fitting {#ssec:ML}
--------------------------
Given either of the models just discussed—or, of course, any other—we wish to estimate a set of parameters for the intrinsic GCLF of a cluster sample using the method of maximum likelihood, following an approach similar to that of Secker & Harris (1993). To do so, we make use of all the observational material described in §\[ssec:gclfhists\].
First, we denote the set of GC magnitudes and uncertainties in any galaxy, in either the $z$ or the $g$ band, by $\{m_i,\epsilon_{m,i}\}$. Second, we write the three-dimensional completeness function discussed above as $f(m,R_h,I_b)$, which again depends not only on GC apparent magnitude but also on a cluster’s half-light radius and the background (“sky” and galaxy) light intensity at the position of the cluster. Third, from our 17 control fields we are able to estimate the luminosity function of contaminants in the field of any ACSVCS galaxy. We call this function $b(m)$, and we determine it by constructing a normal-kernel density estimate, with bandwidth chosen using cross-validation (see Silverman 1986, §§ 2.4, 3.4). Finally, this further allows us to estimate the net fractional contamination in the GC sample of each galaxy: $\widehat{\cal B}=N_C/N$, where $N_C \equiv (1/17) \sum_{i=1}^{17}N_{C,i}$ with $N_{C,i}$ the total number of contaminants present in the $i$-th customized control field, and $N$ is the total number of all GC candidates in the sample.
Now, given this observational input, we assume that an intrinsic GCLF is described by some function $G(m|\Theta)$, where $\Theta$ is the set of model parameters to be fitted. The choices for $G$ that we explore in this paper were discussed in detail in §\[sec:models\]. Thus, for example, for the Gaussian model of equation (\[eq:gauss\]), $\Theta\equiv\{\mu, \sigma_m\}$, while for the evolved Schechter function of equation (\[eq:esmag\]), $\Theta\equiv\{\delta, m_c\}$. We further assume that magnitude measurement errors are Gaussian distributed, so that—in the absence of contamination—the probability of finding an apparent magnitude $m$ for a GC with given effective radius $R_h$, galaxy background $I_b$, and magnitude uncertainty $\epsilon_m$ would be $$G_T(m|\Theta,R_h,I_b,\epsilon_m) =
{\cal A}\ [ h(m|\epsilon_m) \otimes G(m|\Theta)] f(m,R_h,I_b),$$ where $h(m|\epsilon_m)=(2\pi\epsilon_m^2)^{-1/2}\exp (-m^2/2\epsilon_m^2)$; $\otimes$ denotes convolution; and the normalization ${\cal A}$—a function of the GCLF parameters $\Theta$ and the GC properties $R_h$, $I_b$, and $\epsilon_m$—is fixed by requiring that the integral of $G_T$ over the entire magnitude range covered by the observations be unity.
If a fraction ${\cal B}$ of sources in a galaxy are contaminants, then the probability of having a bona fide GC with magnitude $m$ (and given $R_h$, etc.) is reduced to $(1-{\cal B})G_T$, and thus the likelihood that a set of GCLF model parameters $\Theta$ can account for $N$ total objects with observed magnitudes $\{m_i\}$ and properties $\{R_{h, i}, I_{b, i}, \epsilon_{m, i}\}$ is $${\cal L}(\Theta, {\cal B}) = \prod_{i=1}^N \left[
(1-{\cal B})G_T(m_i|\Theta,R_{h,i},I_{b,i},\epsilon_{m,i})
+{\cal B}b(m_i)\right]
\label{eq:lik}$$ in which it is assumed that the luminosity function $b(m)$ of contaminants is also normalized.
For any chosen functional form $G(m|\Theta)$ of the intrinsic GCLF, we specify some initial parameter values $\Theta$, compute $G_T$ and $b$ for each observed object in a galaxy, and maximize on $\Theta$ the product in equation (\[eq:lik\]). In principle, it is possible simultaneously to determine the contamination fraction in this way, but in practice we found this to be a rather unstable procedure (even small inadequacies in the chosen model for $G$ can lead to a maximum-likelihood solution that converges to quite unreasonable values for ${\cal B}$). Thus, we instead made direct use of our prior information from the 17 control fields, and fixed this fraction to the rather precise average $\widehat{\cal B}$ that we have measured for each galaxy.
The uncertainties in the fitted parameters $\Theta$ are estimated by using the covariance matrix calculated at the point of maximum likelihood (e.g., Lupton 1993). These uncertainties include the effects of possible correlation between the parameters, but they do not include the additional, unavoidable uncertainty arising from cosmic variance in the form of $b(m)$ and the expected number $\widehat{\cal B}$ of contaminants in any field. As such, they constitute lower limits to the total uncertainty. This is not a significant issue for GCLF fits to cluster samples combined from several galaxies (see below), but it can be important for fits to individual galaxies.
To deal with this, when we fit any individual GC system, we re-run our maximum-likelihood algorithm 17 times, each time using the background contamination fraction ${\cal B}$ as estimated from a different one of our 17 control fields (versus using $\widehat{\cal B}$ from an average of all control fields to obtain the nominal best fit). We record the different sets of best-fit GCLF parameters obtained in these trials and use the variance in them to evaluate the additional uncertainty arising from cosmic variance of the background contamination.
Bias Tests {#ssec:bias}
----------
Maximum-likelihood estimators are biased in general. It is thus important when deriving conclusions to test the bias properties of the estimator used, under circumstances similar to the ones under study. We have done this specifically for the benchmark case of Gaussian fits to the GCLF. After obtaining mean magnitudes and dispersions from our maximum-likelihood routine for the 89 ACSVCS galaxies, we analyzed 20 simulated datasets per galaxy, using the following procedure. First, we subtracted the number of contaminants $\widehat{\cal B}N$ in the galaxy from the total number $N$ of GC candidates there, to estimate the expected population $N_{\rm GC}$ of bona fide GCs. We then randomly drew a sample of $N_{\rm GC}$ magnitudes from a Gaussian distribution with a mean $\mu$ taken to be the fitted maximum-likelihood estimate for that galaxy, and a dispersion chosen from $\sigma_m=0.4,0.7,1$ or 1.3 mag. (We did 5 simulations for each of these dispersions, giving the total of 20 simulations per galaxy.) The randomly generated objects replaced the $N_{\rm GC}$ objects in that galaxy’s sample with the highest $p_{GC}$ values. The values of $R_h$ and $I_h$ of the latter objects plus the simulated magnitude are used to determine the completeness value $f$ for each source. A uniform random deviate is then computed and if that is larger than the value of $f$ the source is discarded, a new magnitude drawn from the Gaussian and the process repeated until the condition is met. In this way the effects of completeness are taken into account. The maximum-likelihood procedure was finally run on each simulated sample and the output parameters compared to the input ones.
The results of these simulations in the $z$ band are summarized in Figure \[fig:MLbias\]. There may be slight biases in the recovered parameters, with $\langle \Delta\sigma_m/\sigma_m \rangle \approx\! -0.03$ and $\langle \Delta\mu/\sigma_m \rangle \approx\! -0.03$, although there are no significant trends in these average offsets with galaxy luminosity (i.e., sample size). Moreover, the statistical significance of these biases is not high ($< 3 \sigma$), and so we choose not to correct for them. As a result, it is possible that our output best-fit parameters are biased at the level of 3% of the GCLF dispersion; but with the possible exception only of the most populous GC system (that of M87=VCC 1316), this turns out always to be smaller than the formal uncertainties on the GCLF parameters (see §\[ssec:gaussian\]). Note that the [*scatter*]{} of the retrieved parameters compared with the input ones increases towards fainter galaxy magnitudes because the candidate-GC sample size is decreasing, and the variance in the estimates of both $\sigma$ and $\mu$ scales as $\sim\! 1/N$.
Effects of Selection Procedure {#ssec:selec}
------------------------------
As we mentioned in §\[sec:obs\], the procedure we used to construct a sample of GC candidates for each galaxy involved assigning a probability $p_{\rm GC}$ to each source and allowing into the sample only those objects with $p_{\rm GC}\ge 0.5$. This may influence the resulting observed luminosity function and consequently affect the derived parameters of any fitted model. In order to check that we do not unduly bias our GCLF fitting by this selection technique, we also constructed alternate candidate-GC samples that do not use the selection on $p_{\rm GC}$ but only apply a magnitude cut and an upper limit of $R_h\la 5$ pc (cf. the second half of Table \[tab:gclf\_hists\] in §\[ssec:gclfhists\]). The magnitude distributions of such samples are free of any selection effects arising from using the $p_{\rm GC}$ values and are useful for testing the robustness of any result. Thus, when we fit GCLFs to any of our data, we have verified that consistent conclusions are obtained using either of our sample definitions.
Binned Samples {#ssec:binned}
--------------
While we always perform GCLF fits to individual galaxies, some of the fainter systems suffer from small-number statistics and/or excessive contamination. We thus constructed still more GC samples by combining all candidate clusters from as many galaxies as required to reach a total sample size above some minimum. Going down the list of our target galaxies sorted by apparent $B$ magnitude, we accumulate galaxies until the expected number of bona fide GCs (i.e., the total number of candidates minus the number of contaminants estimated from our customized control fields) is $\ga 200$. Although many of the brighter galaxies satisfy this condition by themselves, we refer to the samples defined in this way as “binned” samples. There are 24 of them in all, and they are used in §\[sec:trends\] particularly, to assess trends in GCLF parameters as a function of galaxy luminosity without the significant scatter caused by the small numbers of GCs in faint systems.
Our SBF analysis has shown that some of the ACSVCS galaxies have distance moduli significantly different from the mean $(m-M)_0=31.09$ mag for Virgo (Mei et al. 2007), and thus simply combining the apparent magnitudes of GCs from different galaxies with no correction could artificially inflate the dispersion of any composite GCLF. To avoid this, we do the binning by first using the SBF distances to transform all candidate GC luminosities to the value they would have at a distance of 31.1 mag ($D=16.5$ Mpc).
Model Fits {#sec:results}
==========
In this section we present the results of our maximum-likelihood fitting of Gaussians and evolved Schechter functions to the GCLFs in the Virgo Cluster Survey. Recall that any alternative model may be fit to the GCLF histograms in Table \[tab:gclf\_hists\], which can be downloaded from the electronic edition of the [*Astrophysical Journal*]{}.
Gaussian Fits {#ssec:gaussian}
-------------
The parameter estimates for an intrinsic Gaussian fitted to our 89 individual GCLFs are given in Table \[tab:gclfpars\]. There we list each galaxy’s ID number in the VCC and its total apparent magnitude $B_{\rm gal}$, both taken from Binggeli, Sandage & Tammann (1985). Following this are the maximum-likelihood values of the mean GC magnitude and dispersion and their uncertainties in the $g$-band ($\mu_g, \sigma_g$), the same quantities in the $z$-band ($\mu_z, \sigma_z$), the fraction $\widehat{\cal B}$ of the sample that is expected to be contamination, and the total number $N$ of all objects (including contaminants and uncorrected for incompleteness) in the galaxy’s candidate-GC sample. The last column of Table \[tab:gclfpars\] gives comments on a few galaxies with noteworthy aspects. Note that the uncertainties in the Gaussian parameters include contributions from cosmic variance in the shape and normalization of the contamination luminosity function $b(m)$ (see §\[ssec:ML\]).
In Figure \[fig:gaussfits\] we present histograms of the observed GCLFs along with the best fitting maximum-likelihood models. The galaxies are arranged in order of decreasing apparent $B_{\rm gal}$ magnitude (i.e., the same order as in Table \[tab:gclfpars\]), and there are two panels per galaxy: one presenting the $z$-band data and model fits, and one for the $g$ band. The bin width chosen for display purposes here is not the same for all galaxies, but follows the rule $h=2 (IQR) N^{-1/3}$, where $(IQR)$ is the interquartile range of the magnitude distribution and $N$ is the total number of objects in each GC sample (Izenman 1991).
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Fig. 4. — [*Continued*]{}
There are four curves drawn in every panel of Figure \[fig:gaussfits\]. The long-dashed curve is the best-fit intrinsic Gaussian GCLF, given by equation (\[eq:gauss\]) with the parameters listed in Table \[tab:gclfpars\]. The dotted curve is this intrinsic model multiplied by the completeness function, $f(m, R_h, I_b)$, after marginalizing the latter over the distribution of $R_h$ and $I_b$ for the observed sources in each galaxy. The solid gray curve is our kernel-density estimate of the expected contaminant luminosity function. Finally, the solid black curve is the sum of the solid gray and dotted curves; it is the net distribution for which the likelihood in equation (\[eq:lik\]) above is maximized.
Since we have two realizations of the GCLF for every galaxy—one in the $z$ band and one in the $g$ band—we are able to check the internal consistency of our model parameter estimates. Thus, in Figure \[fig:consistency\] we compare the measured Gaussian means and dispersions in the two bands. The left-hand panel of this plot shows the scatter of $\sigma_z$ vs. $\sigma_g$ about a line of equality, while the right-hand panel shows the difference in fitted means $(\mu_g-\mu_z)$ vs. the average GC $(g-z)$ color in each galaxy (from Peng et al. 2006a), again compared to a line of equality. Both cases show excellent agreement between the maximum-likelihood results for the two bandpasses. We conclude that the measurements are internally consistent and that our uncertainty estimates are reasonable.
Finally, we also fit Gaussians to our 24 “binned” GC samples, constructed by combining the candidates in as many galaxies as necessary to reach net sample sizes of at least 200 (see §\[ssec:binned\]). The IDs and total magnitudes of the galaxies going into each of these bins are summarized in Table \[tab:galbins\], along with the best-fit $z$- and $g$-band Gaussian parameters for each binned GCLF and the best-fit parameters for the evolved Schechter function discussed in §\[sec:models\] (see just below). In Figure \[fig:gaussbin\] we display the binned GCLFs in histogram format, along with a number of curves representing the maximum-likelihood Gaussian fits. The curves in every panel have exactly the same meaning as in the individual GCLF fits of Figure \[fig:gaussfits\]. We additionally show in this Figure (as the crosses in each magnitude bin of each histogram) alternative GCLFs for the binned-galaxy samples, obtained by defining GCs on the basis of absolute magnitude and an upper limit on the half-light radius $R_h$ (§\[ssec:selec\]).
[ ]{}\
Fig. 6. — [*Continued*]{}
\
Fig. 7. — [*Continued*]{}
In §\[sec:trends\] below, we will compare GCLF systematics as a function of galaxy properties for these binned samples vs. the fits to individual galaxies. We also note here, without showing further details, that repeating the exercises of this Section using the samples of GC candidates selected only by magnitude and $R_h$, rather than by a $p_{\rm GC}$ criterion, leads to results that are consistent in all ways with those we present below.
Fits of Evolved Schechter Functions {#ssec:esfits}
-----------------------------------
We have performed fits of the evolved Schechter function in equation (\[eq:esmag\])—or equivalently, the more transparent equation (\[eq:esmass\])—to the GCLFs of our individual galaxies and binned samples. Here we discuss only the results of fitting the 24 binned GC samples, as the results from fitting to all 89 galaxies separately lead to similar conclusions.
In all these fits, we enforced the constraint that the fitted (average) mass loss $\Delta$ be less than ten times the exponential cut-off mass scale $M_c$: $\Delta/M_c < 10$, or $(\delta-m_c)>-2.5$ in magnitude terms. This was done because, as was discussed in §\[ssec:esmod\] (see Figure \[fig:schematic\]), for such large ratios of $\Delta$ to $M_c$ the evolved Schechter function has essentially attained a universal limiting shape. The likelihood surface then becomes very flat for any greater $\Delta/M_c$, and the fitting procedure has difficulty converging if this parameter is allowed to vary to arbitrarily high values. The majority of our evolved Schechter function fits do converge to $\Delta/M_c$ values that satisfy our imposed constraint; in only one case does the “best-fit” model have the limiting $\Delta/M_c=10$.
We show in Figure \[fig:ESbin\] the binned-sample GCLF histograms, along with model curves analogous to those in Figure \[fig:gaussbin\]. Again, then, the intrinsic evolved-Schechter model GCLF is the long-dashed curve; this model multiplied by the marginalized completeness function is the dotted curve; a kernel-density estimate of the contaminant luminosty function is shown as the solid gray curve; and the net best-fitting model (sum of dotted and solid gray curves) is drawn as a solid black curve. Also as in Figure \[fig:gaussbin\], we use crosses in Figure \[fig:ESbin\] to show the GCLFs inferred in every galaxy bin when we define GC samples by simple magnitude cuts and $R_h$ limits, rather than by using our $p_{\rm GC}$ probabilities.
Comparing Figure \[fig:ESbin\] with Figure \[fig:gaussbin\], it is apparent that an evolved Schechter function describes the GCLFs of bright galaxies about as well as a Gaussian does. In some of the fainter galaxies there is possibly a tendency for the Schechter function to overestimate the relative number of faint GCs, but it is difficult to assess how serious this might be. The worst disagreements between the model fits and the data tend to occur in the very faintest extents of the histograms for the handful of the faintest galaxy bins at the end of Figure \[fig:ESbin\]. Indeed, the largest discrepancies appear at magnitudes where contaminants account for $\ga50\%$ of the total observed population. Any impression of success or failure for [*any*]{} model in these extreme regimes of the GCLFs must be tempered by the realization that the fitting itself is something of a challenge under such conditions.
This is further illustrated by contrasting, in both Figures \[fig:ESbin\] and \[fig:gaussbin\], the GCLFs for cluster samples selected by magnitude and $R_h$ only (crosses in the figures), to those for samples selected on the basis of $p_{\rm GC}$ probabilities (bars). The former samples generally tend to put more objects in the faintest GCLF bins, an effect particularly apparent in the faintest galaxies. The low-mass end of the GCLF for faint galaxies is thus not tightly constrained by our observations; there is a fundamental uncertainty, due to contamination, that cannot be overcome by any selection procedure. (Note that some of the more extreme discrepancies between the different GCLF definitions—such as in the faintest magnitude bin of BG 20—are due to the presence in some galaxies of a strong excess of diffuse clusters that are classified as contaminants when using $p_{\rm GC}$ to construct the sample; see Peng et al. 2006b). But it is still worth recalling, in this context, that the “overabundance” of low-mass clusters in the evolved Schechter function, vs. a Gaussian, is in fact a demonstrably better description of the Milky Way GCLF; see Figure \[fig:MWGCLF\].
The fitted magnitude-equivalents $\delta$ and $m_c$ of the mass scales $\Delta$ and $M_c$, in each of the $z$ and $g$ bands, are recorded for each of our binned GCLFs in Table \[tab:galbins\]. In §\[sec:trends\] we discuss in detail the conversion of these to masses and also consider dependences of $\Delta$ and $M_c$ on galaxy luminosity.
Just before looking at these issues, Figure \[fig:gaussVSes\] compares the turnover magnitudes and full-widths at half maximum (FWHM) for the binned $z$-band GCLFs as returned by the fits of evolved Schechter functions (see eq. \[\[eq:es\_magto\]\]), against the same quantities implied by our Gaussian fits. For the turnovers, there is a slight offset, in that the fitted Schechter functions tend to peak at slightly brighter magnitudes (typical difference $\la\! 0.1$ mag, corresponding to a turnover mass scale that is $<\! 10\%$ larger than implied by the Gaussian fits). This is very similar to the offset in the two fitted turnover magnitudes for the Milky Way GCLF in §\[ssec:modcomp\]. As we discussed there, the discrepancy is a result of the intrinsic symmetry assumed in the Gaussian model, vs. the faint-end asymmetry built into the evolved Schechter function.
The FWHMs differ more substantially between the two functional forms, with the evolved-Schechter fits being typically $\simeq\! 0.5$ mag broader (or about 0.2 dex in terms of mass) than the Gaussian fits. But this is again only to be expected from the asymmetry of the former function vs. the symmetry of the Gaussian. As was noted at the end of §\[ssec:esmod\], the shape of the evolved Schechter function is universally flat in terms of $dN/dM$ for low GC masses, or universally $\propto\! 10^{-0.4 m}$ in terms of $dN/dm$ for magnitudes much fainter than the peak of the GCLF. As a result, the faint side of the GCLF is always broader than any Gaussian, and so if the two models give comparable descriptions of the bright halves of all GCLFs, the FWHM of the evolved Schechter functions must always be larger than those of the Gaussian fits. Moreover, for very narrow observed GCLFs, fit by small Gaussian $\sigma_m$ (primarily to reproduce the steepness of the bright side of the GCLF, as discussed below), the evolved Schechter function fits are limited by a minimum FWHM of $\simeq\! 2.66$ mag (§\[ssec:esmod\]), explaining the tendency towards a plateau at the left side of the lower panel of Figure \[fig:gaussVSes\].
Trends Between and Within Galaxies {#sec:trends}
==================================
Having fitted two different GCLF models to each of our individual galaxies and binned samples, we now outline some systematic variations in the properties of GC mass distributions indicated by this work. First, we examine the dependence of GCLF parameters on host galaxy luminosity; then—even though the ACSVCS data are not ideal for this purpose—we look for any evidence of GCLF trends with radius inside the two brightest Virgo galaxies, M49 (VCC 1226) and M87 (VCC 1316).
Variations with Galaxy Luminosity {#ssec:galmb}
---------------------------------
### Gaussian Parameters {#sssec:gausstrend}
Figure \[fig:sigma\_B\] shows one of the main results of this paper: GCLFs are narrower in lower-luminosity galaxies (see also Jordán et al. 2006).
The upper panel of this figure plots the Gaussian dispersion that best fits the $z$-band GCLF, as a function of absolute galaxy magnitude $M_{B,{\rm gal}}$ for our 89 individual galaxies. Filled circles represent galaxies with measured (SBF) distance moduli, while open triangles correspond to galaxies for which no distance modulus is available and for which we assume $(m-M)_0=31.1$ (consistent with the average Virgo distance modulus of Mei et al. 2007) to compute $M_{B,{\rm gal}}$. The lower panel shows the analogous result for our $g$-band GC data. The straight lines drawn in the panels are convenient linear characterizations of the $\sigma_m$–$M_{B,{\rm gal}}$ trends: $$\sigma_z = (1.12\pm0.01) - (0.093 \pm 0.006)(M_{B, {\rm gal}}+20)
\label{eq:sigmaz}$$ and $$\sigma_g = (1.14\pm0.01) - (0.100 \pm 0.007)(M_{B, {\rm gal}}+20)
.
\label{eq:sigmag}$$
While it has been reported before that there is a tendency for the GCLFs in lower luminosity galaxies to show somewhat lower dispersions (e.g., Kundu & Whitmore 2001a), the homogeneity of our sample and analysis make this the most convincing demonstration to date of the existence of a continous trend over a factor of $\approx\! 400$ in galaxy luminosity. It is particularly noteworthy that the fainter galaxies in our sample—all of which are early type—have very modest $\sigma_m \la 1$, values more usually associated with the GCLFs of late-type galaxies. In fact we have also plotted on Figure \[fig:sigma\_B\] the $V$-band GCLF dispersions (Harris 2001) and absolute bulge luminosities of the Milky Way (large filled star at $M_{B, {\rm gal}}=-18.8$; de Vaucouleurs & Pence 1978) and M31 (large filled triangle at $M_{B, {\rm gal}}=-19.2$; from Kent 1989, but assuming a distance of 810 kpc). Clearly these fall well in the midst of our new data, and thus the correlation of $\sigma$ with $M_{B,{\rm gal}}$ would appear to be more fundamental than the older view, that GCLF dispersions depend on galaxy Hubble type (Harris 1991).
At this point it should be noted that the GCs in brighter galaxies are known to have broader color distributions, and hence larger dispersions in metallicity, than those in fainter galaxies (e.g., Peng et al. 2006a). But cluster mass-to-light ratios, $\Upsilon$, are functions of \[Fe/H\] in general, so there will be some galaxy-dependent spread in their values. Since the variance in an observed luminosity distribution is related to that in the mass distribution, by the usual $\sigma^2(\log\,L)=\sigma^2(\log\,M) + \sigma^2(\log\,\Upsilon)$, this then suggests the possibility that the trend we see in the GCLF $\sigma_z$ and $\sigma_g$ vs. galaxy luminosity might result from systematics in $\sigma(\log\,\Upsilon)$ vs. $M_{B,{\rm gal}}$ on top of a more nearly constant $\sigma(\log\,M)$. In fact, this idea was recently invoked by Waters et al. (2006) as a potential explanation for the fact that the $I$-band GCLF of M87 is broader than that of the Milky Way; and by Strader et al. (2006) as a possible reason for the narrower composite GCLF of a subsample of ACSVCS dwarfs versus the GCLFs of Virgo giants. However, neither of those works checked these claims quantitatively. We have done so here (see also Jordán et al. 2006), and we find that the explanation is not tenable.
As we will discuss further in §\[sssec:masstrend\], GC mass-to-light ratios in the longer-wavelength $z$ band vary by less than $\pm10\%$ over the entire range $-2\le {\rm [Fe/H]}\le 0$, which includes the large majority of clusters. Thus $\sigma(\log\,\Upsilon_z) < 0.04$ no matter what the details of the GC metallicity distribution are—making for an utterly negligible “correction” to the observed $\sigma(\log\,L_z)=\sigma_z/2.5$ for all of our GCLFs. In the shorter-wavelength $g$ band, mass-to-light ratios are more sensitive to cluster colors. But here the close agreement of our $g$- and $z$-band GCLF dispersions shows immediately that the former must be reflecting the properties of the GC mass functions just as closely as the latter are. Indeed, more detailed calculations, which include the observed specifics of the color distributions in our galaxies (Peng et al. 2006a), confirm that the spread in expected GC $\Upsilon_g$ values contributes $\sim\! 0.02$ mag to the total observed GCLF dispersion—an amount well within the observational uncertainties on $\sigma_g$ in the first place[^2]. Thus, we proceed knowing that the correlations between GCLF dispersion and galaxy luminosity that we are discussing here are very accurate reflections of equivalent trends in the more fundamental GC mass distributions.
Because of the symmetry assumed in the model, the trend of decreasing Gaussian $\sigma_m$ in Figure \[fig:sigma\_B\] might appear to imply a steepening of the GCLF on both sides of the turnover mass. However, as we have already discussed, if we take the more physically based, evolved-Schechter function of equation (\[eq:esmag\]) or (\[eq:esmass\]) to describe the distribution of GC masses, then [*all*]{} GCLFs must have the same basic shape (and thus half-width) for clusters fainter than about the turnover magnitude—in which case the trends in Figure \[fig:sigma\_B\] can only be driven by systematics in the bright side of the GCLF. Indeed, as was mentioned in §\[sec:models\] above (and discussed at length by, e.g., McLaughlin & Pudritz 1996), it has long been clear that power-law representations of the GC mass function above the turnover mass in the Milky Way and M31 are significantly steeper than those in M87, M49, and other bright ellipticals; there is no “universal” power-law slope for present-day GC mass functions.
Given these points, we have also performed maximum-likelihood fits of pure power-law mass distributions ($dN/dM \propto M^{-\beta}$; or, in terms of magnitude, $dN/dm \propto 10^{0.4 (\beta-1) m}$) to GCs between $\simeq\! 0.5$–2.5 mag brighter than the turnover magnitude in the cluster samples of our individual galaxies. (Such subsamples are both highly complete and essentially uncontaminated in all of our galaxies). The best-fit $\beta$ for the 66 galaxies in which we were able to measure it are presented in Table \[tab:beta\]. The results from fitting to the $z$- and $g$-band data are similar, and thus we show only the former here, in the upper panel of Figure \[fig:power\]. This confirms that the high-mass end of the GCLF steepens systematically for decreasing galaxy luminosity, independently of how the low-mass GC distribution behaves. In Figure \[fig:power\] we also plot a star and triangle showing $\beta$ values for the Milky Way and M31 respectively, measured in the same mass regime using the data from Harris (1996) and Reed et al. (1994) assuming a $V$-band mass-to-light ratio $M/L_V = 2$. The lower panel of Figure \[fig:power\] then plots the fitted power-law exponent for high GC masses against the Gaussian GCLF dispersion from Figure \[fig:sigma\_B\], showing that there is indeed a clear correlation between these two parameters in the sense that a narrower Gaussian $\sigma$ reflects a steeper high-mass power-law $\beta$.
The regularity and the high significance of the narrowing of the GCLF as a function of galaxy luminosity—or the steepening of the mass distribution above the classic turnover point—places a new and stringent constraint on theories of the formation and evolution of the mass function of GCs. In one sense, this is then on a par with the modest amount of variation seen in the turnover mass. An important difference may be that the GCLF turnover could be imprinted to some large extent by long-term dynamical evolution (Fall & Zhang 2001; though see, e.g., Vesperini 2000, 2001, and Vesperini & Zepf 2003 for a differing view, and §\[ssec:evaporation\] below for a discussion of caveats). By contrast, most analyses agree that the shape of $dN/dm$ above the turnover is largely resistant to change by dynamical processes (§\[ssec:dynfric\])—in which case it seems most likely that the systematic variations in Figures \[fig:sigma\_B\] and \[fig:power\] are reflecting a fundamental tendency to form massive star clusters in greater [*relative*]{} numbers in more massive galaxies.
Moving now to the GCLF turnover magnitude, in Figure \[fig:mu\_B\] we show the absolute $\mu_z$ and $\mu_g$ as functions of host galaxy absolute magnitude $M_{B,{\rm gal}}$. In both panels of this figure, horizontal lines are drawn at the levels of the typical turnovers in large ellipticals: excluding VCC 798, which has an anomalously large excess of faint, diffuse star clusters (Peng et al. 2006b), the average Gaussian turnovers for ACSVCS galaxies with $M_{B,{\rm gal}}<-18$ are $$\begin{array}{ccl}
\langle \mu_z \rangle & = & -8.4 \pm 0.2 \\
\langle \mu_g \rangle & = & -7.2 \pm 0.2 \ .
\label{eq:brightmu}
\end{array}
\qquad (M_{B,{\rm gal}} < -18)$$ The turnover in the Milky Way is shown as a large filled star and that in M31 is represented by a large filled triangle, as in Figure \[fig:sigma\_B\]. We estimated these turnovers from the $V$-band values given in Table 13 of Harris (2001), by applying $(g-V)$ and $(V-z)$ colors calculated for 13-Gyr old clusters with ${\rm [Fe/H]}=-1.4$ for the Milky Way (Harris 2001) and ${\rm [Fe/H]}=-1.2$ for M31 (Barmby et al. 2000) using the PEGASE population-synthesis model (Fioc & Rocca-Volmerange 1997).
The $z$-band turnovers in the upper panel of Figure \[fig:mu\_B\] show a tendency to scatter systematically above (fainter than) the bright-galaxy value for systems with $M_{B,{\rm gal}}\ga -18$, but there are no such systematics in the $g$-band turnovers in the lower panel. Interpreting these results is most easily done in terms of equivalent turnover [*mass*]{} scales, and thus we defer further discussion to §\[sssec:masstrend\], where we use the PEGASE model to convert all of our GCLF parameters to their mass equivalents. We note here, however, that the near constancy of $\mu_g$ in Figure \[fig:mu\_B\] is equivalent to the well known “universality” of the GCLF turnover in the more commonly used $V$ band (since our $g$ is the HST F475W filter, which is close to standard $V$).
Before discussing masses in detail, we plot in Figure \[fig:binVSind\] the Gaussian means and dispersions of the $z$-band GCLFs in our 24 binned samples, vs. the average absolute magnitude of the galaxies in each bin (see Table \[tab:galbins\]). The straight lines in each panel are just those from the upper panels of Figs. \[fig:mu\_B\] and \[fig:sigma\_B\], characterizing the fits to all 89 individual galaxies. This comparison shows that the results from our single- and binned-galaxy GC samples are completely consistent, so that our binning process has served—as intended—to decrease the scatter in the observed behavior of $\mu$ and $\sigma$ at low galaxy luminosities. It also confirms the results of our simulations in §\[ssec:bias\] above, which showed that our maximum-likelihood model fitting is not significantly biased by size-of-sample effects. A plot like Figure \[fig:binVSind\], but using our Gaussian fits to the individual and binned $g$-band GCLFs, leads to the same conclusions.
### Mass Scales {#sssec:masstrend}
To better understand the GCLF trends discussed above, and to mesh the Gaussian-based results with those from fits of the more physically motivated evolved-Schechter function, it is advantageous to work in terms of GC mass, rather than $z$ and $g$ magnitudes. To make this switch, we rely on population-synthesis model calculations of $(g-z)$ colors and $g$- and $z$-band mass-to-light ratios as functions of metallicity for “simple” (single-burst) stellar populations.
The model we use is version 2.0 of the PEGASE code (Fioc & Rocca-Volmerange 1997), which we have run by inputting the stellar initial mass function of Kennicutt (1983) to compute cluster masses and $g$ and $z$ luminosities as functions of age for several fixed values of \[Fe/H\]. The results, at an assumed uniform GC age of 13 Gyr, are illustrated in Figure \[fig:upsilon\], which plots the mass-to-light ratios $\Upsilon_g$ and $\Upsilon_z$ in solar units and the $(g-z)$ color against \[Fe/H\]. Given the average $(g-z)$ of the GCs in any of our galaxies (from Peng et al. 2006a), we interpolate on these PEGASE model curves to estimate average $g$ and $z$ mass-to-light ratios. Table \[tab:Ups\] lists the mean GC color in each galaxy and the $M/L$ values we have derived.
It is clear from Figure \[fig:upsilon\] and Table \[tab:Ups\] that the $z$-band mass-to-light ratio varies by only a modest amount for most GCs in our samples: we generally have $0.8\la \langle (g-z) \rangle \la 1.2$ in these cluster systems, and thus $1.45\la \Upsilon_z\la 1.55\ M_\odot\,L_\odot^{-1}$. A $z$-band luminosity is therefore a very good proxy for total cluster mass. By contrast, over the same range of GC color or metallicity, the $g$-band mass-to-light ratio increases monotonically from $\Upsilon_g\simeq1.9\ M_\odot\,L_\odot^{-1}$ to $\Upsilon_g\simeq2.7\ M_\odot\,L_\odot^{-1}$. Note that if any of our GCs were much younger than 13 Gyr, then the numerical values of these mass-to-light ratios would all be lower (by $\sim\! 30\%$–40% at an age of 8 Gyr, for example), but the basic constancy of $\Upsilon_z$ and the systematic increase of $\Upsilon_g$ for redder/more metal-rich GC systems would remain.
This has immediate implications for our plots of the GCLF turnover magnitudes in Figures \[fig:mu\_B\] and \[fig:binVSind\] above. In particular, the GCs are systematically bluer, on average, in lower-luminosity galaxies (e.g., Peng et al. 2006a; see also Table \[tab:Ups\]). Assuming that this reflects a correlation between average cluster metallicity and galaxy luminosity (rather than one between cluster age and $M_{B,{\rm gal}}$), the typical $\Upsilon_g$ must be somewhat lower for GCs in faint galaxies than in bright galaxies, while $\Upsilon_z$ is essentially the same. The fact that the Gaussian GC $\mu_z$ scatters slightly faintward towards fainter $M_{B,{\rm gal}}$ should then reflect a modest downward scatter in the turnover [*mass*]{} scale. But in the $g$ band, this would be balanced to at least some extent by the decrease in mass-to-light ratio, and $\mu_g$ should stay more steady as a function of $M_{B,{\rm gal}}$.
This interpretation of the situation is confirmed in Figure \[fig:MTO\_GZ\], where in the upper panel we plot the Gaussian turnover masses, derived from the $z$- and $g$-band GCLF fits as just described, vs. parent galaxy absolute magnitude. The average turnover magnitudes in equation (\[eq:brightmu\]) and the typical GC mass-to-light ratios in Table \[tab:Ups\] together imply an average turnover mass of $$\langle M_{\rm TO} \rangle = (2.2 \pm 0.4) \times 10^5\ M_\odot
\label{eq:brightmTO}$$ for the brightest ACSVCS galaxies with $M_{B,{\rm gal}} < -18$ (here we have taken the absolute magnitude of the sun to be 4.51 in the $z$-band and 5.10 in $g$). The consistency in most systems between the turnover masses estimated from the two bandpasses shows that, indeed, for $M_{B,{\rm gal}}\ga -18$, there is an overall tendency to find more GC systems with turnover masses somewhat below the average for giant ellipticals, by as much as a factor of 2 in some cases. It also implies that the dependence of GC $\langle (g-z) \rangle$ on galaxy luminosity does primarily reflect metallicity variations, since if GCs had very similar metallicities but much younger ages in fainter galaxies, the $z$- and $g$-band estimates of $M_{\rm TO}$ would differ by as much as the fitted turnover magnitudes in §\[sssec:gausstrend\].
For completeness, in the lower panel of Figure \[fig:MTO\_GZ\], we show the $g$- and $z$-band based estimates of the Gaussian dispersion of logarithmic GC masses. Since $\sigma_M$ does not depend on the cluster mass-to-light ratio, but is just the magnitude dispersion divided by 2.5, this plot is completely equivalent to Figure \[fig:sigma\_B\]. Thus we have also drawn in equation (\[eq:sigmaz\]) above, multiplied by 0.4.
An interesting corollary to all of this is that the reliability of the GCLF as a distance indicator would appear to be somewhat bandpass-dependent, at least when applied to sub-$L_*$ galaxies with $M_{B,{\rm gal}}\ga -19$. We have just argued that the near-universality of the turnover magnitude in the $g$-band—and thus in the very closely related $V$ band—is at some level the fortuitive consequence of quantitatively similar decreases in both the turnover mass and the typical GC mass-to-light ratio in smaller galaxies. At longer wavelengths, however, mass-to-light ratios are not so sensitive to GC metallicity, and variations in the turnover mass carry over more directly into variations in turnover magnitude. We will explore this issue in more detail in future work. However, any such pragmatic concerns about the precision of the GCLF peak magnitude as a standard candle should not detract from the main point of physical interest here: although the differences in GCLF turnover mass that we find are real, they are nevertheless relatively modest. While the galaxies in the ACSVCS range over a factor of $\simeq\!400$ in luminosity, $M_{\rm TO}$ never falls more than $\simeq\! 30\%$–40% away from the (Gaussian) average of $2.2\times10^5\,M_\odot$ for the giant ellipticals.
In the left panel of Figure \[fig:mto\_bd\] we show the turnover masses derived from the Gaussian GCLF means for our binned-galaxy GC samples. This again highlights the tendency to slightly less massive GCLF peaks, on average, in lower-luminosity galaxies. In the right panel of this figure we also show $M_{\rm TO}$ as derived from our fits of an evolved Schechter function to the same GCLFs (see eqs. \[\[eq:es\_magto\]\] and \[\[eq:es\_massto\]\]). The close similarity of the two graphs in Figure \[fig:mto\_bd\] is entirely in keeping with the slight average offset between the Gaussian and extended-Schechter turnover magnitudes in Figure \[fig:gaussVSes\] above. It also illustrates that our main results are not overly dependent on the particular choice of model to fit the GCLFs.
Last, in Figure \[fig:MB\_ES\] we show the GC mass scales $M_c$ (the high-mass exponential cut-off) and $\Delta$ (interpreted as the average mass lost per GC by evaporation) for our fits of evolved Schechter functions to the binned-galaxy GCLFs, as inferred from their magnitude equivalents $m_c$ and $\delta$ in Table \[tab:galbins\]. The upper panel of the figure first plots $M_c$ vs. $M_{B,{\rm gal}}$, using solid points to represent fits to GC samples selected on the basis of our probabilities $p_{\rm GC}$ and open symbols for fits to samples defined only by cuts on magnitude and GC effective radius (see §\[ssec:selec\]). There is a clear, systematic decrease of $M_c$ with decreasing galaxy luminosity. In terms of the structure of the mass function (eq. \[\[eq:esmass\]\]), this corresponds to a steeper fall-off in the frequency of GCs more massive than the turnover point. It is therefore equivalent to our findings in Figures \[fig:sigma\_B\] and \[fig:power\] that the Gaussian $\sigma$ is narrower, and the high-mass power law $\beta$ steeper, for the GCLFs in fainter galaxies. As we discuss in §\[sec:disc\], features such as this likely reflect the initial condition of the GC mass distribution. Thus, if GC systems were indeed born with Schechter-like mass functions, it would seem that the “truncation” mass scale $M_c$ was higher in larger galaxies right from the point of cluster formation.
The graph of $\Delta$ vs. galaxy luminosity in the lower panel of Figure \[fig:MB\_ES\] shows, first, that it is roughly comparable to (though slightly larger than) the GCLF turnover mass in general. This is certainly not unexpected, given the characteristics of the model itself (see the discussion in §\[ssec:esmod\]). In physical terms, though, if the model is taken at face value, the correspondence reflects the fundamental role that evaporation is assumed to play in defining any turnover point at all (see our discussion in §\[sec:models\], and the more detailed exposition of Fall & Zhang 2001). Beyond this, our fits imply that there is a tendency for $\Delta$ to increase as galaxy luminosity decreases, but this is not a particularly regular trend. All in all, there appears to be a fairly narrow range of GC mass loss, $\Delta \approx 2-10 \times 10^5 M_{\odot}$, required to account for our GCLF observations over a large range of galaxy luminosity.
Note that several of the faintest galaxy bins in Figure \[fig:MB\_ES\] have $\Delta/M_c \approx 2$, to be compared with $\Delta/M_c \sim 0.1$ for the brightest systems. This reflects once again the systematic narrowing of the GCLF, due to the steepening of $dN/dM$ for high cluster masses, in fainter galaxies.
In §\[sec:disc\] we will further discuss the variations of $M_{\rm TO}$, $M_C$, and $\Delta$ with galaxy luminosity, and how they relate to questions of GC formation and dynamical evolution.
GCLF Turnovers in the Faintest Galaxies {#ssec:dwarfs}
---------------------------------------
In all of our galaxies there is evidence for the presence of a peak in the GCLF. Recently, van den Bergh (2006) claimed that the combined GCLF for a sample of local dwarf galaxies fainter than $M_{V, {\rm gal}} > -16$ does [*not*]{} show a turnover, but continues to increase to GC masses as low as $\approx\! 10^4\,M_\odot$. These galaxy luminosities translate to $B$-band $M_{B, {\rm gal}} \ga -15.2$, which is essentially the magnitude limit of our ACSVCS sample.
Even though we do not probe down to the galaxy luminosities where van den Bergh (2006) claims a drastically different GCLF behavior, it is nonetheless worth noting that the turnover mass in our faintest galaxies is still fairly close to the “canonical” $M_{\rm TO}\sim 2\times10^5\,M_\odot$. There is no hint of any systematics that would cause $M_{\rm TO}$ to fall to $10^4\,M_\odot$ or less in galaxies just 1 mag fainter than the smallest systems observed here (e.g., see Fig. \[fig:MTO\_GZ\]). It is thus likely relevant that the results of van den Bergh (2006) are based mostly on data from Sharina et al. (2005), who do not account for any potential contamination in their lists of candidate GCs in the local dwarfs. Any GCLF derived from these data must therefore be regarded as quite uncertain, at the faint end especially. Spectroscopic confirmation of the Sharina et al. GC candidates is required.
Variations with Galactocentric Radius {#ssec:gradius}
-------------------------------------
To achieve a fuller understanding of the GCLF, and in particular the competing influences of cluster formation and dynamical evolution on it, we would like to know how it might vary in form as a function of position in its parent galaxy. It has long been understood that the turnover of the Milky Way GCLF is essentially invariant with Galactocentric radius (e.g., Harris 2001), and multiple studies of the M87 GCLF have concluded that its overall shape is basically the same from the center of the galaxy out to several effective radii (McLaughlin, Harris, & Hanes 1994; Harris, Harris, & McLaughlin 1998; Kundu et al. 1999). Beyond this, however, little is known about the generic situation in most galaxies.
For the most part, our data are not well-suited to address this question, due to the small field of view of the ACS. However, we are afforded serendipitously long baselines of galactocentric radius in M87 and M49, by the inclusion in the ACSVCS of a number of low-luminosity galaxies that are projected close to each of these large galaxies. We refer to these galaxies as “companions,” even though they might not be physically associated with their “hosts.” The majority of the GCs observed in the fields of these smaller systems belong to the giants. While each companion does have some GCs of its own, their numbers will be reduced to negligible levels, compared to the M87 or M49 globulars, outside some sufficiently large radius in the low-luminosity galaxy. Thus, we take our original GC samples for the companions present in the survey and consider only those cluster candidates that are found more than 6 effective radii from the companion centers. Since the effective radii of the GC spatial distributions are generally $\approx\! 2$ times larger than those of the underlying galaxy light (Peng et al. 2006, in preparation), this corresponds to excluding sources that are within about 3 GC-system scale radii from the companion centers. This should effectively eliminate $\approx\! 90\%$ of each companion’s native GCs, leaving us with fairly clean samples of extra M49 and M87 globulars, located tens of kpc away from the giant galaxy centers.
We restrict our analysis to companions that have more than 50 GC candidates left after this selection. These are VCC 1199 (companion to M49, projected $4\farcm5$ away); VCC 1192 (M49, $4\farcm2$); VCC 1297 (M87, $7\farcm3$); and VCC 1327 (M87, $7\farcm5$). Note that $1\arcmin=4.8$ kpc for an average distance of $D=16.5$ Mpc to Virgo.
In Figure \[fig:gclf\_ore\] we show the luminosity functions and Gaussian fits for the resulting GC samples in the four fields neighbouring M87 and M49. In Table \[tab:gclfpars\_ore\] we list the best-fit parameters and the mean $(g-z)$ colors and mass-to-light ratios assumed to convert the results to mass. The results are summarized in Figure \[fig:Rgc\], where we show the GCLF turnovers and dispersions as a function of galactocentric distance in M87 and M49 separately. Evidently, none of the Gaussian GCLF parameters shows significant ($>3 \sigma$) variation over the 20–35 kpc baselines probed. Fits of evolved Schechter functions to these GCLFs confirm that $M_{\rm TO}$ in particular does not change. As we discuss further in §\[ssec:evaporation\], this lack of any significant radial trend in $M_{\rm TO}$ with galactocentric distance is hard to reconcile with a picture in which the GCLF turnover is determined solely by dynamical effects (primarily evaporation) acting on a universal power-law like initial cluster mass function evolving in a fixed, time-independent galaxy potential. (In fact, if it varies at all, $M_{\rm TO}$ may even get slightly more massive with increasing radius in Fig. \[fig:Rgc\]. While we do not claim that any such trend is in fact detected here, it would be [*opposite*]{} to naive expectations.)
Discussion {#sec:disc}
==========
We have found interesting trends in three mass scales of physical interest in connection with GC luminosity functions.
The GCLF turnover or peak mass takes a value of $M_{\rm TO}=(2.2\pm0.4)\times 10^5\,M_\odot$ in most bright galaxies, but shows some downward scatter in dwarfs fainter than $M_{B, {\rm gal}} \ga -18$. In M87 and M49, the data are consistent with a more or less constant $M_{\rm TO}$ to projected galactocentric radii of 20–35 kpc.
The higher-mass scale $M_c$ in an evolved Schechter function, which marks the onset of an exponential cut-off in the number of clusters per unit mass, grows steadily [*smaller*]{} in fainter galaxies. This drives a systematic narrowing of the dispersion in more traditional Gaussian fits to the GCLF, or equivalently a steepening of pure power-law fits to the mass function $dN/dM$ at cluster masses $M\ga M_{\rm TO}$.
And the mass $\Delta$ in the evolved Schechter function, which controls the shape of the low-mass end of the GC mass distribution and is instrumental in setting $M_{\rm TO}$, varies by factors of a few—although not entirely monotonically—as a function of galaxy luminosity.
We now discuss these results in terms of their implications for GC formation and dynamical evolution. We begin by focusing on $\Delta$ in the evolved Schechter function, which, in the context of Fall & Zhang’s (2001) dynamical theory for the GCLF, is meant to measure the average amount of mass lost per globular cluster in a galaxy, over a Hubble time of evolution. We then move on to $M_c$ and $M_{\rm
TO}$, asking specifically to what extent the observed variations in these high-mass characteristics of the GCLF might be caused by dynamical friction rather than initial conditions.
Evaporation and the Low-Mass Side of the GCLF {#ssec:evaporation}
---------------------------------------------
The defining feature of the evolved Schechter function in equation (\[eq:esmass\])—which we have found to fit the GC mass distributions of galaxies in the ACSVCS just as well as the traditional, but ad hoc, lognormal form—is the flat shape of $dN/dM$ in the limit of low masses. This asymptotic flatness always follows naturally from a time-independent rate of cluster mass loss, regardless of the assumed initial form of $dN/dM_0$ (Fall & Zhang 2001, and §\[sssec:esevolve\] above). The exact values of the average cumulative mass losses per GC for the galaxies in our sample are, however, more specific to the assumption that $dN/dM_0 \propto M_0^{-2}\,\exp(-M_0/M_c)$—a form chosen to match the observed mass functions of young massive clusters in local mergers and starbursts.
It is worth noting that, even though the average mass loss $\Delta$ in an evolved Schechter function is key to setting the GCLF turnover mass, $M_{\rm TO}$ does not vary as much or as systematically as $\Delta$ does in the ACSVCS sample (cf. Figures \[fig:mto\_bd\] and \[fig:MB\_ES\]). This is because the value of the upper-mass cut-off $M_c$ also influences $M_{\rm TO}$ (see §\[ssec:esmod\]), and $M_c$ varies in such a way as to largely counteract the variation of $\Delta$, keeping $M_{\rm TO}$ more steady as a function of $M_{B,{\rm gal}}$.
Since $M_{\rm TO}$ is observed to be so nearly constant independently of any functional fitting—at least in large galaxies—this balance between variations in $\Delta$ and $M_c$ might be viewed simply as a necessary condition to make evolved Schechter functions match the data at all. But more interesting is that if the physical arguments behind the fitting function are close to correct, our results imply that the near-universality of the GCLF turnover in bright galaxies ($M_{B,{\rm gal}}\la -18$) is in some sense a coincidence resulting from steeper initial $dN/dM$ (with lower $M_c$) in fainter systems being eroded by faster mass-loss rates (yielding larger $\Delta$).
As we discussed in §\[ssec:esmod\], some amount of cluster mass loss may result from tidal shocks, but we expect that in general the largest part comes from two-body relaxation and evaporation, at a rate determined by the mean cluster density inside its half-mass radius: $\mu_{\rm ev}\propto \rho_h^{1/2}$. This basic dependence holds independently of any host galaxy properties, so if the cluster evaporation rate varies systematically as a function of $M_{B,{\rm gal}}$, it presumably reflects systematics in the typical $\rho_h$ of the cluster systems. Then, if GCs are tidally limited, such that their average densities are determined by the galaxy density inside their orbits (e.g., King 1962), variations in their characteristic $\rho_h$ should correspond in some way to variations in the host-galaxy densities. The easiest way to quantify any such connection is to assume a spherical, time-independent galaxy potential with a simple analytical form. Thus, in their models of the Milky Way GC system, Fall & Zhang (2001) relate the $\rho_h$ of individual clusters to their orbital pericenters $r_p$ in a logarithmic potential with a fixed circular speed, $V_c$, so that $\rho_h \propto \rho_{\rm gal}(r_p)\propto V_c^2/r_p^2$. We address the validity of these particular (strong) assumptions about the host galaxy below; but for the moment we follow Fall & Zhang and most other authors (e.g., Vesperini 2000, 2001; Baumgardt & Makino 2003) in making them. What do our fitted $\Delta$ values for the ACSVCS galaxies then imply for the distribution of GC densities and pericenters in these systems?
The evaporation rate of a cluster with observable, [*projected*]{} half-mass radius $R_h$ depends on the density $\rho_h\equiv 3M/(8\pi R_h^3)$ roughly as $$\mu_{\rm ev}({\rm theo}) \simeq 345 \ M_\odot\,{\rm Gyr}^{-1} \
\left(\frac{\rho_h}{M_\odot\,{\rm pc}^{-3}}\right)^{1/2} \ ,
\label{eq:rhohmuev}$$ which again is independent of any assumptions on the host-galaxy potential. However, if $\rho_h$ is taken to be set by a well defined $r_p$ in a steady-state, singular isothermal sphere, then we also have (from eqs. \[4\] and \[15\] of Fall & Zhang 2001) $$\begin{aligned}
\mu_{\rm ev}({\rm theo}) & \simeq & 2.9\times10^4\ M_\odot\,{\rm Gyr}^{-1}\,
\left(r_p/{\rm kpc}\right)^{-1}
\left(V_c/220\ {\rm km~s}^{-1}\right) \nonumber \\
& & \quad \times \left[1-\ln\,(r_p/r_c)\right]^{1/2} \ .
\label{eq:theorate}\end{aligned}$$ In the last term on the right-hand side, which is derived by Innanen, Harris, & Webbink (1983), $r_c$ is the radius of a circular orbit with the same energy as an arbitrary orbit with $r_p\le r_c$.
Now, for the Milky Way, recall from §\[ssec:modcomp\] (eq. \[\[eq:MWesmass\]\]), that we estimate $$\Delta({\rm MW}) = (2.5\pm0.5)\times10^5\,M_\odot
\label{eq:MWdelta}$$ from our fit of an evolved Schechter function to the GCLF. For a GC age of 13 Gyr, this implies a mass-loss rate (averaged over the distribution of cluster $\rho_h$ or, given the assumptions behind eq. \[\[eq:theorate\]\], over all cluster orbits) of $$\langle \mu_{\rm ev}\rangle ({\rm fit)}
= \frac{\Delta({\rm MW})}{13\ {\rm Gyr}}
= (1.9\pm0.4)\times10^4\,M_\odot\,{\rm Gyr}^{-1}\ .
\label{eq:MWrate}$$ Comparing equation (\[eq:MWrate\]) to equation (\[eq:rhohmuev\]) implies an average $\langle \rho_h \rangle \simeq
(3000\pm 600)\,\,M_\odot\,{\rm pc}^{-3}$ for GCs in the Milky Way. This average falls towards the upper end of the range of cluster $\rho_h$ observed today, but it is within a factor of $\simeq\!2$–3 of the mean (e.g., see the data in Harris 1996). Equation (\[eq:theorate\]) further suggests an average pericenter of $\langle r_p\rangle\approx 2$ kpc. This is roughly the same answer found by Fall & Zhang (2001; see their Figure 13), which shows that an evolved Schechter function is a reasonable analytical approximation to their full numerical theory. While such an $\langle r_p \rangle$ is slightly small—just as $\langle \rho_h\rangle$ is slightly high—compared to more direct pericenter estimates for Galactic globulars (cf. Innanen et al. 1983; van den Bergh 1995), it is again within the range of standard values.
It is not at all obvious a priori that average cluster densities and pericenters inferred strictly from fits to the Galactic GC mass function should agree to within factors of 2 or 3 with values estimated by independent methods. The fact that they do is an encouraging sign for the basic picture of evaporation-dominated GCLF evolution. Some residual corrections—downward in “predicted” $\langle \rho_h \rangle$ and up in $\langle r_p \rangle$—are evidently required, but at a level that plausibly could come from straightforward refinements in the various steps leading to equations (\[eq:rhohmuev\]) and (\[eq:theorate\]). For example, there is some room for adjustment of the exact theoretical coefficients for the evaporation rate $\mu_{\rm ev} \propto M/t_{\rm rh} \propto \rho_h^{1/2}$ and the pericenter $r_p\propto \rho_h^{-1/2}$ (see, e.g., the discussions in Fall & Zhang 2001). In addition, we have neglected here any additional mass loss caused by tidal shocks, and we have adopted the idealization of a spherical and time-invariant Galactic potential.
To bring the ACSVCS data into this discussion, we focus on the basic pattern of variation in $\Delta$ as a function of galaxy luminosity, shown in the lower panel of Figure \[fig:MB\_ES\]. First, $\Delta$ increases slightly from the brightest $M_{B,{\rm gal}}\simeq-21.5$ to the fainter $M_{B,{\rm gal}}\simeq-18$. The uncertainties and scatter in $\Delta$ are large, but the mean increase is perhaps a factor of $\approx\!2$–5. Then, at fainter $M_{B,{\rm gal}}\ga -18$, $\Delta$ holds more constant or even decreases again, possibly by as much as a factor of $\approx\!2$–3 by the limiting $M_{B,{\rm gal}}\simeq -16$ of the survey.
If evaporation is responsible for these variations, then we should expect them to be mirrored in the behavior of the average GC half-mass radius as a function of galaxy luminosity: from equation (\[eq:rhohmuev\]), $\langle \mu_{\rm ev} \rangle \propto \langle \rho_h \rangle^{1/2}
\propto \langle R_h \rangle^{-3/2}$, and by definition $\Delta\propto \langle \mu_{\rm ev} \rangle$ for coeval clusters. Globulars in Virgo are marginally resolved with the ACS, and Jordán et al. (2005) have fit PSF-convolved King (1966) models to estimate intrinsic $R_h$ values for individual sources (selected as described in §2 of Jordán et al. 2005) in most of the galaxies that we have dealt with here. The behavior of mean $\langle R_h \rangle$ versus $M_{B,{\rm gal}}$ is shown in Figure 5 of Jordán et al.
A detailed comparison of $\langle R_h \rangle$ and $\Delta$ is not straightforward, since these quantities were estimated separately for GC samples defined differently by Jordán et al. than in this paper. Nevertheless, it is interesting that $\langle R_h \rangle$ can be described as decreasing towards brighter galaxy luminosity in the range $-21.5\la M_{B,{\rm gal}}\la -18$, where $\Delta$ increases, and then turning around to increase somewhat at fainter $M_{B,{\rm gal}}\ga -18$, where $\Delta$ decreases again. The changes in $\langle R_h \rangle$ are—as we would expect—smaller and less clear than those in $\Delta$, but it is just plausible that there is a net decrease of $\simeq\!35\%$ from $M_{B,{\rm gal}}=-21.5$ to $M_{B,{\rm gal}}=-18$ and a slightly larger increase from $M_{B,{\rm gal}}=-18$ to $M_{B,{\rm gal}}=-16$. This would be consistent with the shallowest trends able to fit $\Delta$ versus $M_{B,{\rm gal}}$ in Figure \[fig:MB\_ES\].
We cannot use equation (\[eq:theorate\]) to relate $\Delta$ to typical GC pericenters and average galaxy densities on a case-by-case basis in the ACSVCS sample as in the Milky Way, since $V_c$ observations are not available for all systems. However, scaling relations can be used to some effect here. Large early-type galaxies with $M_{B,{\rm gal}}\la -18$ generally obey $V_c=\sqrt{2}\sigma \propto L_{\rm gal}^{0.25}$ (e.g., Faber & Jackson 1976), $(M/L)_{\rm gal}\propto L_{\rm gal}^{0.2-0.3}$ at optical wavelengths (van der Marel 1991; Cappellari et al. 2006), and thus $R_{\rm eff}\propto L_{\rm gal}^{0.7-0.8}$ by the virial theorem (see also Haşegan et al. 2005). Average mass densities therefore increase towards lower $L_{\rm gal}$, such that equation (\[eq:theorate\]) implies $$\Delta \ \propto \
\left(R_{\rm eff}/\langle r_p \rangle\right)\, R_{\rm eff}^{-1} V_c
\ \propto \
\left(R_{\rm eff}/\langle r_p \rangle\right)\, L_{\rm gal}^{-0.5 \pm 0.05}
\label{eq:deltabright}$$ for bright galaxies. The situation is somewhat different for fainter $M_{B,{\rm gal}}\ga -18$. For Coma Cluster galaxies in this regime, Matković & Guzmán (2005) find $V_c=\sqrt{2}\sigma \propto L_{\rm gal}^{0.5 \pm 0.1}$, while the data in Graham & Guzmán (2003) suggest $R_{\rm eff}\propto L_{\rm gal}^{0.1-0.2}$. If these systems are representative of those in Virgo, then their average densities decrease towards lower $L_{\rm gal}$, and equation (\[eq:theorate\]) leads to $$\Delta \ \propto \
\left(R_{\rm eff}/\langle r_p \rangle\right)\, L_{\rm gal}^{0.35 \pm 0.1}
\label{eq:deltafaint}$$ for faint dwarfs.
The major unknown in equations (\[eq:deltabright\]) and (\[eq:deltafaint\]) is the ratio of galaxy $R_{\rm eff}$ to GC $\langle r_p \rangle$, and how it might or might not vary systematically as a function of galaxy luminosity. If the ratio is constant for all systems, then the dotted lines drawn in the lower panel of Figure \[fig:MB\_ES\] show the expected variation of the mass loss $\Delta$ versus $M_{B,{\rm gal}}$. These lines are normalized to make $\Delta = 2.5 \times10^5\,M_\odot$ at $M_{B, {\rm gal}}=-21$ and to make the bright- and faint-galaxy scalings meet at $M_{B,{\rm gal}}=-18$. The net increase of $L_{\rm gal}^{-0.5}$ from $M_{B,{\rm gal}}=-21.5$ to $M_{B,{\rm gal}}=-18$ is a factor of about 5, while the decrease of $L_{\rm gal}^{0.35}$ from $M_{B,{\rm gal}}=-18$ to $M_{B,{\rm gal}}=-16$ is a factor of approximately 2.
These changes may be somewhat greater than suggested by the actual fitted estimates of $\Delta$. Moreover, an increase of $\Delta$ by a factor of 5 between $M_{B,{\rm gal}}=-21.5$ and $M_{B,{\rm gal}}=-18$ would imply a decrease in $\langle R_h \rangle$ by a factor of $5^{2/3}\approx 3$, which is larger than the measurements of Jordán et al. (2005) support. However, this is clearly not an order-of-magnitude problem. It could easily be alleviated if the galaxy total mass distributions are not isothermal spheres, or if $R_{\rm eff}/\langle r_p \rangle$ depends even weakly on galaxy luminosity, or if uncertainties and scatter in the galaxy scalings result in small deviations from the nominal exponents on $L_{\rm gal}$ in equations (\[eq:deltabright\]) and (\[eq:deltafaint\]). Tidal shocks may also contribute differently to the net $\Delta$ in different galaxies, a complication that we have entirely ignored. Again, then, it is encouraging that these crude relations come as close as they do to explaining the systematics in a cluster mass-loss parameter inferred only from the GCLF—accounting in particular for the change in dependence of $\Delta$ on galaxy luminosity around $M_{B,{\rm gal}}\simeq-18$.
Obviously, more rigorous and detailed analyses of individual galaxies are required to really make (or break) the case in general that the overall form of an evolved Schechter function for the GC mass function, and the parameter $\Delta$ especially, can be interpreted physically and self-consistently as the result of evolution from an initial GC $dN/dM_0\propto M_0^{-2}$ with individual cluster mass-loss rates that are constant in time. From our discussion here, it does seem that this “literal” view of the simple fits to the Milky Way and ACSVCS GCLFs is at least broadly compatible with observations of the cluster densities or radii in these galaxies and with the trends in $\Delta$ vs. $L_{\rm gal}$, if evaporation is the main disruptive process for clusters as massive as $M_{\rm TO}\sim 2\times10^5\,M_\odot$.
Difficulties do arise, however, when considering the additional constraint that the GCLF is invariant over wide ranges of galactocentric radius and GC density in the Milky Way and other large galaxies. As described above, application of equation (\[eq:theorate\]) to the global Galactic GCLF ultimately implies an average GC pericenter of $\langle r_p \rangle \simeq 2$ kpc, corresponding to about half the effective radius of the bulge. Similarly, our normalization of equation (\[eq:deltabright\]) in Figure \[fig:MB\_ES\] implies $\langle r_p \rangle < 0.5\, R_{\rm eff}$ for the brightest early-type galaxies in Virgo. But observationally, the GCLF turnover $M_{\rm TO}$ (and thus $\Delta$) has the same, global value for clusters currently found out to at least 10–15 effective radii in the Milky Way (e.g., Harris 2001) and at least $\simeq 4\,R_{\rm eff}$ in M87 and M49 (§\[ssec:gradius\]). This can only be consistent with evaporation-dominated depletion of an intially steep GC $dN/dM_0\propto M_0^{-2}$ at low masses, and with the additional assumption that the mass loss $\Delta\propto r_p^{-1}$, if cluster orbits are systematically much more elongated at larger galactocentric radius in all these systems.
In fact, for the Milky Way and M87 respectively, Fall & Zhang (2001) and Vesperini et al. (2003) have shown that following this chain of logic leads to the conclusion that globulars should [*initially*]{} have been on predominantly radial orbits outside about one effective radius in each galaxy. On the other hand, the [*present*]{} GC velocity distributions in the Galaxy, in M87, and in M49 are all essentially isotropic—implying orbits with typical axis ratios of only $r_a/r_p\simeq3$—out to the same spatial scales of several $R_{\rm eff}$, over which the observed GCLF is invariant (see, e.g., Dinescu, Girard, & van Altena 1999; Côté et al. 2001, 2003). Fall & Zhang suggest that this difference between (presumed) initial and (observed) present orbital properties might be explained by preferential depletion of GCs on the most radial orbits. But while the idea remains to be tested in detail for the Milky Way, Vesperini et al. (2003) show that—again if the galaxy potential is spherical and time-independent—it does not suffice to account quantitatively for the combined GCLF and kinematics data in M87.
Related to this is the average density, $\langle \rho_h \rangle \simeq 3000\ M_\odot\,{\rm pc}^{-2}$, implied by the more general equation (\[eq:rhohmuev\]) and the required total $\Delta$ for Galactic globulars. A similar $\langle \rho_h \rangle$ is also suggested for GCs in the brightest Virgo galaxies by the $\Delta$ values in Figure \[fig:MB\_ES\]. As we mentioned above, such densities are observed for real clusters; but there is a broad distribution of $\rho_h$, with an average slightly lower than $3000\ M_\odot\,{\rm pc}^{-2}$ and a long tail to much smaller values of $ <\! 100\ M_\odot\,{\rm pc}^{-2}$. More generally, the GCs in most large galaxies have half-mass radii that are largely uncorrelated with cluster mass (e.g., van den Bergh, Morbey, & Pazder 1991; Jordán et al. 2005, and references therein), so that $\rho_h$ apparently always ranges over more than two orders of magnitude. When $\rho_h < 100\ M_\odot\,{\rm pc}^{-2}$, the total evaporative mass loss per cluster over 13 Gyr is $< 5\times10^4\,M_\odot$, well below the typical average $\Delta$ and global $M_{\rm TO}$ for entire GC systems. In the Milky Way at least, the large majority of such low-density GCs are found at Galactocentric distances $r_{\rm gc}\ga 10$ kpc, so in a sense the problem is bound up with the weak radial variation of the GCLF.
These points are important, and they need to be resolved, but they should not be taken as disproof of the idea that long-term dynamical evolution alone might explain the difference between the mass functions of old GCs and young massive clusters. Ultimately, the near-flatness of $dN/dM$ at low masses, which is clearly seen in the Milky Way and is entirely consistent with all of our Virgo GCLFs, only demands that cluster masses decrease linearly in time if the dynamical-evolution hypothesis is correct at all (see Fall & Zhang 2001, and §\[ssec:esmod\] above). It is not absolutely necessary that evaporation account for the full mass-loss rate of every cluster, even though our discussion here has focused on exploring this possibility (and shown that it does come remarkably close, to within factors of 2–3 for the most part). For example, globulars in the extreme low-density tails of $\rho_h$ distributions, mentioned just above, might be much more strongly—and differently—affected by tidal shocks than any previous GCLF calculations have allowed. Such shock-dominated evolution could still lead to a constant mass-loss rate of its own (see Dehnen et al. 2004, and §\[sssec:esevolve\] above), which would add directly to $\mu_{\rm ev}$ without otherwise changing any of the main arguments here.
In more specific terms, the radial invariance of the GCLF might ultimately be explained by modifying a single ancillary assumption in the current dynamical-evolution models rather than discarding the idea altogether. It is the notion of spherical and steady-state galaxy potentials that prompts Fall & Zhang (2001), Vesperini et al. (2003), and almost all other authors to use equations (\[eq:rhohmuev\]) and (\[eq:theorate\]) to tie cluster densities to orbital pericenters in these analyses. But, as Fall & Zhang themselves point out, this is of course an extreme simplification for galaxies that grow through hierarchical merging.
Fall & Zhang suggest, for example, that a major merger could obviate the need for extremely radial orbits to distribute clusters with high mean densities, fixed at small and well defined pericenters, over large volumes in a galaxy. Instead, a merger may efficiently mix two globular cluster systems spatially and isotropize their velocity distribution. This could then work to weaken any radial gradients in the mass loss $\Delta$ and the GCLF turnover mass, which might have resulted from realistic orbital distributions and $\mu_{\rm ev}(r_p)$ dependence like equation (\[eq:theorate\]) in the progenitor galaxies.
In addition to this, multiple minor mergers—which are perhaps more relevant than major mergers for a galaxy like our own—should steadily bring in globulars formed with densities and evaporation rates unrelated, at least initially, to their new orbits in the main galaxy, making the use of equation (\[eq:theorate\]) less than straightforward. In fact, any use of it at all could be questionable in this case, since all clusters would constantly be sampling new pericenters in an evolving potential. Again, then, weak spatial variations in $\Delta$ and $M_{\rm TO}$ need not imply highly radial GC orbits. Prieto & Gnedin (2006) have recently simulated the evolution of the GCLF during the hierarchical growth of a Milky Way-sized galaxy. Starting from an initial cluster mass function $dN/dM_0\propto M_0^{-2}$, which is re-shaped primarily by evaporation—but abandoning equation (\[eq:theorate\]) and instead adopting evaporation rates from GC densities fixed independently of their orbits—they find that it is possible (even without a recent major merger) to produce a final GC system with an isotropic velocity distribution and a radially invariant GCLF similar to the observed Galactic distribution.
A caveat is that the hierarchical-growth simulations most favored by Prieto & Gnedin (2006) are ones in which they assume that all globular clusters have a common mean density inside $R_h$ (just one that is not set by any orbital pericenter). This is still incompatible with the wide range of $\rho_h$ observed for the GCs in many galaxies, and it is furthermore not obvious how the cumulative mass loss $\Delta \propto \langle \rho_h \rangle^{1/2}$ should then vary as a function of galaxy luminosity. On the other hand, Prieto & Gnedin have also run some models allowing for an initial spread of GC densities followed by evaporation at constant $\rho_h$. This is at least more reminiscent of real $\rho_h$ distributions, and it still produces a GCLF that is not too drastically different from the Galactic one. Clearly, more work is required to clarify the dynamical evolution of initial power-law GC mass functions in time-dependent galaxy potentials, with the totality of relevant observational constraints taken into account: a flat $dN/dM$ at low masses; a weak or absent correlation between GC radii and masses; radially invariant GCLFs; currently isotropic velocity distributions; and mass losses $\Delta$ that vary with galaxy luminosity as in Figure \[fig:MB\_ES\].
Should all efforts along these lines fail to explain the combined data, the only option left would seem to be that a peak in the GCLF was established much earlier, by processes more related to cluster formation. One possible scenario has been proposed by Vesperini & Zepf (2003). They suggest that low-mass globulars were initially less concentrated (with a larger ratio of half-mass to tidal radius) than high-mass clusters. The inevitable expansion of all clusters following mass loss driven by stellar evolution would then cause many low-mass clusters preferentially to overflow their tidal radii, leading ultimately to fast disruption times of a few hundred Myr or less (Chernoff & Weinberg 1990). This may turn an initial power-law $dN/dM_0$ at low masses into a roughly flat-topped or even lognormal distribution, with $M_{\rm TO}$ near its current value, very early on. Weaker long-term evaporation (i.e., lower cluster densities or larger and more variable pericenters) could then suffice to explain the residual difference between the initial, steep mass function and the final, observed one, even in a static galaxy potential.
Observations of the young massive clusters in the Antennae galaxies already imply that early disruption is [*independent*]{} of cluster mass, at least for clusters more massive than several $10^4\,M_\odot$ and younger than $\simeq\! 10^8$ yr (Zhang & Fall 1999; Fall et al. 2005). Thus, if the disruption mechanism of Vesperini & Zepf (2003) is to work, the mass-selective aspect of it apparently must be restricted to timescales of $10^8$–$10^9$ yr or so. In any case, the success of this or any similar picture further relies on an appropriately tuned mass dependence in some key GC property being built into cluster systems essentially as an initial condition; but this still requires explanation in itself.
Dynamical Friction and the High-Mass Side of the GCLF {#ssec:dynfric}
-----------------------------------------------------
At GC mass scales $M\ga \Delta$, dynamical friction can in some cases become more important than evaporation or shocks as a cluster destruction mechanism. A point mass $M$ originally on a circular orbit of radius $r$ in a galaxy with a total-mass distribution following a singular isothermal sphere will spiral in to the galaxy center within a time (Binney & Tremaine 1987) $$\tau_{\rm df} \simeq
\frac{5.9\ {\rm Gyr}}{(\ln\Lambda)/10}\left(\frac{r}{{\rm kpc}}\right)^2
\left(\frac{V_c}{220\ {\rm km\ s}^{-1}}\right)
\left(\frac{10^6\, M_\odot}{M}\right) \ ,
\label{eq:df_bt}$$ where $V_c$ is the galaxy’s circular speed and $\ln\Lambda \sim 10$ is the usual Coulomb logarithm.
It is clear from equation (\[eq:df\_bt\]) that dynamical friction cannot be a major factor in deciding the evolution of all but the very most massive tip of the GCLF in $\sim\! L_*$ and brighter galaxies with $V_c \ga 200$ km s$^{-1}$. However, the scaling $\tau_{\rm df}\propto V_c$ implies that the relevance of dynamical friction can increase significantly for lower luminosity galaxies (e.g., Hernandéz & Gilmore 1998; Lotz et al. 2001). It is then reasonable to ask whether a stronger depletion of massive GCs in dwarf galaxies might be able to explain the systematic decrease of $M_c$ versus $M_{B,{\rm gal}}$ in our fits of evolved Schechter functions for these systems, and possibly even the slight decrease in average $M_{\rm TO}$ towards the faintest $M_{B, {\rm gal}}$.
We do not attempt here to find a definitive answer to this question, but only an indication of the ability of dynamical friction to produce the observed trends. One particular subtlety is that the expression for $\tau_{\rm df}$ in equation (\[eq:df\_bt\]) does not allow for clusters to evaporate. But a steadily decreasing cluster mass will lead to a longer total dynamical-friction timescale. We deal with this complication in the simplest way possible: the timescale $\tau_{\rm df}$ for a cluster with initial mass $M_0$ and present mass $M=(M_0-\Delta)$ is approximated by evaluating equation (\[eq:df\_bt\]) at the [*average*]{} mass, $(M+\Delta/2)$.
Let us denote by $\widehat{\Psi}(M,t)$ the GC mass function that would be obtained after a time $t$ of GC evolution in the absence of any dynamical friction. The effects of dynamical friction are easily accounted for by subtracting from $\widehat{\Psi}$ all clusters with instantaneous masses $M$ such that $$(M+\Delta/2) > M_{\rm min}(r,t) \ ,
\label{eq:df_cond.1}$$ where $M_{\rm min}$ follows from equation (\[eq:df\_bt\]) by setting $\tau_{\rm df}<t$: $$M_{\rm min}(r, t) \simeq
\frac{4.5 \times 10^5\,M_\odot}{(\ln\Lambda)/10}
\left(\frac{13\ {\rm Gyr}}{t}\right)
\left(\frac{r}{{\rm kpc}}\right)^2
\left(\frac{V_c}{220\,{\rm km\ s}^{-1}}\right) \ .
\label{eq:df_cond.2}$$ The net, “global” GC mass function (averaged over all GC orbits, or galactocentric radii) at any time $t$ is thus $dN/dM = S(M,t) \times \widehat{\Psi}(M,t)$ where $$S(M,t) = \frac{\int_0^{\infty} \rho_{\rm GC}(r) \,
H[M_{\rm min}(r,t) - (M+\Delta/2)] \,
4\pi r^2 \, dr}
{\int_0^{\infty} \rho_{\rm GC}(r) \, 4\pi r^2 \, dr} \ .
\label{eq:S}$$ Here $\rho_{\rm GC}(r)$ is the space density of GCs (assumed to be independent of cluster mass) and $H$ is the Heaviside step function: $H(x) \equiv 1$ for $x>0$ and $H(x) \equiv 0$ for $x < 0$.
This raises further points to be dealt with in more careful calculations along these lines. First, dynamical friction will clearly affect also the spatial distribution of GCs, so that $\rho_{\rm GC}(r)$ will have a dependence on time, which we ignore. Second, the effects of dynamical friction could introduce some dependence on galactocentric position into the GC mass function, which in a complete treatment would be contrasted with observational limits on any such variations. Third, changing the assumed galaxy potential could significantly affect the derived $\tau_{\rm df}$ (e.g., Hernandez & Gilmore 1998; Read et al. 2006), as could relaxing the unrealistic assumption of strictly circular orbits (e.g., Pesce, Capuzzo-Dolcetta, & Vietri 1992; van den Bosch et al. 1999). Finally, we do not take into account the fact that the ACS has a fixed field of view, and thus we are not always observing truly globally averaged GCLFs—although this point is relevant mainly for the most massive galaxies, where the effects of dynamical friction are expected to be negligible in any case.
These issues notwithstanding, we proceed to estimate the effects of dynamical friction by evaluating $S(M,t)$ as written in equation (\[eq:S\]). We assume that the “friction-free” $\widehat{\Psi}(M,t)$ at the present day is well described by the GCLF of bright ellipticals, where dynamical friction is negligible, and is therefore given by equation (\[eq:esmass\]) with $\Delta=2.6\times 10^5 M_{\odot}$ and $M_c = 3\times 10^6 M_{\odot}$ (see Figure \[fig:MB\_ES\] and Table \[tab:galbins\]). To obtain the final $dN/dM$ including dynamical friction, we then multiply this by the function $S(M, t\equiv13\,{\rm Gyr})$. In doing so, we always take the slowly varying Coulomb logarithm in equation (\[eq:df\_cond.2\]) to be $\ln\Lambda=10$.
We assume that for giant galaxies with $M_B < -18$ we have $V_c \propto \sigma \propto L_{\rm gal}^{0.25}$ (Faber & Jackson 1976), with a zeropoint chosen to give $V_c=484$ km s$^{-1}$ at $M_B=-21.75$, based on the velocity dispersion of M87 (Bender, Saglia & Gerhard 1994). We impose a change in this scaling at $M_B>-18$, so that dwarfs follow $V_c \propto \sigma \propto L_{\rm gal}^{0.5 \pm 0.1}$ (Matković & Guzmán 2005; cf. §\[ssec:evaporation\] above). We can then find $M_{\rm min}(r,t)$ from equation (\[eq:df\_cond.2\]) for any GC in any galaxy.
To specify the spatial distribution of GCs and calculate $S(M,t=13\ {\rm Gyr})$, we estimate the galaxy’s effective radius $R_{\rm eff}$ using the data from Ferrarese et al. (2006a); then we assume that the effective radius of the GC system is just twice $R_{\rm eff}$ (Peng et al. 2006, in preparation). Finally, we assume that $\rho_{\rm GC}(r)$ is given by the density profile of Prugniel & Simien (1997; see also Terzić & Graham 2005), which is an analytical approximation to the deprojection of a Sersic profile ($R^{1/n}$ law), and we let the Sersic index $n$ be determined by $M_{B, {\rm gal}}$ as per equation (25) of Ferrarese et al. (2006a).
The results of the calculations for two representative galaxy magnitudes, $M_{B,{\rm gal}}=-21.75$ and $M_{B,{\rm gal}}=-15.75$, are illustrated in Figure \[fig:dynfric\]. The figure shows both $S(M,t=13\ {\rm Gyr})$ (the monotonically decreasing curves) and the function $dN/d\log M$ (proportional to the GCLF and given by the peaked curves) that follows from dynamical friction acting on the assumed evolved Schechter function. The resulting turnover mass scales are indicated with arrows, which show that the stronger dynamical friction in the fainter galaxy leads to a slightly lower turnover mass scale.
We show the behavior of $M_{\rm TO}$ as a function of $M_{B,{\rm gal}}$ in general, in the upper panel of Figure \[fig:TO\_MB\_mod\] (circles connected by a solid line) and contrast it with the observed variation in our binned-galaxy GC samples (Figure \[fig:mto\_bd\]). The predicted $M_{\rm TO}$ varies quite slowly with $M_{B,{\rm gal}}$, but it ultimately decreases by $\sim 10\%$ from our assumed $2.2\times10^5\,M_\odot$ in the brightest galaxies. This is comparable to the observed decrease of $\approx\! 30\%$ in $M_{\rm TO}$. Thus, dynamical friction may be responsible for some part of the the slow change in GCLF turnover mass with galaxy magnitude.
In the lower panel of Figure \[fig:TO\_MB\_mod\] we show (again, as open circles connected by a solid line) the $M_c$ values inferred by fitting evolved Schechter functions to our model GC mass functions after calculating the effects of dynamical friction. Evidently, we can expect dynamical friction to cause perhaps a $\sim\! 30\%$–40% decrease in the value of $M_c$ from the brightest to the faintest galaxies; but this is altogether too little to account for the factor of $\simeq\! 6$–7 decrease we actually observe. Similarly, if we fit power laws to our dynamical-friction mass functions in the range $3\times10^5\,M_\odot \le M \le 2\times10^6\,M_\odot$, we obtain rather constant powers $\beta\simeq1.7$–1.8 for galaxy magnitudes $-21.75 < M_{B,{\rm gal}} < -15.75$, which is far from being able to explain the observational situation in Figure \[fig:power\] above. We conclude that [*dynamical friction cannot account for more than a small fraction of the observed steepening of the globular cluster mass function above the GCLF turnover*]{}.
These results are essentially in agreement with those of Vesperini (2000), who models the effects of evaporation and dynamical friction on the GCLF, and predicts only slight decreases in the mean $\langle \log\,M \rangle$ and the Gaussian dispersion $\sigma_M$ as galaxy luminosity decreases (see his Figure 6), at levels much smaller than those seen in our data (e.g., Figure \[fig:MTO\_GZ\]). Thus, although we have emphasized the highly simplified nature of our calculations, it nevertheless appears that galaxy-to-galaxy systematics in the cluster formation processes, rather than dynamical evolution, must be largely responsible for the observed variation in the detailed form of the GCLF at high masses.
### Initial Conditions {#sssec:init}
It seems inevitable from the discussion above that the observed steepening or narrowing of the GCLF above the turnover point $M_{\rm TO}\sim
2\times10^5\,M_\odot$ in fainter galaxies—whether this is expressed in terms of smaller Schechter-function mass scales $M_c$ or steeper power-law indices $\beta$ or narrower Gaussian dispersions $\sigma_M$—must reflect non-universal initial conditions in the cluster mass distribution, and therefore some fundamental aspect of the star-formation process.
Observationally, it is known that the luminosity of the brightest young star cluster in a star-forming galaxy scales with the global star formation rate (Billet, Hunter & Elmegreen 2002; Larsen 2002). There has been some discussion as to whether this is just a size-of-sample effect (if more clusters are formed, it is statistically more likely to achieve higher masses by random sampling of an underlying mass distribution that might still be universal) or indicative of a real, physical limit to the initial cluster mass function (Larsen 2002; Weidner, Kroupa & Larsen 2004).
Gieles et al. (2006a, b) argue that there is a physical upper limit, $M_{\rm
max}$, to cluster masses in each of NGC 6946, M51, and the Antennae galaxies (though see Whitmore, Chandar & Fall 2006 for a differing view). The number of clusters found with $M>M_{\rm max}$ falls rapidly to zero in all three cases, but the value of the upper limit is found by Gieles et al. to vary between the galaxies, from $M_{\rm max}\simeq 4$–$10\times10^5\,M_\odot$. Qualitatively, a parameter like $M_{\rm max}$ can be identified with $M_c$ in a Schechter-function description of (initial) GC mass functions. Quantitatively, the range of $M_{\rm max}$ claimed by Gieles et al. for their young systems is very similar indeed to our fitted $M_c$ values for the old GCs in early-type Virgo galaxies (see Figure \[fig:MB\_ES\]).
It will be interesting to explore this possible connection between globulars and young massive clusters in more detail. Possibly one route to take is suggested by the theory of the GCLF developed by Harris & Pudritz (1994), in which a distribution of cluster masses is built up by collisions between gaseous protoclusters. McLaughlin & Pudritz (1996) suggest that the total time required to produce very high-mass clusters may be longer for galaxies in lower-density environments, and this could perhaps be related to our finding of a cut-off at lower $M_c$ (in our current notation) for the initial GC mass functions at fainter $M_{B,{\rm gal}}$. If these types of ideas can be generalized, then both our GCLF observations and the possible existence of an upper mass “limit” in young cluster systems could be reflecting a systematic variation in gas-dynamical timescales as a function of galaxy mass and/or density.
In any case, the fact that dynamical friction is unable to account for the steepening of an initially universal mass function as the mass of the host galaxy decreases, combined with the possible existence in young, relatively unevolved cluster systems of a mass scale similar to $M_c$ in our old GC systems, leads us to favor the view that a significant part of the observed morphology [*at the highest-mass ends*]{} of GCLFs is due to systematics in the initial distributions. The precise extent to which this part of the initial mass function is still reflected in the present-day one is still something of an open question, the answer to which will be a crucial ingredient in our understanding of GC formation and evolution. A detailed understanding of the “microscopic” star-formation processes on rather short timescales in very young clusters could well be key to making much further progress in this direction.
Summary and Conclusions {#sec:conclusions}
=======================
We have presented the GCLFs of 89 early-type galaxies in the Virgo cluster and determined maximum-likelihood estimates for model parameters using fits of Gaussians and a simple “evolved Schechter function” described in §\[ssec:esmod\]. The latter reflects the effects of GC disruption (at a constant rate and presumably due mostly to two-body relaxation and evaporation) on an initial cluster mass distribution that followed a Schechter function with a fixed power-law index of $-2$ at low masses. The evolved mass function tends to a flat shape at low $M$ and is an accurate analytical approximation to the numerical distributions produced in the theory of Fall & Zhang (2001). We have tested the robustness of our results by simulations, by the construction of GCLFs for galaxies binned together to contain a minimum number of clusters, and by using alternate schemes to select GC candidates from catalogues of observed sources. Our main results and conclusions are the following:
1. We find a remarkably regular decrease of the dispersion of the GCLF as the luminosity of the host galaxy decreases (§\[sec:trends\] and Jordán et al. 2006). Quantitatively, the maximum-likelihood estimates of the dispersion $\sigma$ of Gaussian fits to the $z$- and $g$-band data are well described by the linear relations presented in equations (\[eq:sigmaz\]) and (\[eq:sigmag\]). The dispersions for the GCLFs of the Milky Way and M31 fall in the midst of our new data and thus the correlation of $\sigma$ with $M_{B,{\rm gal}}$ would appear to be more fundamental than the older view, that GCLF widths depend on galaxy Hubble type.
This trend reflects a systematic steepening of the GC mass function for [*massive*]{} clusters in particular ($M \ga 3\times10^5 M_{\odot}$, above the peak of the GCLF) as the host galaxy luminosity decreases. When fitting power-law mass functions to this upper cluster mass regime, the power-law exponents in a model of the form $dN/dM \propto M^{-\beta}$ increase from $\beta \la 2$ to $\beta \ga 3 $ over the range of galaxy masses in our sample. This steepening is in turn equivalent to a systematic decrease of the cut-off mass $M_c$ in evolved-Schechter function fits to the GCLFs, from $M_c\simeq 2$–$3\times 10^6\,M_\odot$ in the brightest galaxies to $M_c\simeq\,3$–$4\times10^5\,M_\odot$ in the faintest systems.
2. The GCLF turnover mass $M_{\rm TO}$ is slightly smaller in dwarf systems ($M_B \ga -18$), relative to the same quantity in more massive galaxies. In the latter we have $M_{\rm TO} = (2.2\pm0.4) \times 10^5 M_{\odot}$, decreasing to $M_{\rm TO} \simeq 1.6$–$1.7 \times 10^5 M_{\odot}$ on average for the faintest galaxies in our sample—although individual dwarfs scatter between $1\times 10^5\,M_\odot \la M_{\rm TO} \la 2\times10^5\,M_\odot$ (§\[sec:trends\]). We show that this might be at least partly accounted for by the effects of dynamical friction if all other processes shaping the mass function were to lead to an invariant $M_{\rm TO}$ (§\[ssec:dynfric\]).
3. We explored radial variations of the GCLF over baselines of $20-35$ kpc in M87 (VCC 1316) and M49 (VCC 1226) by studying GCs in the fields of dwarf galaxies close in projection to these giant ellipticals (§\[ssec:gradius\]). We find no evidence for a variation of the turnover mass $M_{\rm TO}$ with galactocentric distance in either galaxy, consistent with previous studies of M87 in particular. This reinforces the importance of the radial invariance of GCLFs as a constraint on models of GCLF formation and dynamical evolution.
4. Our success in fitting evolved Schechter functions to our data (§\[ssec:esfits\]) means that the GC mass functions in early-type Virgo galaxies are consistent with a universally flat shape, $dN/dM \sim {\rm constant}$, in the limit of low masses—as is also found in the Milky Way (§\[ssec:modcomp\] and Fall & Zhang 2001). If this feature is caused by dynamical evolution from a much steeper initial distribution, it requires that cluster masses decrease linearly in time. This can plausibly be expected if evaporation dominates the cluster evolution, although tidal shocks may also lead to similar behavior.
5. Fits of the evolved Schechter function imply that a narrow range of average mass losses per GC—$\Delta \approx (2$–$10) \times 10^5 M_{\odot}$ at the outside—is required in all galaxies to account for our observed GCLFs. Such a range of $\Delta$ across a factor of $\approx\! 400$ in galaxy luminosity is in rough agreement with observed (small) variations in the mean half-mass radii of GCs in the ACSVCS galaxies (Jordán et al. 2005), and with simple scalings of evaporation rate as a function of host-galaxy luminosity (§\[ssec:evaporation\]). However, more work is required to reconcile fully the main idea—that long-term dynamical evolution alone transformed initial Schechter cluster mass functions into the presently observed distributions—with the weak radial variation of GCLFs inside large galaxies and with observations of the orbital distributions and range of mean cluster densities in the same systems.
6. The clear decrease of the GC cut-off mass $M_c$ with galaxy luminosity in evolved-Schechter function descriptions of the GCLF (§\[ssec:esfits\]) is too pronounced to be explained by dynamical friction operating on a universal $dN/dM$ with an initially constant $M_c$ in all galaxies (§\[ssec:dynfric\]). It most likely reflects systematic variations at the high-mass end of the initial GC mass function.
The present-day mass functions of GCs were likely shaped by a variety of processes acting on different timescales, including systematic variations in the initial (proto-)cluster mass function at the high-mass end; long-term dynamical erosion by evaporation, tidal shocks, and dynamical friction; and global relaxation effects in time-varying galaxy potentials (hierarchical merging). It is further possible, though not yet entirely clear, that mass-selective early dissolution of clusters due to stellar evolution may have played some role in defining the observed mass distributions. Future attempts to understand the whole of the GCLF will clearly have to consider all of these processes, and their inevitable interplay, in quite some detail. Such comprehensive modeling will also have to acknowledge the increasingly complex and stringent empirical constraints that follow from combining direct GCLF observations with other GC systematics—such as their structural correlations, and the dynamics of cluster systems—for which data are continually accumulating and improving in quality.
We thank Mike Fall for critical readings of earlier versions of this paper, and for helpful discussions. We also thank M. Kissler-Patig, J. Liske, S. Mieske and S. Zepf for useful discussions and the referee, S[ø]{}ren Larsen, for a careful reading of the manuscript. Support for program GO-9401 was provided through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. S.M. acknowledges additional support from NASA grant NAG5-7697 to the ACS Team. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
[*Facility:*]{}
[00]{} Aguilar, L., Hut, P., & Ostriker, J. P. 1988, , 335, 720 Barmby, P., Huchra, J.P., Brodie, J.P., Forbes, D.A., Schroder, L.L., & Grillmair, C.J. 2000, , 119, 727 Baum, W.A., et al. 1997, , 113, 1483 Baumgardt, H., & makino, J. 2003, , 340, 227 Bender, R., Saglia, R.P., & Gerhard, O.E. 1994, , 269, 785 Billet, O.H., Hunter, D.A., & Elmegreen, B.G. 2002, , 123, 1454 Binggeli, B., Sandage, A., & Tammann, G.A., 1985, , 90, 1681 Binney, J., & Tremaine, S. 1987, Galactic Dynamics (Princeton: Princeton Univ. Press) Burkert, A., & Smith, G. H. 2000, , 542, L95 Cappellari, M., et al. 2006, , 366, 1126 Cen, R. 2001, , 560, 592 Chernoff, D. F., & Weinberg, M. D. 1990, , 351, 121 Côté, P., et al.2001, , 559, 828 Côté, P., McLaughlin, D.E., Cohen, J.G., & Blakeslee, J.P. 2003, , 591, 850 Côté, P., et al. 2004, , 153, 223 (Paper I) Côté, P., et al. 2006, , 165, 57 (Paper VIII) de Grijs, R., Bastian, N., & Lamers, H.J.G.L.M. 2003, , 583, L17 Dehnen, W., Odenkirchen, M., Grebel, E. K., & Rix, H.-W. 2004, , 127, 2753 de Vaucouleurs, G., & Pence, W. D. 1978, , 83, 1163 Dinescu, D. I., Girard, T. M., & van Altena, W. F. 1999, , 117, 1792 Elmegreen, B.G., & Efremov, Y.N. 1997, , 480, 235 Faber, S. M., & Jackson, R. E. 1976, , 204, 668 Fall, S.M., & Rees, M.J. 1977, , 181, 37P Fall, S.M., & Rees, M.J. 1985, , 298, 18 Fall, S.M., & Zhang, Q. 2001, , 561, 751 Fall, S.M., Chandar, R., & Whitmore, B.C. 2005, , 631, L133 Ferrarese, L., et al. 2000 , 128, 431 Ferrarese, L. et al. 2006a, , 164, 334 Ferrarese, L., et al. 2006b, , 644, L21 Fioc, M. & Rocca-Volmerange, B. 1997, , 326, 950 Gieles, M., Larsen, S.S., Scheepmaker, R.A., Bastian, N., Haas, M.R., & Lamers, H.J.G.L.M. 2006a, , 446, L9 Gieles, M., Larsen, S.S., Bastian, N., & Stein, I.T. 2006b, , 450, 129 Gnedin, O.Y., & Ostriker, J.P. 1997, , 474, 223 Gnedin, O.Y., Lee, H. M., & Ostriker, J. P. 1999, , 522, 935 Goudfrooij, P., Gilmore, D., Whitmore, B.C., & Schweizer, F. 2004, , 613, L121 Graham, A.W., & Guzmán, R. 2003, , 125, 2936 Graham, A.W., & Driver, S.P. 2005, PASA, 22, 118 Haşegan et al. 2005, , 627, 203 (paper VII) Harris, W.E. 1991, , 29, 543 Harris, W.E. 1996, , 112, 1487 Harris, W.E. 2001, in Star Clusters (28th Saas-Fee Advanced Course) ed. L. Labhardt & B. Binggeli (Berlin: Springer), 223 Harris, W.E., & Pudritz, R.E. 1994, , 429, 177 Harris, W. E., Harris, G. L. H., & McLaughlin, D. E. 1998, , 115, 1801 Hénon, M. 1961, Ann. d’Astrophys. 24, 369 Hernandez, X., & Gilmore, G. 1998, , 297, 517 Innanen, K. A., Harris, W. E., & Webbink, R. F. 1983, , 88, 338 Izenman, A.J. 1991, Jour. Am. Stat. Assoc., 86, 205 Jordán, A. et al. 2004a, , 154, 509 (Paper II) Jordán, A. et al. 2004b, , 613, 279 (Paper III) Jordán, A. et al. 2005, , 634, 1002 (Paper X) Jordán, A. et al.2006, , 651, L25 Kang, H., Shapiro, P.R., Fall, S.M., & Rees, M.J. 1990, , 363, 488 Kennicutt, R. C. 1983, , 272, 54 Kent, S. M. 1989, , 97, 1614 King, I.R. 1962, , 67, 471 King, I.R. 1966, , 71, 64 Kroupa, P., & Boily, C. M. 2002, , 336, 1188 Kundu, A., Whitmore, B.C., Sparks, W.B., Macchetto, F.D., Zepf, S.E., & Ashman, K.M. 1999, , 513, 733 Kundu, A., & Whitmore, B.C. 2001a, , 121, 2950 Kundu, A., & Whitmore, B.C. 2001b, , 122, 1255 Lamers, H.J.G.L.M., & Gieles, M. 2006, , 455, L17 Larsen, S.S. 2002, , 124, 1393 Larsen, S.S., Brodie, J.P., Huchra, J.P., Forbes, D.A., & Grillmair, C.J. 2001, , 121, 2974 Lotz, J., Telford, R., Ferguson, H.C., Miller, B.W., Stiavelli, M., & Mack, J. 2001, , 552, 572 Lupton, R. 1993, Statistics in Theory and Practice (Princeton: Princeton Univ. Press) Matković, A., & Guzmán, R. 2005, , 362, 289 McLaughlin, D.E. 1994, , 196, 47 McLaughlin, D. E. 2000, , 539, 618 McLaughlin, D.E., & Pudritz, R.E. 1996, , 469, 194 McLaughlin, D. E., & van der Marel, R. P. 2005, , 161, 304 McLaughlin, D. E., Harris, W. E., & Hanes, D. A. 1994, , 422, 486 Mei, S., et al. 2005a, , 156, 113 Mei, S., et al. 2005b, , 625, 121. Mei, S., et al. 2007, , 655, 144 Ostriker, J.P., & Gnedin, O.Y. 1997, , 487, 667 Peebles, P.J.E., & Dicke, R.H. 1968, , 154, 891 Peng, E.W. et al. 2006a, , 639, 95 (Paper IX) Peng, E.W. et al. 2006b, , 639, 838 (Paper XI) Pesce, E., Capuzzo-Dolcetta, R., & Vietri, M. 1992, , 254, 466 Prieto, J. L., & Gnedin, O. Y. 2006, preprint ([astro-ph/0608069]{}) Prugniel, Ph., & Simien, F. 1997, , 321, 111 Read, J.I., Goerdt, T., Moore, B., Pontzen, A.P., Stadel, J., & Lake, G. 2006, [`a`stro-ph/0606636]{} Reed, L.G., Harris, G.L.H., & Harris, W.E. 1994, , 107, 555 Schechter, P.L. 1976, , 203, 297 Secker, J. 1992, , 104, 1472 Secker, J., & Harris, W.E. 1993, , 105, 1358 Sharina, M.E., Puzia, T.H., & Makarov , D.I. 2005, , 442, 85 Silverman, B.W. 1986, Density Estimation for Statistics and Data Analysis (London: Chapman and Hall) Spitzer, L. 1987, Dynamical Evolution of Globular Clusters (Princeton: Princeton Univ. Press) Strader, J., Brodie, J.P., Spitler, L., & Beasley, M.A. 2006, , 132, 2333 Terzić, B., & Graham, A.W. 2005, , 362, 197 Tonry, J.L., & Schneider, D.P. 1988, , 96, 807 van den Bergh, S. 1995, , 110, 1171 van den Bergh, S. 2006, , 131, 304 van den Bergh, S., Morbey, C., & Pazder, J. 1991, , 375, 594 van den Bosch, F. C., Lewis, G. F., Lake, G., & Stadel, J. 1999, , 515, 50 van der Marel, R. P. 1991, , 253, 710 Vesperini, E., & Heggie, D. C. 1998, , 289, 898 Vesperini, E. 2000, , 318, 841 Vesperini, E. 2001, , 322, 247 Vesperini, E., & Zepf, S.E. 2003, , 587, L97 Vesperini, E., Zepf, S. E., Kundu, A., & Ashman, K. M. 2003, , 593, 760 Waters, C.Z., Zepf, S.E., Lauer, T.R., Baltz, E.A., & Silk, J. 2006, , 650, 885 Weidner, C., Kroupa, P., & Larsen, S.S. 2004, , 350, 1503 Whitmore, B.C., Chandar, R., & Fall, S.M., , in press `(astro-ph/0611055)` Zhang, Q., & Fall, S. M. 1999, , 527, L81
[cccccccccccccccccccccc]{} 1226 & 18.0 & 0.4 & 0 & 0.0 & 1.00 & 19.2 & 0.4 & 0 & 0.0 & 1.00 & & 18.0 & 0.4 & 0 & 0.1 & 1.00 & 19.2 & 0.4 & 0 & 0.1 & 1.00\
1226 & 18.4 & 0.4 & 0 & 0.1 & 1.00 & 19.6 & 0.4 & 0 & 0.1 & 1.00 & & 18.4 & 0.4 & 0 & 0.2 & 1.00 & 19.6 & 0.4 & 0 & 0.2 & 1.00\
1226 & 18.8 & 0.4 & 2 & 0.0 & 1.00 & 20.0 & 0.4 & 1 & 0.1 & 1.00 & & 18.8 & 0.4 & 0 & 0.2 & 1.00 & 20.0 & 0.4 & 0 & 0.2 & 1.00\
1226 & 19.2 & 0.4 & 5 & 0.2 & 1.00 & 20.4 & 0.4 & 3 & 0.1 & 1.00 & & 19.2 & 0.4 & 5 & 0.2 & 1.00 & 20.4 & 0.4 & 2 & 0.1 & 1.00\
1226 & 19.6 & 0.4 & 4 & 0.3 & 1.00 & 20.8 & 0.4 & 8 & 0.4 & 1.00 & & 19.6 & 0.4 & 4 & 0.3 & 1.00 & 20.8 & 0.4 & 8 & 0.5 & 1.00\
1226 & 20.0 & 0.4 & 12 & 0.2 & 1.00 & 21.2 & 0.4 & 11 & 0.3 & 1.00 & & 20.0 & 0.4 & 12 & 0.2 & 1.00 & 21.2 & 0.4 & 10 & 0.4 & 1.00\
1226 & 20.4 & 0.4 & 25 & 0.4 & 1.00 & 21.6 & 0.4 & 24 & 0.2 & 1.00 & & 20.4 & 0.4 & 24 & 0.5 & 1.00 & 21.6 & 0.4 & 23 & 0.4 & 1.00\
1226 & 20.8 & 0.4 & 32 & 0.2 & 1.00 & 22.0 & 0.4 & 33 & 0.3 & 1.00 & & 20.8 & 0.4 & 31 & 0.4 & 1.00 & 22.0 & 0.4 & 33 & 0.3 & 1.00\
1226 & 21.2 & 0.4 & 57 & 0.4 & 1.00 & 22.4 & 0.4 & 59 & 0.5 & 1.00 & & 21.2 & 0.4 & 57 & 0.4 & 1.00 & 22.4 & 0.4 & 58 & 0.5 & 1.00\
1226 & 21.6 & 0.4 & 66 & 0.6 & 1.00 & 22.8 & 0.4 & 60 & 0.4 & 1.00 & & 21.6 & 0.4 & 62 & 0.6 & 1.00 & 22.8 & 0.4 & 57 & 0.4 & 1.00\
1226 & 22.0 & 0.4 & 91 & 0.9 & 1.00 & 23.2 & 0.4 & 78 & 0.6 & 1.00 & & 22.0 & 0.4 & 86 & 0.6 & 1.00 & 23.2 & 0.4 & 73 & 0.5 & 1.00\
1226 & 22.4 & 0.4 & 98 & 0.8 & 0.99 & 23.6 & 0.4 & 101 & 1.3 & 0.98 & & 22.4 & 0.4 & 94 & 0.5 & 0.99 & 23.6 & 0.4 & 99 & 0.8 & 0.98\
1226 & 22.8 & 0.4 & 95 & 1.6 & 0.94 & 24.0 & 0.4 & 107 & 1.4 & 0.90 & & 22.8 & 0.4 & 90 & 0.9 & 0.94 & 24.0 & 0.4 & 100 & 0.8 & 0.90\
1226 & 23.2 & 0.4 & 88 & 1.4 & 0.83 & 24.4 & 0.4 & 74 & 1.8 & 0.80 & & 23.2 & 0.4 & 83 & 1.2 & 0.83 & 24.4 & 0.4 & 71 & 1.4 & 0.80\
1226 & 23.6 & 0.4 & 70 & 2.0 & 0.72 & 24.8 & 0.4 & 78 & 2.5 & 0.71 & & 23.6 & 0.4 & 65 & 1.4 & 0.72 & 24.8 & 0.4 & 72 & 2.1 & 0.71\
1226 & 24.0 & 0.4 & 61 & 3.4 & 0.62 & 25.2 & 0.4 & 56 & 2.9 & 0.62 & & 24.0 & 0.4 & 60 & 2.8 & 0.62 & 25.2 & 0.4 & 56 & 2.5 & 0.62\
1226 & 24.4 & 0.4 & 39 & 3.3 & 0.51 & 25.6 & 0.4 & 50 & 2.9 & 0.52 & & 24.4 & 0.4 & 38 & 3.2 & 0.51 & 25.6 & 0.4 & 47 & 2.6 & 0.52\
1226 & 24.8 & 0.4 & 16 & 1.6 & 0.37 & 26.0 & 0.4 & 18 & 1.8 & 0.37 & & 24.8 & 0.4 & 16 & 1.6 & 0.37 & 26.0 & 0.4 & 18 & 1.7 & 0.37\
1226 & 25.2 & 0.4 & 3 & 0.4 & 0.19 & 26.4 & 0.4 & 3 & 0.3 & 0.18 & & 25.2 & 0.4 & 3 & 0.4 & 0.19 & 26.4 & 0.4 & 3 & 0.3 & 0.18\
1316 & 18.0 & 0.4 & 0 & 0.0 & 1.00 & 19.2 & 0.4 & 0 & 0.0 & 1.00 & & 18.0 & 0.4 & 0 & 0.1 & 1.00 & 19.2 & 0.4 & 0 & 0.1 & 1.00\
[ccccccccc]{} & 9.31 & $ 24.105 \pm 0.086 $ & $ 1.366 \pm 0.061 $ &$ 22.789 \pm 0.077 $ & $ 1.321 \pm 0.053 $ & 0.023 & 764 &\
& 9.58 & $ 24.018 \pm 0.049 $ & $ 1.312 \pm 0.035 $ &$ 22.689 \pm 0.041 $ & $ 1.242 \pm 0.030 $ & 0.014 & 1745 &\
& 9.81 & $ 24.062 \pm 0.077 $ & $ 1.340 \pm 0.058 $ &$ 22.747 \pm 0.070 $ & $ 1.316 \pm 0.050 $ & 0.022 & 807 &\
& 10.06 & $ 23.950 \pm 0.097 $ & $ 1.274 \pm 0.075 $ &$ 22.834 \pm 0.093 $ & $ 1.238 \pm 0.071 $ & 0.034 & 367 &\
& 10.09 & $ 25.120 \pm 0.232 $ & $ 1.708 \pm 0.130 $ &$ 23.722 \pm 0.179 $ & $ 1.562 \pm 0.102 $ & 0.016 & 507 & Faint excess\
& 10.26 & $ 23.973 \pm 0.074 $ & $ 1.178 \pm 0.055 $ &$ 22.836 \pm 0.070 $ & $ 1.159 \pm 0.052 $ & 0.035 & 506 &\
& 10.51 & $ 24.403 \pm 0.061 $ & $ 1.207 \pm 0.046 $ &$ 23.211 \pm 0.059 $ & $ 1.199 \pm 0.044 $ & 0.021 & 907 &\
& 10.61 & $ 23.685 \pm 0.097 $ & $ 1.079 \pm 0.076 $ &$ 22.512 \pm 0.092 $ & $ 1.063 \pm 0.067 $ & 0.042 & 244 &\
& 10.76 & $ 23.446 \pm 0.089 $ & $ 1.192 \pm 0.071 $ &$ 22.255 \pm 0.089 $ & $ 1.215 \pm 0.073 $ & 0.046 & 308 &\
& 10.78 & $ 23.951 \pm 0.103 $ & $ 1.423 \pm 0.077 $ &$ 22.717 \pm 0.095 $ & $ 1.390 \pm 0.071 $ & 0.038 & 456 &\
& 11.10 & $ 23.715 \pm 0.090 $ & $ 1.103 \pm 0.072 $ &$ 22.592 \pm 0.089 $ & $ 1.106 \pm 0.069 $ & 0.058 & 254 &\
& 11.18 & $ 24.429 \pm 0.296 $ & $ 1.564 \pm 0.226 $ &$ 23.503 \pm 0.333 $ & $ 1.615 \pm 0.209 $ & 0.076 & 134 & Faint excess\
& 11.37 & $ 23.902 \pm 0.092 $ & $ 0.988 \pm 0.072 $ &$ 22.813 \pm 0.094 $ & $ 1.001 \pm 0.072 $ & 0.065 & 192 &\
& 11.40 & $ 23.687 \pm 0.133 $ & $ 1.218 \pm 0.110 $ &$ 22.548 \pm 0.123 $ & $ 1.203 \pm 0.097 $ & 0.066 & 179 &\
& 11.51 & $ 24.009 \pm 0.198 $ & $ 1.111 \pm 0.176 $ &$ 22.882 \pm 0.186 $ & $ 1.135 \pm 0.148 $ & 0.114 & 92 &\
& 11.80 & $ 23.622 \pm 0.117 $ & $ 1.102 \pm 0.102 $ &$ 22.447 \pm 0.108 $ & $ 1.077 \pm 0.091 $ & 0.068 & 179 & Faint excess\
& 11.80 & $ 23.805 \pm 0.121 $ & $ 1.120 \pm 0.098 $ &$ 22.689 \pm 0.114 $ & $ 1.084 \pm 0.090 $ & 0.067 & 172 &\
& 11.82 & $ 23.872 \pm 0.146 $ & $ 1.073 \pm 0.123 $ &$ 22.831 \pm 0.153 $ & $ 1.120 \pm 0.117 $ & 0.096 & 136 &\
& 11.84 & $ 23.737 \pm 0.098 $ & $ 0.980 \pm 0.078 $ &$ 22.621 \pm 0.098 $ & $ 1.021 \pm 0.076 $ & 0.072 & 176 &\
& 11.94 & $ 23.482 \pm 0.119 $ & $ 1.183 \pm 0.100 $ &$ 22.471 \pm 0.109 $ & $ 1.159 \pm 0.087 $ & 0.071 & 197 &\
& 11.99 & $ 23.692 \pm 0.135 $ & $ 1.248 \pm 0.110 $ &$ 22.584 \pm 0.127 $ & $ 1.213 \pm 0.104 $ & 0.085 & 167 &\
& 12.02 & $ 23.675 \pm 0.121 $ & $ 1.049 \pm 0.094 $ &$ 22.502 \pm 0.110 $ & $ 1.009 \pm 0.086 $ & 0.092 & 146 &\
& 12.03 & $ 23.981 \pm 0.200 $ & $ 0.911 \pm 0.192 $ &$ 23.053 \pm 0.207 $ & $ 0.930 \pm 0.166 $ & 0.194 & 48 &\
& 12.08 & $ 23.721 \pm 0.140 $ & $ 0.868 \pm 0.121 $ &$ 22.712 \pm 0.140 $ & $ 0.893 \pm 0.114 $ & 0.132 & 91 &\
& 12.11 & $ 23.798 \pm 0.145 $ & $ 1.077 \pm 0.123 $ &$ 22.830 \pm 0.140 $ & $ 1.020 \pm 0.130 $ & 0.113 & 101 &\
& 12.15 & $ 23.666 \pm 0.111 $ & $ 1.031 \pm 0.088 $ &$ 22.612 \pm 0.111 $ & $ 1.035 \pm 0.086 $ & 0.097 & 138 &\
& 12.29 & $ 23.672 \pm 0.159 $ & $ 0.798 \pm 0.150 $ &$ 22.615 \pm 0.161 $ & $ 0.871 \pm 0.141 $ & 0.141 & 71 &\
& 12.41 & $ 24.618 \pm 0.364 $ & $ 1.221 \pm 0.250 $ &$ 23.406 \pm 0.239 $ & $ 1.036 \pm 0.168 $ & 0.167 & 62 &\
& 12.50 & $ 24.255 \pm 0.238 $ & $ 1.050 \pm 0.207 $ &$ 23.178 \pm 0.235 $ & $ 1.061 \pm 0.175 $ & 0.165 & 66 &\
& 12.57 & $ 24.135 \pm 0.217 $ & $ 1.106 \pm 0.175 $ &$ 23.066 \pm 0.184 $ & $ 1.064 \pm 0.144 $ & 0.124 & 83 &\
& 12.60 & $ 23.741 \pm 0.130 $ & $ 0.919 \pm 0.115 $ &$ 22.624 \pm 0.126 $ & $ 0.963 \pm 0.101 $ & 0.105 & 116 &\
& 12.67 & $ 24.299 \pm 0.203 $ & $ 0.870 \pm 0.188 $ &$ 23.122 \pm 0.179 $ & $ 0.813 \pm 0.164 $ & 0.178 & 64 &\
& 12.70 & $ 23.688 \pm 0.279 $ & $ 0.977 \pm 0.279 $ &$ 22.789 \pm 0.328 $ & $ 1.143 \pm 0.274 $ & 0.256 & 45 &\
& 12.72 & $ 24.197 \pm 0.215 $ & $ 1.081 \pm 0.166 $ &$ 23.120 \pm 0.204 $ & $ 1.043 \pm 0.147 $ & 0.163 & 74 &\
& 12.84 & $ 24.025 \pm 0.275 $ & $ 0.831 \pm 0.276 $ &$ 23.057 \pm 0.255 $ & $ 0.849 \pm 0.223 $ & 0.198 & 50 &\
& 12.84 & $ 23.817 \pm 0.177 $ & $ 1.042 \pm 0.159 $ &$ 22.800 \pm 0.147 $ & $ 0.902 \pm 0.126 $ & 0.143 & 80 &\
& 12.91 & $ 23.585 \pm 0.163 $ & $ 0.815 \pm 0.147 $ &$ 22.611 \pm 0.165 $ & $ 0.834 \pm 0.132 $ & 0.200 & 54 &\
& 12.91 & $ 24.201 \pm 0.406 $ & $ 1.310 \pm 0.313 $ &$ 23.150 \pm 0.361 $ & $ 1.316 \pm 0.252 $ & 0.217 & 57 &\
& 12.93 & $ 23.895 \pm 0.153 $ & $ 0.901 \pm 0.185 $ &$ 22.757 \pm 0.180 $ & $ 0.892 \pm 0.173 $ & 0.148 & 82 &\
& 13.06 & $ 24.265 \pm 0.125 $ & $ 0.844 \pm 0.117 $ &$ 23.357 \pm 0.148 $ & $ 0.933 \pm 0.116 $ & 0.143 & 104 &\
& 13.10 & $ 23.640 \pm 0.150 $ & $ 0.780 \pm 0.128 $ &$ 22.836 \pm 0.158 $ & $ 0.807 \pm 0.122 $ & 0.176 & 61 &\
& 13.22 & $ 23.764 \pm 0.137 $ & $ 0.750 \pm 0.128 $ &$ 22.749 \pm 0.144 $ & $ 0.759 \pm 0.120 $ & 0.180 & 65 &\
& 13.26 & $ 23.686 \pm 0.133 $ & $ 1.259 \pm 0.107 $ &$ 22.624 \pm 0.118 $ & $ 1.211 \pm 0.094 $ & 0.081 & 173 & VCC1316 Companion\
& 13.30 & $ 23.701 \pm 0.144 $ & $ 0.791 \pm 0.144 $ &$ 22.650 \pm 0.146 $ & $ 0.783 \pm 0.123 $ & 0.179 & 62 &\
& 13.36 & $ 24.094 \pm 0.159 $ & $ 0.999 \pm 0.155 $ &$ 23.239 \pm 0.190 $ & $ 1.112 \pm 0.146 $ & 0.138 & 85 &\
& 13.37 & $ 23.621 \pm 0.148 $ & $ 1.002 \pm 0.114 $ &$ 22.574 \pm 0.129 $ & $ 0.953 \pm 0.093 $ & 0.124 & 90 &\
& 13.45 & $ 24.058 \pm 0.172 $ & $ 0.880 \pm 0.155 $ &$ 23.062 \pm 0.179 $ & $ 0.930 \pm 0.139 $ & 0.170 & 66 &\
& 13.56 & $ 24.038 \pm 0.350 $ & $ 1.146 \pm 0.342 $ &$ 23.058 \pm 0.358 $ & $ 1.246 \pm 0.263 $ & 0.217 & 46 &\
& 13.60 & $ 23.793 \pm 0.096 $ & $ 0.832 \pm 0.071 $ &$ 22.799 \pm 0.089 $ & $ 0.814 \pm 0.064 $ & 0.105 & 119 &\
& 13.64 & $ 23.711 \pm 0.276 $ & $ 0.703 \pm 0.261 $ &$ 22.595 \pm 0.236 $ & $ 0.694 \pm 0.220 $ & 0.256 & 37 &\
& 13.81 & $ 23.481 \pm 0.508 $ & $ 0.976 \pm 0.303 $ &$ 22.444 \pm 0.340 $ & $ 0.893 \pm 0.284 $ & 0.420 & 22 &\
& 13.86 & $ 23.597 \pm 0.739 $ & $ 1.194 \pm 0.588 $ &$ 22.619 \pm 0.690 $ & $ 1.190 \pm 0.581 $ & 0.516 & 18 &\
& 13.93 & $ 23.863 \pm 0.547 $ & $ 1.023 \pm 0.378 $ &$ 22.833 \pm 0.371 $ & $ 0.897 \pm 0.236 $ & 0.246 & 34 &\
& 14.14 & $ 24.952 \pm 0.263 $ & $ 0.558 \pm 0.219 $ &$ 23.881 \pm 0.333 $ & $ 0.316 \pm 0.362 $ & 0.386 & 27 &\
& 14.17 & $ 23.787 \pm 0.237 $ & $ 1.181 \pm 0.198 $ &$ 22.655 \pm 0.215 $ & $ 1.141 \pm 0.185 $ & 0.180 & 60 &\
& 14.20 & $ 24.110 \pm 0.564 $ & $ 0.559 \pm 0.530 $ &$ 23.221 \pm 0.463 $ & $ 0.671 \pm 0.373 $ & 0.487 & 18 &\
& 14.25 & $ 23.886 \pm 0.263 $ & $ 0.922 \pm 0.189 $ &$ 22.797 \pm 0.193 $ & $ 0.870 \pm 0.139 $ & 0.211 & 50 &\
& 14.30 & $ 24.029 \pm 0.321 $ & $ 0.800 \pm 0.281 $ &$ 23.027 \pm 0.300 $ & $ 0.822 \pm 0.196 $ & 0.327 & 29 &\
& 14.31 & $ 24.536 \pm 0.957 $ & $ 1.260 \pm 0.714 $ &$ 23.696 \pm 0.785 $ & $ 1.168 \pm 0.675 $ & 0.468 & 20 &\
& 14.31 & $ 23.741 \pm 0.151 $ & $ 0.929 \pm 0.120 $ &$ 22.722 \pm 0.139 $ & $ 0.900 \pm 0.119 $ & 0.162 & 68 &\
& 14.33 & $ 23.401 \pm 0.119 $ & $ 1.140 \pm 0.097 $ &$ 22.298 \pm 0.106 $ & $ 1.083 \pm 0.086 $ & 0.092 & 152 & VCC1316 Companion\
& 14.37 & $ 23.688 \pm 0.293 $ & $ 1.042 \pm 0.243 $ &$ 22.603 \pm 0.225 $ & $ 0.953 \pm 0.187 $ & 0.233 & 49 &\
& 14.39 & $ 23.908 \pm 0.235 $ & $ 0.701 \pm 0.177 $ &$ 22.844 \pm 0.195 $ & $ 0.646 \pm 0.139 $ & 0.330 & 28 &\
& 14.51 & $ 24.132 \pm 0.190 $ & $ 1.050 \pm 0.169 $ &$ 23.112 \pm 0.199 $ & $ 1.088 \pm 0.149 $ & 0.158 & 71 &\
& 14.51 & $ 23.552 \pm 0.149 $ & $ 0.717 \pm 0.119 $ &$ 22.621 \pm 0.136 $ & $ 0.702 \pm 0.115 $ & 0.221 & 49 &\
& 14.53 & $ 24.408 \pm 0.461 $ & $ 0.957 \pm 0.481 $ &$ 23.462 \pm 0.558 $ & $ 1.093 \pm 0.434 $ & 0.380 & 22 &\
& 14.54 & $ 24.116 \pm 0.268 $ & $ 0.700 \pm 0.246 $ &$ 22.953 \pm 0.160 $ & $ 0.500 \pm 0.209 $ & 0.332 & 28 &\
& 14.54 & $ 23.942 \pm 0.198 $ & $ 0.782 \pm 0.169 $ &$ 23.063 \pm 0.179 $ & $ 0.845 \pm 0.140 $ & 0.229 & 50 &\
& 14.55 & $ 23.543 \pm 0.255 $ & $ 0.858 \pm 0.258 $ &$ 22.612 \pm 0.236 $ & $ 0.849 \pm 0.213 $ & 0.303 & 34 &\
& 14.69 & $ 24.471 \pm 0.326 $ & $ 0.672 \pm 0.463 $ &$ 23.578 \pm 0.402 $ & $ 0.825 \pm 0.359 $ & 0.379 & 25 &\
& 14.74 & $ 24.362 \pm 0.684 $ & $ 0.938 \pm 0.822 $ &$ 24.366 \pm 1.728 $ & $ 1.460 \pm 0.956 $ & 0.478 & 17 &\
& 14.75 & $ 24.332 \pm 0.802 $ & $ 1.427 \pm 0.802 $ &$ 23.293 \pm 0.701 $ & $ 1.350 \pm 0.478 $ & 0.351 & 26 &\
& 14.76 & $ 24.137 \pm 0.421 $ & $ 0.573 \pm 0.364 $ &$ 23.030 \pm 0.539 $ & $ 0.511 \pm 0.392 $ & 0.471 & 19 &\
& 14.94 & $ 24.496 \pm 0.691 $ & $ 1.352 \pm 0.608 $ &$ 23.806 \pm 0.674 $ & $ 1.325 \pm 0.387 $ & 0.271 & 35 &\
& 14.96 & $ 24.079 \pm 0.178 $ & $ 0.884 \pm 0.175 $ &$ 23.148 \pm 0.178 $ & $ 0.894 \pm 0.152 $ & 0.189 & 63 &\
& 15.04 & $ 23.777 \pm 0.091 $ & $ 1.070 \pm 0.072 $ &$ 22.660 \pm 0.086 $ & $ 1.049 \pm 0.067 $ & 0.064 & 213 & VCC1226 Companion\
& 15.08 & $ 23.514 \pm 0.240 $ & $ 0.553 \pm 0.247 $ &$ 22.522 \pm 0.197 $ & $ 0.515 \pm 0.226 $ & 0.378 & 26 &\
& 15.20 & $ 24.280 \pm 0.278 $ & $ 0.887 \pm 0.251 $ &$ 23.298 \pm 0.228 $ & $ 0.826 \pm 0.175 $ & 0.259 & 38 &\
& 15.20 & $ 23.957 \pm 0.218 $ & $ 0.545 \pm 0.198 $ &$ 23.099 \pm 0.336 $ & $ 0.581 \pm 0.319 $ & 0.274 & 38 &\
& 15.20 & $ 23.900 \pm 0.436 $ & $ 0.281 \pm 0.536 $ &$ 22.964 \pm 0.389 $ & $ 0.304 \pm 0.370 $ & 0.459 & 20 &\
& 15.30 & $ 23.508 \pm 0.267 $ & $ 0.493 \pm 0.212 $ &$ 22.674 \pm 0.247 $ & $ 0.501 \pm 0.177 $ & 0.495 & 17 &\
& 15.33 & $ 23.806 \pm 0.250 $ & $ 0.701 \pm 0.265 $ &$ 22.757 \pm 0.283 $ & $ 0.664 \pm 0.329 $ & 0.355 & 27 &\
& 15.49 & $ 24.449 \pm 0.144 $ & $ 0.666 \pm 0.145 $ &$ 23.468 \pm 0.150 $ & $ 0.747 \pm 0.120 $ & 0.186 & 60 &\
& 15.49 & $ 23.027 \pm 0.984 $ & $ 0.967 \pm 1.086 $ &$ 21.565 \pm 0.520 $ & $ 0.463 \pm 0.566 $ & 0.622 & 14 &\
& 15.50 & $ 23.828 \pm 0.102 $ & $ 1.163 \pm 0.084 $ &$ 22.679 \pm 0.092 $ & $ 1.123 \pm 0.072 $ & 0.060 & 228 & VCC1226 Companion\
& 15.68 & $ 23.810 \pm 0.213 $ & $ 0.826 \pm 0.214 $ &$ 22.821 \pm 0.207 $ & $ 0.901 \pm 0.163 $ & 0.275 & 43 &\
& 15.68 & $ 23.840 \pm 0.197 $ & $ 0.691 \pm 0.137 $ &$ 22.910 \pm 0.159 $ & $ 0.639 \pm 0.113 $ & 0.292 & 33 &\
& 15.89 & $ 23.977 \pm 0.439 $ & $ 0.378 \pm 0.260 $ &$ 23.156 \pm 0.381 $ & $ 0.482 \pm 0.526 $ & 0.417 & 22 &\
& 15.97 & $ 24.177 \pm 0.154 $ & $ 0.225 \pm 0.201 $ &$ 23.059 \pm 0.417 $ & $ 0.615 \pm 0.336 $ & 0.477 & 19 &\
[cccccccccccccc]{} 0 & 1 (1226) & -21.8 & -21.8 & -21.8 & 746 & $ -7.025 \pm 0.086 $ & $ 1.366 \pm 0.061 $ & $ -8.341 \pm 0.077 $ & $ 1.321 \pm 0.053 $ & $ -7.150 \pm 0.133 $ & $ -10.045 \pm 0.362 $ & $ -8.465 \pm 0.132 $ & $ -11.257 \pm 0.360 $\
1 & 1 (1316) & -21.5 & -21.5 & -21.5 & 1721 & $ -7.104 \pm 0.049 $ & $ 1.312 \pm 0.035 $ & $ -8.433 \pm 0.041 $ & $ 1.242 \pm 0.030 $ & $ -7.287 \pm 0.089 $ & $ -9.850 \pm 0.232 $ & $ -8.690 \pm 0.092 $ & $ -10.911 \pm 0.232 $\
2 & 1 (1978) & -21.3 & -21.3 & -21.3 & 789 & $ -7.014 \pm 0.077 $ & $ 1.340 \pm 0.058 $ & $ -8.329 \pm 0.070 $ & $ 1.316 \pm 0.050 $ & $ -7.265 \pm 0.137 $ & $ -9.750 \pm 0.356 $ & $ -8.617 \pm 0.146 $ & $ -10.928 \pm 0.381 $\
3 & 1 (881) & -21.2 & -21.2 & -21.2 & 355 & $ -7.334 \pm 0.097 $ & $ 1.274 \pm 0.075 $ & $ -8.450 \pm 0.093 $ & $ 1.238 \pm 0.071 $ & $ -7.533 \pm 0.198 $ & $ -9.877 \pm 0.525 $ & $ -8.607 \pm 0.221 $ & $ -11.043 \pm 0.647 $\
4 & 1 (763) & -21.1 & -21.1 & -21.1 & 488 & $ -7.385 \pm 0.074 $ & $ 1.178 \pm 0.055 $ & $ -8.522 \pm 0.070 $ & $ 1.159 \pm 0.052 $ & $ -7.786 \pm 0.201 $ & $ -9.371 \pm 0.460 $ & $ -8.955 \pm 0.200 $ & $ -10.499 \pm 0.457 $\
5 & 1 (731) & -21.3 & -21.3 & -21.3 & 888 & $ -7.431 \pm 0.061 $ & $ 1.207 \pm 0.046 $ & $ -8.623 \pm 0.059 $ & $ 1.199 \pm 0.044 $ & $ -7.651 \pm 0.137 $ & $ -9.789 \pm 0.339 $ & $ -8.818 \pm 0.134 $ & $ -11.011 \pm 0.337 $\
6 & 1 (1903) & -20.1 & -20.1 & -20.1 & 294 & $ -7.434 \pm 0.089 $ & $ 1.192 \pm 0.071 $ & $ -8.625 \pm 0.089 $ & $ 1.215 \pm 0.073 $ & $ -7.641 \pm 0.209 $ & $ -9.971 \pm 0.611 $ & $ -8.776 \pm 0.188 $ & $ -11.418 \pm 0.576 $\
7 & 1 (1632) & -20.3 & -20.3 & -20.3 & 439 & $ -7.089 \pm 0.103 $ & $ 1.423 \pm 0.077 $ & $ -8.323 \pm 0.095 $ & $ 1.390 \pm 0.071 $ & $ -7.198 \pm 0.152 $ & $ -10.393 \pm 0.474 $ & $ -8.443 \pm 0.154 $ & $ -11.516 \pm 0.469 $\
8 & 1 (1231) & -19.8 & -19.8 & -19.8 & 239 & $ -7.221 \pm 0.090 $ & $ 1.103 \pm 0.072 $ & $ -8.344 \pm 0.089 $ & $ 1.106 \pm 0.069 $ & $ -7.841 \pm 0.311 $ & $ -8.888 \pm 0.669 $ & $ -9.013 \pm 0.319 $ & $ -10.002 \pm 0.680 $\
9 & 2 & -19.6 & -19.7 & -19.5 & 347 & $ -7.196 \pm 0.074 $ & $ 1.102 \pm 0.057 $ & $ -8.308 \pm 0.076 $ & $ 1.103 \pm 0.057 $ & $ -7.749 \pm 0.279 $ & $ -8.909 \pm 0.635 $ & $ -8.946 \pm 0.294 $ & $ -9.957 \pm 0.647 $\
10 & 2 & -19.4 & -19.5 & -19.2 & 248 & $ -7.282 \pm 0.088 $ & $ 1.111 \pm 0.069 $ & $ -8.444 \pm 0.089 $ & $ 1.103 \pm 0.069 $ & $ -8.276 \pm 0.424 $ & $ -8.597 \pm 0.838 $ & $ -9.726 \pm 0.557 $ & $ -9.586 \pm 1.040 $\
11 & 2 & -19.4 & -19.4 & -19.4 & 283 & $ -7.341 \pm 0.086 $ & $ 1.092 \pm 0.066 $ & $ -8.426 \pm 0.089 $ & $ 1.096 \pm 0.068 $ & $ -8.485 \pm 0.517 $ & $ -8.478 \pm 0.974 $ & $ -9.431 \pm 0.453 $ & $ -9.696 \pm 0.879 $\
12 & 2 & -19.1 & -19.3 & -18.9 & 347 & $ -7.380 \pm 0.072 $ & $ 1.091 \pm 0.055 $ & $ -8.447 \pm 0.071 $ & $ 1.092 \pm 0.055 $ & $ -8.262 \pm 0.323 $ & $ -8.767 \pm 0.652 $ & $ -9.410 \pm 0.336 $ & $ -9.814 \pm 0.667 $\
13 & 2 & -19.0 & -19.0 & -19.0 & 212 & $ -7.333 \pm 0.084 $ & $ 0.982 \pm 0.065 $ & $ -8.446 \pm 0.083 $ & $ 0.968 \pm 0.066 $ & $ -8.321 \pm 0.519 $ & $ -8.467 \pm 1.013 $ & $ -9.596 \pm 0.602 $ & $ -9.482 \pm 1.131 $\
14 & 2 & -19.1 & -19.1 & -19.0 & 214 & $ -7.459 \pm 0.084 $ & $ 1.051 \pm 0.065 $ & $ -8.479 \pm 0.085 $ & $ 1.042 \pm 0.068 $ & $ -8.028 \pm 0.335 $ & $ -9.134 \pm 0.766 $ & $ -9.212 \pm 0.443 $ & $ -9.943 \pm 0.964 $\
15 & 4 & -18.5 & -18.6 & -18.3 & 283 & $ -6.908 \pm 0.088 $ & $ 1.042 \pm 0.067 $ & $ -7.996 \pm 0.086 $ & $ 1.032 \pm 0.066 $ & $ -8.117 \pm 0.643 $ & $ -7.887 \pm 1.186 $ & $ -9.330 \pm 0.806 $ & $ -8.931 \pm 1.463 $\
16 & 5 & -18.3 & -18.5 & -18.1 & 257 & $ -7.081 \pm 0.089 $ & $ 1.012 \pm 0.070 $ & $ -8.146 \pm 0.087 $ & $ 0.969 \pm 0.068 $ & $ -8.399 \pm 0.648 $ & $ -7.972 \pm 1.158 $ & $ -9.413 \pm 0.642 $ & $ -9.057 \pm 1.157 $\
17 & 4 & -18.2 & -18.3 & -18.0 & 208 & $ -7.338 \pm 0.087 $ & $ 0.949 \pm 0.072 $ & $ -8.330 \pm 0.087 $ & $ 0.945 \pm 0.068 $ & $ -9.619 \pm 2.113 $ & $ -7.835 \pm 6.880 $ & $ -9.929 \pm 0.949 $ & $ -9.095 \pm 1.615 $\
18 & 3 & -17.8 & -18.0 & -17.6 & 205 & $ -7.263 \pm 0.082 $ & $ 0.951 \pm 0.062 $ & $ -8.276 \pm 0.085 $ & $ 0.961 \pm 0.065 $ & $ -8.610 \pm 0.675 $ & $ -8.111 \pm 1.202 $ & $ -9.882 \pm 0.802 $ & $ -9.041 \pm 1.371 $\
19 & 3 & -17.7 & -17.8 & -17.7 & 197 & $ -7.409 \pm 0.079 $ & $ 0.901 \pm 0.059 $ & $ -8.421 \pm 0.081 $ & $ 0.905 \pm 0.060 $ & $ -10.101 \pm 1.986 $ & $ -7.839 \pm 7.800 $ & $ -10.042 \pm 0.896 $ & $ -9.133 \pm 1.512 $\
20 & 8 & -17.1 & -17.5 & -16.8 & 196 & $ -7.149 \pm 0.099 $ & $ 0.953 \pm 0.080 $ & $ -8.216 \pm 0.094 $ & $ 0.927 \pm 0.072 $ & $ -8.247 \pm 0.680 $ & $ -8.025 \pm 1.253 $ & $ -9.371 \pm 0.630 $ & $ -9.161 \pm 1.154 $\
21 & 6 & -16.6 & -16.8 & -16.5 & 222 & $ -7.217 \pm 0.080 $ & $ 0.943 \pm 0.060 $ & $ -8.240 \pm 0.080 $ & $ 0.916 \pm 0.060 $ & $ -8.343 \pm 0.591 $ & $ -8.166 \pm 1.103 $ & $ -9.609 \pm 0.656 $ & $ -9.079 \pm 1.155 $\
22 & 9 & -16.4 & -16.7 & -16.1 & 193 & $ -7.072 \pm 0.086 $ & $ 0.875 \pm 0.068 $ & $ -8.043 \pm 0.096 $ & $ 0.921 \pm 0.071 $ & $ -9.383 \pm 1.795 $ & $ -7.437 \pm 4.766 $ & $ -8.974 \pm 0.505 $ & $ -9.135 \pm 0.951 $\
23 & 10 & -15.7 & -16.0 & -15.4 & 201 & $ -7.133 \pm 0.072 $ & $ 0.749 \pm 0.058 $ & $ -8.090 \pm 0.077 $ & $ 0.762 \pm 0.062 $ & $ -9.792 \pm 0.088 $ & $ -7.292 \pm 1.956 $ & $ -10.815 \pm 0.092 $ & $ -8.315 \pm 5.536 $\
[rcccrcccrcc]{} 1226 & $ 1.80 \pm 0.11 $ & $ 1.72 \pm 0.11 $ & & 654 & $ 2.58 \pm 0.76 $ & $ 2.73 \pm 0.72 $ & & 1178 & $ 2.42 \pm 0.40 $ & $ 1.82 \pm 0.38 $\
1316 & $ 1.75 \pm 0.07 $ & $ 1.79 \pm 0.07 $ & & 944 & $ 2.50 \pm 0.44 $ & $ 2.55 \pm 0.46 $ & & 1283 & $ 2.85 \pm 0.61 $ & $ 2.90 \pm 0.61 $\
1978 & $ 1.84 \pm 0.11 $ & $ 1.77 \pm 0.11 $ & & 1938 & $ 2.65 \pm 0.45 $ & $ 2.83 \pm 0.45 $ & & 1261 & $ 1.63 \pm 0.55 $ & $ 2.01 \pm 0.52 $\
881 & $ 1.80 \pm 0.16 $ & $ 1.79 \pm 0.16 $ & & 1279 & $ 1.87 \pm 0.28 $ & $ 1.85 \pm 0.28 $ & & 698 & $ 2.42 \pm 0.35 $ & $ 2.38 \pm 0.33 $\
798 & $ 2.15 \pm 0.15 $ & $ 1.95 \pm 0.15 $ & & 1720 & $ 2.69 \pm 0.47 $ & $ 2.97 \pm 0.50 $ & & 1422 & $ 2.95 \pm 0.94 $ & $ 4.07 \pm 1.17 $\
763 & $ 1.85 \pm 0.14 $ & $ 1.87 \pm 0.14 $ & & 355 & $ 3.50 \pm 0.97 $ & $ 2.75 \pm 0.79 $ & & 2048 & $ 1.44 \pm 0.87 $ & $ 1.26 \pm 0.82 $\
731 & $ 1.71 \pm 0.10 $ & $ 1.77 \pm 0.10 $ & & 1619 & $ 2.46 \pm 0.72 $ & $ 2.25 \pm 0.71 $ & & 9 & $ 2.42 \pm 0.82 $ & $ 2.20 \pm 0.77 $\
1535 & $ 2.03 \pm 0.20 $ & $ 1.94 \pm 0.20 $ & & 1883 & $ 3.18 \pm 0.60 $ & $ 2.85 \pm 0.56 $ & & 1910 & $ 2.35 \pm 0.52 $ & $ 2.07 \pm 0.51 $\
1903 & $ 1.87 \pm 0.18 $ & $ 2.03 \pm 0.18 $ & & 1242 & $ 3.25 \pm 0.45 $ & $ 3.06 \pm 0.45 $ & & 856 & $ 1.84 \pm 0.63 $ & $ 1.70 \pm 0.64 $\
1632 & $ 1.89 \pm 0.16 $ & $ 1.83 \pm 0.16 $ & & 784 & $ 3.77 \pm 1.14 $ & $ 3.23 \pm 1.01 $ & & 140 & $ 3.55 \pm 1.31 $ & $ 2.51 \pm 1.21 $\
1231 & $ 2.22 \pm 0.23 $ & $ 2.13 \pm 0.23 $ & & 1537 & $ 2.13 \pm 0.62 $ & $ 2.48 \pm 0.67 $ & & 1087 & $ 2.73 \pm 0.54 $ & $ 2.50 \pm 0.52 $\
2095 & $ 1.79 \pm 0.34 $ & $ 1.85 \pm 0.33 $ & & 778 & $ 2.07 \pm 0.48 $ & $ 2.00 \pm 0.48 $ & & 1861 & $ 2.52 \pm 0.71 $ & $ 1.92 \pm 0.62 $\
1154 & $ 1.81 \pm 0.28 $ & $ 2.02 \pm 0.28 $ & & 1321 & $ 3.67 \pm 1.29 $ & $ 4.99 \pm 1.97 $ & & 1431 & $ 2.55 \pm 0.57 $ & $ 2.69 \pm 0.59 $\
1062 & $ 2.13 \pm 0.26 $ & $ 1.99 \pm 0.26 $ & & 828 & $ 2.28 \pm 0.45 $ & $ 2.48 \pm 0.42 $ & & 1528 & $ 2.19 \pm 0.70 $ & $ 2.56 \pm 0.67 $\
2092 & $ 2.30 \pm 0.41 $ & $ 2.35 \pm 0.41 $ & & 1250 & $ 1.92 \pm 0.52 $ & $ 2.09 \pm 0.49 $ & & 437 & $ 3.55 \pm 0.91 $ & $ 4.04 \pm 1.10 $\
369 & $ 2.14 \pm 0.25 $ & $ 2.13 \pm 0.25 $ & & 1630 & $ 1.91 \pm 0.52 $ & $ 1.99 \pm 0.50 $ & & 2019 & $ 2.22 \pm 0.70 $ & $ 3.17 \pm 0.81 $\
759 & $ 2.32 \pm 0.27 $ & $ 2.19 \pm 0.27 $ & & 1146 & $ 2.33 \pm 0.52 $ & $ 2.47 \pm 0.50 $ & & 21 & $ 2.32 \pm 0.88 $ & $ 2.76 \pm 0.93 $\
1692 & $ 1.93 \pm 0.29 $ & $ 2.40 \pm 0.29 $ & & 1025 & $ 3.09 \pm 0.75 $ & $ 2.81 \pm 0.66 $ & & 1499 & $ 2.75 \pm 0.90 $ & $ 2.69 \pm 0.88 $\
1030 & $ 1.93 \pm 0.25 $ & $ 2.13 \pm 0.25 $ & & 1303 & $ 2.55 \pm 0.59 $ & $ 2.48 \pm 0.54 $ & & 1545 & $ 2.61 \pm 0.74 $ & $ 2.57 \pm 0.73 $\
2000 & $ 2.07 \pm 0.25 $ & $ 2.16 \pm 0.24 $ & & 1913 & $ 3.03 \pm 0.58 $ & $ 2.57 \pm 0.58 $ & & 1075 & $ 3.94 \pm 1.31 $ & $ 3.85 \pm 1.28 $\
685 & $ 1.71 \pm 0.25 $ & $ 1.71 \pm 0.24 $ & & 1125 & $ 2.78 \pm 0.57 $ & $ 2.37 \pm 0.58 $ & & 1539 & $ 3.14 \pm 0.92 $ & $ 2.69 \pm 0.85 $\
1664 & $ 1.85 \pm 0.29 $ & $ 1.66 \pm 0.28 $ & & 1475 & $ 2.37 \pm 0.54 $ & $ 2.55 \pm 0.55 $ & & 1185 & $ 5.63 \pm 1.62 $ & $ 5.56 \pm 1.59 $\
[rccccrccccrccc]{} 1226 & 1.24 & 1.47 & 2.69 & &1242 & 1.11 & 1.49 & 2.44 & &1297 & 1.05 & 1.50 & 2.33\
1316 & 1.23 & 1.47 & 2.67 & & 784 & 1.14 & 1.48 & 2.50 & &1861 & 1.00 & 1.50 & 2.24\
1978 & 1.25 & 1.47 & 2.72 & &1537 & 1.00 & 1.50 & 2.24 & & 543 & 0.94 & 1.51 & 2.12\
881 & 1.09 & 1.49 & 2.41 & & 778 & 1.04 & 1.50 & 2.31 & &1431 & 1.00 & 1.50 & 2.24\
798 & 1.14 & 1.48 & 2.50 & &1321 & 1.04 & 1.50 & 2.31 & &1528 & 0.95 & 1.51 & 2.14\
763 & 1.11 & 1.49 & 2.44 & & 828 & 1.00 & 1.50 & 2.24 & &1695 & 1.01 & 1.50 & 2.26\
731 & 1.19 & 1.47 & 2.59 & &1250 & 0.98 & 1.51 & 2.20 & &1833 & 1.01 & 1.50 & 2.26\
1535 & 1.18 & 1.48 & 2.57 & &1630 & 1.10 & 1.49 & 2.42 & & 437 & 0.90 & 1.52 & 2.05\
1903 & 1.18 & 1.48 & 2.57 & &1146 & 1.20 & 1.47 & 2.61 & &2019 & 0.90 & 1.52 & 2.05\
1632 & 1.21 & 1.47 & 2.63 & &1025 & 0.97 & 1.51 & 2.18 & & 200 & 0.82 & 1.54 & 1.91\
1231 & 1.12 & 1.48 & 2.46 & &1303 & 0.94 & 1.51 & 2.12 & & 571 & 0.92 & 1.52 & 2.09\
2095 & 1.07 & 1.49 & 2.37 & &1913 & 1.02 & 1.50 & 2.27 & & 21 & 0.88 & 1.52 & 2.01\
1154 & 1.12 & 1.48 & 2.46 & &1327 & 1.06 & 1.49 & 2.35 & &1488 & 0.87 & 1.52 & 1.99\
1062 & 1.14 & 1.48 & 2.50 & &1125 & 0.93 & 1.51 & 2.11 & &1499 & 0.93 & 1.51 & 2.11\
2092 & 1.13 & 1.48 & 2.48 & &1475 & 0.94 & 1.51 & 2.12 & &1545 & 0.93 & 1.51 & 2.11\
369 & 1.15 & 1.48 & 2.52 & &1178 & 1.06 & 1.49 & 2.35 & &1192 & 1.10 & 1.49 & 2.42\
759 & 1.10 & 1.49 & 2.42 & &1283 & 1.03 & 1.50 & 2.29 & &1075 & 0.93 & 1.51 & 2.11\
1692 & 1.08 & 1.49 & 2.39 & &1261 & 1.05 & 1.50 & 2.33 & &1440 & 0.98 & 1.51 & 2.20\
1030 & 1.14 & 1.48 & 2.50 & & 698 & 1.00 & 1.50 & 2.24 & & 230 & 0.92 & 1.52 & 2.09\
2000 & 1.05 & 1.50 & 2.33 & &1422 & 1.09 & 1.49 & 2.41 & &2050 & 0.89 & 1.52 & 2.03\
685 & 1.07 & 1.49 & 2.37 & &2048 & 1.01 & 1.50 & 2.26 & & 751 & 0.85 & 1.53 & 1.96\
1664 & 1.18 & 1.48 & 2.57 & &1871 & 0.96 & 1.51 & 2.16 & &1828 & 0.88 & 1.52 & 2.01\
654 & 0.99 & 1.50 & 2.22 & & 9 & 1.01 & 1.50 & 2.26 & &1407 & 1.02 & 1.50 & 2.27\
944 & 1.06 & 1.49 & 2.35 & & 575 & 1.00 & 1.50 & 2.24 & &1886 & 0.80 & 1.55 & 1.90\
1938 & 0.99 & 1.50 & 2.22 & &1910 & 1.06 & 1.49 & 2.35 & &1199 & 1.13 & 1.48 & 2.48\
1279 & 1.04 & 1.50 & 2.31 & &1049 & 0.97 & 1.51 & 2.18 & &1539 & 0.97 & 1.51 & 2.18\
1720 & 1.08 & 1.49 & 2.39 & & 856 & 1.02 & 1.50 & 2.27 & &1185 & 0.92 & 1.52 & 2.09\
355 & 1.09 & 1.49 & 2.41 & & 140 & 1.00 & 1.50 & 2.24 & &1489 & 0.98 & 1.51 & 2.20\
1619 & 1.06 & 1.49 & 2.35 & &1355 & 0.92 & 1.52 & 2.09 & &1661 & 0.95 & 1.51 & 2.14\
1883 & 1.06 & 1.49 & 2.35 & &1087 & 0.94 & 1.51 & 2.12 & && & &\
[ccccccccccc]{} 1327 & $ 23.886 \pm 0.182 $ & $ 1.288 \pm 0.144 $ &$ 22.777 \pm 0.164 $ & $ 1.224 \pm 0.129 $ & 0.070 & 93 & VCC1316 & 1.104 & 2.43 & 1.49\
1199 & $ 23.805 \pm 0.127 $ & $ 1.224 \pm 0.102 $ &$ 22.631 \pm 0.117 $ & $ 1.175 \pm 0.091 $ & 0.053 & 151 & VCC1226 & 1.100 & 2.42 & 1.49\
1192 & $ 23.643 \pm 0.102 $ & $ 1.013 \pm 0.080 $ &$ 22.540 \pm 0.099 $ & $ 1.011 \pm 0.076 $ & 0.064 & 144 & VCC1226 & 1.155 & 2.54 & 1.48\
1297 & $ 23.410 \pm 0.183 $ & $ 1.208 \pm 0.142 $ &$ 22.320 \pm 0.166 $ & $ 1.138 \pm 0.130 $ & 0.103 & 67 & VCC1316 & 1.090 & 2.40 & 1.49\
[^1]: We use the distances obtained using the polynomial calibration presented in Mei et al. 2007.
[^2]: We note that the median value of $(\sigma_g - \sigma_z)$ for our sample galaxies is 0.02 mag.
|
---
abstract: |
We consider the extinction regime in the spatial stochastic logistic model in ${{\mathbb{R}^d}}$ (a.k.a. Bolker–Pacala–Dieckmann–Law model of spatial populations) using the first-order perturbation beyond the mean-field equation. In space homogeneous case (i.e. when the density is non-spatial and the covariance is translation invariant), we show that the perturbation converges as time tends to infinity; that yields the first-order approximation for the stationary density. Next, we study the critical mortality—the smallest constant death rate which ensures the extinction of the population—as a function of the mean-field scaling parameter ${\varepsilon}>0$. We find the leading term of the asymptotic expansion (as ${\varepsilon}\to0$) of the critical mortality which is apparently different for the cases $d\geq3$, $d=2$, and $d=1$.
**Keywords:** extinction threshold, spatial logistic model, mean-field equation, population density, perturbation, correlation function, asymptotic behaviour
**2010 Mathematics Subject Classification:** 34E10, 92D25, 34D10, 58C15
author:
- 'Dmitri Finkelshtein[^1]'
title: 'Extinction threshold in spatial stochastic logistic model: Space homogeneous case'
---
Introduction
============
The spatial stochastic logistic model was introduced in 1997 by B.Bolker and S.W.Pacala, [@BP1997], and it has had a continual interest since then in both population ecology, e.g. [@MDL2004; @LMD2003; @DL2000; @LD2000; @OC2006], and (pure) mathematics, e.g. [@FM2004; @Eth2004; @FKK2009; @KK2016; @SW2015; @BKK2019; @BMW2014; @BMW2018]. The model describes spatial branching of individuals in a population with a density dependent death rate. We consider it in the following notations.
We fix $m>0$, the *mortality* constant, and two functions, $a_{\varepsilon}^+$ and $a_{\varepsilon}^-$, the *dispersion* and the *competition* kernels, respectively. Here ${\varepsilon}>0$ is an artificial scaling parameter: $$\label{eq:aeps}
a^\pm_{\varepsilon}(x):={\varepsilon}^d a^\pm({\varepsilon}x), \qquad x\in{{\mathbb{R}^d}},$$ where $d\geq1$ and $a^\pm$ are fixed nonnegative integrable functions on ${{\mathbb{R}^d}}$, which are assumed to be non-degenerate: $$\label{eq:nondegen}
\varkappa^\pm:=\int_{{\mathbb{R}^d}}a^\pm(x)\,dx=\int_{{\mathbb{R}^d}}a_{\varepsilon}^\pm(x)\,dx >0, \qquad {\varepsilon}>0.$$
Let ${\gamma}_{t,{\varepsilon}}\subset{{\mathbb{R}^d}}$ denote a discrete random set of positions of individuals at a moment of time $t\geq0$. The set may be finite or locally finite (the latter means that it has a finite number of points in each compact set from ${{\mathbb{R}^d}}$). For an infinitesimally small $\delta>0$, there happens, with the probability $1-o(\delta)$, exactly one out of two possible events within the time-interval $[t,t+\delta)$: either the individual placed at an $x\in{\gamma}_{t,{\varepsilon}}$ sends an off-spring to an area $\Lambda\subset{{\mathbb{R}^d}}$ with the probability $$\delta \, \int_{\Lambda}a_{\varepsilon}^+(x-y)\,dy+o(\delta);$$ or the individual placed at an $x\in{\gamma}_{t,{\varepsilon}}$ dies with the probability $$\delta \biggl(m+\sum_{y\in{\gamma}_{t,{\varepsilon}}\setminus\{x\}} a_{\varepsilon}^-(x-y)\biggr) + o(\delta).$$
In population ecology, one of the fundamental questions relates to the persistence of populations, or conversely to the possibility of their extinction. The latter can be defined through the equation $$\label{eq:extinction}
\lim_{t\to\infty}k_{t,{\varepsilon}}(x)=0, \quad x\in{{\mathbb{R}^d}}$$ (we write henceforth $x\in{{\mathbb{R}^d}}$ instead of ‘for a.a. $x\in{{\mathbb{R}^d}}$’), where $k_{t,{\varepsilon}}(x)\geq0$ denotes the local population density given through the equality $$\label{eq:kt1def}
{{\mathbb{E}}}\bigl[|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}|\bigr] = \int_{\Lambda}k_{t,{\varepsilon}}(x)\,dx$$ which should hold for each compact $\Lambda\subset{{\mathbb{R}^d}}$. Henceforth, ${{\mathbb{E}}}[\zeta]$ denotes the expected value of a random variable $\zeta$ (with respect to the distribution of ${\gamma}_{t,{\varepsilon}}$), and $|\eta|$ denotes number of points in a finite subset $\eta\subset{{\mathbb{R}^d}}$.
It can be shown, see Section \[sec:prelim\] below for details, that, for ${\varepsilon}\to0$, $$\label{eq:meanfielddens}
k_{t,{\varepsilon}}(x)=q_t({\varepsilon}x)+o(1),$$ where $q_t(x)\geq 0$ solves the so-called *mean-field*, or *kinetic*, nonlinear equation, see below. Moreover, $$\lim_{t\to \infty} q_t({\varepsilon}x)=0, \quad x\in{{\mathbb{R}^d}}, \ {\varepsilon}>0,$$ if and only if $m\geq {{\varkappa^+}}$, cf. . It is natural to expect, however, that may take place for smaller value of $m$, because of the term $o(1)$ in which naturally depends on $x\in{{\mathbb{R}^d}}$ and $t\geq0$. To discuss this, one needs the next term of the expansion , using the approach considered in [@OFKCBK2014; @OC2006; @CSFSO2019]. It yields that, for ${\varepsilon}\to0$, $$\label{eq:beyondmeanfielddens}
k_{t,{\varepsilon}}(x)=q_t ({\varepsilon}x)+{\varepsilon}^d p_t ({\varepsilon}x)+o({\varepsilon}^d),$$ where $p_t(x)$ can be obtained from a coupled system of linear nonhomogeneous and nonautonomous equations – (see Section \[sec:prelim\] for details).
In the present paper, we consider the space homogeneous regime, when $k_{t,{\varepsilon}}, q_t, p_t$ do not depend on the space variable. Then both $q_t$ and $p_t$ satisfy ordinary differential equations , , respectively, with $$\lim_{t\to\infty} q_t=\frac{{{\varkappa^+}}- m}{{{\varkappa^-}}}=:q^*>0,$$ for $m<{{\varkappa^+}}$. The limit $p^*:=\lim\limits_{t\to\infty} p_t$ is found in Theorem \[thm:limits\] below. Assuming that that the last term in also has a limit as $t\to\infty$ of the same order of ${\varepsilon}$, we get that the extinction, in the space homogeneous case, takes place iff $$\label{eq:exteqnintro}
q^*+{\varepsilon}^d p^*+o({\varepsilon}^d)=0.$$
Note that the conditions we imposed typically lead to $p^*<0$, that explains why should take place for $m<{{\varkappa^+}}$. To formalise this, we replace $m$ by $m({\varepsilon})<{{\varkappa^+}}$ and reveal the asymptotics of $m({\varepsilon})$ from . We show that (Theorems \[thm:assympd3\], \[thm:assympd2\], \[thm:assympd1\]), $$\label{eq:mainresult}
q^*({\varepsilon}):=\frac{{{\varkappa^+}}-m({\varepsilon})}{{{\varkappa^-}}}=\begin{cases}
{\lambda}_3 {\varepsilon}^d +o({\varepsilon}^d), & d\geq3,\\[2mm]
{\lambda}_2 {\varepsilon}^2 W({\varepsilon}^{-2})+ o\bigl({\varepsilon}^2 W({\varepsilon}^{-2})\bigr), & d=2,\\[2mm]
{\lambda}_1 {\varepsilon}^{\frac{2}{3}}+o\bigl( {\varepsilon}^{\frac{2}{3}}\bigr), & d=1,
\end{cases}$$ where ${\lambda}_3,{\lambda}_2,{\lambda}_1$ are explicit positive constants dependent on $a^+$ and $a^-$. Here $W(x)$ denotes the Lambert W function that solves $W(x)e^{W(x)}=x$ for $x\geq0$; using its known asymptotics we also get that, for the case $d=2$, $$\frac{{{\varkappa^+}}-m({\varepsilon})}{{{\varkappa^-}}}=-2 {\lambda}_2{\varepsilon}^2\log{\varepsilon}+o({\varepsilon}^2\log{\varepsilon}).$$
In other words, we show that the mortality needed to ensure that the population (statistically) will extinct as time tends to infinity is less than ${{\varkappa^+}}$, namely, $$m({\varepsilon})={{\varkappa^+}}-{{\varkappa^-}}q^*({\varepsilon}),$$ where $q^*({\varepsilon})>0$ is given by .
It is worth noting that the orders of the leading terms in the asymptotics coincide, for all $d\geq1$, with the asymptotics of the critical branching parameter for a lattice contact model considered in [@Dur1999; @DP1999; @BDS1989], where ${\varepsilon}$ was the mesh size of the lattice. We expect to discuss a connection between two models as well as to consider the space non-homogeneous case in forthcoming papers.
The paper is organised as follows. In Section \[sec:prelim\], we describe further details about the spatial and stochastic logistic model, and discuss how one can derive equations on $q_t$ and $p_t$. In Section \[sec:sphomcase\], we explain the specific of the space-homogeneous case and prove the existence of the limit $p^*=\lim\limits_{t\to\infty}p_t$. In Section \[sec:critmort\], we introduce $m({\varepsilon})$ and discuss the limits of $q^*({\varepsilon})$ and $p^*({\varepsilon})$ depending on the dimension $d$. Finally, in Sections \[sec:asympd3\]–\[sec:asympd1\] we find the asymptotics of $q^*({\varepsilon})$ (and hence of ${m_\mathrm{cr}}({\varepsilon})$) for $d\geq3$, $d=2$, and $d=1$, respectively.
Spatial and stochastic logistic model {#sec:prelim}
=====================================
We consider dynamics of a system consisting of indistinguishable individuals. Each individual is fully characterized by its position $x\in{{\mathbb{R}^d}}$, $d\geq1$. We will always assume that there are not two or more individuals at the same position.
Let ${{{{\mathcal{B}}}_\mathrm{c}}}({{\mathbb{R}^d}})$ denote the set of all Borel subsets of ${{\mathbb{R}^d}}$ with compact closure. We will consider discrete systems only, finite or locally finite. The latter means that, if ${\gamma}_{t,{\varepsilon}}=\{x\}$ is a system of individuals at some moment of time $t\geq0$, then we assume that $|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}|<\infty$ for all ${\Lambda}\in{{{{\mathcal{B}}}_\mathrm{c}}}({{\mathbb{R}^d}})$. In particular, of course, a finite ${\gamma}_{t,{\varepsilon}}$ is also locally finite. We will call such ${\gamma}_{t,{\varepsilon}}$ a (finite or locally finite) *configuration*.
The individuals of a configuration are *random*, hence we will speak about random configurations ${\gamma}_{t,{\varepsilon}}$ with respect to (w.r.t. henceforth) a probability distribution. Let ${\Gamma}$ denote the space of locally finite configurations. We fix the $\sigma$-algebra ${{\mathcal{B}}}({\Gamma})$ on ${\Gamma}$ generated by all mapppings ${\Gamma}\ni{\gamma}\mapsto |{\gamma}\cap{\Lambda}|\in{\mathbb{N}}_0:={\mathbb{N}}\cup\{0\}$, ${\lambda}\in{{{{\mathcal{B}}}_\mathrm{c}}}({{\mathbb{R}^d}})$.
The dynamics of configurations in time $t$ is defined through the dynamics of their distributions. Heuristically, the scheme is as follows. We consider, for an ${\varepsilon}\in(0,1)$, a mapping on measurable functions $F:{\Gamma}\to{{\mathbb{R}}}$ given by $$\begin{aligned}
\notag
(L_{\varepsilon}F)({\gamma})&=\sum_{x\in{\gamma}} \biggl( m+\sum_{y\in{\gamma}\setminus\{x\}} a_{\varepsilon}^-(x-y)\biggr)\Bigl(F \bigl({\gamma}\setminus\{x\}\bigr) - F\bigl({\gamma}\bigr)\Bigr)\\
&\quad +\sum_{x\in{\gamma}}\int_{{\mathbb{R}^d}}a_{\varepsilon}^+(x-y) \Bigl(F \bigl({\gamma}\cup\{y\}\bigr) - F\bigl({\gamma}\bigr)\Bigr)dy.\label{eq:genL}\end{aligned}$$ Recall that $m>0$ is a constant and functions $a_{\varepsilon}^\pm$ are defined through , where $0\leq a^+,a^-\in L^1({{\mathbb{R}^d}})$ and holds.
Operator has two properties: 1) $L_{\varepsilon}1=0$ and 2) if, for a given function $F$, a configuration ${\gamma}^* $ is such that $F({\gamma}^* )\geq F({\gamma})$ for all ${\gamma}\in{\Gamma}$ (i.e. if ${\gamma}^* $ is a global maximum for $F$), then $(L_{\varepsilon}F)({\gamma}^* )\leq 0$. Hence, formally, $L_{\varepsilon}$ is a *Markov generator*.
The dynamics of ${\gamma}_{t,{\varepsilon}}$ if defined then through the differential equation: $$\label{eq:stochdyn}
{\dfrac{d}{d t}}{{\mathbb{E}}}\bigl[F({\gamma}_{t,{\varepsilon}})\bigr] = {{\mathbb{E}}}\bigl[(L_{\varepsilon}F)({\gamma}_{t,{\varepsilon}})\bigr]$$ which should be satisfied for a large class of functions $F$.
A function $k_{t,{\varepsilon}}:{{\mathbb{R}^d}}\to{{\mathbb{R}}}_+:=[0,\infty)$ is said to be *the first order correlation function* (for the distribution of ${\gamma}_{t,{\varepsilon}}$), if for any function $g(x)\geq0$, $$\label{eq:cfi}
{{\mathbb{E}}}\Bigl[ \sum_{x\in{\gamma}_{t,{\varepsilon}}} g(x)\Bigr]=\int_{{\mathbb{R}^d}}g(x) k_{t,{\varepsilon}}(x)\,dx.$$
The function $k_{t,{\varepsilon}}(x)$ is also called *the density* of individuals of the configuration ${\gamma}_{t,{\varepsilon}}$, since, taking $g(x)={1\!\!1}_{\Lambda}(x)$ for a ${\Lambda}\in{{{{\mathcal{B}}}_\mathrm{c}}}({{\mathbb{R}^d}})$, we get from that holds.
A symmetric function $k_{t,{\varepsilon}}^{(2)}:({{\mathbb{R}^d}})^2\to{{\mathbb{R}}}_+$ is called *the second-order correlation function*, if, for any symmetric function $g_2:({{\mathbb{R}^d}})^2\to{{\mathbb{R}}}_+$, $$\label{eq:cfij}
{{\mathbb{E}}}\Bigl[ \sum_{\substack{x \in{\gamma}_{t,{\varepsilon}}\\ y \in{\gamma}_{t,{\varepsilon}}\\ x \neq y }} g_2(x ,y )\Bigr]=\int_{{\mathbb{R}^d}}\int_{{\mathbb{R}^d}}g_2(x ,y ) k_{t,{\varepsilon}}^{(2)}(x ,y )\,dx dy .$$
Combining with , we can also write, $$\begin{aligned}
\notag
{{\mathbb{E}}}\Bigl[ \sum_{\substack{x \in {\gamma}_{t,{\varepsilon}}\\y \in {\gamma}_{t,{\varepsilon}}}} g_2(x ,y )\Bigr]&=\int_{{\mathbb{R}^d}}\int_{{\mathbb{R}^d}}g_2(x ,y ) k_{t,{\varepsilon}}^{(2)}(x ,y )\,dx dy \\&\quad +
\int_{{\mathbb{R}^d}}g_2(x,x) k_{t,{\varepsilon}}(x)\,dx.\label{eq:cfijgen}
\end{aligned}$$ Substituting to the symmetric function $$\label{eq:specg2}
g_2(x ,y )=\frac{1}{2}\Bigl({1\!\!1}_{{\Lambda}_1}(x ){1\!\!1}_{{\Lambda}_2}(y )+{1\!\!1}_{{\Lambda}_1}(y ){1\!\!1}_{{\Lambda}_2}(x )\Bigr),$$ where ${\Lambda}_1,{\Lambda}_2\in{{{{\mathcal{B}}}_\mathrm{c}}}({{\mathbb{R}^d}})$, we get $$\begin{aligned}
{{\mathbb{E}}}\bigl[|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_1|\,
|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_2|\bigr] &= \int_{{\Lambda}_1} \int_{{\Lambda}_2} k_{t,{\varepsilon}}^{(2)}(x ,y )\,dx dy +
\int_{{\Lambda}_1\cap{\Lambda}_2} k_{t,{\varepsilon}} (x)\,dx,\end{aligned}$$ and hence the *covariance* between random numbers $|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_1|$ and $|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_2|$ is given by $$\begin{aligned}
&{{\mathbb{E}}}\Bigl[\Bigl(|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_1|-{{\mathbb{E}}}\bigl[|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_1|\bigr]\Bigr)\,
\Bigl(|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_2|-{{\mathbb{E}}}\bigl[|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_2|\bigr]\Bigr)\Bigr] \label{eq:covar}\\
&= {{\mathbb{E}}}\bigl[|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_1|\,
|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_2|\bigr]-{{\mathbb{E}}}\bigl[|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_1|\bigr]\,{{\mathbb{E}}}\bigl[
|{\gamma}_{t,{\varepsilon}} \cap{\Lambda}_2|\bigr]\notag\\
&= \int_{{\Lambda}_1} \int_{{\Lambda}_2} \Bigl(k_{t,{\varepsilon}}^{(2)} (x ,y )-k_{t,{\varepsilon}} (x )\,k_{t,{\varepsilon}} (y )\Bigr)\,dx dy +
\int_{{\Lambda}_1\cap{\Lambda}_2} k_{t,{\varepsilon}} (x)\,dx.\label{eq:covarans}\end{aligned}$$
Substituting into and using and , we obtain that $k_{t,{\varepsilon}}(x)$ satisfies the following equation $$\begin{aligned}
{\dfrac{\partial}{\partial t}}k_{t,{\varepsilon}}(x)&=\int_{{\mathbb{R}^d}}a_{\varepsilon}^+(x-y)k_{t,{\varepsilon}}(y)\,dy - m k_{t,{\varepsilon}}(x)\notag\\
&\quad - \int_{{\mathbb{R}^d}}a_{\varepsilon}^-(x-y)k_{t,{\varepsilon}}^{(2)}(x,y)\,dy,
\end{aligned}$$ see e.g. [@FKK2009; @FKK2011a] for details. Similarly, the evolution of $k_{t,{\varepsilon}}^{(2)}(x,y)$ depends on the third order correlation function and so on.
It can be shown, see [@FM2004; @FKK2011a; @FKKozK2014], that then, for ${\varepsilon}\to0$, $$\label{eq:meanfield}
\begin{aligned}
k_{t,{\varepsilon}}(x)&=q_t({\varepsilon}x)+o(1), \\
k_{t,{\varepsilon}}^{(2)}(x ,y )&=q_t({\varepsilon}x )q_t({\varepsilon}y )+o(1),\end{aligned}$$ where $q_t $ solves the following *mean-field*, or *kinetic*, equation $$\begin{aligned}
{\dfrac{\partial}{\partial t}}q_t (x) &=\int_{{\mathbb{R}^d}}a^+(x-y)q_t(y)\,dy - m q_t(x)\notag\\
&\quad - q_t(x) \int_{{\mathbb{R}^d}}a^-(x-y)q_t(y)\,dy.\label{eq:fkpp}\end{aligned}$$ Note that it was shown using another scaling, which apparently is equivalent to the considered one, see [@OFKCBK2014] for details. For various properties of solutions to , see [@FT2017c; @FKT100-1; @FKT100-2; @FKT100-3; @FKMT2017; @KT2019; @FKT2019md].
The asymptotics however does not describe effectively the covariance between random numbers $|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_1|$ and $|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_2|$, especially in the case of disjoint ${\Lambda}_1,{\Lambda}_2\in{{{{\mathcal{B}}}_\mathrm{c}}}({{\mathbb{R}^d}})$, since then, by and , $${{\mathbb{E}}}\Bigl[\Bigl(|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_1|-{{\mathbb{E}}}\bigl[|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_1|\bigr]\Bigr)\,
\Bigl(|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_2|-{{\mathbb{E}}}\bigl[|{\gamma}_{t,{\varepsilon}}\cap{\Lambda}_2|\bigr]\Bigr)\Bigr] = o(1).$$
To partially reveal the covariance above, one needs hence an enhanced asymptotics . A mathematical approach for this was proposed in [@OFKCBK2014], justifying the heuristic considerations in the early publication [@OC2006]; the approach has been recently generalised in [@CSFSO2019]. Namely, it was shown that $$\label{eq:beyondmeanfield}
\begin{aligned}
k_{t,{\varepsilon}}(x)&=q_t ({\varepsilon}x)+{\varepsilon}^d p_t ({\varepsilon}x)+o({\varepsilon}^d),\\
k_{t,{\varepsilon}}^{(2)} (x ,y )&=q_t ({\varepsilon}x )q_t ({\varepsilon}y )+{\varepsilon}^d g_t ({\varepsilon}x ,{\varepsilon}y )+o({\varepsilon}^d),
\end{aligned}$$ where $$\begin{aligned}
{\dfrac{\partial}{\partial t}}p_t(x) &=\int_{{\mathbb{R}^d}}a^+ (x- y) p_t(y) \,dy-m p_t(x) - q_t(x)\int_{{\mathbb{R}^d}}a^- (x-y) p_t(y) \,dy\notag \\& \ \ - p_t(x) \int_{{\mathbb{R}^d}}a^- (x-y) q_t(y) \,dy -\int_{{\mathbb{R}^d}}g_t(x, y) a^- (x-y)\,dy; \label{eq:eqnptx}\end{aligned}$$ and $$\begin{aligned}
{\dfrac{\partial}{\partial t}}g_t(x,y)&=
\int_{{\mathbb{R}^d}}[ g_t(x,z) a^+ (y- z)+ g_t(z,y) a^+ (x-z)]\,dz \notag -2mg_t(x,y) \\&\ \
-g_t(x,y)\int_{{\mathbb{R}^d}}[ a^- (x-z)+ a^- (y-z)] q_t(z)\,dz
\notag \\ &\ \ + a^+ (x-y)[ q_t(x) + q_t(y) ] -2 a^- (y-x) q_t(y) q_t(x) \notag \\
&\ \ - \int_{{\mathbb{R}^d}}[ a^- (x-z) q_t(x) g_t(z,y)+ a^- (y-z) q_t(y) g_t(x, z)]\,dz .
\label{eq:eqngtx} \end{aligned}$$
Space-homogeneous case {#sec:sphomcase}
======================
Let henceforth, for an integrable function $f$ on ${{\mathbb{R}^d}}$, $\hat{f}$ denote its *unitary* Fourier transform given by $$\label{eq:fourier}
\hat{f}(\xi):=\int_{{\mathbb{R}^d}}f(x) e^{- 2 i \pi x\cdot \xi}\,d\xi,$$ where $x\cdot \xi$ denotes the standard dot-product in ${{\mathbb{R}^d}}$. Note that $$\label{eq:fourest}
|\hat{f}(\xi)|\leq \int_{{\mathbb{R}^d}}|f(x)|\,dx, \qquad \xi\in{{\mathbb{R}^d}}.$$
We formulate now our basic assumptions on the kernels $a^\pm:{{\mathbb{R}^d}}\to [0,\infty)$: $$\tag{\textbf{A1}}\label{eq:newA1}
\begin{gathered}
a^\pm\in L^1({{\mathbb{R}^d}})\cap L^\infty({{\mathbb{R}^d}}); \qquad \hat{a}^\pm\in L^1({{\mathbb{R}^d}})\\
a^\pm(-x)=a^\pm(x),\quad x\in{{\mathbb{R}^d}}.
\end{gathered}$$ Note that , together with , imply that $a^\pm,\hat{a}^\pm\in L^2({{\mathbb{R}^d}})$. It is also well-known that $\hat{a}^\pm$ are (uniformly) continuous functions on ${{\mathbb{R}^d}}$.
Equation has two constant stationary solutions $q_t(x)=0$ and $q_t(x)=q^*$, where $$\label{eq:theta}
q^*:=\frac{{{\varkappa^+}}-m}{{{\varkappa^-}}}.$$ We will always assume that $$\label{eq:nondeg}
{{\varkappa^+}}>m, \tag{\textbf{A2}}$$ i.e. that $ q^* >0$; otherwise, the solution to with $q_0(x)\geq0$, $x\in{{\mathbb{R}^d}}$, would uniformly degenerate as $t\to\infty$. We assume also that $$\label{eq:compar}
J^* (x):= a^+(x)- q^* a^-(x)\geq 0, \qquad x\in{{\mathbb{R}^d}}.\tag{\textbf{A3}}$$ The reason for this restriction is as follows. By , assumption yields $$|\hat{J}^* (\xi)|\leq \int_{{\mathbb{R}^d}}|J^* (x)|\,dx
=\int_{{\mathbb{R}^d}}J^* (x)\,dx ={{\varkappa^+}}- q^* {{\varkappa^-}}=m. \label{eq:lessm}$$ Next $a^\pm(-x)=a^\pm(x)$ for $x\in{{\mathbb{R}^d}}$ implies that $\hat{a}^\pm(\xi)\in{{\mathbb{R}}}$ for $\xi\in{{\mathbb{R}^d}}$, and therefore, $$\label{eq:posFT}
{{\varkappa^+}}-\hat{J}^* (\xi)\geq {{\varkappa^+}}-m>0, \qquad \xi\in{{\mathbb{R}^d}}.$$ If fails, then (under further assumptions) there exists an infinite family of non-constant (in space) stationary solutions to , see [@KT2019].
Under assumption , if $q_0(x)=q_0$ for all $x\in{{\mathbb{R}^d}}$, then, by [@FKT100-1 Proposition 2.7], the solution to is also space homogeneous: $q_t(x)=q_t$, where $q_t$ solves the logistic differential equation $$\label{eq:logistic}
{\dfrac{d}{d t}}q_t= {{\varkappa^+}}q_t - mq_t - {{\varkappa^-}}q_t^2={{\varkappa^-}}q_t(q^*-q_t).$$ It is straightforward to check that then $$\label{eq:logisticsol}
q_t =\frac{ q^* q_0}{q_0+( q^* -q_0) e^{-({{\varkappa^+}}-m) t}},$$ hence $$\label{eq:limqt}
\lim_{t\to\infty} q_t=q^*.$$
We will assume henceforth that $$\label{eq:q0small}
0<q_0< q^* ,$$ then, by , $$\label{eq:qtbelow}
0<q_t< q^*, \qquad t>0.$$ Note that then, by , ${\dfrac{d}{d t}}q_t > 0$ for $t>0$, i.e. $q_t$ is (strictly) increasing.
Equations and are linear, and it is straightforward to check that, in the space-homogeneous case, when $p_0(x)= p_0$, $g_0(x,y)=g_0(x-y)$ for all $x,y\in{{\mathbb{R}^d}}$, this property will be preserved in time, so that takes the form $$\begin{aligned}
k_{t,{\varepsilon}} (x)&=k_{t,{\varepsilon}} =q_t +{\varepsilon}^d p_t +o({\varepsilon}^d),\label{eq:beyondmfhom1} \\
k_{t,{\varepsilon}} (x,y)& = k_{t,{\varepsilon}} (x-y)= q_t q_t + {\varepsilon}^d g_t (x-y)+o({\varepsilon}^d),\notag\end{aligned}$$ where, recall $q_t$ solves and hence is given by , and the equations for $p_t,g_t(x)$ take the following form: $$\begin{aligned}
{\dfrac{d}{d t}}p_t &= {{\varkappa^+}}p_t -m p_t - 2 {{\varkappa^-}}q_t p_t -\int_{{\mathbb{R}^d}}g_t(y) a^- (y)\,dy;\label{eq:ptfirst}\\
{\dfrac{\partial}{\partial t}}g_t(x)&=2\int_{{\mathbb{R}^d}}a^+ (x- y) g_t(y)\,dy -2 {{\varkappa^-}}q_t g_t(x) -2mg_t(x)\notag\\&\quad +2 a^+ (x) q_t-2 a^- (x) q_t ^2-2 q_t \int_{{\mathbb{R}^d}}a^- (x- y) g_t(y)\,dy.\label{eq:gtfirst}\end{aligned}$$
For $q_0\in(0,q^*)$, holds, and hence $$\label{eq:defJt}
J_t(x):=a^+(x)-q_t a^-(x)> J^* (x)\geq0.$$ One can rewrite then : $$\label{eq:gt}
{\dfrac{\partial}{\partial t}}g_t(x)=2\int_{{\mathbb{R}^d}}J_t (x- y) g_t(y)\,dy
-2 ({{\varkappa^-}}q_t+m) g_t(x)
+2 q_t J_t (x) .$$
It is straightforward to check that if $g_0\in L^1({{\mathbb{R}^d}})\cap L^\infty({{\mathbb{R}^d}})$, then $g_t\in L^1({{\mathbb{R}^d}})\cap L^\infty({{\mathbb{R}^d}})$ for all $t>0$. One can apply then the Fourier transform to both parts of to get $$\label{eq:gtF}
{\dfrac{\partial}{\partial t}}\hat{g}_t(\xi) = 2 \bigl(\hat{J}_t(\xi)-{{\varkappa^-}}q_t-m\bigr)\hat{g}_t(\xi)+ 2 q_t \hat{J}_t (\xi).$$ By the above, $g_t,a^-\in L^2({{\mathbb{R}^d}})$, $t\geq0$, and since we have chosen the unitary Fourier transform , we can rewrite , by using the Parseval identity, as follows: $$\label{eq:ptF}
{\dfrac{d}{d t}}p_t = {{\varkappa^+}}p_t -m p_t - 2 {{\varkappa^-}}q_t p_t -\int_{{\mathbb{R}^d}}\hat{g}_t(\xi) \hat{a}^-(\xi)\,d\xi.$$
We are going to find limits of $\hat g_t$ and $p_t$ as $t\to\infty$. To this end, we prove an abstract lemma which is actually an adaptation of e.g. [@Paz1983 Theorem 5.8.2] to the case of bounded operators (that apparently requires wicker conditions).
\[le:tobeused\] Let $\bigl(X,\lVert\cdot\rVert_X\bigr)$ be a Banach space, and let $\bigl(\mathcal{L}(X),\lVert\cdot\rVert
\bigr)$ denote the Banach space of linear bounded operators on $X$. Let $A\in C\bigl([0,\infty)\to\mathcal{L}(X)\bigr)$ be a continuous operator-valued function. Suppose that there exists $c,\nu>0$ such that, for all $t\geq s\geq0$, the operator $$U(t,s):=\exp\biggl( \int_s^t A(\tau)\,d\tau \biggr)\in\mathcal{L}(X)$$ satisfies $$\label{eq:Uexp}
\bigl\lVert U(t,s)\bigr\rVert \leq c e^{-\nu(t-s)}.$$ Let $f\in C\bigl([0,\infty),X\bigr)$ be a continuous $X$-valued function. Suppose that $f(t)$ converges in $X$ to some $f(\infty)\in X$ and $A(t)$ strongly converges to some $A(\infty)\in \mathcal{L}(X)$ as $t\to\infty$. Finally, suppose that $A(\infty)$ is an invertible operator, i.e. that there exists $A(\infty)^{-1}\in\mathcal{L}(X)$. Then the unique classical solution to the following non-homogeneous Cauchy problem in $X$: $$\label{eq:Cauchy}
\dfrac{d}{dt} u(t) = A(t)u(t)+f(t), \qquad u(0)=u_0\in{{\mathbb{R}^d}},$$ converges in $X$, as $t\to\infty$, to $$-A(\infty)^{-1}f(\infty)\in X.$$
Since $A\in C_b\bigl([0,\infty)\to\mathcal{X}\bigr)$, the unique classical solution $u\in C_b\bigl([0,\infty),X\bigr)$ to (i.e. such that $u\in C^1\bigl((0,\infty),X\bigr)$) is given by $$\label{eq:wfewrew}
u(t)=U(t,0)u(0)+\int_0^t U(t,s) f(s)\,ds,$$ see e.g. [@DK1974 Chapter 3] (all integrals are in the sense of Bochner henceforth). By , $$\label{eq:3r325532}
\|U(t,0)u(0)\|_X\leq e^{-\nu t}\|u(0)\|_X\to0,\qquad t\to\infty.$$
Suppose, firstly, that $f(\infty)=0$. Since then $f(s)\to0$ in $X$ as $s\to\infty$, one gets that, for any ${\varepsilon}>0$, there exists $T=T({\varepsilon})>0$ such that $\|f(s)\|\leq {\varepsilon}$ for all $s\geq T$. Since $f\in C\bigl([0,\infty),X\bigr)$, one can define $\|f\|_T:=\sup\limits_{t\in[0,T]}\|f(t)\|_{X}<\infty$. Then $$\begin{aligned}
\biggl\lvert \int_0^t U(t,s) f(s)\,ds\biggr\rVert&\leq
\int_0^T \lVert U(t,s)\rVert \|f(s)\|_X\,ds
+{\varepsilon}\int_T^t \lVert U(t,s)\rVert \,ds\\
&\leq c\|f\|_T \int_0^T e^{-\nu(t-s)}\,ds+
c\,{\varepsilon}\int_T^t e^{-\nu(t-s)}\,ds\\
&\leq c\|f\|_T\frac{1}{\nu}e^{-\nu(t-T)}+\frac{c \,{\varepsilon}}{\nu},
\end{aligned}$$ and combining this with , one gets that $u(t)\to 0=u(\infty)$ in $X$ as $t\to\infty$.
For a general $f(\infty)\in X$, denote $v(t):=u(t)-u(\infty)$. Since $A(\infty)$ is invertible, one can define $$u(\infty):=-A(\infty)^{-1} f(\infty)\in X.$$ Then $$\dfrac{d}{dt} v(t)=\dfrac{d}{dt}u(t)=A(t)u(t)+f(t)
=A(t)v(t)+f(t)+A(t)u(\infty).$$ We set $g(t):=f(t)+A(t)u(\infty)$, $t\geq0$. By the assumptions on $f$ and $A$, $g\in C\bigl([0,\infty),X\bigr)$ and $$\lim_{t\to\infty} g(t)=f(\infty)+A(\infty)\bigl( -A(\infty)^{-1}f(\infty) \bigr) =0,$$ where the limit is in $X$. Then, by the proved above, $v(t)\to 0$ in $X$, and hence $u(t)\to u(\infty)$ in $X$.
\[thm:limits\] Let – hold. Let $q_0$ satisfies and $g_0, \hat{g}_0\in L^1({{\mathbb{R}^d}})\cap L^\infty({{\mathbb{R}^d}})$. Then there exist limits $$\begin{aligned}
\hat{g}^* (\xi):&=\lim_{t\to\infty} \hat{g}_t(\xi)=\frac{q^* \hat{J}^* (\xi)}{{{\varkappa^+}}-\hat{J}^* (\xi)}\leq \frac{m}{{{\varkappa^-}}}, \qquad \xi\in{{\mathbb{R}^d}}, \label{eq:gfstat}\\
p^*:&=\lim_{t\to\infty} p_t=-\frac{1}{{{\varkappa^-}}}\int_{{\mathbb{R}^d}}\frac{\hat{J}^* (\xi)}{{{\varkappa^+}}-\hat{J}^* (\xi)}\hat{a}^-(\xi)\,d\xi\in{{\mathbb{R}}}.
\label{eq:pstat}\end{aligned}$$ Moreover, the convergence in takes place in the norms of both $L^1({{\mathbb{R}^d}})$ and $L^\infty({{\mathbb{R}^d}})$. As a result, $g_t$ converges, as $t\to\infty$, in $L^\infty({{\mathbb{R}^d}})$ to $g^*$, the inverse Fourier transform of $\hat{g}^*$.
We will actually use in the proof a part of the estimate only, rather than the more strict assumption . More precisely, it is easy to check that all arguments of the proof remain correct if the assume, instead of , that, for some $\alpha\in(0,{{\varkappa^+}})$, $${{\varkappa^+}}-\hat{J}^* (\xi)\geq \alpha, \qquad \xi\in{{\mathbb{R}^d}};$$ the uppear bound in will be then replaced by $q^*\frac{{{\varkappa^+}}- \alpha}{\alpha}$.
We denote $$\label{eq:defj}
j(\xi,t):=\hat{J}_t(\xi)-{{\varkappa^-}}q_t-m, \qquad \xi\in{{\mathbb{R}^d}},\ t\geq0,$$ and apply Lemma \[le:tobeused\] to equation , where $X=L^1({{\mathbb{R}^d}})$ or $X=L^\infty({{\mathbb{R}^d}})$, $A(t)$ is the multiplication operator by the function $2j(\xi,t)$, and $f(t,\xi)=2q_t\hat{J}_t(\xi)$. Note that, for any $t\geq s\geq0$, $\xi\in{{\mathbb{R}^d}}$, we have, by , , , $$\begin{aligned}
\notag
\hat{J}_t(\cdot), j(\cdot,t)&\in L^1({{\mathbb{R}^d}})\cap L^\infty({{\mathbb{R}^d}}),\\
|j(\xi,t)-j(\xi,s)|&\leq 2|\hat{J}_t(\xi)-\hat{J}_s(\xi)|+2{{\varkappa^-}}|q_t-q_s| \leq 4{{\varkappa^-}}\lvert q_t-q_s\rvert,\notag\\
|f(t,\xi)-f(s,\xi)|&\leq 2|q_t-q_s||\hat{J}_t(\xi)|
+2q_s|\hat{J}_t(\xi)-\hat{J}_s(\xi)|\notag\\
&\leq 2\bigl(|\hat{a}^+(\xi)|+q^*(q^*+1)|\hat{a}^-(\xi)| \bigr)|q_t-q_s|.\label{eq:AFR3WRT}\end{aligned}$$ Therefore, $A\in C\bigl([0,\infty)\to\mathcal{L}(X)\bigr)$ and $f\in C\bigl([0,\infty),X\bigr)$ for both $X=L^1({{\mathbb{R}^d}})$ and $X=L^\infty({{\mathbb{R}^d}})$. Note that, by , $$\hat{g}_t(\xi) = \exp\biggl( 2 \int_0^t j(\xi,\tau)\,d \tau\biggr)\hat{g}_0(\xi)
+\int_0^t\exp\biggl( 2 \int_s^t j(\xi,\tau)\,d\tau\biggr) q_s \hat{J}_s (\xi)\,ds,$$ and then, by the Riemann–Lebesgue lemma, $\hat{g}_t(\cdot)\in C({{\mathbb{R}^d}})$, $t\geq0$.
By , we also have that $f(t,\xi)$ converges, in the norm of either of $X$ to $2q^*\hat{J}^*(\xi)$; and also $A(t)$ strongly converges to the operator $A(\infty)$ of the multiplication by $$2\lim_{t\to\infty} \hat{J}_t(\xi)=2(\hat{J}^*(\xi)-{{\varkappa^-}}q^*-m)=2(\hat{J}^*(\xi)-{{\varkappa^+}}).$$ By , operator $A(\infty)$ is invertible.
Next, for all $t> s\geq0$, $\int_s^t A(\tau)\,d \tau$ is the operator of multiplication by $2\int_s^t j(\cdot,\tau)\, d \tau$. We have $$\int_s^t j(\xi,\tau)\,d \tau=\bigl(\hat{a}^+(\xi)-m\bigr)(t-s)-(\hat{a}^-(\xi)+{{\varkappa^-}})\int_s^t q_\tau\,d \tau.\label{eq:intermed}$$ Since, by , $${\dfrac{d}{d t}}\log q_t =\frac{1}{ q_t }{\dfrac{d}{d t}}q_t ={{\varkappa^+}}-m - {{\varkappa^-}}q_t,$$ we get $$\log q_t-\log q_s = \int_s^t \frac{d}{d \tau}\log q_\tau d \tau= ({{\varkappa^+}}-m )(t-s)- {{\varkappa^-}}\int_s^t q_\tau d \tau,$$ and hence $$\label{eq:intofqt}
\int_s^t q_\tau d \tau = q^* (t-s)-\frac{1}{{{\varkappa^-}}}\log \frac{q_t}{q_s}.$$ Substituting into , and using and that $q_t$ is increasing and $|\hat{a}^-(\xi)|\leq {{\varkappa^-}}$ holds, we get $$\int_s^t j(\xi,\tau)\,d \tau= -\bigl({{\varkappa^+}}-\hat{J}^* (\xi)\bigr) (t-s)+\frac{\hat{a}^-(\xi)+{{\varkappa^-}}}{{{\varkappa^-}}}\log \frac{q_t}{q_s}.\label{eq:return}$$
Therefore, cosnidering a multiplication operator $U(t,s)=\exp\Bigl(\int_s^t A(\tau)\,d \tau \Bigr)$ and using and that $$|\hat{a}^-(\xi)| \leq {{\varkappa^-}}, \qquad 0<q_0\leq q_s<q_t<q^*, \quad t>s>0,$$ we get from that, in either of spaces $X$, $$\|U(t,s)\|=\sup_{\xi\in{{\mathbb{R}^d}}}\exp\biggl(2\int_s^t j(\xi,\tau)\,d \tau\biggr)\leq \biggl(\frac{q^*}{q_0}\biggr)^4 e^{-2({{\varkappa^+}}-m)(t-s)}. \label{eq:intest0}$$
Therefore, by Lemma \[le:tobeused\], $$\hat{g}_t(\xi)\to - A(\infty)^{-1}f(\infty)
=-\frac{1}{2(\hat{J}^*(\xi)-{{\varkappa^+}})}2q^*\hat{J}^*(\xi)=
\frac{q^*\hat{J}^*(\xi)}{{{\varkappa^+}}-\hat{J}^*(\xi))}=:\hat{g}^*(\xi)$$ in the sense of norm in both $L^1({{\mathbb{R}^d}})$ and $L^\infty({{\mathbb{R}^d}})$ (and, in particular, pointwise). We also have, by , that $$|\hat{g}^*(\xi)|\leq q^*\frac{m}{{{\varkappa^+}}-m}=\frac{m}{{{\varkappa^-}}}, \qquad \xi\in{{\mathbb{R}^d}},$$ that finishes the proof of .
Since $\hat{g}_t\in L^1({{\mathbb{R}^d}})$, its inverse Fourier transform coinsides with $g_t$ a.e.; in particular, they coincide as elements of $L^\infty({{\mathbb{R}^d}})$. Hence if $g^*$ is the inverse Fourier transform of $\hat{g}^*\in L^1({{\mathbb{R}^d}})$, then, by , $$\|g_t-g^*\|_{L^\infty({{\mathbb{R}^d}})}\leq \|\hat g_t-\hat g^*\|_{L^1({{\mathbb{R}^d}})}\to0,\qquad t\to\infty,$$ that proves the last statement of Theorem \[thm:limits\].
We are going to apply now Lemma \[le:tobeused\] to equation , with $X={{\mathbb{R}}}$. Since $\hat{g}_t\to \hat{g}^*$ in $L^\infty({{\mathbb{R}^d}})$ and $\hat{g}_t$ is a classical solution to in $L^\infty({{\mathbb{R}^d}})$ (i.e. a continuous mapping from $[0,\infty)$ to $L^\infty({{\mathbb{R}^d}})$), function $\hat{g}_t(\xi)$ is globally bounded in $t\geq0$ and $\xi\in{{\mathbb{R}^d}}$. Then $$-\int_{{\mathbb{R}^d}}\hat{g}_t(\xi) \hat{a}^-(\xi)\,d\xi\to-\int_{{\mathbb{R}^d}}\hat{g}^*(\xi) \hat{a}^-(\xi)\,d\xi, \qquad t\to\infty,$$ by the dominated convergence theorem. $A(t)$ is given now through the multiplication by $c_t:={{\varkappa^+}}-m - 2 {{\varkappa^-}}q_t$, and, by , $$\begin{aligned}
\int_s^t c_\tau d \tau&=({{\varkappa^+}}-m)(t-s)- 2 {{\varkappa^-}}q^* (t-s)+2 \log \frac{q_t}{q_s}\notag\\
&=-({{\varkappa^+}}-m)(t-s)+2 \log \frac{q_t}{q_s}\leq-({{\varkappa^+}}-m)(t-s)+2 \log \frac{q^*}{q_0}.\end{aligned}$$ Hence, by the same arguments as before, we can apply Lemma \[le:tobeused\]: since, by , $$\lim_{t\to\infty} c_t={{\varkappa^+}}-m - 2 {{\varkappa^-}}q^*=
-{{\varkappa^-}}q^*,$$ we get $$p_t\to -\frac{1}{{{\varkappa^-}}q^*}\int_{{\mathbb{R}^d}}\hat{g}^*(\xi) \hat{a}^-(\xi)\,d\xi,$$ that implies .
Critical mortality {#sec:critmort}
==================
We are going to discuss now the extinction regime. Recall that $o({\varepsilon}^d)$ in depends on $t$, so we have $$k_{t,{\varepsilon}}=q_t+{\varepsilon}^d p_t+o_t({\varepsilon}^d),$$ where, for each $t>0$, $$\lim_{{\varepsilon}\to0}\frac{o_t({\varepsilon}^d)}{{\varepsilon}^d}=0.$$ We will assume that $$\lim_{t\to\infty}o_t({\varepsilon}^d)=o({\varepsilon}^d).$$ Then, the extinction takes place if and only if holds.
We fix an $m\in(0,{{\varkappa^+}})$ for which holds. We consider a function ${m_\mathrm{cr}}:(0,1)\to (m,{{\varkappa^+}})$ and set, cf. , for ${\varepsilon}\in(0,1)$, $$\begin{aligned}
\label{eq:qepsbdd}
q^*({\varepsilon}):&=\frac{{{\varkappa^+}}-{m_\mathrm{cr}}({\varepsilon})}{{{\varkappa^-}}}\in\Bigl( 0,\frac{{{\varkappa^+}}-m}{{{\varkappa^-}}}\Bigr),\\
\intertext{and also, cf. \eqref{eq:compar},} \label{eq:defjeps}
J_{\varepsilon}(x):&=a^+ (x)- q^*({\varepsilon}) a^- (x)\geq0,
\\
\intertext{because of \eqref{eq:compar}, \eqref{eq:qepsbdd}. Finally, we set, cf. \eqref{eq:pstat}, for ${\varepsilon}\in(0,1)$,}
\label{eq:pepsint}
p^*({\varepsilon}):&=-\frac{1}{{{\varkappa^-}}}\int_{{\mathbb{R}^d}}\frac{\hat{J}_{\varepsilon}(\xi)}{{{\varkappa^+}}-\hat{J}_{\varepsilon}(\xi)}\hat{a}^-(\xi)\,d\xi.\end{aligned}$$ Note that, by , , $$\label{eq:ineqhatJeps}
|\hat{J}_{\varepsilon}(\xi)|\leq \int_{{\mathbb{R}^d}}\bigl\lvert {J}_{\varepsilon}(x)\bigr\rvert\,dx=\int_{{\mathbb{R}^d}}{J}_{\varepsilon}(x) \,dx = {m_\mathrm{cr}}({\varepsilon}),$$ and hence $p^*({\varepsilon})$ is well-defined: and yield $$|p^*({\varepsilon})|\leq \frac{1}{{{\varkappa^-}}}\frac{{m_\mathrm{cr}}({\varepsilon})}{{{\varkappa^+}}-{m_\mathrm{cr}}({\varepsilon})}\int_{{\mathbb{R}^d}}\bigl\lvert \hat{a}^-(\xi)\bigr\rvert\,d\xi<\infty, \qquad {\varepsilon}\in(0,1).$$
In the rest of the paper, our main object of interest will be the following equation, cf. , $$\label{eq:mcrchar}
q^*({\varepsilon})+ {\varepsilon}^d p^*({\varepsilon})+o({\varepsilon}^d)=0.$$
\[prop:evidentlimit\] Let – hold. If holds and there exists $\lim\limits_{{\varepsilon}\to0}{m_\mathrm{cr}}({\varepsilon})$, then $$\label{eq:mcritconv}
\lim_{{\varepsilon}\to0}{m_\mathrm{cr}}({\varepsilon})={{\varkappa^+}}, \qquad
\lim_{{\varepsilon}\to0}q^*({\varepsilon})=0.$$
Clearly, ${m_\mathrm{cr}}(0):=\lim\limits_{{\varepsilon}\to0}{m_\mathrm{cr}}({\varepsilon})\leq {{\varkappa^+}}$. Suppose that ${m_\mathrm{cr}}(0)<{{\varkappa^+}}$. Let $\alpha\in(0,1)$ be such that $\alpha {{\varkappa^+}}>{m_\mathrm{cr}}(0)$. Then there exists ${\varepsilon}_\alpha\in(0,1)$ such that ${m_\mathrm{cr}}({\varepsilon})<\alpha {{\varkappa^+}}$ for all $0<{\varepsilon}<{\varepsilon}_\alpha$. Therefore, by , $$|\hat{J}_{\varepsilon}(\xi)| <\alpha {{\varkappa^+}}, \qquad \xi\in{{\mathbb{R}^d}}, \ 0<{\varepsilon}<{\varepsilon}_\alpha.$$ Then, by , , $$|p^*({\varepsilon})|\leq \frac{1}{{{\varkappa^-}}}\int_{{\mathbb{R}^d}}\frac{\alpha {{\varkappa^+}}}{{{\varkappa^+}}-\alpha {{\varkappa^+}}}\hat{a}^-(\xi)\,d\xi<\infty,$$ and hence ${\varepsilon}^d p^*({\varepsilon})\to0$, ${\varepsilon}\to0$. Therefore, by , we get that $q^*({\varepsilon})\to0$ and hence, by , ${m_\mathrm{cr}}({\varepsilon})\to{{\varkappa^+}}$ that contradicts the assumption. The statement is proved.
The behaviour of $p^*({\varepsilon})$ as ${\varepsilon}\to0$ depends on the dimension $d\in{\mathbb{N}}$: the limit (as ${\varepsilon}\to0$) of the integrand in is equal to, because of , $\dfrac{\hat{a}^+(\xi)\hat{a}^-(\xi)}{{{\varkappa^+}}-\hat{a}^+(\xi)}$ that has a singularity at the origin, which is, in general, non-integrable for $d<3$. We discuss this under the following additional assumption: $$\tag{\textbf{A4}}\label{eq:secmom}
\int_{{\mathbb{R}^d}}|x|^2a^+(x)\,dx<\infty.$$ Since $a^+ \in L^\infty({{\mathbb{R}^d}})$, the inequality in implies that $\int_{{\mathbb{R}^d}}|x|a^+(x)\,dx<\infty$, and using that $a^+(-x)=a^+(x)$, $x\in{{\mathbb{R}^d}}$, we get $$\int_{{\mathbb{R}^d}}x a^+ (x)\,dx=0\in{{\mathbb{R}^d}}.$$ Then, by the Taylor expansion for $\hat{a}^+(\xi)$ defined by , we get (cf. [@Sas2013 Corollary 1.2.7] for another coefficients of the Fourier transform) that $$\hat{a}^+(\xi)=\hat{a}^+(0)-2\pi^2\sum_{i,j=1}^d a^+_{i,j} \xi_i\xi_j+ o(|\xi|^2)={{\varkappa^+}}-2\pi^2 A^+\xi\cdot\xi+ o(|\xi|^2),$$ for $\xi\to0$, where $$\label{eq:defofaplus}
a^+_{i,j} := \int_{{\mathbb{R}^d}}x_ix_j a^+ (x)\,dx, \qquad 1\leq i,j\leq d,$$ and hence $A^+:=\bigl( a_{i,j} \bigr)_{i,j=1}^d$ is a Hermitian (strictly) positive definite matrix. Then there exists a Hermitian (strictly) positive definite matrix $B^+$ such that $(B^+)^2=A^+$, and hence $$\label{eq:Texpcf}
{{\varkappa^+}}- \hat{a}^+(\xi)=2\pi^2|B^+ \xi|^2 + o(|\xi|^2), \qquad \xi\to0.$$
Under assumptions –, assumption holds with $a^+$ replaced by $a^-$ and hence by $J_{\varepsilon}$ or $J^*$. Let $B^-,B_{\varepsilon},B^*$ be the Hermitian positive definite matrices corresponding to the functions $a^-,J_{\varepsilon},J^*$, respectively. Then, the corresponding analogues of hold, with, in particular, ${{\varkappa^+}}$ replaced by ${{\varkappa^-}}=\hat{a}^\pm(0)$, ${m_\mathrm{cr}}({\varepsilon})=\hat{J}_{\varepsilon}(0)$, $m=\hat{J}^*(0)$, respectively. It is easy to see also that $$\label{eq:soineedthis}
|B_{\varepsilon}\xi|^2=|B^+\xi|^2-q^*({\varepsilon})|B^-\xi|^2, \qquad \xi\in{{\mathbb{R}^d}}, \ {\varepsilon}\in(0,1).$$
Next, for any invertible matrix $B$, $$\label{eq:dblest}
\bigl( \bigl\lVert (B )^{-1}\bigr\rVert \bigr)^{-1}|\xi| \leq |B \xi| \leq \|B \||\xi|, \qquad \xi\in{{\mathbb{R}^d}},$$ Then, for small enough $\delta>0$, $$\label{eq:dblestlemma}
\frac{\pi^2}{\lVert (B^\pm)^{-1} \rVert^2}\lvert \xi\rvert ^2 \leq {{\varkappa^\pm}}-\hat{a}^\pm(\xi)\leq 3\pi^2\lVert B^\pm\rVert^2 \lvert \xi\rvert ^2, \qquad |\xi|\leq \delta.$$ The corresponding double inequalities can be also obtained for $J_{\varepsilon}$ and $J^*$.
\[eq:limitofpstar\] Let – hold. Let also ${m_\mathrm{cr}}:(0,1)\to (m,{{\varkappa^+}})$ and $p^*({\varepsilon})$, defined through –, be such that holds. Then, for $d\geq3$, $$\label{eq:pstar0}
\lim_{{\varepsilon}\to0}p^*({\varepsilon})=-\frac{1}{{{\varkappa^-}}}\int_{{\mathbb{R}^d}}\frac{\hat{a}^+(\xi)\hat{a}^-(\xi)}{{{\varkappa^+}}-\hat{a}^+(\xi)}\,d\xi=:p^*(0)\in{{\mathbb{R}}},$$ whereas, for $d\leq 2$, $\lim\limits_{{\varepsilon}\to0}p^*({\varepsilon})=-\infty$.
Note that we do not need to assume to get the statement.
For each $\delta>0$, one can expand $p^*({\varepsilon})$ as follows $$\begin{aligned}
p^*({\varepsilon})&=p_{\leq \delta}^*({\varepsilon})+p_{\geq \delta}^*({\varepsilon})\\ &=:
-\frac{1}{{{\varkappa^-}}}\int_{|\xi|\leq \delta}\frac{\hat{J}_{\varepsilon}(\xi)}{{{\varkappa^+}}-\hat{J}_{\varepsilon}(\xi)}\hat{a}^-(\xi)\,d\xi-\frac{1}{{{\varkappa^-}}}\int_{|\xi|\geq \delta} \frac{\hat{J}_{\varepsilon}(\xi)}{{{\varkappa^+}}-\hat{J}_{\varepsilon}(\xi)}\hat{a}^-(\xi)\,d\xi.\end{aligned}$$ To estimate $p_{\geq \delta}^*({\varepsilon})$, we verify firsly the following inequality: $$\label{eq:inequnif}
{{\varkappa^+}}- \hat{J}_{\varepsilon}(\xi)> m-\hat{J}^*(\xi)\geq0, \qquad \xi\in{{\mathbb{R}^d}}.$$ Namely, by , , the first inequality in is equivalent to $${{\varkappa^+}}-m> \bigl(q^*-q^*({\varepsilon})\bigr)\hat{a}^-(\xi), \qquad \xi\in{{\mathbb{R}^d}},$$ that is true since $|\hat{a}^-(\xi)|\leq {{\varkappa^-}}$, $\xi\in{{\mathbb{R}^d}}$, and $q^*-q^*({\varepsilon})<q^*=\frac{{{\varkappa^+}}-m}{{{\varkappa^-}}}$. The second inequality in is just .
Next, since the function $$m-\hat{J}^*(\xi)\geq m-\hat{J}^*(0)=0$$ is continuous in $\xi\in{{\mathbb{R}^d}}$, we conclude, cf. , that, for any $\delta>0$, there exists $\mu_\delta>0$, such that $$m-\hat{J}^*(\xi)\geq \mu_\delta, \qquad |\xi|\geq \delta,$$ and hence $$\label{eq:estextball}
|p_{\geq \delta}^*({\varepsilon})|\leq
\frac{1}{{{\varkappa^-}}} \frac{{{\varkappa^+}}}{\mu_\delta} \int_{{\mathbb{R}^d}}\bigl\lvert\hat{a}^-(\xi)\bigr\rvert\,d\xi<\infty.$$
Next, by an analogue of for $J_{\varepsilon}$, we get, for small enough $\delta>0$, $${{\varkappa^+}}-\hat{J}_{\varepsilon}(\xi)\geq {m_\mathrm{cr}}({\varepsilon})-\hat{J}_{\varepsilon}(\xi)\geq \frac{\pi^2}{\lVert B_{\varepsilon}^{-1}\rVert^2}|\xi|^2,$$ and hence, $$|p^*_{\leq \delta}({\varepsilon})|\leq {\mathrm{const}\,}\int_{|\xi|\leq \delta} \frac{1}{|\xi|^2}\,d\xi<\infty \qquad \text{for } d\geq3.$$ Combining the latter estimate with , we get that, for $d\geq3$, holds by and the dominated convergence theorem.
Let now $d\leq 2$. By , one can always choose $\delta_0>0$ small enough to ensure that, for $\delta<\delta_0$, $$\hat{a}^-(\xi)\geq {{\varkappa^-}}- 3\pi^2\lVert B^-\rVert^2 \lvert \xi\rvert ^2>\frac{{{\varkappa^-}}}{2}>0, \qquad |\xi|\leq \delta.$$ Then $$\hat{J}_{\varepsilon}(\xi)=\hat{a}^+(\xi) - q^*({\varepsilon})\hat{a}^-(\xi)\geq \hat{J}^*(\xi), \qquad |\xi|\leq \delta<\delta_0,$$ and, possibly redefining $\delta_0$, we similarly get that $$\hat{J}^*(\xi)\geq m - 3\pi^2\lVert B^*\rVert^2 \lvert \xi\rvert ^2>\frac{m}{2}>0, \qquad |\xi|\leq \delta<\delta_0.$$ Next, by applied to $a=J_{\varepsilon}$, we get $$\begin{aligned}
{{\varkappa^+}}-\hat{J}_{\varepsilon}(\xi)&={{\varkappa^+}}-{m_\mathrm{cr}}({\varepsilon})+{m_\mathrm{cr}}({\varepsilon})-\hat{J}_{\varepsilon}(\xi)\\&\leq {{\varkappa^+}}-{m_\mathrm{cr}}({\varepsilon})+3\pi^2\|B_{\varepsilon}\|^2 |\xi|^2\\
&\leq {{\varkappa^+}}-{m_\mathrm{cr}}({\varepsilon})+3\pi^2\|B^+\|^2 |\xi|^2, \qquad |\xi|\leq \delta,\end{aligned}$$ where we used . Combining the previous inequalities, we get that, for a fixed $\delta<\delta_0$, $$\begin{aligned}
-p_{\leq \delta}^*({\varepsilon})&=
\frac{1}{{{\varkappa^-}}}\int_{|\xi|\leq \delta}\frac{\hat{J}_{\varepsilon}(\xi)}{{{\varkappa^+}}-\hat{J}_{\varepsilon}(\xi)}\hat{a}^-(\xi)\,d\xi\\
&\geq \frac{m}{4}\int_{|\xi|\leq \delta}\frac{1}{{{\varkappa^+}}-{m_\mathrm{cr}}({\varepsilon})+3\pi^2\|B^+\|^2 |\xi|^2}\,d\xi.\end{aligned}$$
Therefore, for $d=2$, we get, by passing to polar coordinates, that $$-p_{\leq \delta}^*({\varepsilon})\geq c_1\log\biggl(1+\frac{c_2}{{{\varkappa^+}}-{m_\mathrm{cr}}({\varepsilon})}\biggr);$$ and, for $d=1$, we get that $$-p_{\leq \delta}^*({\varepsilon})\geq \frac{c_3}{\sqrt{{{\varkappa^+}}-{m_\mathrm{cr}}({\varepsilon})}} \arctan\frac{c_4}{\sqrt{{{\varkappa^+}}-{m_\mathrm{cr}}({\varepsilon})}},$$ for certain $c_1,c_2,c_3,c_4>0$ (with $c_2,c_4$ depending on the fixed $\delta$). Since, by , ${{\varkappa^+}}-{m_\mathrm{cr}}({\varepsilon})\to0$ as ${\varepsilon}\to0$, the statement is proved.
Asymptotics of the critical mortality: d>=3 {#sec:asympd3}
==============================================
We are going to reveal the asymptotic of ${m_\mathrm{cr}}({\varepsilon})$ (or, equivalently, $q^*({\varepsilon})$) assuming that does hold. We start with the case $d\geq3$.
If, additionally to – and , we assume that the limit $\lim\limits_{{\varepsilon}\to0}{m_\mathrm{cr}}({\varepsilon})$ exists, then, by Proposition \[prop:evidentlimit\], holds, and hence, by Proposition \[eq:limitofpstar\], we get . Then implies $$\lim_{{\varepsilon}\to0}\frac{q^*({\varepsilon})}{{\varepsilon}^d}=-\lim_{{\varepsilon}\to0}p^*({\varepsilon})-\lim_{{\varepsilon}\to0}\frac{o({\varepsilon}^d)}{{\varepsilon}^d}=- p^*(0)=\frac{1}{{{\varkappa^-}}}\int_{{\mathbb{R}^d}}\frac{\hat{a}^+(\xi)\hat{a}^-(\xi)}{{{\varkappa^+}}-\hat{a}^+(\xi)}\,d\xi.$$ Since $q^*({\varepsilon})>0$, we will get that $I\geq0$, where $$\label{eq:int}
I:=\int_{{\mathbb{R}^d}}\frac{\hat{a}^+(\xi)}{ {{\varkappa^+}}-\hat{a}^+(\xi)}\hat{a}^-(\xi)\,d\xi.$$
The first statement of the following theorem shows that one can replace the requiremnt about existence of the limit of ${m_\mathrm{cr}}({\varepsilon})$ by the continuity of the function $$\label{r1eps}
r({\varepsilon}):=\frac{o({\varepsilon}^d)}{{\varepsilon}^d}$$ in a neighbourhood of $0$, where $o({\varepsilon}^d)$ is from . Then, we reveal the next term of the asymptotic under additional smoothness of $r({\varepsilon})$.
\[thm:assympd3\] Let $d\geq3$ and –, hold. Let $r$ given by be continuous for small ${\varepsilon}>0$. Let $I\neq0$, where $I$ is given by . Then $I>0$ and $$\label{eq:asymptdgeq3}
q^*({\varepsilon})= \frac{I}{ {{\varkappa^-}}} {\varepsilon}^{d} +o({\varepsilon}^{d}).$$ As a result, $${m_\mathrm{cr}}({\varepsilon})={{\varkappa^+}}-{\varepsilon}^d I +o({\varepsilon}^d).\label{eq:mcrdgeq3}$$
If, additionally, $r({\varepsilon})$ is continuously differentiable for small ${\varepsilon}>0$ and if $r'(0):=\lim\limits_{{\varepsilon}\to0+}r'({\varepsilon})<\infty$, then it determines the next term of the asymptotics, namely, in , $$\label{eq:32235352dfgfdg}
o({\varepsilon}^{d})= - r'(0){\varepsilon}^{d+1} +o({\varepsilon}^{d+1}).$$
We denote $$\label{eq:4343regf536}
{\lambda}({\varepsilon}):={\varepsilon}^{-d} q^* ({\varepsilon}).$$ One can rewrite then as follows $$\label{eq:2345gsdx436}
{\lambda}({\varepsilon})-\frac{1}{ {{\varkappa^-}}}
\int_{{\mathbb{R}^d}}\frac{\hat{a}^+(\xi)-{\varepsilon}^d {\lambda}({\varepsilon})\hat{a}^-(\xi)}{ {{\varkappa^+}}-\hat{a}^+(\xi)+{\varepsilon}^d {\lambda}({\varepsilon})\hat{a}^-(\xi)}\hat{a}^-(\xi)\,d\xi+r({\varepsilon})=0.$$
*Step 1.* Note that, since $d\geq3$, we have $|I|<\infty$, by the arguments above. We set $${\lambda}_3:=\frac{|I|}{{{\varkappa^-}}}\in(0,\infty),$$ Let $\delta\in\bigl(0,\min\{{\lambda}_3,1\}\bigr)$ be such that, cf. , $$\label{eq:condondelta3}
({\lambda}_3+\delta)\delta^d<\frac{{{\varkappa^+}}-m}{{{\varkappa^-}}},$$ and let also $r({\varepsilon})$, given by , be continuous on $(0,\delta)$. We set then $$\label{eq:sarqr33434}
r(0):=0=\lim\limits_{{\varepsilon}\to0+}r({\varepsilon}), \qquad r(-{\varepsilon}):=r({\varepsilon}), \quad {\varepsilon}\in(0,\delta).$$
For $$({\lambda},{\varepsilon})\in E_\delta:=({\lambda}_3- \delta,{\lambda}_3 +\delta)\times (- \delta,\delta),$$ we consider the function $$F({\lambda},{\varepsilon}):=\frac{1}{ {{\varkappa^-}}}
\int_{{\mathbb{R}^d}}\frac{\hat{a}^+(\xi)-|{\varepsilon}|^d {\lambda}\, \hat{a}^-(\xi)}{ {{\varkappa^+}}-\hat{a}^+(\xi)+|{\varepsilon}|^d {\lambda}\, \hat{a}^-(\xi)}\hat{a}^-(\xi)\,d\xi-{\lambda}{\,\mathrm{sgn}}(I) -r({\varepsilon}).$$ Henceforth ${\,\mathrm{sgn}}(I)=1$ for $I>0$ and ${\,\mathrm{sgn}}(I)=-1$ for $I<0$.
For $({\lambda},{\varepsilon})\in E_\delta$ and $\delta$ as in , we have that $a^+(x)-{\lambda}|{\varepsilon}|^d a^-(x)\geq 0$, $x\in{{\mathbb{R}^d}}$. Hence one can apply Proposition \[eq:limitofpstar\] for a (new) function ${m_\mathrm{cr}}({\varepsilon})$ such that $|{\varepsilon}|^d {\lambda}=\frac{{{\varkappa^+}}-{m_\mathrm{cr}}({\varepsilon})}{{{\varkappa^-}}}$. It yields then $$\label{eq:Fla0}
\lim_{{\varepsilon}\to0}F({\lambda},{\varepsilon})=F({\lambda},0)=\frac{1}{{{\varkappa^-}}}I-{\lambda}{\,\mathrm{sgn}}(I),$$ and since $|I|{\,\mathrm{sgn}}(I)=I$, we get $$\label{eq:evneed}
F({\lambda}_3,0)=0.$$
*Step 2.* By and the dominated convergence theorem, $F$ is continuous on $E_\delta$, for small enough $\delta>0$. Prove that $\frac{\partial }{\partial {\lambda}}F({\lambda},{\varepsilon})$ is continuously differentiable on $E_\delta$, for small enough $\delta>0$. We have $$\label{eq:sagewetew}
\frac{\partial }{\partial {\lambda}}F({\lambda},{\varepsilon})=-\frac{1}{ {{\varkappa^-}}}
\int_{{\mathbb{R}^d}}\frac{|{\varepsilon}|^d \, \hat{a}^-(\xi)}{ \bigl({{\varkappa^+}}-\hat{a}^+(\xi)+|{\varepsilon}|^d {\lambda}\, \hat{a}^-(\xi)\bigr)^2}\hat{a}^-(\xi)\,d\xi-{\,\mathrm{sgn}}(I),$$ that is continuous for $({\lambda},{\varepsilon})\in E_\delta$, ${\varepsilon}\neq0$.
For ${\varepsilon}=0$, ${\lambda}\in({\lambda}_3- \delta, {\lambda}_3+\delta)$, we have by , $$\label{eq:partder}
\frac{\partial }{\partial {\lambda}}F({\lambda},0)=\lim_{h\to0}\frac{F({\lambda}+h,0)-F({\lambda},0)}{h}=-{\,\mathrm{sgn}}(I)\neq0.$$
By , the denominator in behaves as $|\xi|^4$ near the origin that is integrable for $d\geq5$ only. In the latter case, using again and the dominated convergence theorem, we get from that $$\label{eq:r23233524}
\lim_{{\varepsilon}\to0} \frac{\partial }{\partial {\lambda}}F({\lambda},{\varepsilon})=-{\,\mathrm{sgn}}(I),$$ hence, by , $\frac{\partial }{\partial {\lambda}}F$ is continuous at $({\lambda},0)$ for all ${\lambda}\in({\lambda}_3- \delta,{\lambda}_3 + \delta)$.
Let now $d=3,4$. Find $\lim_{{\varepsilon}\to0} \frac{\partial }{\partial {\lambda}}F({\lambda},{\varepsilon})$. By the same arguments as for getting , we obtain that $$\int_{{{\mathbb{R}^d}}\setminus \Delta_\delta}\frac{\bigl(\hat{a}^-(\xi)\bigr)^2}{ \bigl({{\varkappa^+}}-\hat{a}^+(\xi)+|{\varepsilon}|^d {\lambda}\, \hat{a}^-(\xi)\bigr)^2}\,d\xi<\infty,$$ for any neighbourhood $\Delta_\delta$ of the origin, $0\in\Delta_\delta\subset{{\mathbb{R}^d}}$, of a positive Lebesgue measure. Therefore, for any such $\Delta_\delta$ with small enough $\delta>0$, $$\lim_{{\varepsilon}\to0}
\frac{\partial }{\partial {\lambda}}F({\lambda},{\varepsilon})=-\frac{1}{ {{\varkappa^-}}}
\lim_{{\varepsilon}\to0} |{\varepsilon}|^d h({\varepsilon},\delta)-{\,\mathrm{sgn}}(I),$$ where $$h({\varepsilon}, \delta):=
\int_{\Delta_\delta}
\frac{\bigl(\hat{a}^-(\xi)\bigr)^2}{ \bigl({{\varkappa^+}}-\hat{a}^+(\xi)+|{\varepsilon}|^d {\lambda}\, \hat{a}^-(\xi)\bigr)^2}\,d\xi.$$ We define now $$\label{eq:Deltadelta}
\Delta_\delta:=\{\xi\in{{\mathbb{R}^d}}: |B^+\xi|\leq \delta\},$$ that is just an image of the ball $\{|\xi|\leq \delta\}$ under the mapping generated by a Hermitian positive definite matrix $D:=(B^+)^{-1}$. Note that $\det(D)>0$. Since $D $ generates a bounded continuous linear mapping on ${{\mathbb{R}^d}}$, $\Delta_\delta$ is a bounded neighbourhood of the origin. Then $$h({\varepsilon}, \delta)=
\int_{\{|\xi|\leq \delta\}}
\frac{\bigl(\hat{a}^-(D \xi)\bigr)^2}{ \bigl({{\varkappa^+}}-\hat{a}^+(D \xi)+|{\varepsilon}|^d {\lambda}\, \hat{a}^-(D \xi)\bigr)^2}\,\det(D )d\xi\geq0.$$ The inequality , applied for $B=D $, implies that $o(|D \xi|^2)=o(|\xi|^2)$ for $|\xi|\to0$. Then, by , for small enough $\delta>0$ and $|\xi|\leq \delta$, there exist $c_1,c_2>0$ $$\label{eq:estorignew}
\begin{gathered}
c_1|\xi|^2\leq {{\varkappa^+}}-\hat{a}^+(D \xi)=2\pi^2|\xi|^2+o(|\xi|^2)\leq c_2 |\xi|^2,\\
\frac{{{\varkappa^-}}}{2}\leq \hat{a}^-(D \xi)={{\varkappa^-}}+o(1)\leq {{\varkappa^-}}.
\end{gathered}$$ Then, there exist $C_1,C_2,C_3,C_4>0$, such that $$\begin{aligned}
h({\varepsilon},\delta)&\leq \int_{\{|\xi|\leq \delta\}}
\frac{C_1}{ \bigl(|\xi|^2+|{\varepsilon}|^d {\lambda}C_2\bigr)^2}\,d\xi
\\&\leq C_3\int_0^\delta \frac{r^{d-1}}{ \bigl(r^2+|{\varepsilon}|^d {\lambda}C_2\bigr)^2}\,dr\leq C_3 \delta^{d-3}\int_0^\delta \frac{r^2 }{ \bigl(r^2+|{\varepsilon}|^d {\lambda}C_2\bigr)^2}\,dr\\
&=C_4
\biggl(\frac{1}{\sqrt{|{\varepsilon}|^d {\lambda}C_2}}
\arctan\Bigl( \frac{\delta}{\sqrt{|{\varepsilon}|^d {\lambda}C_2}}\Bigr)-
\frac{\delta}{ (|{\varepsilon}|^d {\lambda}C_2+\delta^2)}\biggr).\end{aligned}$$ As a result, $|{\varepsilon}|^d h({\varepsilon},\delta)\leq C_5|{\varepsilon}|^{\frac{d}{2}}$ for some $C_5>0$, and hence holds. Therefore, $\frac{\partial }{\partial {\lambda}}F({\lambda},{\varepsilon})$ is continuous at $({\lambda},0)$ as well.
*Step 3.* As a result, $F({\lambda},{\varepsilon})$ is continuous on $E_\delta$ and continuously differentiable in ${\lambda}$ for small enough $\delta>0$. Since also holds, we conclude, by the implicit function theorem, that there exists (possibly smaller) $\delta>0$ and a unique function ${\lambda}={\lambda}({\varepsilon})$, ${\varepsilon}\in(-\delta,\delta)$, such that ${\lambda}(0)={\lambda}_3$ and $$\label{eq:23523tgmdsd}
F({\lambda}({\varepsilon}),{\varepsilon})=0, \qquad {\varepsilon}\in(-\delta,\delta);$$ moreover, ${\lambda}({\varepsilon})$ is continuous in ${\varepsilon}\in(-\delta,\delta)$. The latter implies $${\lambda}({\varepsilon})={\lambda}_3+o(1), \quad {\varepsilon}\to0.$$ Since implies , we get that $$q^*({\varepsilon})= \frac{|I|}{ {{\varkappa^-}}} {\varepsilon}^{d} +o({\varepsilon}^{d}).$$ In particular, holds, and then, by , we get from that $|I|=I$, i.e. $I>0$. Thus, one has and, by , we get also .
*Step 4.* Assume now, additionally, that $r({\varepsilon})$ is continuously differentiable for ${\varepsilon}\in(0,\delta)$ and that $r'(0):=\lim_{{\varepsilon}\to0+}r'({\varepsilon})<\infty$. Then extends this to ${\varepsilon}\in(-\delta,\delta)$. We have then that $$\frac{\partial }{\partial {\varepsilon}}F({\lambda},{\varepsilon})=-\frac{d}{ {{\varkappa^-}}}
\int_{{\mathbb{R}^d}}\frac{{\varepsilon}|{\varepsilon}|^{d-2} \,{\lambda}\, \hat{a}^-(\xi)}{ \bigl({{\varkappa^+}}-\hat{a}^+(\xi)+|{\varepsilon}|^d {\lambda}\, \hat{a}^-(\xi)\bigr)^2}\hat{a}^-(\xi)\,d\xi-r'({\varepsilon}),$$ is continuous for $({\lambda},{\varepsilon})\in E_\delta$, ${\varepsilon}\neq0$. The same arguments as above show that, for $d\geq5$, $\frac{\partial }{\partial {\varepsilon}}F$ is continuous at $({\lambda},0)$, ${\lambda}\in({\lambda}_3- \delta,{\lambda}_3+\delta)$ with $$\frac{\partial }{\partial {\varepsilon}}F({\lambda},0)=-r'(0).$$ For $d=3,4$, we have, by the same arguments as the above, $$\label{eq:adsqwwqr}
\lim_{{\varepsilon}\to0} \frac{\partial }{\partial {\varepsilon}}F({\lambda},{\varepsilon})=C_6\lim_{{\varepsilon}\to0}
{\varepsilon}|{\varepsilon}|^{d-2}O\bigl(|{\varepsilon}|^{-\frac{d}{2}}\bigr)-r'(0)=-r'(0).$$ as $d-1>\frac{d}{2}$ for $d\geq3$. Next, $$\begin{gathered}
\frac{\partial }{\partial {\varepsilon}}F({\lambda},0)=\lim_{{\varepsilon}\to0}\frac{F({\lambda},{\varepsilon})-F({\lambda},0)}{{\varepsilon}}=\frac{1}{ {{\varkappa^-}}}\lim_{{\varepsilon}\to0}f({\lambda},{\varepsilon})-r'(0),
\shortintertext{where}
\begin{aligned}
f({\lambda},{\varepsilon}):&=\frac{1}{ {\varepsilon}}
\int_{{\mathbb{R}^d}}\biggl(\frac{\hat{a}^+(\xi)-|{\varepsilon}|^d {\lambda}\, \hat{a}^-(\xi)}{ {{\varkappa^+}}-\hat{a}^+(\xi)+|{\varepsilon}|^d {\lambda}\, \hat{a}^-(\xi)}-
\frac{\hat{a}^+(\xi)}{ {{\varkappa^+}}-\hat{a}^+(\xi)}\biggr)
\hat{a}^-(\xi)\,d\xi \\
&=-\frac{{{\varkappa^+}}{\lambda}|{\varepsilon}|^d}{ {\varepsilon}}
\int_{{\mathbb{R}^d}}b({\lambda},{\varepsilon},\xi) \,d\xi,
\end{aligned}
\\\shortintertext{and}
b({\lambda},{\varepsilon},\xi):=\frac{\hat{a}^-(\xi)}{\bigl({{\varkappa^+}}-\hat{a}^+(\xi)+|{\varepsilon}|^d {\lambda}\, \hat{a}^-(\xi)\bigr)\bigl( {{\varkappa^+}}-\hat{a}^+(\xi)\bigr)}.\end{gathered}$$ Since $d>1$, by re-choosing $\delta>0$, we get, similarly to the arguments above, that $$\begin{aligned}
\lim_{{\varepsilon}\to0}f({\lambda},{\varepsilon})=-{{\varkappa^+}}{\lambda}\lim_{{\varepsilon}\to0}\frac{|{\varepsilon}|^d}{ {\varepsilon}}
\int_{\Delta_\delta} b({\lambda},{\varepsilon},\xi)\,d\xi,
\end{aligned}$$ where $\Delta_\delta$ is given by . By the change of variables and , we have, for small enough $\delta>0$, and some constants $c_1,c_2,c_3,c_4>0$, $$\begin{aligned}
&\quad \biggl\lvert \frac{|{\varepsilon}|^d}{ {\varepsilon}}
\int_{\Delta_\delta} b({\lambda},{\varepsilon},\xi)\,d\xi \biggr\rvert \\&\leq |{\varepsilon}|^{d-1}{{\varkappa^-}}\det (D) \int_{\{|\xi|\leq d\}}
\frac{1}{\bigl(c_1|\xi|^2+\frac{{{\varkappa^-}}}{2}|{\varepsilon}|^d {\lambda}\bigr)\bigl( c_1|\xi|^2\bigr)}\,d\xi\\
&\leq |{\varepsilon}|^{d-1} c_2 \int_{\{|\xi|\leq d\}}
\frac{1}{\bigl(|\xi|^2+c_3{\lambda}|{\varepsilon}|^d \bigr)|\xi|^2}\,d\xi=|{\varepsilon}|^{d-1} c_4 \int_0^\delta
\frac{r^{d-1}}{\bigl(r^2+c_3{\lambda}|{\varepsilon}|^d \bigr)r^2}\,d\xi\\
&\leq |{\varepsilon}|^{d-1} c_4 \delta^{d-3}\int_0^\delta
\frac{1}{\bigl(r^2+c_3{\lambda}|{\varepsilon}|^d \bigr)}\,d\xi\\
&=c_4\delta^{d-3}|{\varepsilon}|^{d-1}\frac{1}{\sqrt{c_3 {\lambda}|{\varepsilon}|^d}}\arctan\frac{\delta}{\sqrt{c_3 {\lambda}|{\varepsilon}|^d}}\to0, \qquad {\varepsilon}\to0.\end{aligned}$$ Therefore, for $d=3,4$, we also have $$\frac{\partial }{\partial {\varepsilon}}F({\lambda},0)=-r'(0)$$ and hence $\frac{\partial }{\partial {\varepsilon}}F$ is continuous at $({\lambda},0)$ for ${\lambda}\in({\lambda}_3- \delta,{\lambda}_3+\delta)$.
As a result, both partial derivatives $\frac{\partial }{\partial {\lambda}}F$ and $\frac{\partial }{\partial {\varepsilon}}F$ are continuous on $E_\delta$, hence $F$ is continuously differentiable on $E_\delta$.
Then, the implicit function theorem ensures that the unique function ${\lambda}={\lambda}({\varepsilon})$, ${\varepsilon}\in(-\delta,\delta)$, such that ${\lambda}(0)={\lambda}_3$ and holds, is continuously differentiable in ${\varepsilon}$. Differentiating in ${\varepsilon}$, we get $${\lambda}'({\varepsilon})\frac{\partial }{\partial {\lambda}}F({\lambda}({\varepsilon}),{\varepsilon})+\frac{\partial }{\partial {\varepsilon}}F({\lambda}({\varepsilon}),{\varepsilon})=0,$$ and hence, by , $$\begin{aligned}
{\lambda}'(0)&=\frac{\partial }{\partial {\varepsilon}}F({\lambda}({\varepsilon}),{\varepsilon})\biggr\rvert_{{\varepsilon}=0}=-r'(0).\end{aligned}$$ As a result, $${\lambda}({\varepsilon})= {\lambda}_3 - r'(0){\varepsilon}+o({\varepsilon}),$$ that, by implies .
\[eq:remindecoef\] Note that ${\lambda}_3=\frac{I}{{{\varkappa^+}}}$, where $I$ is given by , does not actually depend on ${{\varkappa^+}}$ nor ${{\varkappa^-}}$.
Asymptotics of the critical mortality: d=2 {#sec:asympd2}
==========================================
Let $d=2$ and hold. We start with some heuristic arguments. Let, additionally, hold; then, by Proposition \[eq:limitofpstar\], $\lim_{{\varepsilon}\to0} p^*({\varepsilon})=-\infty$. By , , we can expect then that $$\label{eq:anz}
q^*({\varepsilon})\approx{\varepsilon}^2 l({\varepsilon}),$$ where $l({\varepsilon})\sim p^*({\varepsilon})$, ${\varepsilon}\to0$, and therefore, $$\label{eq:heur}
\lim_{{\varepsilon}\to0} l({\varepsilon})=\infty, \qquad \lim_{{\varepsilon}\to0} {\varepsilon}^2 l({\varepsilon})=0.$$ Then, by , the anzatz implies $$\begin{aligned}
l({\varepsilon})&\sim \frac{1}{ {{\varkappa^-}}}
\int_{{{\mathbb{R}}}^2}\frac{\hat{a}^+(\xi)-{\varepsilon}^2 l({\varepsilon})\, \hat{a}^-(\xi)}{ {{\varkappa^+}}-\hat{a}^+(\xi)+{\varepsilon}^2 l({\varepsilon})\, \hat{a}^-(\xi)}\hat{a}^-(\xi)\,d\xi\\&=\frac{{{\varkappa^+}}}{{{\varkappa^-}}}
\int_{{{\mathbb{R}}}^2} \frac{1}{{{\varkappa^+}}-\bigl(\hat{a}^+ (\xi)- {\varepsilon}^2 l({\varepsilon}) \hat{a}^- (\xi)\bigr)}\hat{a}^-(\xi)\,d\xi-\frac{1}{{{\varkappa^-}}}
\int_{{{\mathbb{R}}}^2} \hat{a}^-(\xi)\,d\xi.\end{aligned}$$
By the same arguments as above, the singularity of the latter expression as ${\varepsilon}\to0$ is fully determined by the integral $$\sigma(\delta,{\varepsilon}) :=
\frac{{{\varkappa^+}}}{{{\varkappa^-}}}
\int_{\Delta_\delta} \frac{1}{{{\varkappa^+}}-\bigl(\hat{a}^+ (\xi)- {\varepsilon}^2 l({\varepsilon}) \hat{a}^- (\xi)\bigr)}\hat{a}^-(\xi)\,d\xi$$ for small enough $\delta>0$, where $\Delta_\delta$ is given by . By and the change of variables as above, $$\begin{aligned}
\sigma(\delta,{\varepsilon})\sim{\mathrm{const}\,}\cdot
\int_{|\xi|\leq \delta} \frac{1}{|\xi|^2+ {{\varkappa^-}}{\varepsilon}^2 l({\varepsilon})}\,d\xi={\mathrm{const}\,}\cdot\int_0^\delta \frac{r}{r^2+ {{\varkappa^-}}{\varepsilon}^2 l({\varepsilon})}\,dr.\end{aligned}$$ Integrating, we conclude that, heuristically, for some $c_1,c_2>0$, $$\begin{aligned}
l({\varepsilon})&\sim c_1 \log\biggl(1+\frac{c_2}{{\varepsilon}^2 l({\varepsilon})}\biggr)\sim c_1\log \frac{c_2}{{\varepsilon}^2 l({\varepsilon})}, \end{aligned}$$ by , i.e. $\frac{l({\varepsilon})}{c_1} e^{\frac{l({\varepsilon})}{c_1}}\sim \frac{c_2}{c_1{\varepsilon}^2}$. Therefore, $$l({\varepsilon})\sim c_1 W\biggl( \frac{c_2}{c_1{\varepsilon}^2} \biggr),$$ where $W(z)$, $z>0$, is the unique solution to the equation $ye^y=z$, $y>0$, (the principal branch of) of the so-called Lambert $W$ function. It is well-known that $$W(z)= \log z - \log \log z+o(1), \qquad z\to+\infty.$$ Therefore, $$W\biggl( \frac{c_2}{c_1{\varepsilon}^2} \biggr)=-2\log{\varepsilon}-\log(-\log{\varepsilon})+O(1),$$ in other words, two leading terms of the asymptotics of $q^*({\varepsilon})\approx c'{\varepsilon}^2 W(c{\varepsilon}^{-2})$ depend on $c'$ but not on $c$. This gives a hint to define ${\lambda}({\varepsilon})$ in the proof of the following theorem.
\[thm:assympd2\] Let $d=2$ and – hold. Let hold with $o({\varepsilon}^2)$ such that the function $$r({\varepsilon}):=\frac{o({\varepsilon}^2)}{{\varepsilon}^2 W({\varepsilon}^{-2})} \label{r2eps}$$ is continuous for small ${\varepsilon}>0$. Then $$\label{eq:asymptford2}
q^*({\varepsilon})={\lambda}_2{\varepsilon}^2W({\varepsilon}^{-2})+o\bigl({\varepsilon}^2 W({\varepsilon}^{-2})\bigr),$$ where $$\label{eq:la2}
{\lambda}_2:=\dfrac{{{\varkappa^+}}}{2\pi \sqrt{a_{11}a_{22}-a_{12}^2}},$$ and $a_{ij}$, $1\leq i,j\leq 2$, are given by .
\[eq:remindecoef2\] Note that ${\lambda}_2$ does not actually depend on ${{\varkappa^+}}$ (cf. Remark \[eq:remindecoef\]).
We define ${\lambda}({\varepsilon})>0$, ${\varepsilon}\in(0,1)$, through the equality $$\label{eq:expofqlog}
q^*({\varepsilon})={\lambda}({\varepsilon}){\varepsilon}^2W({\varepsilon}^{-2}).$$ One can rewrite then as follows $${\lambda}({\varepsilon})-\frac{1}{ {{\varkappa^-}}W({\varepsilon}^{-2})}
\int_{{{\mathbb{R}}}^2}\frac{\hat{a}^+(\xi)- {\varepsilon}^2W({\varepsilon}^{-2}){\lambda}({\varepsilon})\, \hat{a}^-(\xi)}{ {{\varkappa^+}}-\hat{a}^+(\xi)+{\varepsilon}^2W({\varepsilon}^{-2}) {\lambda}({\varepsilon})\, \hat{a}^-(\xi))}\hat{a}^-(\xi)\,d\xi+r({\varepsilon})=0,$$ where $r({\varepsilon})\to0$, ${\varepsilon}\to0$, is given by
Let $\delta\in\bigl(0,\min\{{\lambda}_2,1\}\bigr)$ be such that, cf. , , $$\label{eq:condondelta}
({\lambda}_2+\delta)\delta^2 W(\delta^{-2})<\frac{{{\varkappa^+}}-m}{{{\varkappa^-}}},$$ and let also $r({\varepsilon})$ be continuous on $(0,\delta)$. Note that the function in the left hand side of is increasing in $\delta>0$. We define then $r$ on $(-\delta,0]$ as in . For ${\lambda}\in ({\lambda}_2- \delta,{\lambda}_2 +\delta)$, ${\varepsilon}\in (- \delta,\delta)$, ${\varepsilon}\neq0$, we consider the function $$\begin{aligned}
F({\lambda},{\varepsilon}):&=\frac{1}{ {{\varkappa^-}}W({\varepsilon}^{-2})}
\int_{{{\mathbb{R}}}^2}\frac{\hat{a}^+(\xi)-{\lambda}{\varepsilon}^2 W({\varepsilon}^{-2}) \, \hat{a}^-(\xi)}{ {{\varkappa^+}}-\hat{a}^+(\xi)+{\lambda}{\varepsilon}^2 W({\varepsilon}^{-2}) \, \hat{a}^-(\xi)}\hat{a}^-(\xi)\,d\xi-{\lambda}-r({\varepsilon})\notag\\
&=\frac{{{\varkappa^+}}}{ {{\varkappa^-}}W({\varepsilon}^{-2})}
\int_{{{\mathbb{R}}}^2}\frac{1}{ {{\varkappa^+}}-\hat{a}^+(\xi)+{\lambda}{\varepsilon}^2 W({\varepsilon}^{-2})\, \hat{a}^-(\xi)}\hat{a}^-(\xi)\,d\xi \notag\\&\quad -\frac{1}{ {{\varkappa^-}}W({\varepsilon}^{-2}) } \int_{{{\mathbb{R}}}^2} \hat{a}^-(\xi)\,d\xi-{\lambda}-r({\varepsilon}).\label{eq:defFleps2}\end{aligned}$$ Clearly, $F$ is continuous for ${\lambda}\in ({\lambda}_2- \delta,{\lambda}_2 +\delta)$, ${\varepsilon}\in (- \delta,\delta)$, ${\varepsilon}\neq0$.
By the same arguments as above, $$\begin{gathered}
\label{eq:expchm}
\lim_{{\varepsilon}\to0}F({\lambda},{\varepsilon})=-{\lambda}+\lim_{{\varepsilon}\to0}\frac{1}{W({\varepsilon}^{-2})}\sigma({\varepsilon},\delta,{\lambda}),
\\\intertext{where, for a small enough $\delta>0$,}
\sigma({\varepsilon},\delta,{\lambda}):=\frac{{{\varkappa^+}}}{{{\varkappa^-}}}\int_{\Delta_\delta}\frac{1}{ {{\varkappa^+}}-\hat{a}^+(\xi)+{\lambda}{\varepsilon}^2 W({\varepsilon}^{-2})\, \hat{a}^-(\xi)}\hat{a}^-(\xi)\,d\xi,\notag\end{gathered}$$ and $\Delta_\delta$ is given by .
Recall that the inequality , applied for $B=D $, implies that $o(|D \xi|^2)=o(|\xi|^2)$ for $|\xi|\to0$. Then, cf. , by , for any $\rho\in(0,1)$ there exists $\delta_\rho>0$ small enough such that, for $|\xi|\leq\delta<\delta_\rho$, $$\label{eq:estorig}
\begin{gathered}
2\pi^2(1- \rho)|\xi|^2\leq {{\varkappa^+}}-\hat{a}^+(D \xi)=2\pi^2|\xi|^2+o(|\xi|^2)\leq 2\pi^2(1+ \rho) |\xi|^2,\\
(1-\rho){{\varkappa^-}}\leq \hat{a}^-(D \xi)={{\varkappa^-}}+o(1)\leq {{\varkappa^-}}.
\end{gathered}$$
By change of variables and , for each $\rho\in(0,1)$, there exists $\delta_\rho<{\lambda}_2$, such that, for all $\delta\in(0,\delta_\rho)$, $$\label{eq:dblestsigma}
\frac{1}{1-\rho}\tau({\varepsilon},\delta,{\lambda})\geq \sigma({\varepsilon},\delta,{\lambda})\geq \frac{1-\rho}{1+\rho} \tau({\varepsilon},\delta,{\lambda}),$$ where, for $D=(B^+)^{-1}$, $$\begin{aligned}
\tau({\varepsilon},\delta,{\lambda}):&= {{\varkappa^+}}\det(D )\int_{|\xi|\leq \delta}\frac{1}{ 2\pi^2 |\xi|^2+{{\varkappa^-}}{\lambda}{\varepsilon}^2 W({\varepsilon}^{-2})}\,d\xi\\&=2{{\varkappa^+}}\pi \det(D ) \int_0^\delta \frac{1}{ 2\pi^2 r^2+{{\varkappa^-}}{\lambda}{\varepsilon}^2 W({\varepsilon}^{-2})}\,r\, dr\\
&=\frac{{{\varkappa^+}}}{2\pi} \det(D ) \log\biggl(1 + \frac{2\pi^2\delta^2}{{{\varkappa^-}}{\lambda}{\varepsilon}^2 W({\varepsilon}^{-2})} \biggr).\end{aligned}$$ Note that $W({\varepsilon}^{-2})e^{W({\varepsilon}^{-2})}={\varepsilon}^{-2}$, i.e. ${\varepsilon}^2 W({\varepsilon}^{-2})=e^{-W({\varepsilon}^{-2})}$. Set $R:=W({\varepsilon}^{-2})\to+\infty$, ${\varepsilon}\to0$. Then $$\begin{aligned}
&\quad \frac{1}{W({\varepsilon}^{-2})}\tau({\varepsilon},\delta,{\lambda})
=\frac{ {{\varkappa^+}}\det(D ) }{2\pi R}\log\biggl(1 + \frac{2\pi^2\delta^2}{{{\varkappa^-}}{\lambda}} e^{R}\biggr)\\
&=\frac{{{\varkappa^+}}\det(D ) }{2\pi R}\Biggl(\log\frac{2\pi^2\delta^2}{{{\varkappa^-}}{\lambda}}+R+\log\biggl(1 + \frac{{{\varkappa^-}}{\lambda}}{2\pi^2\delta^2} e^{-R}\biggr)\Biggr)\to \frac{ {{\varkappa^+}}\det(D ) }{2\pi},\end{aligned}$$ as $R\to+\infty$, i.e. as ${\varepsilon}\to0$. Combining this with and , we conclude that, for each $\rho\in(0,1)$, $$\lim_{{\varepsilon}\to0}F({\lambda},{\varepsilon})+{\lambda}\in\biggl( \frac{{{\varkappa^+}}\det(D )}{2\pi}\frac{1-\rho}{1+\rho}, \frac{{{\varkappa^+}}\det(D )}{2\pi} \frac{1}{1-\rho}\biggr).$$ By sending $\rho$ to $0$, we get $$\lim_{{\varepsilon}\to0}F({\lambda},{\varepsilon})=-{\lambda}+\frac{{{\varkappa^+}}\det(D )}{2\pi} =-{\lambda}+\frac{{{\varkappa^+}}}{2\pi\det(B^+)}=-{\lambda}+{\lambda}_2,$$ where ${\lambda}_2$ is given by , since $(B^+)^2=A^+$ implies $\det(B^+)=\sqrt{\det(A^+)}$.
Therefore, if we set $$F({\lambda},0):={\lambda}_2-{\lambda}, \qquad{\lambda}\in({\lambda}_2- \delta,{\lambda}_2+\delta),$$ then $F({\lambda},{\varepsilon})$ becomes a continuous function on $$E_\delta:=({\lambda}_2- \delta,{\lambda}_2 +\delta)\times (- \delta,\delta)$$ with a small enough $\delta\in(0,{\lambda}_2)$. Moreover, $F({\lambda}_2,0)=0$. Next, since $$\frac{F({\lambda}+h,0)-F({\lambda},0)}{h}=-1, \qquad {\lambda},{\lambda}+h\in({\lambda}_2- \delta,{\lambda}_2+\delta),$$ we have $$\frac{\partial}{\partial {\lambda}}F({\lambda},0)=-1\neq0, \qquad {\lambda}\in({\lambda}_2- \delta,{\lambda}_2+\delta).$$ Next, for $({\lambda},{\varepsilon})\in E_\delta$, ${\varepsilon}\neq0$, we have, by , $$\begin{aligned}
\frac{\partial}{\partial {\lambda}}F({\lambda},{\varepsilon})&=-1- \frac{1}{ {{\varkappa^-}}W({\varepsilon}^{-2})}
\int_{{{\mathbb{R}}}^2}\frac{ {\varepsilon}^2 W({\varepsilon}^{-2})\, \hat{a}^-(\xi)}{ \bigl({{\varkappa^+}}-\hat{a}^+(\xi)+{\lambda}{\varepsilon}^2 W({\varepsilon}^{-2}) \, \hat{a}^-(\xi)\bigr)^2}\hat{a}^-(\xi)\,d\xi\\
&=-1- \frac{ {\varepsilon}^2 }{ {{\varkappa^-}}}
\int_{{{\mathbb{R}}}^2}\frac{ \bigl(\hat{a}^-(\xi)\bigr)^2}{ \bigl({{\varkappa^+}}-\hat{a}^+(\xi)+{\lambda}{\varepsilon}^2 W({\varepsilon}^{-2})\, \hat{a}^-(\xi)\bigr)^2}\,d\xi.\end{aligned}$$ By the same arguments as above, $$\begin{gathered}
\lim_{{\varepsilon}\to0}
\frac{\partial }{\partial {\lambda}}F({\lambda},{\varepsilon})=-1-\frac{1}{ {{\varkappa^-}}}
\lim_{{\varepsilon}\to0} {\varepsilon}^2 h({\varepsilon},\delta),
\shortintertext{where}
h({\varepsilon}, \delta)=
\int_{\Delta_\delta}
\frac{\bigl(\hat{a}^-(\xi)\bigr)^2}{ \bigl(|B^+\xi|^2+o(|\xi|^2)+{\lambda}{\varepsilon}^2 W({\varepsilon}^{-2}) ({{\varkappa^-}}+o(1))\bigr)^2}\,d\xi,\end{gathered}$$ where $\Delta_\delta$ is given by . By change and variables and , we get that, for some $C_1,C_2,C_3>0$ and for small enough $\delta>0$, $$\begin{aligned}
0<h({\varepsilon},\delta)&\leq \int_{\{|\xi|\leq \delta\}}
\frac{C_1}{ \bigl(|\xi|^2+ {\lambda}{\varepsilon}^2 W({\varepsilon}^{-2}) C_2\bigr)^2}\,d\xi\\
&\leq C_3\int_0^\delta \frac{r }{ \bigl(r^2+ {\lambda}{\varepsilon}^2 W({\varepsilon}^{-2})C_2\bigr)^2}\,dr\\
&=\frac{C_3}{2C_2}\frac{\delta^2}{ {\lambda}{\varepsilon}^2 W({\varepsilon}^{-2}) \bigl(\delta^2+ {\lambda}{\varepsilon}^2 W({\varepsilon}^{-2}) C_2\bigr)},\end{aligned}$$ and hence $ {\varepsilon}^2 h({\varepsilon},\delta)\to0$, ${\varepsilon}\to0$, that yields $$\lim_{{\varepsilon}\to0}
\frac{\partial }{\partial {\lambda}}F({\lambda},{\varepsilon})=-1,$$ and therefore, $\frac{\partial }{\partial {\lambda}}F$ is continuous on $E_\delta$.
Again, the implicit function theorem states that there exists a unique function ${\lambda}={\lambda}({\varepsilon})$, ${\varepsilon}\in(-\delta,\delta)$ (with, possibly, smaller $\delta$), such that ${\lambda}(0)={\lambda}_2$ and $$F({\lambda}({\varepsilon}),{\varepsilon})=0, \qquad{\varepsilon}\in(-\delta,\delta);$$ moreover, ${\lambda}({\varepsilon})$ is *continuous* in ${\varepsilon}\in(-\delta,\delta)$. Therefore, ${\lambda}({\varepsilon})={\lambda}_2+o(1)$, ${\varepsilon}\to0$; hence, from and , we get .
If function $a^+$ in Theorem \[thm:assympd2\] is radially symmetric, i.e. $a^+(x)=b^+(|x|)$ for some $b^+:{{\mathbb{R}}}\to{{\mathbb{R}}}$, then $a_{12}=a_{21}=0$ and $$a_{11}=a_{22}=\int_{{{\mathbb{R}}}^2} x_1^2 a^+(x)\,dx=\frac{1}{2}\int_{{{\mathbb{R}}}^2}|x|^2 a^+(x)\,dx,$$ so that $${\lambda}_2=\frac{{{\varkappa^+}}}{\pi \int_{{{\mathbb{R}}}^2}|x|^2 a^+(x)\,dx}.$$
\[rem:nondif2\] It can be checked that $F({\lambda},{\varepsilon})$ defined by is not continuously differentiable in ${\varepsilon}$ at ${\varepsilon}=0$ (even if we assume that $r$ is); hence, in general, one can not expect that ${\lambda}({\varepsilon})$ is continuously differentiable at ${\varepsilon}=0$. Hence, the question about the next term of the assymptotic in remains open.
Asymptotics of the critical mortality: d=1 {#sec:asympd1}
==========================================
Let $d=1$ and hold. We firstly proceed again heuristically. Similarly to the arguments at the beginning of Section \[sec:asympd2\], if holds then we may expect, for ${\varepsilon}\to0$, $$\begin{gathered}
\label{eq:anz1}
q^*({\varepsilon})\approx{\varepsilon}l({\varepsilon}), \\
\shortintertext{where}
l({\varepsilon})\sim p^*({\varepsilon})\to\infty, \qquad {\varepsilon}l({\varepsilon})\to 0.\notag \end{gathered}$$ Then, by , the ansatz implies $$\begin{aligned}
l({\varepsilon})&\sim \frac{1}{ {{\varkappa^-}}}
\int_{{{\mathbb{R}}}}\frac{\hat{a}^+(\xi)-{\varepsilon}l({\varepsilon})\, \hat{a}^-(\xi)}{ {{\varkappa^+}}-\hat{a}^+(\xi)+{\varepsilon}l({\varepsilon})\, \hat{a}^-(\xi)}\hat{a}^-(\xi)\,d\xi\\&=\frac{1}{{{\varkappa^+}}{{\varkappa^-}}}
\int_{{{\mathbb{R}}}} \hat{a}^-(\xi)\,d\xi-
\frac{1}{{{\varkappa^-}}}
\int_{{{\mathbb{R}}}} \frac{1}{{{\varkappa^+}}-\bigl(\hat{a}^+ (\xi)- {\varepsilon}l({\varepsilon}) \hat{a}^- (\xi)\bigr)}\hat{a}^-(\xi)\,d\xi.\end{aligned}$$
By the same arguments as above, the singularity of the latter expression as ${\varepsilon}\to0$ is fully determined by the integral $$\begin{aligned}
&\quad
\frac{1}{{{\varkappa^-}}}
\int_{-\delta}^\delta \frac{1}{{{\varkappa^+}}-\bigl(\hat{a}^+ (\xi)- {\varepsilon}l({\varepsilon}) \hat{a}^- (\xi)\bigr)}\hat{a}^-(\xi)\,d\xi\\
&\sim c_1 \cdot\int_0^\delta \frac{1}{r^2+ c_2 {\varepsilon}l({\varepsilon})}\,dr=\frac{c_3}{\sqrt{{\varepsilon}l({\varepsilon})}}\arctan \frac{\delta}{\sqrt{c_2 {\varepsilon}l({\varepsilon})}},\end{aligned}$$ for small enough $\delta>0$ and some $c_1,c_2,c_3>0$; here we used . As a result, heuristically, $$l({\varepsilon})\sqrt{{\varepsilon}l({\varepsilon})}\approx {\mathrm{const}\,},$$ and hence $l({\varepsilon})\approx {\mathrm{const}\,}{\varepsilon}^{-\frac13}$, ${\varepsilon}\to0$. Again, it gives us a hint to define ${\lambda}({\varepsilon})$ in the proof of the following theorem.
\[thm:assympd1\] Let $d=1$ and – hold. Let hold with $o({\varepsilon})$ such that the function $$r({\varepsilon}):={\varepsilon}^{-\frac{2}{3}}o({\varepsilon}) \label{r1eps1}$$ is continuous for small ${\varepsilon}>0$. Then $$\label{eq:asymptford1}
q^*({\varepsilon})={\lambda}_1{\varepsilon}^{\frac{2}{3}}+o\bigl({\varepsilon}^{\frac{2}{3}}\bigr),$$ where $${\lambda}_1:=\Biggl(\dfrac{({{\varkappa^+}})^2}{ 2{{\varkappa^-}}\displaystyle\int_{{\mathbb{R}}}x^2 a^+(x)\,dx}\Biggr)^{\frac13}.$$
Note that, in contrast to the cases $d\geq 3$ and $d=2$, cf. Remarks \[eq:remindecoef\], \[eq:remindecoef2\], ${\lambda}_1$ depends effectively on (the ratio of) ${{\varkappa^+}}$ and ${{\varkappa^-}}$.
We set ${\lambda}({\varepsilon}):=q^*({\varepsilon}){\varepsilon}^{-\frac{2}{3}}$ for ${\varepsilon}\in(0,1)$, and then rewrite as follows $${\lambda}({\varepsilon})-\frac{{\varepsilon}^{\frac{1}{3}}}{ {{\varkappa^-}}}
\int_{{\mathbb{R}}}\frac{\hat{a}^+(\xi)- {\varepsilon}^{\frac{2}{3}}{\lambda}({\varepsilon})\, \hat{a}^-(\xi)}{ {{\varkappa^+}}-\hat{a}^+(\xi)+{\varepsilon}^{\frac{2}{3}}{\lambda}({\varepsilon})\, \hat{a}^-(\xi))}\hat{a}^-(\xi)\,d\xi+r({\varepsilon})=0,$$ where $r({\varepsilon})\to0$, ${\varepsilon}\to0$, is given by
Let $\delta\in\bigl(0,\min\{{\lambda}_1,1\}\bigr)$ be such that, cf. , , , $$({\lambda}_1+\delta)\delta^{\frac{2}{3}}<\frac{{{\varkappa^+}}-m}{{{\varkappa^-}}},$$ and let also $r({\varepsilon})$ be continuous on $(0,\delta)$. We define then $r$ on $(-\delta,0]$ as in . For ${\lambda}\in ({\lambda}_1- \delta,{\lambda}_1 +\delta)$, ${\varepsilon}\in (- \delta,\delta)$, ${\varepsilon}\neq0$, we consider the function $$\begin{aligned}
F({\lambda},{\varepsilon}):&=\frac{{\varepsilon}^{\frac{1}{3}}}{ {{\varkappa^-}}}
\int_{{{\mathbb{R}}}^2}\frac{\hat{a}^+(\xi)-{\lambda}{\varepsilon}^{\frac{2}{3}} \, \hat{a}^-(\xi)}{ {{\varkappa^+}}-\hat{a}^+(\xi)+{\lambda}{\varepsilon}^{\frac{2}{3}} \, \hat{a}^-(\xi)}\hat{a}^-(\xi)\,d\xi-{\lambda}-r({\varepsilon})\notag\\
&=\frac{{{\varkappa^+}}{\varepsilon}^{\frac{1}{3}}}{{{\varkappa^-}}}
\int_{{{\mathbb{R}}}^2}\frac{1}{ {{\varkappa^+}}-\hat{a}^+(\xi)+{\lambda}{\varepsilon}^{\frac{2}{3}}\, \hat{a}^-(\xi)}\hat{a}^-(\xi)\,d\xi \notag\\&\quad -\frac{{\varepsilon}^{\frac{1}{3}}}{ {{\varkappa^-}}} \int_{{{\mathbb{R}}}^2} \hat{a}^-(\xi)\,d\xi-{\lambda}-r({\varepsilon}).\label{eq:defFleps1}\end{aligned}$$ Clearly, $F$ is continuous for ${\lambda}\in ({\lambda}_1- \delta,{\lambda}_1+\delta)$, ${\varepsilon}\in (- \delta,\delta)$, ${\varepsilon}\neq0$.
Let $$B:=\int_{{\mathbb{R}}}|x|^2 a^+(x)\,dx=2\int_0^\infty x^2 a^+(x)\,dx.$$ By and the same arguments as in the proof of Theorem \[thm:assympd2\], $$\begin{aligned}
\lim_{{\varepsilon}\to0}F({\lambda},{\varepsilon})&=-{\lambda}+\frac{{{\varkappa^+}}}{{{\varkappa^-}}}\lim_{{\varepsilon}\to0}{\varepsilon}^{\frac{1}{3}}\int_{-\delta}^\delta\frac{1}{ {{\varkappa^+}}-\hat{a}^+(\xi)+{\lambda}{\varepsilon}^{\frac{2}{3}}\, \hat{a}^-(\xi)}\hat{a}^-(\xi)\,d\xi\\
&=-{\lambda}+2{{\varkappa^+}}\lim_{{\varepsilon}\to0}{\varepsilon}^{\frac{1}{3}}\int_{0}^\delta\frac{1}{ 2\pi^2 B r^2+{{\varkappa^-}}{\lambda}{\varepsilon}^{\frac{2}{3}}}\,dr\\
&= - {\lambda}+\frac{\sqrt{2}{{\varkappa^+}}}{\pi\sqrt{{\lambda}{{\varkappa^-}}B}}\lim_{{\varepsilon}\to0}
\arctan\frac{\sqrt{2B}\pi \delta}{\sqrt{{{\varkappa^-}}{\lambda}}{\varepsilon}^{\frac13}}\\
&= - {\lambda}+\frac{{{\varkappa^+}}}{\sqrt{2{\lambda}{{\varkappa^-}}B}}.\end{aligned}$$
Therefore, if we set $$F({\lambda},0):= - {\lambda}+\frac{{{\varkappa^+}}}{\sqrt{2{\lambda}{{\varkappa^-}}B}}, \qquad{\lambda}\in({\lambda}_1- \delta,{\lambda}_1+\delta),$$ then $F({\lambda},{\varepsilon})$ becomes a continuous function on $$E_\delta:=({\lambda}_1- \delta,{\lambda}_1 +\delta)\times (- \delta,\delta)$$ with a small enough $\delta\in(0,{\lambda}_1)$. Moreover, it is straightforward to check that $$F({\lambda}_1,0)=0.$$
For ${\lambda}\in({\lambda}_1- \delta,{\lambda}_1+\delta)$, we have $$\frac{\partial}{\partial {\lambda}}F({\lambda},0)=\lim_{h\to0}\frac{F({\lambda}+h,0)-F({\lambda},0)}{h}=-1-\frac{{{\varkappa^+}}}{2\sqrt{2 {{\varkappa^-}}B}}{\lambda}^{-\frac{3}{2}}< 0,$$ and also $$\frac{\partial}{\partial {\lambda}}F({\lambda}_1,0)=
-1-\frac{{{\varkappa^+}}}{2\sqrt{2 {{\varkappa^-}}B}}
\Bigl(({{\varkappa^+}})^{\frac23}(2{{\varkappa^-}}B)^{-\frac13}\Bigr)^{-\frac{3}{2}}=-\frac32\neq0.$$
Next, for $({\lambda},{\varepsilon})\in E_\delta$, ${\varepsilon}\neq0$, we have, by , $$\frac{\partial}{\partial {\lambda}}F({\lambda},{\varepsilon})=-1- \frac{ {\varepsilon}}{ {{\varkappa^-}}}
\int_{{{\mathbb{R}}}}\frac{ \bigl(\hat{a}^-(\xi)\bigr)^2}{ \bigl({{\varkappa^+}}-\hat{a}^+(\xi)+{\lambda}{\varepsilon}^{\frac{2}{3}}\, \hat{a}^-(\xi)\bigr)^2}\,d\xi.$$ By the same arguments as above, $$\begin{aligned}
\lim_{{\varepsilon}\to0}
\frac{\partial }{\partial {\lambda}}F({\lambda},{\varepsilon})&
=-1-\frac{{{\varkappa^+}}}{ {{\varkappa^-}}}
\lim_{{\varepsilon}\to0} {\varepsilon}\int_{-\delta}^\delta
\frac{\bigl(\hat{a}^-(\xi)\bigr)^2}{ \bigl({{\varkappa^+}}-\hat{a}^+(\xi)+{\lambda}{\varepsilon}^{\frac{2}{3}}\, \hat{a}^-(\xi)\bigr)^2}\,d\xi\\
&= -1-{{\varkappa^+}}{{\varkappa^-}}\lim_{{\varepsilon}\to0} {\varepsilon}\int_{-\delta}^\delta
\frac{1}{ \bigl(2\pi^2 B|\xi|^2+{\lambda}{{\varkappa^-}}{\varepsilon}^{\frac{2}{3}} \bigr)^2}\,d\xi\\
\intertext{and by straightforward integration, one gets}
&= -1- {{\varkappa^+}}\lim_{{\varepsilon}\to0}
\frac{\delta {\varepsilon}^{\frac13}}{2\pi^2 B \delta^2 {{\varkappa^-}}{\lambda}+{\varepsilon}^{\frac23}({{\varkappa^-}})^2{\lambda}^2}\\&\quad
-{{\varkappa^+}}\lim_{{\varepsilon}\to0}\frac{1}{\pi \sqrt{2B{{\varkappa^-}}}{\lambda}^{\frac32}}\arctan\biggl(\sqrt{\frac{2 B}{{{\varkappa^-}}{\lambda}}}\pi \delta {\varepsilon}^{-\frac13}\biggr)
\Biggr)\\
&=-1-\frac{{{\varkappa^+}}}{2\sqrt{2B{{\varkappa^-}}}}{\lambda}^{-\frac32}=\frac{\partial}{\partial {\lambda}}F({\lambda},0).\end{aligned}$$ Therefore, $\frac{\partial }{\partial {\lambda}}F$ is continuous on $E_\delta$.
Again, the implicit function theorem states that there exists a unique continuous function ${\lambda}={\lambda}({\varepsilon})$ such that ${\lambda}(0)={\lambda}_1$ and $F({\lambda}({\varepsilon}),{\varepsilon})=0$, ${\varepsilon}\in(-\delta,\delta)$ ${\varepsilon}\in(-\delta,\delta)$ (with, possibly, smaller $\delta$). Therefore, ${\lambda}({\varepsilon})={\lambda}_1+o(1)$, ${\varepsilon}\to0$, that yields .
Similarly to the case $d=2$, see Remark \[rem:nondif2\], function $F({\lambda},{\varepsilon})$ defined by is not continuously differentiable in ${\varepsilon}$ at ${\varepsilon}=0$, hence the next term of the assymptotic in remains an open problem.
Acknowledgements {#acknowledgements .unnumbered}
================
I thank Otso Ovaskainen and Panu Somervuo for discussions that motivated the analyses presented in this paper.
[10]{} M. Bessonov, S. Molchanov, and J. Whitmeyer. [A mean field approximation of the [B]{}olker-[P]{}acala population model](http://math-mprf.org/journal/articles/id1335/). , 200 (2):0 329–348, 2014.
M. Bessonov, S. Molchanov, and J. Whitmeyer. [A multi-class extension of the mean field [B]{}olker-[P]{}acala population model](http://dx.doi.org/10.1515/rose-2018-0013). , 260 (3):0 163–174, 2018.
V. Bezborodov, Y. Kondratiev, and O. Kutoviy. [Lattice birth-and-death processes](http://dx.doi.org/10.17323/1609-4514-2019-19-1-7-36). , 190 (1):0 7–36, 2019.
B. Bolker and S. W. Pacala. [Using moment equations to understand stochastically driven spatial pattern formation in ecological systems](http://dx.doi.org/10.1006/tpbi.1997.1331). , 520 (3):0 179–197, 1997.
M. Bramson, R. Durrett, and G. Swindle. Statistical mechanics of crabgrass. , 170 (2):0 444–481, 1989.
S. Cornell, Y. Suprunenko, D. Finkelshtein, P. Somervuo, and O. Ovaskainen. [A unified framework for analysis of individual-based models in ecology and beyond](http://dx.doi.org/10.1038/s41467-019-12172-y). , 100 (1), 2019.
J. L. Daletskii and M. G. Krein. . American Mathematical Society, Providence, R.I., 1974. vi+386 pp. Translated from the Russian by S. Smith, Translations of Mathematical Monographs, Vol. 43.
U. Dieckmann and R. Law. Relaxation projections and the method of moments. In [*The Geometry of Ecological Interactions*]{}, pages 412–455. Cambridge University Press, Cambridge, UK, 2000.
R. Durrett. [Stochastic spatial models](http://dx.doi.org/10.1137/S0036144599354707). , 410 (4):0 677–718, 1999.
R. Durrett and E. A. Perkins. [Rescaled contact processes converge to super-brownian motion in two or more dimensions](http://dx.doi.org/10.1007/s004400050228). , 1140 (3):0 309–399, June 1999.
A. M. Etheridge. [Survival and extinction in a locally regulated population](http://dx.doi.org/10.1214/aoap/1075828051). , 140 (1):0 188–214, Feb. 2004.
D. Finkelshtein, Y. Kondratiev, Y. Kozitsky, and O. Kutoviy. [The statistical dynamics of a spatial logistic model and the related kinetic equation](http://dx.doi.org/10.1142/S0218202515500128). , 250 (2):0 343–370, 2015.
D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. [Individual based model with competition in spatial ecology](http://dx.doi.org/10.1137/080719376). , 410 (1):0 297–317, 2009.
D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. [Semigroup approach to birth-and-death stochastic dynamics in continuum](http://dx.doi.org/10.1016/j.jfa.2011.11.005). , 2620 (3):0 1274–1308, 2012.
D. Finkelshtein, Y. Kondratiev, S. Molchanov, and P. Tkachov. [Global stability in a nonlocal reaction-diffusion equation](http://dx.doi.org/10.1142/S0219493718500375). , 180 (5):0 1850037 (15 pages), 2018.
D. Finkelshtein, Y. Kondratiev, and P. Tkachov. [Accelerated front propagation for monostable equations with nonlocal diffusion: multidimensional case](http://dx.doi.org/10.1007/s41808-019-00045-w). , 50 (2):0 423–471, Dec. 2019.
D. Finkelshtein, Y. Kondratiev, and P. Tkachov. [Doubly nonlocal [F]{}isher–[KPP]{} equation: Front propagation](http://dx.doi.org/10.1080/00036811.2019.1643011). , pages 1–24, 2019.
D. Finkelshtein, Y. Kondratiev, and P. Tkachov. [Doubly nonlocal [F]{}isher–[KPP]{} equation: [S]{}peeds and uniqueness of traveling waves](http://dx.doi.org/10.1016/j.jmaa.2019.02.010). , 4750 (1):0 94–122, 2019.
D. Finkelshtein, Y. Kondratiev, and P. Tkachov. [Existence and properties of traveling waves for doubly nonlocal [F]{}isher–[KPP]{} equations](http://dx.doi.org/10/abstr). , 20190 (10):0 1–27, 2019.
D. Finkelshtein and P. Tkachov. [Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line](http://dx.doi.org/10.1080/00036811.2017.1400537). , 980 (4):0 756–780, 2019.
N. Fournier and S. M[é]{}l[é]{}ard. [A microscopic probabilistic description of a locally regulated population and macroscopic approximations](http://dx.doi.org/10.1214/105051604000000882). , 140 (4):0 1880–1919, 2004.
Y. Kondratiev and Y. Kozitsky. [The evolution of states in a spatial population model](http://dx.doi.org/10.1007/s10884-016-9526-6). , pages 1–39, 2016.
C. Kuehn and P. Tkachov. [Pattern formation in the doubly-nonlocal fisher-kpp equation](http://dx.doi.org/10.3934/dcds.2019087). , 390 (4):0 2077, 2019.
R. Law and U. Dieckmann. [A dynamical system for neighborhoods in plant communities](http://dx.doi.org/10.1890/0012-9658(2000)081[2137:ADSFNI]2.0.CO;2). , 810 (8):0 2137–2148, 2000.
R. Law, D. J. Murrell, and U. Dieckmann. [Population growth in space and time: spatial logistic equations](http://dx.doi.org/10.1890/0012-9658(2003)084[0252:PGISAT]2.0.CO;2). , 840 (1):0 252–262, 2003.
D. J. Murrell, U. Dieckmann, and R. Law. [On moment closures for population dynamics in continuous space](http://dx.doi.org/10.1016/j.jtbi.2004.04.013). , 2290 (3):0 421–432, 2004.
O. Ovaskainen and S. Cornell. Space and stochasticity in population dynamics. , 103:0 12781–12786, 2006.
O. Ovaskainen, D. Finkelshtein, O. Kutoviy, S. Cornell, B. Bolker, and Y. Kondratiev. [A mathematical framework for the analysis of spatial-temporal point processes](http://dx.doi.org/10.1007/s12080-013-0202-8). , 70 (1):0 101–113, 2014.
A. Pazy. , volume 44 of [*Applied Mathematical Sciences*]{}. Springer-Verlag, New York, 1983. ISBN 0-387-90845-5. viii+279 pp.
Z. Sasv[á]{}ri. , volume 50. Walter de Gruyter, 2013.
M. Savov and S.-D. Wang. [Fluctuation limits of a locally regulated population and generalized [L]{}angevin equations](http://dx.doi.org/10.1142/S0219025715500095). , 180 (2):0 1550009, 27, 2015.
[^1]: Department of Mathematics, Swansea University, Bay Campus, Fabian Way, Swansea SA1 8EN, U.K. ([d.l.finkelshtein@swansea.ac.uk]{}).
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abstract: 'Observations of the Sun at millimeter and submillimeter wavelengths offer a unique probe into the structure, dynamics, and heating of the chromosphere; the structure of sunspots; the formation and eruption of prominences and filaments; and energetic phenomena such as jets and flares. High-resolution observations of the Sun at millimeter and submillimeter wavelengths are challenging due to the intense, extended, low-contrast, and dynamic nature of emission from the quiet Sun, and the extremely intense and variable nature of emissions associated with energetic phenomena. The Atacama Large Millimeter/submillimeter Array (ALMA) was designed with solar observations in mind. The requirements for solar observations are significantly different from observations of sidereal sources and special measures are necessary to successfully carry out this type of observations. We describe the commissioning efforts that enable the use of two frequency bands, the 3 mm band (Band 3) and the 1.25 mm band (Band 6), for continuum interferometric-imaging observations of the Sun with ALMA. Examples of high-resolution synthesized images obtained using the newly commissioned modes during the solar commissioning campaign held in December 2015 are presented. Although only 30 of the eventual 66 ALMA antennas were used for the campaign, the solar images synthesized from the ALMA commissioning data reveal new features of the solar atmosphere that demonstrate the potential power of ALMA solar observations. The ongoing expansion of ALMA and solar-commissioning efforts will continue to enable new and unique solar observing capabilities.'
address:
- 'National Astronomical Observatory of Japan (NAOJ), 2-21-1, Osawa, Mitaka, Tokyo 181-8588, Japan'
- 'Department of Astronomical Science, The Graduate University for Advanced Studies (SOKENDAI), 2-21-1, Osawa, Mitaka, Tokyo 181-8588, Japan'
- 'National Radio Astronomy Observatory (NRAO), 520 Edgemont Road, Charlottesville, VA 22903, USA'
- 'Joint ALMA Observatory (JAO), Alonso de Córdova 3107, Vitacura 763-0355, Santiago, Chile'
- 'Space Vehicles Directorate, Air Force Research Laboratory, 3550 Aberdeen Avenue SE, Kirtland AFB, NM 87117-5776, USA'
- 'Institute for Space-Earth Environmental Research, Nagoya University, Furo, Chikusa, Nagoya, 464-8601, Japan'
- 'National Institute of Information and Communications Technology, 4-2-1, Nukui-Kitamachi, Koganei 184-8795, Tokyo, Japan'
- 'Astrophysics Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK'
- 'European Southern Observatory, Alonso de Córdova 3107, Vitacura 763-0355, Santiago, Chile'
- 'European Southern Observatory (ESO), Karl-Schwarzschild-Strasse 2, 85748 Garching bei München, Germany '
- 'Hvar Observatory, Faculty of Geodesy, University of Zagreb, Kačićeva 26, HR-10000, Zagreb, Croatia'
- 'Astronomical Institute, Academy of Sciences, Fričova 298, 251 65 Ondřejov, Czech Republic'
- 'Korea Astronomy and Space Science Institute, 776, Daedeokdae-ro, Yuseong-gu, Daejeon, 305-348, Republic of Korea'
- 'School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, Scotland、UK'
- 'Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way, Berkeley, CA 94720, USA'
- 'Institute of Theoretical Astrophysics, University of Oslo, Postboks 1029 Blindern, N-0315 Oslo, Norway'
- 'Center For Solar-Terrestrial Research, New Jersey Institute of Technology, Newark, NJ 07102, USA'
- 'Lockheed Martin Solar & Astrophysics Lab, Org. A021S, Bldg. 252, 3251 Hanover Street, Palo Alto, CA 94304, USA'
- 'Max-Planck-Institut for Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany'
- 'Astronomical Institute, St.Petersburg University, Universitetskii pr. 28, 198504 St.Petersburg, Russia'
- 'Center for Space Plasma and Aeronomic Research, Univ. of Alabama Huntsville, Huntsville, AL 35899, USA'
- 'National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District, Beijing 100012,China'
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title: 'Observing the Sun with Atacama Large Millimeter/submillimeter Array (ALMA): High Resolution Interferometric Imaging'
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Introduction
============
The Atacama Large Millimeter/submillimeter Array (ALMA) is a powerful, general purpose radio telescope designed to address a broad program of forefront astrophysics at millimeter and submillimeter (mm/submm) wavelengths [@5136193; @2010SPIE.7733E..17H]. Briefly, ALMA is an interferometric array that will ultimately be comprised of 66 antennas: $50 \times 12$ m antennas (the 12-m array); $12 \times 7$ m antennas (the 7-m array); and $4\times 12$ m “total power" antennas (the TP array)[^1]. All antennas are capable of observing continuum and spectral line radiation at frequencies ranging from 35 – 950 GHz (or wavelengths of 0.32 – 8.6 mm). The 12-m array is reconfigurable, with the distance between two antennas (the antenna baseline) ranging from 15 m up to 16 km, thereby providing great flexibility in angular resolution and surface brightness sensitivity. The 7-m array is a compact fixed array with baselines ranging from 9 m to 50 m. The four TP antennas are used as single dishes to measure emission on the broadest angular scales. The 7 m antennas bridge the gap between the angular scales measured by the TP antennas and those measured by the 12-m array. ALMA is located on the Chajnantor plain of the Chilean Andes at an elevation of 5000 m, an exceptional site for mm/submm observations.
ALMA science operations began with Cycle 0 in October 2011 with limited numbers of antennas and capabilities. Both the instrument and its capabilities have been expanding steadily since then and for ALMA observing Cycle 4, beginning in October 2016, at least $40\times 12$ m antennas, $10\times 7$ m antennas, and 3 TP antennas were available for scientific observations. The 12-m array supported nine antenna configurations and a total of seven frequency bands was available for scientific use on all antennas in Cycle 4. Additional technical details are available in the ALMA Cycle 4 Technical Handbook[^2][@ALMAC4Tech]. A more complete description of solar observing modes supported by ALMA in Cycle 4 is given in Section 5.
Solar physics has been an important component of the ALMA science program since its inception. Continuum and spectral-line radiation from the Sun at mm/submm wavelengths offers a unique probe of chromospheric structure and dynamics; the structure and dynamics of sunspots; of the formation and eruption of prominences and filaments; and of energetic phenomena such as jets and flares [see, @2002AN....323..271B; @2011SoPh..268..165K]. Particularly powerful are submm/mm observations carried out jointly with optical/IR and UV/EUV observations. A recent and comprehensive overview of solar science with ALMA is presented by @2016SSRv..200....1W.
Although the antennas of ALMA were carefully designed and constructed for observing the Sun, it is not possible to observe the Sun using the same observing modes that are employed for other astronomical objects. Since the other components of ALMA are optimized to observe faint objects (high-z galaxies), the mm/sub-mm wave radiation from the Sun is outside the normal operating parameter range. In addition, in the case of ALMA, the Sun is significantly bigger than the field of view (primary beam) of the ALMA 7 m and 12 m antennas, and the field of view is filled with complex solar structures that occupy a wide range of spatial frequencies. For these reasons, special measures must be taken to observe the Sun with ALMA.
To open solar observing to researchers, the ALMA solar development team, the joint ALMA observatory (JAO), and ALMA regional centers (ARC) of East Asia, Europe, and North America have been developing and commissioning solar observing modes that exploit both single-dish total-power maps of the Sun and interferometric observations of the Sun. Six solar commissioning campaigns were conducted from 2011 – 2015, culminating with the release of ALMA Band 3 and Band 6 to the solar community for continuum observing in the Cycle 4 proposal cycle [October 2016 – September 2017, @ALMAC4PropG].
In this article we present an overview of the observing modes available for interferometric solar observations with the Band 3 and Band 6 receivers [@2008SPIE.7020E..1BC; @2004stt..conf..181E] of ALMA in Cycle 4. A companion article by @White17 presents techniques developed for rapidly mapping the Sun using the TP antennas. In Section 2, we explain the challenges of observing the Sun with ALMA and the steps taken to address them. In Section 3 we discuss calibration procedures developed for solar observing. In Section 4 we present some Scientific Verification data (SV data) of the Sun that were released by the JAO in January 2017. The SV data were obtained during the sixth ALMA solar commissioning campaign, held in December 2015. We conclude in Section 5 with a brief discussion of future solar capabilities with ALMA.
Solar Observing with ALMA
=========================
In this section, the particular problems posed by observing the Sun with ALMA are outlined and their resolution is discussed. To include the Sun as part of ALMA’s scientific program meant designing telescope hardware, electronics, and computing systems that could achieve high performance across an extraordinary range of spatial, spectral, temporal, and intensity scales without compromising the performance of any mechanical, electrical, or optical element along the signal path. A critical problem for observing the Sun with a precision telescope like ALMA is the potential thermal load on the antennas imposed by the optical and infrared (OIR) radiation from the Sun. The issue was considered carefully during the design phase of ALMA, and it was mitigated by ensuring that the dish panel surfaces are rough enough at OIR wavelengths to scatter the bulk of the OIR radiation out of the optical path [@2004SPIE.5489.1085U; @2006PASP..118.1257M; @5136193] while maximizing the antenna efficiency at mm/submm wavelengths (better than 25 $\mu$m rms surface accuracy). Therefore, we do not discuss the issue further in this article.
Reduction of the Solar Signal at mm/submm Wavelengths
-----------------------------------------------------
The Sun is an intense mm/submm source, orders of magnitude more intense than cosmic sources that ALMA is optimized to observe. The brightness temperature of the Quiet sun is 5000 – 7000 K in the ALMA frequency, range and active-Sun phenomena can produce much higher brightness temperatures. ALMA receivers are designed for a maximum signal corresponding to an effective brightness of about 800 K at the receiver input, thereby limiting their dynamic range. Therefore, the solar signal must be attenuated or the receiver gain must be reduced to ensure that receivers remain linear, or nearly so. Two approaches to this problem were developed and tested during the commissioning phase: i) to attenuate the signal with a solar filter placed in the optical path in front of the receiver; or ii) to reduce the receiver gain to provide it with greater dynamic range. We discuss each approach in turn.
### ALMA Solar Filters
The initial solution adopted by ALMA to manage the input signal was the use of a solar filter (SF) that is mounted on the Amplitude Calibration Device (ACD) of each antenna [@ALMAC4Tech]. When placed in the optical path the solar filter is required to attenuate the signal by $4+2\lambda_{\rm mm}$ dB with a return loss of -25 dB (-20 dB for $\nu> 400$ GHz) and a cross polarization induced by the filter of -15 dB, or less. There are several drawbacks to this solution [see @6665775]:
- The hot and ambient calibration loads on the ACD cannot be observed when the SF is in the optical path, making amplitude calibration difficult.
- The SNR on calibrator sources is greatly reduced, not just by the attenuation introduced by the SF, but by the thermal noise that is added to the system temperature by the SF itself.
- The SF introduces frequency-dependent (complex) gain changes that may be time dependent and must be calibrated.
- The SFs introduce significant wave-front errors into the illumination pattern on the antenna, resulting in distortions to the beam shape and increased sidelobes.
- The Water Vapor Radiometers (WVRs) are blocked by the ACD for many bands when the SF is inserted into the optical path and phase corrections based on WVR measurements are therefore not possible in these bands.
Some of these difficulties have been overcome, the complex gains of antennas outfitted with SFs were measured during the third solar observing campaign in 2013, and interferometric imaging with solar filters has been demonstrated. In fact, the SFs will likely be used for observations of solar flares at some future time. Nevertheless, the disadvantages to the use of solar filters are significant. They must be moved out of the beam when observing calibrators, thereby increasing operational overhead. Since they introduce frequency-dependent and possibly time-dependent gains, they must be measured for every filter and frequency setting. Other calibrations including pointing, focus, and beam-shape measurements need to have the filters in place. The reduced SNR makes such measurements more difficult and time consuming.
### Receiver Gain Reduction
While the use of solar filters has been demonstrated to work, their use introduces enough disadvantages to consider whether an alternative approach may be more attractive. @6665775 pointed out that the ALMA Superconductor-Insulator-Superconductor (SIS) mixers could be de-tuned or de-biased to reduce the mixer gain. Since the dynamic range scales roughly inversely with gain, these settings can handle larger signal levels before saturating, potentially allowing solar observing without the use of the SFs, at least for non-flaring conditions on the Sun. This idea is illustrated in Figure 1, which shows the SIS current (left axis) and conversion gain (right axis) plotted against the SIS voltage bias for the ALMA Bands 3 and 6 receivers. The normal voltage bias tuning is on the first photon step below the gap where the gain conversion has a maximum. However, the mixer still operates at other voltage bias settings. These settings produce lower conversion gain, but since the dynamic range scales roughly inversely with gain, these settings can handle larger signal levels before saturating. In addition to the SIS bias voltage, the local oscillator (LO) power can be altered in order to further modify the receiver performance although that has not been explored in detail.
![SIS current and conversion gain as a function of voltage setting for the ALMA Band 3 (left) and Band 6 (right) receivers. The arrowed ellipses indicate the relevant ordinate: left for the SIS current and right for the conversion gain. \[fig:fig1\]](INT_Fig1_Band3.eps "fig:") ![SIS current and conversion gain as a function of voltage setting for the ALMA Band 3 (left) and Band 6 (right) receivers. The arrowed ellipses indicate the relevant ordinate: left for the SIS current and right for the conversion gain. \[fig:fig1\]](INT_Fig1_Band6.eps "fig:")
Tests conducted in 2014 showed that for Band 3 the second photon step below the gap has the flattest gain response as a function of SIS bias voltage as well as better linearity and sensitivity than the first step above the gap. This second photon step below the gap is suitable for “quiet" Sun observations and is referred to as “Mixer-Detuned mode 1", or MD1. For “active" Sun observations, further gain reduction is achieved by tuning to the second photon step above the gap, referred to as MD2. For Band 6 receivers it was found that the second photon step below the gap did not always provide a flat and stable gain response (at least with nominal LO power). Moreover, the receiver gain compression is moderate on the quiet Sun even at nominal receiver settings (first step below the gap). Therefore, no change from nominal settings is recommended for “quiet" Sun observing (MD1). For “active" Sun observations, tuning to the first photon step above the gap is recommended (MD2). It is seen that for both Band 3 and Band 6 receivers the MD2 mode provides reduced gain and better linearity for quiet Sun inputs. However, the improved dynamic range comes at the cost of higher system temperature due to increased receiver noise. This is not a problem when pointing at the Sun, for which the antenna temperature is significantly larger than the system temperature (see Section 3.1).
----------------------- -------------------- -------------------- -------------------- --------------------
[**Band 3**]{} 2nd Step Below Gap 1st Step Below Gap 1st Step Above Gap 2nd Step Above Gap
(MD1) (nominal) (MD2)
Receiver noise \[K\] $\approx$50 $<$41 $\approx$200 $\approx$800
Estimated compression $\approx$10 % $\approx$35 % $\approx$15 % (a few %)
(Quiet Sun input)
[**Band 6**]{} 2nd Step Below Gap 1st Step Below Gap 1st Step Above Gap -
(nominal/MD1) (MD2)
Receiver noise \[K\] $\approx$200 $<$83 $\approx$1000 -
Estimated compression $<$5 % $\approx$10 % (a few %) -
(Quiet Sun input)
----------------------- -------------------- -------------------- -------------------- --------------------
: SIS Mixer Settings for ALMA Cyle 4 solar observations
The analyses by @Yagoubov16 [@Iwai15; @Iwai16] report that the gain compression, an indicator of non-linear response of the receiver, $\approx$10 % at quiet sun and $\approx$15 % at active regions for the MD1 mode in both the Band 3 and Band 6 receivers. Considering the specification of the receivers, the decreasing of the sensitivity, and the brightness temperature range of the Sun, the receivers with the MD2 mode are believed to respond linearly. However, we cannot exclude the possibility that there is small amount of gain compression, which could possibly be a few %. Taking account of the current precision of solar visibilities obtained with ALMA, the influence of the nonlinear response with the MD2 mode is sufficiently small as to be neglected. On the other hand, if the MD1 mode is used to observe active regions and flares, the mildly nonlinear response will reduce the accuracy of measured brightness temperatures. If the MD2 mode is used to observe flares it, too, may suffer significant gain compression.
In assessing the two approaches to managing solar signals, it was concluded that the use of MD modes is preferable over the use of SFs for the non-flaring Sun because of the greater simplicity of their implementation and the associated calibration procedures, as we discuss further below. That said, it is likely that the use of SFs will be necessary to observe solar flares at mm/submm wavelengths.
Managing Signal Power Prior to Digitization
-------------------------------------------
Calibration of the ALMA antenna gains is discussed in greater detail in Section 3. Briefly, a calibrator source with known properties is observed by the array and the complex gains are deduced. The phase solutions are then transferred to the source data. For reasons discussed in Section 3, it is both possible and desirable to observe both the Sun and calibrator sources in a fixed MD mode. While both calibrators and the Sun can be observed in an MD mode, the power entering the system when pointing to cold sky when observing a calibrator and the power entering the system when observing the Sun are vastly different. ALMA employs two stages of heterodyne frequency conversion to shift the observed (sky) frequency down to a frequency range where system electronics can be used to digitize the analog signals and then correlate them. First, the signals at the observed radio frequency on the sky are mixed with a reference frequency (local oscillator (LO)) to an intermediate frequency (IF). The resulting IF frequency bands lie above and below the LO frequency (upper sideband and lower sideband). These are further subdivided and down-converted with a second LO in the IF Processor to four basebands that lie within the 2 – 4 GHz band. For continuum observations in ALMA Bands 3 and 6, a total of four 2 GHz bands are processed. These are referred to as spectral windows (SPWs). The continuum spectral windows observed by ALMA in Band 3 and Band 6 for Cycle 4 are detailed in Table 2.
SPW 1 SPW 2 SPW 3 SPW 4
-------- --------------- --------------- --------------- ---------------
Band 3 92 – 94 GHz 94 – 96 GHz 104 – 106 GHz 106 – 108 GHz
Band 6 229 – 231 GHz 231 – 233 GHz 245 – 247 GHz 247 – 249 GHz
: Continuum frequencies for ALMA Cycle 4 solar observations
The baseband signals are then digitized and correlated. The analog-to-digital converters (ADCs) are sensitive to input power; an important consideration given the difference in input power when observing calibrators and the Sun because the difference exceeds the dynamic range of the ADCs by a large margin. To adjust the input levels to the ADC to optimum values it is necessary to adjust signal power levels through the use of two stepped attenuators under digital control. One stepped attenuator is in the IF Switch, which controls which receiver signal enters the IF Processor; the other stepped attenuator is in the IF Processor itself [@ALMAC4Tech].
The solar development team carried out extensive test observations in October and November 2014 to determine the appropriate attenuator values. The stepped attenuators were set to values that optimized ADC signal input levels when observing the Sun. However, when the attenuation levels configured in the IF Switch and IF Processor are optimized for the Sun, they are non-optimum for calibrator sources. It is necessary to reduce the attenuation levels relative to the solar values when observing phase and bandpass calibrators. The recommended input level to the ADCs is 3.8 dBm. By adjusting IF Switch and IF Processor attenuation levels for calibrator observations relative to those used for observations of the Sun the input levels into the ADCs for observations of both the Sun and calibrators are near the recommended value (Table 3).
---------- --------- ----------- --------- -------------------- --------------------
Receiver MD mode IF Switch IF Proc Sun Calibrator (sky)
Band 3 MD1 -8 dB -10 dB $\approx\!3.5$ dBm $\approx\!3.5$ dBm
MD2 -8 dB 0 dB $\approx\!3$ dBm $\approx\!4$ dBm
Band 6 MD1 -10 dB -10 dB $\approx\!3.5$ dBm $\approx\!2.5$ dBm
MD2 -8 dB 0 dB $\approx\!4$ dBm $\approx\!4.5$ dBm
---------- --------- ----------- --------- -------------------- --------------------
: Differences of the attenuation levels for the calibrators from those for the Sun
Solar Data Calibration
======================
An interferometric measurement of the source at a particular time, frequency, and polarization by a pair of antennas is referred to as a “visibility". It is a complex quantity characterized by an amplitude and a phase and may be thought of as a single spatial Fourier component of the brightness distribution of the source. The measurement is made in the aperture plane : the -plane. The objective is to sample the -plane with sufficient density to recover the brightness distribution of source through Fourier inversion of the visibilities coupled with deconvolution techniques. Two key calibrations of ALMA visibility data are to measure the (time variable) complex gain of each antenna (amplitude and phase calibration) and to place the measurements on an absolute flux scale (flux calibration). Many additional calibrations are routinely required: antenna baselines, delay, frequency bandpass, polarization, etc. Those possibly affected by solar observing are touched on below.
Gain Calibration
----------------
Normally, amplitude and phase calibration of the antenna gains are performed by observing strong mm/ submm sources with accurately known positions, structure, and flux densities. Most flux calibrators are strong quasars or planets while phase calibrators are usually quasars that are point-like to the antenna array. By observing a phase-calibrator source every few minutes, the complex antenna gain (amplitude and phase) is deduced as a function of time. The gain solutions are then interpolated to the times at which the solar source is being observed and applied to the source data. The overall flux scale is determined by scaling the visibilities to Kelvin using measurements of the System Equivalent Flux Density (see Section 3.2) and further referencing the scaled visibility measurements to those of the flux calibrator [see Section 10.5 of @ALMAC4Tech]. However, the calibration of solar observations differs in key respects from those of faint, non-solar sources as we now discuss.
![Upper panel: The distribution of the quasars brighter than 0.5 Jy in Band 6. The color and size of the circle indicates the flux of a quasar. The black line indicates the track of the Sun. Lower panel: The separation angle between the Sun and possible calibrator sources ($>\!1$ Jy). The color indicates the flux of a quasar (same as that used in the upper panel).\[fig:fig2\]](INT_Fig2.eps)
When the SIS mixers are de-tuned to an MD mode the dynamic range of the receivers can accommodate the strong signal input from the Sun in a (nearly) linear fashion. Adopting the so-called MD mode for solar observing comes with two penalties: first, by tuning away from the nominal bias voltage in the SIS mixer, the MD mode introduces an unknown, but stable, gain change to the signal. This can either be measured for each antenna, frequency, and polarization or it can be ignored by observing both the source and the calibrator using the MD mode, in which case the gain change cancels out. The latter approach has been taken. Second, the use of MD modes results in an increase in receiver noise and a corresponding reduction in sensitivity. While the MD1 mode results in only a modest increase in the receiver noise ($\approx 20\ \%$ in band 3) the use of MD2 mode results in a much more significant increase of the receiver noise: characterized in terms of the receiver temperature, it increases from $\approx 50$ K to $\approx 1000$ K. This is not a problem as long as sufficiently strong calibrator sources are available that can overcome the reduced sensitivity. In practice, strong calibrators become increasingly sparse, especially at higher frequencies and so care must be taken in identifying a suitable calibrator source when using the MD2 mode. Using the ALMA Calibrator Source Catalogue, Figure 2 shows the distribution of possible calibrators that can be observed with the MD2 mode in Band 6 as a function of their flux density and position relative to the Sun (solid black line). There is a period in early July when there is no suitable calibrator within $20^\circ$ of the Sun, which leads to degraded transfer of phases during calibration (see Section 4). The situation is similar for Band 3; hence, observations of the Sun with the frequency bands higher than Band 3 are not recommended in early July.
As described in Section 2, steps have been taken to ensure that input signals remain nearly linear and within power limits to ensure optimum system performance when observing the Sun and calibrator sources. However, in addition to maintaining signal levels one must ensure that the signal phase is maintained. Phase differences between calibrator and solar-source scans are avoided by using the same MD mode to observe both. However, an additional concern is whether the stepped IF Switch and IF Proc attenuators themselves introduce unacceptable system temperature changes and/or differential phase variation between the Sun and calibrator settings, thereby corrupting phase calibration referenced against suitable calibrator sources. The variation in system temperature caused by the stepped attenuators is negligibly small, so it is not necessary to correct for their influence on flux calibration. On the other hand, the stepped attenuators do introduce significant phase shifts, depending on the difference in attenuation introduced for solar and calibrator scans. If the values of the phase shifts in all of the antennas are identical, however, the phase shift will be differenced out and the transfer of phase calibration from a calibrator to the solar source can proceed without the added complexity of measuring and applying differential phase corrections to account for phase errors introduced by the IF Switch and IF Processor attenuators. To check this, the bright quasar 3C279 was observed during the commissioning campaign in December 2015 while systematically changing IF Switch and IF Proc attenuator states on all antennas. Figure 3 shows an example of the differential phase variations caused by changing the attenuation levels. The channel-averaged value of the phase variation in a spectral window is very close to 0, and its standard deviation across the spectral window is 0.3 degrees for the attenuator in the IF Switch, and 0.6 degrees in the IF Processor. Moreover, there is no significant change of the phase variation during the campaign. These results indicates that the characteristics of the stepped attenuators are uniform and stable, and the phase variation caused in one antenna is almost canceled out by that in the other antenna. Therefore, there is no need to carry out additional calibration for the phase variation caused by changing the attenuation levels.
![The (differential) phase variation in a Band 6 spectral window. Left: The case of changing the attenuator in IF Switch -8 dB from the solar setting. Right: The case of changing the attenuator in the IF Processor by -10 dB. Colors indicate the observing day; red: 14, orange: 15, green: 16, dark green: 17, blue:18, purple:20 December 2015. \[fig:fig3\]](INT_Fig3.eps)
As we shall see below, amplitude and flux calibration referenced to standard source such as strong quasars or planets do not apply to solar data. The reason is that, in contrast to the vast majority of sidereal sources, the “antenna temperature" $T_{\rm ant}$, which indicates the input power from an observing target (Sun) in equivalent temperature scale, is significantly larger than the “system temperature" $T_{\rm sys}$ that indicates the system noise due to the receiver, other electronics, and spurious signals. In addition, the Sun is obviously not point-like; it fills the field of view of both the 7 m and 12 m antennas and their sidelobes. To properly calibrate visibility amplitudes and place them on a common flux scale it is necessary to measure both $T_{\rm sys}$ and $T_{\rm ant}$.
Flux Calibration
----------------
When an astronomical object is normally observed with ALMA, the output from the correlator is a normalized cross-correlation coefficient $\rho_{\rm mn}$ for a pair of antennas $m$ and $n$, is written as
$$\rho_{\rm mn} = {\sqrt{T_{\rm corr_m}T_{\rm corr_n}}
\over
{\sqrt{(T_{\rm ant_m}+T_{\rm sys_m})(T_{\rm ant_n}+T_{\rm sys_n})} } }$$
where $T_{\rm ant}$ is the antenna temperature, $T_{\rm sys}$ is the system temperature, and $T_{\rm corr}$ is the temperature of the correlated component of $T_{\rm ant}+T_{\rm sys}$. The relation between antenna temperature \[in units of K\] and flux density $S$ \[units W Hz$^{-1}$ m$^{-2}$\] is
$$T_{ant} = { {SA_{\rm e}} \over {2k} }$$
where $k$ is the Boltzmann constant and $A_e$ is the effective antenna collecting area \[$m^2$\]. The relation is also valid for the correlated component. From Eqs. (1) and (2), the amplitude of a visibility measurement is
$$S_{\rm corr_{mn}} = 2k {
{\sqrt{(T_{\rm ant_m}+T_{\rm sys_m}) (T_{\rm ant_n}+T_{\rm sys_n})}}
\over
{\sqrt{A_{\rm e_m} A_{\rm e_n}}}
} \rho_{\rm mn}$$
A System Equivalent Flux Density (SEFD) is defined as
$$SEFD = 2k { {T_{\rm sys}} \over {A_{\rm e}} }$$
Then, the amplitude of a visibility is written as
$$S_{\rm corr_{mn}} = \rho_{\rm mn}\sqrt{SEFD_{\rm m}\ SEFD_{\rm n}}\sqrt{(1+q_{\rm m})(1+q_{\rm n})}$$
where $q=T_{\rm ant}/T_{\rm sys}$. The antenna temperature of most celestial sources is much smaller than the system temperature, $T_{\rm ant} \ll T_{\rm sys}$, and $q = 0$. This is the case for calibrator sources which only need measurements of $T_{\rm sys}$ to scale the visibilities. In contrast, when observing the Sun $T_{\rm ant}>T_{\rm sys}$, and it is therefore necessary to measure both $T_{\rm sys}$ and $T_{\rm ant}$ in order to correctly scale the visibility measurements. The procedure for measuring $T_{\rm ant}$ and $T_{\rm sys}$ is described in detail by @White17 in the context of single dish observations of the Sun. Briefly, the antenna temperature is measured using the ACD on which “hot load" and “cold load" reference inputs are available. The following measurements are performed before each source scan:
- a cold-load observation $P_{\rm cold}$ (also known as the ambient load), in which an absorber at the temperature of the thermally controlled receiver cabin (nominally $15\ --\ 18^\circ$ C) fills the beam path;
- a hot-load observation $P_{\rm hot}$, in which an absorber heated to about $85^\circ$ C fills the beam path
- a sky observation $P_{\rm sky}$, offset from the Sun (typically by two degrees) and at the same elevation. The attenuation levels of the attenuators in IF chain are the same as that for the measurement of $P_{\rm cold}$ and $P_{\rm hot}$.
- an off observation $P_{\rm off}$, which is the same as the $P_{\rm sky}$, except the attenuation levels are set to the values optimized for the Sun
- a Sun observation $P_{\rm sun}$, which is at the attenuation levels of the target (Sun)
- a zero level measurement $P_{\rm zero}$, which reports the levels in the detectors when no power is being supplied.
The autocorrelation data output from the correlator cannot be used for these measurements because the correlator has insufficient dynamic range to measure $P_{\rm off}$. Instead, the measurements rely on the total-power data obtained by the baseband square-law detectors. The antenna temperature of the science target on the Sun is given by:
$$T_{\rm ant}=\frac{P_{\rm sky}-P_{\rm zero}}{P_{\rm off}-P_{\rm zero}} \frac{P_{\rm sun}-P_{\rm off}}{P_{\rm hot}-P_{\rm cold}} (T_{\rm hot}-T_{\rm cold})$$
and the system temperature is given from the online measurements [see Section 10.4, @ALMAC4Tech]. Additional details regarding flux calibration are provided in @White17.
To derive $T_{\rm ant}$ in practice requires modifying the standard ALMA observing sequence. There are three major differences, as shown schematically in Figure 4. The first is that subscans are needed for observing the sky near the Sun at the start and end of a science-target scan, the reason being that the $P_{\rm off}$ measurement has to be carried out with the attenuator levels set for observing the Sun. Hence, $P_{\rm off}$ is measured by the first and last subscans within the science target scan. The duration of the subscans used for measuring $P_{\rm off}$ is currently set to a few seconds. The second difference from standard procedures is that an atmospheric calibration is not carried out for each calibrator scan because it introduces too long a delay (many minutes) between source scans, to the possible detriment of a given observer’s scientific objectives. Instead, the system temperature derived from the atmospheric calibration near the Sun is applied to phase-calibrator data. The third modification is that the measurement of $P_{\rm zero}$ is carried out at the beginning of a solar observation. The value of $P_{\rm zero}$ is found to be very stable for a given antenna and frequency band, so we do not need to carry out the measurement frequently. Since the subscan duration of the SV data is less than 30 seconds, a measurement of $T_{\rm ant}$ and the amplitude calibration of the visibilities are done for every subscan within a science target scan. Considering dynamic solar phenomena with short temporal scales (flares), the short integration time for calculating $T_{\rm ant}$ is suitable for science although significant computer resources are needed for the amplitude calibration. Figure 5 shows an example of the time variation of $T_{\rm ant}$ for a mosaic observation, where an image is constructed from a pattern of discrete antenna pointings (see Section 4). It is clear that $T_{\rm ant}$ varies as ALMA points to different locations on the Sun.
Bandpass Calibration
--------------------
Continuum observations are performed in four spectral windows. In fact, the observations in each SPW are coarsely channelized and corrected for the variation in phase and amplitude across the frequency band, a process that is referred to as bandpass calibration. Following bandpass calibration the channels may be summed and imaged as continuum emission. Bandpass calibration is carried out in the usual manner even when solar MD observing modes are used: a strong calibrator is observed in an MD mode with the attenuator levels optimized for the Sun and the bandpass solution is obtained. The bandpass shape and stability were checked for the MD modes and attenuator states. It was found that the perturbations to bandpass amplitudes and phases were small. For the IF-switch and IF-processor-attenuator settings adopted for observations with an MD mode, it was found that the RMS difference between bandpass phases for an attenuator state and the nominal attenuator state was generally a fraction of a degree for both the Band 3 and Band 6 receivers, the maximum being 1.2 degrees. Similarly, the normalized amplitude difference was typically a fraction of 1 %. We conclude that no explicit correction is needed to normal bandpass calibration as a result of using MD modes or different attenuator states when observing calibrator sources and the Sun.
Additional Considerations
-------------------------
The primary beams of the main 12 m ALMA antennas are small compared with the Sun ($\approx 58$ at band 3, $\approx 24$ at band 6), and solar structures have various spatial scales. Therefore, to synthesize the solar brightness distribution visibility measurements should be distributed uniformly with spatial frequency in the aperture plane (the -plane). The -coverage can be improved by employing the Earth rotation synthesis technique, but this is only scientifically useful for slowly-varying, stationary structures, while many solar structures are dynamic in nature and vary on short time scales ($<$ one minute).
In Cycle 4, $40\times 12$ m antennas and $10\times 7$ m antennas were available for solar observing. The distribution of the 12 m antennas on the Chajnantor Plateau (array configuration) varies throughout the cycle from compact configurations to high-resolution long-baseline configurations. The proposal guide lists the configurations available for observing extended sources [see Table A-2 of @ALMAC4PropG]. The table reveals that multiple configurations of the 12-m array are needed to observe extended sources in configurations larger than C40-4, as more extended configurations undersample the Sun’s brightness distribution. Different configurations of the 12-m array cannot be realized at the same time. Therefore, solar observations must be carried out with the compact-array configurations. In Cycle 4, only the three most compact antenna configurations are available for solar observing: C40-1, C40-2, and C40-3.
A second reason that solar observations are restricted to compact array configurations is that higher angular resolution requires longer antenna baselines, and longer baselines are susceptible to phase fluctuations caused by precipitable water vapor (PWV) in the atmosphere overlying the array. For non-solar observing, it is possible to estimate the amount of precipitable water vapor along the line of sight of each antenna using Water Vapor Radiometers [WVR: @ALMAC4Tech]. Such measurements are essential for the phase calibration of long baselines, especially for higher frequencies. However, the WVR system is not available for solar observations because the WVRs saturate when the antennas point at the Sun. The issue of phase fluctuations on longer baselines and/or at higher observing frequencies will need to be confronted as new solar observing capabilities are made available – for example, the use of band 7 (275 – 373 GHz) and band 9 (602 – 720 GHz).
Although solar observing is confined to compact array configurations, the spatial-frequency coverage of the -plane from the 12 m antennas alone is still not adequate for synthesizing solar images. It is essential to observe the Sun with the 7-m array and 12-m array simultaneously. For non-solar observations, the 7-m array is operated with the ACA correlator [@2012PASJ...64...29K]. Simultaneous observations with the 7-m array and the 12-m array are not performed in general. However, since solar imaging requires the short baseline coverage provided by the 7-m array together with the longer baselines provided by the 12-m array both the 7m and 12m antennas are connected to the 64-input baseline correlator , and the ACA correlator is not used for solar interferometric observations. In the other words, solar observations with ALMA are carried out with a heterogeneous array.
To synthesize solar images calibrated to absolute brightness temperatures, the data that are obtained from the heterogeneous array are still not complete because angular scales greater than those measured by the shortest antenna baselines are not available. These are measured by the TP antennas using fast-scan mapping techniques described by @White17. In Cycle 4, a solar observation with TP antennas are carried out with a solar interferometric observation simultaneously to enable the two types of measurement to be combined as appropriate. When the elevation of the Sun is higher than 70$^\circ$, we cannot observe the Sun with the fast-scanning mode of the TP array. On the other hand, to avoid shadowing, solar observations with the fixed 7-m array cannot be performed when the elevation of the Sun is lower than 40$^\circ$. Moreover, the heterogeneous array also cannot observe the Sun when the elevation is higher than 82$^\circ$. Considering these elevation limitations the temporal range for solar observations in a day is limited, as shown in Figure 6.
ALMA Solar Imaging Examples
===========================
The solar commissioning campaign for verifying the Cycle 4 solar observing modes described above was held from 14 – 21 December 2015. The specific modes and capabilities offered in Cycle 4, and verified during the campaign, are as follows:
- Band 3 and Band 6 continuum observations of the Sun will be supported
- Solar observing will only be offered for the most compact array configurations
- Both 7 m and 12 m antennas will be correlated by the 64-input baseline correlator
- Both single pointing and mosaic (up to 150 pointings) interferometric observations of target sources will be supported
- Observations with the interferometer will be supported by fast-scanning total power (TP) maps of the full disk of the Sun
A number of solar targets was observed: active regions, quiet sun, solar limb, and a prominence above the limb. Only $\approx$30 antennas, including $9\times 7$ m antennas were typically available for the campaign. Therefore, the quality of the solar images presented in this article is not as good as those expected in Cycle 4 because of the larger number of antennas available in Cycle 4. Most of the data obtained from the December 2015 campaign were released by JAO as Scientific Verification (SV) data on 18 January 2017. The solar SV data can be downloaded from the ALMA Science Portal web site hosted by each ARC.
Data and Image Synthesis
------------------------
In order to introduce solar images synthesized from ALMA observations, we use the SV data listed in Table 4. The observing period given in the table includes all calibrations required to execute the observation; the bandpass and flux calibrations before the scientific scans. All of the examples given used the MD2 observing mode and the imaging employed the mosaic technique, in which a grid of discrete antenna pointings is used to image a much larger field of view than is available with a single pointing. For the examples presented here, the maximum number of mosaic pointings currently supported by the instrument were used: 149 pointings. The ICRS reference coordinates refer to the RA/Dec coordinates of the center of field of view at the reference time. The integration time for each MOSAIC pointing is 6.048 seconds, and the angular separation of points is 11.2 for Band 6, and 24.1 for Band 3; Nyquist sampling in each Band.
---------- -------------------- ----------- ------------ ----------------- ------------ ------------------------------------------
Data Set Execution Block ID Frequency Target Observing Reference ICRS Reference
Band Period Time Coordinate
1 3 AR 12470 18:01 – 18:48UT 18:32:41UT 17$^{\rm h}$35$^{\rm m}$32.218$^{\rm s}$
16 Dec 2015 -23$^{\rm d}$1623.843
2 6 AR 12470 19:15 – 20:08UT 19:49:00UT 17$^{\rm h}$44$^{\rm m}$10.112$^{\rm s}$
18 Dec 2015 -23$^{\rm d}$1930.632
3 6 South Pole 13:31 – 14:32UT 14:09:37UT 17$^{\rm h}$51$^{\rm m}$46.086$^{\rm s}$
20 Dec 2015 -23$^{\rm d}$4133.229
---------- -------------------- ----------- ------------ ----------------- ------------ ------------------------------------------
: Science Verification Data Used
The Common Astronomy Software Applications (CASA) package [@2007ASPC..376..127M], which is the standard reduction/imaging/analysis software for ALMA data, was used to calibrate and image the SV data. CASA can deal with data obtained with a heterogeneous array represented by the use of both 12m and 7m antennas. Hence, ALMA standard calibrating method is used for solar data, except for the amplitude calibration steps described in Section 3.2. When we use the [clean]{} task of CASA for synthesizing a solar image, the [mosaic]{} option for the [imagemode]{} parameter has to be used even for the data of single-pointing observations, to deal with the heterogeneous-array nature of the data. For mosaic observations, the coordinate of each pointing has to be re-calculated relative to the center of the FOV using ALMA pointing data. This is necessary because the heliocentric coordinate frame is moving relative to the RA/Dec coordinate frame during an observation. The reference time in Table 4 indicates the time used to define the reference position of the Sun. To improve image quality, we include the data from all four SPWs for synthesizing one solar image in this article. Therefore, the observing frequency of the solar images shown in this article is the same as the frequency of the first LO: 100 GHz for Band 3, 239 GHz for Band 6.
Figure 7 shows the images of the leading sunspot in AR 12470 on 16 December 2015 synthesized from the 149-point mosaic observation in Band 3 (example 1 in Table 4). The default visibility weighting option of the CASA [clean]{} task is to set the Briggs robust weighting parameter [@Briggs95] to zero. We note that artifacts appear in the image in the form of fine stripes, as evident in panels a and c. We attribute this to undue weight being given to longer interferometric baselines. In particular the locations of the centers of 12-m array and 7-m array are not the same; the distance between them is about 200m. Hence, data on baselines longer than 200 m are always included in solar data, even when the observation is done with the most compact configuration of the 12-m array, as was the case in December 2015. The resulting baseline distribution is non-optimum and the (nonlinear) image deconvolution process is subject to instability. The weighting of these longer baselines can be reduced by applying more nearly “natural weighting" [@2001isra.book.....T] by setting the Briggs robust weighting parameter to unity. When this is done, the artificial stripes disappear in the image (panels b and d of Figure 7). Hence, in this article, we always set the robust parameter to 1.0. The value is not fully optimized, and the most suitable value might depends on the target and array configuration. The angular resolution of the images shown in Figure 7b, d – the dimension of the synthesized beam - is 4.9$\times$2.2.
The synthesized solar images include pixels with negative values. The negative values have physical meaning, because the interferometric data does not include the DC component of the brightness distribution in the field of view. Therefore, simultaneous single-dish observations are essential for obtaining absolute brightness temperatures from ALMA data. Figure 7e is the result of combining the synthesized image and the full-Sun map constructed from the simultaneous single-dish mapping data. The full-Sun map is created with CASA using the reduction & imaging script included in the SV-data package. @White17 pointed out that a correction factor has to be applied to any map created with CASA. The factor is applied to the full-Sun map used for creating the combined image shown in Figure 7e. We note that the correction factor is not applied to the full-Sun images of the SV data released on 18 January 2017.
For the combination, we use the default parameters of the [feather]{} task in CASA. We found that the averaged brightness temperature of the combined image is always larger (5$\ --\ $10%) than the temperature brightness at the same position in the single-dish map even though the values should be similar. This means that the parameters of the [feather]{} task will need to be tuned in order to obtain consistent images, before using combined images for precise discussion of the absolute Tb.
Figure 8 presents solar images synthesized from a 149-point mosaic observation in Band 6 using the MD2 mode. Panel a shows the leading sunspot in AR 12470 (example 2 in Table 4) on 18 December 2015, and the panel b shows the solar limb near the South Pole (example 3 in Table 4). The calibration and synthesis imaging process are the same as those employed for Band 3, except for the observing frequency. Note that here the single dish data was not used for the images in Figure 8. The synthesized beams are $2.4\arcsec\times 0.9\arcsec$ for the sunspot image and $1.7\arcsec\times 1.0\arcsec$ deg for the South Pole image. We note that the narrow bright limb seen in the Figure 8b does not indicate “limb brightening" that can be seen in a direct full-Sun image with radio. The value in a synthesized image instead indicates the derivation from the average brightness of the FoV that is determined mainly by the beam shape of an antenna ($\approx$25 at 239 GHz), even when we observe the target in the MOSAIC mode. The deviation at the solar disk near the limb appears anomalously large, because the brightness changes suddenly from the quiet sun level to the sky level. Thus, such a narrow bright limb appears only in the synthesized image. It should not be present in a combined image that is created from the synthesized image and full-Sun map.
Estimating the Noise Level of Solar Synthesized images
------------------------------------------------------
The noise level of a synthesized image may be determined from the value of the brightness on blank sky. However, this method cannot be applied to solar synthesized images because the primary beam of ALMA antennas is significantly smaller than the Sun in all frequency bands. Solar emission therefore completely fills the field of view in most cases, complicating the task of estimating noise. We therefore use an alternate method. ALMA is designed to support full polarimetry. To measure the Stokes-polarization parameters, the Band 3 and Band 6 receiver cartridges contain two complete receiver systems sensitive to orthogonal linear polarizations [@ALMAC4Tech]. We call one polarization X and the other one Y. The 64-input baseline correlator enables us to calculate four cross-correlations (XX, YY, XY, and YX) from the X- and Y-signals for each antenna baseline. However, only XX- and YY-cross-correlations are useful for solar observations in Cycle 4 because ALMA support of full Stokes polarimetry is not yet offered as a scientific capability. Nevertheless, we can synthesize images using XX- and YY-data that are observed simultaneously. In the absence of any flare emission, as was the case for the examples presented, solar mm/sub-mm emission is thermal emission from optically thick plasma . Although there is possibility that the thermal emission is circularly polarized due to the presence of strong magnetic fields [@2004ASSL..314...71G; @2016ApJ...818....8M; @Loukitcheva17], net linear polarization should be absent due to differential Faraday rotation, and we can assume that any such polarization at 100 GHz and 239 GHz is negligibly small in comparison with the precision of current ALMA solar observations. The crosstalk of the polarizations in the receiver system can be also neglected [@2006stt..conf..154C]. Therefore, the difference between the solar images synthesized from XX- and YY- data should be zero in principle, and the difference between the two polarizations can therefore be used as a proxy for the noise level in the final images (see Appendix A).
![(a) and (b) The Band 3 sunspot images synthesized from the data of XX and YY respectively; (c) The difference image of (a) and (b); (d) The pixel distributions of brightness in (a) \[Black\] and (b) \[Red\]; (e) the pixel distribution function of the difference image (c). The red line on (e) indicates the Gaussian function fit to the distribution.](INT_Fig9.eps)
Figure 9 shows estimations of the noise-level from maps formed using the XX- and YY-correlations. From the width of the Gaussian function fitted to the distribution of the differential (Figure 9e), the noise level of the Band 3 synthesized image of the sunspot (Figure 7b) is 3.7 K when the integration time is six seconds and the integration bandwidth is 8 GHz. We also apply the method to the sunspot image observed with Band 6 (Figure 8a), and estimate the noise level to be 9.8 K. The integration time and bandwidth of the Band 6 image are the same as those of the Band 3 image.
Imaging Artifacts Above the Solar Limb
--------------------------------------
In addition to thermal noise, imaging artifacts may be present in a synthesis image as a result of incomplete sampling of the -plane, non-optimum weighting of the visibility data (cf. Section 4.1), source variability, or other factors. An example of an artifact resulting from incomplete sampling and possibly non-optimum weighting is shown in Figure 10, in which a detail of the mosaic image of the South Pole is shown. Figure 10a shows a map made using the heterogeneous array comprised of 7 m antennas and 12 m antennas, as also shown in Figure 8b. Figure 10b shows the same image using only the 12 m antennas and Figure 10c shows the same image using only 7 m antennas. A stripe of negative flux density appears above the limb in Figure 10a and a stripe of positive flux density is seen even higher above the limb. The stripes are non-physical artifacts due to incomplete sampling of the “step function" represented by the bright solar disk falling off to cold sky. The interferometric array shows a “ringing" or “overshoot" response as a result. In the image synthesized from only 7m antennas the positive enhancement is very weak (Figure 10c and the blue lines in Figure 10d, e) although the negative stripe persists. The image synthesized from only 12m antennas shows a stronger enhancement with a peak located about 20 above the limb (Panel b and the red lines in panels d and e of Figure 10). We note that the shortest baseline of the 12-m array observing the solar limb is 12.9 m, so the largest angular scale measured is 20.1 at 239 GHz. For the heterogeneous array the shortest baseline measured is 7.6 m, corresponding to an angular scale of 34.3 at 239 GHz. In principle, inclusion of the 7 m antennas should bridge the gap between the single-dish total-power map (resolution 24.4) and the largest angular scale measured by the 12-m array.
![The solar-limb images synthesized from the data of the (a) heterogeneous array, (b) 12-m array, and (c) 7-m array. The red and blue contours in the panels indicate +20 K level of 12-m array and 7-m array respectively. (d) and (e) show the brightness profiles as a function of the distance from the solar limb. Black: Heterogeneous array, Red: 12-m array, Blue: 7-m array. The difference of (d) and (e) is the range of [*y*]{}-axis.](INT_Fig10.eps)
A possible problem is mis-matched cross-calibration between 7 m and 12 m antennas. CASA currently supports two approaches to calibrating visibilities obtained with a heterogeneous array. In one, the data are jointly calibrated and in the other the data obtained with the 12-m array and 7-m array are calibrated independently and then combined. We carried out the calibration of the data using both methods, and compared the resulting images. However, we cannot find any significant difference. Another possibility is that the relative weighting of the visibility baselines is incorrect: a careful assessment of the weights assigned to 7 m–7 m, 7 m–12 m, and 12 m–12 m baselines, as well as the weight given to the single dish total power map is needed. A final possibility is insufficient numbers of short antenna baselines. The 7-m array provides short baselines, and the visibilities of the baselines should suppress the sidelobes created by the 12-m array. In our case, we can see the suppression of the sidelobe by 7 m antennas (see the difference of the red and black lines in Figure 10). The remaining enhancement in the image synthesized from the data with the 7 m + 12 m heterogeneous array might indicate the lack of the short baselines. The commissioning observation is carried out using 9 $\times$ 7 m antennas and 21 $\times$ 12 m antennas. The number of the antennas is smaller than that for Cycle 4 observations and so there will be opportunities to better understand and resolve the issue.
Co-alignment between ALMA and other instruments
-----------------------------------------------
To maximize the scientific impact of ALMA data, it is very important to compare ALMA images with those obtained by instruments operating at other wavelengths with similar angular resolution. Direct comparisons require that ALMA images are accurately co-aligned with those produced by other instruments. ALMA operates in a geocentric coordinate frame using Right Ascension and Declination while heliocentric coordinate are usually used for solar imaging data. Therefore, ALMA images must be converted from RA/Dec coordinates to a heliocentric coordinate frame.
The precision of the absolute pointing of the ALMA antennas is better than 2 [@ALMAC4Tech]. Figure 11 shows the result of co-alignment between the sunspot image with Band 6, UV continuum, and EUV images obtained with Solar Dynamics Observatory/Atmospheric Imaging Assembly [SDO/AIA: @2012SoPh..275...17L]. For the co-alignments, we do not make any adjustment except for the coordinate conversion. It is hard to verify the co-alignment rigorously, because it is hard to find counterparts of the Band 6 images in the AIA images. The bright structure above the remnant of the light bridge in the AIA 304 Å image is very similar to that in the Band 6 image. In comparing the edge of the structure in the umbra (yellow arrow in Figure 11a) the precision of the co-alignment appears to be better than the size of the synthesized beam (Figure 10e, g).
Similarly, Figure 12 shows the result of the co-alignment between the Band 6 image and a Mg [ii]{} k$_{\rm 2v}$ image obtained with the Interface Region Imaging Spectrograph [IRIS: @2014SoPh..289.2733D]. In this case, we can easily identify the counterparts of the Band 6 image in the IRIS image. Therefore, the co-alignment is done only by the visual inspection. The similarity between the images suggests that Band 6 and Mg [ii]{} k$_{\rm 2v}$ line emissions are formed within approximately the same range of heights.
Concluding Remarks
==================
To conclude, this article summarizes the development and science-verification efforts leading up to the release of solar-observing modes by ALMA for Cycle 4 in 2016 – 2017. While current capabilities remain limited, they represent a major advance over observational capabilities previously available at mm/submm wavelengths. Coupled with exciting space-based observations obtained by, , SDO, and IRIS; and ground based observations at, National Solar Observatory, Big Bear Solar Observatory, Tenerife, and La Palma, ALMA opens a new window on contemporary scientific problems in solar physics.
Current ALMA capabilities are summarized at the beginning of Section 4. Looking forward, additional capabilities are planned in support of solar observing that will greatly expand ALMA’s science capabilities. It is planned that the following new capabilities will be available in the near future:
- Band 7 (275 – 373 GHz: 850 $\mu$m) and Band 9 (602 – 720 GHz: 450 $\mu$m) continuum observations of the Sun will be supported, in addition to Bands 3 and 6
- Low-resolution spectroscopy (TDM mode) in Bands 3, 6, 7, and 9
- Support of full Stokes polarimetry
- Support of sub-second integration times
In the longer term, additional ALMA frequency bands will become available for use by the solar community. A number of other capabilities are under consideration, but the timing of their availability has not yet been established. These include the use of science subarrays, where the ensemble of 66 ALMA antennas can be divided into two or more independent arrays to perform multi-band or multi-target observations; band switching observations where an observer can change frequency bands on short times scales; fast-scan single dish mapping of small regions of the Sun – an active region – on short time scales (tens of seconds); larger mosaics to enable imaging of larger regions on the Sun. The solar community will be informed about new capabilities for solar observing when the call for proposals is issued by the Joint ALMA Observatory each year.
The ALMA solar commissioning effort was supported by ALMA Development grants from NAOJ (for the East Asia contribution), NRAO (for the North American contribution), and ESO (for the European contribution). The help and cooperation of engineers, telescope operators, astronomers–on–duty, Extension and Optimization of Capabilities (EOC; Formerly Commissioning and Science Verification) team, and staff at the ALMA Operations Support Facility was crucial for the success of solar commissioning campaigns in 2014 and 2015. We are grateful to the ALMA project for making solar observing with ALMA possible. This article makes use of the following ALMA data: ADS/JAO.ALMA\#2011.0.00020.SV, ADS/JAO.ALMA\#2011.0.00001.CAL. ALMA is a partnership of ESO (representing its member states), NSF (USA), NINS (Japan), together with NRC (Canada), NSC, ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. SDO is the first mission to be launched for NASA’s Living With a Star (LWS) Program. IRIS is a NASA small explorer mission developed and operated by LMSAL with mission operations executed at NASA Ames Research center and major contributions to downlink communications funded by ESA and the Norwegian Space Centre. This work was partly carried out on the solar data analysis system and common-use data-analysis computer system operated by Astronomy Data Center of NAOJ. M. Shimojo was supported by JSPS KAKENHI Grant Number JP17K05397. R. Brajša acknowledges partial support of this work by Croatian Science Foundation under the project 6212 Solar and Stellar Variability and by the European Commission FP7 project SOLARNET (312495, 2013 - 2017), which is an Integrated Infrastructure Initiative (I3) supported by the FP7 Capacities Programme. G.D. Fleishiman acknowledges support from NSF grants AGS-1250374 and AGS-1262772. The trip of Y. Yan to 2015 ALMA Solar Campaign was partially supported by NSFC grant 11433006. S. Wedemeyer acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 682462)
ALMA antennas measure the two orthogonal linear polarizations X and Y, and the 64-input baseline correlator measures the products of the linearly polarized antenna voltages. For a pair of antennas, $m$ and $n$, the correlation products are $v_{\rm x_mx_n}$, $v_{\rm y_my_n}$, $v_{\rm x_my_n}$, and $v_{\rm y_mx_n}$. For well-designed antenna feeds and weakly polarized emission [@1999ASPC..180..111C], the response of the interferometer can be expressed as $$\begin{aligned}
v'_{\rm xx} &=& g_{\rm mx} g^\ast_{\rm nx} (I + Q \cos 2\chi + U \cos 2\chi) + \sigma'_{\rm xx} \\
v'_{\rm xy} &=& g_{\rm mx} g^\ast_{\rm ny} ((d_{\rm mx}-d^\ast_{\rm ny})I - Q \cos 2\chi + U \cos 2\chi + jV) + \sigma'_{\rm xy} \\
v'_{\rm yx} &=& g_{\rm my} g^\ast_{\rm nx} ((d^\ast_{\rm nx}-d_{\rm my})I - Q \cos 2\chi + U \cos 2\chi - jV) + \sigma'_{\rm yx} \\
v'_{\rm yy} &=& g_{\rm my} g^\ast_{\rm ny} (I - Q \cos 2\chi - U \cos 2\chi) + \sigma'_{\rm yy}\end{aligned}$$ where $I$ is Stokes parameter describing the total intensity of the radiation, $Q$ and $U$ are the Stokes parameters characterizing linearly polarized radiation, and $V$ is the Stokes parameter characterizing circularly polarized radiation. The parallactic angle \[$\chi$\] includes the effects of rotation of the alt–az ALMA antennas as viewed from the source. The $g$-factors are complex gain factors established by calibration and the $d$-terms represent polarization “leakage" which, by careful design, are small but measurable complex numbers, also determined by calibration. The noise in each correlation measurement is represented by $\sigma'$. At present ALMA does not support full Stokes polarimetry and in particular, measurements of Stokes-$V$, which requires calibration of the complex leakage terms. However, it is expected that support of full Stokes polarimetry will be implemented soon, thereby enabling a powerful new probe of chromospheric magnetic fields. For the present purpose, however, only the parallel correlations are of interest here. Rearranging $v_{xx}$ and $v_{yy}$ we have $$\begin{aligned}
(I + Q \cos 2\chi + U \cos 2\chi) &=& v'_{\rm xx}/(g_{\rm mx} g^\ast_{\rm nx}) + \sigma_{\rm xx}'/(g_{\rm mx} g^\ast_{\rm nx}) = v_{\rm xx} + \sigma_{\rm xx}\\
(I - Q \cos 2\chi - U \cos 2\chi) &=& v'_{\rm yy}/(g_{\rm my} g^\ast_{\rm ny}) + \sigma_{\rm yy}'/(g_{\rm my} g^\ast_{\rm ny}) = v_{\rm yy} + \sigma_{\rm yy}\end{aligned}$$ where the unprimed quantities represent calibrated measurements. Summing and differencing these quantities and propagating the noise terms yields $$\begin{aligned}
I &=& {1\over 2}(v_{\rm xx} + v_{\rm yy}) + \sigma_{\rm I} \\
Q \cos 2\chi &+& U \cos 2\chi = {1\over 2}(v_{\rm xx} - v_{\rm yy})+ \sigma_{\rm I}\end{aligned}$$ where $\sigma_{\rm I}=\sqrt{\sigma^2_{\rm xx}+\sigma^2_{\rm yy}}/2$. It is seen that the sum of the calibrated correlation products $v_{\rm xx}$ and $v_{\rm yy}$ for a given antenna pair represents the interferometer’s response to Stokes $I$. While the Stokes-$V$ parameter may be non-zero the Stokes-$Q$ and $U$ parameters are expected to be zero for thermal solar emission and so $(v_{\rm xx} - v_{\rm yy})/2 = \sigma_{\rm I}$. Note further that for emission that is not linearly polarized, the calibrated noise terms are such that $\sigma_{\rm xx}=\sigma_{\rm yy}$ and so $\sigma_{\rm I}=\sigma_{\rm xx}/\sqrt{2} = \sigma_{\rm yy}/\sqrt{2}$. Since synthesis maps represent a linear superposition of interferometric measurements, the same relation holds true for synthesis images.
[30]{} \#1[ISBN \#1]{}\#1[\#1]{}\#1[\#1]{}\#1[\#1]{}\#1[*\#1*]{}\#1[**\#1**]{}\#1[\#1]{}\#1[\#1]{}\#1[\#1]{}\#1[\#1]{}\#1\#1\#2\#1[\#1]{}\#1[\#1]{}\#1[*\#1*]{}\#1[\#1]{}\#1[**\#1**]{}\#1[\#1]{}\#1[*\#1*]{}\#1[\#1]{}\#1[\#1]{}\#1[\#1]{}\#1\#1[\#1]{}\#1[\#1]{} \#1[[](http://dx.doi.org/#1)]{} \#1[[](http://arxiv.org/abs/#1)]{} \#1[[](http://adsabs.harvard.edu/abs/#1)]{}\#1\#1\#1\#1\#1[\#1]{}\#1[\#1]{}\#1[\#1]{}\#1[\#1]{}
, , , , , , , , : 2016, *[ALMA Cycle 4 Technical Handbook]{}*, http://almascience.org/documents-and-tools/cycle4/alma-technical-handbook. 978-3-923524-66-2.
, , , , , , : 2016, *[ALMA Cycle 4 Proposer’s Guide]{}*. ALMA, Doc. 4.2 v1.0.
: , . , . . .
: 1995, High fidelity deconvolution of moderately resolved sources. PhD thesis, The New Mexico Institue of Mining and Technology, Socorro New Mexico.
, , , , , , , , , , , , , , , , , , : , . In: , , , , , , , (eds.) , . .
, , , , , , , , , , , , , , , , , : , . In: , , , (eds.) , , . . .
: , , , , , . .
, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , : , . , . . .
: , . , . . .
, , , , , , , , , , , , : , . In: (ed.) , . .
, , , , , , , , , , , : , . , . . .
, : , . In: , (eds.) , , . . .
, , : , . In: , , . . .
, , , , , , , , , , , , , , , , , : , . , . . .
: 2015, [in ALMA Solar Observing I Test and Validation]{}. Technical report, ALMA Solar Comissioning Report. Joint ALMA Observatory, Chile.
: 2016, [Nonlinearity of ALMA Antennas in Detuning Mode 1.]{} Technical report, CSV-3246 Report. Joint ALMA Observatory, Chile.
, , , , , , , , , , : , . , . . .
, , , : , . , . . .
, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , : , . , . . .
, , , , : 2017, [Millimeter radiation from a 3D model of the solar atmosphere. II. Chromospheric magnetic field]{}. *[ [*Astron. Astrophys.*]{}]{}* **in press**.
, , , , , , : , . , . . .
, , , , : , , , , , . .
, , , , : , . , . . .
, , : , , , . .
, , , , , , , : , . In: (ed.) , , . . .
, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , : , . , . . .
, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , : 2017, [Observing the Sun with ALMA: Fast-scan Single-dish Mapping]{}. *[[*Solar Phys.*]{}]{}*, in press.
, : , . (), . .
: 2016, [Recommendations on mixer de-tuning for Solar observations with ALMA]{}. Technical report, CSV-3245 Report, Joint ALMA Observatory, Chile.
: , . In: , , , . .
[^1]: Atacama Compact Array (ACA: also known as the Morita Array) is a short-spacing imaging system consisting of the TP array and 7-m array [@2009PASJ...61....1I].
[^2]: http://almascience.org/documents-and-tools/cycle4/alma-technical-handbook
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abstract: 'Let $\mathcal{M}(\Omega, \mu)$ denote the algebra of all scalar-valued measurable functions on a measure space $(\Omega, \mu)$. Let $B \subset \mathcal{M}(\Omega, \mu)$ be a set of finitely supported measurable functions such that the essential range of each $f \in B$ is a subset of $\{ 0,1 \}$. The main result of this paper shows that for any $p \in (0, \infty)$, $B$ has strict $p$-negative type when viewed as a metric subspace of $L_{p}(\Omega, \mu)$ if and only if $B$ is an affinely independent subset of $\mathcal{M}(\Omega, \mu)$ (when $\mathcal{M}(\Omega, \mu)$ is considered as a real vector space). It follows that every two-valued (Schauder) basis of $L_{p}(\Omega, \mu)$ has strict $p$-negative type. For instance, for each $p \in (0, \infty)$, the system of Walsh functions in $L_{p}[0,1]$ is seen to have strict $p$-negative type. The techniques developed in this paper also provide a systematic way to construct, for any $p \in (2, \infty)$, subsets of $L_{p}(\Omega, \mu)$ that have $p$-negative type but not $q$-negative type for any $q > p$. Such sets preclude the existence of certain types of isometry into $L_{p}$-spaces.'
address:
- 'Department of Mathematics and Statistics Canisius College Buffalo, NY 14208 USA'
- 'Department of Decision Sciences University of South Africa PO Box 392, UNISA 0003 SOUTH AFRICA'
author:
- Anthony Weston
title: 'The geometry of two-valued subsets of $L_{p}$-spaces'
---
Introduction {#sec:1}
============
A set $B \subset L_{p}(\Omega, \mu)$ is *two-valued* if $|B| > 1$ and the essential range of each $f \in B$ is a subset of $\{ 0,1 \}$. Our interest in two-valued subsets of $L_{p}(\Omega, \mu)$ was piqued by the following elegant theorem.
\[hyp:cube\] Suppose $k, n \geq 1$. A subset $B = \{ \mathbf{x}_{0}, \mathbf{x}_{1}, \ldots, \mathbf{x}_{k} \}$ of the Hamming cube $\{ 0, 1\}^{n} \subset \ell_{1}^{(n)}$ is affinely independent if and only if $B$ has strict $1$-negative type.
The two-valued subsets of $\ell_{1}^{(n)}$ are precisely those $B \subseteq \{ 0, 1\}^{n}$ that have at least two elements. A notable feature of Theorem \[hyp:cube\] is that the subsets of the Hamming cube that have strict $1$-negative type are characterized solely in terms of the vector space structure of $\mathbb{R}^{n}$. In this paper we generalize Theorem \[hyp:cube\] to the setting of arbitrary two-valued subsets of $L_{p}$-spaces. Our results are valid for all $p \in (0, \infty)$. In order to proceed it is helpful to recall some basic information about classical and strict negative type.
Negative type was originally studied in relation to the problem of isometrically embedding metric and normed spaces into $L_{p}$-spaces, $0 < p \leq 2$. Cayley [@Cay], Menger [@Men], Schoenberg [@Sc3] and Bretagnolle *et al.* [@Bre] authored pivotal papers on this embedding problem. The closely related notion of generalized roundness was developed by Enflo [@En2] to study universal uniform embedding spaces. These notions may be defined in the following manner.
\[neg:gen\] Let $p \geq 0$ and let $(X,d)$ be a metric space with $|X| > 1$.
1. $(X,d)$ has $p$-[*negative type*]{} if and only if for all integers $n \geq 2$, all finite subsets $\{z_{1}, \ldots , z_{n} \} \subseteq X$, and all scalar $n$-tuples $\boldsymbol{\zeta} = (\zeta_{1}, \ldots, \zeta_{n}) \in \mathbb{R}^{n}$ that satisfy $\zeta_{1} + \cdots + \zeta_{n} = 0$, we have $$\begin{aligned}
\label{p:neg}
\sum\limits_{j,i =1}^{n} d(z_{j},z_{i})^{p} \zeta_{j} \zeta_{i} & \leq & 0.\end{aligned}$$
2. $(X,d)$ has *strict* $p$-[*negative type*]{} if and only if it has $p$-negative type and the inequalities (\[p:neg\]) are strict except in the trivial case $\boldsymbol{\zeta} = \boldsymbol{0}$.
3. The *generalized roundness* of $(X,d)$, denoted by $\wp_{(X, d)}$ or simply $\wp_{X}$, is the supremum of the set of all $p$ for which $(X,d)$ has $p$-negative type.
Most studies of negative type properties of subsets and subspaces of $L_{p}$-spaces have focussed on the case $0 < p \leq 2$. There are a number of reasons why this restriction on the range of $p$ is observed in the literature. For instance, Schoenberg [@Sc3] noted that if $0 < p \leq 2$ and if $(\Omega, \mu)$ is a measure space such that $L_{p}(\Omega, \mu)$ is at least two-dimensional, then $L_{p}(\Omega, \mu)$ has $p$-negative type but it does not have $q$-negative type for any $q > p$. This ensures that non-empty subsets of $L_{p}(\Omega, \mu)$ have at least $p$-negative type provided $0 < p \leq 2$. The situation is vastly different if $p > 2$. In this case, if $(\Omega, \mu)$ is a measure space such that $L_{p}(\Omega, \mu)$ is at least three-dimensional, then $L_{p}(\Omega, \mu)$ does not have $q$-negative type for any $q > 0$. This implication follows from theorems of Bretagnolle *et al.* [@Bre], Dor [@Dor], Misiewicz [@Mis] and Koldobsky [@Kol]. One effect of the latter result is that it makes it considerably more difficult to determine which normed spaces embed linearly and isometrically into some $L_{p}$-space when $p > 2$ (because $p$-negative type is no longer a necessary or sufficient condition).
In spite of this apparent roadblock when $p > 2$, it nevertheless seems to be an intriguing project to determine subsets of $L_{p}(\Omega, \mu)$ that have $s$-negative type for some $s = s(p) > 0$ that depends upon $p$. This is one avenue of inquiry in this paper: for each $p \in (0, \infty)$ we identify interesting subsets of $L_{p}(\Omega, \mu)$ that have $p$-negative type but not $q$-negative type for any $q > p$. For the reasons we have indicated, this type of result is rare in the case $p > 2$.
For $p \in (0, 2)$, Kelleher *et al*. [@Kel] have shown that if a subset $B$ of $L_{p}(\Omega, \mu)$ is affinely independent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space), then $B$ has strict $p$-negative type. The converse of this statement is true when $p = 2$ but not when $p < 2$ (see Theorem \[thm:inner\] and Remark \[rem:ex\]). However, in the case $p =1$, we see from Theorem \[hyp:cube\] that the converse statement can hold under additional assumptions. Murugan’s proof of Theorem \[hyp:cube\] relies on novel properties of vertex transitive graphs and the special geometry of the Hamming cube. Another way to prove Theorem \[hyp:cube\] is to apply Nickolas and Wolf [@Nic Theorem 3.4 (3)]. In a second avenue of inquiry in this paper, we obtain a significant generalization of Murugan’s theorem. It turns out that the only thing one really needs to know about the Hamming cube is that it is a two-valued set in an $L_{p}$-space, $0 < p < \infty$. In fact, the two avenues of inquiry that we have mentioned converge to one and the same thing: the geometry of two-valued sets in $L_{p}$-spaces.
The results in this paper are obtained by studying the nature of $p$-polygonal equalities in two-valued subsets of $L_{p}$-spaces. Such equalities are completely characterized in terms of balanced simplices in Theorem \[thm:1\]. This is an interesting result because balanced simplices, being defined purely in terms of the underlying vector space (Definition \[def:bal\]), do not depend upon $p$. It follows, at once, that each two-valued set $B \subset L_{p}(\Omega, \mu)$ has $p$-negative type. Furthermore, $B$ is seen to have strict $p$-negative type if and only if $B$ is affinely independent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space). These results are valid for all $p \in (0, \infty)$ and they give noteworthy information about the metric geometry of two-valued sets in $L_{p}$-spaces. For instance, it follows that the system of Walsh functions in $L_{p}[0,1]$ has strict $p$-negative type for all $p \in (0, \infty)$. We also see, for the first time in the literature, a systematic way to construct subsets of $L_{p}(\Omega, \mu)$ that have generalized roundness $p$ for $p > 2$.
Throughout we assume that all measures are non-trivial and positive. We further assume that $0 < p < \infty$ (unless stated otherwise) and that $(\Omega, \mu)$ is a measure space for which $L_{p}(\Omega, \mu)$ is at least two-dimensional. If $p \in (0,1)$, then $L_{p}(\Omega, \mu)$ is endowed with the usual quasi-norm. We also let $\mathcal{M}(\Omega, \mu)$ denote the vector space of all scalar-valued measurable functions on $(\Omega, \mu)$. It is always the case that sums indexed over the empty set are taken to be zero. When we refer to a subset of a metric space as having $p$-negative type we mean, of course, the subset together with the metric that it inherits from the ambient space.
Signed simplices and polygonal equalities in $L_{p}$-spaces {#sec:2}
===========================================================
Signed simplices and $p$-polygonal equalities provide an effective alternative means for studying classical and strict $p$-negative type.
\[S1\] Let $X$ be a set and suppose that $s,t > 0$ are integers. A *signed $(s,t)$-simplex in $X$* is a collection of (not necessarily distinct) points $x_{1}, \ldots, x_{s}, y_{1}, \ldots, y_{t} \in X$ together with a corresponding collection of real numbers $m_{1}, \ldots, m_{s}, n_{1}, \ldots, n_{t}$ that satisfy $m_{1} + \cdots + m_{s}
= n_{1} + \cdots + n_{t}$. Such a configuration of points and real numbers will be denoted by $D = [x_{j}(m_{j});y_{i}(n_{i})]_{s,t}$ and will hereafter simply be called a *simplex*.
A simplex $D = [x_{j}(m_{j});y_{i}(n_{i})]_{s,t}$ in the real line will be called a *two-valued simplex* if $x_{j}, y_{i} \in \{ 0,1 \}$ for all $j, i$. A simplex $D = [x_{j}(m_{j});y_{i}(n_{i})]_{s,t}$ in $L_{p}(\Omega, \mu)$ will be called a *two-valued simplex* if $\{ x_{j}, y_{i} \}$ is a two-valued set in $L_{p}(\Omega, \mu)$.
\[S3\] Given a signed $(s,t)$-simplex $D = [x_{j}(m_{j});y_{i}(n_{i})]_{s,t}$ in a set $X$ we denote by $S(D)$ the set of distinct points in $X$ that appear in $D$. For each $z \in S(D)$ we define the *repeating numbers* ${\mathbf{m}}(z)$ and ${\mathbf{n}}(z)$ as follows: $${\mathbf{m}}(z) = \sum\limits_{j : z = x_{j}} m_{j} \mbox{ and }
{\mathbf{n}}(z) = \sum\limits_{i : z = y_{i}} n_{i}.$$ We say that the simplex $D$ is *degenerate* if ${\mathbf{m}}(z) = {\mathbf{n}}(z)$ for all $z \in S(D)$.
Informally, a simplex $D$ is degenerate if each $z \in S(D)$ is equally represented in both halves of the simplex. Degenerate simplices in a metric space $(X, d)$ equate in a precise way to the trivial case $\boldsymbol{\zeta} =
\boldsymbol{0}$ in Definition \[neg:gen\].
\[S4\] Let $p \geq 0$ and let $(X,d)$ be a metric space. For each signed $(s,t)$-simplex $D = [x_{j}(m_{j});y_{i}(n_{i})]_{s,t}$ in $X$ we define $$\begin{aligned}
\gamma_{p}(D) & = &
\sum\limits_{j,i = 1}^{s,t} m_{j}n_{i}d(x_{j},y_{i})^{p} -
\sum\limits_{ 1 \leq j_{1} < j_{2} \leq s} m_{j_{1}}m_{j_{2}}d(x_{j_{1}},x_{j_{2}})^{p} \\
& ~ & - \sum\limits_{ 1 \leq i_{1} < i_{2} \leq t} n_{i_{1}}n_{i_{2}}d(y_{i_{1}},y_{i_{2}})^{p}.\end{aligned}$$ We call $\gamma_{p}(D)$ the *$p$-simplex gap of $D$ in $(X,d)$*.
The fundamental relationships between negative type and simplex gaps are expressed in the following theorem.
\[2.3\] Let $p > 0$ and let $(X,d)$ be a metric space. Then the following conditions are equivalent:
1. $(X,d)$ has $p$-negative type.
2. $\gamma_{p}(D) \geq 0$ for each signed simplex $D$ in $X$.
Moreover, $(X,d)$ has strict $p$-negative type if and only if $\gamma_{p}(D) > 0$ for each non-degenerate signed simplex $D$ in $(X,d)$.
Theorem \[2.3\] is a variation of themes developed in Weston [@We1], Lennard *et al*.[@Ltw], and Doust and Weston [@Dou]. The motivation for all such studies has stemmed from Enflo’s original definition of generalized roundness that was given in [@En2]. Theorem \[2.3\] motivates the notion of a (non-trivial) $p$-polygonal equality. The nature of such equalities in two-valued subsets of $L_{p}(\Omega, \mu)$ will be central to the development of our main results in Section \[sec:3\].
\[S6\] Let $p \geq 0$ and let $(X,d)$ be a metric space. A *$p$-polygonal equality* in $(X, d)$ is an equality of the form $\gamma_{p}(D) = 0$ where $D$ is a simplex in $X$. If, moreover, the simplex $D$ is non-degenerate, we will say that the $p$-polygonal equality $\gamma_{p}(D) = 0$ is *non-trivial*.
For $p \in (0, 2)$, the non-empty subsets of $L_{p}(\Omega, \mu)$ that have strict $p$-negative type may be characterized in terms of so-called virtually degenerate simplices.
\[v:degenerate\] A non-degenerate simplex $D = [x_{j}(m_{j});y_{i}(n_{i})]_{s,t}$ in $L_{p}(\Omega, \mu)$ is said to be *virtually degenerate* if the family of evaluation simplices $D(\omega) = [x_{j}(\omega)(m_{j});y_{i}(\omega)(n_{i})]_{s,t}$, $\omega \in \Omega$, are degenerate in the scalar field of $L_{p}(\Omega, \mu)$ $\mu$-almost everywhere.
It is not known to what extent the following theorem may be generalized to the case $p \in (2, \infty)$.
\[Lp\] Suppose $p \in (0,2)$. A non-empty subset of $L_{p}(\Omega, \mu)$ has strict $p$-negative type if and only if it does not admit any virtually degenerate simplices.
A difficulty with Theorem \[Lp\] is that the description of the subsets of $L_{p}(\Omega, \mu)$ that have strict $p$-negative type is not purely geometric. In practice, it is a hard problem to decide if a set $B \subset L_{p}(\Omega, \mu)$ admits a virtually degenerate simplex. However, virtually degenerate simplices satisfy the following definition.
\[def:bal\] Let $D = [x_{j}(m_{j});y_{i}(n_{i})]_{s,t}$ be a simplex in a vector space $X$. We say that $D$ is *balanced* if $\sum m_{j}x_{j} = \sum n_{i}y_{i}$.
Informally, a simplex $D$ in a vector space $X$ is balanced if the two halves of the simplex have the same center of gravity. In a real or complex vector space there is a direct link between non-degenerate balanced simplices and affinely dependent subsets. This theorem underpins the main considerations of this paper.
\[thm:B\] Let $n \geq 1$ be an integer and let $X$ be a real or complex vector space. Then a subset $Z = \{ z_{0}, z_{1}, \ldots z_{n} \}$ of $X$ admits a non-degenerate balanced simplex if and only if the set $Z$ is affinely dependent (when $X$ is considered as a real vector space).
Theorem \[thm:B\] leads to a complete description of the subsets of inner product spaces that have strict $2$-negative type.
\[thm:inner\] A non-empty subset $Z$ of a real or complex inner product space $X$ has strict $2$-negative type if and only if $Z$ is an affinely independent subset of $X$ (when $X$ is considered as a real vector space).
A notable feature of Theorem \[thm:inner\] is that the vector space structure of an inner product space $X$ completely determines the subsets of $X$ that have strict $2$-negative type. In the next section we will encounter a similar phenomenon for two-valued subsets of $L_{p}$-spaces, $0 < p < \infty$.
The geometry of two-valued subsets of $L_{p}(\Omega, \mu)$ {#sec:3}
==========================================================
We begin this section by classifying the (non-trivial) $p$-polygonal equalities in two-valued subsets of $L_{p}$-spaces, $0 < p < \infty$.
\[lemma:1\] Let $p \in (0, \infty)$ be given. If $D = [x_{j}(m_{j});y_{i}(n_{i})]_{s,t}$ is a two-valued simplex in the real line, then $$\gamma_{p}(D) = \gamma_{1}(D) = \biggl| \sum\limits_{j} m_{j}x_{j} - \sum\limits_{i} n_{i}y_{i} \biggl|^{2}.$$
Let $p > 0$ be given. Suppose that $D = [x_{j}(m_{j});y_{i}(n_{i})]_{s,t}$ is a two-valued simplex in the real line. By definition, we have $x_{j}, y_{i} \in \{ 0, 1 \}$ for all $j$ and $i$. Thus all distances appearing in the definition of $\gamma_{p}(D)$ are $0$ or $1$. As a result, we see that $\gamma_{p}(D) =
\gamma_{1}(D)$. This establishes the first equality of the lemma.
To establish the second equality, we create a new simplex by “compressing” the original simplex $D$. We do this by setting $x_{1}^{\ast} = y_{1}^{\ast} = 0$ and $x_{2}^{\ast} = y_{2}^{\ast} = 1$. Weights for these vertices are then given by the corresponding repeating numbers from the original simplex $D$: $m_{1}^{\ast} = {\mathbf{m}}(0), m_{2}^{\ast} = {\mathbf{m}}(1), n_{1}^{\ast} = {\mathbf{n}}(0)$, and $n_{2}^{\ast} = {\mathbf{n}}(1)$.
Notice that $m_{1}^{\ast} + m_{2}^{\ast} = m_{1} + \cdots m_{s} = n_{1} + \cdots n_{t} = n_{1}^{\ast} + n_{2}^{\ast}$. The compressed simplex is then simply defined to be $D^{\ast} = [x_{j}^{\ast}(m_{j}^{\ast});y_{i}^{\ast}(n_{i}^{\ast})]_{2,2}$. It easy to verify that $\gamma_{p}(D) = \gamma_{1}(D) = \gamma_{1}(D^{\ast})$.
By definition of $D$ and $D^{\ast}$, $$\begin{aligned}
\label{Leq:1}
\biggl| \sum\limits_{j} m_{j}x_{j} - \sum\limits_{i} n_{i}y_{i} \biggl|^{2} & = &
\biggl|\sum\limits_{j} m_{j}^{\ast}x_{j}^{\ast} - \sum\limits_{i} n_{i}^{\ast}y_{i}^{\ast} \biggl|^{2} \nonumber \\
& = & (m_{2}^{\ast} - n_{2}^{\ast})^{2}.\end{aligned}$$
On the other hand, $$\begin{aligned}
\label{Leq:2}
\gamma_{p}(D) & = & \gamma_{1}(D^{\ast}) \nonumber \\
& = & m_{1}^{\ast}n_{2}^{\ast} + m_{2}^{\ast}n_{1}^{\ast} - m_{1}^{\ast}m_{2}^{\ast} - n_{1}^{\ast}n_{2}^{\ast} \nonumber \\
& = & (n_{1}^{\ast} - m_{1}^{\ast})(m_{2}^{\ast} - n_{2}^{\ast}) \nonumber \\
& = & (m_{2}^{\ast} - n_{2}^{\ast})^{2}.\end{aligned}$$ The lemma now follows from (\[Leq:1\]) and (\[Leq:2\]).
\[thm:1\] Let $p \in (0, \infty)$ be given. If $D = [x_{j}(m_{j});y_{i}(n_{i})]_{s,t}$ is a two-valued simplex in $L_{p}(\Omega, \mu)$, then $$\begin{aligned}
\label{Teq:1}
\gamma_{p}(D) & = & \biggl\| \sum\limits_{j} m_{j}x_{j} - \sum\limits_{i} n_{i}y_{i} \biggl\|_{2}^{2}.\end{aligned}$$ In particular, $\gamma_{p}(D) = 0$ if and only if the simplex $D$ is balanced.
Let $p > 0$ be given. Suppose that $D = [x_{j}(m_{j});y_{i}(n_{i})]_{s,t}$ is a two-valued simplex in $L_{p}(\Omega, \mu)$. Except on a set of $\mu$-measure zero, the evaluation simplices $D(\omega) = [x_{j}(\omega)(m_{j});y_{i}(\omega)(n_{i})]_{s,t}$, $\omega \in \Omega$, are two-valued in the real line. So, by Lemma \[lemma:1\], we have $$\begin{aligned}
\gamma_{p}(D(\omega)) & = &
\biggl| \sum\limits_{j} m_{j}x_{j}(\omega) - \sum\limits_{i} n_{i}y_{i}(\omega) \biggl|^{2},\end{aligned}$$ $\mu$-almost everywhere. Integrating over $\Omega$ with respect to $\mu$ gives the desired conclusion.
It is an immediate consequence of Theorem \[thm:1\] that two-valued subsets of $L_{p}(\Omega, \mu)$ have $p$-negative type, $0 < p < \infty$. In fact, the following considerably stronger statement applies.
\[cor:1\] Let $B \subset L_{p}(\Omega, \mu)$ be a two-valued set. Then:
1. $B$ has $p$-negative type.
2. $B$ has strict $p$-negative type if and only if $B$ is affinely independent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space).
3. If $B$ is affinely dependent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space), then $B$ has generalized roundness $p$.
For any simplex $D = [x_{j}(m_{j});y_{i}(n_{i})]_{s,t}$ in $B$ we have $\gamma_{p}(D) \geq 0$ by (\[Teq:1\]). Thus $B$ has $p$-negative type by part (1) of Theorem \[2.3\].
To prove (2) we argue contrapositively. Suppose that $B$ does not have strict $p$-negative type. Then, by part (2) of Theorem \[2.3\], there must be a non-degenerate simplex $D$ in $B$ such that $\gamma_{p}(D) = 0$. Hence $D$ is balanced by the second statement of Theorem \[thm:1\]. Since $D$ is non-degenerate and balanced it follows from Theorem \[thm:B\] that $S(D)$ is affinely dependent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space). As $S(D) \subseteq B$, this shows that $B$ is affinely dependent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space).
Now suppose that $B$ is affinely dependent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space). Then $B$ admits a non-degenerate balanced simplex $D$ by Theorem \[thm:B\]. Thus $\gamma_{p}(D) = 0$ by the second statement of Theorem \[thm:1\]. As $D$ is also non-degenerate we conclude from part (2) of Theorem \[2.3\] that $S(D)$, and hence $B$, does not have strict $p$-negative type. Moreover, since $B$ does not have strict $p$-negative type, it cannot have $q$-negative type for any $q > p$ by Li and Weston [@Hli Theorem 5.4]. However, $B$ has $p$-negative type by (1). Thus $\wp_{B} = p$. This completes the proof of (2) and (3).
\[rem:ex\] For $p \not= 2$ the forward implication of Corollary \[cor:1\] (2) does not necessarily hold for subsets of $L_{p}(\Omega, \mu)$ that take three or more values. For example, consider the points $z_{0} = (0,0), z_{1} = (1,1), z_{2} = (2,1)$ and $z_{3} = (2,0) \in \ell_{p}^{(2)}$ with $0 < p < 2$. The set $Z$ takes the values $\{ 0, 1, 2 \}$ and is affinely dependent. Moreover, it is easy to see that $Z$ does not admit any virtually degenerate simplices. Thus $Z$ has strict $p$-negative type by Theorem \[Lp\]. An even simpler example can be constructed in the case $p > 2$ by considering any three-valued set in $\ell_{p}^{(2)}$ that contains distinct points $x, y$ and $z$ such that $z = (x + y)/2$. The set $\{ x,y,z \} \subset \ell_{p}^{(2)}$ does not have $q$-negative type for any $q > 2$ because it contains a metric midpoint.
If $0 < p \leq 2$, Corollary \[cor:1\] (1) holds for any subset of $L_{p}(\Omega, \mu)$. However, if $2 < p < \infty$ and $L_{p}(\Omega, \mu)$ is at least three-dimensional, then $L_{p}(\Omega, \mu)$ does not have $q$-negative type for any $q > 0$. In this instance, Corollary \[cor:1\] (1) is quite striking since it identifies subsets of $L_{p}(\Omega, \mu)$ that have, at least, $p$-negative type.
The next corollary is an immediate consequence of Corollary \[cor:1\] (2) because (Schauder) bases are linearly independent.
\[cor:1.5\] Every two-valued (Schauder) basis of $L_{p}(\Omega, \mu)$ has strict $p$-negative type.
It is also the case that Corollary \[cor:1\] (2) prohibits the existence of certain types of isometry.
\[cor:2\] Let $B$ be any two-valued subset of $L_{p}(\Omega, \mu)$ that is affinely dependent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space). Then:
1. No metric space that has strict $p$-negative type is isometric to $B$. In particular, no ultrametric space is isometric to $B$.
2. If $0 < p < q \leq 2$, no subset of any $L_{q}$-space is isometric to $B$.
$B$ does not have strict $p$-negative type by Corollary \[cor:1\] (2). So no metric space that has strict $p$-negative type can be isometric to $B$. The second statement of part (1) is then immediate because ultrametric spaces have strict $q$-negative type for all $q > 0$ by Faver *et al*. [@Fav Corollary 5.3].
Now suppose that $0 < p < 2$. If $A$ is a subset of an $L_{q}$-space such that $|A| > 1$ and $q \in (p, 2]$, then $A$ has $q$-negative type. Thus $A$ has strict $p$-negative type by Li and Weston [@Hli Theorem 5.4]. Hence $A$ is not isometric to $B$ by part (1).
It is possible, and worthwhile, to recast Corollary \[cor:1\] (2) in terms of $\mathcal{M}(\Omega, \mu)$. Recall that $\mathcal{M}(\Omega, \mu)$ denotes the vector space of all scalar-valued measurable functions on $(\Omega, \mu)$. We will say that a function $f \in \mathcal{M}(\Omega, \mu)$ is *finitely supported* if $\mu (\operatorname{supp}(f)) < \infty$, and that a set $B \subset \mathcal{M}(\Omega, \mu)$ is *finitely supported* if each $f \in B$ is finitely supported.
At the beginning of Section \[sec:1\] we stated that a set $B \subset L_{p}(\Omega, \mu)$ is *two-valued* if $|B| > 1$ and the essential range of each $f \in B$ is a subset of $\{ 0,1 \}$. More generally, we will say that a set $B \subset \mathcal{M}(\Omega, \mu)$ is *two-valued* if $|B| > 1$ and the essential range of each $f \in B$ is a subset of $\{ 0,1 \}$. Clearly, the essential range of a function $f \in \mathcal{M}(\Omega, \mu)$ is a subset of $\{ 0,1 \}$ if and only if $f = \chi_{\operatorname{supp}(f)}$ $\mu$-almost everywhere. Consequently, for any $p \in (0, \infty)$ and any function $f \in \mathcal{M}(\Omega, \mu)$ whose essential range is a subset of $\{ 0,1 \}$, we see that $f \in L_{p}(\Omega, \mu)$ if and only if $f$ is finitely supported. This observation implies the following useful lemma.
\[lem:2\] Let $B$ be a two-valued set in $\mathcal{M}(\Omega, \mu)$. Then, for any $p \in (0, \infty)$, the following statements are equivalent:
1. $B$ is finitely supported.
2. $B$ is a two-valued set in $L_{p}(\Omega, \mu)$.
A key feature of Corollary \[cor:1\] (2) is the strong interplay that it illustrates between the metric geometry of $L_{p}(\Omega, \mu)$ and the underlying vector space. The property of strict $p$-negative type depends explicitly upon $p$, whereas the property of affine independence depends only on the underlying vector space. However, for any $p \in (0, \infty)$, the underlying vector space is a vector subspace of $\mathcal{M}(\Omega, \mu)$. This is important because $\mathcal{M}(\Omega, \mu)$ is completely independent of $p$. Thus we obtain a generalization of Theorem \[hyp:cube\] that is valid for any $p \in (0, \infty)$.
\[cor:3\] Let $B$ be a finitely supported two-valued set in $\mathcal{M}(\Omega, \mu)$. Then, for any $p \in (0, \infty)$, the following statements are equivalent:
1. $B$ has strict $p$-negative type when viewed as a metric subspace of $L_{p}(\Omega, \mu)$.
2. $B$ is an affinely independent subset of $\mathcal{M}(\Omega, \mu)$ (when $\mathcal{M}(\Omega, \mu)$ is considered as a real vector space).
Immediate from Corollary \[cor:1\] and Lemma \[lem:2\].
Corollary \[cor:3\] has interesting outcomes in the finite-dimensional setting of (real) $\ell_{p}^{(n)}$. In this case, $\mathcal{M}(\Omega, \mu) = \mathbb{R}^{n}$. Letting $d_{p}$ denote the metric on $\mathbb{R}^{n}$ induced by the $p$-norm we deduce the following consequences of Corollary \[cor:3\].
\[cor:4\] Let $B$ be any two-valued set in $\mathbb{R}^{n}$. If $|B| > n + 1$, then the generalized roundness of the metric space $(B, d_{p})$ is $p$ for all $p \in (0, \infty)$.
Let $B = \{ \mathbf{x}_{0}, \mathbf{x}_{1}, \ldots, \mathbf{x}_{k} \}$ be a given two-valued set in $\mathbb{R}^{n}$ such that $k > n$. Let $p \in (0, \infty)$ be given. By Corollary \[cor:1\] (1), $(B, d_{p})$ has $p$-negative type. However, the vectors $\{ \mathbf{x}_{1} - \mathbf{x}_{0}, \mathbf{x}_{2} - \mathbf{x}_{0}, \ldots, \mathbf{x}_{k} - \mathbf{x}_{0} \}$ are linearly dependent because $k > n$. So it follows from Corollary \[cor:3\] that $(B, d_{p})$ does not have strict $p$-negative type. This precludes $(B, d_{p})$ from having $q$-negative type for any $q > p$ by Li and Weston [@Hli Theorem 5.4]. Thus $\wp_{(B, d_{p})} = p$, as asserted.
\[cor:5\] Let $B$ be any two-valued set in $\mathbb{R}^{n}$ and let $(X, d)$ be a $k$-point metric space that has strict $p$-negative type for some $p \in (0, \infty)$. If $(X, d)$ embeds isometrically into $(B, d_{p})$, then $n \geq k - 1$.
Let $B$ be a given two-valued set in $\mathbb{R}^{n}$ and let $(X, d)$ be a $k$-point metric space that has strict $p$-negative type for some $p \in (0, \infty)$. If $(X, d)$ isometrically embeds into $(B, d_{p})$, then there is a (necessarily two-valued) subset $\tilde{X}$ of $B$ such that $(\tilde{X}, d_{p})$ has strict $p$-negative type. Thus $\tilde{X}$ is an affinely independent subset of $\mathbb{R}^{n}$ by Corollary \[cor:3\]. Therefore $k = |\tilde{X}| \leq n + 1$.
We remark that Corollary \[cor:4\] and Corollary \[cor:5\] generalize similar results for the Hamming cube $\{ 0,1 \}^{n} \subset \ell_{1}^{(n)}$ given in Murugan [@Mur].
It is worth noting that the results of this section hold for any $\{ \alpha, \beta \}$-valued subset $B$ of $L_{p}(\Omega, \mu)$, $\alpha \not= \beta$. Indeed, we may assume that $\alpha = 0$ by a translation. Then, given any $p > 0$ and any $\{ 0, \beta\}$-valued simplex $D = [x_{j}(m_{j});y_{i}(n_{i})]_{s,t}$ in $L_{p}(\Omega, \mu)$, it follows that $$\begin{aligned}
\gamma_{p}(D) & = & |\beta|^{p-2} \cdot \biggl\| \sum\limits_{j} m_{j}x_{j} - \sum\limits_{i} n_{i}y_{i} \biggl\|_{2}^{2}\end{aligned}$$ by slightly modifying the statement and proof of Lemma \[lemma:1\]. So, for any $p \in (0, \infty)$, we see that $\gamma_{p}(D) = 0$ if and only if the simplex $D$ is balanced. For example, by Corollary \[cor:1\], it then follows that the classical system $B$ of Walsh functions in $L_{p}[0,1]$ forms a set of strict $p$-negative type.
We conclude this paper by commenting briefly on the case $p = \infty$. If $B$ is a two-valued subset of $\mathcal{M}(\Omega, \mu)$, then it is automatically a subset of $L_{\infty}(\Omega, \mu)$. Moreover, as a metric subspace of $L_{\infty}(\Omega, \mu)$, $B$ is an ultrametric space. Therefore $B$ has strict $p$-negative type for all $p \in (0, \infty)$ by Faver *et al*.[@Fav Corollary 5.3].
Acknowledgments {#acknowledgments .unnumbered}
===============
I am indebted to the referees for their thoughtful comments on the preliminary version of this paper. The outcome is a far more general and appealing paper. I would also like to thank the CCRDS research group in the Department of Decision Sciences at the University of South Africa for their kind support and insightful input during the preparation of this paper.
[Ar1]{}
J. Bretagnolle, D. Dacunha-Castelle and J. L. Krivine, *Lois stables et espaces $L^{p}$*, Ann. Inst. H. Poincaré Sect. B (N.S.) **2** (1966), 231–259.
A. Cayley, *On a theorem in the geometry of position*, Cambridge Mathematical Journal **II** (1841), 267–271.
L. E. Dor, *Potentials and isometric embeddings in $L_{1}$*, Israel J. Math. **24** (1976), 260–268.
I. Doust and A. Weston, *Enhanced negative type for finite metric trees*, J. Funct. Anal. **254** (2008), 2336–2364.
P. Enflo, *On a problem of Smirnov*, Ark. Mat. **8** (1969), 107–109.
T. Faver, K. Kochalski, M. Murugan, H. Verheggen, E. Wesson and A. Weston, *Roundness properties of ultrametric spaces*, Glasgow Math. J. **56** (2014), 519–535.
C. Kelleher, D. Miller, T. Osborn and A. Weston, *Polygonal equalities and virtual degeneracy in $L_{p}$-spaces*, J. Math. Anal. Appl. [**415**]{} (2014), 247–268.
A. Koldobsky, *Generalised Lévy representation of norms and isometric embeddings into $L_{p}$-spaces*, Ann. Inst. H. Poincaré, [**28**]{} (1992), pp. 335-353.
C. J. Lennard, A. M. Tonge and A. Weston, *Generalized roundness and negative type*, Michigan Math. J. **44** (1997), 37–45.
H. Li and A. Weston, *Strict p-negative type of a metric space*, Positivity **14** (2010), 529–545.
K. Menger, *Die Metrik des Hilbert-Raumes*, Akad. Wiss. Wien Abh. Math.-Natur. K1 **65** (1928), 159–160.
J. Misiewicz, *Positive definite functions on $\ell_{\infty}^{(n)}$*, Statist. and Prob. Letters [**8**]{} (1989), pp. 255-260.
M. Murugan, *Supremal $p$-negative type of vertex transitive graphs*, J. Math. Anal. Appl. **391** (2012), 376–381.
P. Nickolas and R. Wolf, *Distance geometry in quasihypermetric spaces. III*, Math. Nachr. **284** (2011), 747–760.
I. J. Schoenberg, *Metric spaces and positive definite functions*, Trans. Amer. Math. Soc. **44** (1938), 522–536.
A. Weston, *On the generalized roundness of finite metric spaces*, J. Math. Anal. Appl. **192** (1995), 323–334.
|
---
author:
- Mark Blumstein
title: Length and Multiplicities in Graded Commutative Algebra
---
Introduction
============
This paper is a synthesis of the main ideas from the author’s matster’s thesis. The author would like to thank Jeanne Duflot for her steady guidance and dedication as advisor. This paper came to be via the study of the commutative algebra of equivariant cohomology rings $H^*_G(X)$ associated to a group $G$ acting on a topological space $X$, which are of course naturally graded. This study was really begun about 50 years ago by Quillen [@Quillen1], [@Quillen2] who described the (Krull) dimension of these graded rings and gave a decomposition of their spectra (in the sense of algebraic geometry). Many authors have followed with their own studies of these rings from the point of view of commutative algebra; recent contributions include the work of Symonds [@Symonds], Lynn [@Lynn] and Duflot [@Duflot3].
We were attempting to generalize the work of Lynn [@Lynn], resulting in the paper [@BD], and found that we needed a careful exploration of various notions of multiplicity for graded rings which were nonstandard in two ways: not positively graded (for example, they might be graded localizations) and/or not generated by elements of degree 1. Since, as algebraic topologists, the degree of homogeneous elements in these rings can have geometric or representation-theoretic meaning to us, we were not comfortable with using the geometer’s trick of the Veronese embedding to get around the second problem.
In this paper, there is nary a word about equivariant cohomology or algebraic topology. It is all about graded commutative algebra and much of it is expository. The main results of interest to us are the theorems about length, multiplicity and a second notion of “degree" (which is really another sort of multiplicity) for the Poincarè series of a graded ring. When we were able to write results for rings from a larger collection than simply those of cohomology type, we tried to do so. We hope that workers in fields other than algebraic topology find this exposition useful.
A Review of Standard Definitions and Facts in the Graded Category {#Introductory Results in the Graded Context}
=================================================================
We consider only strictly commutative $\mathbb{Z}$-graded rings and modules in this paper and use the standard notation: if $A$ is a graded ring, $M$ is a graded $A$-module and $n \in \mathbb{Z}$, $M_n$ is the set of homogeneous elements of degree $n$ (although “degree" will also have another meaning here); for every $x \in M$, $x$ may be written uniquely as $x= \Sigma_{n \in \mathbb{Z}}x_n$, where $x_n \in M_n$ and $x_j = 0$ for all but finitely many $j$, the element $x_n$ is the homogeneous component of $x$ of degree $n$. An element $x \in M$ is a homogeneous element if and only if $x$ has at most one nonzero homogeneous component. If $d \in \mathbb{Z}$, we use the following convention for the suspended $A$-module $M(d)$: $$M(d)_j \doteq M_{d+j},$$ for every $j \in \mathbb{Z}$; also, if $M$ and $N$ are graded $A$-modules, and $\psi: M \to N$ is an $A$-module homomorphism, then $\psi$ is a *graded homomorphism of degree $d$* if for every integer $n$, $\psi(M_n) \subseteq N_{n+d}$.
Suppose $A$ is a graded ring. The category ${\mathfrak{grmod}}(A)$ has objects finitely generated graded $A$-modules. The morphisms of ${\mathfrak{grmod}}(A)$ are the $A$-module homomorphisms which are graded of degree zero (i.e. degree-preserving).
Recall that a submodule $N$ of a graded $A$-module $M$ is a graded submodule if and only if it is generated over $A$ by homogeneous elements; this is equivalent to the condition that for every element of $N$, all of its homogeneous components are in $N$.
Whether or not $M$ and $A$ are graded, the set of associated primes for $M$ in $A$ is denoted by $Ass_A(M)$ and the support of an $A$-module $M$ is the set $Supp_A(M) \doteq \{ {{\mathfrak{p}}}\in Spec(A) : M_{{\mathfrak{p}}}\neq 0 \}$. For $M$ finitely generated over $A$, ${{\mathfrak{p}}}\in Supp_A(M)$ if and only if $Ann_A(M) \subseteq {{\mathfrak{p}}}$. A prime ideal of $A$ that contains $Ann_A(M)$, and is minimal amongst all primes containing $Ann_A(M)$ is called a *minimal prime* for $M$. If $M= A/{{\mathcal{I}}}$ for an ideal ${{\mathcal{I}}}$ of $A$, then a minimal prime for $M$ is called a minimal prime in $A$ over ${{\mathcal{I}}}$. Note that $Ass_A(M) \subseteq Supp_A(M)$.
The graded support of $M$, $*Supp_A(M)$, is the set of all graded prime ideals in the support of $M$. If ${{\mathcal{I}}}$ is a graded ideal in $A$, the graded variety of ${{\mathcal{I}}}$, $*V({{\mathcal{I}}})$, is the set of all graded primes in $A$ containing ${{\mathcal{I}}}$. (Recall that if $\mathcal{J}$ is any ideal in $A$, graded or not, $V(\mathcal{J})$ is the set of prime ideals in $A$ containing $\mathcal{J}$.)
We collect some standard results about $Ann_A(M)$ and $Ass_A(M)$ for the graded category below.
\[lemma Ass is graded\] Let $A$ be a graded ring with $M$ a graded $A$-module.
- $Ann_A(M)$ is a graded ideal in $A$ and $Ann_A(M) = Ann_A(M(d))$ for every $d \in \mathbb{Z}$.
- If ${{\mathfrak{p}}}\in Ass_A(M)$, then ${{\mathfrak{p}}}$ is a graded ideal of $A$ and is the annihilator of a homogeneous element in $A$.
- Therefore, if ${{\mathcal{I}}}$ is a graded ideal in $A$, all primes in $Ass_A(A/{{\mathcal{I}}})$ are graded.
- If ${{\mathfrak{p}}}$ is a minimal prime for $M$, then ${{\mathfrak{p}}}\in Ass_A(M)$; thus, all minimal primes for $M$ are graded.
\[lemma minimal prime\]
Finally, for an ideal ${{\mathcal{I}}}$ in a graded ring, graded or not, ${{\mathcal{I}}}^*$ is defined as the largest, graded ideal contained in ${{\mathcal{I}}}$; i.e. ${{\mathcal{I}}}^*$ is the ideal generated by all homogeneous elements of ${{\mathcal{I}}}$; if ${{\mathfrak{p}}}$ is a prime ideal in $A$, ${{\mathfrak{p}}}^*$ is also a prime ideal in $A$.
Noetherian graded rings
-----------------------
When we say that a graded $A$-module $M$ is a Noetherian $A$-module, we mean that it is Noetherian in the usual sense, forgetting the grading.
One can show [@brhe] that the following conditions on $A$ are equivalent:
- $A$ is Noetherian.
- Every graded ideal in $A$ is generated by a finite set of homogeneous elements.
- $A_0$ is Noetherian and $A$ is a finitely generated $A_0$-algebra by a set of homogeneous elements.
So, if $M$ is a finitely generated graded $A$-module, and $A$ is Noetherian, then $M$ is Noetherian, and
- every $A$-submodule $N$ of $M$ is finitely generated over $A$, and if $N$ is graded, it is generated over $A$ by a finite set of homogeneous elements;
- for every $j$, $M_j$ is a Noetherian $A_0$-module and so every $A_0$-submodule of $M_j$ is finitely generated: if one has an ascending chain $X_1 \subseteq X_2 \subseteq \cdots $ of $A_0$-submodules of $M_j$, then letting $AX_i$ be the (graded) $A$-submodule generated by $X_i$, we must have $AX_i = AX_{i+1}$ for all $i$ greater than or equal to some fixed $N$. By degree considerations, $X_i = AX_i \cap M_j$ for every $i$, so $X_i = X_{i+1}$ for $i \geq N$.
If $A$ is a graded Noetherian ring and $M \in {\mathfrak{grmod}}(A)$,
- $ Supp_A(M) = V(Ann_A(M)) \doteq V(M)$, so that $*Supp_A(M) = *V(Ann_A(M)) \doteq *V(M).$
- If ${{\mathcal{I}}}$ is a graded ideal in $A$, $*V(M/{{\mathcal{I}}}M) = *V(M) \cap *V({{\mathcal{I}}}) = *V(Ann_A(M) + {{\mathcal{I}}})$.
- If ${{\mathcal{I}}}$ and $\tilde{{{\mathcal{I}}}}$ are two graded ideals in $A$ then their radicals are also graded, and $*V({{\mathcal{I}}}) = *V(\tilde{{{\mathcal{I}}}})$ if and only if $\sqrt{{{\mathcal{I}}}} = \sqrt{\tilde{{{\mathcal{I}}}}}$.
The proof of $a.$ can be found in [@se]; also [@se] tells us that $V(M/{{\mathcal{I}}}M) = V(M) \cap V({{\mathcal{I}}})= V(Ann_A(M) + {{\mathcal{I}}})$ and so $b.$ follows from this. For c., the forward implication follows since all minimal primes over ${{\mathcal{I}}}$ are graded, thus occur as minimal elements both in $V({{\mathcal{I}}})$ and $*V({{\mathcal{I}}})$, and $\sqrt{{{\mathcal{I}}}}$ is the intersection of the (finite number of) minimal primes over ${{\mathcal{I}}}$.
The type of filtration described in the following lemma will be used several times in this paper; and we provide a brief discussion of its proof.
\[corollary the filtration graded\] If $A$ is a Noetherian graded ring and $M \in {\mathfrak{grmod}}(A)$ is nonzero, there exists a finite filtration $M^{\bullet}$ of $M$ by graded submodules ($M^0 = 0; M^{n} = M$), integers $d_j$ and graded primes ${{\mathfrak{p}}}_j \in Spec(A)$ with graded isomorphisms of graded $A$-modules, $M^{i+1}/M^{i} \cong A / {{\mathfrak{p}}}_{i+1} (-d_{i+1})$. Furthermore, given a finite list of graded primes $({{\mathfrak{p}}}_j \mid 1 \leq j \leq n )$ in $Spec(A)$ (not necessarily distinct), and a graded filtration $M^{\bullet}$ of $M$ by graded submodules as above, we must have $$Ass_A(M) \subseteq \{{{\mathfrak{p}}}_j \mid 1 \leq j \leq n \} \subseteq *Supp_A(M)$$ and these three sets must have the same minimal elements, the set of which consists of the minimal primes of $M$. Finally, if ${{\mathfrak{p}}}$ is a minimal prime for $M$, forgetting all gradings and using the fact that the ordinary localization $M_{{{\mathfrak{p}}}}$ is a finitely generated Artinian $A_{{{\mathfrak{p}}}}$-module, the number of times that $A/{{\mathfrak{p}}}$, possibly shifted, occurs as a graded $A$-module isomorphic to a subquotient of $M^{\bullet}$ is always equal to the length of $M_{{{\mathfrak{p}}}}$ as an $A_{{{\mathfrak{p}}}}$-module and is thus independent of the choice of the graded filtration $M^{\bullet}$.
We remind the reader of the proof of the first statement, adapted to the graded case: Using the Noetherian hypothesis, since $M \neq 0$, $Ass_A(M) \neq \emptyset$, so we may pick an element ${{\mathfrak{p}}}_1 \in Ass_A(M)$. Then ${{\mathfrak{p}}}_1$ is graded and there exists a homogeneous element $m_ 1\in M$ such that ${{\mathfrak{p}}}_1 = ann_A(m_1)$. Suppose $\deg(m_1) = d_1$, then $A/{{\mathfrak{p}}}_1(-d_1)$ is graded isomorphic to a graded $A$-submodule of $M$ which we call $M^1$.
If $M^1 = M$, we are done. If not, we take the $A$-module $M/M^1$, notice that it is nonzero, and produce an associated prime ${{\mathfrak{p}}}_2 \in Ass_A(M/M^1)$. Since $M/M^1$ is a graded $A$-module ${{\mathfrak{p}}}_2$ is also graded. Suppose ${{\mathfrak{p}}}_2 = Ann_A(\overline{m}_2)$ where $m_2 \notin M^1$ is a homogeneous element in $M$ and $\deg(m_2) = d_2$; $\overline{m}_2$ is the coset of $m_2$ in $M/M^1$. Thus there is a graded submodule $M^1 \subseteq M^2$ such that $M^2/M^1$ is graded isomorphic to $A/{{\mathfrak{p}}}_2(-d_2)$. At some point there must be a smallest $n\geq 1$ with $M^n = M$, since otherwise the Noetherian hypothesis would be violated.
For the last two statements, we refer to [@se].
Graded Length
=============
We’ve already started using the notation “$^*P$" for a modification of a property or definition $P$ in the ungraded category to obtain a property or definition in the graded category, and we continue it in this section. From now on, unless stated otherwise, all modules and rings are graded, although we will sometimes redundantly restate this.
If $A$ is a graded ring, a graded ideal $\mathcal{N}$ is \*maximal if and only if $\mathcal{N} \neq A$ and $\mathcal{N}$ is a maximal element in the set of all proper graded ideals of $A$.
A \*simple $A$-module is a nonzero graded $A$-module with no nonzero proper graded submodules. A \*composition series for a graded module $M \in {\mathfrak{grmod}}(A)$ is a chain of graded $A$-submodules of $M$, $0=M^0 \subset \cdots \subset M^n = M$ such that each successive quotient $M^i/ M^{i-1}$ is isomorphic as a graded $A$-module to a \*simple module. The \*length of the \*composition series $0=M^0 \subset \cdots \subset M^n = M$ is defined to be $n$.
The fundamental theorem about \*composition series mirrors that in the ungraded case. The proof of the following is nearly identical to the ungraded case ([@Eis],Theorem 2.13), with only minor adjustments made to account for the grading, and we leave this effort to the reader.
Suppose for $M \in {\mathfrak{grmod}}(A)$ that a \*composition series of length $n$ for $M$ exists. Then, every chain of graded submodules of $M$ has length $\leq n$, and can be refined to a \*composition series of length $n$. Every \*composition series for $M$ has length $n$.
If $M$ has a \*composition series as an $A$-module, the \*length of $M \in {\mathfrak{grmod}}(A)$ is defined to be the length of a \*composition series for $M$. We use the notation $*\ell_A(M)$ for this number; as usual, we say $*\ell_A(M) = \infty$ if $M$ does not have a \*composition series.
If we forget all gradings on $A$ and $M$, $\ell_A(M)$ denotes the usual length of $M$ as an $A$-module.
Some properties of $*\ell_A$ are as expected:
- If $0 \rightarrow M \rightarrow N \rightarrow P \rightarrow 0$ is an exact sequence in ${\mathfrak{grmod}}(A)$, then $N$ has a \*composition series if and only if both $M$ and $P$ do; and in this case, $*\ell_A(N) = *\ell_A(M) + *\ell_A(P)$.
- If $d \in \mathbb{Z}$, then $*\ell_A(M(d)) = *\ell_A(M)$.
The only simple $A$-modules in the ungraded case are $A$-modules of the form $A/{{\mathfrak{m}}}$, where ${{\mathfrak{m}}}$ is a maximal ideal of $A$ (recall all rings are commutative). Thus we are led to define graded fields; these are the rings of \*length zero as modules over themselves.
[@Eis] \[theorem gr field\] Let $F$ be a graded ring. The following are equivalent:
1. Every nonzero homogeneous element in $F$ is invertible.
2. $F_0$ is a field and either $F = F_0$, or there exists a $d>0$ and an $x \in F_d$ such that $F \cong F_0[x,x^{-1}]$ as a graded ring. In fact, in this last case, $d>0$ is the smallest positive degree with $F_d \neq 0$.
3. The only graded ideals in $F$ are $F$ and $0$.
A ring satisfying any of these three equivalent conditions is called a **graded field**.
\[example graded field\] If $F$ is a graded field with a nonzero positive degree element, then $F$ is \*simple as a module over itself, but it is not simple as such. To see this write $F = F_0[t,t^{-1}]$, with $\deg(t) = d >0$, and $F_0$ a field. So $F$ is certainly \*simple, but if $\mathcal{J}$ is the ungraded ideal generated by $t+1$, $\mathcal{J}$ is a nonzero proper $F$-submodule of $F$, so $F$ is not simple. Furthermore, $F$ has a unique \*maximal ideal, the zero ideal, but has as least as many ungraded nonzero maximal ideals as the nonzero elements of $F$. While $F$ has a \*composition series, it has no composition series.
Similarly to the ungraded case, $M$ is a \*simple $A$-module if and only if there exists a \*maximal ideal $\mathcal{N}$ of $A$, an integer $d$ and a graded $A$-module isomorphism $M \cong (A/\mathcal{N})(d)$: if $M$ is \*simple, let $x$ be any nonzero homogeneous element of $M$, say $\deg(x) = -d$. Then, the submodule of $M$ generated by $x$ is nonzero and graded, so must be all of $M$. The homomorphism $A(d) \rightarrow M$ of graded $A$-modules defined by $a \mapsto ax$ is thus surjective; its kernel is a graded ideal in $A(d)$ of the form $\mathcal{N}(d)$ for some graded ideal $\mathcal{N}$ of $A$; since $M$ is \*simple, $\mathcal{N}$ must be \*maximal. The converse is left to the reader.
Other facts parallel to the ungraded case include: 1) for every proper graded ideal $\mathcal{I}$ in $A$, there exists a \*maximal ideal $\mathcal{N}$ containing $\mathcal{I}$; 2) if $\mathcal{N}$ is a proper graded ideal of $A$, then $\mathcal{N}$ is \*maximal if and only if $A/\mathcal{N}$ is a graded field. Thus, every \*maximal ideal in $A$ is a graded prime ideal. Furthermore, if $\mathcal{N}$ is \*maximal in $A$, then $\mathcal{N}_0$ is a maximal ideal in $A_0$.
The structure of finitely generated graded modules over graded fields mirrors that for the ungraded category:
Suppose that $M$ is a finitely generated graded module over a graded field $F = F_0[t,t^{-1}]$, where $t$ has positive degree $d$ and $F_0$ is a field. Then
1. $M$ is a free graded $F$-module, of finite rank, on a set of homogeneous generators.
2. $M_0$ is a finite-dimensional vector space over $F_0$ of $F_0$-dimension less than or equal to the rank of $M$ over $F$.
Assume $M \neq 0$. Say $M$ is finitely generated over $F$ by homogeneous elements $e_1, \ldots, e_r$, where $r\geq 1$ is the minimal number for a homogeneous generating set for $M$ as an $F$-module. Then, $M$ is free on the $e_js$: certainly this set spans $M$ over $F$. Suppose that there is a relation $\sum_j \alpha_je_j = 0,$ with $\alpha_j \in F$. We may assume that all the $\alpha_js$ are homogeneous. If $\alpha_r \neq 0$, then it is invertible in $F$, so $\sum_{j=1}^{r-1} \alpha_r^{-1}\alpha_je_j +e_r= 0$, implying that $r$ is not minimal. Therefore $\alpha_r = 0$; and continuing the process, $\alpha_j=0$ for every $j$.
Set $d_j = \deg e_j$. Now, note that $X \doteq \{t^{-d_j/d}e_j \mid 1 \leq j \leq r \;\mbox{and}\; d \; \mbox{divides} \; d_j \}$ is a basis for $M_0$ over $F_0$; of course, if $d$ does not divide any $d_j$, then $M_0 = 0$. To see this, note that $X$ is linearly independent over $F_0$, since the $e_js$ are linearly independent over $F$. If $x \in M_0$, then $x = \sum_j \alpha_je_j$, where $\alpha_j$ is a homogeneous element of $F$ and $\deg \alpha_j + d_j = 0, \forall j$. Now, if $\alpha_j \neq 0$, $d$ divides its degree, by definition of $F$. Thus, $d$ divides $d_j$ for every $j$ such that $\alpha_j \neq 0$. If $d$ divides $d_j$, then $\alpha_j = \beta_j t^{-d_j/d},$ where $\beta_j \in F_0$. Thus $x$ is in the $F_0$-span of $X$.
$M \in {\mathfrak{grmod}}(A)$ is said to be a \*Artinian module if $M$ satisfies DCC on all chains of graded $A$-submodules of $M$.
Unlike the Noetherian case, an $A$-module $M$ can be \*Artinian without being Artinian: an example is given by $A=M$, where $A$ is a graded field with a nonzero positive degree element.
Similarly to the ungraded case, we have
Suppose that $A$ is a graded Noetherian ring and $M \in {\mathfrak{grmod}}(A)$. Then the following are equivalent:
- $M$ is \*Artinian.
- $*\ell_A(M) < \infty.$
- $*V(M)$ consists of a finite number of \*maximal ideals.
The proof of the equivalence of a) and b) in the ungraded case, as in [@AtMac], adapts in a straightforward way to the graded case. Note that the proof of “b) implies a)" does not require $A$ to be Noetherian.
To see how b) implies c), assume that $M$ has a \*composition series $$0=M^0 \subset M^1 \subset \cdots \subset M^{n-1} \subset M^n=M;$$ the \*simplicity of the subquotients means that there are \*maximal graded ideals ${{\mathfrak{m}}}_i$ of $A$ and integers $d_i$ such that $M^i/M^{i-1} \cong (A/{{\mathfrak{m}}}_i)(d_i)$ as graded $A$-modules. Thus, ${{\mathfrak{m}}}_1 {{\mathfrak{m}}}_2 \cdots {{\mathfrak{m}}}_n \subseteq Ann_A(M)$. If ${{\mathfrak{p}}}$ is a prime minimal over $Ann_A(M)$, then we have seen that ${{\mathfrak{p}}}$ is graded. Since ${{\mathfrak{m}}}_1 \cdots {{\mathfrak{m}}}_n \subseteq {{\mathfrak{p}}}$, we must have ${{\mathfrak{m}}}_i \subseteq {{\mathfrak{p}}}$ for at least one $i$. But ${{\mathfrak{m}}}_i$ is \*maximal, so ${{\mathfrak{m}}}_i ={{\mathfrak{p}}}$. Therefore $*V(M) \subseteq \{{{\mathfrak{m}}}_1, \ldots, {{\mathfrak{m}}}_n \}$.
For c) implies b), since $\sqrt{Ann_A(M)}$ is the intersection of the primes minimal over $Ann_A(M)$, and there are a finite number of these, all graded, the hypothesis implies that this finite list of primes consists entirely of \*maximal ideals; say these ideals are ${{\mathfrak{m}}}_1, \ldots, {{\mathfrak{m}}}_n$. Thus, there is an $N$ such that $({{\mathfrak{m}}}_1 \cdots {{\mathfrak{m}}}_n)^N \subseteq Ann_A(M)$ and there is a sequence $\tilde{{{\mathfrak{m}}}}_1, \ldots, \tilde{{{\mathfrak{m}}}}_{nN}$ of \*maximal ideals in $A$, not necessarily distinct, whose product is contained in $Ann_A(M)$. Analogously to the ungraded case, one can then construct a \*composition series for $M$.
If $V$ is a graded vector space over an ordinary field $k$, where $k$ is regarded as a graded ring concentrated in degree zero (this means that all nonzero elements have degree zero), then $${\mathrm{vdim}}_k(V) \doteq \; \mbox{the dimension of } \; V \; \mbox{as a vector space over} \; k.$$
\[lemma length of tensor\] Let $A$ be a graded Noetherian ring which is a finitely generated graded algebra over a field $k \subseteq A_0$, $M \in {\mathfrak{grmod}}(A)$, $V$ a graded finite dimensional vector space over $k$, and say that ${\mathrm{vdim}}_k(V) = d$. If $a \in A$, $m \otimes v \in M \otimes_k V$, then give $M \otimes_k V$ an $A$-module structure by $a \cdot (m \otimes v) \doteq (a \cdot m) \otimes v$, and grade $M \otimes_k V$ in the usual way. Then $$*\ell_A(M\otimes_k V) = *\ell_A(M)\cdot d.$$
Since $V$ is finite dimensional over $k$, we may suspend $V$ appropriately and assume, without loss of generality, that $V_j = 0$ for $j <0$; in this case, there exists an $n$ such that $j > n$ implies $V_j =0$. Define a graded filtration of $M\otimes_k V$ by graded $A$-modules: $\mathcal{F}^i \doteq M \otimes_k (V_0 \oplus \cdots \oplus V_{n-i})$ for $0 \leq i \leq n$, and $\mathcal{F}^{n+1} \doteq 0$. Consider that $\mathcal{F}^i/\mathcal{F}^{i+1} \cong M \otimes_k V_{n-i}$, and the additive property of length allows $*\ell_A(M\otimes_k V) = \sum_{i=0}^n *\ell_A(\mathcal{F}^i/\mathcal{F}^{i+1}) = \sum_{i=0}^n*\ell_A(M \otimes_k V_{n-i})$.
By hypothesis, each graded component $V_j$ of $V$ is a finite dimensional graded vector space concentrated in degree $j$. Thus, there is a graded isomorphism for each $j$, $V_j \cong k^{f(j)}(-j)$ where $f(j)$ is a function giving the vector space dimension of $V_j$. Since $M \otimes_k V_j \cong M \otimes_k k^{f(j)} \cong \oplus_1^{f(j)} M(-j)$, we have that $*\ell_A(M \otimes_A V_j) = *\ell_A(M)\cdot f(j)$. By hypothesis, $\sum_{j=0}^n f(j) = d$, the total vector space dimension of $V$, and finally $*\ell_A(M \otimes_k V) = \sum_{i=0}^n*\ell_A(M \otimes_k V_{n-i}) = \sum_{i=0}^n *\ell_A(M)\cdot f(n-i) = *\ell_A(M) \sum_{i=0}^n f(n-i) = *\ell_A(M) \cdot d$.
### Positively or Negatively Graded Rings
A graded ring $S$ is positively (resp. negatively) graded if and only if $S_i = 0$ for $i <0$ (resp. $i>0$). The graded ideal $S_+$ (resp. $S_-$) of $S$ is defined as $\oplus_{i >0} S_i$ (resp. $\oplus_{i<0} S_i$). Note that if $M \in {\mathfrak{grmod}}(S)$, since $S$ is positively (resp. negatively) graded, there exists an integer $e$ such that $M_i =0$ for all $i <e$ (resp. $i >e$). Also, for a proper, graded ideal ${{\mathfrak{m}}}$ of $S$, the following are equivalent:
- ${{\mathfrak{m}}}$ is \*maximal in $S$.
- ${{\mathfrak{m}}}= {{\mathfrak{m}}}_0 \oplus S_+,$ (resp. ${{\mathfrak{m}}}_0 \oplus S_-$) and ${{\mathfrak{m}}}_0$ (the degree zero elements of ${{\mathfrak{m}}}$) is a maximal ideal in $S_0$.
- $S/{{\mathfrak{m}}}$ is a graded field, concentrated in degree zero; i.e. $S/{{\mathfrak{m}}}$ is an ordinary field.
- ${{\mathfrak{m}}}$ is a maximal ideal in $S$.
For positively or negatively graded rings, there is no difference between \*length and length:
\[lemma pos graded length equals \*length\] Suppose that $S$ is a positively or negatively graded Noetherian ring, and $M \in {\mathfrak{grmod}}(S)$ is such that $*\ell_S(M) < \infty$. Then, $*\ell_S(M) = \ell_S(M)$.
Since $*V(M)$ consists of a finite number of \*maximal ideals, there is a sequence of graded $S$-modules $$0=M^0 \subset M^1 \subset \cdots \subset M^{n-1} \subset M^n=M,$$ \*maximal graded ideals ${{\mathfrak{m}}}_i$ of $S$ and integers $d_i$ such that $M^i/M^{i-1} \cong (S/{{\mathfrak{m}}}_i)(d_i)$ as graded $S$-modules. By the remark above, $S/{{\mathfrak{m}}}_i$ is concentrated in degree 0 and each ${{\mathfrak{m}}}_i$ is a maximal ideal in $S$. So, forgetting gradings everywhere, the given \*composition series is a composition series.
Even in the cases where \*length and length coincide, we’ll usually just talk about \*length, emphasizing constructions using graded modules only. For example,
\[lemma \*length of components\] Suppose $S$ is a positively graded ring and $X \in {\mathfrak{grmod}}(S)$.
- If $*\ell_S(X) < \infty$, there exists an integer $J$ such that if $j > J$, then $X_j= 0$.
- If $S_i$ is finitely generated as an $S_0$-module for every $i$, then $X_j$ is a finitely generated $S_0$-module, for every $j$.
- Suppose $S_0$ is Artinian, $S_i$ is finitely generated as an $S_0$-module for every $i$, and there exists an integer $J$ such that if $j >J$, then $X_j = 0$. Then, $\ell_{S_0}(X_j) < \infty$ for every $j$, and $*\ell_S(X) = \ell_{S_0}(X) < \infty$, where $\ell_{S_0}(X) \doteq \sum_j \ell_{S_0}(X_j)$ is the (total) $S_0$-length of $X$.
For every $t \in \mathbb{Z}$ define $X_{\geq t} \doteq \oplus_{s \geq t} X_s$. Since $S$ is positively graded, $X_{\geq t}$ is a graded $S$-submodule of $X$. Since $X$ is finitely generated over $S$, and $S$ is positively graded, there exists a $t_0 \in \mathbb{Z}$ such that $X_{\geq t_0} = X$. So we have a descending chain of graded $S$-submodules of $X$ $$\cdots \subseteq X_{\geq t_0 +k} \subseteq X_{\geq t_0 + k-1} \subseteq \cdots \subseteq X_{\geq t_0 + 1} \subseteq X_{\geq t_0} = X. (*)$$
For a), if $*\ell_S(X) < \infty, $ $X$ is \*Artinian, so this chain stabilizes. By definition, this means that there exists an $J \geq t_0$ such that $X_j = 0$ for $j >J$.
For b), let $t_0$ be defined as in the first paragraph above; assume that $X_{t_0} \neq 0$. Then, one can prove, by induction on $j$, that each $X_j$ is finitely generated over $S_0$ as follows. If $j = t_0$, then since $X$ is finitely generated as an $S$-module, say by $x_1, \ldots x_N$, if $\beta_{t_0} = \{ x_i \mid \deg(x_i) = t_0\}$, $X_{t_0}$ must be generated by $\beta_{t_0}$ as an $S_0$-module. Assume that $j > t_0$ and $X_u$ is finitely generated over $S_0$ for $u <j$. Then, $X_{<j} = \oplus_{u=t_0}^{j-1} X_u = \oplus_{u<j}X_j$, is finitely generated over $S_0$. Choose a finite set $\beta_{<j}$ of homogeneous elements that generate $X_{<j}$ over $S_0$. Choose finite generating sets $\alpha_u$ for each $S_u$ over $S_0$. Let $\beta_j = \{ x_i \mid \deg(x_i) = j \}$. The claim is that the finite set $B_j \doteq \{ a e \mid a \in \alpha_u, e \in \beta_{<j} \; \mbox{and} \; u+ \deg(e) = j \} \cup \beta_j$ spans $ X_j$ over $S_0$: if $x \in X_j$, then $x = \sum_i a_i x_i$, with $a_i$ homogeneous in $S$ for every $i$, and if $a_ix_i \neq 0$, $\deg(a_i) + \deg(x_i) = j$; from now on we’ll just talk about the indices $i$ such that $a_ix_i \neq 0$. If $\deg(a_i) = 0$, then $x_i \in \beta_j \subseteq B_j$ and $a_i \in S_0$. If $\deg(a_i) >0$, then $\deg(x_i)$ is strictly less than $j$ so that $x_i$ is in the $S_0$-span of $\beta_{<j}$; certainly, $a_i$ is in the $S_0$-span of $\alpha_{\deg(a_i)}$, so $a_ix_i$ is in the $S_0$-span of $ B_j$.
For c), given $J$ such that $X_J = 0$ for $j >J$, and choosing $t_0 \leq J$ such that $X_j = 0$ for $j < t_0$, the chain (\*) terminates at the left in 0, and has successive quotients isomorphic to a graded $S$-module $X_j$ (concentrated in degree $j$), where the $S$-module structure is determined by $rx = 0$ if $r \in S_+$. Since b) says that each $X_j$ is finitely generated over $S_0$, and $S_0$ is Artinian, the chain (\*) may be refined to a \*composition series of $X$, of length equal to $\sum_j \ell_{S_0}(X_j)$.
Graded Localization
===================
Localizing in the graded category can be done in a few ways. We may localize as usual, forgetting the graded structures, we may localize at sets consisting of homogeneous elements, or as in Grothendieck [@GRO], consider the degree zero part of this last localized module. In this section we make the relevant definitions, and compare the different methods.
Let $T$ be a multiplicatively closed subset (MCS) consisting entirely of homogeneous elements of $A$. We’ll call this a “GMCS". Since $T$ is an MCS we may construct the localization $T^{-1}M$ as usual. By definition, $T^{-1}M$ is graded by: $(T^{-1}M)_i \doteq \{ \frac{m}{t} \in T^{-1}M \mid m \; \mbox{is homogeneous and} \; \deg m - \deg t = i \}$.With this grading, $T^{-1}M$ becomes a graded $T^{-1}A$-module. In the case where ${{\mathfrak{p}}}\in Spec(A)$, and $T$ is the set of homogeneous elements of $ A-{{\mathfrak{p}}}$, we use the notation $M_{[{{\mathfrak{p}}}]}$ to denote the localization $T^{-1}M$, graded as above.
For a GMCS $T$, we’ll assume from now on that $1 \in T$ and $0 \notin T$.
The following list of lemmas collect some facts about graded localizations; we leave the proofs to the reader.
Let ${{\mathfrak{p}}}\in Spec(A)$. The set of homogeneous elements in $A-{{\mathfrak{p}}}$ is equal to the set of homogeneous elements in $A- {{\mathfrak{p}}}^*$. Therefore, $M_{[{{\mathfrak{p}}}]} = M_{[{{\mathfrak{p}}}^*]}$
\[lemma graded localize the quotient neq 0\] Let ${{\mathfrak{p}}}$ and ${{\mathfrak{q}}}$ be prime ideals of $A$, with ${{\mathfrak{q}}}$ graded. Then, $(A/{{\mathfrak{q}}})_{[{{\mathfrak{p}}}]} \neq 0$ if and only if ${{\mathfrak{q}}}\subseteq {{\mathfrak{p}}}^*$. If ${{\mathfrak{p}}}$ is a minimal prime of $A$, then $(A/{{\mathfrak{q}}})_{[{{\mathfrak{p}}}]} \neq 0$ if and only if ${{\mathfrak{q}}}= {{\mathfrak{p}}}$.
\[lemma props of graded localization\]If $M \in {\mathfrak{grmod}}(A)$, and $T$ is a GMCS in $A$, then
1. $T^{-1}M \in {\mathfrak{grmod}}(T^{-1}A).$
2. If $A$ is a Noetherian ring then $T^{-1}A$ is a Noetherian ring and $T^{-1}M \in {\mathfrak{grmod}}(T^{-1}A)$.
3. There is a one-one, inclusion-preserving correspondence between the prime ideals in $A$ that are disjoint from $T$, and the prime ideals in $T^{-1}A$ given by ${{\mathfrak{p}}}\mapsto T^{-1}{{\mathfrak{p}}}$; moreover this correspondence restricts to a one-one correspondence between the graded prime ideals in $A$ disjoint from $T$ and the graded prime ideals in $T^{-1}A$, and further restricts to a one-one correspondence between the ideals (all graded) in $Ass_A(M)$ that are disjoint from $T$, and the ideals (also all graded) in $Ass_{T^{-1}A}T^{-1}M$.
\[lemma graded iso of localizing a shifted graded module\] Let $M \in {\mathfrak{grmod}}(A)$, $T$ a GMCS in $A$, and let $d$ be any integer. Then there is a graded isomorphism of graded $T^{-1}A$-modules $T^{-1}(M(d)) \cong (T^{-1}M)(d)$.
Suppose that $M \in {\mathfrak{grmod}}(A)$.
1. ${{\mathfrak{p}}}\in Supp_A(M)$ if and only if $M_{[{{\mathfrak{p}}}^*]} \neq 0$ if and only if ${{\mathfrak{p}}}^* \in *V(M)$. Therefore, $$*V(M) = *Supp_A(M) =
\{ {{\mathfrak{q}}}\in *V(A) \mid M_{[{{\mathfrak{q}}}] } \neq 0\}.$$
2. If $0 \rightarrow M \rightarrow N \rightarrow P \rightarrow 0$ is a short exact sequence in ${\mathfrak{grmod}}{A}$, then $*V(N) = *V(M) \cup *V(P).$
Since $V(N) = V(M) \cup V(P)$, b) follows.
For $a)$, it’s straightforward to see that the ungraded object $M_{{{\mathfrak{p}}}} \neq 0$ implies that $M_{[{{\mathfrak{p}}}^*]} \neq 0$. If $M_{[{{\mathfrak{p}}}^*]} \neq 0$, and $Ann_A(M)$ is not contained in ${{\mathfrak{p}}}^*$, then since both are graded ideals, there exists a homogeneous element $r \in Ann_A(M)$ such that $r \notin {{\mathfrak{p}}}^*$. But then, $m/t = 0/r = 0$ for every $m \in M$ and homogeneous $t \notin {{\mathfrak{p}}}^*$. Finally, suppose that $Ann_A(M) \subseteq {{\mathfrak{p}}}^*$, yet $M_{{{\mathfrak{p}}}} = 0$. If $x_1, \ldots, x_j$ are homogeneous elements of $M$ generating $M$ as an $A$-module, since $x_i/1= 0$ for every $i$, there exist $s_i \notin {{\mathfrak{p}}}$ such that $s_ix_i=0$ for each $i$. We may assume that each $s_i$ is homogeneous, since $x_i$ is. Since $s_i \notin {{\mathfrak{p}}}$, $s_i \notin {{\mathfrak{p}}}^*$, so that $s = s_1s_2\cdots s_j \notin {{\mathfrak{p}}}^*$ and is homogeneous. Furthermore, $sm= 0$ for every $m \in M$, so $s \in {{\mathfrak{p}}}^*$, a contradiction.
For ${{\mathfrak{p}}}$ a graded prime in $A$, $T$ a GMCS, $(T^{-1} {{\mathfrak{p}}})_0=T^{-1}{{\mathfrak{p}}}\cap (T^{-1}A)_0$, and if ${{\mathfrak{p}}}\cap T = \emptyset$ then $(T^{-1}{{\mathfrak{p}}})_0$ is a prime ideal in $(T^{-1}A)_0$.
[@GRO] If ${{\mathfrak{p}}}\in Spec(A)$, then we denote the degree $0$ part of $M_{[{{\mathfrak{p}}}]}$ by $M_{({{\mathfrak{p}}})}$.
If $M$ is an $A$-module, $M_{({{\mathfrak{p}}})}$ is an $A_{({{\mathfrak{p}}})}$-module.
If $A$ is a graded ring, ${{\mathfrak{p}}}$ is a graded prime ideal in $A$, and $T$ is the GMCS consisting of the homogeneous elements of $A-{{\mathfrak{p}}}$, then $T^{-1}{{\mathfrak{p}}}\doteq {{\mathfrak{p}}}_{[{{\mathfrak{p}}}]}$ is a \*maximal ideal in $T^{-1}A \doteq A_{[{{\mathfrak{p}}}]}$ and ${{\mathfrak{p}}}_{({{\mathfrak{p}}})} = ({{\mathfrak{p}}}_{[{{\mathfrak{p}}}]})_0$ is a maximal ideal in $A_{({{\mathfrak{p}}})}$.
Now, if $M$ is a graded $A$-module and ${{\mathfrak{p}}}$ is a graded prime ideal, we know that the standard localization $M_{{{\mathfrak{p}}}}$ isn’t usually graded as we allow inhomogeneous elements of $A$ not in ${{\mathfrak{p}}}$ to be inverted. If ${{\mathfrak{p}}}$ is a minimal prime ideal for $M$, it must be graded, as we have seen, and from ordinary commutative algebra, we know that $M_{{{\mathfrak{p}}}}$ has finite length as an $A_{{{\mathfrak{p}}}}$-module. But we can also consider the graded localization $M_{[{{\mathfrak{p}}}]}$ and the comparison between length and \*length:
\[theorem \*composition series exists for Mgrp when p is minimal\] Suppose that $A$ is a Noetherian graded ring. Let $M \in {\mathfrak{grmod}}(A)$, and ${{\mathfrak{p}}}$ be a prime minimal over the graded ideal $Ann_A(M)$. Then, a \*composition series exists for the graded $A_{[{{\mathfrak{p}}}]}$-module $M_{[{{\mathfrak{p}}}]}$. Moreover, $$*\ell_{A_{[{{\mathfrak{p}}}]}}(M_{[{{\mathfrak{p}}}]}) =\ell_{A_{{{\mathfrak{p}}}}}(M_{{{\mathfrak{p}}}}).$$
We will produce a \*composition series for $M_{[{{\mathfrak{p}}}]}$, as an $A_{[{{\mathfrak{p}}}]}$-module and calculate its length.
Construct a graded filtration $M^{\bullet}$ as in Lemma \[corollary the filtration graded\], and then localize this filtration using the graded localization. We now have a filtration of $M_{[{{\mathfrak{p}}}]}$ by graded $A_{[{{\mathfrak{p}}}]}$-submodules which looks like $0=(M^{0})_{[{{\mathfrak{p}}}]} \subseteq (M^1)_{[{{\mathfrak{p}}}]} \subseteq \cdots \subseteq (M)_{[{{\mathfrak{p}}}]}$. By exactness of localization and the condition on successive quotients of $M^{\bullet}$ we have that $(M^{i+1}/M^{i})_{[{{\mathfrak{p}}}]} \cong (A/{{\mathfrak{p}}}_{i+1}(-d_{i+1}))_{[{{\mathfrak{p}}}]}$ is a graded isomorphism of $A_{[{{\mathfrak{p}}}]}$-modules, for appropriate integers $d_i$, where the graded primes ${{\mathfrak{p}}}_i$ are as in \[corollary the filtration graded\].
There is a graded isomorphism $((A/{{\mathfrak{p}}}_i)(-d_i))_{[{{\mathfrak{p}}}]} \cong (A/{{\mathfrak{p}}}_i)_{[{{\mathfrak{p}}}]}(-d_i)$, and $$(A/{{\mathfrak{p}}}_i)_{[{{\mathfrak{p}}}]}(-d_i) \neq 0 \; \mbox{ if and only if} \; {{\mathfrak{p}}}= {{\mathfrak{p}}}_i$$ (by minimality of ${{\mathfrak{p}}}$).
In the case that ${{\mathfrak{p}}}\neq {{\mathfrak{p}}}_i$, $(A/{{\mathfrak{p}}}_i)_{[{{\mathfrak{p}}}]} =0$ and we have $(M^{i})_{[{{\mathfrak{p}}}]} = (M^{i-1})_{[{{\mathfrak{p}}}]}$. Now throw away all such submodules $(M^i)_{[{{\mathfrak{p}}}]}$ which are equal to the submodule $(M^{i-1})_{[{{\mathfrak{p}}}]}$ to get a reduced filtration $((\overline{M}^j)_{[{{\mathfrak{p}}}]})$ of $M_{[{{\mathfrak{p}}}]}$, where for each $j$, $(\overline{M}^{j})_{[{{\mathfrak{p}}}]} \subset (\overline{M}^{j+1})_{[{{\mathfrak{p}}}]}$ is a strict inclusion, $(\overline{M}^{s})_{[{{\mathfrak{p}}}]} = M_{[{{\mathfrak{p}}}]}$ for some $s$, and the zeroth term of the filtration is zero. The claim is that this reduced filtration forms a \*composition series for $M_{[{{\mathfrak{p}}}]}$ of \*length equal to the number of times that $A/{{\mathfrak{p}}}$, shifted, appeared as a successive quotient in the original filtration $M^{\bullet}$.
For each $j$ the successive quotient $(\overline{M}^{j+1})_{[{{\mathfrak{p}}}]} / (\overline{M}^{j})_{[{{\mathfrak{p}}}]}$ is graded isomorphic to $(A/{{\mathfrak{p}}})_{[{{\mathfrak{p}}}]}(-d_{j+1})$, as an $A_{[{{\mathfrak{p}}}]}$-module. But $(A/{{\mathfrak{p}}})_{[{{\mathfrak{p}}}]}$ is a graded field, since $A_{[{{\mathfrak{p}}}]}$ has a unique graded prime ideal ${{\mathfrak{p}}}_{[{{\mathfrak{p}}}]}$; thus, $(\overline{M}^{j+1})_{[{{\mathfrak{p}}}]}/(\overline{M}^{j})_{[{{\mathfrak{p}}}]} \cong (A/{{\mathfrak{p}}})_{[{{\mathfrak{p}}}]}(-d_{j+1})$ is a \*simple $A_{[{{\mathfrak{p}}}]}$-module for each $j$.
Going back to the original filtration $M^{\bullet}$ and forgetting the grading everywhere, recall that the number of times that $A/{{\mathfrak{p}}}$ appears as a successive quotient in any finite filtration of $M$ which has successive quotients isomorphic to $A/{{\mathfrak{q}}}$ for some prime ${{\mathfrak{q}}}$, graded or not, is always the same, and is equal to $\ell_{A_{{{\mathfrak{p}}}}}(M_{{{\mathfrak{p}}}})$.
\*Local rings
-------------
If $A$ is a graded ring, then $A$ is \*local if and only if there is one and only one \*maximal ideal of $A$.
Some examples of \*local rings are immediate. For example, a graded field is always \*local, with unique \*maximal ideal $0$. This shows that generally, a \*maximal ideal of a graded ring $A$ may not be a maximal ideal of $A$. If ${{\mathfrak{p}}}$ is a graded prime in $A$, then $A_{[{{\mathfrak{p}}}]}$ is a \*local ring with unique \*maximal ideal ${{\mathfrak{p}}}_{[{{\mathfrak{p}}}]}$. The end of this section gives a partial characterization of \*local rings.
Also, as one might expect, if $A$ is a \*local graded ring, with unique \*maximal ideal $\mathcal{N}$, then
- For every proper ideal $\mathcal{I}$ (graded or not) of $A$, $\mathcal{I}^* \subseteq \mathcal{N}$.
- Every homogeneous element of $A-\mathcal{N}$ is invertible: i.e., for every $x \in A-\mathcal{N}$ with $\deg x = d$, there exists a $y \in A-\mathcal{N}$ of degree $-d$ such that $xy = 1 \in A_0$.
- $A/\mathcal{N}$ is a graded field; also, for every $y \in \mathcal{N}_j$ and every $x \in \mathcal{A}_{-j}$, $1-xy \in A_0$ is a unit in $A_0$.
(Graded Nakayama’s lemma) \[lemma graded nakayama\]Suppose $(A,\mathcal{N})$ is a \*local ring and $M$ is a finitely generated graded $A$-module with $N$ a graded $A$-submodule of $M$. If ${{\mathfrak{q}}}$ is a proper graded ideal in $M$, then $N+ {{\mathfrak{q}}}M = M$ implies that $M = N$.
(Slight variation of proof of Nakayama’s lemma in [@AtMac].) We may assume $N= 0$ by passing to $M/N$. Say $M \neq 0$; choose a homogeneous generating set $x_1, \ldots, x_r$ for $M$ over $A$ with a minimal number $r \geq 1$ of nonzero homogeneous elements. Suppose that ${{\mathfrak{q}}}M = M$; then there are homogeneous elements $\alpha_j \in {{\mathfrak{q}}}\subseteq \mathcal{N}$ such that $x_r = \alpha_1x_1 + \cdots + \alpha_r x_r$; we must have $\deg \alpha_j + \deg x_j = \deg x_r$ for every $j$ such that $\alpha_j x_j \neq 0$. By minimality, $\alpha_r x_r\neq 0$ and so $\deg \alpha_r = 0$. Using the remarks above, $1-\alpha_r$ is an invertible element of $A_0$. Thus, we may write $x_r$ as an $A$-linear combination of $x_1, \ldots, x_{r-1}$, contradicting the minimality of $r$.
\[proposition maximal equals minimal\] If $A$ is \*local and Noetherian with unique \*maximal ideal $\mathcal{N}$, and $M$ is a nonzero finitely generated graded $A$-module with $\mathcal{N}$ a minimal prime over $Ann_A(M)$ (equivalently, $*V(M) = \{ \mathcal{N} \}$), then $M$ is a \*Artinian $A$-module, and for each $j \in \mathbb{Z}$, $M_j$ is an Artinian $A_0$-module. Furthermore, for each $j \in \mathbb{Z}$, $\ell_{A_0}M_j \leq *\ell_{A}M.$ If, in addition, there is a homogeneous element of degree 1 (or, equivalently, -1) in $A - \mathcal{N}$, $\ell_{A_0}M_j = *\ell_{A}M$ for every $j$.
$M$ is \*Artinian, since $*V(M) = \{\mathcal{N}\}$. In fact, in this case, $M$ has a \*composition series with the property that each successive quotient is annihilated by $\mathcal{N}$ and is also free of rank one over the graded field $A/\mathcal{N}$. Taking the degree $j$ part of each module in this \*composition series, we get a chain of $A_0$-submodules of $M_j$ and the dimension of each successive quotient over the field $K \doteq (A/\mathcal{N})_0 \doteq A_0/\mathcal{N}_0$ is either zero or 1. Thus, since $\mathcal{N}_0$ also annihilates each successive quotient in this “degree j" filtration, we see that we can make appropriate deletions in the “degree j" part of the \*composition series for $M$ to yield a composition series for $M_j$ over $A_0$ of length less than or equal to $*\ell_A(M)$.
For the last statement, supposing that there is a homogeneous element of degree 1 in $A-\mathcal{N}$, then there are nonzero elements of every degree in the graded $A$-module $(A/\mathcal{N})(d)$, for every $d \in \mathbb{Z}$; to see this, note that $A/\mathcal{N}$ is a graded field, equal to $K[T, T^{-1}]$, where $T \neq 0$ has least positive degree in $A/\mathcal{N}$, namely degree 1. So, each successive quotient in the \*composition series for $M$ is nonzero in every degree. After taking the “degree j" part of this \*composition series, each quotient must be of rank 1 over $K$. Thus the equality holds.
\[corollary length and \*length at minimal prime\]Suppose that $A$ is a Noetherian graded ring. Let $M \in {\mathfrak{grmod}}(A)$, and ${{\mathfrak{p}}}$ be a prime minimal over $Ann_A(M)$, necessarily graded. Then, $M_{[{{\mathfrak{p}}}]}$ is an \*Artinian $A_{[{{\mathfrak{p}}}]}$-module and $M_{({{\mathfrak{p}}})}$ is an Artinian $A_{({{\mathfrak{p}}})}$-module. Also, $$\ell_{A_{({{\mathfrak{p}}})}} ( M_{({{\mathfrak{p}}})} )\leq *\ell_{A_{[{{\mathfrak{p}}}]}}(M_{[{{\mathfrak{p}}}]}) = \ell_{A_{{{\mathfrak{p}}}}}(M_{{{\mathfrak{p}}}}).$$ In addition, if there is a homogeneous element of degree 1 (or -1) in $A-{{\mathfrak{p}}}$, then $$\ell_{A_{({{\mathfrak{p}}})}} ( M_{({{\mathfrak{p}}})} )= *\ell_{A_{[{{\mathfrak{p}}}]}}(M_{[{{\mathfrak{p}}}]}) = \ell_{A_{{{\mathfrak{p}}}}}(M_{{{\mathfrak{p}}}}).$$
In ending this section, we point out that \*local graded rings are often graded localizations of positively (or negatively) graded rings at graded prime ideals.
Suppose that $(A, \mathcal{N})$ is a \*local ring. Then, there exists a homogeneous element of strictly positive degree in $A-\mathcal{N}$ if and only if there exists a homogeneous element of strictly negative degree in $A-\mathcal{N}$: if $s \in A-\mathcal{N}$ is homogeneous of degree $d >0$ then, since every homogenous element of $A-\mathcal{N}$ is invertible, there exists a $t \in A-\mathcal{N}$ that is homogeneous and $st=1 \in A_0$. Necessarily, the degree of $t$ is $-d$. Since the argument is reversible, we have the conclusion.
Thus, we have alternatives:
- There exist homogeneous elements of $A-\mathcal{N}$ in at least one strictly positive degree and at least one strictly negative degree. The analysis of this alternative is given below and we see that $A$ is a graded localization of a positively graded ring at a graded prime ideal.
- $A$ is a positively or negatively graded ring: in this case, $A$ is the localization of itself (a positively or negatively graded ring) at the \*maximal ideal $\mathcal{N}$ since we’ve seen that $\mathcal{N}_0$ is the unique maximal ideal in $A_0$ and $A_e = \mathcal{N}_e$ for all $e \neq 0$.
- $A$ has nonzero elements of both positive and negative degree, $\mathcal{N}_d = A_d$ for all $d \neq 0$ and $\mathcal{N}_0$ is the unique maximal ideal of $A_0$. In this case, since $\mathcal{N}$ is an ideal, we must have $A_d A_{-d} \subseteq \mathcal{N}_0$, for all $d \neq 0$. As an example, consider $A = k[s,t]/(st)$ where $k$ is a field (all elements of degree 0), the degree of $s$ is one and the degree of $t$ is -1. In this case, one might not be able to obtain $A$ as a graded localization of a positively (or negatively) graded object at a graded prime ideal. But we don’t fully analyze this case here.
Anyway, in the case of the first of the alternatives, let $$S(A) = \oplus_{d \geq 0} A_d$$ be the “positive part" of $A$, graded with the natural grading; this too is a graded ring, and it is certainly positively graded. Considering the graded abelian subgroup $S(\mathcal{N}) = \oplus_{d \geq 0} \mathcal{N}_d$ of $S(A)$, we see that it is a graded prime ideal in $S(A)$. We claim that $S(A)_{[S(\mathcal{N})]}$ is isomorphic as a graded ring to $A$, with $\mathcal{N}$ corresponding to $S(\mathcal{N})_{[S(\mathcal{N})]}$, under the well-defined injective homomorphism of graded rings defined by $a/b \mapsto ab^{-1}$, if $a \in S(A)_d$ and $b \in A_e-\mathcal{N}_e$ for $d, e \geq 0$. To see that the homomorphism is surjective, suppose that $x$ is a homogeneous element of degree $j$ in $A$. If $j \geq 0$, $x/1 \mapsto x$, and $x/1 \in S(A)_{[S(\mathcal{N})]}$. If $j<0$, the assumption of the first alternative says that there is a homogeneous element $t \in A-\mathcal{N}$ with $\deg(t) = k >0$. Then, there is a positive integer $l$ such that $lk+j >0$ so that $t^lx/t^l \in S(A)_{[S(\mathcal{N})]}$ and $t^lx/t^l \mapsto x$.
Krull Dimension in grmod(A)
===========================
From now on, we assume that $A$ is a Noetherian graded ring, unless explicitly stated otherwise.
\[Dimension in C(A)\]
The height of a prime ideal ${{\mathfrak{p}}}$ of $A$, graded or not, is defined as usual: ${{\mathrm{ht}}}({{{\mathfrak{p}}}})$ is the longest length $n$ (which always exists, using the Noetherian hypothesis) of a chain of primes ${{\mathfrak{p}}}_0 \subset \cdots \subset {{\mathfrak{p}}}_n={{\mathfrak{p}}}$; thus we define the graded height ${{^*\mathrm{ht}}}({{{\mathfrak{p}}}})$ of a graded prime ideal ${{\mathfrak{p}}}$ in the ring $A$, as the longest length $m$ (which always exists, using the Noetherian hypothesis) of a chain of graded primes ${{\mathfrak{p}}}_0 \subset \cdots \subset {{\mathfrak{p}}}_m={{\mathfrak{p}}}$. For every graded prime ${{\mathfrak{p}}}$, ${{\mathrm{ht}}}({{\mathfrak{p}}}) \geq {{^*\mathrm{ht}}}({{\mathfrak{p}}})$.
Forgetting the grading on $A$ and $M$, one defines the Krull dimension of a graded $A$-module $M$ as usual; here this is denoted by $\dim_A(M)$. As usual, $\dim(A) \doteq \dim_A(A).$ The **graded Krull dimension** of a graded $A$-module $M$, ${{\phantom{}^*\mathrm{dim}}}_A(M)$, is the greatest $D$ such that there exists a strictly increasing chain $$\mathfrak{p}_0 \subset \ldots \subset \mathfrak{p}_D$$ of graded prime ideals in $A$ such that $Ann_A(M) \subseteq \mathfrak{p}_0$. If no such greatest $D$ exists, $M$ has infinite graded Krull dimension. For the zero module, we define ${{\phantom{}^*\mathrm{dim}}}(0) = -\infty.$ By definition, ${{\phantom{}^*\mathrm{dim}}}(A) \doteq {{\phantom{}^*\mathrm{dim}}}_A(A).$ For any graded $A$-module $M$,
- ${{\phantom{}^*\mathrm{dim}}}_A(M) \leq \dim_A(M)$.
Also, since $Ann_A(M) = Ann_A(M(n))$, for every $n \in \mathbb{Z}$,
- $\dim_A(M) = \dim_A(M(n)), $ for every $n \in \mathbb{Z}$.
- ${{\phantom{}^*\mathrm{dim}}}_A(M) = {{\phantom{}^*\mathrm{dim}}}_A(M(n)),$ for every $n \in \mathbb{Z}$.
\[example graded Krull dimension of Laurent polynomial ring\]
Let $F$ be a graded field of the form $F \cong F_0[t,t^{-1}]$, where $\deg(t)>0$. The only graded prime in $F$ is $0$, so that ${{\phantom{}^*\mathrm{dim}}}(F)=0$. On the other hand, $\dim(F)=1.$ \[laurentex\]
More generally,
If $A$ is has only one graded prime ideal $\mathcal{N}$, then $A$ is a \*local, \*Artinian ring with ${{\phantom{}^*\mathrm{dim}}}(A) = 0$, and $A_0$ is an Artinian ring of Krull dimension zero with unique nilpotent maximal ideal $\mathcal{N}_0$.
Most of the proofs for the following lemma may be found in [@brhe].
Suppose that ${{\mathfrak{p}}}\in$ Spec($A$). (${{\mathfrak{p}}}$ may or may not be graded.) We know that ${{\mathfrak{p}}}$ has finite height; say ${{\mathrm{ht}}}({{\mathfrak{p}}})=d$. \[mainlemma6.2\]
- If ${{\mathfrak{q}}}\in Spec(A)$ and ${{\mathfrak{p}}}^* \subseteq {{\mathfrak{q}}}\subseteq {{\mathfrak{p}}}$ then either ${{\mathfrak{q}}}= {{\mathfrak{p}}}$ or ${{\mathfrak{q}}}= {{\mathfrak{p}}}^*$.
- There exists a chain of primes ${{\mathfrak{q}}}_0 \subset \cdots \subset {{\mathfrak{q}}}_d = {{\mathfrak{p}}}$, such that ${{\mathfrak{q}}}_0, \cdots , {{\mathfrak{q}}}_{d-1}$ are all graded.
- If ${{\mathfrak{p}}}$ is graded then there exists a chain of graded prime ideals such that $${{\mathfrak{p}}}_0 \subset {{\mathfrak{p}}}_1 \subset \cdots \subset {{\mathfrak{p}}}_d = {{\mathfrak{p}}}\text{,}$$ so that ${{\mathrm{ht}}}({{\mathfrak{p}}}) = {{^*\mathrm{ht}}}({{\mathfrak{p}}})$.
- If ${{\mathfrak{p}}}$ is not graded $({{\mathfrak{p}}}^*$ is a proper subset of ${{\mathfrak{p}}})$, then $${{\mathrm{ht}}}({{\mathfrak{p}}}) = {{\mathrm{ht}}}({{\mathfrak{p}}}^*)+1 = {{^*\mathrm{ht}}}({{\mathfrak{p}}}^*)+1 \text{.}$$
\[corollary difference Krull dim Z graded\] If $A$ is a graded Noetherian ring, then $${{\phantom{}^*\mathrm{dim}}}(A) \leq \dim(A) \leq {{\phantom{}^*\mathrm{dim}}}(A) +1;$$therefore if $M \in {\mathfrak{grmod}}(A)$, $${{\phantom{}^*\mathrm{dim}}}_A(M) \leq \dim_A(M) \leq {{\phantom{}^*\mathrm{dim}}}_A(M) +1.$$
If $A$ has finite Krull dimension, the first inequality is always true; also, there is a maximal ideal ${{\mathfrak{m}}}$ of $A$, not necessarily graded, such that ${{\mathrm{ht}}}({{\mathfrak{m}}}) = \dim(A)$. But then $\dim(A) = {{\mathrm{ht}}}({{\mathfrak{m}}}) \leq {{^*\mathrm{ht}}}({{\mathfrak{m}}}^*) +1 \leq {{\phantom{}^*\mathrm{dim}}}(A) +1.$ If $A$ does not have finite Krull dimension, then for every positive integer $e$ there is a prime ideal ${{\mathfrak{p}}}$ of height larger than $e$. But then ${{^*\mathrm{ht}}}({{\mathfrak{p}}}^*)$ is larger than $e-1$, so ${{\phantom{}^*\mathrm{dim}}}(A) $ is infinite as well.
### Krull dimension for modules over positively graded rings {#Krull dimension for modules over positively graded rings}
The results in this section also hold for negatively graded rings, after changing definitions appropriately.
If $S$ is a positively graded ring, $$Proj(S) \doteq \{ {{\mathfrak{p}}}\in \mbox{Spec}(S) \mid {{\mathfrak{p}}}\; \mbox{is graded and} \; S_+ \not \subseteq {{\mathfrak{p}}}\}.$$
Note that if ${{\mathfrak{p}}}\in Proj(S)$, then the set of homogeneous elements of $S-{{\mathfrak{p}}}$ has at least one nonzero element of strictly positive degree. We’ve noted that $\mathcal{N}$ is a \*maximal ideal in $S$ if and only if $\mathcal{N} = \mathcal{N}_0 \oplus S_+$, with $\mathcal{N}_0$ a maximal ideal in $S_0$. Thus, $Proj(S)$ contains no \*maximal ideals.
For positively graded rings, there is no difference between ${{\phantom{}^*\mathrm{dim}}}$ and $\dim$:
Let the ring $S$ be a positively graded ring of finite Krull dimension and $M \in {\mathfrak{grmod}}(S)$. Then,
- $\dim(S) = {{\phantom{}^*\mathrm{dim}}}(S)$; therefore,
- $\dim_S(M) = {{\phantom{}^*\mathrm{dim}}}_S(M)$.
\[maintheorem6.2\]
For graded localizations of positively graded rings, the following is well-known:
\[theorem krull dimension localized Z graded\] Suppose that $S$ is a positively graded ring of finite Krull dimension, and ${{\mathfrak{p}}}$ is a graded prime ideal of $S$. Then, if $S_+ \subseteq {{\mathfrak{p}}}$, $\dim(S_{[{{\mathfrak{p}}}]}) = {{\phantom{}^*\mathrm{dim}}}(S_{[{{\mathfrak{p}}}]})$, and if $S_+ \not \subseteq {{\mathfrak{p}}}$, $\dim(S_{[{{\mathfrak{p}}}]}) = {{\phantom{}^*\mathrm{dim}}}(S_{[{{\mathfrak{p}}}]}) + 1.$
Since $S_{[p]}$ is a localization of $S$, it is Noetherian. Ignoring the grading and recalling the standard order-preserving correspondence between the set of all primes of $S$ disjoint from $T$ and the prime ideals of of $T^{-1}S$, for any MCS or GMCS $T$ in $S$. So $\infty > \dim(S) \geq \dim(S_{[{{\mathfrak{p}}}]}).$
We have already seen, then, that $ {{\phantom{}^*\mathrm{dim}}}(S_{[{{\mathfrak{p}}}]}) \leq \dim(S_{[{{\mathfrak{p}}}]}) \leq {{\phantom{}^*\mathrm{dim}}}(S_{[{{\mathfrak{p}}}]}) + 1$.
Now let $T$ be the GMCS consiting of all homogeneous elements of $S$ not in ${{\mathfrak{p}}}$.
In the case where $S_+ \subseteq {{\mathfrak{p}}}$, we must have ${{\mathfrak{p}}}= ({{\mathfrak{p}}}\cap S_0) \oplus S_+.$ For any element $t \in T$, this forces $\deg t = 0$. Thus, $S_{[{{\mathfrak{p}}}]}$ is a positively graded ring of finite Krull dimension, so $\dim(S_{[{{\mathfrak{p}}}]}) = {{\phantom{}^*\mathrm{dim}}}(S_{[{{\mathfrak{p}}}]}).$
Now, $S_{[{{\mathfrak{p}}}]}/{{\mathfrak{p}}}_{[{{\mathfrak{p}}}]} = (S/{{\mathfrak{p}}})_{[{{\mathfrak{p}}}]}$ is a graded field, and it does have a positive degree element since $S_+ \not \subseteq {{\mathfrak{p}}}$: Choose any homogeneous $t \in S_+$, $t \notin {{\mathfrak{p}}}$. Then $t \in T$, and has positive degree, thus $(t+{{\mathfrak{p}}})/1$ is a nonzero, positive degree element of $(S/{{\mathfrak{p}}})_{[{{\mathfrak{p}}}]}$. Forgetting the grading, this domain has dimension 1. Thus, there must exist a prime ${{\mathfrak{q}}}$, necessarily ungraded, of $S_{[{{\mathfrak{p}}}]}$ such that $${{\mathfrak{p}}}_{[{{\mathfrak{p}}}]} \subset {{\mathfrak{q}}}.$$ Therefore $$\dim(S_{[{{\mathfrak{p}}}]}) \geq {{\mathrm{ht}}}({{\mathfrak{p}}}_{[{{\mathfrak{p}}}]}) +1 = {{^*\mathrm{ht}}}({{\mathfrak{p}}}_{[{{\mathfrak{p}}}]}) +1= {{\phantom{}^*\mathrm{dim}}}(S_{[{{\mathfrak{p}}}]})+1,$$ yielding the conclusion.
The following lemma establishes a relationship between primes in the localized ring and primes in the degree $0$ part of the localization, the ideas are implicit in [@GRO].
Suppose that $S$ is a positively graded ring, and $T$ is any GMCS that contains at least one element of positive degree. \[lemma existence of degree 0 localization correspondence\] If ${{\mathfrak{q}}}$ is a prime ideal in $(T^{-1}S)_0$, then there exists a unique graded prime ${{\mathfrak{p}}}\in Proj(S)$, disjoint from $T$, such that ${{\mathfrak{q}}}= (T^{-1}{{\mathfrak{p}}})_0$.
Uniqueness is left to the reader. To establish existence, let ${{\mathfrak{q}}}\in Spec(T^{-1}S)_0$. Define for $i \geq 0$, $${{\mathfrak{p}}}_i \doteq \{x \in S_i\; | \; \exists j >0, t \in T_j \text{ s.t. } \frac{x^j}{t^i} \in {{\mathfrak{q}}}\},$$ so that, since ${{\mathfrak{q}}}$ is prime, $${{\mathfrak{p}}}_0 = \{r \in S_0 \; | \; \frac{r}{1} \in {{\mathfrak{q}}}\}.$$ Define ${{\mathfrak{p}}}\doteq \oplus_{i\geq 0} {{\mathfrak{p}}}_i$, we will show that ${{\mathfrak{p}}}$ satisfies the required conditions.
First, each ${{\mathfrak{p}}}_i$ is an abelian group with respect to $+$. For if $x,y \in {{\mathfrak{p}}}_i$, $i \geq 0$, then there exists a $k_1,k_2 >0$ and $s \in T_{k_1}, t \in T_{k_2}$ such that $\frac{x^{k_1}}{s^i}$ and $\frac{y^{k_2}}{t^i}$ are in ${{\mathfrak{q}}}$. Then, $(x+y)^{k_1+k_2} = \sum_{\alpha+\beta=k_1+k_2}c_{(\alpha,\beta)}x^\alpha y^\beta$, for the binomial coefficient $c_{(\alpha,\beta)} \in S_0$. Now, either $\alpha \geq k_1$ or $\beta \geq k_2$. If $\alpha \geq k_1$, then $\frac{x^\alpha y^\beta}{s^i t^i} = \frac{x^{k_1}}{s^i} \cdot \frac{x^{\alpha-k_1}y^\beta}{t^i}$. This is a product of an element in ${{\mathfrak{q}}}$ with an element in $(T^{-1}S)_0$, so it must be in ${{\mathfrak{q}}}$. A similar computation handles the case that $\beta \geq k_2$. Therefore, $\frac{(x+y)^{k_1+k_2}}{(st)^i} \in {{\mathfrak{q}}}$, and so $x + y \in {{\mathfrak{p}}}_i$.
To show that that ${{\mathfrak{p}}}$ is an ideal in $S$, one needs only to show that $S_i {{\mathfrak{p}}}_j \subseteq {{\mathfrak{p}}}_{i+j}$ for every $i,j$. Suppose $s \in S_i$ and $x \in {{\mathfrak{p}}}_j$. There exists $k>0, t \in T_k$ with $\frac{x^k}{t^j} \in {{\mathfrak{q}}}$. Then, $\frac{(sx)^k}{t^{j+i}} = \frac{s^k}{t^i} \cdot \frac{x^k}{t^j}$, the product of an element in $(T^{-1}S)_0$ with an element in ${{\mathfrak{q}}}$, and therefore $sx \in {{\mathfrak{p}}}_{i+j}$ so that ${{\mathfrak{p}}}$ is a graded ideal in $S$.
Furthermore, ${{\mathfrak{p}}}\cap T = \emptyset$: if not, choose a $t \in {{\mathfrak{p}}}_i \cap T$. So, there exists a $k >0$ and an $s \in T_k$ such that $\frac{t^k}{s^i} \in {{\mathfrak{q}}}$. However, the product $\frac{s^i}{t^k}\cdot \frac{t^k}{s^i}$ must also be in ${{\mathfrak{q}}}$, which contradicts that $1 \not \in {{\mathfrak{q}}}$. Since $T$ has at least one nonzero element of positive degree, and ${{\mathfrak{p}}}\cap T = \emptyset,$ $S_+ \not \subseteq {{\mathfrak{p}}}$.
To verify that ${{\mathfrak{p}}}$ is prime, suppose that $f \in S_n$, $g\in S_m$, and $fg \in {{\mathfrak{p}}}_{n+m}$. There exists a $k >0$ and $t \in T_k$ such that $\frac{(fg)^k}{t^{m+n}} \in {{\mathfrak{q}}}$. Now, $\frac{(fg)^k}{t^{m+n}} = \frac{f^k}{t^n} \cdot \frac{g^k}{t^m} \in {{\mathfrak{q}}}$, and by primality of ${{\mathfrak{q}}}$, together with the definition of ${{\mathfrak{p}}}$, either $f \in {{\mathfrak{p}}}_n$ or $g\in {{\mathfrak{p}}}_m$.
We have established that ${{\mathfrak{p}}}\in Proj(S)$, and it only remains to show that ${{\mathfrak{q}}}= (T^{-1}{{\mathfrak{p}}})_0$. Suppose that $\xi \in {{\mathfrak{q}}}$, so $\xi$ may be written as $\frac{x}{t}$, with $x \in S_i$, $t \in T_i$. If $i >0$ , then $x^i/t^i = \xi^i \in {{\mathfrak{q}}}$, so $x \in {{\mathfrak{p}}}_i$ by definition, and $ \xi = \frac{x}{t} \in (T^{-1}{{\mathfrak{p}}})_0$. If $i = 0$, then $\frac{t}{1}\xi = \frac{x}{1} \in {{\mathfrak{q}}}$, so $x \in {{\mathfrak{p}}}_0$ and $\xi =\frac{x}{t} \in (T^{-1}{{\mathfrak{p}}})_0$.
On the other hand, suppose that $\frac{x}{t} \in (T^{-1}{{\mathfrak{p}}})_0$, $x \in {{\mathfrak{p}}}_i$, $t \in T_i$. By definition, there exists a $k >0$, and an $s \in T_k$ such that $\frac{x^k}{s^i} \in {{\mathfrak{q}}}$. Then, $\frac{s^i}{t^k}\cdot \frac{x^k}{s^i} \in {{\mathfrak{q}}}$ since $\frac{s^i}{t^k} \in (T^{-1}S)_0$ and $\frac{x^k}{s^i} \in {{\mathfrak{q}}}$. Of course, $\frac{s^i}{t^k}\cdot \frac{x^k}{s^i} = \left(\frac{x}{t}\right)^k$, and by primality of ${{\mathfrak{q}}}$, $\frac{x}{t} \in {{\mathfrak{q}}}$.
Thus we have
Suppose $S$ is a positively graded ring, and $T$ is a GMCS in $S$ containing at least one element of positive degree. Then, there exists a one-to-one inclusion-preserving correspondence $$\{{{\mathfrak{p}}}\in \mbox{Proj}(S) \;| \;{{\mathfrak{p}}}\cap T = \emptyset \} \leftrightarrow
\{{{\mathfrak{q}}}\in Spec (T^{-1}S)_0 \};$$ this correspondence takes ${{\mathfrak{p}}}$ to $(T^{-1}{{\mathfrak{p}}})_0.$
Using the correspondence of the above theorem, we have, as expected,
Suppose $S$ is a positively graded ing of finite Krull dimension. Let ${{\mathfrak{p}}}\in Proj(S)$. Then $S_{({{\mathfrak{p}}})}$ is a local Noetherian ring of finite Krull dimension and $$\dim(S_{({{\mathfrak{p}}})}) = {{^*\mathrm{ht}}}({{\mathfrak{p}}}) = {{^*\mathrm{ht}}}({{\mathfrak{p}}}_{[{{\mathfrak{p}}}]})= {{\phantom{}^*\mathrm{dim}}}(S_{[{{\mathfrak{p}}}]})= \dim(S_{[{{\mathfrak{p}}}]}) -1.$$
### Poincaré series and dimension for positively graded rings
The ring $\mathbb{Z}[[t]][t^{-1}]$ is denoted by $\mathbb{Z}((t))$; thus an element of $\mathbb{Z}((t))$ is a formal Laurent series $f(t)$ with integer coeffiicients; there always exists an $n \in \mathbb{Z}$ with $t^nf(t) \in \mathbb{Z}[[t]].$
In order to define the Poincaré series for a graded abelian group $M$, we assume, in addition, that 1) each $M_j$ is a finitely generated module over a ordinary commutative Artinian ring $S_0$ and 2) $M_j = 0$ for $j <<0$. Whenever we write down a Poincaré series for a graded abelian group $M$, we will make these assumptions.
For example, if $S$ is a positively graded Noetherian ring and $M \in {\mathfrak{grmod}}(S)$, then as long as $S_0$ is Artinian, 1) and 2) hold.
Suppose that $M$ is a graded abelian group and $S_0$ is an Artinian ring satisfying 1) and 2) above. Then the **Poincaré series** of $M$ is the formal Laurent series with integer coefficients $$P_M(t) = \sum_{i \in \mathbb{Z}} \ell_{S_0}(M_i)t^i,$$ where $\ell_{S_0}(M_i)$ is the length of the finitely generated module $M_i$ over the Artinian ring $S_0$.
Sometimes the Poincaré series is called the Hilbert series, or the Hilbert-Poincaré series.
\[theorem Hilbert-Serre\](The Hilbert-Serre Theorem) [@AtMac] Let $S$ be a positively graded Noetherian ring with $S_0$ Artinian, $M \in {\mathfrak{grmod}}(S)$. Suppose that $S$ is generated as a $S_0$-algebra by elements $x_1, \ldots , x_n$ of positive degrees $d_1, \ldots , d_n$. Then, $$P_M(t) = \frac{q(t)}{\prod_{i=1}^{n}(1-t^{d_i})},$$ where $q(t) \in \mathbb{Z}[t, t^{-1}]$.
Furthermore, if $M$ has no elements of negative degree, $q(t) \in \mathbb{Z}[t]$.
From now on, we assume that $S$ is a positively graded Noetherian ring of finite (Krull) dimension, with $S_0$ Artinian.
Some facts to note about Poincaré series:
- Let $\hat{S}$ be another positively graded ring. Assume also that the graded abelian group $M$ is in ${\mathfrak{grmod}}(S)$, $S_0 = \hat{S}_0$ is Artinian and $M$ is also a graded $\hat{S}$-module (but not necessarily finitely generated as such). Then whether we consider $M$ as an $S$-module or as a $\hat{S}$-module, its Poincaré series does not change. For example, let $y_1, \ldots, y_s \in S_+$ be homogeneous. Define $\hat{S} = S_0\langle y_1, \ldots, y_s \rangle$ to be the graded subring of $S$ generated by $S_0$ and $y_1, \ldots, y_s$. Now, whether we consider $M$ as an $S$-module, or as an $\hat{S}$-module, its Poincaré series is the same.
- If $M$ has a Poincaré series with respect to $S$, then so does $M(n)$, for every $n \in \mathbb{Z}$, and $$P_{M(n)}(t) = t^{-n}P_M(t).$$
- If $0 \rightarrow P \rightarrow M \rightarrow N \rightarrow 0$ is a short exact sequence in ${\mathfrak{grmod}}(S)$, then $$P_M(t) = P_P(t) + P_N(t).$$
- If $M, N \in {\mathfrak{grmod}}(S)$, then $P_{M \otimes_{S_0}N}(t) = P_M(t)P_N(t),$ if $M \otimes_{S_0} N$ is given the usual grading.
We end this section with a brief discussion of the connection of the Poincaré series with (Krull) dimension.
Let $M$ be in ${\mathfrak{grmod}}(S)$, $M \neq 0$.
- If $M \in {\mathfrak{grmod}}(S)$, $d_1(M)$ is the least $j$ such that there exist positive integers $f_1, \ldots, f_j$ with $$(\prod_{i=1}^j (1-t^{f_i}))P_M(t) \in \mathbb{Z}[t, t^{-1}].$$ By definition, $d_1(M)= 0$ if and only if $P_M(t)$ is in $\mathbb{Z}[t, t^{-1}].$ Note that the Hilbert-Serre theorem shows that $d_1(M) < \infty$; also, $d_1(M)$ is the order of the pole at $t=1$ for $P_M(t).$
- $s_1(M)$ is the least $s$ such that there exist homogeneous elements $y_1, \ldots, y_s \in S_+$ with $M$ finitely generated over $S_0 \langle y_1, \ldots , y_s \rangle \subseteq S$. By definition, $s_1(M) = 0$ if and only if $M$ is a finitely generated graded $S_0$-module. Note that for a finite set of homogeneous generators for $S_+$, the number of elements in that set is an upper bound for $s_1(M)$.
- $d_1(0)=s_1(0) = -\infty.$
Note that if $n \in \mathbb{Z}$, then $d_1(M(n)) = d_1(M)$, since $P_{M(n)}(t) = t^{-n}P_M(t)$. Also, $s_1(M(n)) = s_1(M)$ by definition. The following theorem and proposition could be considered “folklore", but the paper of Smoke cited is, as far as we know, the first appearance of these statements in the literature.
**Smoke’s Dimension Theorem** (Theorem 5.5 of [@sm])\
\[theorem Smoke dimension\] Suppose that $S$ is a positively graded ring of finite Krull dimension, with $S_0$ Artinian. Let $M \in {\mathfrak{grmod}}(S)$. If $d_1(M), s_1(M)$ are defined as above, we have $$d_1(M) = s_1(M) = {{\phantom{}^*\mathrm{dim}}}_S(M) < \infty \text{ .}$$
Under the hypotheses of the theorem, we’ve already seen that ${{\phantom{}^*\mathrm{dim}}}_S(M) = \dim_S(M)$, so all of these numbers equal $\dim_S(M)$ as well.
Graded ideals of definition and graded systems of parameters
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Returning to the more general case a graded ring $A$, not necessarily positively graded, we define analogously to Serre, a graded ideal of definition and a graded system of parameters.
Let $A$ be a graded ring and $M \in {\mathfrak{grmod}}(A)$. A proper, graded ideal ${{\mathcal{I}}}$ of $A$ such that $*\ell_A(M/{{\mathcal{I}}}M) <\infty$ is called a graded ideal of definition for $M$ (a GIOD for $M$).
(This is a little different from Serre’s definition [@se] of an ideal of definition in the ungraded case.) Lemmas 2.4 and 3.9 say that ${{\mathcal{I}}}$ is a graded ideal of definition for $M$ if and only if all graded primes containing ${{\mathcal{I}}}+ Ann_A(M)$ are \*maximal.
Let $A$ be a Noetherian ring of finite Krull dimension, and also assume that $A$ is either a positively graded ring or a \*local ring with unique \*maximal ideal $\mathcal{N}$. Define ${{\mathfrak{m}}}$ to be the graded ideal $A_+$ in the first case, and the ideal $\mathcal{N}$ in the second. Suppose $M \neq 0$ is in ${\mathfrak{grmod}}(A)$. A sequence $y_1, \ldots , y_D$ of homogeneous elements of ${{\mathfrak{m}}}$, such that
- the graded $A$-module $M/(y_1, \ldots, y_D)M$ has finite \*length over $A$ and
- $D = {{\phantom{}^*\mathrm{dim}}}_A(M)$
is called a graded system of parameters (GSOP) for the $A$-module $M$.
Note that by definition, a GSOP (or a GIOD) for $M$ is also a GSOP (resp. GIOD) for $M(n)$, for every $n \in \mathbb{Z}$ (and vice versa).
In the positively graded case, an alternative characterization of some GIODs (and thus some GSOPs) is given by:
\[lemma finite generation gsop\] Suppose that $S$ is a positively graded Noetherian ring of finite Krull dimension, with $S_0$ Artinian, and $y_1, \ldots, y_u$ are homogeneous elements of $S_+$. Let $M \in {\mathfrak{grmod}}(S)$. Then, $M/(y_1, \ldots, y_u)M$ has finite \*length over $S$ if and only if $M$ is a finitely generated $S_0\langle y_1, \ldots, y_u \rangle$-module.
Recall that $S_0\langle y_1, \ldots, y_u \rangle$ is the subring of $S$ generated by $S_0$ and $y_1, \ldots, y_u$. Let $t_0 \in \mathbb{Z}$ be chosen such that $M_j = 0$ for $j < t_0$. Suppose that $X \doteq M/(y_1, \ldots, y_u)M$ has finite \*length over $S$. We’ve seen that there exists an integer $t_0 \leq t_1$ such that $X_j = 0$ if $j > t_1$. Using Lemma \[remark \*Samuel equals Samuel\], $M_j$ is finitely generated over $S_0$, so for every $j$ such that $t_0 \leq j \leq t_1$ we may choose a finite set $E_j$ of generators, possibly empty, for $M_j$ over $S_0$. Then, we prove that $M$ is generated by the finite set $E \doteq \cup_{j=t_0}^{t_1} E_j$ over $S_0\langle y_1, \ldots, y_u \rangle$; to do this we show, using induction on $\deg(z)$, that a homogeneous element $z$ of $M$ is in the submodule of $M$ generated by $E$ over $S_0\langle y_1, \ldots, y_u \rangle$. To start the induction, note that if $\deg(z) \leq t_1$, the claim is certainly true. Let $s > t_1$ and suppose that the inductive hypothesis holds for every homogeneous $w$ of degree strictly less than $s$. Let $z$ be a homogeneous element of $M$ of degree $s$. Since $s > t_1, (M/(y_1, \ldots, y_u)M))s = 0$, so $s \in (y_1, \ldots, y_u)M.$ Write $z = \sum_{\alpha = 1}^{u} y_{\alpha} m_{\alpha}$. Since $\deg(y_{\alpha}) + \deg(m_{\alpha}) = s$ for every $\alpha$ such that $y_{\alpha}m_{\alpha} \neq 0$, and $\deg(y_{\alpha}) >0$ for every such $\alpha$, we must have $\deg(m_{\alpha})<s$ for every $\alpha$ with $y_{\alpha}m_{\alpha} \neq 0$. Thus by induction, $m_{\alpha}$ is a linear combination of elements of $E$, with coefficients in $S_0\langle y_1, \ldots, y_u \rangle$. Clearly, then, so is $z$. Note that this part of the proof never used that $S_0$ is Artinian.
Conversely, suppose $M$ is generated by a finite set $E$ of nonzero homogeneous elements as a graded $S_0\langle y_1, \ldots, y_u \rangle$-module. Set ${{\mathcal{I}}}\doteq (y_1, \ldots, y_u)$. Let $t = \max \{ \deg(e) \mid e \in E \} $. Then, for $j > t$, $(M/{{\mathcal{I}}}M)_j = 0$: If $x \in M_j$, $j >t$, write $x = \sum_{e \in E} f_e e$, where $f_e \in S_0\langle y_1, \ldots, y_u \rangle$ is homogeneous. If $\deg(f_e) \neq 0$, and $f_e e \neq 0$, then $f_e e \in {{\mathcal{I}}}M$. Therefore, $x$ is equivalent to $
\sum_{f_e \neq 0, \deg(f_e) = 0} f_e e $ mod ${{\mathcal{I}}}M$. However, for every summand in this last sum, we must have $\deg(e) = \deg(x) >t$ if $f_e e \neq 0$, a contradiction. Thus, $x$ is equivalent to $0$ mod ${{\mathcal{I}}}M$. Lemma \[lemma \*length of components\] tells us that since $S_0$ is Artinian, $*\ell_S(M/{{\mathcal{I}}}M) < \infty$.
The following proposition is another part of the “folklore" knowledge, but the citation is the first that we know of in the literature.
(Theorem 6.2 of [@sm]) \[proposition algebraic independence\] Suppose that $S$ is a positively graded Noetherian ring of finite Krull dimension, with $S_0$ Artinian, and $M \neq 0$ is in ${\mathfrak{grmod}}(S)$. Let $D(M)\doteq d_1(M) = s_1(M) = {{\phantom{}^*\mathrm{dim}}}_S(M) = \dim_S(M)$, so Theorem \[theorem Smoke dimension\] and Lemma \[lemma finite generation gsop\] tell us that a GSOP exists for $M$. Moreover, $D(M)$ is the length of any GSOP and if $y_1, \ldots ,y_{D(M)} \in S_+$ is a GSOP for $M$, $y_1, \ldots, y_{D(M)}$ are algebraically independent over $S_0$.
Multiplicities for graded modules
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In this section, we define the \*Samuel multiplicity and \*Koszul multiplicity for modules in ${\mathfrak{grmod}}(A)$. All of this work is done analogously to Serre [@se], and since we only give brief discussions/proofs here, if the reader does not have in mind the development of multiplicities in [@se], it’s advised to have a copy of [@se] at hand. Another treatment of multiplicity in the graded case is given in [@pr].
The \*Samuel multiplicity is explored using the tools of the graded category which we have developed thus far: \*length, \*dimension, graded localization, etc. The \*Koszul multiplicity is defined using tools from homological algebra. In each case, to adapt the theory from the ungraded case, we have the added complication of our objects being bi-graded - the internal grading that the module inherits from ${\mathfrak{grmod}}(A)$, and an external grading coming from either the associated graded module in the case of Samuel multiplicities, or the complex grading for Koszul multiplicities. Keeping track of the bi-grading, all morphisms respect both gradings, and as one might expect, the bi-grading does not cause any problems. We show, as in the ungraded case, the two multiplicities (\*Koszul and \*Samuel) agree.
Finally, we show that the graded multiplicity theory agrees with the ungraded theory by simply forgetting the grading, when we work over positively graded rings. This is to be expected, for we have shown that \*length and length agree in the positively graded case.
In the following, we will consider filtrations of $A$-modules; as previously, we will use upper indices for filtrations, whether working in a graded or an ungraded category. We’ll use notations like $M^{\bullet}$ or often $\mathcal{F}(M)$ for filtrations of $M$ by $A$-modules. Filtrations will be indexed in different ways, according to convention.
Suppose that ${{\mathcal{I}}}$ is an ideal in $A$. A filtration $\mathcal{F}(M)$ with $\mathcal{F}^{i+1}(M) \subseteq \mathcal{F}^i(M)$ for every $i \geq 0$, is called ${{\mathcal{I}}}$-bonne if ${{\mathcal{I}}}\mathcal{F}^n(M) \subseteq \mathcal{F}^{n+1}(M)$, for every $n\geq 0 $, and with equality for $n>>0$.
If ${{\mathcal{I}}}$ is an ideal in $A$, the ${{\mathcal{I}}}$-adic filtration $\cdots \subseteq {{\mathcal{I}}}^{j+1}M \subseteq {{\mathcal{I}}}^jM \subseteq \cdots \subseteq {{\mathcal{I}}}M \subseteq M$ is ${{\mathcal{I}}}$-bonne.
If $A$ is a graded ring, and $M$ a graded $A$-module, a filtration $\mathcal{F}(M)$ is graded if and only if all the submodules $\mathcal{F}^j(M)$ are graded submodules; if ${{\mathcal{I}}}$ is a graded ideal, the definition of an ${{\mathcal{I}}}$-bonne graded filtration remains the same as in the ungraded case.
The Ungraded Case
-----------------
We begin by outlining the procedure for defining the Hilbert and Samuel polynomials in the ungraded case (see [@se] for full discussion/proofs).
Suppose that $H$ is a positively graded ring with $H_0$ Artinian, and that $H$ is generated as an $H_0$-algebra by a finite number of homogeneous elements $x_1, \ldots, x_u$ in $H_1$. Such a ring $H$ is then called a “standard" graded ring. For any finitely generated, positively graded $H$-module $M$, $M_n$ is a finitely generated $H_0$-module for every $n$. Since $H_0$ is Artinian, the Hilbert function, $n \mapsto \ell_{H_0}(M_n)$, is defined for all integers $n \geq 0$. Using induction on the number of generators for $H$ as an $H_0$-algebra, and the additivity of length over exact sequences, one may prove that the Hilbert function is polynomial-like; in other words there is a unique polynomial $f$ with rational coefficients such that $f(n) = \ell_{H_0}(M_n)$ for all $n$ sufficiently large. The polynomial describing the function $n \mapsto \ell_{H_0}(M_n)$ is called the Hilbert polynomial of $M$ (over $H$).
Recall the delta notation from the theory of polynomial-like functions: if $f$ is a function with an integer domain, then $\Delta f$ is the function defined by $\Delta f(n) \doteq f(n+1)-f(n)$. Then, we know that $f$ is polynomial-like if and only if $\Delta f$ is polynomial-like. We may iterate the operator “$\Delta$" on integer domain functions, obtaining operators $\Delta^r$, for $r \geq 0$.
We review the definition of a Samuel polynomial and Samuel multiplicity in the ungraded case. So, for this and the next two paragraphs, suppose that $A$ is an ungraded Noetherian ring, $M$ an ungraded finitely generated $A$-module, and ${{\mathcal{I}}}$ is an ideal of $A$ such that $M/{{\mathcal{I}}}M$ has finite length over $A$; this last is true if and only if $V({{\mathcal{I}}}+ Ann_A(M))$ consists of a finite number of maximal ideals in $A$.
Summarizing the discussion in [@se], given an ideal ${{\mathcal{I}}}$ with $\ell_A(M/{{\mathcal{I}}}M) < \infty$ and an ${{\mathcal{I}}}$-bonne filtration $\mathcal{F}(M)$, $\ell_A(M/\mathcal{F}^n(M))$, is well-defined. Now, $V(M/{{\mathcal{I}}}M) = V(Ann_A(M) + {{\mathcal{I}}})$ consists of a finite number of maximal ideals; without loss of generality we may assume that $Ann_A(M) = 0$ and $V(M/{{\mathcal{I}}}M) = V({{\mathcal{I}}})$ consists of a finite number of maximal ideals, so that $A/{{\mathcal{I}}}$ is an Artinian ring. The positively graded associated graded module $gr(M) = \oplus_{n \geq 0} \mathcal{F}^n(M)/\mathcal{F}^{n+1}(M)$ is finitely generated over the positively graded associated graded ring $gr(A) = \oplus_{n \geq 0} {{\mathcal{I}}}^n/{{\mathcal{I}}}^{n+1}$. Furthermore $gr(A)$ is generated over $gr(A)_0 = A/{{\mathcal{I}}}$, an Artinian ring, by elements of degree one, and the Hilbert polynomial for $gr(M)$ as a $gr(A)$-module exists.
Then, $n \mapsto \ell_A(M/\mathcal{F}^{n+1}(M)) - \ell_A(M/ \mathcal{F}^n(M)) = \ell_A(\mathcal{F}^n(M)/\mathcal{F}^{n+1}(M))$ is polynomial-like, and the general theory of polynomial-like functions tells us that the Samuel function $n \mapsto \ell_A(M/\mathcal{F}^n(M))$ is also polynomial-like. The polynomial describing this function is called the Samuel polynomial $p(M, \mathcal{F}, n)$ of the $A$-module $M$ with respect to the filtration $\mathcal{F}$ and the ideal $\mathcal{I}$.
The Graded Case
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We make new, similar definitions in the graded category, now assuming $A$ is a graded Noetherian ring and $M \in {\mathfrak{grmod}}(A)$. We do not assume that $A$ is positively graded, nor that it is generated by elements of degree 1.
To define the \*Hilbert polynomial, start with certain bigraded objects: Suppose that $H$ is a bigraded ring such that $H_{i,j} = 0$ for $i <0$, $H_{0,*} \doteq \oplus_{j \in \mathbb{Z}} H_{0,j}$ is a graded ring that is \*Artinian and $H$ is generated as an bigraded algebra over the graded ring $H_{0,*}$ by a finite number of elements in $H_{1,*} \doteq \oplus_{j \in \mathbb{Z}} H_{1,j}$. $M$ is taken to be a bigraded $H$-module such that $M_{i,j} = 0$ for $i<0$ and $M$ is generated as an $H$-module by a finite number of bi-homogeneous elements. Then, for each $k \geq 0$, $M_{k,*} \doteq \oplus_{j \in \mathbb{Z}} M_{k,j}$ is a finitely generated graded $H_{0,*}$-module, so $*\ell_{H_{0,*}}(M_{k,*})$ is well-defined for every $k \geq 0$. Furthermore, the function $k \mapsto *\ell_{H_{0,*}}(M_{k,*})$ is polynomial like. To see this, following the argument in [@se] for the ungraded case, use induction on the number of bihomogeneous generators (taken from $H_{1,*}$) for $H$ as an $H_{0,*}$-algebra, and additivity of $*\ell$ over exact sequences of graded modules. The exact sequence used in Theorem II.B.3.2 of [@se] becomes an exact sequence of graded modules, with middle map multiplication by a generator of bidegree $(1,d)$: $$0 \rightarrow N_{n,*} \rightarrow M_{n,*}(-d) \rightarrow M_{n+1,*} \rightarrow R_{n+1,*} \rightarrow 0;$$ there is a shift for the second graded degree in the second term, and the rest of proof is the same otherwise with length replaced by \*length. Furthermore, using the argument of Theorem II.B.3.2 of [@se] for the ungraded case, we see that if $H$ is generated as a bigraded algebra over $H_{0,*}$ by $r$ elements of bidegree $(1,-)$, then the \*Hilbert polynomial has degree less than or equal to $r-1$.
In the same spirit, we define a \*Samuel function by making appropriate changes to consider the grading, as follows.
Suppose $A$ is a graded ring, ${{\mathcal{I}}}$ is a graded ideal in $A$ and $\mathcal{F}(M)$ is a graded ${{\mathcal{I}}}$-bonne filtration of $M$.
Note that if ${{\mathcal{I}}}$ is a graded ideal in $A$, $\mathcal{F}(M)$ is a graded ${{\mathcal{I}}}$-bonne filtration of $M$, and $d \in \mathbb{Z}$ is a fixed integer, we may shift degrees by $d$ throughout the filtration yielding an ${{\mathcal{I}}}$-bonne filtration $\mathcal{F}(d)$ of $M(d)$: $\mathcal{F}(d)^n(M(d)) \doteq (\mathcal{F}^n(M))(d)$. To see that this filtration is also ${{\mathcal{I}}}$-bonne, just compute that ${{\mathcal{I}}}(\mathcal{F}(d)^n(M(d))) = ({{\mathcal{I}}}\mathcal{F}^n( M))(d)$ as follows. Suppose that $x \in ({{\mathcal{I}}}\mathcal{F}^n(M))(d)_j =( {{\mathcal{I}}}\mathcal{F}^n(M))_{d+j}$, so that $x = \sum_t \alpha_t m_t$, where $\alpha_t \in {{\mathcal{I}}}, m_t \in \mathcal{F}^n(M)$ are all homogeneous and $\deg(\alpha_t) + \deg(m_t) = d+j$ whenever $\alpha_tm_t \neq 0$. Thus, $\deg(m_t) = d + (j-\deg(\alpha_t))$ for all such $t$, so that $m_t \in (\mathcal{F}(d)^n)(M(d))_{j-\deg(\alpha_t)}$, $\alpha_t m_t \in {{\mathcal{I}}}(\mathcal{F}(d)^n(M(d)))_j$ for every $t$ and $x \in {{\mathcal{I}}}(\mathcal{F}(d)^n(M(d)))_j.$ The converse is similarly proved. In particular, the $d$-suspension of the ${{\mathcal{I}}}$-adic filtration on $M$ is the ${{\mathcal{I}}}$-adic filtration on $M(d)$.
Given a GIOD ${{\mathcal{I}}}$ for $M$, and a graded ${{\mathcal{I}}}$-bonne filtration $\mathcal{F}(M)$, $*\ell_A(M/\mathcal{F}^n(M)) < \infty$. Passing without loss of generality to the case $Ann_A(M) = 0$ as in the ungraded case, we see that $A/{{\mathcal{I}}}$ is a \*Artinian ring and that the associated bigraded module $gr(M) = \oplus_{n \geq 0} \mathcal{F}^n(M)/\mathcal{F}^{n+1}(M)$, where $gr(M)_{n,j} \doteq \mathcal{F}^n(M)_j/\mathcal{F}^{n+1}(M)_j$, is finitely generated over the associated bigraded ring $gr(A) = \oplus_{n \geq 0} {{\mathcal{I}}}^n/{{\mathcal{I}}}^{n+1}$ (where $gr(A)_{n,j} \doteq ( {{\mathcal{I}}}^n)_j/({{\mathcal{I}}}^{n+1})_j$). Note that $gr(A)$ is generated by elements of bidegree $(1,-)$, as an algebra over the \*Artinian graded ring $A/{{\mathcal{I}}}$ and thus the \*Hilbert polynomial for $gr(M)$ as a $gr(A)$-module exists.
Suppose that ${{\mathcal{I}}}$ is a GIOD for $M \in {\mathfrak{grmod}}{A}$ and $\mathcal{F}$ is an ${{\mathcal{I}}}$-bonne filtration of $M$. The \*Samuel function with respect to $\mathcal{F}$ and ${{\mathcal{I}}}$ is defined on the nonnegative integers by $n \mapsto *\ell_A(M/\mathcal{F}^{n}(M))$.
Since $*\ell_A(M/\mathcal{F}^{n+1}(M)) - *\ell_A(M/\mathcal{F}^n(M)) = *\ell_A(\mathcal{F}^n(M)/\mathcal{F}^{n+1}(M))$, the $\Delta$ operator applied to the \*Samuel function is polynomial-like, so
If $M \in {\mathfrak{grmod}}(A)$ and ${{\mathcal{I}}}$ is a GIOD for $M$, the \*Samuel function for the graded ${{\mathcal{I}}}$-bonne filtration $\mathcal{F}(M)$ is polynomial-like.
To set notation, the polynomial that calculates $*\ell_A(M/\mathcal{F}^n(M))$ for $n >>0$ will be called $*p(M,\mathcal{F},n)$, and if $\mathcal{F}$ is the ${{\mathcal{I}}}$-adic filtration on $M$, we will instead write $*p(M,{{\mathcal{I}}},n).$
The following lemma incorporates graded versions of results in II.B.4 of [@se].
\[lemma Samuel props\] Suppose that $M \in {\mathfrak{grmod}}(A)$ and $\mathcal{F}(M)$ is a graded ${{\mathcal{I}}}$-bonne filtration of $M$ for some GIOD ${{\mathcal{I}}}$ for $M$. Then
1. For every $d \in \mathbb{Z}$, ${{\mathcal{I}}}$ is a GIOD for $M(d)$, $\mathcal{F}(d)(M(d))$ is an ${{\mathcal{I}}}$-bonne filtration of the graded $A$-module $M(d)$ and $*p(M(d), \mathcal{F}(d), n) = *p(M,\mathcal{F},n)$.
2. $*p(M, {{\mathcal{I}}},n) = *p(M, \mathcal{F},n) + R(n)$, where $R$ is a polynomial with nonnegative leading coefficient and degree strictly less than that of the degree of $*p(M, {{\mathcal{I}}},n)$.
3. If $(Ann_A(M) + {{\mathcal{I}}})/Ann_A(M)$ is generated by $r$ homogeneous elements, then the degree of $*p(M, {{\mathcal{I}}},n)$ is less than or equal to $r$, and $\Delta^r(*p)$ is a constant less than or equal to $*\ell_A(M/{{\mathcal{I}}}M).$
4. If $0 \rightarrow N \rightarrow M \rightarrow P \rightarrow 0$ is a short exact sequence in ${\mathfrak{grmod}}(A)$, and ${{\mathcal{I}}}$ is a GIOD for $M$, then ${{\mathcal{I}}}$ is a GIOD for both $N$ and $P$ and $$*p(M, {{\mathcal{I}}}, n) + R(n) = *p(N, {{\mathcal{I}}}, n) + *p(P, {{\mathcal{I}}}, n),$$ where $R$ is a polynomial with nonnegative leading coefficient and degree strictly less than that of $*p(N, {{\mathcal{I}}},n)$.
5. If ${{\mathcal{I}}}$ and $\hat{{{\mathcal{I}}}}$ are two GIODs for $M$ such that $*V({{\mathcal{I}}}+ Ann_A(M)) = *V(\hat{{{\mathcal{I}}}} + Ann_A(M))$, then the degree of $*p(M, {{\mathcal{I}}},n)$ equals the degree of $*p(M, \hat{{{\mathcal{I}}}}, n).$
We’ve already noted that ${{\mathcal{I}}}(\mathcal{F}(d)^n(M(d))) = ({{\mathcal{I}}}\mathcal{F}^n( M))(d)$; so that $\mathcal{F}(d)(M(d))$ is an ${{\mathcal{I}}}$-bonne filtration of $M(d)$. The \*Samuel polynomials are identical since $M(d)/\mathcal{F}(d)^n(M(d)) = M(d)/ (\mathcal{F}^n(M)(d))=( M/(\mathcal{F}^n(M))(d),$ for every $n$. The proofs of b)-e) follow exactly the proofs in Section II.B.4 of Lemma 3 and Propositions 10 and 11 of [@se], adapted with clear notational changes to the graded case, and are not given here.
Since we will be interested in the leading coefficient of \*Samuel polynomials, b) above tells us that we may as well just consider ${{\mathcal{I}}}$-adic filtrations and suppress all talk about ${{\mathcal{I}}}$-bonne filtrations; the need to consider general ${{\mathcal{I}}}$-bonne filtrations $\mathcal{F}$ is indicated in the proof of d), even though we haven’t given it, since the proof of d) uses the Artin-Rees lemma, which also holds in the graded context.
Suppose that $M \in {\mathfrak{grmod}}(A)$, ${{\mathcal{I}}}$ is a GIOD for $M$ and $d \in \mathbb{Z}$, $d \geq \deg(*p(M,{{\mathcal{I}}},n))$. The **\*Samuel multiplicity** of $M$ with respect to ${{\mathcal{I}}}$ is defined as $$*e(M,{{\mathcal{I}}},d) \doteq\Delta^d (*p(M,{{\mathcal{I}}},n)).$$
By properties of the finite difference operator $\Delta$, we see that $*e(M,{{\mathcal{I}}},d) = 0$ whenever $d > \deg(*p(M,{{\mathcal{I}}},n))$. When $d= \deg(*p(M,{{\mathcal{I}}},n))$, $*e(M,{{\mathcal{I}}},d)$ is a positive integer, and one may compute that $$*p(M,{{\mathcal{I}}},n) = \frac{*e(M,{{\mathcal{I}}},d)}{d!} n^{d} + \text{lower order terms.}$$
Using Lemma \[lemma Samuel props\]d), we see that if $$0 \rightarrow N \rightarrow M \rightarrow P \rightarrow 0$$ is a short exact sequence in ${\mathfrak{grmod}}(A)$, ${{\mathcal{I}}}$ is a GIOD for $M$ and $d \geq \deg(*p(M, {{\mathcal{I}}}, n))$, then both $*e(N,{{\mathcal{I}}}, d)$ and $*e(P,{{\mathcal{I}}},d)$ exist and $$*e(M, {{\mathcal{I}}}, d) = *e(N, {{\mathcal{I}}}, d) + *e(P, {{\mathcal{I}}},d).$$
Therefore, using Lemma \[lemma Samuel props\]a) as well, we have
\[corollary sum decomp for multiplicity\]Suppose that $M \in {\mathfrak{grmod}}(A)$, ${{\mathcal{I}}}$ is a GIOD for $M$ and $M^{\bullet}$ is a graded filtration of $M$ such that $0 = M^0 \subset M^1 \subset \cdots M^{N-1} \subset M^N = M,$ and, for each $N \geq i \geq 1$, there are graded prime ideals ${{\mathfrak{p}}}_i$ in $A$, integers $d_i$ and graded isomorphisms of $A$-modules $(A/{{\mathfrak{p}}}_i )(d_i) \cong M^{i}/M^{i-1}$. Then,
- ${{\mathcal{I}}}$ is a GIOD for $A/{{\mathfrak{p}}}_i$ and $*p(A/{{\mathfrak{p}}}_i, {{\mathcal{I}}},n)$ exists, for $1 \leq i \leq N$.
- If $D \doteq \max\{ \deg(*p(A/{{\mathfrak{p}}}_i, {{\mathcal{I}}}, n)) \doteq d_i \mid 1 \leq i \leq N\}$ and $\mathcal{D}(M^{\bullet}) \doteq \{{{\mathfrak{p}}}_j \mid d_j = D \}$, $$*e(M, {{\mathcal{I}}}, D) = \sum_{{{\mathfrak{p}}}\in \mathcal{D}(M^{\bullet})} n_{{{\mathfrak{p}}}}(M^{\bullet}) (*e(A/{{\mathfrak{p}}}, {{\mathcal{I}}}, D)),$$ where $n_{{{\mathfrak{p}}}}(M^{\bullet})$ is equal to the number of times $A/{{\mathfrak{p}}}$, possibly suspended, occurs as an $A$-module isomorphic to a subquotient of the filtration $M^{\bullet}$. Furthermore, all of the integers on both sides of the equation are strictly positive.
Finally, we point out some scenarios in which \*Samuel multiplicities equal those computed in the ungraded category.
\[theorem star e equals e, pos graded\] **The positively graded case.** Suppose that $S$ is a positively graded Noetherian ring with $S_0$ Artinian, $M \in {\mathfrak{grmod}}(S)$ and ${{\mathcal{I}}}$ a GIOD for $M$. Then, the “ungraded" Samuel polynomial $p(M, {{\mathcal{I}}}, n)$ exists, $ *p(M, {{\mathcal{I}}}, n) = p(M, {{\mathcal{I}}}, n)$ and, for every $d$, $*e(M, {{\mathcal{I}}}, d) = e(M, {{\mathcal{I}}}, d).$
Lemma \[lemma pos graded length equals \*length\] tells us that, when we forget the grading, ${{\mathcal{I}}}$ has the property that $\ell_S(M/\mathcal{I}^nM) = *\ell_S(M/\mathcal{I}^nM) < \infty$. Therefore, the “ungraded" Samuel polynomial $p(M, {{\mathcal{I}}}, n)$ exists ($p(M, {{\mathcal{I}}}, n)$ is computed after forgetting the grading) and $ *p(M, {{\mathcal{I}}}, n) = p(M, {{\mathcal{I}}}, n)$ . So, if $d \geq \deg(*p(M, {{\mathcal{I}}},n)) = \deg(p(M, {{\mathcal{I}}}, n))$, $*e(M, {{\mathcal{I}}}, d)$ is the exact same multiplicity $e(M, {{\mathcal{I}}}, d)$ defined in [@se], after forgetting the grading.
\[theorem star e equals e, element of degree 1\] **The \*local case in which $A-\mathcal{N}$ has a homogeneous element of degree 1.** Suppose that $(A, \mathcal{N})$ is a \*local Noetherian ring, $M \in {\mathfrak{grmod}}(A)$ and $A-\mathcal{N}$ has a homogeneous element of degree 1. Then, ${{\mathcal{I}}}_0$ is such that $\ell_{A_0}(M_0/({{\mathcal{I}}}_0)^n M_0) < \infty$, $*p(M, {{\mathcal{I}}},n) = p(M_0, {{\mathcal{I}}}_0 ,n)$ and for every $d$, $e(M_0, {{\mathcal{I}}}_0,d) = *e(M, {{\mathcal{I}}}, d).$
First, note that for any graded ideal $\mathcal{J}$ in $A$, and every $X \in {\mathfrak{grmod}}(A)$, it turns out in this case that $(\mathcal{J}X)_0 = \mathcal{J}_0 X_0$: the containment “$\supseteq$" is clear. For the remaining containment, let $T \in A-\mathcal{N}$ be any element of degree one. Now, every element of $(\mathcal{J}X)_0$ has the form $\sum_j a_jx_j$ where $a_j \in \mathcal{J}$ and $x_j \in X$ and $\deg(a_j) + \deg(x_j) = 0$ whenever $a_jx_j \neq 0$. However, $T$ is invertible in $A$, and $\sum_j a_jx_j = \sum_j (a_j T^{-\deg(a_j)})(T^{\deg(a_j)}x_j) \in \mathcal{J}_0X_0$. Using this result for powers of $\mathcal{J}$, and induction, we see that $$(\mathcal{J}^n)_0 = (\mathcal{J}_0)^n,$$ for every $n \geq 1$.
Recall that if $X \in {\mathfrak{grmod}}(A)$ is such that $*V(X) = \{\mathcal{N} \}$ (or equivalently, $*\ell_A(X) < \infty$), since $A-\mathcal{N}$ has a homogeneous element of degree 1, Proposition \[proposition maximal equals minimal\] tells us that for every $j$, $\ell_{A_0}(X_j) = *\ell_{A}(X)$ for every $j$.
Putting all this together, if ${{\mathcal{I}}}$ is a GIOD for $M$, and $X = M/{{\mathcal{I}}}^nM$, then we have $X_0 = (M/{{\mathcal{I}}}^nM)_0 = M_0/({{\mathcal{I}}}^n)_0 M_0 = M_0/({{\mathcal{I}}}_0)^nM_0$ and thus $\ell_{A_0}(M_0/{{\mathcal{I}}}_0^nM_0) = *\ell_A(M/{{\mathcal{I}}}^nM) < \infty$ for every $n$. Therefore, ${{\mathcal{I}}}_0$ is an ideal such that the ordinary Samuel polynomial $p(M_0, {{\mathcal{I}}}_0, n)$, constructed in the ungraded case for the $A_0$-module $M_0$, is defined and $*p(M, {{\mathcal{I}}},n) = p(M_0, {{\mathcal{I}}}_0, n)$. Therefore, in this case, for $d \geq \deg(*p(M, {{\mathcal{I}}},n)) = \deg(p(M, {{\mathcal{I}}}, n))$, $*e(M, {{\mathcal{I}}}, d)$ is equal to the multiplicity $e(M_0, {{\mathcal{I}}}_0, d)$ defined in the ungraded case.
We do not make a comparison if $(A, \mathcal{N})$ is a \*local Noetherian ring with no homogeneous elements of degree 1 in $A-\mathcal{N}$.
\*Dimension, \*Samuel polynomials and GSOPs for \*local rings
-------------------------------------------------------------
In this section, $A$ is a \*local Noetherian graded ring with unique \*maximal graded ideal $\mathcal{N}$. Here we present an analogue in the graded category to the fundamental theorem of dimension theory for local rings. This theorem shows the relationship between \*Krull dimension, graded systems of parameters, and the degree of the \*Samuel polynomial. Applying the results to the category ${\mathfrak{grmod}}(R)$, for $R$ positively graded and $R_0$ a field, we combine the fundamental dimension theorem for \*local rings to Smoke’s dimension theorem (\[theorem Smoke dimension\]). In this case, the order of the pole of the Poincare series at $t=1$, equals the measures from the fundamental \*local dimension theorem, which in turn equal the ungraded Krull dimension. This is summarized in corollary \[corollary fun thm dimension theory pos graded\]. Returning to the theory of multiplicities, we conclude the section with a sum decomposition of the \*Samuel multiplicity by minimal primes (corollary \[corollary \*Samuel sum decomposition\]).
We start by supposing that $\mathcal{I}$ is a GIOD for $M$; since $A$ is \*local, we’ve seen that this is true if and only if $*V(M/{{\mathcal{I}}}M) = \{ \mathcal{N} \}.$ The previous section shows that the degree of the \*Samuel polynomial of $M$ with respect to the ${{\mathcal{I}}}$-adic filtration does not depend on the choice of ${{\mathcal{I}}}$. We call this degree $*d(M)$. Of course, $\mathcal{N}$ is always a GIOD for $M$.
If $M \in {\mathfrak{grmod}}(A)$, $M \neq 0$, $*s(M)$ is defined to be the least $s$ such that there exist homogeneous elements $w_1, \ldots , w_s \in \mathcal{N}$ such that the graded $A$-module $M/(w_1, \ldots , w_s)M$ has finite \*length over $A$. Note that $*s(M) = 0$ if and only if $*\ell_A(M) < \infty$.
The fundamental theorem for \*local dimension theory is:
\[theorem fun thm graded dimension theory\] If $(A,\mathcal{N})$ is a \*local Noetherian ring and $M \in {\mathfrak{grmod}}(A)$, then $$*\dim_A(M) = *d(M) = *s(M).$$
The proof of this mimics the proof of the analogous theorem in the ungraded, local case given in [@se] in Section III.B.2, Theorem 1, but we give a sketch anyway. First, if $x$ is a homogeneous element of $\mathcal{N}$, let $_xM $ be the graded $A$-module consisting of all elements $m$ of $M$ such that $xm=0$. If $\deg(x) = d$, then there are short exact sequences in ${\mathfrak{grmod}}(A)$ $$0 \rightarrow _xM(-d) \rightarrow M(-d) \stackrel{\cdot x}{\rightarrow} xM \rightarrow 0,$$ $$0 \rightarrow xM \rightarrow M \rightarrow M/xM \rightarrow 0.$$ If ${{\mathcal{I}}}$ is a GIOD for $M$, it is also a GIOD for every module in the exact sequences above. Furthermore, the short exact sequences and Lemma \[lemma Samuel props\] say that $*p(_xM, {{\mathcal{I}}}, n) - *p(M/xM, {{\mathcal{I}}}, n)$ is a polynomial of degree strictly less than $*d(M)$. It’s straightforward to see that $*s(M) \leq *s(M/xM)+1$.
We may as well assume that the GIOD we are using to calculate $*d(M)$ is $\mathcal{N}$.
Next, set $\mathcal{D}(M)$ to be the (finite) set of all ${{\mathfrak{p}}}$ in $*V(M)$ with the property that ${{\phantom{}^*\mathrm{dim}}}_A(M) = {{\phantom{}^*\mathrm{dim}}}_A(A/{{\mathfrak{p}}}) = {{\phantom{}^*\mathrm{dim}}}(A/{{\mathfrak{p}}});$ it’s important to note that $\mathcal{D}(M)$ could also be defined as the set of all primes in $V(M)$ with $\dim_A(M) = \dim_A(A/{{\mathfrak{p}}})$ since the minimal elements in the sets $*V(M)$ and $V(M)$ are exactly the same. If a homogeneous element $x$ is not in any prime of $\mathcal{D}(M)$, then ${{\phantom{}^*\mathrm{dim}}}_A(M/xM) < {{\phantom{}^*\mathrm{dim}}}_A(M);$ this is true for exactly the same reason as in the ungraded case: $*V(M/xM) = *V((x) + Ann_A(M))$.
Finally, one proceeds to the proof by first arguing that ${{\phantom{}^*\mathrm{dim}}}_A(M) \leq *d(M)$, then $*d(M) \leq *s(M)$, and lastly, $*s(M) \leq {{\phantom{}^*\mathrm{dim}}}_A(M)$.
For the first inequality one uses induction on $*d(M)$. Note that $*d(M) = 0$ means that there is a $q$ such that $*\ell_A(M/\mathcal{N}^iM) = *\ell_A(M/\mathcal{N}^{i+1}M)$ for all $i \geq q$. But this forces $\mathcal{N}^qM = \mathcal{N}^{q+1}M$ and graded Nakayama’s lemma says that $\mathcal{N}^qM = 0$, so that $*V(M)$ has exactly one ideal, $\mathcal{N}$ in it. By definition, ${{\phantom{}^*\mathrm{dim}}}_A(M) = 0$. Supposing that $*d(M) \geq 1$, as in [@se], we reduce to the case $M = A/{{\mathfrak{p}}}$ for some graded prime ideal ${{\mathfrak{p}}}$ properly contained in $\mathcal{N}$. Taking a chain of graded prime ideals ${{\mathfrak{p}}}\doteq {{\mathfrak{p}}}_0 \subset {{\mathfrak{p}}}_1 \subset \cdots \subset {{\mathfrak{p}}}_n$ in $A$, we may suppose that $n \geq 1$, and thus may choose a homogeneous element $x$ in ${{\mathfrak{p}}}_1$ that is not in ${{\mathfrak{p}}}$. Since $x \notin {{\mathfrak{p}}}$, but $x \in {{\mathfrak{p}}}_1$, the chain ${{\mathfrak{p}}}_1 \subset \cdots \subset {{\mathfrak{p}}}_n$ corresponds to a chain of primes in $*V(M/xM)$. Since $M = A/{{\mathfrak{p}}}$, and $x \notin {{\mathfrak{p}}}$, $_xM = 0$, so that $*p(M/xM, \mathcal{N},n)$ has degree strictly less than $*d(M)$, and by induction, ${{\phantom{}^*\mathrm{dim}}}_A(M/xM) \leq *d(M/xM)$. Thus, $n-1 \leq {{\phantom{}^*\mathrm{dim}}}_A(M/xM) \leq *d(M)-1$ and $n \leq *d(M)$. This forces ${{\phantom{}^*\mathrm{dim}}}_A(M) \leq *d(M)$.
For the second inequality, if $x_1, \ldots, x_k$ is a list of homogeneous elements of $\mathcal{N}$ that generate a GIOD ${{\mathcal{I}}}$ for $M$, we must have that $*V({{\mathcal{I}}}+Ann_A(M))$ contains only $\mathcal{N}$, so that $*p(M, {{\mathcal{I}}}, n)$ and $*p(M, \mathcal{N}, n)$ have the same degree $*d(M)$. But, Lemma \[lemma Samuel props\] says that $*p(M, {{\mathcal{I}}}, n)$ has degree less than or equal to $k$. Thus, $*d(M) \leq *s(M)$.
For the third inequality, use induction on ${{\phantom{}^*\mathrm{dim}}}_A(M)$, which we may assume to be at least 1, since ${{\phantom{}^*\mathrm{dim}}}_A(M) = 0$ if and only if $\mathcal{N}$ is the only prime in $*V(M)$, so that $M$ has finite \*length and $*s(M)= 0$ by definition. If ${{\phantom{}^*\mathrm{dim}}}_A(M) \geq 1$, none of the primes in $\mathcal{D}(M)$ are \*maximal, so there is a homogeneous element $x \in \mathcal{N}$ such that $x$ is not in any of the primes in $\mathcal{D}(M)$. We’ve noted above that $*s(M) \leq *s(M/xM)+1$ and ${{\phantom{}^*\mathrm{dim}}}_A(M) \geq {{\phantom{}^*\mathrm{dim}}}_A(M/xM) +1. $ These inequalities plus the induction hypotheses give us the result.
If $R$ is a positively graded ring with $R_0 =k$ a field, then $(R,R_+)$ is a \*local ring, so we may apply the fundamental theorem for \*local dimension. On the other hand, recall Smoke’s dimension theorem (theorem \[theorem Smoke dimension\]). For any $M \in {\mathfrak{grmod}}(R)$ the hypotheses for Smoke’s dimension theorem are satisfied, and we may therefore combine the two dimension theorems.
\[corollary fun thm dimension theory pos graded\]If $R$ is a positively graded Noetherian ring with $R_0$ a field and $M \in {\mathfrak{grmod}}(R)$, then $$*\dim_R(M) = *d(M) = *s(M)= s_1(M) = d_1(M) = \dim_R(M).$$
Going back to the definition of a GSOP for the $A$-module $M$, as a corollary to Theorem \[theorem fun thm graded dimension theory\]. we have
If $(A, \mathcal{N})$ is a \*local graded Noetherian ring and $M \in {\mathfrak{grmod}}(A)$, then a GSOP exists for $M$, and the length of every GSOP is equal to $*\dim_A(M) = *d(M) = *s(M).$ Moreover, if $A-\mathcal{N}$ has a homogeneous element $T$ of degree 1, necessary invertible, and $d(M_0)$ is the degree of the ordinary Samuel polynomial $p(M_0, \mathcal{N}_0, n)$, then $d(M_0) = *d(M)$, and if $x_1, \ldots, x_D$ is a GSOP for $M$, where $D = {{\phantom{}^*\mathrm{dim}}}_A(M) = *d(M) = d(M_0)$, then $x_1T^{-e_1}, \ldots, x_DT^{-e_D}$ is an ordinary system of parameters for $M_0$ as an $A_0$-module, if $e_i = \deg(x_i)$.
The first statement is clear using \[theorem fun thm graded dimension theory\]; for the second use Theorem \[theorem star e equals e, element of degree 1\] to see that $*d(M)= d(M_0)$; if $x_1, \ldots, x_D$ generate a GIOD ${{\mathcal{I}}}$ for $M$, then ${{\mathcal{I}}}_0$ is generated by $x_1T^{-e_1}, \ldots, x_DT^{-e_D}$. Therefore, the ungraded dimension theorem ensures that $D = d(M_0) = \dim_{A_0}(M_0)$, so $x_1T^{-e_1}, \ldots, x_DT^{-e_D}$ is an ordinary system of parameters for $M_0$.
We also have a corollary to Corollary \[corollary sum decomp for multiplicity\]; here $\mathcal{D}(M)$ is defined as the set of minimal primes of maximal dimension (as in the proof of Theorem \[theorem fun thm graded dimension theory\])
\[corollary \*Samuel sum decomposition\] Suppose that $(A, \mathcal{N})$ is a \*local graded Noetherian ring, $M \in {\mathfrak{grmod}}(A)$, ${{\mathcal{I}}}$ is a GIOD for $M$ and $M^{\bullet}$ is a graded filtration of $M$ such that $0 = M^0 \subset M^1 \subset \cdots M^{N-1} \subset M^N = M,$ and, for each $N \geq i \geq 1$, there are graded prime ideals ${{\mathfrak{p}}}_i$ in $A$, integers $d_i$ and graded isomorphisms of $A$-modules $(A/{{\mathfrak{p}}}_i )(d_i) \cong M^{i}/M^{i-1}$. Then, if $D \doteq {{\phantom{}^*\mathrm{dim}}}_A(M)$,
$$*e(M, {{\mathcal{I}}}, D) = \sum_{{{\mathfrak{p}}}\in \mathcal{D}(M)} *\ell_{A_{[{{\mathfrak{p}}}]}}(M_{[{{\mathfrak{p}}}]}) (*e(A/{{\mathfrak{p}}}, {{\mathcal{I}}}, D)).$$
Lemma \[corollary the filtration graded\] tells us that, for every minimal prime ${{\mathfrak{p}}}$ for $M$, there is at least one subquotient of the filtration isomorphic to the graded $A$-module $A/{{\mathfrak{p}}}$, possibly shifted. Therefore, adding the \*local hypothesis,
- $\mathcal{D}(M^{\bullet}) = \mathcal{D}(M)$ since we now know that, for every shift $d$, the degree of $*p((A/{{\mathfrak{p}}})(d), \mathcal{I}, n)= *p(A/{{\mathfrak{p}}},\mathcal{I},n)$ is independent of the choice of $\mathcal{I}$ and is equal to ${{\phantom{}^*\mathrm{dim}}}_A(A/{{\mathfrak{p}}})$. Theorem \[theorem fun thm graded dimension theory\] also tells us that the $D$ in this corollary is exactly the same $D$ as in Corollary \[corollary sum decomp for multiplicity\].
- Moreover, for every prime ${{\mathfrak{p}}}$ in $\mathcal{D}(M)$, $A/{{\mathfrak{p}}}$, possibly shifted, occurs exactly $*\ell_{A_{[{{\mathfrak{p}}}]}}(M_{[{{\mathfrak{p}}}]}) = \ell_{A_{{{\mathfrak{p}}}}}(M_{{{\mathfrak{p}}}})$ times (using Theorem \[theorem \*composition series exists for Mgrp when p is minimal\] and Corollary \[corollary sum decomp for multiplicity\]) as a subquotient of the filtration $M^{\bullet}$, so that $n_{{{\mathfrak{p}}}}(M^{\bullet}) = *\ell_{A_{[{{\mathfrak{p}}}]}}(M_{[{{\mathfrak{p}}}]}) = \ell_{A_{{{\mathfrak{p}}}}}(M_{{{\mathfrak{p}}}})$.
Koszul complexes in grmod(A) and \*Koszul multiplicities
========================================================
In this section, $A$ is any graded Noetherian ring and $M \in {\mathfrak{grmod}}(A)$. We again follow the discussion in [@se].
The definition of a complex of modules in ${\mathfrak{grmod}}(A)$ is as usual: this is a sequence $(\mathbf{M}, \partial) $ $$\cdots \rightarrow M_j \stackrel{\partial_j}{\rightarrow} M_{j-1} \stackrel{\partial_{j-1}}{\rightarrow} \cdots \rightarrow \stackrel{\partial_1}{\rightarrow} M_0$$ of objects and morphisms in ${\mathfrak{grmod}}(A)$, such that $\partial \partial = 0$ everywhere. The sequence of morphisms $\partial$ above is called the differential for the complex.
The subscripts $j$ seem assigned ambiguously, but here’s what we mean: If $(\mathbf{M},\partial)$ is a complex in ${\mathfrak{grmod}}(A)$ as above, then the set of elements of $M_j$ of degree $i$ is equal to $$(M_j)_i \doteq M_{j,i}.$$
In other words, when speaking of a complex in ${\mathfrak{grmod}}(A)$, a single integer subscript denotes the sequential index of the complex, and a doubly-indexed subscript is read as “first index is the complex index, second is the graded-module index". We will often suppress the internal gradings, so if there is just one subscript, it refers to the “complex index". Hopefully this won’t be too confusing.
To further set notation, we will regard any $M$ in ${\mathfrak{grmod}}(A)$ as a “complex concentrated in degree 0"–this is the complex with all differentials equal to zero, with $M_i = 0$, if the “complex-index" $i \neq 0$, and $M_0 = M$, for “complex-index " $0$.
The homology groups of a complex $(\mathbf{M}, \partial)$ are defined as $``\ker \partial/ \mbox{im} \;\partial"$ of course, and are also in ${\mathfrak{grmod}}(A)$:
$$H_j(\mathbf{M})_i = \ker(\partial:M_{j,i} \rightarrow M_{j-1,i})/\mbox{im}(\partial:M_{j+1,i} \rightarrow M_{j,i}).$$
Morphisms of graded complexes and short exact sequences of graded complexes are defined in the usual way.
A short exact sequence of graded complexes in ${\mathfrak{grmod}}(A)$ gives rise to a long exact sequence on homology: if $0 \rightarrow \mathbf{A} {\xrightarrow}{\alpha} \mathbf{B} {\xrightarrow}{\beta} \mathbf{C} \rightarrow 0$ is a short exact sequence of graded complexes in ${\mathfrak{grmod}}(A)$, there exists a graded morphism $\mathbf{\omega}$ of complex degree $-1$ such that the sequence $$\cdots {\xrightarrow}{\omega_{j+1}} H_j(\mathbf{A}) {\xrightarrow}{\alpha_*} H_j(\mathbf{B}) {\xrightarrow}{\beta_*} H_j(\mathbf{C}) {\xrightarrow}{\omega_j} H_{j-1}(\mathbf{A}) {\xrightarrow}{\alpha_*} \cdots$$ is an exact sequence in ${\mathfrak{grmod}}(A)$.
The **\*Euler characteristic** of a complex $(\mathbf{M}, \partial)$ in ${\mathfrak{grmod}}(A)$ is defined when $(\mathbf{M}, \partial)$ is such that each $A$-module $M_i$ has $*\ell_A(M_i) < \infty$ and for all but finitely many $i$, $*\ell_A(M_i) = 0$ . Given these conditions, the following sum is well defined: $$*\chi(\mathbf{M}) \doteq \Sigma_i (-1)^i *\ell_A(M_i).$$
Since $*\ell$ sums over short exact sequences, we get the following lemma.
\[lemma euler splits over ses\] Let $0 \rightarrow \mathbf{A} \rightarrow \mathbf{B} \rightarrow \mathbf{C} \rightarrow 0$ be a short exact sequence of graded complexes in ${\mathfrak{grmod}}(A)$. Then, the \*Euler characteristic of $\mathbf{B}$ is defined if and only if the \*Euler characteristics of $\mathbf{A}$ and $\mathbf{C}$ are, and $*\chi(\mathbf{B}) = *\chi(\mathbf{A}) + *\chi(\mathbf{C})$.
If the two conditions for a well-defined \*Euler characteristic of a complex are not met, there may be a way to salvage the situation by passing to homology.
Let $(\mathbf{M}, \partial)$ be a complex in ${\mathfrak{grmod}}(A)$ such that for every $i$, $H_i(\mathbf{M})$ has finite \*length over $A$, and for $i >> 0$ $*\ell(H_i(\mathbf{M})) = 0$. We define the \*Euler characteristic of the homology to be $*\chi(\mathbf{H(M)}) \doteq \Sigma_i (-1)^i *\ell_A(H_i (\mathbf{M}))$.
With the proof exactly analogous to that in the ungraded case, we have
\[theorem Euler of homology equals Euler of complex\] When the \*Euler characteristic $*\chi(\mathbf{M})$ is defined, then $*\chi(\mathbf{H}(\mathbf{M}))$ is also defined, and we have that $*\chi(\mathbf{M}) = *\chi(\mathbf{H(M)})$.
Note however that the converse is not necessarily true; i.e. $*\chi$ of the homology may be defined but $*\chi$ of the complex not.
Using the additivity of $*\ell$, and the long exact sequence on homology, if $\mathbf{A} \rightarrowtail \mathbf{B} \twoheadrightarrow \mathbf{C}$ is a short exact sequence of graded complexes in ${\mathfrak{grmod}}(A)$ such that the \*Euler characteristic of the homology of each complex is defined, then, $$*\chi(\mathbf{H(B)}) = *\chi(\mathbf{H(A)}) + *\chi(\mathbf{H(C)}).$$
If $A$ is a graded ring, we may do homological algebra in ${\mathfrak{grmod}}(A)$ quite analogously to how it’s done in the ungraded case. In particular, graded $A$-modules $Tor_i^A(M,N) \in {\mathfrak{grmod}}(A)$ for every $i \geq 0$, and $M,N \in {\mathfrak{grmod}}(A)$ may be defined mimicking the constructions and definitions in the ungraded category: beginning with the graded tensor product $M \otimes_A N$. (For the definition of the graded tensor product of graded modules over a graded ring, see [@GRO].) The tensor product $M \otimes_A N$ has a natural grading on it: if $m \in M_i$ and $n \in N_j$ are homogeneous elements, then $\deg(m\otimes n) = i + j$. Then, one proceeds to talk about projective resolutions, and arrives at the definition of $Tor_i^A(M,N) \in {\mathfrak{grmod}}(A)$. We do not give further details here.
The Koszul complex
------------------
Standard properties of the Koszul complex in the ungraded case may be found in [@se], Chapter IV. We use Serre’s notation: if $\bar{x} \doteq x_1, \ldots, x_u$ is a sequence of elements in $A$, then the Koszul complex is denoted by $\mathbf{K}(\bar{x},A)$.
If we pass to the graded category, with $A$ a graded ring, and choose a sequence $\bar{x} \doteq x_1, \ldots, x_u$ of homogeneous elements of $A$, the definition of the graded Koszul complex is briefly summarized as follows. Recall that the tensor product of graded complexes $\mathbf{C} \otimes_A \mathbf{D}$ is defined exactly analogously to the ungraded case, and is again a graded complex; keep in mind in particular the definition of the differential of a tensor product of complexes: if $c \in C_i$ and $d \in D_j$, then $\partial_{C \otimes_A D}(c \otimes d) = \partial_C(c) \otimes d + (-1)^i c \otimes \partial_D(c).$ Starting with the case $u=1$, $\mathbf{K}(x_1,A)$ is the two-term complex in ${\mathfrak{grmod}}(A)$ $$K_1(x_1, A) = A(-d) \stackrel{\cdot x_1}{\rightarrow} K_0(x_1,A) = A,$$ where $d$ is the degree of $x_1$. Then, if $\bar{x} = x_1, \ldots, x_u$ is a sequence of homogeneous elements in $A$, $$\mathbf{K}(\bar{x},A) \doteq \mathbf{K}(x_1,A) \otimes_A \cdots \otimes_A \mathbf{K}(x_u,A).$$
If $M \in {\mathfrak{grmod}}(A)$, the Koszul complex associated to the graded $A$-module $M$ and $\bar{x}$ is: $$\mathbf{K}^A(\bar{x},M) \doteq \mathbf{K}(\bar{x},A) \otimes_A M.$$ If we’re always regarding a graded abelian group $M$ as an $A$-module, for a fixed graded ring $A$, we will often delete the superscript $A$.
Setting notation, $K_0(x_i, A)$ is identified with $A$ as a free, graded $A$-module (in other words, the free generator lies in degree zero, and is identified with $1\in A_0$). For $K_1$, choose $e_{x_i}$ of $\deg(x_i)$ and identify $K_1(x_i,A)$ with the free graded $A$-module generated by the homogeneous element $e_{x_i}$. Then, $K_p(\bar{x}, A)$ is the free graded $A$-module isomorphic to the free graded $A$-module generated by the homogeneous elements $e_{x_{i_1}} \otimes \cdots \otimes e_{x_{i_p}}$ of degree $\deg(x_{i_1}) + \cdots +\deg(x_{i_p})$ where $i_1 < \cdots < i_p$, so is isomorphic to the graded exterior product $$\Lambda^p(A(-\deg(x_1)) \oplus \cdots \oplus A(-\deg(x_u))).$$ In addition, both the $p$th part of the Koszul complex $\mathbf{K}^A(\bar{x}, M)$, and its differential have exactly the same form as described in [@se], IV.A.2 in the ungraded case. A particular consequence is that, as a graded $A$-module, $K_p^A(\bar{x}, M)$ is a direct sum of ${u \choose p}$ copies of $M$, each shifted: the copy associated to the multi-index $i_1 < \cdots i_p$ looks like $M(-(\deg(x_{i_1}) + \cdots +\deg(x_{i_p})))$; if ${{\mathcal{I}}}$ is the graded ideal of $A$ generated by $x_1, \ldots, x_u$, and $k \geq 0$, then $K_p^A(\bar{x}, M)/ {{\mathcal{I}}}^k K_p^A(\bar{x}, M)$ is, as a graded $A$-module, isomorphic to ${u \choose p}$ copies of $M/{{\mathcal{I}}}^kM$, each shifted as described above.
The $p$th homology group of the graded Koszul complex $\mathbf{K}^A(\bar{x},M)$ is denoted by $H_{p}(\bar{x}, M),$ or $H^A_p(\bar{x}, M)$ if we need to emphasize the role of $A$. These homology groups are also graded $A$-modules.
Suppose that $x_1, \ldots , x_u$ is a sequence of nonzero nonunit homogeneous elements in $A$ and $M \in {\mathfrak{grmod}}(A)$. This sequence is a $M$-sequence if and only if $x_1$ is not a zero-divisor on $M$, and for each $i>1$, $x_i$ is not a zero-divisor on $M/(x_1, \ldots, x_{i-1})M$.
The following may all be proved as in the ungraded case (see [@se], Chapter IV):
Let $A$ be a graded ring and $M \in {\mathfrak{grmod}}(A)$. If $\bar{x}$ is a $M$-sequence, then the Koszul complex $\mathbf{K}^A(\bar{x},M)$ is acyclic. As in the ungraded case, $H_0^A(\bar{x},M) = M/(x_1,\ldots,x_u)M$.
Conversely, in the \*local Noetherian case one has
If $(A, \mathcal{N})$ is a \*local Noetherian ring, and $M \in {\mathfrak{grmod}}(A)$, then the following are equivalent, for a sequence of homogeneous elements $\bar{x} \doteq x_1, \ldots, x_u$ of $\mathcal{N}$:
- $H_p^A(\bar{x}, M) = 0,$ for $p \geq 1$.
- $ H_1^A(\bar{x}, M) = 0.$
- $\bar{x}$ is an $M$-sequence in $A$.
The proofs of the above Propositions are exactly analogous as that of IV.A.2, Propositions 2, 3 in [@se], replacing any use of Nakayama’s lemma with the graded Nakayama’s lemma (Lemma \[lemma graded nakayama\]); similarly, IV.A.2, Corollary 2 yields, in the graded case,
If $(A, \mathcal{N})$ is a \*local Noetherian ring, $M \in {\mathfrak{grmod}}(A)$, and $\bar{x} = x_1, \ldots, x_u$ are homogeneous elements of $\mathcal{N}$ that form an $A$-sequence for $A$, then there is a natural isomorphism of graded $A$-modules $$\psi: H_i^A(\bar{x}, M) \rightarrow Tor_i^A(A/(\bar{x}), M).$$
Finally, IV.A.2, Proposition 4, has the analogous
Suppose that $(A, \mathcal{N})$ is a \*local graded Noetherian ring and $M \in {\mathfrak{grmod}}(A)$. If $x_1, \ldots, x_u$ are homogeneous elements of $\mathcal{N}$, then $(\bar{x}) + Ann_A(M) \subseteq Ann_A(H_i^A(\bar{x}, M)).$
As a corollary,
\[corollary homology fdvs\] Suppose that $(A, \mathcal{N})$ is a \*local graded Noetherian ring , and $M \in {\mathfrak{grmod}}(A)$. Let ${{\mathcal{I}}}$ be a GIOD for $M$, generated by the homogeneous sequence $\bar{x} = x_1, \ldots , x_u \in \mathcal{N}$. Then, $H_j^A(\bar{x}, M) $ has finite \*length over $A$ for every $j \geq 0$.
Since ${{\mathcal{I}}}+ Ann_A(M) \subseteq Ann_A(H_j^A(\bar{x}, M))$, and $\{ \mathcal{N}\} = *V({{\mathcal{I}}}+ Ann_A(M))$, if $H_j^A(\bar{x}, M) \neq 0, \{ \mathcal{N} \} = *V(Ann_A(H_j^A(\bar{x}, M))).$
Thus, in the \*local case the \*Euler characteristic of the homology of the graded Koszul complex is well defined:
Suppose that $(A, \mathcal{N})$ is a \*local, Noetherian graded ring and $M \in {\mathfrak{grmod}}(A)$. Let ${{\mathcal{I}}}$ be a GIOD for $M \in {\mathfrak{grmod}}(A)$ generated by a homogeneous sequence $\bar{x}=x_1, \ldots, x_u$. We define the \***Koszul multiplicity** $*\chi^A(\bar{x},M)$ to be the \*Euler characteristic of the homology of the graded Koszul complex: $$*\chi^A(\bar{x},M) = \sum_{i=1}^u(-1)^i *\ell_A(H_i^A(\bar{x},M)).$$
Equality of \*Samuel and \*Koszul multiplicities
------------------------------------------------
As in the ungraded case [@se], IV.A.3, the \*Koszul multiplicity is equal to a certain \*Samuel multiplicity. This section concludes our account of the theory of multiplicities adapted to the $\mathbb{Z}$-graded category.
Let $(A, \mathcal{N})$ be a \*local, Noetherian graded ring, $M \in {\mathfrak{grmod}}{A}$ and $\bar{x} =x_1, \ldots, x_u$ a sequence of homogeneous elements contained in $\mathcal{N}$. If ${{\mathcal{I}}}$ is the graded ideal of $A$ generated by $\bar{x}$, suppose also that ${{\mathcal{I}}}$ is a GIOD for $M$.
One then filters the graded Koszul complex, yielding graded complexes $\mathcal{F}^i\mathbf{K}$ for every $i$, with $\mathcal{F}^iK_p \doteq {{\mathcal{I}}}^{i-p}K_p$ for every $p$ (we’ve dropped the arguments $\bar{x}, M$ for expediency). Notice we have three indices now: the filtration index, the complex index and the internal gradings of the various $A$-modules involved. We are suppressing the internal grading. This filtration defines the associated graded complex $gr(\mathbf{K}) \doteq \oplus_i \mathcal{F}^i\mathbf{K}/ \mathcal{F}^{i+1} \mathbf{K}$.
If $gr(A)$ is the associated bigraded ring to the ${{\mathcal{I}}}$-adic filtration, then denote the images of $x_1, \ldots, x_u$ in $gr(A)_{1,*}$ by $\xi_1, \ldots, \xi_u$. Let $gr(M)$ be the bigraded $gr(A)$-module associated with the ${{\mathcal{I}}}$-adic filtration of $M$. Then, there is an isomorphism of graded objects $gr(\mathbf{K}) \cong \mathbf{K}(\bar{\xi}, gr(M))$. Moreover, one argues that the homology modules $H_p(\bar{\xi},gr(M))$ have finite \*length over $gr(A)$, for all $p$, since $A/({{\mathcal{I}}}+ Ann_A(M))$ is \*Artinian. This in turn, enables one to argue that there exists an $m \geq u$ such that the graded homology groups of the complex $\mathcal{F}^i\mathbf{K}/ \mathcal{F}^{i+1} \mathbf{K}$ all vanish for $i >m$, and so one sees that the graded homology groups the complex $\mathcal{F}^i\mathbf{K}$ all vanish if $i >m$.
Continuing as in [@se], IV.A.3, (which is really a spectral sequence argument), this means there is an $m$ such that if $i >m$, then $H_p(\mathbf{K}) \cong H_p(\mathbf{K}/\mathcal{F}^i \mathbf{K})$ for $i >m$ and for all $p$.
Using the fact that \*Euler characteristics don’t change when passing to homology, $*\chi(\bar{x},M) = \sum_p (-1)^p *\ell(H_p(\mathbf{K}/\mathcal{F}^i(\mathbf{K})) = *\chi(\mathbf{K}/\mathcal{F}^i\mathbf{K})$, for $i >m$. As noted in the previous section, $(\mathbf{K}/\mathcal{F}^i\mathbf{K})_p$ is isomorphic as a graded $A$-module to a direct sum of ${u \choose p}$ copies of $M/{{\mathcal{I}}}^{i-p}M$, shifted appropriately, and since the \*length of a shifted $A$-module $M(d)$ is the same as the \*length of $M$, the remainder of the proof is argued exactly as in [@se], IV.A.3, with length replaced by \*length, $p$ (a Samuel polynomial) replaced by $*p$ and $e$ replaced by $*e$.
Thus, we have
\[theorem koszul equals samuel\] Let $(A, \mathcal{N})$ be a \*local Noetherian ring. Let $x_1, \ldots, x_u \in \mathcal{N}$ be homogeneous elements generating a graded ideal of definition ${{\mathcal{I}}}$ for $M \in {\mathfrak{grmod}}(A)$. Then, $$*\chi^A(\bar{x},M) = *e(M,{{\mathcal{I}}},u),$$ so $*\chi^A(\bar{x}, M)$ is a strictly positive integer if ${{\phantom{}^*\mathrm{dim}}}_A(M) = u$, and $*\chi^A(\bar{x}, M) = 0$ if $u > {{\phantom{}^*\mathrm{dim}}}_A(M)$.
Multiplicities and Degree for Positively Graded Rings {#section Multiplicities and Degree for Positively Graded Rings}
=====================================================
In this section, we specialize to the case of a positively graded Noetherian ring $R$ with $R_0$ a field $k$; all graded modules are in ${\mathfrak{grmod}}(R).$ Set ${{\mathfrak{m}}}= R_+$ and note that $(R, {{\mathfrak{m}}})$ is then a \*local graded Noetherian ring. We do not want to make the assumption that $R$ is generated by elements in degree 1.
In this chapter we may use the \* notation even though we could just as well omit the \* (e.g. If $M \in {\mathfrak{grmod}}(R)$, then $*\dim_R(M)=\dim_R(M)$.) This is done to emphasize the fact that all computations may be done in the graded category using the theory developed in the previous two chapters (which is often simpler than the ungraded theory.)
We introduce the *degree* of a graded module, show how it relates to \*multiplicity (Theorem \[theorem main theorem\]), and give a sum decomposition of degree by a certain set of minimal primes (Theorem \[theorem sum formula for degree and multiplicity\].)
Since $R_0=k$, $\ell_k(M_i) = {\mathrm{vdim}}_k(M_i)$ for every $i$, so the Poincaré series for $M$ is equal to $$P_M(t) = \sum_{i \in \mathbb{Z}} {\mathrm{vdim}}_k(M_i) t^i.$$ Furthermore, this Laurent series has a pole at $t=1$ using the Hilbert-Serre theorem, and the order of the pole $d_1(M)$ at $t=1$ is, by Smoke’s dimension theorem, is exactly ${{\phantom{}^*\mathrm{dim}}}_R(M)$.
This leads to the definition of $\deg_R(M)$:
If $R$ is a positively graded Noetherian ring with $R_0=k$ a field, $M\in {\mathfrak{grmod}}(R), \newline M \neq 0$ and $D(M) = {{\phantom{}^*\mathrm{dim}}}_R(M)$, then $$\deg_R(M) \doteq \lim_{t \rightarrow 1} (1-t)^{D(M)}P_M(t)$$ is a well-defined, strictly positive, rational number. For convenience, define $\deg_R(0) = 0.$
Often we delete the subscript $R$ and just write $\deg(M)$. We use the (somewhat ambiguous) name of “degree" for this rational number in deference to the nomenclature already used in [@Bensonea]. For equivariant cohomology, this “degree" was first studied by Maiorana [@ma].
Multiplicities and Euler-Poincaré series
========================================
If $X \in {\mathfrak{grmod}}(R)$ has finite \*length as an $R$-module, since each $R_i$ is finite-dimensional as a vector space over $k$, we may use Lemmas \[lemma pos graded length equals \*length\] and \[lemma \*length of components\] to conclude that $\ell_R(X) = *\ell_R(X) = {\mathrm{vdim}}_k(X)$, where ${\mathrm{vdim}}_k(X)$ is the total dimension $\sum_j {\mathrm{vdim}}_k(X_j)$ of the graded $k$-vector space $X$. We may then prove:
Suppose $R$ is a positively graded Noetherian ring with $R_0 = k$, a field, and $X \in {\mathfrak{grmod}}(R)$ is such that $*\ell_R(X) < \infty$. If $B$ is a graded subring of $R$, Noetherian or not, with $B_0 = k = R_0$, then $X \in {\mathfrak{grmod}}(B)$, $*\ell_B(X) < \infty$ and $*\ell_B(X) = *\ell_R(X) = \ell_R(X) = {\mathrm{vdim}}_k(X) < \infty$.
For, using Lemma \[lemma \*length of components\] applied to $R$, each $X_i$ is finite-dimensional over $k$, and there are integers $t_0$ and $J$ such that $t_0 \leq J$ with $X= \oplus_{j=t_0}^J X_j$. Also, $*\ell_R(X) = {\mathrm{vdim}}_k(X)$. However, since $k \subseteq B \subseteq R$, $X$ is a finitely generated $B$-module. Whether $B$ is Noetherian or not, since $B_j \subseteq R_j$ for every $j$, and $R_j$ is finite-dimensional over $k$, so is $B_j$. Thus, using Lemma \[lemma \*length of components\] applied to $B$, $*\ell_B(X) = {\mathrm{vdim}}_k(X)$ as well.
If $M \in {\mathfrak{grmod}}(R)$, then ${{\mathfrak{m}}}$ is a GIOD for $M$, and we may then calculate a \*Samuel polynomial $*p_R(M, {{\mathfrak{m}}}, n)$ for $M$; Theorem \[theorem star e equals e, pos graded\] says that this is the ordinary Samuel polynomial $p_R(M, {{\mathfrak{m}}}, n)$; Corollary \[corollary fun thm dimension theory pos graded\] says that the degree of this polynomial is $$D(M) \doteq *d(M) = *s(M) = s_1(M) = d_1(M) = {{\phantom{}^*\mathrm{dim}}}_R(M) = \dim_R(M).$$
Now, suppose that $\bar{x} = x_1, \ldots, x_{D(M)}$ is a GSOP for the $R$-module $M$. If ${{\mathcal{I}}}$ is the graded ideal in $R$ generated by $\bar{x}$, ${{\mathcal{I}}}$ is a GIOD for $M$. We can change rings to $B \doteq k\langle x_1, \ldots, x_{D(M)} \rangle$, note that this is a graded polynomial ring over $k$ in the indicated variables (Proposition \[proposition algebraic independence\]). The ideal $\hat{{{\mathcal{I}}}}$ generated by $\bar{x}$ in $B$ is also a GIOD in $B$ since clearly $\hat{{{\mathcal{I}}}}^n M = {{\mathcal{I}}}^nM$ for every $n$. Therefore Theorem \[theorem star e equals e, pos graded\] and the previous lemma guarantee that, for every $n$, the polynomials below are all equal, as indicated: $$*p_R(M, {{\mathcal{I}}}, n) = p_R(M, {{\mathcal{I}}}, n) = p_B(M, \hat{{{\mathcal{I}}}}, n) = *p_B(M, \hat{{{\mathcal{I}}}}, n);$$ in particular, they all have the same degree $D(M)$, and the following positive integers are also all equal: $$*e_R(M, {{\mathcal{I}}}, D(M)) = e_R(M, {{\mathcal{I}}}, D(M)) = e_B(M, \hat{{{\mathcal{I}}}}, D(M)) = *e_B(M, \hat{{{\mathcal{I}}}}, D(M)).$$
Euler-Poincaré series
---------------------
The following lemma is found in Avramov and Buchweitz [@avbu]; [@sm] contains a similar result.
(Lemma 7 of [@avbu]) \[corollary Euler-Poincare series props\]If $M,N \in {\mathfrak{grmod}}(R)$, then
1. For each $i$, the graded $R$-module $Tor_i^R(M,N)$ has finite dimensional (over $k = R_0$) homogenous components $Tor_i^R(M,N)_j$, for every $j$; also, for every $i$, $Tor_i^R(M,N)_j = 0$ for $j <<0$. Thus one may form the Laurent series $$P_{Tor_i^R(M,N)}(t) \doteq \sum_{j \in \mathbb{Z}} {\mathrm{vdim}}_k(Tor_i^R(M,N))_jt^j.$$
2. Furthermore,the alternating sum $$\chi_R(M,N)(t) \doteq \sum_{i \geq 0} (-1)^i P_{Tor_i^R(M,N)}(t),$$ which is by definition the Euler-Poincaré series of $M,N$, is a well-defined Laurent series with integer coefficients and $$P_R(t)\chi_R(M,N)(t) = P_M(t)P_N(t).$$
\[section Multiplicities and the Euler-Poincare Series\] If a GSOP $\bar{x}$ is given for $M \in {\mathfrak{grmod}}(R)$, $B \doteq k \langle \bar{x} \rangle \subseteq R$ is then a graded polynomial ring over $k$ (Proposition \[proposition algebraic independence\]), and $M \in {\mathfrak{grmod}}(B)$, using Lemma \[lemma finite generation gsop\]. Whether we consider $M \in {\mathfrak{grmod}}(R)$, or $M \in {\mathfrak{grmod}}(B)$, the Poincarè series of $M$ does not change.
Hilbert’s Syzygy Theorem tells us that the graded Koszul complex $\mathbf{K}^{k\langle \bar{x} \rangle}(\bar{x}, k)$ is acyclic, thus is a free, finite graded resolution of $k$ as a graded $k\langle \bar{x} \rangle$-module. In particular, we may tensor this resolution with $M$ and use it to compute $Tor_i^{k\langle \bar{x} \rangle}(k,M) = Tor_i^{k\langle \bar{x} \rangle}(M,k),$ showing that $$Tor_i^{k\langle \bar{x} \rangle}(M,k) = H^{k\langle \bar{x} \rangle}_i(\bar{x}, M).$$
\[star chi equals chi\] Let $\bar{x} = x_1, \ldots, x_{D(M)}$ be a GSOP for $M \in {\mathfrak{grmod}}(R)$, and let ${{\mathcal{I}}}$ be the graded ideal in $R$ generated by $\bar{x}$. For every $i$, $ P_{Tor_i^{k\langle \bar{x} \rangle}(M,k)}(t) \in \mathbb{Z}[t, t^{-1}],$ and therefore $\chi_{k\langle \bar{x} \rangle}(M,k)(t) \in \mathbb{Z}[t,t^{-1}].$ Furthermore, $$\chi_{k\langle \bar{x} \rangle}(M,k)(t) = \sum_{j=0}^{D(M)} (-1)^jP_{H_j^{k\langle \bar{x} \rangle}(\bar{x},M)}(t),$$ and evaluating this Laurent polynomial at $t=1$, we compute $$\chi_{k\langle \bar{x} \rangle}(M,k)(1) = *\chi^{k\langle \bar{x} \rangle}(\bar{x}, M) = *e_R({{\mathcal{I}}}, M, D(M))= *\chi^R(\bar{x}, M),$$ where $D(M) = {{\phantom{}^*\mathrm{dim}}}_R(M).$
Lemma \[corollary Euler-Poincare series props\] shows the first part of the statement, and since the resolution $\mathbf{K}^{k\langle \bar{x} \rangle}(\bar{x}, k)$ is in any case zero for complex degree larger than $D(M)$, $Tor_i^{k\langle \bar{x} \rangle}(M,k)$ is also zero for $i > D(M)$, so $$\chi_{k \langle \bar{x} \rangle}(M,k)(t) \doteq \sum_{j \geq 0} (-1)^j P_{Tor_j^{k \langle \bar{x} \rangle}(M,k)}(t) = \sum_{j=0}^{D(M)} (-1)^jP_{H_j^{k\langle \bar{x} \rangle}(\bar{x},M)}(t),$$ being a finite sum of Laurent polynomials, is a Laurent polynomial.
Setting $B \doteq k\langle \bar{x} \rangle$ yields, $$*\ell_B(H_i^B(\bar{x}, M)) = \ell_B(H_i^B(\bar{x},M)) = {\mathrm{vdim}}_k(H_i^B(\bar{x}, M)),$$ so $$*\chi^B(\bar{x}, M) = \sum_{j=0}^{D(M)} (-1)^j \ell_B(H_j^B(\bar{x}, M))= \sum_{j=0}^{D(M)} (-1)^j {\mathrm{vdim}}_k(H_j^B(\bar{x}, M)) \doteq \chi_B(M,k)(1).$$
As noted at the beginning of this section, if $\hat{{{\mathcal{I}}}}$ is the ideal generated by $\bar{x}$ in $B$, then $*e_B(M, \hat{{{\mathcal{I}}}}, D(M)) = *e_R(M, {{\mathcal{I}}}, D(M))$ and Theorem \[theorem koszul equals samuel\] says that $$*\chi^B(\bar{x}, M) = *e_B(M, \hat{{{\mathcal{I}}}}, D(M)) = *e_R(M, {{\mathcal{I}}}, D(M)) = *\chi^R(\bar{x}, M) .$$
Degree of a Graded Module in ${\mathfrak{grmod}}(R)$
----------------------------------------------------
Given $M \in {\mathfrak{grmod}}(R)$, $\deg(M) >0$, if $M \neq 0$, we can read off the degree of a module directly from the Poincare series if we expand it as a Laurent series about $t=1$: $$P_R(t) = \frac{\deg(M)}{(1-t)^{D(M)}} + \text{"higher order terms"}.$$
Suppose that $0 \rightarrow N \rightarrow M \rightarrow P \rightarrow 0$ is an exact sequence in ${\mathfrak{grmod}}(R)$. Then,
- $D(M) = \max \{D(N), D(P) \}. $
- If $D(N) < D(M),$ then $\deg(M) = \deg(P).$
- If $D(P) <D(M),$ then $\deg(M)=\deg(N).$
- If $D(P)=D(N)=D(M)$, then $\deg(M)=\deg(N) + \deg(P).$
- $\deg(M(d)) = \deg(M)$, for every integer $d$.
This immediately yields, as in [@Bensonea]:
\[theorem sum formula for degree and multiplicity\] Let $M \in {\mathfrak{grmod}}(R)$, and $\mathcal{D}(M)$ be defined as in Theorem \[theorem fun thm graded dimension theory\]: this is the set of prime ideals ${{\mathfrak{p}}}$ in $R$, necessarily minimal primes for $M$ and graded, such that ${{\phantom{}^*\mathrm{dim}}}_R(R/{{\mathfrak{p}}}) = {{\phantom{}^*\mathrm{dim}}}_R(M)$. Then, $$\deg(M) = \sum_{{{\mathfrak{p}}}\in \mathcal{D}(M)}*\ell_{R_{[{{\mathfrak{p}}}]}}(M_{[{{\mathfrak{p}}}]})\cdot \deg(R/{{\mathfrak{p}}}).$$
Choose a graded filtration $M^{\bullet}$ of $M$ of the form in Lemma \[corollary the filtration graded\] We know that if ${{\mathfrak{p}}}\in \mathcal{D}(M)$, then the graded $R$-module $R/{{\mathfrak{p}}}$, possibly shifted, occurs exactly $*\ell_{R_{[{{\mathfrak{p}}}]}}(M_{[{{\mathfrak{p}}}]}) = \ell_{R_{{{\mathfrak{p}}}}}(M_{{{\mathfrak{p}}}})$ times (using Theorem \[theorem \*composition series exists for Mgrp when p is minimal\]) as a subquotient in the filtration. The lemma above then gives the result.
We want to compare degree to our previously studied multiplicities.
Letting $\bar{x}$ be a GSOP for $M \in {\mathfrak{grmod}}(R)$, we’ve seen that $k\langle \bar{x} \rangle$ is a graded polynomial ring, and one directly calculates that $$P_{k\langle \bar{x} \rangle}(t) = \frac{1}{\prod_{i=1}^{D(M)} (1-t^{d_i})},$$ where $d_i$ is the degree of the homogeneous element $x_i$.
Now, using $M \in {\mathfrak{grmod}}(k\langle \bar{x} \rangle)$, and recalling that $P_M(t)$ is the same whether we consider $M \in {\mathfrak{grmod}}(R)$ or $M \in {\mathfrak{grmod}}(k \langle \bar{x} \rangle)$, Lemma \[corollary Euler-Poincare series props\] gives that $$P_k(t)P_M(t) = P_{k \langle \bar{x} \rangle}(t) \chi_{k \langle \bar{x} \rangle}(M,k)(t).$$ Also, $\chi_{k\langle \bar{x} \rangle}(M,k)(t) \in \mathbb{Z}[t,t^{-1}]$. Since $P_k(t) = 1$, we have
If $M \in {\mathfrak{grmod}}(R)$ and $\bar{x}$ is a GSOP for $M$, then $$P_M(t) = \frac{\chi_{k\langle \bar{x} \rangle}(M,k)(t)}{\prod_{i=1}^{D(M)} (1-t^{d_i})},$$ with $\chi_{k\langle \bar{x} \rangle}(M,k)(t) \in \mathbb{Z}[t,t^{-1}].$
Since $$(1-t)^{D(M)}P_M(t) = \frac{\chi_{k\langle \bar{x} \rangle}(M,k)(t) }{\prod_{i=1}^{D(M)}(1+t + \cdots + t^{d_i-1})},$$ taking the limit as $t$ approaches 1, and using Lemma \[star chi equals chi\], yields:
\[theorem main theorem\] If $M \neq 0$ is in ${\mathfrak{grmod}}(R)$, and $x_1, \ldots , x_{D(M)}$ of degrees $d_1, \ldots, d_{D(M)}$ form a GSOP for $M$, generating the graded ideal ${{\mathcal{I}}}$ of $R$, then $$\deg(M) = \frac{*e_R(M, {{\mathcal{I}}}, D(M))}{d_1 \cdots d_{D(M)}} = \frac{*\chi^R(\bar{x}, M)}{d_1 \cdots d_{D(M)}}.$$
Thus, the ratio $$\frac{*e_R(M, {{\mathcal{I}}}, D(M))}{d_1 \cdots d_{D(M)}}$$ is independent of the choice of system of parameters $x_1, \ldots, x_{D(M)}$ for $M$.
Note that we can delete the “stars" in the equalities of the above theorem and retain the equalities, using Theorem \[theorem star e equals e, pos graded\]. The reader should compare this result to Proposition 5.2.2 of [@pr], which states a similar result for rings with standard grading.
[XXXXXNNN]{} L. Avramov and R. Buchweitz, *Lower Bounds for Betti Numbers*, Compositio Mathematica, **86** (1993).
M.F Atiyah and I.G. Macdonald, *Introduction to Commutative Algebra*, Addison-Wesley, Reading, MA, 1969.
D.J. Benson and W.W. Crawley-Boevey, *A Ramification Formula for Poincarè Series and a Hyperplane Formula for Modular Invariants*, Bull. London Math. Soc, **27** (1995), 435-440. M. Blumstein and J. Duflot, *Commutative Algebra and Degree Formulas for Equivariant Cohomology Rings*, in preparation. W. Bruns and J. Herzog, *Cohen-Macaulay rings. Revised Edition,* Cambridge University Press, Cambridge, U.K., 1998.
J. Duflot, *Numerical Invariants for Equivariant Cohomology,* The Quarterly Journal of Mathematics, **64** (2013), 665-720.
D. Eisenbud, *Commutative Algebra with a View Toward Algebraic Geometry*, Springer-Verlag, New York, 1995.
A. Grothendieck, *Elements de geometrie algebrique (rediges avec la collaboration de Jean Dieudonne) : II,* Etude globale elementaire, de quelques classes de morphismes, Publications mathematiques de lI.H.E.S. 8 (1961).
R. Lynn, *A Degree Formula for Equivariant Cohomology,* Transactions of the American Mathematical Society, **366** 2014.
J.A. Maiorana, *Smith Theory for p-Groups,* Transactions of the American Mathematical Society, **223** 1976, 253$-$266.
D. Quillen, *The spectrum of an equivariant cohomology ring: I.* Annals of Mathematics, **94** 1971, 459-572.
D. Quillen, *The spectrum of an equivariant cohomology ring: II.* Annals of Mathematics, **94** 1971, 573-602. P. Roberts, *Multiplicities and Chern classes in local algebra.* Cambridge University Press, Cambridge, U.K. (1998). W. Smoke, *Dimension and Multiplicity for Graded Algebras,* Journal of Algebra, **21** 1972.
J.P. Serre, *Local Algebra,* Springer-Verlag, Berlin (2000).
P. Symonds, *On the Castelnuovo-Mumford Regularity of the Cohomology Ring of a Group,* Journal of the American Mathematical Society, **23** No. 4, 2010, 1159$-$1173.
|
---
author:
- 'Gentaro <span style="font-variant:small-caps;">Watanabe</span>$^{1,2}$'
title: 'Simulating pasta phases by molecular dynamics and cold atoms — Formation in supernovae and superfluid neutrons in neutron stars '
---
Introduction
============
Collapse driven supernova explosion, an explosion in the death of a massive star, is one of the most dramatic phenomena in the universe and has been a long-standing mystery in astrophysics [@colgate]. One of the key ingredients to understand the mechanism of the supernova explosion is the study of matter in the core of supernovae [@bethe].
State of matter in supernova cores also undergoes dramatic changes in the process of the collapse. Matter experiences an adiabatic compression, in which the density in the central region of the core increases from $\sim 10^{9}$ g cm$^{-3}$ in the beginning of the collapse and finally reaches around the normal nuclear density $\sim
3\times 10^{14}$ g cm$^{-3}$ (corresponding to the number density of nucleons $\rho_{0}=0.165$ fm$^{-3}$) just before the bounce. It is predicted that, in the final stage of the collapse, nuclei are rod-like or slab-like rather than (roughly) spherical in the central region of the core [@rpw; @hashimoto]. These non-spherical nuclei are collectively called nuclear “pasta”.
Recently, nuclear pasta attracts much attention of astrophysicists and nuclear physicists \[see, e.g., Refs. \] since it has been pointed out that the nuclear pasta can occupy 10 – 20 $\%$ of the mass of supernova cores [@opacity] and thus might have influences on the dynamics of supernova explosions [@gentaro2; @horowitz; @opacity]. However, a fundamental problem of whether or not and how pasta phases are formed during the collapse is still unclear. In the present work [@qmd_formation], we succeed in numerically simulating the formation of a trianglar lattice of rod-like nuclei from a bcc lattice of spherical nuclei in collapsing cores and solve the above long-standing problem. In addition, we discover that the formation process is very different from a generally accepted scenario based on an instability with respect to nuclear fission. Our work provides a solid basis for understanding materials in supernova cores, which is indispensable for solving a long-standing problem of the mechanism of supernova explosions.
Fission Instability?
====================
In a generally accepted conjecture, based on the Bohr-Wheeler condition for vanishing the fission barrier derived for isolated nuclei, it is predicted that the formation of the pasta phases are triggered by the fission instability with respect to the quadrupolar deformation of spherical nuclei [@review]. The essential point of this prediction is that, at higher densities, the effect of the Coulomb repulsion between protons in nuclei, which tends to make a nucleus deform, becomes dominant over the effect of the surface tension of the nuclei, which favors a spherical nucleus. However, Refs. and have pointed out that the background electrons, which have been ignored in the above prediction, suppress the effect of the Coulomb repulsion between protons in nuclei and the fission barrier never vanishes in the relevant density region. These findings cast a doubt on the above conjecture based on the fission instability.
Theoretical Framework
=====================
Since formation of the pasta phases is accompanied by dynamical and drastic changes of the nuclear structure, an [*ab-initio*]{} approach is called for. The quantum molecular dynamics (QMD) [@aichelin], which can properly incorporate the thermal fluctuations [@qmd_hot] and enables us to simulate large systems with $\sim 10^3$ nucleons or more [@qmd], is a suitable approach for the present purpose.
We use the QMD Hamiltonian of Ref. . In our simulations, we consider a system with protons, neutrons, and charge-neutralizing background electrons in a cubic box with periodic boundary condition. Here, we shall focus on the case of the proton fraction $x\simeq 0.39$ (total number of nucleons $N=3328$ with 1312 protons and 2016 neutrons).
We simulate the compression of the bcc phase of spherical nuclei in the collapse \[see Ref. for detailed procedures\]. Starting from $\rho=0.15\rho_0$, we increase the density by changing the box size $L$ (the particle positions are rescaled at the same time). Here the average rate of the compression is $\lesssim\mathcal{O}(10^{-6})\ \rho_0/($fm$/c)$ yielding the time scale of $\gtrsim 10^{5}$ fm$/c$ to reach the typical density region of the phase with rod-like nuclei. This is much larger than the time scale of the change of nuclear shape and thus the dynamics observed in our simulation is determined by the intrinsic physical properties of the system. We perform adiabatic compression and isothermal compression at various temperatures \[see Ref. for details\]. In all the cases, we observe the formation of rod-like nuclei; here we show a typical example in which we obtain a clear lattice structure of the rod-like nuclei.
Results
=======
Figure \[snapshot\] shows the snapshots of the formation process of the pasta phase in adiabatic compression. Here, we start from initial condition at $\rho=0.15 \rho_0$ and $T=0.25$ MeV ($t=0$ fm$/c$) \[Fig. \[snapshot\](a)\]. At $\rho\simeq 0.243 \rho_0$ \[Fig. \[snapshot\](c)\], the first pair of two nearest-neighbor nuclei start to touch and fuse (dotted circle), and then form an elongated nucleus \[see, e.g., Fig.\[snapshot\](d)\]. After multiple pairs of nuclei become such elongated nuclei, we observe zigzag structure as shown in Fig.\[snapshot\](d). Then these elongated nuclei stick together \[see Figs. \[snapshot\](e) and (f)\], and all the nuclei fuse to form rod-like nuclei as shown in Fig. \[snapshot\](g). Finally, we obtain a triangular lattice of rod-like nuclei after relaxation \[Figs. \[snapshot\](h-1) and (h-2)\].
Note that before nuclei deform to be elongated due to the fission instability, they stick together keeping their spherical shape \[see Fig. \[snapshot\](c)\]. Besides, in the middle of the transition process, pair of spherical nuclei get closer to fuse in a way such that the resulting elongated nuclei take a zigzag configuration and they further connect to form wavy rod-like nuclei. This feature is observed in all the other cases in which we obtain a clear lattice structure of rod-like nuclei, and the above scenario of the transition process is qualitatively the same also for those cases. It is very different from a generally accepted picture that all the nuclei elongate in the same direction along the global axis of the resulting rod-like nuclei and they join up to form straight rod-like nuclei [@review].
Instead of such a scenario, we have shown that the pasta phases are formed in the following process \[see Ref. for details\]. When nearest neighbor nuclei are so close that the tails of their density profile overlap with each other, net attractive interaction between these nuclei starts to act. This internuclear attraction leads to the spontaneous breaking of the bcc lattice and triggers the formation of the pasta phases.
Simulating Pasta Phases by Ultracold Fermi Gases
================================================
In the remaining part of this article, we shall discuss the connection between the pasta phases and the ultracold atomic Fermi gases. Pasta phases can exist also in crusts of neutron stars. There, pasta nuclei are immersed in background electrons and a gas of dripped neutrons, which is regarded to be in a superfluid state.
It has been pointed out that the low density neutron matter shows the same universal property as unitary Fermi gases provided that the interparticle separation $r_s$ between neutrons is much smaller than the absolute value of the scattering length $a\simeq -18.5$ fm and is much larger than the effective range $r_e\simeq 2.7$ fm [@bertsch; @baker] \[see also Refs. \]. Here, “universal” means that properties do not depend on details of the nuclear potential.
In dripped neutron gas in the pasta phases, the density is $\rho_n
=k_{\rm F}^3/(3\pi^2) \gtrsim 0.05$ fm$^{-3}$ and thus the interparticle separation $r_s\gtrsim 1.7$ fm is comparable to $r_e$ (here, $k_{\rm F}$ is the Fermi wave vector of the ideal Fermi gas with the same density). However, Schwenk and Pethick [@schwenk; @pethick] have shown that in such a region of $k_{\rm
F}r_e\sim 1$, corrections from the universality in the equation of state is accurately described within the effective range expansion and no further details of nuclear forces are not needed. The energy per particle $E/N$ can be expressed in the same form as the universal case but with a $r_e$-dependent factor $\xi(k_{\rm F}r_e)$: $E/N =
\xi(k_{\rm F}r_e)\ (3/5) E_{\rm F}$ with $E_{\rm F}\equiv
\hbar^2k_{\rm F}^2/2m$ and $m$ being the fermion (neutron, in the present case) mass. This means that the dripped neutron gas in the pasta phases can be simulated by cold atomic Fermi gases with a narrow Feshbach resonance. At qualitative level, we can also expect that unitary Fermi gases would provide a suggestive guidance for exploring interesting physical effects related to the dripped superfluid neutrons in the pasta phases.
In Refs. and , we have studied superfluid unitary Fermi gases in a one-dimensional (1D) periodic potential $V_{\rm ext}(z)=s E_{\rm R} \sin^2{(q_{\rm B}z)}$, where $s$ is the laser intensity, $E_{\rm R}=\hbar^2q_{\rm B}^2/2m$ is the recoil energy, $q_{\rm B}=\pi/d$ is the Bragg wave vector, and $d$ is the lattice constant. This setup resembles superfluid neutrons in the pasta phase with slab-like nuclei, where $\rho_n\simeq 0.5\rho_0 \simeq
0.08$ fm$^{-3}$ and $d\simeq 15$ – 20 fm and thus $E_{\rm F}/E_{\rm
R}=(k_{\rm F}/q_{\rm B})^2 \sim 40$ – 70. The strength $V_0\equiv s E_{\rm R}$ of the optical lattice corresponds to the depth of the bottom of the conduction band of neutrons measured from the average potential energy outside nuclei. Since the neutron Fermi energy $E_{{\rm F}, n}$ for the density of neutrons inside nuclei is $E_{\rm F,n}\sim 35$ – 40 MeV and the neutron chemical potential $\mu_n$ inside nuclei is $\mu_n \sim 10$ – 15 MeV, we estimate $V_0 \sim |\mu_n-E_{{\rm F}, n}| \sim 25$ MeV. Thus, we obtain $s\sim 25$ – 50.
In Fig. \[fig\_kinv\_meff\_uni\], we show the incompressibility $\kappa^{-1}=n\partial_n^2 e$ and the effective mass $m^*=n(\partial_P^2 e)^{-1}$ of the unitary Fermi gas. Here $e=e(n,P)$ is the energy density averaged over the unit cell as a function of the average (coarse-grained) density $n$ and the quasimomentum $P$ of the superflow in the $z$ direction.
The rapid reduction of $\kappa^{-1}$ with decreasing $E_{\rm F}/E_{\rm
R}$ at small $E_{\rm F}/E_{\rm R}$ is due to the formation of bosonic molecules induced by the periodic potential. Maximum of $\kappa^{-1}$ and $m^*$ around $E_{\rm F}/E_{\rm R}\sim 1$ is the effect of the energy band gap. Since the tunneling rate through the barriers, which relates to $m^*$, depends on the barrier height exponentially, $m^*$ increases drastically for larger $s$. The enhancement of $m^*$ has been found also in dripped neutron gas in the crust of neutron stars [@chamel].
Taking account of the density dependence of $\xi$ in the case of the large effective range, we derive $\kappa^{-1}$ and $m^*$ for $sE_{\rm
R}/E_{\rm F} \ll 1$ using the equation of state, $E/N = (3/5) \xi
E_{\rm F}$, within the hydrodynamic theory. In the relevant region of $k_{\rm F}r_e \gtrsim 3$, $\partial_n^2\xi$ is negligible, and keeping up to the first order of $\partial_n\xi$ ($n\xi^{-1}\partial_n\xi
\lesssim 0.1$ in this region [@schwenk]), we obtain $$\kappa^{-1} \simeq \frac{2}{3}\xi
E_{\rm F} \left\{ 1+\frac{1}{32} \xi^{-2}
\left(\frac{sE_{\rm R}}{E_{\rm F}}\right)^2
+ 3n\xi^{-1}\partial_n\xi
\left[1+\frac{1}{32}\left(\frac{sE_{\rm R}}{E_{\rm F}}\right)^2\right]
\right\},
\label{expansionk-1}$$ and $$\frac{m^*}{m} \simeq 1+\frac{9}{32} \xi^{-2} %(k_{\rm F}r_e)
\left(\frac{s E_{\rm R}}{E_{\rm F}}\right)^2
\ .
\label{expansionm}$$ Since $\xi$ is a monotonically increasing function of $k_{\rm F}r_e$ [@schwenk], $m^*$ for non-zero $r_e$ is smaller than that of unitary Fermi gases.
Acknowledgements {#acknowledgements .unnumbered}
================
Works reported in this paper have been done in collaborations with H. Sonoda, K. Sato, K. Yasuoka, T. Ebisuzaki, F. Dalfovo, G. Orso, F. Piazza, L. P. Pitaevskii, and S. Stringari. We used MDGRAPE-2 and -3 of the RIKEN Super Combined Cluster System, WIGLAF at the University of Trento, and BEN at ECT$^*$. We were supported by the JSPS, by the MEXT through Grant No. 19104006, by EuroQUAM-FerMix, by MiUR, by CNR, and by the EC Sixth Framework Programme.
[99]{} S. A. Colgate and R. H. White, Astrophys. J. [**143**]{} (1966), 626.
H. A. Bethe, Rev. Mod. Phys. [**62**]{} (1990), 801.
D. G. Ravenhall, C. J. Pethick, and J. R. Wilson, Phys. Rev. Lett. **50** (1983), 2066.
M. Hashimoto, H. Seki, and M. Yamada, Prog. Theor. Phys. **71** (1984), 320.
K. Oyamatsu, contribution to this volume.
K. Nakazato, contribution to this volume.\
K. Nakazato, K. Oyamatsu, and S. Yamada, Phys. Rev. Lett. [**103**]{} (2009), 132501.
H. Sonoda [*et al.*]{}, Phys. Rev. C [**75**]{} (2007), 042801(R).
G. Watanabe, K. Iida, and K. Sato, Nucl. Phys. [**A687**]{} (2001), 512.
C. J. Horowitz, M. A. Pérez-García, and J. Piekarewicz, Phys. Rev. C [**69**]{} (2004), 045804.\
C. J. Horowitz [*et al.*]{}, [*ibid.*]{} [**70**]{} (2004), 065806.
G. Watanabe [*et al.*]{}, Phys. Rev. Lett. **103** (2009), 121101.
C. J. Pethick and D. G. Ravenhall, Annu. Rev. Nucl. Part. Sci. [**45**]{} (1995) 429.
S. Brandt, Master thesis, Copenhagen Univ. (1985).
T. J. Bürvenich, I. N. Mishustin, and W. Greiner, Phys. Rev. C [**76**]{} (2007), 034310.
J. Aichelin, Phys. Rep. [**202**]{} (1991), 233.
G. Watanabe [*et al.*]{}, Phys. Rev. C [**69**]{} (2004), 055805.\
H. Sonoda [*et al.*]{}, Phys. Rev. C [**77**]{} (2008), 035806.
G. Watanabe [*et al.*]{}, Phys. Rev. C [**66**]{} (2002), 012801(R); Phys. Rev. C [**68**]{} (2003), 035806; Phys. Rev. Lett. [**94**]{} (2005), 031101.
T. Maruyama [*et al.*]{}, Phys. Rev. C [**57**]{} (1998), 655.
G. F. Bertsch, in the announcement of the Tenth International Conference on Recent Progress in Many-Body Theories, (1999) unpublished.
G. A. Baker, Jr., Phys. Rev. C [**60**]{} (1999), 054311.
A. Gezerlis and J. Carlson, Phys. Rev. C [**77**]{} (2008), 032801(R).
T. Abe and R. Seki, Phys. Rev. C [**79**]{} (2009), 054003.
C. J. Pethick, contribution to this volume.
A. Schwenk and C. J. Pethick, Phys. Rev. Lett. [**95**]{} (2005), 160401.
G. Watanabe [*et al.*]{}, Phys. Rev. A [**78**]{} (2008), 063619.
G. Watanabe [*et al.*]{}, Phys. Rev. A [**80**]{} (2009), 053602.
N. Chamel, .
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---
abstract: 'We present preliminary numerical evidence that the physical conditions in high-mass star forming regions can arise from global gravitational infall, with the velocity dispersions being caused primarily by infall motions rather than random turbulence. To this end, we study the clumps and cores appearing in the region of central collapse in a numerical simulation of the formation, evolution, and subsequent collapse of a dense cloud out of a transonic compression in the diffuse atomic ISM. The clumps have sizes $\sim 1$ pc, masses of several hundred $M_\odot$, and three-dimensional velocity dispersions $\sim 3$ km s$^{-1}$, in agreement with typical observed values for such structures. The clumps break down into massive cores of sizes $\sim 0.1$ pc, densities $\sim 10^5$, masses 2-300 $M_\odot$, with distributions of these quantities that peak at the same values as the massive core sample in a recent survey of the Cygnus X molecular cloud complex. Although preliminary, these results suggest that high-mass star forming clumps may be in a state of global gravitational collapse rather than in equilibrium supported by strong turbulence.'
author:
- 'E. Vázquez-Semadeni, J. Ballesteros-Paredes, R. S. Klessen, A. K. Jappsen'
title: 'Massive Star Forming Regions: Turbulent Support or Global Collapse?'
---
Introduction {#sec:intro}
============
High-mass star forming regions are characterized by more extreme physical conditions than their low-mass counterparts [e.g. @GL99; @Kurtz_etal00; @Beuther_etal07], having “clumps” of sizes 0.2–0.5 pc, mean densities $n \sim 10^5$ cm$^{-3}$, masses between 100 and 1000 $M_\odot$, and velocity dispersions ranging between 1.5 and 4 km s$^{-1}$. In turn, these clumps break down into even denser “cores” that are believed to be the immediate precursors of single or gravitationally bound multiple massive protostars. The high velocity dispersions of these clumps are generally interpreted as strong turbulence that manages to support the clumps against gravity [e.g., @GL99; @MT03]. However, the notion of “turbulent support” is difficult to maintain at the scales of these cores. Turbulence is a flow regime in which the largest velocity differences are associated with the largest separations [e.g. @Frisch95], and moreover in the case of supersonic turbulence the clumps are expected to be formed by large-scale compressive motions, so that the turbulence is likely to have a strong compressive component [@HF82; @BVS99; @BP_etal08; @VS_etal08]. So, it is difficult to imagine a turbulent velocity field inside the clumps that is completely random in such a way as to only provide support against gravity – the compressive component may rather foster core contraction. In addition, numerical simulations of cloud formation in the diffuse atomic ISM [e.g. @VPP95; @VPP96; @PVP95; @BVS99; @BHV99; @AH05; @Heitsch_etal05; @Heitsch_etal06; @VS_etal06; @VS_etal07; @HA07] and of star formation in turbulent, self-gravitating clouds [e.g. @KHM00; @HMK01; @BBB03; @VKSB05; @VS_etal07] show that the velocity fields are organized at all scales, exhibiting a continuity from the large scales outside the clumps all the way to their interiors.
The large-scale compressions can be of turbulent origin (e.g., passing spiral arm shocks, supernova shells, or simply the general transonic turbulence in the diffuse medium) or of gravitational origin [e.g., large-scale gravitational or magneto-gravitational instabilities; @Elm91; @KOS02; @FBK08]. The self-gravitating simulations mentioned above exhibit gravitationally driven motions up to the largest scales. In particular, @VS_etal07 [hereafter Paper I] have presented simulations of molecular cloud formation by generic compressions in the diffuse atomic ISM and of its subsequent collapse and star-forming stage, using the SPH/N-body code GADGET [@Springel_etal01], complemented with parameterized heating and cooling taken from @KI02 and a sink-particle prescription [@Jappsen_etal05]. In these simulations, the clouds that formed acquired their initial turbulence from instabilities of the compressed layer [@Vishniac94; @WF98; @KI02; @Heitsch_etal05; @Heitsch_etal06; @VS_etal06], but soon they became gravitationally unstable and began contracting. This contraction phase was nevertheless characterized by a virial-like energy balance with $|E_{\rm grav}| \sim 2 E_{\rm kin}$, which was however due to the gravitational contraction, not to virial equilibrium. During the global contraction, clumps produced by the initial turbulence proceeded to collapse on their own, forming what resembled low-mass star forming regions.
In Paper I, we speculated that the global contraction might be halted by stellar energy feedback before the global collapse was completed. However, here we forgo that speculation, and take the simulation at face value, presenting a preliminary study of the physical conditions in the region where the global collapse finally converges, showing that they resemble the physical conditions of high-mass star forming regions, thus suggesting that such regions may be in generalized gravitational collapse rather than in a state of turbulent “support”.
The numerical model {#sec:model}
===================
The simulation we consider is the one labeled L256$\Delta v$0.17 in Paper I. We refer the reader to that paper for details. Here we just mention that it is an SPH simulation with self-gravity, parameterized heating and cooling implying a thermally bistable medium, using $3.24
\times 10^6$ particles, using sink particles, and initially set up to produce a collision of streams of diffuse gas (at the same density as their surroundings) that induces a transition to the cold, dense phase and the generation of turbulence in the dense gas. The size of the numerical box was 256 pc, and the inflow velocity of the colliding streams was 1.25 times the sound speed in the ambient gas, which had $T
= 5000$ K and a mean density of $n = 1$ cm$^{-3}$. The turbulent cloud eventually reached densities typical of molecular gas, and began to contract gravitationally. The cloud had a flattened shape during most of its evolution, and star formation began at $t \sim 17$ Myr in the periphery of the cloud, where secondary compression produced by the gas squirting off the collision site produced the highest initial densities. Animations showing the large-scale evolution of this simulation can be found in the electronic edition of Paper I.
By $t = 23.4$ Myr, the global collapse is completed, although the residual turbulence causes the motions to have a random component, so that the collapse center spans several parsecs across. Figure \[fig1\] shows a column density of the central 50 pc of this simulation in the $y$-$z$ plane at $t=23.4$ Myr, integrating over the central 8 pc along the $x$ direction. The dense cloud is seen near the center of the image, with streams of gas still infalling onto it. A region 8 pc on a side containing the cloud is indicated by the square, for which animations can be found at [http://www.astrosmo.unam.mx/$\sim$e.vazquez/turbulence/movies.html]{}. These animations show the evolution of the central 8 pc of the simulation for $22.1 \le t \le 24.7$ Myr, showing a violent collapsing evolution in which infalling clumps of gas interact but do not entirely merge, but rather undergo shredding and distortion. It is this violent region that we analyze in the next section.
Physical conditions of clumps and cores in the collapse center {#sec:phys_cond_ctr}
==============================================================
Figures \[fig2\]a and \[fig2\]b show two views of two parsec-sized clumps within the 8-pc cloud, to we refer as “Clump A” and “Clump B”. Note that these are just cubic boxes enclosing the dense clumps, rather than actual clumps defined by any clump-finding algorithm. Their properties can be compared with the observed typical properties of clumps in high-mass star forming regions. We do this by interpolating the SPH data for the central 8-pc region into a fixed grid with a resolution of $256^3$. Clump A, which has a size of 1.5 pc per side, has a mass ${\cal M} = 1400 M_\odot$, a mean density of $\langle n
\rangle = 1.27 \times 10^4$ cm$^{-3}$, and a three-dimensional velocity dispersion $\sigma = 3.6$ km s$^{-1}$. Clump B, in turn, has a linear size of 0.8 pc, a mass ${\cal M} = 300 M_\odot$, a mean density of $\langle n \rangle = 1.72 \times 10^4$ cm$^{-3}$, and a velocity dispersion $\sigma = 2.8$ km s$^{-1}$. So, in general these properties compare well with those quoted in §\[sec:intro\], except perhaps for slightly lower mean densities than typical, which can be understood as a consequence of our usage of cubic boxes rather than clumps. The boxes include some lower-density gas. On the other hand, our densities fare in well with those reported for massive starless clumps in the Cygnus X molecular complex by @Motte_etal07.
Within these clumps, we identify cores by applying a simple clump-finding algorithm based on finding connected sets of grid points whose densities are above a certain threshold. For this preliminary analysis, we consider a single threshold $n_{\rm thr} = 5 \times 10^4$ cm$^{-3}$, leaving us with 20 cores, 14 of which have $M > 4 M_\odot$. For each core, we measure its mass, mean density, and velocity dispersion, and estimate its size as $R
\approx (3 V/4 \pi)^{1/3}$, where $V$ is its volume. These properties can be compared with those reported by recent surveys of cores in high-mass star forming regions, such as that by @Motte_etal07 for the Cygnus X region. These authors give these data for a sample of 129 dense cores with sizes $\sim 0.1$ pc, masses 4–950 $M_\odot$, and mean densities $\sim 10^5$ cm$^{-3}$. In order to compare their data to the cores in our clumps, we fortuitously select the first 57 cores (those in the first page) in their Table 1 and show the distribution of their properties in the histograms presented in Fig. \[fig3\] by the [*dotted*]{} lines. Superimposed on these histograms, the [*solid*]{} lines show the corresponding distributions for the 14 cores more massive than $4 M_\odot$ in Clumps A and B of our simulation. We see that, although the distributions of the Cygnus X cores are in general broader than those of the cores in our simulation, the peaks of the distributions match for size, mass, and mean density.
The fact that the distribution of densities for the cores in the simulation is narrower than the Cygnus X one is most likely a result of our sample being significantly smaller than that for Cygnus X, and of our having considered only a single density threshold for defining the cores. Our previous experience with this clump-finding method is that the mean density of the cores found is generally within less than one order of magnitude of the threshold density. This limitation is also the probable cause of the absence of the small-size tail of the cores in the simulation, since smaller cores should appear when higher density thresholds are considered. So, in order to obtain a wider range of densities, masses and sizes, it is necessary to consider a suite of density thresholds. We plan to do it in the final study, to be presented elsewhere.
Conclusions {#sec:conclusions}
===========
In this contribution we have presented preliminary numerical evidence that the physical conditions in high-mass star forming regions can arise from global gravitational infall, with the velocity dispersions being caused primarily by infall motions rather than random turbulence. The evidence comes from the first study of core properties in a simulation of the entire evolution of a molecular cloud, from its formation in the diffuse atomic ISM to its gravitational collapse. Although the analysis presented here is only preliminary, it is consistent with recent suggestions, based on comparisons between simulations and observations, that molecular clouds [e.g., @HB07] and clumps [e.g. @PHA07] may be in a state of gravitational collapse. If confirmed, these suggestions point towards a return to the original suggestion by @GK74 that the observed linewidths in molecular clouds are due primarily to gravitational contraction. This suggestion was dismissed by @ZP74 through the argument that this would imply a much larger average star formation rate in the Galaxy than observed. However, it is possible that this criticism may be overcome if magnetic field fluctuations in the clouds imply that some parts of them are magnetically supported so that only the non-supported parts parts of the clouds undergo collapse [@HBB01; @Elm07], and the star formation rate is then regulated by stellar feedback, with globally collapsing motions arising only for those regions that manage to “percolate” through the field fluctuations and the stellar-feedback motions. We plan to perform simulations including magnetic support and stellar feedback in future studies to investigate this possibility.
The numerical simulation was performed in the cluster at CRyA-UNAM acquired with CONACYT grants to E.V.-S. 36571-E and 47366-F. The visualization was produced using the SPLASH code [@Price07]. We also thankfully acknowledge financial support from grants UNAM-PAPIIT 110606 to J. B.-P., and SFB 439, “Galaxies in the Young Universe”, funded by the German Science Foundation (DFG), to R.S.K.
Audit, E. & Hennebelle, P. 2005, A&A 433, 1
Ballesteros-Paredes, J., Hartmann, L. & Vázquez-Semadeni, E. 1999b, ApJ 527, 285
Ballesteros-Paredes, J., Vázquez-Semadeni, E., & Scalo, J. 1999a, ApJ, 515, 286
Ballesteros-Paredes, J., Gazol, A., Vázquez-Semadeni, E., & Kim, J. 2008, to be submitted to MNRAS
Bate, M. R., Bonnell, I. A., & Bromm, V. 2003, MNRAS 336, 705
Beuther , H., Churchwell, E. B., McKee, C. F., Tan, J. C. 2007, in Protostars and Planets V, eds. B. Reipurth, D. Jewitt, and K. Keil (Tucson: University of Arizona Press), 165
Elmegreen, B. G. 1991, in The Physics of Star Formation and Early Stellar Evolution, ed. C.J. Lada and N. D. Kylafis (Dordrecht: Kluwer), 35
Elmegreen, B. 2007, ApJ, 668, 1064
Field, G., Blackman, E. G, Keto, E. R. 2008, MNRAS, submitted (astro-ph/0601574v4)
Frisch, U. 1995, Turbulence. The legacy of A.N. Kolmogorov (Cambridge: Cambridge University Press)
Garay, G. & Lizano, S. 1999, PASP, 111, 1049
Goldreich, P., & Kwan, J. 1974, ApJ 189, 441
Hartmann, L., Ballesteros-Paredes, J., & Bergin, E. A. 2001, ApJ, 562, 852
Hartmann, L. & Burkert, A. 2007, ApJ, 654, 988
Heitsch, F., Burkert, A., Hartmann, L., Slyz, A. D. & Devriendt, J. E. G. 2005, ApJ 633, L113
Heitsch, F., Slyz, A., Devriendt, J., Hartmann, L., & Burkert, A. 2006, ApJ 648, 1052
Heitsch, F., Mac Low, M. M., & Klessen, R. S. 2001, ApJ, 547, 280
Hennebelle, P. & Audit, E. 2007, A&A, 465, 431
Hunter, J. H., Jr., & Fleck, R. C., Jr. 1982, ApJ, 256, 505
Jappsen, A.-K., Klessen, R. S., Larson, R. B., Li, Y., and Mac Low, M.-M. 2005, A&A 435, 611
Kim, W.-T., Ostriker, E. C. & Stone, J. M. 2002, ApJ, 581, 1080
Klessen, R. S., Heitsch, F., & MacLow, M. M. 2000, ApJ, 535, 887
Koyama, H. & Inutsuka, S.-I. 2002, ApJ, 564, L97
Kurtz, S., Cesaroni, R., Churchwell, E., Hofner, P., Walmsley, C. M. 2000, in Protostars and Planets IV, eds. V. Mannings, A. P. Boss, S. S. Russell (Tucson: University of Arizona Press), 299
McKee, C. F., Tan, J. C. 2003, ApJ, 585, 850
Motte, F., Bontemps, S., Schilke, P., Schneider, N., Menten, K. M., Broguière, D. 2007, A&A, 476, 1243
Passot, T., Vázquez-Semadeni, E., & Pouquet, A. 1995,
Peretto, N., Hennebelle, P., André, P. 2007, A&A, 464, 983
Price, D. J. 2007, PASA, 24, 159
Springel, V., Yoshida, N., White, S. D. M. 2001, New Astron., 6, 79
Vázquez-Semadeni, E., Gómez, G. C., Jappsen, A. K., Ballesteros-Paredes, J., González, R. F., & Klessen, R. S. 2007, ApJ, 657, 870 (Paper I)
Vázquez-Semadeni, E., González, R. F., Ballesteros-Paredes, J., Kim, J. 2008, MNRAS, to be submitted
Vázquez-Semadeni, E., Kim, J., Shadmehri, M. & Ballesteros-Paredes, J. 2005, ApJ, 618, 344
Vázquez-Semadeni, E., Passot, T., & Pouquet, A. 1995, ApJ, 441, 702
Vázquez-Semadeni, E., Passot, T., & Pouquet, A. 1996, ApJ, 473, 881
Vázquez-Semadeni, E., Ryu, D., Passot, T., González, R. F., & Gazol, A.., 2006, ApJ, 643, 245
Vishniac, E. T. 1994, ApJ, 428, 186
Walder, R. & Folini, D. 1998 A&A, 330, L21
Zuckerman, B. & Palmer, P. 1974, ARA&A, 12, 279
|
---
abstract: 'We investigate the importance of the pseudo-continuum bound-free opacity from hydrogen atoms in the atmospheres of cool white dwarfs. This source of absorption, when calculated by the occupation probability formalism applied in the modeling of white dwarf atmospheres with $T_{\rm eff}\rm <17000 \, K$, dominates all other sources of opacity at optical wavelengths. This is unrealistic and not observed. On the other hand, a significant flux suppression in the blue part of the spectra of cool white dwarfs has been reported, and mainly interpreted as a result of the pseudo-continuum absorption from atomic hydrogen. We investigate this problem by proposing a new, more realistic approach to calculating this source of opacity. We show that this absorption is orders of magnitude smaller than that predicted by current methods. Therefore, we rule out the pseudo-continuum opacity as a source of the flux deficiency observed in the spectra of cool white dwarfs.'
author:
- 'Piotr M. Kowalski'
title: 'The Pseudo-continuum Bound-free Opacity of Hydrogen and its Importance in Cool White Dwarf Atmospheres'
---
Introduction
============
Cool white dwarfs are among the oldest objects in the Universe. The detection of large populations of these old stars in our Galaxy gives us a unique opportunity to use them as chronometers. To achieve this goal, we need reliable atmosphere models for cool white dwarfs. Such models would allow for a better determination of the atmospheric parameters of observed white dwarf stars, tracking the evolution of their atmospheric composition, and to obtain better boundary conditions for modeling internal structures and the cooling of these stars [@FN00]. One problem in modeling cool white dwarf atmospheres is the implementation of the Däppen-Anderson-Mihalas (hereafter DAM) model [@DAM] in calculating the pseudo-continuum bound-free opacity from the ground state of atomic hydrogen. When this source of opacity is calculated following the DAM prescription with the occupation probabilities of Hummer & Mihalas (1988, hereafter HM) for white dwarfs with $T_{\rm eff}\rm<17000 \, K$, it dominates all other absorption processes at optical wavelengths. Such strong absorption is completely unrealistic and is not observed [@Bergeron01; @Bergeron97]. Nonetheless, this process may still be significant in white dwarf atmospheres. In fact, @Bergeron97 suggested that this absorption mechanism may be responsible for the observed flux suppression in the blue part of the spectra of cool white dwarfs. Following that idea, @Bergeron01 introduced an arbitrary damping function to the DAM pseudo-continuum opacity to obtain good fits of the models to the photometry of selected stars. However, the damping function must be adjusted for each star, indicating how unsuitable the DAM model is. As a result, the pseudo-continuum bound-free opacity of hydrogen atoms is ignored in the modeling of cool white dwarf atmospheres. In this paper we readdress this issue by computing this absorption mechanism with a much more realistic physical model to assess its importance in cool white dwarf atmospheres.
The pseudo-continuum bound-free opacity results from the perturbation of hydrogen atoms in their ground state by their interaction with other particles in a dense medium [@DAM]. Such perturbations result in a lowering of the ionization barrier of some of the hydrogen atoms and the possibility of a bound-free transition caused by photon with energies that are smaller than the ionization potential of the isolated hydrogen atom. DAM consider this process in the framework of the occupation probability formalism proposed by HM. They constructed a simple model for the optical properties of an interacting hydrogen medium. In the weakly ionized atmospheres of cool white dwarfs, the hydrogen atoms in higher excited states are perturbed mostly by neutral particles [@BWF91] and eventually destroyed (pressure ionized) by excluded volume interactions. The effect is stronger for excited states because the size of a hydrogen atom is a monotonically increasing function of the principal quantum number. This model for the pseudo-continuum opacity is not very precise, as it treats the inter-particle interactions and the associated lowering of the ionization potential only in terms of the average sizes of the particles. We will show that a quantum mechanical description of the interactions between hydrogen atoms and perturbers, absent in the simple DAM model, is necessary to obtain a physically realistic value of the pseudo-continuum opacity of hydrogen atoms in the atmospheres of white dwarf stars.
To accurately account for the effect of lowering the hydrogen ionization barrier in a dense medium, we consider short-range, binary collisions. The ionization energy of the perturbed hydrogen atom is given by the ionization energy of the H-perturber pair. This energy is given by the difference between the exact interaction energy curves calculated quantum mechanically of the neutral and singly ionized dimers (interacting pairs). The probability of having a pair colliding with a given collision distance can be determined from the interaction potential of the neutral dimer. The resulting probabilities of lowered hydrogen ionization potential for the physical conditions of white dwarfs atmospheres are orders of magnitude smaller than those obtained from the DAM model with the HM occupation probabilities. In the next section we discuss the DAM approach to the pseudo-continuum opacity. Our new calculations of this absorption mechanism and its application to the physical conditions found in the atmospheres of cool white dwarfs are presented in sections 3 and 4, respectively.
The pseudo-continuum opacity model of Däppen-Anderson-Mihalas
=============================================================
The DAM model is based on the occupation probability formalism introduced by HM for calculating thermodynamical properties of a non-ideal gas. HM introduced the non-ideal effects on the equation of state by modifying the internal partition functions of bound species. The modified partition function of the hydrogen atom is given by $$Z_{\rm H}=\sum_{i}g_{i}\omega_{i}e^{-\epsilon_{i}/k_{B}T}=\sum_{i}\omega_{i}\gamma_{i},$$ where $k_{B}$ is the Boltzmann constant, $i$ indicates the atomic level, $g_i$ is the statistical weight, $\epsilon_{i}$ is the energy of hydrogen atom counted from the energy of the ground state ($i\rm =0$), $T$ is the temperature, and $\omega_i$ is the occupation probability. In the HM model, $\omega_{i}$ represents the strength of non-ideal gas perturbations on an atomic level *i . Its value for the physical conditions found inside weakly ionized cool white dwarf atmospheres is given by the excluded volume interactions model [@BWF91] and decreases for higher principal quantum number (HM, Equation 3.4). The number of hydrogen atoms in the excited state $i$ is expressed as $$\frac{n_{i}}{n_{tot}}=\frac{\omega_{i}\gamma_{i}}{Z_{\rm H}}, \label{3}$$ where $n_{tot}$ is the total number density of atomic hydrogen.*
DAM extended this model for the calculation of the optical properties of hydrogen atoms in the partially ionized plasma. They interpreted the factor $1-\omega_{i}$ as the fraction of hydrogen atoms with atomic level $i$ being “dissolved," i.e. sufficiently perturbed to describe an unbound electron and ion. They assigned this value to the fraction of free-electrons states available for a bound-free transition to level $i$. DAM used this interpretation to derive that for the absorption of a photon associated with the transition from level $i$ to level $j$, the probability that level $j$ is bound, and the transition is “bound-bound," is $\omega_j/\omega_i<1$. Therefore, the probability of having a “bound-free" transition is $$P_{bf}=1-\frac{\omega_j}{\omega_i}.$$ To describe the pseudo-continuum absorption to the continuum energy levels localized between two discrete but “dissolved" levels of the hydrogen atom, DAM introduced the pseudo-level $n^{*}$ with occupation probability $\omega_{n^{*}}$. The latter is calculated by the interpolation between the $\omega_{i}$’s. Therefore, the probability that, during the $i$ to $n^{*}$ transition, the absorbing electron goes into the continuum is given by (Eq. 32 of DAM) $$P_{bf}=1-\frac{\omega_{n^{*}}}{\omega_i}. \label{9a}$$
The dominant perturbers in the atmospheres of cool white dwarfs are neutral particles. Therefore in calculating the DAM pseudo-continuum opacity we used the hard sphere model for $\omega_i$ of HM (Section IIIa). In Figure 1 we present the main opacity sources at the photosphere of white dwarf star LP 380-5 considered for the potential presence of pseudo-continuum opacity by @Bergeron01. The DAM pseudo-continuum opacity is much too high and dominates all other opacity sources by orders of magnitude. This is unrealistic and simply not observed. Therefore, the DAM model highly overestimates the probabilities of pseudo-continuum bound-free absorption (Eq. \[9a\]) in the atmospheres of cool white dwarf stars. In view of the difficulty with this application of the DAM model of the pseudo-continuum opacity, we propose a more realistic approach to calculate this source of absorption.
A new model for the pseudo-continuum opacity of H
=================================================
The bound-free absorption process results in the ionization of a hydrogen atom. The lowering of the hydrogen ionization potential arises from the interaction of the hydrogen atom with neighboring particles [@DAM]. However, we are interested in the pseudo-continuum opacity far from the Lyman edge, i.e. $\rm \lambda\wig>1500 \, \AA$. A significant lowering of the hydrogen ionization barrier, by more than $\sim 5 \, \rm eV$, is required for bound-free absorption at these wavelengths. Such a situation occurs in the case of rare, close collisions for which the inter-particle collision distances, $r_{c}$, are small enough that the probability ($P_c$) of finding a colliding pair with an inter-particle separation smaller than $r_{c}$ is much smaller than unity. In this case, multi-particle collisions are insignificant, as the probability of having a close $N$-particle collision is roughly $\sim P_c^{N-1}$. Therefore, it is sufficient to consider the interaction between a $\rm H$ atom and its closest neighbor only. @Allard used this approximation to successfully explain the complex shape of the Lyman $\rm \alpha$ line wings detected in the spectra of white dwarfs with $T_{\rm eff}\rm \sim 12000 \, K$.
For a given colliding pair the change in the ionization energy results from the formation of a temporary dimer, whose ionization energy differs from that of the isolated hydrogen atom, $I_0\rm =13.598 \, eV$. This modified ionization barrier $I_p$ is given by the ionization energy of a dimer calculated at a fixed inter-particle separation $r_{c}$: $$I_p(r_c)=V_{\rm H^+-pert}(r_c)-V_{\rm H-pert}(r_c),$$ where $V_{\rm H-pert}(r_c)$ and $V_{\rm H^+-pert}(r_c)$ are the energies of the neutral and singly ionized dimers, respectively. This picture is in the spirit of the Franck-Condon principle, which states that as a result of large differences between the mass of the absorbing electron and the nuclear mass, the nuclei remain fixed during the absorption/emission of a photon [@Davydov]. The differential probability of finding such a dimer with an inter-particle separation between $r_c$ and $r_c+dr_c$ in the low density medium ($\rho \wig< 0.1 \rm \, g/cm^3$) is given by [@MS].
$$dP(r_c)=n_{pert}\left(\int_{\theta,\phi}{e^{-V_{\rm H-pert}(r_c,\theta,\phi)/k_BT} \sin \theta d \theta d \phi}\right)r_c^2dr_c , \label{11}$$
where $n_{pert}$ is a number density of perturbers, $V_{\rm H-pert}$ is the interaction energy between a hydrogen atom and the perturber localized at the position $(r_c,\theta,\phi)$ in relation to the hydrogen atom, which is assumed to be at the origin of the coordinate system. The dominant species, and therefore the main perturbers, in the atmospheres of cool white dwarfs are $\rm H$, $\rm H_2$, and $\rm He$. The sources for the potentials for the interaction of hydrogen atom with perturbers are chosen to be: $\rm H-H$ \[@KW\], $\rm H-H_2$ \[@BOO\], and $\rm H-He$ \[@SH\]. The corresponding potential curves for the singly ionized dimers are chosen to be: $\rm H_2^{+}$ \[@BR\], $\rm H_3^{+}$ \[@PROS\], and $\rm HeH^{+}$ \[@GR\]. These potentials are plotted on Fig. 2. As a result of the *gerade/ungerade symmetry in the $\rm H-H$ interaction, we have to consider both the *bound and the *anti-bound $\rm H-H$ potential energy curves. For the singly ionized dimers we choose the ground state potential energy curves because the upper energy curves (like *anti-bound state for $\rm H_2^+$, Fig. 2.) would result in a much smaller decrease in the ionization energy for a given $r_c$. This decrease occurs with much smaller probability than the same change in the ionization potential that results from the ionic ground states at a much larger $r_{c}$, and therefore can be neglected. The resulting ionization energy $I_p(r_c)$ for a given dimer as a function of the collision distance are plotted on Fig. 3. The ionization energy can decrease to $\rm \sim 5 \, eV$, which occurs at very small $r_{c}$. Due to the strong repulsion in the neutral dimer at short separation (Fig. 2), such a large lowering of the ionization energy occurs with very small probability, as for small $r_c$ the Boltzmann factor $e^{-V_{\rm H-pert}/k_BT}<<1$ in Eq. (\[11\]). Moreover, in the case of the $\rm H-H$ interaction, only the *anti-bound potential leads to a significant decrease of the ionization barrier (Fig. 2).*****
The probability of a hydrogen atom having an ionization energy $I_p$ sufficiently smaller for photo-ionization to be caused by a photon of frequency $\nu$ is $$P(\nu)=\int_{I_p(r_c)<h\nu} dP(r_c),$$ where the integration is performed over the range of separations $r_c$ such that $I_p(r_c)<h\nu$. The resulting probabilities, as a function of photon wavelength $\rm \lambda$, of having a bound-free transition for $\rm H$ for two temperatures and three different compositions are shown in Fig. 4. The corresponding probabilities from the DAM model (also shown in Fig. 4) are a few orders of magnitude larger. We find that for a given density, collisions with $\rm H_2$ are most effective at lowering the ionization energy of $\rm H$. This is because for a given value of $I_p(r_c)$, the interaction potential $\rm V_{H-H_2}(r_c)$ computed for the orientation angle $\sim 90^{o}$ (see caption of Fig 2. for definition) is much less repulsive than $\rm V_{H-H}(r_c)$ and $\rm V_{H-He}(r_c)$.
The pseudo-continuum opacity is then obtained with $$\kappa_{bf}(\nu)=\frac{n_H}{\rho} P(\nu)\sigma^0_{bf}(\nu) \ \ \ (h\nu< \rm 13.598 \, eV)\label{333}$$ where $\sigma^0_{bf}(\nu)$ is the bound-free cross-section of the isolated hydrogen atom extrapolated beyond the Lyman edge (DAM), $n_{\rm H}$ is the number density of hydrogen atoms, and $\rho$ is the mass density. A proper calculation would consider the bound-free cross-section of the dimer as a function of inter-particle separation. To our knowledge this information is not available. However, as the photo-ionization cross-sections for $\rm H$ [@PL] and $\rm H_2$ [@FORD] differ by less than a factor of $3$ and the cross-section is the effective size of the absorbing system as seen by the photon, we estimate that the extrapolation of the bound-free cross-section of the isolated hydrogen atom beyond Lyman edge introduces an uncertainty no larger than an order of magnitude on $\kappa_{bf}(\nu)$ calculated by Eq. \[333\]. In helium-rich atmospheres, where the density can be as high as $\rm 2-3 \, g/cm^3$, equation (\[11\]) should be corrected when $\rho \rm \wig>0.1\,g/cm^3$, by a factor $e^{\rm w_{\rm H-pert}}$ [@MS], where $\rm w_{\rm H-pert}$ is a thermal potential that arises from the correlations in the dense fluid. Solving the Ornstein-Zernike equation in the Percus-Yevick approximation [@MS] for fluid $\rm He$, we have verified that in the helium-rich atmospheres of cool white dwarfs, $e^{\rm w_{\rm H-pert}}<100$. Therefore, due to correlations in the most extreme case, Eq. \[333\] leads to an underestimate of $\kappa_{bf}$ of no more than two orders of magnitude. We will see below that this does not affect our conclusions.
Importance of the pseudo-continuum opacity in cool white dwarf atmospheres
==========================================================================
Our goal is to investigate the suggestion of @Bergeron97 and @Bergeron01 that the pseudo-continuum bound-free absorption by atomic hydrogen beyond the Lyman edge may be the missing source of opacity in the atmosphere models of cool white dwarfs $(T_{\rm eff}\rm \wig< 6000 \, K)$. For this purpose, we have computed opacities for temperatures, densities and $\rm He/H$ composition that are representative of the photospheres of these stars. The pseudo-continuum opacity calculated with our model dominates all other sources of opacity at wavelengths shorter than $\rm \sim 2000\,\AA$ (Fig. 5). For these cool stars the flux at these wavelengths is extremely small and completely negligible (see Fig. 5 of @Bergeron01). We also note that the uncertainty in the photo-ionization cross-section and our neglect of the correlations in dense helium-rich atmospheres do not alter the pseudo-continuum opacity enough to make it important at $\rm \lambda>2000\,\AA$.
Fig. 1. reproduces the opacity plot of @Bergeron01 (Fig. 4 in that paper). This figure represents the main sources of opacity for the physical conditions found at the photosphere of the white dwarf star LP 380-5. Our pseudo-continuum opacity is several orders of magnitude smaller than that necessary to fit the blue spectrum ($\rm 3000-4000\,\AA$) of this star [@Bergeron01], and is completely insignificant at $\rm \lambda>2000 \, \AA$.
Conclusions
===========
The possible presence of an unknown absorption mechanism in the atmospheres of cool white dwarfs has been reported by @Bergeron01 and @Bergeron97. This opacity source has been attributed to the pseudo-continuum bound-free opacity from hydrogen atoms in their ground state that results from a lowering of the ionization potential because of inter-particle interactions in the gas. Opacity models based on the occupation probability formalism highly overestimate this source of absorption in the atmospheres of cool white dwarfs. For this reason, it is usually omitted in models. In this paper, we presented a realistic model for this absorption mechanism based on binary collisions, exact pair interaction potentials, and the ionization energy of the colliding pair. We find that the pseudo-continuum bound-free opacity decreases very rapidly with wavelength beyond the Lyman edge and becomes completely negligible beyond $\lambda \sim 2000 \, \rm \AA$ over the entire range of temperature, density and $\rm H/He$ composition relevant to cool white dwarf atmospheres. Therefore, another absorption mechanism must be invoked to explain the flux excess of the models in the blue part of the spectrum of these stars.
I thank D. Saumon for useful discussions and comments on this manuscript. This research was supported by the United States Department of Energy under contract W-7405-ENG-36.
Allard, N. F., Kielkopf, J. F., & Loeillet, B. 2004, , 424, 347 Bates, D. R., & Reid, H. G. 1968, Adv. Atom. and Molec. Phys. 4, 13 Bergeron, P. 2001, , 558, 369 Bergeron, P., Ruiz, M. T., & Leggett, S. K. 1997, , 108, 339 Bergeron, P., Wesemael, F., & Fontaine, G. 1991, , 367, 253 Boothroyd, A. I., Keogh, W. J., Martin, P. G., Peterson, M. R. 1991, J. Chem. Phys., 95, 4343 Däppen, W., Anderson, L. & Mihalas, D. 1987, , 319, 195 (DAM) Davydov, A. S. 1965, Quantum Mechanics (Oxford: Addison-Wesley), Chap. 12, Sec. 123 Fontaine, G., Brassard, P. & Bergeron, P. 2000, PASP, 113, 409 Ford, A. L., Kate, K. D. & Dalgarno, A. 1975, , 195, 819 Green, T. A. et al. 1974, J. Chem. Phys., 61, 5186 Hummer, D. G., & Mihalas, D. 1988, , 331, 794 (HM) Kolos, W., & Wolniewicz, L. 1965, J. Chem. Phys. 43, 2429 Martynov, G. A. 1992, Fundamental theory of liquids (Bristol: Adam Hilger), Chap. 5 Palenius, H. P., Kohl, J. L., & Parkinson, W. H. 1975 Phys. Rev. A. 13, 1805 Prosmiti, R. Polyansky, O. L. and Tennyson, J. 1997, Chem. Phys. Lett. 273, 107 Shalabi, A. S., Eid, Kh. M., Kamel, M. A., & El-Barbary, A. A. 1998, Physics Letters A, 239, 87
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abstract: 'Say that an edge of a graph $G$ dominates itself and every other edge adjacent to it. An edge dominating set of a graph $G=(V,E)$ is a subset of edges $E'' \subseteq E$ which dominates all edges of $G$. In particular, if every edge of $G$ is dominated by exactly one edge of $E''$ then $E''$ is a dominating induced matching. It is known that not every graph admits a dominating induced matching, while the problem to decide if it does admit it is NP-complete. In this paper we consider the problems of finding a minimum weighted dominating induced matching, if any, and counting the number of dominating induced matchings of a graph with weighted edges. We describe an exact algorithm for general graphs that runs in $O^*(1.1939^n)$ time and polynomial (linear) space. This improves over any existing exact algorithm for the problems in consideration.'
author:
- 'Min Chih Lin [^1] , Michel J. Mizrahi$^\star$'
- |
\
Jayme L. Szwarcfiter [^2]
title: 'An $O^*(1.1939^n)$ time algorithm for minimum weighted dominating induced matching'
---
Introduction
============
Under the widely accepted assumption that $P \ne NP$ there are several problems with important applications for which no polynomial algorithm exists. The need to get an exact solution for many of those problems has lead to a growing interest in the area of design and analysis of exact exponential time algorithms for NP-Hard problems [@Fo-Kr; @Woeginger:exactalgorithms]. Even a slight improvement of the base of the exponential running time may increase the size of the instances being tractable. There has been many new and promising advances in recent years towards this direction [@Bjorklund2010; @Bjorklund2007].
In this paper we give an exact algorithm for the weighted and counting version of the NP-Hard problem Dominating Induced Matching (also known as DIM or Efficient Edge Domination) which has been extensively studied [@Ca-Ce-De-Si; @Do-Lo; @Korpel; @Lu-Ko-Ta; @Lu-Ta; @Br-Le-Ra; @Br-Mo; @Br-Hu-Ne; @Ca-Ko-Lo]. Further notes about this problem and some applications related to encoding theory, network routing and resource allocation can be found in [@Gr-Sl-Sh-Ho; @Livingston].
The unweighted version of the dominating induced matching problem is known to be NP-complete [@Gr-Sl-Sh-Ho], even for planar bipartite graphs of maximum degree 3 [@Br-Hu-Ne] or regular graphs [@Ca-Ce-De-Si]. There are polynomial time algorithms for some classes, such as chordal graphs [@Lu-Ko-Ta], generalized series-parallel graphs [@Lu-Ko-Ta] (both for the weighted problem), claw-free graphs [@Ca-Ko-Lo], graphs with bounded clique-width [@Ca-Ko-Lo], hole-free graphs [@Br-Hu-Ne], convex graphs [@Korpel], dually-chordal graphs [@Br-Le-Ra], $P_7$-free graphs [@Br-Mo], bipartite permutation graphs [@Lu-Ta] (see also [@Br-Lo]).
If $P \ne NP$ it is not possible to solve this problem in polynomial time, hence it becomes important to improve the exponential algorithm in order to identify the instances that can be solved within reasonable time.
A straightforward brute-force algorithm to solve weighted DIM can be achieved in $O^*(2^n)$ time and polynomial space.
The minimum weighted DIM problem can be expressed as an instance of the maximum weighted independent set problem on the square of the line graph $L(G)$ of $G$, and also as an instance of the minimum weighted dominating set problem on $L(G)$, by slightly way described in [@Br-Le-Ra; @Milanic] for unweighted DIM problem.
The minimum weighted dominating set can be solved in $O^*(1.5780^n)$ time [@Fomin05boundingthe], while the maximum weight independent set can be solved in $O^*(1.4423^n)$ time by enumeration of all maximal independent sets [@Ts]. To the best of our knowledge there are no better method to obtain the maximum weighted independent set (a better algorithm $O^*(1.2209^n)$ for unweighted maximum independent set is due by [@Fo-Gr-Kr]). Hence the DIM problem for a graph $G$ can be solved by using this algorithm in $L^2(G)$, which runs in $O^*(1.4423^m)$ time.
For the minimum weighted DIM this algorithm behaves better than the brute-force alternative whenever $1.4423^m < 2^n$, this is, $m < 1.8926n$.
The paper [@DBLPMin] shows how to solve the DIM problem in $O^*(1.7818^n)$ time and polynomial space while the same algorithm runs in $O(n+m)$ time if the graph has a fixed dominating set. In the same work another approach based on enumerating maximal independent sets was developed and allows to solve both DIM problems (minimum weighted problem and counting problem) in $O^*(1.4423^n)$ time and polynomial space. For the counting problem, there exists algorithms such as [@Dahllof2002] which can be used to count the number of MWIS’s in $O^*(1.3247^n)$ time, leading an $O^*(1.3247^m)$ time and polynomial space algorithm to count the numbers of DIM’s.
Comparing with the straightforward brute-force algorithm, it is convenient to use it as long as $1.3247^m < 2^n$, and this occurs whenever $m < 2.4650n$.
There are NP-complete instances of the DIM problem where the number of edges in G is relatively low such as for planar bipartite graphs of maximum degree 3 [@Br-Hu-Ne], where $m \leq 1.5n$. Therefore using transformations and exact algorithms for MWIS or for counting MWIS’s is better than using the brute-force algorithm. Note however that cases where brute-force algorithm is not convenient strongly relies on the number edges. For instance, for a graph with $O(n)$ edges such that $m \approx 3n$ the MWIS algorithm behaves better than the brute-force one.
In this paper, we propose an algorithm for solving the weighted DIM problem and the counting DIM’s problem, in $O(m \cdot 1.1939^n) \in O^*(1.1939^n)$ time and $O(m)$ space in general graphs which improves over the existing algorithms for these problems. We employ techniques described in [@Fo-Kr] for the analysis of our algorithm, and as such we use their terminology.
The proposed algorithm was designed using the [*branching $\&$ reduce*]{} paradigm. More information about this design technique as well as the running time analysis for this kind of algorithms can be found in [@Fo-Kr].
Preliminaries
=============
By $G(V,E)$ we denote a *simple undirected graph* with vertex set $V$ and edge set $E$, $n=|V|$ and $m=|E|$. We consider $G$ as a *weighted* graph, that is, one in which there is a non-negative real value, denoted $weight(vw)$ assigned to each edge $vw$ of $G$. If $v \in V$ and $V' \subseteq V$, then denote by $N(v)$, the set of vertices adjacent (neighbors) to $v$, denote $d(v) = |N(v)|$ the degree of the vertex, denote by $G[V']$ the subgraph of $G$ induced by $V'$, and write $N_{V'}(v)=N(v) \cap V'$. Some special graphs or vertices are of interest for our purposes. A graph formed by two triangles having a common edge is called a [*diamond*]{}. By removing an edge incident to a vertex of degree 2 of a diamond, we obtain a [*paw*]{}. Finally, a vertex of degree 1 is called [*pendant*]{}.
Given an edge $e \in E$, say that $e$ *dominates* itself and every edge sharing a vertex with $e$. Subset $E'\subseteq E$ is an *induced matching* of $G$ if each edge of $G$ is dominated by at most one edge in $E'$. A *dominating induced matching (DIM)* of $G$ is a subset of edges which is both dominating and an induced matching. Not every graph admits a DIM, and the problem of determining whether a graph admits it is also known in the literature as *efficient edge domination problem*. The weighted version of DIM problem is to find a DIM such that the sum of weights of its edges is minimum among all DIM’s, if any. The counting version of the problem consists on counting the amount of DIM’s a graph has. It is easy to see that the weighted version and the counting version of the problem are harder than the unweighted one. If the graph $G$ has negative weights the problem can be solved using the same algorithm that solves the problem for non-negative weights. Let $-M$ be the minimum weight among all edges of $G$, modify the weights of $G$ by adding $M$ to the weights of all edges. It is not hard to see that every DIM is a maximum induced matching, and hence the number of edges of every DIM in $G$ is the same. Therefore the optimal solution for the modified graph is the same that the optimal solution for the original graph.
We assume the graph $G$ to be connected, otherwise, the DIM of $G$ is the union of the DIM’s of its connected component, and so we can restrict to the connected case.
We will use an alternative definition [@Do-Lo] of the problem of finding a dominating induced matching. It asks to determine if the vertex set of a graph $G$ admits a partition into two subsets. The vertices of the first subset are called [*white*]{} and induce an independent set of the graph, while those of the second subset are named [*black*]{} and induce an 1-regular graph.
A straightforward brute-force algorithm for finding the DIM of a graph $G$ consists in finding all bipartitions of $V(G)$, color one of the parts as [*white*]{}, the other as [*black*]{}, and checking if the result is a valid DIM. The complexity of this algorithm is $O(2^n \cdot m) \in O^*(2^n)$.
Extensions of Colorings {#sec-extensions}
=======================
Assigning one of the two possible colors, white or black, to vertices of $G$ is called a coloring of $G$. A coloring is *partial* if only part of the vertices of G have been assigned colors, otherwise it is *total*. A black vertex is [*single*]{} if it has no black neighbor, and is paired if it has exactly one black neighbor. Each coloring, partial or total, can be [*valid*]{} or [*invalid*]{}.
Next, we describe the natural conditions for determining if a coloring is valid or invalid.
:\[lemma-validation\] RULES FOR VALIDATING COLORINGS:\
The following are necessary and sufficient conditions for a coloring to be valid:
A partial coloring is valid whenever:
V1.\[itm:validI\]
: No two white vertices are adjacent, and
V2.\[itm:validII\]
: Each black vertex is either single or paired. Each single vertex has some uncolored neighbor.
A total coloring is valid whenever:
V3,\[itm:valid1\]
: No two white vertices are adjacent, and
V4.\[itm:valid2\]
: Each black vertex is paired.
\[cor-totalvalid\] There is a one-to-one correspondence between total valid colorings and dominating induced matchings of a graph.
[*Proof:*]{} It follows from the definitions. $_\triangle$
For a coloring $C$ of the vertices of $G$, denote by $C^{-1}(white)$ and $C^{-1}(black)$, the subsets of vertices colored white and black. A coloring $C'$ is an [*extension*]{} of a $C$ if $C^{-1}(black)\subseteq C'^{-1}(black)$ and $C^{-1}(white) \subseteq C'^{-1}(white)$. For $V', V'' \subset
V(G)$ if $C'$ is obtained from $C$ by adding to it the vertices of $V'$ with the color black and those of $V''$ with the color white then write $C = C' \cup BLACK(V') \cup WHITE(V'')$. Note that a [*total valid*]{} coloring can be only an extension of [*partial valid*]{} colorings and itself.
Given a partial coloring $C$, the basic idea of the algorithm is to iteratively find extensions $C'$ of $C$, until eventually a total valid coloring is reached. It follows from the validation rules that if $C$ is invalid, so is $C'$. Therefore, the algorithm keeps checking for validation, and would discard an extension whenever it becomes invalid.
Basically, there are two different ways of possibly extending a coloring. First, there are partial colorings $C$ which force the colors of some of the so far uncolored vertices, leading to an extension $C'$ of $C$. In this case, say that $C'$ has been obtained from $C$ by [*propagation*]{}. The following is a convenient set of rules, whose application may extend $C$, in the above described way.
:\[lemma-propagation\] RULES FOR PROPAGATING COLORS:\
The following are forced colorings for the extensions of a partial coloring of $G$.
P1.\[itm:ruleDiamond\]
: In an induced diamond, degree-3 vertices must be black and the remaining ones must be white
P2.\[itm:rulePendant\]
: The neighbor of a pendant vertex must be black
P3.\[itm:ruleWhiteN\]
: Each neighbor of a white vertex must be black
P4.\[itm:rulePaired\]
: Except for its pair, the neighbors of a paired (black) vertex must be white
P5.\[itm:ruleTwoB\]
: Each vertex with two black neighbors must be white
P6.\[itm:ruleOneN\]
: If a single black vertex has exactly one uncolored neighbor then this neighbor must be black
P7.\[itm:rulePaw\]
: In an induced paw, the two odd-degree vertices must have different colors
P8.\[itm:ruleC4\]
: In an induced $C_4$, adjacent vertices must have different colors
P9.\[itm:ruleContained\]
: If the neighborhood of any uncolored neighbor of a single (black) vertex $s$ is contained in the neighborhood of $s$ then the uncolored neighbor $v$ of $s$ minimizing weight(sv) must be black. If there are several options for vertex $v$, choose any one of them. We require rules P1 and P8 to be applied before P9.
The rules P1, P7, P8 follows from [@Br-Hu-Ne]. while rules P3, P4, P5, P6 follows from [@Do-Lo].
The rule P2 follows from the coloring definition since each black vertex must be paired in order for the coloring to be [*valid*]{}. Finally, for P9, let $s$ be a single vertex. Suppose the neighborhood of any uncolored neighbor of $s$ lies within the neighborhood of $s$. Then the choice of the vertex to become the pair of $s$ is independent of the choices for the remaining single vertices of the graph. Therefore, to obtain a minimum weighted dominating induced matching of $G$, the neighbor $v$ of $s$ minimizing $weight(sv)$ must be black. $_\triangle$
\[lem-K4\] [@Br-Hu-Ne] If $G$ contains a $K_4$ then $G$ has no DIM.
Say that a coloring $C$ is [*empty*]{} if all vertices are uncolored.
Let $C$ be a valid coloring and $C'$ an extension of it, obtained by the application of the propagation rules. If $C = C'$ then $C$ is called [*stable*]{}. On the other hand, if $C \neq C'$ then $C'$ is not necessarily valid. Therefore, after applying iteratively the propagation rules, we reach an extension which is either invalid or stable.
In order to possibly extend a stable coloring $C$, we apply [*bifurcation rules*]{}. Any coloring directly obtained by these rules is not forced. Instead, in each of the these rules, there are two possibly conflicting alternatives leading to distinct extensions $C'_1, C'_2$ of $C$. Each of $C'_1$ or $C'_2$ may be independently valid or invalid. The next lemma describes the bifurcation rules. We remark that there exist simpler bifurcation rules. However, using the rules below we obtain a sufficient number of vertices that get forced colorings, through the propagation which follow the application of any bifurcation rule, so as to guarantee a decrease of the overall complexity of the algorithm. The complexity obtained relies heavily on this fact.
In general, we adopt the following notation. If $C$ is a stable coloring then S denotes the set of single vertices of it , $U$ is the set of uncolored vertices and $T = U \setminus \cup_{s \in S} N_U(s)$.
\[lemma-bifurcation\]: BIFURCATION RULES\
Let $C$ be a partial (valid) stable coloring of a graph $G$. At least one of the following alternatives can be applied to define extensions $C'_1,
C'_2$ of $C$.
B1.\[itm:ruleEmpty\]
: If $C$ is an [*empty*]{} coloring: choose an arbitrary vertex $v$ then $C'_1 := C \cup
BLACK(\{v\})$ and $C'_2 := C \cup WHITE(\{v\})$\
B2.\[itm:ruleExEdge\]
: If $\exists$ edge $vw$ s.t. $v \in N_U(s)$ and $w \in
N_U(s')$, for some $s,s' \in S, s \ne s'$ then $C'_1 := C \cup
BLACK(\{v\})$ and $C'_2 := C \cup WHITE(\{v\})$\
B3.\[itm:ruleSomes\]
: For some $s \in S$, if $\exists v \in N_U(s)$ s.t. $\exists w \in N_T(v)$:
(a)\[itm:Bif3a\]
: If $|N_U(s)| \neq 3 \vee d(w) \neq 3 \vee |N_T(v)| \geq 2$ then $C'_1 := C \cup BLACK(\{v\})$ and $C'_2 := C \cup WHITE(\{v\})$.\
(b)\[itm:Bif3b\]
: If $|N_U(s)| = 3 \wedge d(w) = 3 \wedge N_T(v) = \{w\}$, let $N_U(s) = \{v,v',v''\}$.\
i.
: If $N_U(v') = N_U(v'') = \emptyset$ then $C'_1 := C \cup BLACK(\{v\})$ and $C'_2 := C \cup WHITE(\{v\})$\
ii.
: If $N_U(v') \neq \emptyset$, let $w' \in N_T(v')$, with $w' \neq w$. If $|N(w)\cup N(w')| > 5$ or $ww' \notin E(G)$ then $C'_1 := C \cup
BLACK(\{v\})$ and $C'_2 := C \cup WHITE(\{v\})$\
iii.\[itm:B3b.iii\]
: If $N_U(v') \neq \emptyset$, let $w' \in N_T(v')$, with $w' \neq w$. If $ww' \in E(G)$ and $z \in N(w) \cap N(w')$ then $C'_1 := C \cup BLACK(\{v''\})$, while if $weight(sv) + weight(w'z) \leq weight(sv') + weight(wz)$ then $C'_2 := C \cup BLACK(\{v\})$, otherwise $C'_2 := C \cup BLACK(\{v'\})$\
Each rule is applied after the previous rule, that is, if the condition of the previous case is not verified in the entire graph. Note that this applies to subitems of case B3. [*Proof*]{} If $C$ is an [*empty*]{} coloring then the rule B1 is applied.\
If $C$ is not an [*empty*]{} coloring and $C$ is not a [*total*]{} coloring then $S \ne \emptyset$. Since $C$ is not [*total*]{} and the graph is connected then there is at least one edge $sv$ where $v$ is uncolored. If $s$ is white then $v$ must be black P3 else if $s$ is a [*paired*]{} vertex then $v$ must be white P4 . Therefore $s$ must be a single black vertex, hence $S \ne \emptyset$. Let $s \in S$. Since $C$ is valid then $N_U(s) \ne \emptyset$ by V2 and since is [*stable*]{} $|N_U(s)| \ne 1$ by P6 Therefore $|N_U(s)| \geq 2$. Moreover rule P9 can not be applied, therefore $\exists v \in N_U(s)$ s.t. $|N_U(v) \setminus N(s)| > 0$, let $w \in N_U(v) \setminus N(s)$. If $\exists s' \in S, s \ne s'$ s.t. $w \in N_U(s')$ then rule B2 is applied.\
Suppose that rule P2 can not be applied. Then $w \in N_T(v) (|N_T(v)| \ge 1)$. Clearly, $d(w) \ne 1$, otherwise, rule P2 must be applied and $v$ must get color black.
In case $|N_U(s)| \ne 3$ or $d(w) \ne 3$ or $|N_T(v)| \geq 2$ we apply rule B3(a). Otherwise: $|N_U(s)|=3,d(w)=3,|N_T(v)|=1$. Note that in B3(b) whenever we refer to $v'w'$ it behaves symmetric to $vw$ since otherwise $v'w'$ were found in step B3(a) replacing $vw$.\
In the first subcase of B3(a) the case analyzed is whenever $N_U(v') = N_U(v'') = \emptyset$, while in the second and third the algorithm handle the cases when at least one of them has uncolored neighbors.\
Suppose w.l.o.g. $N_U(v') \ne \emptyset$ where $w' \in N_T(v')$. It is easy to see that $w \ne w'$ since otherwise $svwv'$ is a $C_4$ and therefore $w$ can’t be uncolored by rule P8. Now there are three cases which lead to two possible outcomes from the algorithm: In case $ww' \in E(G)$ or $|N_U(v) \cup N_U(w)|>5$ then the result of the algorithm is given by the second subcase (ii), else it is given by the third subcase (iii). $_\triangle$
The Algorithm {#sec-algorithm}
=============
The lemmas described in the last section lead to an exact algorithm for finding a minimum weight DIM of a graph $G$, if any, which we describe below.
In the initial step of the algorithm, we find the set containing the $K_4$’s of $G$. If $K4 \ne \emptyset$, by lemma \[lem-K4\], $G$ does not have DIM’s, and terminate the algorithm. Otherwise, define the set $COLORINGS$ to contain through the process the candidates colorings to be examined and eventually extended. This set should be implemented using a $LIFO$ (Last In First Out) data structure which achieves linear space complexity of the algorithm because the number of colorings in $COLORINGS$ is at most $n+1$, and each coloring needs $O(1)$ space. We give more detailed explanation in the next section. Next, include in COLORINGS an [*empty*]{} coloring. In the general step, we choose any coloring $C$ from $COLORINGS$ and remove it from this set. Then iteratively propagate the coloring by Lemma \[lemma-propagation\] into an extension $C'$ of it, and validate the extension by Lemma \[lemma-validation\]. The iterations are repeated until one of the following situations is reached: $C'$ is invalid, $C'$ is a total valid coloring, or a partial stable (valid) coloring. In the first alternative, $C'$ is discarded and a new coloring from $COLORINGS$ is chosen. If $C'$ is a a total valid coloring, then sum the amount of valid DIMs related to this coloring, find its weight and if smaller than the least weight so far obtained, it becomes the current candidate for the minimum weight of a DIM if $G$. Finally, when $C'$ is stable we extended it by bifurcation rules: choose the first rule of Lemma \[lemma-bifurcation\] satisfying $C'$, compute the extensions $C'$ and $C''$, insert them in $COLORINGS$, select a new coloring from $COLORINGS$ and repeat the process.
The formulation below describes the details of the method. The propagation and validation of a coloring $C$ is done by the procedure $PROPAGATE-VALIDATE(C, RESULT)$. At the end, the returned coloring corresponds to the extension $C'$ of $C$, after iteratively applying propagation. The variable RESULT indicates the outcome of the validation analysis. If $C'$ is invalid then $RESULT$ is returned as ‘invalid’; if $C'$ is a valid total coloring then it contains ‘total’, and otherwise $RESULT$ equals ‘partial’. Finally, $BIFURCATE(C, C'_1,C'_2)$ computes the extensions $C'_1$ and $C'_2$ of $C$.\
[**Algorithm Minimum Weighted DIM / Counting DIM**]{}
[**procedure**]{} $PROPAGATE-VALIDATE (C, RESULT)$
Correctness and Complexity {#sec-correctness-complexity}
==========================
It is easy to see that our algorithm uses the [*branch $\&$ reduce*]{} paradigm since [*propagation rules*]{} can be mapped to [*reduction rules*]{} since are used to simplify the problem instance or halt the algorithm and the [*bifurcation rules*]{} can be mapped to [*branching rules*]{} since are used to solve the problem instance by recursively solving smaller instances of the problem.
\[thm-correctness\] The algorithm described in the previous section correctly computes the minimum weight of a dominating induced matching of a graph $G$.
[*Proof*]{}: The correctness of the algorithm follows from the fact that the algorithm considers all the cases that need to be considered, this is, any coloring that represents a DIM must be explored. Lemmas \[lemma-propagation\] and \[lemma-bifurcation\] ensures that the simplifications of the instances are valid, [*invalid*]{} colorings are discarded, some valid colorings can not be explored only if other valid coloring representing a better DIM (with less weight) is explored.
For proving the worst-case running time upperbound for the algorithm we will use the following useful definition and theorem.
\[def:Branch\] [@Fo-Kr] Let $b$ a branching rule and $n$ the size of the instance. Suppose rule $b$ branches the current instance into $r \geq 2$ instances of size at most $n-t_1,n-t_2, \ldots, n-t_r$, for all instances of size $n \geq max \{t_i:i = 1, 2, \ldots, r\}$. Then we call $b = (t_1,t_2,\ldots, t_r)$ the [*branching vector*]{} of branching rule $b$.
The branching vector $b = (t_1, t_2, \ldots, t_r)$ implies the linear recurrence $T(n) \leq T(n-t_1) + T(n-t_2) + \ldots, T(n-t_r)$.
\[thm:Branch\] [@Fo-Kr] Let $b$ be a branching rule with branching vector $(t_1, t_2, \ldots ,t_r)$. Then the running time of the branching algorithm using only branching rule $b$ is $O^*(\alpha^n)$, where $\alpha$ is the unique positive real root of
$x^n - x^{n-t_1} - x^{n-t_2} - \ldots - x^{n-t_r} = 0$
The unique positive real root $\alpha$ is the [*branching factor*]{} of the branching vector $b$.
We denote the branching factor of $(t_1, t_2, \ldots, t_r)$ by $\tau(t_1, t_2, \ldots, t_r)$.
Therefore for analyzing the running time of a branching algorithm we can compute the factor $\alpha_i$ for every branch rule $b_i$, and an upper bound of the running time of the branching algorithm is obtained by taking $\alpha = max_i \alpha_i$ and the result is an upper bound for the running time of $O^*(\alpha^n)$.
The upper bound comes from counting the leaves of the search tree given by the algorithm, using the fact that each leave can be executed in polynomial time. The complexity of the algorithm without hiding the polynomial depends on the upperbound time for the execution of each branch in the search tree.
Further notes on this topic can be found in [@Fo-Kr]
\[thm-complexity\] The algorithm above described requires $O^*(1.1939^n)$ time and $O(n+m)$ space for completion.
[*Proof*]{}:
Using the definition \[def:Branch\] and the theorem \[thm:Branch\] the calculation of the upper bound time is reduced to calculation of the [*branching vector*]{} for each branching rule (i.e. bifurcation rules in our algorithm) and obtain the associated [*branching factor*]{} for each case. Then the bound is given by the maximum [*branching factor*]{}. Note that to use this we must observe that the [*reduction rules*]{} (i.e. propagation rules in our algorithm) can be computed in polynomial time and leads to at most one valid extension of the considered coloring. So, the propagation rules do not affect the exponential factor of the algorithm. Moreover, each branch of the algorithm has cost $O(n+m)$ in time and space. This is easy to note since from the [*empty*]{} coloring up to any [*total*]{} coloring each vertex $v$ is painted once and the cost in time incurred for painting each vertex is given by the updating of the color of the vertex and updating this information for the neighborhood, hence $
|N(v)|$ times a constant operation for updating a counter with amount of black/white/uncolored neighbors. Therefore, the total cost for each branch is $O(n+m)$.
Let’s analyse each bifurcation rule to obtain the maximum [*branching factor*]{}:
1.
: If $C$ is an [*empty*]{} coloring: choose an arbitrary vertex $v$ then $C'_1 := C \cup BLACK(\{v\})$ and $C'_2 := C \cup WHITE(\{v\})$: It is easy to see that this rule is executed once, after that, the coloring is never empty again. Since this rule bifurcation opens two branchs then we can upper bound the time of the algorithm by $2$ times the complexity of the algorithm executed in an instance of size $n-1$. Therefore the asymptotic behavior of the algorithm is not affected.
2.
: [*If $\exists$ edge $vw$ s.t. $v \in N_U(s)$ and $v' \in N_U(s')$, for some $s,s' \in S, s \ne s'$ then $C'_1 := C \cup BLACK(\{v\})$ and $C'_2 := C \cup WHITE(\{v\})$.*]{}
Here we extend the original coloring $C'$ to $C'_1$ and $C'_2$ by coloring the vertex $v$ with black and white respectively. Recall that exists an edge $vw$ such that $v \in N_U(s)$, $w \in N_U(s')$. If $v$ is black then $N_U(s) \setminus v$ are white, while $v'$ is white. On the other hand, if $v$ is white then $w$ is black and $N_U(s') \setminus w$ are white. Therefore the size of uncolored vertices is reduced for each branch (i.e. for each new coloring). The associated branching vector is $(1 + |N_U(s)|, 1 + |N_U(s')|)$. By rule P2 $|N_U(s)| \geq 2$ and $|N_U(s')| \geq 2$. The following observation turns out to be useful: If $|N_U(s_i)| = 2$ then $N_U(s_i)$ can be totally painted wether $v$ is black or white. The case with $N_u(s')$ is symmetric. Therefore the branching vector with biggest branching factor is (3,5) $(\tau(3,5)\approx 1.1939)$.
3.
: For some $s \in S$, if $\exists v \in N_U(s)$ s.t. $\exists w \in N_T(v)$:
Note that if $\not \exists w \in N_T(v)$ for any $v \in N_U(s)$ then either the propagating rule P9 or P5 can be applied to get an extension of the coloring.
(a)
: If $|N_U(s)| \neq 3 \vee d(w) \neq 3 \vee |N_T(v)| \geq 2$ then $C'_1 := C \cup BLACK(\{v\})$ and $C'_2 := C \cup WHITE(\{v\})$.\
Since $v$ is uncolored then $w$ is not a [*pendant*]{} vertex, $d(w)>1$. Since $w$ is uncolored then it has nor white nor paired black neighbor. Moreover, if $w$ has a single black neighbor then this is the case analyzed above. Therefore $w$ has uncolored neighbors and let $x$ be one of them.\
(a.1)
: $|N_T(v)| \geq 2$: Let $v' \in N_T(v)$. In $C'_1$ $\{v,x\}$ will be black while $\{v_1,v',w\}$ will be white. In $C'_2$ $\{v\}$ will be white while $\{v',w\}$ will be black. This lead to the branching vector (3,5).\
(a.2)
: $d(w) \neq 3$. If $d(w)=2$ then in $C'_1$ the vertices $N_U(s) \cup \{w,x\}$ will be colored and in $C'_2$ the vertices $\{v,x\}$ will be black while $\{w\}$ will be white. Therefore the branching vector will be at least (3,5).\
Else if $d(w) > 3$ then in $C'_1$ the vertices $N_U(s) \cup N_U[w]$ will be colored and in $C'_2$ the vertices $\{v,w\}$ will be colored. In case $|N_U(s)|=2$ then $v_1$ will be colored too. Therefore the branching vector (2,7) $(\tau(2,7)=1.1908)$.\
(a.3)
: $|N_U(s)|=2$: Let $N_U(s) = \{v,v_1\}$ and $N(w) = \{v,x,x'\}$. In $C'_1$ after applying propagation rules the vertices $\{v,x,x'\}$ will be black while $\{v_1,w\}$ will be white. In $C'_2$ after applying propagation rules the vertices $\{v_1,w\}$ will be black while $\{v\}$ will be white. The result is the branching vector (3,5).\
(a.4)
: $|N_U(s)|>3$: Let $\{v_1,v_2,v_3\} \in N_U(s)$ and $N(w) = \{v,x,x'\}$. In $C'_1$ after applying propagation rules the vertices $\{v,x,x'\}$ will be black while $\{v_1,v_2,v_3,w\}$ will be white. In $C'_2$ after applying propagation rules the vertices $\{w\}$ will be black while $\{v\}$ will be white. The result is the branching vector (2,7)\
(b)
: [*If $|N_U(s)| = 3 \ \wedge \ d(w) = 3$ where $N_U(w)=\{v,x,x'\}, N_U(s) = \{v,v',v''\}$*]{} Note that $\{x,x'\} \cap \{v,v',v''\} = \emptyset$ since otherwise at least one of them must be colored by rule P8.
(b.1)
: [*If $N_U(v') = N_U(v'') = \emptyset$ then\
$C'_1 := C \cup BLACK(\{v\})$ and $C'_2 := C \cup WHITE(\{v\})$* ]{}:\
Suppose w.l.o.g. $weight(sv') \leq weight (sv'')$, then\
In $C'_1$ after applying propagation rules the vertices $\{v,w',w''\} $ will be black while $\{v',v'',w\} $ will be white. In $C'_2$ after applying propagation rules the vertices $\{v',w\} $ will be black while $\{v,v''\} $ will be white. The result is the branching vector (4,6) $(\tau(4,6)=1.1510)$.\
(b.2)
: [*If $N_U(v') \neq \emptyset$, let $w' \in N_T(v')$, with $w' \neq w$:\
If $|N_T[w] \cup N_T[w']| > 5$*]{} then\
$C'_1 := C \cup BLACK(\{v\})$ and $C'_2 := C \cup WHITE(\{v\})$\
Note that if $d(w') \neq 3$ then $v'w'$ satisfies the properties of an already analized case, hence $C'_1 := C \cup BLACK(\{v'\})$ and $C'_2 := C \cup WHITE(\{v'\})$.
Since $d(w) = d(w') = 3$ and $|N_T[w] \cup N_T[w']| > 5$, then $\exists x,y$ s.t. $x \in N_T(w), x \notin N_T(w')$ and $y \in N_T(w'), y \notin N_T(w)$. In $C'_1$ after applying propagation rules the vertices $\{v,x,x',w'\}$ will be black while $ \{v',v'',w\}$ will be white. If $x'=w'$ then $y$ must be black by rule P6. In $C'_2$ the vertex $\{w\}$ will be black while the vertex $\{v\}$ will be white. The result is the branching vector (2,7)\
(b.3)
: [*If $N_U(v') \neq \emptyset$, let $w' \in N_T(v')$, $w' \neq w$\
If $|N_T[w] \cup N_T[w']| \leq 3$ and $z \in N(w) \cap N(w')$*]{} then\
$C'_1 := C \cup BLACK(\{v''\})$,\
if $weight(sv) + weight(w'z) \leq weight(sv') + weight(wz)$ then\
$C'_2 := C \cup BLACK(\{v\})$\
otherwise $C'_2 := C \cup BLACK(\{v'\})$\
Since $d(w) = d(w') = 3$ then $ww' \in E(G)$ and $\exists z \in N_T(v) \cap N_T(w)$, otherwise the case is one of the above.
In both colorings, $C'_1$ and $C'_2$ the vertices $\{v,v',v'',w,w',z\}$ will be colored. The branching vector is (6,6). $(\tau(6,6)=1.1225)$.
The worst branching factor is $\tau(3,5)\approx 1.1939$. In consequence, the time complexity of this algorithm is $O*(1.1939^n)$.
To achieve linear space complexity, we use a stack to store the coloring sequence of the current branch. The only additional space is needed for $COLORINGS$ and extra information to restore the initial condition for each coloring. For each coloring $c \in COLORINGS$ extended from a bifurcation rule we store the number of colored vertices before the bifurcation, the vertex colored during bifurcation and its color. These elements are sufficient to restore the initial condition. $_\triangle$
The analysis can be extended for the case of non-connected graphs. It is easy to obtain the same upper bound after separating the cases where each connected component of four or less vertices is solved in constant time.
Counting the number of DIM’s
============================
The previous algorithm can be easily adapted to count the number of DIM’s. The number of DIM’s is the number of [*total valid*]{} colorings. Given a coloring $C$ we define $TVC(C)$ the number of [*total valid*]{} colorings that can be extended from $C$. If we apply any propagation rule to coloring $C$ we obtain a coloring $C'$. Clearly $TVC(C) = TVC(C')$, except for rule P9. In the later case $TVC(C) = TVC(C') \cdot |N_U(s)|$ where $s$ is the single vertex chosen to apply the rule.
If we apply any bifurcation rule to coloring $C$ we obtain two extended colorings $C'_1$ and $C'_2$. Clearly $TVC(C) = TVC(C'_1) + TVC(C'_2)$, except for rule B3(b)iii. In the later case $TVC(C) = TVC(C'_1) + 2 \cdot TVC(C'_2)$.
Using the above facts it is trivial to modify the algorithm to solve the counting problem.
Conclusions
===========
We have developed a new exact exponential algorithm for an extensively studied problem. Moreover the developed algorithm is practical since there are no big constants or polynomials hidden in the upper-bound and it is straightforward to implement it.
Problem Previous results New results
-------------- --------------------------------------------------------------------------------------- -----------------------
Weighted DIM $O^*(1.4423^m)$ [@Br-Le-Ra; @Milanic; @Ts], $O(1.4423^n \cdot m)$ [@DBLPMin] $O(1.1939^n \cdot m)$
Counting DIM $O^*(1.3247^m)$ [@Br-Le-Ra; @Milanic; @Dahllof2002], $O(1.4423^n \cdot m)$ [@DBLPMin] $O(1.1939^n \cdot m)$
[99]{}
Andreas Bjorklund. Determinant sums for undirected hamiltonicity. In [*Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science*]{}, FOCS ’10, pages 173–182, Washington, DC, USA, 2010. IEEE Computer Society.
Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets möbius: fast subset convolution. In [*Proceedings of the thirty-ninth annual ACM symposium on Theory of computing*]{}, STOC ’07, pages 67–74, New York, NY, USA, 2007. ACM.
A. Brandstädt, C. Hundt and R. Nevries, Efficient edge domination on hole-free graphs in polynomial time. [*Proceedings of LATIN’ 2010*]{}, [*Lecture Notes in Computer Science*]{}, pp. 650-651.
Andreas Brandst[ä]{}dt, Arne Leitert, and Dieter Rautenbach. Efficient dominating and edge dominating sets for graphs and hypergraphs. In [*ISAAC*]{}, pages 267–277, 2012.
Andreas Brandst[ä]{}dt and Raffaele Mosca. Dominating induced matching for $P_7$-free graphs in linear time. [*Proceedings ISAAC’ 2011*]{}, pp. 100-109.
A. Brandstädt and V. V. Lozin, On the linear structure and clique-width of bipartite permutation graphs, [*Ars Combin.*]{} 67 (2003), 273-281.
D. M. Cardoso, N. Korpelainen and V. V. Lozin, On the complexity of the induced matching problem in hereditary classes of graphs, [*Discrete Appl. Math.*]{} 159 (2011), 521-531.
D. M. Cardoso and V. V. Lozin, Dominating induced matchings, [*Lecture Notes in Comput. Sci.*]{} 5420 (2009), 77-86
D. M. Cardoso, J. O. Cerdeira, C. Delorme and P. C. Silva, Efficient edge domination in regular graphs, [*Discrete Appl. Math.*]{} 156 (2008), 3060-3065.
Vilhelm Dahllöf and Peter Jonsson. An algorithm for counting maximum weighted independent sets and its applications. In [*Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms*]{}, SODA ’02, pages 292–298, Philadelphia, PA, USA, 2002. Society for Industrial and Applied Mathematics.
F. V. Fomin, F. Grandoni and D. Kratsch, Measure and conquer: a simple $O^*(1.220^n)$ independent set algorithm, [*Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms (SODA’06)*]{}, pp. 18-25, New York, N.Y., USA, 2006.
Fedor V. Fomin, Fabrizio Grandoni, Artem Pyatkin, and Alexey A. Stepanov. Bounding the number of minimal dominating sets: a measure and conquer approach. In [*IN PROCEEDINGS OF THE 16TH ANNUAL INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION (ISAAC 2005*]{}, pages 573–582. Springer, 2005.
F. V. Fomin, D. Kratsch, [*Exact Exponential Algorithms*]{}, EACTS Series in Theoretical Computer Science, Springer, Berlin, 2010.
D. L. Grinstead, P. J. Slater, N. A. Sherwani and N.D. Holmes, Efficient edge domination problems in graphs, [*Inform. Process. Lett.*]{} 48 (1993), 221-228
Y. Iwata, A faster algorithm for dominating set analyzed by the potential method, [*Proceedings of the 6th Internnational Conference on Parameterized and Exact Computation IPEC’ 11*]{}, pp. 41-54, Springer-Verlag, Berlin, 2012.
N. Korpelainen, A polynomial-time algorithm for the dominating induced matching problem in the class of convex graphs, [*Electron. Notes Discrete Math.*]{} 32 (2009), 133-140
Min Chih Lin, Michel J. Mizrahi, and Jayme Luiz Szwarcfiter. Exact algorithms for dominating induced matchings. , abs/1301.7602, 2013.
M. Livingston and Q. F. Stout. Distributing resources in hypercube computers. In [*Proceedings of the third conference on Hypercube concurrent computers and applications: Architecture, software, computer systems, and general issues - Volume 1*]{}, C3P, pages 222–231, New York, NY, USA, 1988. ACM.
C. L. Lu, M.-T. Ko and C. Y. Ta, Perfect edge domination and efficient edge domination in graphs, [*Discrete Appl. Math.*]{} 119 (2002), 227-250.
C. L. Lu and C. Y. Ta, Solving the weighted efficient edge domination problem on bipartite permutation graphs, [*Discrete Appl. Math.*]{} 87 (1998), 203-211.
Martin Milanic. Hereditary efficiently dominatable graphs. , pages n/a–n/a, 2012.
J. W. Moon and L. Moser, On cliques of graphs, [*Israel Journal of Mathematics*]{} 3 (1965), 23-28.
S. Tsukiyama, M. Ide, H. Ariyoshi and I. Shirakawa, A new algorithm for generating all maximal independent sets, [*SIAM J. Comput.*]{} 6 (1979), 505-579.
Gerhard J. Woeginger. Exact algorithms for np-hard problems: A survey. , pages 185–207.
[^1]: Partially supported by UBACyT Grants 20020100100754 and 20020090100149, PICT ANPCyT Grant 1970 and PIP CONICET Grant 11220100100310.
[^2]: Partially supported by CNPq, CAPES and FAPERJ, research agencies. Presently visiting the Instituto Nacional de Metrologia, Qualidade e Tecnologia, Brazil.
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abstract: |
Tasks such as social network analysis, human behavior recognition, or modeling biochemical reactions, can be solved elegantly by using the probabilistic inference framework. However, standard probabilistic inference algorithms work at a propositional level, and thus cannot capture the symmetries and redundancies that are present in these tasks.
Algorithms that exploit those symmetries have been devised in different research fields, for example by the lifted inference-, multiple object tracking-, and modeling and simulation-communities. The common idea, that we call *state space abstraction*, is to perform inference over compact representations of sets of symmetric states. Although they are concerned with a similar topic, the relationship between these approaches has not been investigated systematically.
This survey provides the following contributions. We perform a systematic literature review to outline the state of the art in probabilistic inference methods exploiting symmetries. From an initial set of more than 4,000 papers, we identify 116 relevant papers. Furthermore, we provide new high-level categories that classify the approaches, based on common properties of the approaches. The research areas underlying each of the categories are introduced concisely. Researchers from different fields that are confronted with a state space explosion problem in a probabilistic system can use this classification to identify possible solutions. Finally, based on this conceptualization, we identify potentials for future research, as some relevant application domains are not addressed by current approaches.
author:
- |
Stefan Lüdtke stefan.luedtke2@uni-rostock.de\
Institute of Computer Science\
University of Rostock, Germany Max Schröder max.schroeder@uni-rostock.de\
Frank Krüger frank.krueger@uni-rostock.de\
Institute of Communications Engineering\
University of Rostock, Germany Sebastian Bader sebastian.bader@uni-rostock.de\
Thomas Kirste thomas.kirste@uni-rostock.de\
Institute of Computer Science\
University of Rostock, Germany
bibliography:
- 'aaa-complete-results-theoretical.bib'
- 'ref.bib'
- 'Reviews.bib'
- 'Excluded-complete.bib'
- 'further-manual-citations.bib'
title: |
State-Space Abstractions for Probabilistic Inference:\
A Systematic Review
---
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to the three anonymous reviewers for their extensive comments and suggestions, which vastly improved the quality of the paper.
|
---
abstract: |
We derive frequency moment sum rules for the retarded electronic Green’s function and self-energy of the Holstein model for both equilibrium and nonequilibrium cases. We also derive sum rules for the phonon propagator in equilibrium and nonequilibrium. These sum rules allow one to benchmark nonequilibrium calculations and help with interpreting the behavior of electrons driven out of equilibrium by an applied electric field. We exactly evaluate the sum rules when the system is in the atomic limit. We also discuss the application of these sum rules to pump/probe experiments like time-resolved angle-resolved photoemission spectroscopy.
Key Words: nonequilibrium, electron-phonon problem, sum rules, Green functions, self-energy.
author:
- |
J. K. Freericks$^1$,\
Khadijeh Najafi$^1$\
A. F. Kemper$^2$,\
and T. P. Devereaux$^{3,4}$\
\
$^1$Department of Physics, Georgetown University,\
37th and O Sts. NW, Washington, DC 20057, USA\
$^2$Lawrence Berkeley National Laboratory,\
1 Cyclotron Road, Berkeley, CA 94720, USA\
$^3$Stanford Institute for Materials and Energy Science,\
SLAC National Accelerator Laboratory,\
Menlo Park, CA 94025, USA\
$^4$Geballe Laboratory for Advanced Materials,\
Stanford University, Stanford, CA 94305, USA
bibliography:
- 'freericks\_feis.bib'
title: Nonequilibrium sum rules for the Holstein model
---
Introduction
============
In recent years, we have seen significant advances in time-resolved experiments on systems that have strong electron-phonon interactions [@berkeley; @bovensiepen]. These experiments study how energy is transferred between the electronic and phononic parts of the system. One of the interesting effects that has been seen in these experiments is the so-called phonon-window effect [@prx], where electrons with energies farther than the phonon frequency from the Fermi level relax quickly back to equilibrium after the pulsed field is applied, but those close to the Fermi level relax on a much longer time scale, because their relaxation involves multiparticle processes due to a restricted phase space. It is clear that this experimental and theoretical work is just starting to analyze electron-phonon interacting systems in the time domain. Hence, any exact results that can be brought to bear on this problem will be important.
In this work, we derive sum rules for the zeroth and first two moments of the retarded electronic Green’s function and for the zeroth moment of the retarded self-energy. The moment sum rules have already been derived in equilibrium [@kornilovitch; @rosch], but they actually hold true, unchanged, in nonequilibrium as well [@sumrules1; @sumrules2; @sumrules3; @sumrules4]. With these sum rules, one can understand how the electron-phonon interaction responds to nonequilibrium driving, and how different response functions will behave.
We start with the so-called Holstein model [@holstein1; @holstein2], given by the following Hamiltonian in the Schroedinger representation: $$\mathcal{H}(t)=-\sum_{ij\sigma}t_{ij}(t)c^\dagger_{i\sigma}c^{}_{j\sigma}+\sum_{i\sigma}[g(t) x_i-\mu] c^\dagger_{i\sigma}c^{}_{i\sigma}+\sum_i\frac{p_i^2}{2m}+\frac{1}{2}\kappa\sum_ix_i^2
\label{eq: ham}$$ where $c^\dagger_{i\sigma}$ ($c^{}_{i\sigma}$) are the fermionic creation (annihilation) operators for an electron at lattice site $i$ with spin $\sigma$ (with anticommutator $\{c^{}_{i\sigma},c^\dagger_{j\sigma\prime}\}_+=\delta_{ij}\delta_{\sigma\sigma'}$), and $x_i$ and $p_i$ are the phonon coordinate and momentum (with commutator $[x_i,p_j]_-=i\hbar\delta_{ij}$), respectively. The hopping $-t_{ij}(t)$ between lattice sites $i$ and $j$ can be time dependent \[for example, an applied electric field corresponds to the Peierls’ substitution [@peierls]\], $\mu$ is the chemical potential for the electrons, $g(t)$ is the time-dependent electron-phonon interaction, $m$ is the mass of the optical (Einstein) phonon and $\kappa$ is the corresponding spring constant. The frequency of the phonon is $\omega=\sqrt{\kappa/m}$. It is often convenient to also express the phonon degree of freedom in terms of the raising and lowering operators $a^\dagger_i$ and $a^{}_i$ (with commutator $[a^{}_i,a^\dagger_j]_-=\delta_{ij}$) with $x_i=(a^\dagger_i+a^{}_i)\sqrt{\hbar/(2m\omega)}$ and $p_i=(-a^\dagger_i+a^{}_i)\sqrt{\hbar m\omega/2}/i$. This Hamiltonian involves electrons that can hop between different sites on a lattice and interact with harmonic Einstein phonons that have the same phonon frequency for every lattice site. The hopping and the electron-phonon coupling are taken to be time dependent for the nonequilibrium case. We set $\hbar=1$ and $k_B=1$ for the remainder of this work.
Formalism for the electronic sum rules
======================================
Since we will be working in nonequilibrium, we need to allow the time to lie somewhere on the Kadanoff-Baym-Keldysh contour, which runs in the positive time direction from $t_{min}$ to $t_{max}$, back to $t_{min}$ and down the imaginary axis to $t_{min}-i\beta$, with $\beta=1/T$ the inverse temperature (it is assumed the system is in equilibrium at time $t_{min}$). Allowing the time to be chosen anywhere on the contour, the contour-ordered electronic Green’s function is defined by $$G^c_{ij\sigma}(t,t')=-i{\rm Tr}\mathcal{T}_c e^{-\beta\mathcal{H}(t_{min})}c^{}_{i\sigma}(t)c^\dagger_{j\sigma}(t')/\mathcal{Z},
\label{eq: g_contour}$$ where $\mathcal{T}_c$ denotes time ordering along the contour, and $\mathcal{Z}={\rm Tr}\exp[-\beta \mathcal{H}(t_{min})]$, with the system in equilibrium at the initial time $t_{min}$ at a temperature $T=1/\beta$. The Fermi operators are written in the Heisenberg representation $c_{i\sigma}(t)=U^\dagger(t,t_{min})c_{i\sigma}U(t,t_{min})$, with $U(t,t')$ the evolution operator from time $t'$ to time $t$. The evolution operator satisfies $idU(t,t')/dt=\mathcal{H}(t)U(t,t')$ and $U(t,t)=1$. From the contour-ordered Green’s function, one can extract all of the needed Green’s functions, like the so-called lesser Green’s function and the retarded Green’s function, which we consider in detail here, and which is defined by $$G^R_{ij\sigma}(t,t')=-i\theta(t-t'){\rm Tr}e^{-\beta\mathcal{H}(t_{min})}\{c^{}_{i\sigma}(t),c^\dagger_{j\sigma}(t')\}_+/\mathcal{Z},
\label{eq: g_retarded}$$ where $\{.,.\}_+$ denotes the anticommutator. Converting to Wigner’s average and relative times $t_{ave}=(t+t')/2$ and $t_{rel}=t-t'$, we can find the frequency-dependent retarded Green’s function for each average time via $$G^R_{ij\sigma}(t_{ave},\omega)=\int_0^\infty dt_{rel} e^{i\omega t_{rel}}G^R_{ij\sigma}(t_{ave}+\frac{1}{2}t_{rel},t_{ave}-\frac{1}{2}t_{rel}).
\label{eq: gret_w}$$ The $n$th spectral moment in real space is then defined via $$\mu^{Rn}_{ij\sigma}(t_{ave})=-\frac{1}{\pi}\int_{-\infty}^{\infty}d\omega \,\omega^n{\rm Im}G^R_{ij\sigma}(t_{ave},\omega).
\label{eq: moment_def}$$ The moments are more convenient to evaluate as derivatives in time $$\begin{aligned}
&~&\mu^{Rn}_{ij\sigma}(t_{ave})=\\
&~&{\rm Im}\Biggr\{
\frac{i^{n+1}}{\mathcal{Z}}\frac{d^n}{dt_{rel}^n} {\rm Tr}e^{-\beta \mathcal{H}(t_{min})}
\{c^{}_{i\sigma}(t_{ave}+\frac12 t_{rel}),c^\dagger_{j\sigma}(t_{ave}-\frac12 t_{rel})\}_+\Biggr|_{t_{rel}=0^+}\Biggr\}.\nonumber
\label{eq: moment_deriv}\end{aligned}$$ These time derivatives can be replaced by partial time derivatives with respect to time-dependent terms in the Hamiltonian plus commutators with the Hamiltonian. In particular, we find that $$\mu^{R0}_{ij\sigma}(t_{ave})={\rm Tr}e^{-\beta \mathcal{H}(t_{min})}
\{c^{}_{i\sigma}(t_{ave}),c^\dagger_{j\sigma}(t_{ave})\}_+/\mathcal{Z}$$ for the zeroth moment, $$\begin{aligned}
\mu^{R1}_{ij\sigma}(t_{ave})&=&-\frac12\langle \{[\mathcal{H}_H(t_{ave}),c^{}_{i\sigma}(t_{ave})]_-,c^\dagger_{j\sigma}(t_{ave})\}_+\rangle\nonumber\\
&+&\frac12\langle\{c^{}_{i\sigma}(t_{ave}),[\mathcal{H}_H(t_{ave}),c^\dagger_{j\sigma}(t_{ave})]_-\}_+\rangle,\end{aligned}$$ for the first moment, where the angle brackets denote the trace over all states weighted by the density matrix ($\langle O\rangle={\rm Tr}\exp[-\beta\mathcal{H}(t_{min})]O/\mathcal{Z}$), the symbol $[.,.]_-$ denotes the commutator, and the subscript $H$ on the Hamiltonian indicates that it is in the Heisenberg representation. The second moment is more complicated and satisfies $$\begin{aligned}
\mu^{R2}_{ij\sigma}(t_{ave})&=&-\frac14\langle\{[\mathcal{H}_H(t_{ave}),[\mathcal{H}_H(t_{ave}),c^{}_{i\sigma}(t_{ave})]_-]_-,c^\dagger_{j\sigma}(t_{ave})\}_+\rangle\nonumber\\
&-&\frac12\langle\{[\mathcal{H}_H(t_{ave}),c^{}_{i\sigma}(t_{ave})]_-,[\mathcal{H}_H(t_{ave}),c^\dagger_{j\sigma}(t_{ave})]_-\}_+\rangle\nonumber\\
&+&\frac14 \langle\{c^{}_{i\sigma}(t_{ave}),[\mathcal{H}_H(t_{ave}),[\mathcal{H}_H(t_{ave}),c^\dagger_{j\sigma}(t_{ave})]_-]_-\}_+\rangle\nonumber\\
&+&\frac14{\rm Im}\langle \{[\mathcal{H}^\prime_H(t_{ave}),c^{}_{i\sigma}(t_{ave})]_-,c^\dagger_{j\sigma}(t_{ave})\}_+\rangle\nonumber\\
&+&\frac14{\rm Im}\langle\{c^{}_{i\sigma}(t_{ave}),[\mathcal{H}^\prime_H(t_{ave}),c^\dagger_{j\sigma}(t_{ave})]_-\}_+\rangle,
\label{eq: second_formal}\end{aligned}$$ where the prime indicates it is the Heisenberg representation of the time derivative of the Schroedinger representation Hamiltonian \[[*i. e.*]{}, $\mathcal{H}_{H}^\prime(t_{ave})= $\
$U^\dagger(t_{ave},-\infty)\partial \mathcal{H}_S(t)/\partial t|_{t=t_{ave}} U(t_{ave},-\infty)$\]. One can directly see that the two terms with the derivative of the Hamiltonian \[last two lines of Eq. (\[eq: second\_formal\])\] are equal and opposite and hence cancel.
These moments can now be evaluated straightforwardly, although the higher the moment is the more work it takes. We find the well-known result $$\mu^{R0}_{ij\sigma}(t_{ave})=\delta_{ij}$$ for the zeroth moment. The first moment satisfies $$\mu^{R1}_{ij\sigma}(t_{ave})=-t_{ij}(t_{ave})-\mu\delta_{ij}+g(t_{ave})\langle x_i(t_{ave})\rangle\delta_{ij}$$ and the second moment becomes $$\begin{aligned}
\mu^{R2}_{ij\sigma}(t_{ave})&=&\sum_kt_{ik}(t_{ave})t_{kj}(t_{ave})+2\mu t_{ij}(t_{ave})+\mu^2\delta_{ij}\\
&-&t_{ij}(t_{ave})g(t_{ave})\langle x_i(t_{ave})+x_j(t_{ave})\rangle-2\mu g(t_{ave})\langle x_i(t_{ave})\rangle\delta_{ij}\nonumber\\
&+&g^2(t_{ave})\langle x_i^2(t_{ave})\rangle\delta_{ij}.
\nonumber\end{aligned}$$ Unlike in the case of the Hubbard or Falicov-Kimball model, where the sum rules relate to constants or simple expectation values [@sumrules1; @sumrules2; @sumrules3], one can see here that one needs to know things like the average phonon coordinate and its fluctuations in order to find the moments. We will discuss this further below.
Our next step, is to calculate the self-energy moments, which are defined via $$C^{Rn}_{ij\sigma}(t_{ave})=-\frac{1}{\pi}\int d\omega \,\omega^n{\rm Im}\Sigma^R_{ij\sigma}(t_{ave},\omega).$$ Note that the self-energy is defined via the Dyson equation $$G_{ij\sigma}^R(t,t')=G_{ij\sigma}^{R0}(t,t')+\sum_{kl}\int d\bar t\int d\bar t' G^{R0}_{ik\sigma}(t,\bar t)\Sigma^R_{kl\sigma}(\bar t,\bar t')G^{R}_{lj\sigma}(\bar t',t'),$$ where $G^{R0}$ is the noninteracting Green’s function and the time integrals run from $-\infty$ to $\infty$. The strategy for evaluating the self-energy moments is rather simple. First, one writes the Green’s function and self-energy in terms of the respective spectral functions $$G^R_{ij\sigma}(t_{ave},\omega)=-\frac1\pi\int\frac{{\rm Im} G^R_{ij\sigma}(t_{ave},\omega^\prime)}{\omega-\omega^\prime+i0^+}d\omega^\prime$$ and $$\Sigma^R_{ij\sigma}(t_{ave},\omega)=\Sigma^R_{ij\sigma}(t_{ave},\infty)-\frac1\pi\int \frac{{\rm Im}\Sigma^R_{ij\sigma}(t_{ave},\omega^\prime)}{\omega-\omega^\prime+i0^+}d\omega^\prime.$$ Next, one substitutes those spectral representations into the Dyson equation that relates the Green’s function and self-energy to the noninteracting Green’s function. By expanding all functions in a series in $1/\omega$ for large $\omega$, one finds the spectral formulas involve summations over the moments. By employing the exact values for the Green’s function moments, one can extract the moments for the self-energy. Details for the formulas appear elsewhere [@sumrules2]. The end result is $$\Sigma^R_{ij\sigma}(t_{ave},\infty)=g(t_{ave})\langle x_i(t_{ave})\rangle \delta_{ij}$$ and $$C^{R0}_{ij\sigma}(t_{ave})=g^2(t_{ave})[
\langle x_i^2(t_{ave})\rangle -\langle x_i(t_{ave})\rangle^2].$$ So, the total strength (integrated weight) of the self-energy depends on the fluctuations of the phonon field.
Formalism for the phononic sum rules
====================================
The retarded phonon Green’s function is defined in a similar way, via $$D_{ij}^R(t,t')=-i\theta(t-t'){\rm Tr}e^{-\beta\mathcal{H}(t_{min})}[x_i(t),x_j(t')]_-/\mathcal{Z},$$ with the operators in the Heisenberg representation. The moments are defined in the same way as before. First one converts to the average and relative time coordinates and Fourier transforms with respect to the relative coordinate $$D_{ij}^R(t_{ave},\omega)=\int dt_{rel}e^{i\omega t_{rel}}D_{ij}^R(t_{ave}+\frac12t_{rel},t_{ave}-\frac12 t_{rel}),
\label{eq: phonon}$$ and then one computes the moments via $$m^{Rn}_{ij}(t_{ave})=-\frac{1}{\pi}\int d\omega \,\omega^n {\rm Im} D_{ij}^R(t_{ave},\omega).$$ The zeroth moment vanishes because $x_i$ commutes with itself at equal times. For the higher moments, we also derive a formula similar to what was used for the electronic Green’s functions. In particular, we have $$m^{R1}_{ij}(t_{ave})=-\frac12{\rm Im}\Big\{\langle [x^\prime_i(t_{ave}),x_j(t_{ave})]_-\rangle-
\langle [x_i(t_{ave}),x_j^\prime(t_{ave})]_-\rangle\Big\}$$ for the first moment. But $x^\prime_i(t_{ave})=-i[x_i(t_{tave}),\mathcal{H}_H(t_{ave})]_-=p_i(t_{ave})/m$, so we find $$m^{R1}_{ij}(t_{ave})=\frac{1}{m}\delta_{ij}.$$ Similarly, $$\begin{aligned}
m^{R2}_{ij}(t_{ave})&=&-\frac14{\rm Im}i\Big\{\langle [x^{\prime\prime}_i(t_{ave}),x_j(t_{ave})]_-\rangle-2\langle[x^\prime_j(t_{ave}),x^\prime_j(t_{ave})]_-\rangle\nonumber\\
&+&\langle [x_i(t_{ave}),x^{\prime\prime}_j(t_{ave})]_-\rangle\Big\}.\end{aligned}$$ Using the fact that $x^{\prime\prime}_i(t_{ave})=-i[p_i(t_{ave}),\mathcal{H}_H(t_{ave})]_-=-g(t_{ave})(n_{i\uparrow}(t_{ave})+n_{i\downarrow}(t_{ave}))-\kappa x_i(t_{ave})$, then shows that $m^{R2}_{ij}(t_{ave})=0$, since all commutators vanish. We don’t analyze the phonon self-energy here. Unlike the electronic moments, the phononic moments, are much simpler, and do not require any expectation values to evaluate them.
We end this section by showing that the imaginary part of the retarded phonon Green’s function is an odd function of $\omega$, which explains why all the even moments vanish. If one evaluates the complex conjugate of the retarded phonon Green’s function, one finds $$D_{ij}^R(t,t')^*=i\theta(t-t'){\rm Tr} [x_j(t'),x_i(t)]_-e^{-\beta \mathcal{H}(t_{min})}/\mathcal{Z}=D_{ij}^R(t,t')$$ where the last identity follows by switching the order of the operators in the commutator and the invariance of the trace under a cyclic permutation. Hence, the phonon propagator in the time representation is real. Evaluating the frequency-dependent propagator, then shows that $D_{ij}^{R*}(t_{ave},\omega)=D_{ij}^R(t_{ave},-\omega)$ by taking the complex conjugate of Eq. (\[eq: phonon\]). Hence the real part of the retarded phonon propagator in the frequency representation is an even function of frequency while the imaginary part is an odd function of frequency, and therefore all even moments vanish.
Atomic limit of the Holstein model
==================================
To get an idea of the phonon expectation values and the fluctuations, we solve explicitly for the expectation values for the Holstein model in the atomic limit, where $t_{ij}(t)=0$ and we can drop the site index from all operators. In this limit, one can exactly determine the Heisenberg representation operator $x(t)$ by solving the equation of motion for the Heisenberg representation operators $a(t)$ and $a^\dagger(t)$. This yields $$x(t)=\frac{ae^{-i\omega t}+a^\dagger e^{i\omega t}}{\sqrt{2m\omega}}-2{\rm Re}\left \{ie^{-\omega t}\int_0^tdt'e^{i\omega t'}g(t')\right \}\frac{n_\uparrow+n_\downarrow}{2m\omega}
\label{eq: x_heisenberg}$$ where the electronic number operators commute with $\mathcal{H}$ now, so they have no time dependence. Since the atomic sites are decoupled from one another, we can focus on just a single site. The partition function for a single site can be evaluated directly by employing standard raising and lowering operator identities. To begin, we note that the Hilbert space is composed of a direct product of the harmonic oscillator states $$|n\rangle=\frac{1}{\sqrt{n!}}\left ( a^\dagger \right )^n |0\rangle$$ and the fermionic states $$|0\rangle,\quad |\uparrow\rangle=c^\dagger_\uparrow|0\rangle,\quad |\downarrow\rangle=c^\dagger_\downarrow|0\rangle,\quad
|\uparrow\downarrow\rangle=c^\dagger_\uparrow c^\dagger_\downarrow |0\rangle.$$ The partition function satisfies $$\mathcal{Z}_{at}=\sum_{0,\uparrow,\downarrow,\uparrow\downarrow}\sum_{n^b=0}^\infty\langle n^b,n^f|\exp[-\beta \{
(gx-\mu)(n^f_\uparrow+n^f_\downarrow)+\omega (n^b+\frac12)\}]|n^b,n^f\rangle,$$ where $n^f$ denotes the Fermi number operator and $n^b$ the Boson number operator (we will drop the $\exp[\beta\omega/2]$ term which provides just a constant). Since the product states are not eigenstates of $\mathcal{H}$, we cannot immediately evaluate the partition function. Instead, we need to first go to the interaction representation with respect to the bosonic Hamiltonian in imaginary time (and we drop the constant term from the Hamiltonian), to find that $$\begin{aligned}
\mathcal{Z}_{at}&=&{\rm Tr}_f{\rm Tr}_be^{-\beta\omega n^b}\mathcal{T}_{\tau} \exp\left [ -\int_0^\beta d\tau'\left \{\frac{e^{-\omega\tau'}a+e^{\omega\tau'}a^\dagger}{\sqrt{2m\omega}}g(t_{min}
)-\mu\right \}n^f\right ],\nonumber\\
&=&{\rm Tr}_f{\rm Tr}_b e^{\beta\omega n^b}U_I(\beta)
\label{eq: partition}\end{aligned}$$ where the time-ordering operator is with respect to imaginary time and the time-ordered product is the evolution operator in the interaction representation and denoted by $U_I(\tau)$. Because the only operators that don’t commute in the evolution operator are $a$ and $a^\dagger$, and their commutator is a c-number, one can get an exact representation for the evolution operator via the Magnus expansion [@magnus], as worked out in the Landau and Lifshitz [@landau] or Gottfried [@gottfried] texts. The end result for the time-ordered product in Eq. (\[eq: partition\]) becomes $$\begin{aligned}
U_I(\beta)&=&\exp\left [ -\frac{g(t_{min})n^f}{\sqrt{2m\omega^3}}\left (1-e^{-\beta\omega}\right )a\right ]
\exp\left [ -\frac{g(t_{min})n^f}{\sqrt{2m\omega^3}}\left (e^{\beta\omega}-1\right )a^\dagger\right ]\nonumber\\
&\times&\exp\left [-\frac{g^2(t_{min})n^{f2}}{\sqrt{2m\omega^3}}\left ( e^{\beta\omega}-1-\beta\omega+\mu n^f\right )\right ],\end{aligned}$$ which used the Campbell-Baker-Hausdorff theorem $$e^{A+B}=e^Be^Ae^{\frac12[A,B]_-}$$ for the case when the commutator $[A,B]_-$ is a number, not an operator, to get the final expression. We substitute this result for the evolution operator into the trace over the bosonic states, expand the exponentials of the $a$ and $a^\dagger$ operators in a power series, and evaluate the bosonic expectation value to find $$\begin{aligned}
\mathcal{Z}_{at}&=&{\rm Tr}_f\sum_{m=0}^\infty\sum_{n=0}^\infty \frac{(n+m)!}{n!m!m!}\left (\frac{g^2(t_{min})n^{f2}}{2m\omega^3}\left (e^{\beta\omega}-1+e^{-\beta\omega}-1\right )\right )^m\nonumber\\
&\times&\exp\left [-\beta\omega n+\beta\mu n^f-\frac{g^2(t_{min})n^{f2}}{2m\omega^3}\left ( e^{\beta\omega}-1-\beta\omega\right )\right ].\end{aligned}$$ Next, we use Newton’s generalized binomial theorem $$\sum_{n=0}^\infty\frac{(n+m)!}{n!m!}z^n=\frac{1}{(1-z)^{m+1}},$$ to simplify the expression for the partition function to $$\mathcal{Z}_{at}={\rm Tr}_f\exp\left [\frac{\beta g^2(t_{min})n^{f2}}{2m\omega^2}+\beta\mu n^f\right ]\frac{1}{1-e^{-\beta\omega}}.$$ Performing the trace over the fermionic states then yields $$\mathcal{Z}_{at}=\frac{1}{1-e^{-\beta\omega}}\left \{1+2e^{\beta\mu}\exp\left [\frac{\beta g^2(t_{min})}{2m\omega^2}\right ]
+e^{2\beta\mu}\exp\left [\frac{2\beta g^2(t_{min})}{m\omega^2}\right ]\right \}.$$
We calculate expectation values following the same procedure, but inserting the relevant operators in the Heisenberg representation at the appropriate place, and carrying out the remainder of the derivation as done for the partition function. For example, since the Fermi number operators commute with the atomic Hamiltonian, they are the same operator in the Heisenberg and Schroedinger representations, and we immediately find that the electron density is a constant in time and is given by $$\langle n_\uparrow+n_\downarrow\rangle=\frac{2e^{\beta\mu}\exp\left [\frac{\beta g^2(t_{min})}{2m\omega^2}\right ]
+2e^{2\beta\mu}\exp\left [\frac{2\beta g^2(t_{min})}{m\omega^2}\right ]}{1+2e^{\beta\mu}\exp\left [\frac{\beta g^2(t_{min})}{2m\omega^2}\right ]+e^{2\beta\mu}\exp\left [\frac{2\beta g^2(t_{min})}{m\omega^2}\right ]}.$$
We next want to calculate $\langle x(t)\rangle$ and $\langle x^2(t)\rangle-\langle x(t)\rangle^2$, where the operator is in the Heisenberg representation, and given in Eq. (\[eq: x\_heisenberg\]). It is straightforward but tedious to calculate the averages. After much algebra, we find $$\langle x(t)\rangle=\langle n_\uparrow+n_\downarrow\rangle\left (-\frac{g(t_{min})}{m\omega^2}\cos\omega t-{\rm Re}\left \{ie^{-i\omega t}\int_0^tdt'e^{i\omega t'}\frac{g(t')}{m\omega}\right \}\right ).$$ (Note that if we are in equilibrium, so $g(t)=g$ is a constant, then one finds $\langle x(t)\rangle=-\langle n_\uparrow+n_\downarrow\rangle g/m\omega^2$, which has no time dependence, as expected.) The fluctuation satisfies $$\begin{aligned}
\langle x^2(t)\rangle-\langle x(t)\rangle^2&=&[\langle n_\uparrow\rangle(1-\langle n_\uparrow\rangle)+\langle n_\downarrow\rangle(1-\langle n_\downarrow\rangle)+2\langle n_\uparrow n_\downarrow\rangle-2\langle n_\uparrow\rangle\langle n_\downarrow\rangle ]\nonumber\\
&\times&\left [\frac{g(t_{min})}{m\omega^2}\cos\omega t+{\rm Re}\left \{ie^{-i\omega t}\int_0^tdt'e^{i\omega t'}\frac{g(t')}{m\omega}\right \}\right ]^2\nonumber\\
&+&\frac{1}{2m\omega}{\rm coth}\left (\frac{\beta\omega}{2}\right ),\end{aligned}$$ which consists of two terms: a time-dependent piece (which becomes a constant when $g$ is a constant) that represents the quantum fluctuations due to the electron-phonon interaction and a phonon piece that varies with temperature (and is independent of $g$). The latter piece becomes large when $T\rightarrow\infty$ (being proportional to $T$ at high $T$), which tells us that fluctuations generically grow with increasing the temperature of the system, so that one expects the zeroth moment of the self energy to increase as the temperature increases, or if the system is heated up by being driven by a large electric field. Note that if one expands the self-energy perturbatively, as in Migdal-Eliashberg theory, then only the term independent of $g$ survives, as the other term is higher order in $g$ and lies outside of the Migdal-Eliashberg result [@lex].
Discussion and applications of the sum rules
============================================
One of the most important recent experiments in electron-phonon interacting systems involves time-resolved angle-resolved photoemission (tr-ARPES), which can be analyzed in such a way that one can extract information about the electronic self-energy [@prx; @lex]. If one assumes that the phonons form an infinite heat capacity bath, then they are not changed by the excitation of the electrons, and the fluctuations of the phonon field remain a constant as a function of time. This leads to a self-energy that can transiently change shape as a function of time, but does not change its spectral weight. Recent calculations show precisely this behavior [@prx; @lex]. One can also understand it from the perturbation theory expansion, where a direct evaluation of the diagrams for the self-energy, and the sum rules for the electronic Green’s functions, establish that the zeroth moment of the retarded electronic self-energy is a constant [@lex]. What is perhaps more interesting, is when one treats a fully self-consistent system where the electrons and phonons both can exchange energy with one another, and the phonon bath properties change transiently. In this case, one has to examine the self-consistency for both the electrons and the phonons within the perturbation theory, and the general form of the sum-rules hold. The calculations shown above in the atomic limit indicate that it is likely that adding energy into the phonon system increases the phonon fluctuations and thereby creates a stronger electronic self-energy. One would expect there to be oscillations of the spectral weight as well. It is also likely that these ideas can be incorporated into the quantitative analysis of experiments that we expect to see occur over the next few years.
Conclusions and future work
===========================
In this work, we have shown the simplest sum rules for electrons interacting with phonons. These sum rules have been established in equilibrium for some time now, but our work shows that they directly extend to nonequilibrium. We also established new sum rules for the phonon propagator. In general, these sum rules are complicated to use, because they require one to determine both the average phonon expectation value and its fluctuations, so they might find their most important application to numerics as benchmarking, assuming one can calculate the relevant expectation values with the numerical techniques employed to solve the problem. But they also allow us to examine the physical behavior we expect to see if we look at how the moments might change in time due to the effect of a transient light pump applied to the system. For example, we expect that as energy is exchanged from electrons to phonons, the electron self-energy should increase its spectral weight, with the opposite occuring as the phonons transfer energy back to the electrons. This result is not one that could have been easily predicted without the sum rules.
In the future, there are a number of ways these sum rules can be extended. One can examine more realistic models, like the Hubbard-Holstein model and find those sum rules. One can look into the effects of anharmonicity on the sum rules, and finally, one can carry out the calculations to higher order, to examine more moments. We plan to work on a number of these problems in the future.
Acknowledgments
===============
This work was supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering (DMSE) under Contracts No. DE-AC02-76SF00515 (Stanford/SIMES), No. DE-FG02-08ER46542 (Georgetown), and No. DE-SC0007091 (for the collaboration). J.K.F. was also supported by the McDevitt bequest at Georgetown
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---
abstract: 'Properties of compositions and convex combinations of averaged nonexpansive operators are investigated and applied to the design of new fixed point algorithms in Hilbert spaces. An extended version of the forward-backward splitting algorithm for finding a zero of the sum of two monotone operators is obtained.'
author:
- |
Patrick L. Combettes$^1$ and Isao Yamada$^2$\
$\!^1$Sorbonne Universités – UPMC Univ. Paris 06\
UMR 7598, Laboratoire Jacques-Louis Lions\
F-75005 Paris, France\
\
$\!^2$Tokyo Institute of Technology\
Department of Communications and Computer Engineering\
Tokyo 152-8550, Japan\
\
date:
title: |
Compositions and Convex Combinations of\
Averaged Nonexpansive Operators[^1]
---
[**Keywords.**]{} averaged operator $\cdot$ fixed-point algorithm $\cdot$ forward-backward splitting $\cdot$ monotone operator $\cdot$ nonexpansive operator
Introduction
============
Since their introduction in [@Bail78], averaged nonexpansive operators have proved to be very useful in the analysis and the numerical solution of problems arising in nonlinear analysis and its applications; see, e.g., [@Bail92; @Bail14; @Baus96; @Livre1; @Byrn04; @Cegi12; @Opti04; @Cond13; @Crom07; @Ogur02; @Ragu13; @Weny08; @Yama01; @Yama04].
\[d:averaged\] Let ${\ensuremath{{\mathcal H}}}$ be a real Hilbert space, let $D$ be a nonempty subset of ${\ensuremath{{\mathcal H}}}$, let $\alpha\in{\ensuremath{\left]0,1\right[}}$, and let $T\colon D\to{\ensuremath{{\mathcal H}}}$ be a nonexpansive (i.e., 1-Lipschitz) operator. Then $T$ is averaged with constant $\alpha$, or $\alpha$-averaged, if there exists a nonexpansive operator $R\colon D\to{\ensuremath{{\mathcal H}}}$ such that $T=(1-\alpha){\ensuremath{\mathrm{Id}}\,}+\alpha R$.
As discussed in [@Livre1; @Opti04; @Ogur02], averaged operators are stable under compositions and convex combinations and such operations form basic building blocks in various composite fixed point algorithms. The averagedness constants resulting from such operations determine the range of the step sizes and other parameters in such algorithms. It is therefore important that they be tight since these parameters have a significant impact on the speed of convergence.
In this paper, we discuss averagedness constants for compositions and convex combinations of averaged operators and construct novel fixed point algorithms based on these constants. In particular, we obtain a new version of the forward-backward algorithm with an extended relaxation range and iteration-dependent step sizes.
Throughout the paper, ${\ensuremath{{\mathcal H}}}$ is a real Hilbert space with scalar product ${{\left\langle{{\cdot}\mid{\cdot}}\right\rangle}}$ and associated norm $\|\cdot\|$. We denote by ${\ensuremath{\mathrm{Id}}\,}$ the identity operator on ${\ensuremath{{\mathcal H}}}$ and by $d_S$ the distance function to a set $S\subset{\ensuremath{{\mathcal H}}}$; ${\ensuremath{\:\rightharpoonup\:}}$ and $\to$ denote, respectively, weak and strong convergence in ${\ensuremath{{\mathcal H}}}$.
Compositions and convex combinations of averaged operators {#sec:2}
==========================================================
We first recall some characterizations of averaged operators (see [@Opti04 Lemma 2.1] or [@Livre1 Proposition 4.25]).
\[p:av1\] Let $D$ be a nonempty subset of ${\ensuremath{{\mathcal H}}}$, let $T\colon D\to{\ensuremath{{\mathcal H}}}$ be nonexpansive, and let $\alpha\in{\ensuremath{\left]0,1\right[}}$. Then the following are equivalent:
1. \[p:av1i\] $T$ is $\alpha$-averaged.
2. \[p:av1i’\] $(1-1/\alpha){\ensuremath{\mathrm{Id}}\,}+(1/\alpha)T$ is nonexpansive.
3. \[p:av1ii\] $(\forall x\in D)(\forall y\in D)$ $\|Tx-Ty\|^2{\ensuremath{\leqslant}}\|x-y\|^2-\displaystyle{\frac{1-\alpha}{\alpha}}
\|({\ensuremath{\mathrm{Id}}\,}-T)x-({\ensuremath{\mathrm{Id}}\,}-T)y\|^2$.
4. \[p:av1iii\] $(\forall x\in D)(\forall y\in D)$ $\|Tx-Ty\|^2+(1-2\alpha)\|x-y\|^2{\ensuremath{\leqslant}}2(1-\alpha){{\left\langle{{x-y}\mid{Tx-Ty}}\right\rangle}}$.
The next result concerns the averagedness of a convex combination of averaged operators.
\[p:av2\] Let $D$ be a nonempty subset of ${\ensuremath{{\mathcal H}}}$, let $(T_i)_{i\in I}$ be a finite family of nonexpansive operators from $D$ to ${\ensuremath{{\mathcal H}}}$, let $(\alpha_i)_{i\in I}$ be a family in ${\ensuremath{\left]0,1\right[}}$, and let $(\omega_i)_{i\in I}$ be a family in $]0,1]$ such that $\sum_{i\in I}\omega_i=1$. Suppose that, for every $i\in I$, $T_i$ is $\alpha_i$-averaged, and set $T=\sum_{i\in I}\omega_i T_i$ and $\alpha=\sum_{i\in I}\omega_i\alpha_i$. Then $T$ is $\alpha$-averaged.
For every $i\in I$, there exists a nonexpansive operator $R_i\colon D\to{\ensuremath{{\mathcal H}}}$ such that $T_i=(1-\alpha_i){\ensuremath{\mathrm{Id}}\,}+\alpha_i R_i$. Now set $R=(1/\alpha)\sum_{i\in I}\omega_i\alpha_i R_i$. Then $R$ is nonexpansive and $$\sum_{i\in I}\omega_iT_i=\sum_{i\in I}\omega_i(1-\alpha_i){\ensuremath{\mathrm{Id}}\,}+\sum_{i\in I}\omega_i\alpha_i R_i=(1-\alpha){\ensuremath{\mathrm{Id}}\,}+\alpha R.$$ We conclude that $T$ is $\alpha$-averaged.
\[raQg507\] In view of [@Cegi12 Corollary 2.2.17], Proposition \[p:av2\] is equivalent to [@Cegi12 Theorem 2.2.35], and it improves the averagedness constant of [@Opti04 Lemma 2.2(ii)] which was $\alpha=\text{\rm max}_{i\in I}\alpha_i$. In the case of two operators, Proposition \[p:av2\] can be found in [@Ogur02 Theorem 3(a)].
Next, we turn our attention to compositions of averaged operators, starting with the following result, which was obtained in [@Ogur02 Theorem 3(b)] with a different proof.
\[paQg507-03\] Let $D$ be a nonempty subset of ${\ensuremath{{\mathcal H}}}$, let $(\alpha_1,\alpha_2)\in{\ensuremath{\left]0,1\right[}}^2$, let $T_1\colon D\to D$ be $\alpha_1$-averaged, and let $T_2\colon D\to D$ be $\alpha_2$-averaged. Set $$\label{XAre2e26a}
T=T_1 T_2\quad\text{and}\quad
\alpha=\frac{\alpha_1+\alpha_2-2\alpha_1\alpha_2}
{1-\alpha_1\alpha_2}.$$ Then $\alpha\in{\ensuremath{\left]0,1\right[}}$ and $T$ is $\alpha$-averaged.
Since $\alpha_1(1-\alpha_2)<(1-\alpha_2)$, we have $\alpha_1+\alpha_2<1+\alpha_1\alpha_2$ and, therefore, $\alpha\in{\ensuremath{\left]0,1\right[}}$. Now let $x\in D$, let $y\in D$, and set $$\label{Gt^reeb8718g}
\tau=\frac{1-\alpha_1}{\alpha_1}+
\frac{1-\alpha_2}{\alpha_2}.$$ It follows from Proposition \[p:av1\] that $$\begin{aligned}
\label{e:TfgtF5e9-14a}
\|T_1 T_2x-T_1 T_2 y\|^2
&{\ensuremath{\leqslant}}\|T_2 x-T_2 y\|^2-\frac{1-\alpha_1}{\alpha_1}
\left\|({\ensuremath{\mathrm{Id}}\,}-T_1)T_2 x-({\ensuremath{\mathrm{Id}}\,}-T_1)T_2 y\right\|^2\nonumber\\
&{\ensuremath{\leqslant}}\|x-y\|^2-\frac{1-\alpha_2}{\alpha_2}
\left\|({\ensuremath{\mathrm{Id}}\,}-T_2)x-({\ensuremath{\mathrm{Id}}\,}-T_2)y\right\|^2\nonumber \\
&\quad\;-\frac{1-\alpha_1}{\alpha_1}
\left\|({\ensuremath{\mathrm{Id}}\,}-T_1)T_2 x-({\ensuremath{\mathrm{Id}}\,}-T_1)T_2 y\right\|^2.\end{aligned}$$ Moreover, by [@Livre1 Corollary 2.14], we have $$\begin{aligned}
\label{e:TfgtF5e9-14b}
&\hskip -8 mm
\frac{1-\alpha_1}{\tau\alpha_1}\left\|({\ensuremath{\mathrm{Id}}\,}-T_1)T_2
x-({\ensuremath{\mathrm{Id}}\,}-T_1)T_2 y\right\|^2+\frac{1-\alpha_2}{\tau\alpha_2}
\left\|({\ensuremath{\mathrm{Id}}\,}-T_2)x-({\ensuremath{\mathrm{Id}}\,}-T_2)y\right\|^2\nonumber\\
&=\left\|\frac{1-\alpha_1}{\tau\alpha_1}
\big(({\ensuremath{\mathrm{Id}}\,}-T_1)T_2 x-({\ensuremath{\mathrm{Id}}\,}-T_1)T_2 y\big)+
\frac{1-\alpha_2}{\tau\alpha_2}\big(({\ensuremath{\mathrm{Id}}\,}-T_2)x-({\ensuremath{\mathrm{Id}}\,}-T_2)y\big)
\right\|^2\nonumber\\
&\quad\;+\frac{(1-\alpha_1)(1-\alpha_2)}{\tau^2\alpha_1\alpha_2}
\left\|(x-y)-\left(T_{1}T_2x-T_{1}T_2y\right)
\right\|^2\nonumber \\
&{\ensuremath{\geqslant}}\frac{(1-\alpha_1)(1-\alpha_2)}{\tau^2\alpha_1\alpha_2}
\left\|({\ensuremath{\mathrm{Id}}\,}-T_1T_2)x-({\ensuremath{\mathrm{Id}}\,}-T_1T_2)y\right\|^2. \end{aligned}$$ Combining , , and yields $$\begin{aligned}
\|T_1 T_2 x-T_1 T_2 y\|^2
&{\ensuremath{\leqslant}}\|x-y\|^2-\frac{(1-\alpha_1)(1-\alpha_2)}
{\tau\alpha_1\alpha_2}
\left\|\left({\ensuremath{\mathrm{Id}}\,}-T_1T_2\right)x-\left({\ensuremath{\mathrm{Id}}\,}-T_1T_2\right)y\right\|^2
\nonumber\\
&=\|x-y\|^2-\frac{1-\alpha_1-\alpha_2+\alpha_1\alpha_2}
{\alpha_1+\alpha_2-2\alpha_1\alpha_2}
\left\|\left({\ensuremath{\mathrm{Id}}\,}-T_1T_2\right)x-\left({\ensuremath{\mathrm{Id}}\,}-T_1T_2\right)y\right\|^2
\nonumber\\
&=\|x-y\|^2-\frac{1-\alpha}{\alpha}
\left\|\left({\ensuremath{\mathrm{Id}}\,}-T_1T_2\right)x-\left({\ensuremath{\mathrm{Id}}\,}-T_1T_2\right)y\right\|^2.\end{aligned}$$ In view of Proposition \[p:av1\], we conclude that $T$ is $\alpha$-averaged.
In [@Cegi12 Theorem 2.2.37], the averagedness constant of was written as $$\label{XAre2e26b}
\alpha=\dfrac{1}{1+\dfrac{1}{
\dfrac{\alpha_1}{1-\alpha_1}+\dfrac{\alpha_2}{1-\alpha_2}}}.$$ By induction, it leads to the following result for the composition of $m$ averaged operators, which was obtained in [@Cegi12] (combine [@Cegi12 Theorem 2.2.42] and [@Cegi12 Corollary 2.2.17]).
\[paQg507-03’\] Let $D$ be a nonempty subset of ${\ensuremath{{\mathcal H}}}$, let $m{\ensuremath{\geqslant}}2$ be an integer, and set $$\label{hhghwedw710f27}
\phi\colon{\ensuremath{\left]0,1\right[}}^m\to{\ensuremath{\left]0,1\right[}}\colon
(\alpha_1,\ldots,\alpha_m)\mapsto\dfrac{1}{1+\dfrac{1}{{\ensuremath{\displaystyle\sum}}_{i=1}^m
\dfrac{\alpha_i}{1-\alpha_i}}}.$$ For every $i\in\{1,\ldots,m\}$, let $\alpha_i\in{\ensuremath{\left]0,1\right[}}$ and let $T_i\colon D\to D$ be $\alpha_i$-averaged. Set $$\label{XAre2e25}
T=T_1\cdots T_m\quad\text{and}\quad
\alpha=\phi(\alpha_1,\ldots,\alpha_m).$$ Then $T$ is $\alpha$-averaged.
We proceed by induction on $k\in\{2,\ldots,m\}$. To this end, let us set $(\forall k\in\{2,\ldots,m\})$ $\beta_k=[1+[\sum_{i=1}^k\alpha_i/(1-\alpha_i)]^{-1}]^{-1}$. By Proposition \[paQg507-03\] and , the claim is true for $k=2$. Now assume that, for some $k\in\{2,\ldots,m-1\}$, $T_1\cdots T_{k}$ is $\beta_k$-averaged. Then we deduce from Proposition \[paQg507-03\] and that the averagedness constant of $(T_1\cdots T_{k})T_{k+1}$ is $$\label{XAre2e26c}
\dfrac{1}{1+\dfrac{1}{
\dfrac{1}{\beta_k^{-1}-1}+\dfrac{\alpha_{k+1}}{1-\alpha_{k+1}}}}
=\dfrac{1}{1+\dfrac{1}{\bigg({\ensuremath{\displaystyle\sum}}_{i=1}^k\dfrac{\alpha_i}
{1-\alpha_i}
\bigg)+\dfrac{\alpha_{k+1}}{1-\alpha_{k+1}}}}=\beta_{k+1},$$ which concludes the induction argument.
The following result provides alternative expressions for the averagedness constant $\alpha$ of .
\[paQg507-03”\] Let $m{\ensuremath{\geqslant}}2$ be an integer, let $\phi$ be as in , let $(\alpha_i)_{1{\ensuremath{\leqslant}}i{\ensuremath{\leqslant}}m}\in{\ensuremath{\left]0,1\right[}}^m$, and let $(\sigma_j)_{1{\ensuremath{\leqslant}}j{\ensuremath{\leqslant}}m}$ the elementary symmetric polynomials in the variables $(\alpha_i)_{1{\ensuremath{\leqslant}}i{\ensuremath{\leqslant}}m}$, i.e., $$\label{Bv54Ewe309-27b}
(\forall j\in\{1,\ldots,m\})\quad
\sigma_j=\sum_{1{\ensuremath{\leqslant}}i_1<\cdots<i_j{\ensuremath{\leqslant}}m}
\prod_{l=1}^j\alpha_{i_l}.$$ Then the following hold:
1. \[paQg507-03”ii\] $\phi(\alpha_1,\ldots,\alpha_m)=\big[
{\sum_{l=1}^{{\ensuremath{{+\infty}}}}\sum_{i=1}^{m}\alpha_{i}^{l}}\big]\big/\big[
{1+\sum_{l=1}^{{\ensuremath{{+\infty}}}}\sum_{i=1}^{m}\alpha_{i}^{l}}\big]$.
2. \[paQg507-03”iii\] $\phi(\alpha_1,\ldots,\alpha_m)=
\big[{\sum_{j=1}^m(-1)^{j-1}j\sigma_j}\big]\big/
\big[{1+\sum_{j=2}^{m} (-1)^{j-1}(j-1)\sigma_j}\big]$.
3. \[paQg507-03”i\] $\phi(\alpha_1,\ldots,\alpha_m)>\text{\rm max}_{1{\ensuremath{\leqslant}}i{\ensuremath{\leqslant}}m}
\alpha_i$.
\[paQg507-03”ii\]: Indeed, yields $$\phi(\alpha_1,\ldots,\alpha_m)=
\frac{{\ensuremath{\displaystyle\sum}}_{i=1}^{m}\left(\frac{1}{1-\alpha_i}-1\right)}
{1+{\ensuremath{\displaystyle\sum}}_{i=1}^{m}\left(\frac{1}{1-\alpha_i}-1\right)},\quad
\text{where}\quad(\forall i\in\{1,\ldots,m\})\quad
\frac{1}{1-\alpha_i}-1=\sum_{l=1}^{{\ensuremath{{+\infty}}}}\alpha_i^{l}.$$
\[paQg507-03”iii\]: Using the inductive argument of the proof of Proposition \[paQg507-03’\] and , we observe that $\phi(\alpha_1,\ldots,\alpha_m)$ can be defined via the recursion $$\label{Gt^reeb8718f}
\begin{array}{l}
\left\lfloor
\begin{array}{l}
\beta_1=\alpha_1\\
\text{for}\;k=1,\ldots,m-1\\
\begin{array}{l}
\left\lfloor
\begin{array}{l}
\beta_{k+1}={\displaystyle{\frac{\alpha_{k+1}+\beta_k-2\alpha_{k+1}\beta_k}{1-\alpha_{k+1}\beta_k}}}
\end{array}
\right.\\[2mm]
\end{array}\\
\phi(\alpha_1,\ldots,\alpha_m)=\beta_m.
\end{array}
\right.
\end{array}$$ Set $$\label{e:BnBn6Tf3419b}
(\forall k\in\{1,\ldots,m \})\quad
\begin{cases}
s_{0}(k)=1\\
s_{k+1}(k)=0\\
(\forall j\in\{1,\ldots, k\})\quad
s_{j}(k)=\sum_{1{\ensuremath{\leqslant}}i_1<\cdots<i_j{\ensuremath{\leqslant}}k}
\prod_{l=1}^j\alpha_{i_l}.
\end{cases}$$ We have $(\forall j\in\{1,\ldots, m\})$ $\sigma_j=s_j(m)$. Furthermore, $$\label{e:BnBn6Tf3418b}
(\forall k\in\{1,\ldots,m-1\})(\forall j\in\{0,\ldots,k\})\quad
s_{j+1}(k+1)=s_{j+1}(k)+\alpha_{k+1}s_{j}(k).$$ Let us show by induction that, for every $k\in \{2,\ldots, m\}$, $$\label{e:n6Tk13420a}
\beta_{k}=\frac{\sum_{j=1}^k(-1)^{j-1}js_j(k)}{1+\sum_{j=2}^{k}
(-1)^{j-1}(j-1)s_{j}(k)}.$$ Since $s_{1}(2)=\alpha_1+\alpha_2$ and $s_2(2)=\alpha_1\alpha_2$, yields $$\beta_2=\frac{\alpha_2+\beta_1-2\alpha_2\beta_1}{1-\alpha_2 \beta_1}
=\frac{\alpha_2+\alpha_1-2\alpha_2\alpha_1}{1-\alpha_2\alpha_1}
=\frac{s_{1}(2)-2s_2(2)}{1-s_2(2)}.$$ This establishes for $k=2$. Now suppose that holds for some $k\in \{2,\ldots, m-1\}$. We derive from and that $$\begin{aligned}
\label{e:BnBn6Tf3418d}
&\hskip -6mm
\big(\alpha_{k+1}+\beta_k-2\alpha_{k+1}\beta_k\big)
\Bigg(1+\sum_{j=2}^{k}(-1)^{j-1}(j-1)s_{j}(k)\Bigg)\nonumber\\
&=\alpha_{k+1}\bigg(1+\sum_{j=2}^{k}(-1)^{j-1}(j-1)s_{j}(k)\bigg)
\nonumber\\
&\quad\;+\sum_{j=1}^{k}(-1)^{j-1}js_{j}(k)-2\alpha_{k+1}
\bigg(s_{1}(k)+\sum_{j=2}^{k}(-1)^{j-1}js_{j}(k)\bigg)\nonumber\\
&=\alpha_{k+1}+s_{1}(k)+
\sum_{j=2}^{k}(-1)^{j-1}js_{j}(k)\nonumber\\
&\quad\;+\alpha_{k+1}\bigg(\sum_{j=2}^{k}(-1)^{j-1}(j-1)s_{j}(k)
-2\sum_{j=2}^{k}(-1)^{j-1}js_{j}(k)-2s_{1}(k)\bigg)
\nonumber\\
&=s_{1}(k+1)+\sum_{j=2}^{k}(-1)^{j-1}js_{j}(k)
-\alpha_{k+1}\bigg(\sum_{j=2}^{k}(-1)^{j-1}(j+1)s_{j}(k)
+2s_{1}(k)\bigg)\nonumber\\
&=s_{1}(k+1)+\sum_{j=2}^{k}(-1)^{j-1}js_{j}(k)
-\alpha_{k+1}\sum_{j=1}^{k}(-1)^{j-1}(j+1)s_{j}(k)\nonumber\\
&=s_{1}(k+1)+\sum_{j=2}^{k}(-1)^{j-1}js_{j}(k)
-\alpha_{k+1}\sum_{j=1}^{k-1}(-1)^{j-1}(j+1)s_{j}(k)\nonumber\\
&\quad\; -(-1)^{k-1}(k+1)\alpha_{k+1}s_{k}(k)\nonumber\\
&=s_{1}(k+1)+\sum_{j=2}^{k}(-1)^{j-1}js_{j}(k)
+\alpha_{k+1}\sum_{j=2}^{k}(-1)^{j-1}js_{j-1}(k)
+(-1)^{k}(k+1)s_{k+1}(k+1)\nonumber\\
&=s_{1}(k+1)+\sum_{j=2}^{k}(-1)^{j-1}j\big(s_{j}(k)
+\alpha_{k+1}s_{j-1}(k)\big)
+(-1)^{k}(k+1)s_{k+1}(k+1)\nonumber\\
&=s_{1}(k+1)+\sum_{j=2}^{k}(-1)^{j-1}js_{j}(k+1)
+(-1)^{k}(k+1)s_{k+1}(k+1)\nonumber\\
&=\sum_{j=1}^{k+1}(-1)^{j-1}js_{j}(k+1)\end{aligned}$$ and that $$\begin{aligned}
\label{e:BnBn6Tf3418c}
&\hskip -7mm
\big(1-\alpha_{k+1}\beta_k\big)
\Bigg(1+\sum_{j=2}^{k}(-1)^{j-1}(j-1)s_{j}(k)\Bigg)\nonumber\\
&=\Bigg(1-\frac{\sum_{j=1}^{k}(-1)^{j-1}js_{j}(k)\alpha_{k+1}}
{1+\sum_{j=2}^{k}(-1)^{j-1}(j-1)s_{j}(k)}\Bigg)\Bigg(
1+\sum_{j=2}^{k}(-1)^{j-1}(j-1)s_{j}(k)\Bigg)\nonumber \\
&=1+\sum_{j=2}^{k}(-1)^{j-1}(j-1)s_{j}(k)-\sum_{j=1}^{k}(-1)^{j-1}j
s_{j}(k)\alpha_{k+1}\nonumber \\
&=1+\sum_{j=2}^{k}(-1)^{j-1}(j-1)s_{j}(k)+\sum_{j=1}^{k}(-1)^{j-1}j
\big(s_{j+1}(k)-s_{j+1}(k+1)\big)\nonumber\\
&=1+\sum_{j=2}^{k}(-1)^{j-1}(j-1)s_{j}(k)-
\sum_{j=2}^{k+1}(-1)^{j-1}(j-1)s_{j}(k)+\sum_{j=2}^{k+1}
(-1)^{j-1}(j-1)s_{j}(k+1)\nonumber\\
&=1+\sum_{j=2}^{k+1}(-1)^{j-1}(j-1)s_{j}(k+1). \end{aligned}$$ Taking the ratio of and yields $$\label{ewope36}
\beta_{k+1}={\displaystyle{\frac{\alpha_{k+1}+\beta_k-2\alpha_{k+1}\beta_k}{1-\alpha_{k+1}\beta_k}}}=\frac{\sum_{j=1}^{k+1}(-1)^{j-1}js_j(k+1)}
{1+\sum_{j=2}^{k+1}(-1)^{j-1}(j-1)s_{j}(k+1)}.$$ This shows that holds for every $k\in \{2,\dots,m\}$.
\[paQg507-03”i\]: We need to consider only the case when $m=2$ since the general case will follow from by induction. We derive from that $$\label{e:m=2}
\beta_2=\frac{\alpha_1+\alpha_2-2\alpha_1\alpha_2}
{1-\alpha_1\alpha_2}.$$ Since $\beta_2-\alpha_1=\alpha_2(1-\alpha_1)^{2}/(1-\alpha_1\alpha_2)>0$ and $\beta_2-\alpha_2=\alpha_1(1-\alpha_2)^{2}/(1-\alpha_1\alpha_2)>0$, we have $\beta_2>\text{max}\{\alpha_1,\alpha_2\}>0$.
\[KWghy5-f-09\] Let us compare the averagedness constant of Proposition \[paQg507-03’\] with alternative ones. Set $$\label{e:opti2004}
\widetilde{\phi}\colon{\ensuremath{\left]0,1\right[}}^m\to{\ensuremath{\left]0,1\right[}}\colon
(\alpha_1,\ldots,\alpha_m)\mapsto
{\displaystyle{\frac{m\,\text{max}\{\alpha_1,\ldots,\alpha_m\}}{(m-1)\text{max}\{\alpha_1,\ldots,\alpha_m\}+1}}},$$ and let $(\alpha_i)_{1{\ensuremath{\leqslant}}i{\ensuremath{\leqslant}}m}\in{\ensuremath{\left]0,1\right[}}^m$.
1. \[KWghy5-f-09i\] The averagedness constant of Proposition \[paQg507-03’\] is sharper than that of [@Opti04 Lemma 2.2(iii)], namely $$\label{e:opti04}
\phi(\alpha_1,\ldots,\alpha_m){\ensuremath{\leqslant}}\widetilde{\phi}(\alpha_1,\ldots,\alpha_m).$$
2. \[r:iuiiipii\] ${\phi}(\alpha_1,\ldots,\alpha_m)=
\widetilde{\phi}(\alpha_1,\ldots,\alpha_m)$ if $\alpha_1=\cdots=\alpha_m$ and, in particular, if all the operators are firmly nonexpansive, i.e., $\alpha_1=\cdots=\alpha_m=1/2$.
3. \[KWghy5-f-09iii\] If $m=2$, the averagedness constant of Proposition \[paQg507-03’\] is strictly sharper than that of [@Weny08 Lemma 3.2], namely (see also [@Cegi12 Remark 2.2.38]) $$\label{e:chin08}
\phi(\alpha_1,\alpha_2)<\widehat{\phi}(\alpha_1,\alpha_2),
\quad\text{where}\quad
\widehat{\phi}(\alpha_1,\alpha_2)=\alpha_1+\alpha_2-\alpha_1\alpha_2.$$ In addition, $\phi(\alpha_1,\alpha_1)=
\widetilde{\phi}(\alpha_1,\alpha_1)<
\widehat{\phi}(\alpha_1,\alpha_1)$ while, for $\alpha_1={3}/{4}$ and $\alpha_2={1}/{8}$, $\widehat{\phi}(\alpha_1,\alpha_2)={25}/{32}<
{6}/{7}=\widetilde{\phi}(\alpha_1,\alpha_2)$, which shows that $\widetilde{\phi}$ and $\widehat{\phi}$ cannot be compared in general.
\[KWghy5-f-09i\]: Combine [@Cegi12 Theorem 2.2.42], and [@Cegi12 Corollary 2.2.17].
\[r:iuiiipii\]: Set $\beta_1=\delta_1=\alpha_1$ and $$\label{e:BnBn6Tf3421b}
(\forall k\in\{2,\ldots,m\})\quad
\begin{cases}
\beta_k=\dfrac{1}{1+\dfrac{1}{{\ensuremath{\displaystyle\sum}}_{i=1}^k
\dfrac{\alpha_i}{1-\alpha_i}}},\\[18mm]
\delta_k={\displaystyle{\frac{k\,\text{max}\{\alpha_1,\ldots,\alpha_k\}}{(k-1)\text{max}\{\alpha_1,\ldots,\alpha_k\}+1}}}.
\end{cases}$$ Then yields $$\label{Bv54Ewe36-9a}
(\forall k\in\{1,\ldots,m\})\quad
\delta_k=\frac{k\alpha_1}{(k-1)\alpha_1+1}.$$ Let us show by induction that $$\label{jIv5989e36-9b}
(\forall k\in\{1,\ldots,m\})\quad \beta_k=\delta_k.$$ We have $\beta_1=\delta_1=\alpha_1$. Next, suppose that, for some $k\in\{1,\ldots,m-1\}$, $\beta_k=\delta_k$. Then $\alpha_{k+1}=\alpha_1$, while and yield $$\label{e:case4}
\beta_{k+1}=\dfrac{1}{1+\dfrac{1}{
\dfrac{1}{\beta_k^{-1}-1}+\dfrac{\alpha_1}{1-\alpha_1}}}
=\dfrac{1}{1+\dfrac{1}{
\dfrac{1}{\delta_k^{-1}-1}+\dfrac{\alpha_1}{1-\alpha_1}}}
=\frac{(k+1)\alpha_1}{k\alpha_1+1}=\delta_{k+1}.$$ This establishes .
\[KWghy5-f-09iii\]: This inequality was already obtained in [@Cegi12 Remark 2.2.38]. It follows from the fact that $$\widehat{\phi}(\alpha_1,\alpha_2)-
\phi(\alpha_1,\alpha_2)=\frac{\alpha_1\alpha_2
(1-\alpha_1)(1-\alpha_2)}{1-\alpha_1\alpha_2}>0.$$ The remaining assertions are easily verified.
Algorithms {#sec:3}
==========
We present applications of the bounds discussed in Section \[sec:2\] to fixed point algorithms. Henceforth, we denote the set of fixed points of an operator $T\colon{\ensuremath{{\mathcal H}}}\to{\ensuremath{{\mathcal H}}}$ by ${\ensuremath{\text{\rm Fix}\,}}T$.
As a direct application of Proposition \[p:av2\] and Proposition \[paQg507-03’\], we first consider so-called “string-averaging” iterations, which involve a mix of compositions and convex combinations of operators. In the case of projection operators, such iterations go back to [@Cens01].
\[p:nZZy418\] Let $(T_i)_{i\in I}$ be a finite family of nonexpansive operators from ${\ensuremath{{\mathcal H}}}$ to ${\ensuremath{{\mathcal H}}}$ such that $\bigcap_{i\in I}{\ensuremath{\text{\rm Fix}\,}}T_i\neq{\ensuremath{{\varnothing}}}$, and let $(\alpha_i)_{i\in I}$ be real numbers in $\left]0,1\right[$ such that, for every $i\in I$, $T_i$ is $\alpha_i$-averaged. Let $p$ be a strictly positive integer, for every $k\in\{1,\ldots,p\}$ let $m_k$ be a strictly positive integer and let $\omega_k\in\left]0,1\right]$, and suppose that ${\ensuremath{\mathrm i}}\colon{\big\{{(k,l)}~\big |~{k\in\{1,\ldots,p\},\,l\in
\{1,\ldots,m_k\}}\big\}}\to I$ is surjective and that $\sum_{k=1}^{p}\omega_k=1$. Define $$\label{e:nZZy418}
T=\sum_{k=1}^{p}\omega_kT_{{\ensuremath{\mathrm i}}(k,1)}\cdots T_{{\ensuremath{\mathrm i}}(k,m_k)}.$$ Then the following hold:
1. \[p:nZZy418i\] Set $$\label{e:nZZy418b}
\alpha=\sum_{k=1}^{p}\dfrac{\omega_k}{1+\dfrac{1}
{{\ensuremath{\displaystyle\sum}}_{i=1}^{m_k}\dfrac{\alpha_{{\ensuremath{\mathrm i}}(k,i)}}{1-\alpha_{{\ensuremath{\mathrm i}}(k,i)}}}}$$ Then $T$ is $\alpha$-averaged and ${\ensuremath{\text{\rm Fix}\,}}T=\bigcap_{i\in I}{\ensuremath{\text{\rm Fix}\,}}T_i$.
2. \[p:nZZy418ii\] Let $(\lambda_n)_{n\in {\ensuremath{\mathbb N}}}$ be a sequence in $\left]0,1/\alpha\right[$ such that $\sum_{n\in{\ensuremath{\mathbb N}}}\lambda_n(1/\alpha-\lambda_n)={\ensuremath{{+\infty}}}$. Furthermore, let $x_0\in {\ensuremath{{\mathcal H}}}$ and set $$(\forall n\in {\ensuremath{\mathbb N}})\quad
x_{n+1}=x_n+\lambda_n\big(Tx_n-x_n\big).$$ Then $(x_n)_{n\in {\ensuremath{\mathbb N}}}$ converges weakly to a point in $\bigcap_{i\in I}{\ensuremath{\text{\rm Fix}\,}}T_i$.
\[p:nZZy418i\]: The $\alpha$-averagedness of $T$ follows from Propositions \[p:av2\] and \[paQg507-03’\]. The remaining assertions follow from [@Livre1 Proposition 4.34 and Corollary 4.37].
\[p:nZZy418ii\]: This follows from \[p:nZZy418i\] and [@Livre1 Proposition 5.15(iii)].
Proposition \[p:nZZy418\] improves upon [@Livre1 Corollary 5.18], where the averagedness constant $\alpha$ of was replaced by $$\label{e:2009-03-21a}
\alpha'=\underset{1{\ensuremath{\leqslant}}k{\ensuremath{\leqslant}}p}{\text{\rm max}}\:
\rho_k,\quad\text{with}\quad
(\forall k\in\{1,\ldots,p\})\quad
\rho_k=\frac{m_k}{m_k-1+\displaystyle{\frac{1}{\text{\rm max}
\big\{\alpha_{{\ensuremath{\mathrm i}}(k,1)},\ldots,\alpha_{{\ensuremath{\mathrm i}}(k,m_k)}\big\}}}}.$$ In view of Remarks \[raQg507\] and \[KWghy5-f-09\]\[KWghy5-f-09i\], $\alpha'{\ensuremath{\geqslant}}\alpha$ and therefore $\alpha$ provides a larger range for the relaxation parameters $(\lambda_n)_{n\in{\ensuremath{\mathbb N}}}$.
The subsequent applications require the following technical fact.
[[@Poly87 Lemma 2.2.2]]{} \[l:7\] Let $(\alpha_n)_{n\in{\ensuremath{\mathbb N}}}$, $(\beta_n)_{n\in{\ensuremath{\mathbb N}}}$, and $(\varepsilon_n)_{n\in{\ensuremath{\mathbb N}}}$ be sequences in ${\ensuremath{\left[0,+\infty\right[}}$ such that $\sum_{n\in{\ensuremath{\mathbb N}}}\varepsilon_n<{\ensuremath{{+\infty}}}$ and $(\forall n\in{\ensuremath{\mathbb N}})$ $\alpha_{n+1}{\ensuremath{\leqslant}}\alpha_n-\beta_n
+\varepsilon_n$. Then $(\alpha_n)_{n\in{\ensuremath{\mathbb N}}}$ converges and $\sum_{n\in{\ensuremath{\mathbb N}}}\beta_n<{\ensuremath{{+\infty}}}$.
Next, we introduce a general iteration process for finding a common fixed point of a countable family of averaged operators which allows for approximate computations of the operator values.
\[paQg507-04a\] For every $n\in{\ensuremath{\mathbb N}}$, let $\alpha_n\in{\ensuremath{\left]0,1\right[}}$, let $\lambda_n\in\left]0,1/\alpha_n\right[$, let $e_n\in{\ensuremath{{\mathcal H}}}$, and let $T_n\colon{\ensuremath{{\mathcal H}}}\to{\ensuremath{{\mathcal H}}}$ be an $\alpha_n$-averaged operator. Suppose that $S=\bigcap_{n\in{\ensuremath{\mathbb N}}}{\ensuremath{\text{\rm Fix}\,}}T_n\neq{\ensuremath{{\varnothing}}}$ and that $\sum_{n\in{\ensuremath{\mathbb N}}}\lambda_n\|e_n\|<{\ensuremath{{+\infty}}}$. Let $x_0\in{\ensuremath{{\mathcal H}}}$ and set, for every $n\in{\ensuremath{\mathbb N}}$, $$\label{eaQg507-01a}
x_{n+1}=x_n+\lambda_n\big(T_nx_n+e_n-x_n\big).$$ Then the following hold:
1. \[paQg507-04a-i\] Let $n\in{\ensuremath{\mathbb N}}$, let $x\in S$, and set $\nu=\sum_{k\in{\ensuremath{\mathbb N}}}\lambda_k\|e_k\|+2\sup_{k\in{\ensuremath{\mathbb N}}}\|x_k-x\|$. Then $\nu<{\ensuremath{{+\infty}}}$ and $$\begin{aligned}
\|x_{n+1}-x\|^2
&{\ensuremath{\leqslant}}\|x_n+\lambda_n(T_nx_n-x_n)-x\|^2+\nu\lambda_n\|e_n\|
\label{fjskd43h8X4a}\\
&{\ensuremath{\leqslant}}\|x_n-x\|^2-\lambda_n(1/\alpha_n-\lambda_n)
\|T_nx_n-x_n\|^2+\nu\lambda_n\|e_n\|.
\label{eaQg507-04p}\end{aligned}$$
2. \[paQg507-04a-ii\] $\sum_{n\in{\ensuremath{\mathbb N}}}\lambda_n(1/\alpha_n-\lambda_n)
\|T_nx_n-x_n\|^2<{\ensuremath{{+\infty}}}$.
3. \[paQg507-04a-iii\] $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ converges weakly to a point in $S$ if and only if every weak sequential cluster point of $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ is in $S$. In this case, the convergence is strong if ${\ensuremath{\text{\rm int}\,}}S\neq{\ensuremath{{\varnothing}}}$.
4. \[paQg507-04a-iv\] $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ converges strongly to a point in $S$ if and only if $\varliminf d_S(x_n)=0$.
\[paQg507-04a-i\]: Set $$\label{eaQg507-04u}
R_n=(1-1/\alpha_n){\ensuremath{\mathrm{Id}}\,}+(1/\alpha_n)T_n
\quad\text{and}\quad\mu_n=\alpha_n\lambda_n.$$ Then ${\ensuremath{\text{\rm Fix}\,}}R_n={\ensuremath{\text{\rm Fix}\,}}T_n$ and, by Proposition \[p:av1\], $R_n$ is nonexpansive. Furthermore, can be written as $$\label{eaQg507-04a}
x_{n+1}=x_n+\mu_n\big(R_nx_n-x_n\big)
+\lambda_ne_n,\quad\text{where}\quad\mu_n\in{\ensuremath{\left]0,1\right[}}.$$ Now set $z_n=x_n+\mu_n(R_nx_n-x_n)$. Since $x\in{\ensuremath{\text{\rm Fix}\,}}R_n$ and $R_n$ is nonexpansive, we have $$\begin{aligned}
\label{eaQg507-04t}
\|z_n-x\|
&=\|(1-\mu_n)(x_n-x)+\mu_n(R_nx_n-R_nx)\|\nonumber\\
&{\ensuremath{\leqslant}}(1-\mu_n)\|x_n-x\|+\mu_n\|R_nx_n-R_nx\|\nonumber\\
&{\ensuremath{\leqslant}}\|x_n-x\|.\end{aligned}$$ Hence, yields $$\begin{aligned}
\|x_{n+1}-x\|
&{\ensuremath{\leqslant}}\|z_n-x\|+\lambda_n\|e_n\|
\label{eaQg507-04c}\\
&{\ensuremath{\leqslant}}\|x_n-x\|+\lambda_n\|e_n\|
\label{eaQg507-04b}\end{aligned}$$ and, since $\sum_{k\in{\ensuremath{\mathbb N}}}\lambda_k\|e_k\|<{\ensuremath{{+\infty}}}$, it follows from Lemma \[l:7\] that $$\label{e:mu}
\nu=\sum_{k\in{\ensuremath{\mathbb N}}}\lambda_k\|e_k\|+
2 \underset{k\in{\ensuremath{\mathbb N}}}{\rm\text{sup}\,}\|x_k-x\|<{\ensuremath{{+\infty}}}.$$ Moreover, using , , and [@Livre1 Corollary 2.14], we can write $$\begin{aligned}
\|x_{n+1}-x\|^2
&{\ensuremath{\leqslant}}\|z_n-x\|^2+(2\|z_n-x\|+\lambda_n\|e_n\|)
\lambda_n\|e_n\|\nonumber\\
&{\ensuremath{\leqslant}}\|z_n-x\|^2+(2\|x_n-x\|+\lambda_n\|e_n\|)
\lambda_n\|e_n\|\nonumber\\
&{\ensuremath{\leqslant}}\|(1-\mu_n)(x_n-x)+\mu_n(R_nx_n-x)\|^2+
\nu\lambda_n\|e_n\|\label{eaQg507-04s}\\
&=(1-\mu_n)\|x_n-x\|^2+\mu_n\|R_nx_n-x\|^2\nonumber\\
&\quad\;-\mu_n(1-\mu_n)\|R_nx_n-x_n\|^2+\nu\lambda_n\|e_n\|
\nonumber\\
&=(1-\mu_n)\|x_n-x\|^2+\mu_n\|R_nx_n-R_nx\|^2\nonumber\\
&\quad\;-\mu_n(1-\mu_n)\|R_nx_n-x_n\|^2+\nu\lambda_n\|e_n\|
\nonumber\\
&{\ensuremath{\leqslant}}\|x_n-x\|^2-\mu_n(1-\mu_n)\|R_nx_n-x_n\|^2+\nu\lambda_n\|e_n\|
\nonumber\\
&=\|x_n-x\|^2-\lambda_n(1/\alpha_n-\lambda_n)
\|T_nx_n-x_n\|^2+\nu\lambda_n\|e_n\|
\label{eaQg507-04i}\\
&{\ensuremath{\leqslant}}\|x_n-x\|^2+\nu\lambda_n\|e_n\|.
\label{eaQg507-04h}\end{aligned}$$ Thus, follows from and , and provides .
\[paQg507-04a-ii\]: This follows from , , and Lemma \[l:7\].
\[paQg507-04a-iii\]: The weak convergence statement follows from , , and [@Else01 Theorem 3.8], while the strong convergence statement follows from [@Else01 Proposition 3.10].
\[paQg507-04a-iv\]: By [@Livre1 Corollary 4.15], the sets $({\ensuremath{\text{\rm Fix}\,}}T_n)_{n\in{\ensuremath{\mathbb N}}}$ are closed, and so is therefore their intersection $S$. Hence, the result follows from , , \[paQg507-04a-ii\], and [@Else01 Theorem 3.11].
The main result of this section is the following.
\[cr7seGhn3243gd4\] Let $\varepsilon\in\left]0,1/2\right[$, let $m{\ensuremath{\geqslant}}2$ be an integer, let $x_0\in{\ensuremath{{\mathcal H}}}$, and define $\phi$ as in . For every $i\in\{1,\ldots,m\}$ and every $n\in{\ensuremath{\mathbb N}}$, let $\alpha_{i,n}\in{\ensuremath{\left]0,1\right[}}$, let $T_{i,n}\colon{\ensuremath{{\mathcal H}}}\to{\ensuremath{{\mathcal H}}}$ be $\alpha_{i,n}$-averaged, and let $e_{i,n}\in{\ensuremath{{\mathcal H}}}$. For every $n\in{\ensuremath{\mathbb N}}$, let $\lambda_n\in\left]0,(1-\varepsilon)(1+\varepsilon
\phi(\alpha_{1,n},\ldots,\alpha_{m,n}))/
\phi(\alpha_{1,n},\ldots,\alpha_{m,n})\right]$ and set $$\label{e:main}
x_{n+1}=x_n+\lambda_n\bigg(T_{1,n}\bigg(T_{2,n}
\big(\cdots T_{m-1,n}(T_{m,n}x_n+e_{m,n})
+e_{m-1,n}\cdots\big)+e_{2,n}\bigg)
+e_{1,n}-x_n\bigg).$$ Suppose that $$\label{eaQg507-04y}
S=\bigcap_{n\in{\ensuremath{\mathbb N}}}{\ensuremath{\text{\rm Fix}\,}}(T_{1,n}\cdots T_{m,n})\neq{\ensuremath{{\varnothing}}}\quad{and}\quad
(\forall i\in\{1,\ldots,m\})\quad
\sum_{n\in{\ensuremath{\mathbb N}}}\lambda_n\|e_{i,n}\|<{\ensuremath{{+\infty}}},$$ and define $$\label{eaQg507-04o}
(\forall i\in\{1,\ldots,m\})(\forall n\in{\ensuremath{\mathbb N}})\quad T_{i+,n}=
\begin{cases}
T_{i+1,n}\cdots T_{m,n},&\text{if}\;\;i\neq m;\\
{\ensuremath{\mathrm{Id}}\,},&\text{if}\;\;i=m.
\end{cases}$$ Then the following hold:
1. \[cr7seGhn3243gd4i\] $\sum_{n\in{\ensuremath{\mathbb N}}}\lambda_n(1/\phi(\alpha_{1,n},\ldots,\alpha_{m,n})
-\lambda_n)\|T_{1,n}\cdots T_{m,n}x_n-x_n\|^2<{\ensuremath{{+\infty}}}$.
2. \[cr7seGhn3243gd4ii\] $(\forall x\in S)$ $\underset{1{\ensuremath{\leqslant}}i{\ensuremath{\leqslant}}m}{\text{\rm max}}{\ensuremath{\displaystyle\sum}}_{n\in{\ensuremath{\mathbb N}}}
{\displaystyle{\frac{\lambda_n(1-\alpha_{i,n})}{\alpha_{i,n}}}}
\left\|({\ensuremath{\mathrm{Id}}\,}-T_{i,n})T_{i+,n}x_{n}-({\ensuremath{\mathrm{Id}}\,}-T_{i,n})T_{i+,n}x
\right\|^2<{\ensuremath{{+\infty}}}$.
3. \[cr7seGhn3243gd4iii\] $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ converges weakly to a point in $S$ if and only if every weak sequential cluster point of $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ is in $S$. In this case, the convergence is strong if ${\ensuremath{\text{\rm int}\,}}S\neq{\ensuremath{{\varnothing}}}$.
4. \[cr7seGhn3243gd4iv\] $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ converges strongly to a point in $S$ if and only if $\varliminf d_S(x_n)=0$.
Let $n\in{\ensuremath{\mathbb N}}$ and let $x\in S$. We can rewrite as an instance of , namely $$\label{e:zn}
x_{n+1}=x_n+\lambda_n\big(T_nx_n+e_n-x_n\big),$$ where $$T_n=T_{1,n}\cdots T_{m,n}$$ and $$\label{e:5thonf}
e_n=T_{1,n}\bigg(T_{2,n}\big(\cdots
T_{m-1,n}(T_{m,n}x_n+e_{m,n})+e_{m-1,n}\cdots\big)+e_{2,n}\bigg)
+e_{1,n}-T_{1,n}\cdots T_{m,n}x_n.$$ It follows from Proposition \[paQg507-03’\] that $$\label{eaQg507-04x}
T_n\;\text{is $\alpha_n$-averaged, where}\;
\alpha_n=\phi(\alpha_{1,n},\ldots,\alpha_{m,n}).$$ Since $\alpha_n\in{\ensuremath{\left]0,1\right[}}$, $$\label{e:w}
\frac{(1-\varepsilon)(1+\varepsilon\alpha_n)}{\alpha_n}
<\frac{(1-\varepsilon)(1+\varepsilon)}{\alpha_n}
=\frac{1-\varepsilon^2}{\alpha_n}
<\frac{1}{\alpha_n}$$ and therefore $\lambda_n\in\left]0,1/\alpha_n\right[$, as required in Proposition \[paQg507-04a\].
\[cr7seGhn3243gd4i\]: Using the nonexpansiveness of the operators $(T_{i,n})_{1{\ensuremath{\leqslant}}i{\ensuremath{\leqslant}}m}$, we derive from that $$\begin{aligned}
\label{e:qf8}
\|e_n\|
&{\ensuremath{\leqslant}}\|e_{1,n}\|+\nonumber\\[2mm]
&\quad\;\bigg\|T_{1,n}\bigg(T_{2,n}\big(\cdots T_{m-1,n}(T_{m,n}x_n
+e_{m,n})+e_{m-1,n}\cdots\big)+e_{2,n}\bigg)-T_{1,n}\cdots
T_{m,n}x_n \bigg\|\nonumber\\[2mm]
&{\ensuremath{\leqslant}}\|e_{1,n}\|+\nonumber\\[2mm]
&\quad\;\bigg\|T_{2,n}\bigg(T_{3,n}\big(\cdots T_{m-1,n}
(T_{m,n}x_n+e_{m,n})+e_{m-1,n}\cdots\big)+e_{3,n}\bigg)+e_{2,n}-
T_{2,n}\cdots T_{m,n}x_n\bigg\|\nonumber\\[2mm]
&{\ensuremath{\leqslant}}\|e_{1,n}\|+\|e_{2,n}\|+\nonumber\\[2mm]
&\quad\;\bigg\|T_{3,n}\bigg(T_{4,n}\big(\cdots T_{m-1,n}
(T_{m,n}x_n+e_{m,n})+e_{m-1,n}\cdots\big)+e_{4,n}\bigg)+e_{3,n}-
T_{3,n}\cdots T_{m,n}x_n\bigg\|\nonumber\\[2mm]
&\;\;\vdots\nonumber\\
&{\ensuremath{\leqslant}}\sum_{i=1}^m\|e_{i,n}\|.\end{aligned}$$ Accordingly, yields $$\label{e:efwrf7}
\sum_{k\in{\ensuremath{\mathbb N}}}\lambda_k\|e_k\|<{\ensuremath{{+\infty}}}.$$ Hence, we deduce from Proposition \[paQg507-04a\]\[paQg507-04a-i\] that $$\label{e:mu'}
\nu=\sum_{k\in{\ensuremath{\mathbb N}}}\lambda_k\|e_k\|+
2\underset{k\in{\ensuremath{\mathbb N}}}{\rm\text{sup}\,}\|x_k-x\|<{\ensuremath{{+\infty}}}$$ and from Proposition \[paQg507-04a\]\[paQg507-04a-ii\] that $$\label{e:2011-11-23}
\sum_{k\in{\ensuremath{\mathbb N}}}\lambda_k\Big(\frac{1}{\alpha_k}-\lambda_k\Big)
\|T_kx_k-x_k\|^2<{\ensuremath{{+\infty}}}.$$
\[cr7seGhn3243gd4ii\]: We derive from Proposition \[p:av1\] that $$\begin{aligned}
\label{eaQg507-05a}
(\forall i\in\{1,\ldots,m\})(\forall (u,v)\in{\ensuremath{{\mathcal H}}}^2)\nonumber\\
\|T_{i,n}u-T_{i,n}v\|^2
&{\ensuremath{\leqslant}}\|u-v\|^2-\frac{1-\alpha_{i,n}}{\alpha_{i,n}}
\|({\ensuremath{\mathrm{Id}}\,}-T_{i,n})u-({\ensuremath{\mathrm{Id}}\,}-T_{i,n})v\|^2.\end{aligned}$$ Using this inequality $m$ times leads to $$\begin{aligned}
\label{eaQg507-04q}
\|T_nx_n-x\|^2
&=\left\|T_{1,n}\cdots T_{m,n}x_n-T_{1,n}\cdots T_{m,n}x\right\|^2
\nonumber\\
&{\ensuremath{\leqslant}}\|x_n-x\|^2-\sum_{i=1}^{m}\frac{1-\alpha_{i,n}}{\alpha_{i,n}}
\left\|({\ensuremath{\mathrm{Id}}\,}-T_{i,n})T_{i+,n}x_n-({\ensuremath{\mathrm{Id}}\,}-T_{i,n})T_{i+,n}x\right\|^2
\nonumber\\
&{\ensuremath{\leqslant}}\|x_n-x\|^2-\frac{\beta_n}{\lambda_n},\end{aligned}$$ where $$\label{Kiuxv532}
\beta_n=\lambda_n
\underset{1{\ensuremath{\leqslant}}i{\ensuremath{\leqslant}}m}{\text{\rm max}}
\bigg(\frac{1-\alpha_{i,n}}{\alpha_{i,n}}
\left\|({\ensuremath{\mathrm{Id}}\,}-T_{i,n})T_{i+,n}x_n-({\ensuremath{\mathrm{Id}}\,}-T_{i,n})T_{i+,n}x\right\|^2
\bigg).$$ Note also that $$\begin{aligned}
\label{fjskd43h8X5a}
\lambda_n{\ensuremath{\leqslant}}\frac{(1-\varepsilon)(1+\varepsilon\alpha_n)}{\alpha_n}
&\Rightarrow&
\lambda_n{\ensuremath{\leqslant}}\frac{1+\varepsilon\alpha_n}
{(1+\varepsilon)\alpha_n}\nonumber\\
&\Leftrightarrow&
\bigg(1+\frac{1}{\varepsilon}\bigg)\lambda_n{\ensuremath{\leqslant}}\frac{1}{\varepsilon\alpha_n}+1\nonumber\\
&\Leftrightarrow&
\lambda_n-1{\ensuremath{\leqslant}}\frac{1}{\varepsilon}
\bigg(\frac{1}{\alpha_n}-\lambda_n\bigg).\end{aligned}$$ Thus, Proposition \[paQg507-04a\]\[paQg507-04a-i\], , and [@Livre1 Corollary 2.14] yield $$\begin{aligned}
\label{eaQg507-04r}
\|x_{n+1}-x\|^2
&{\ensuremath{\leqslant}}\|(1-\lambda_n)(x_n-x)+\lambda_n(T_nx_n-x)\|^2+
\nu\lambda_n\|e_n\|\nonumber\\
&=(1-\lambda_n)\|x_n-x\|^2+\lambda_n\|T_nx_n-x\|^2
+\lambda_n(\lambda_n-1)\|T_nx_n-x_n\|^2+\nu\lambda_n\|e_n\|
\nonumber\\
&{\ensuremath{\leqslant}}(1-\lambda_n)\|x_n-x\|^2+\lambda_n\|T_nx_n-x\|^2+\varepsilon_n,\end{aligned}$$ where $$\label{eaQg507-03i}
\varepsilon_n=\frac{\lambda_n}{\varepsilon}
\bigg(\frac{1}{\alpha_n}-\lambda_n\bigg)
\|T_nx_n-x_n\|^2+\nu\lambda_n\|e_n\|.$$ On the one hand, it follows from , , and that $$\label{eaQg507-05c}
\sum_{k\in{\ensuremath{\mathbb N}}}\varepsilon_k<{\ensuremath{{+\infty}}}.$$ On the other hand, combining and , we obtain $$\label{eaQg507-05b}
\|x_{n+1}-x\|^2{\ensuremath{\leqslant}}\|x_n-x\|^2-\beta_n+\varepsilon_n.$$ Consequently, Lemma \[l:7\] implies that $\sum_{k\in{\ensuremath{\mathbb N}}}\beta_k<{\ensuremath{{+\infty}}}$.
\[cr7seGhn3243gd4iii\]–\[cr7seGhn3243gd4iv\]: These follow from their counterparts in Proposition \[paQg507-04a\].
Theorem \[cr7seGhn3243gd4\] extends the results of [@Opti04 Section 3], where the relaxations parameters $(\lambda_n)_{n\in{\ensuremath{\mathbb N}}}$ cannot exceed 1. Since these parameters control the step-lengths of the algorithm, the proposed extension can result in significant accelerations.
Application to forward-backward splitting {#sec:4}
=========================================
The forward-backward algorithm is one of the most versatile and powerful algorithm for finding a zero of the sum of two maximally monotone operators (see [@Svva10; @Opti14] and the references therein for historical background and recent developments). In [@Opti04], the first author showed that the theory of averaged nonexpansive operators provided a convenient setting for analyzing this algorithm. In this section, we exploit the results of Sections \[sec:2\] and \[sec:3\] to further extend this analysis and obtain a new version of the forward-backward algorithm with an extended relaxation range.
Let us recall a few facts about monotone set-valued operators and convex analysis [@Livre1]. Let $A\colon{\ensuremath{{\mathcal H}}}\to 2^{{\ensuremath{{\mathcal H}}}}$ be a set-valued operator. The domain, the graph, and the set of zeros of $A$ are respectively defined by ${\ensuremath{\text{\rm dom}\,}}A={\big\{{x\in{\ensuremath{{\mathcal H}}}}~\big |~{Ax\neq{\ensuremath{{\varnothing}}}}\big\}}$, ${\ensuremath{\text{\rm gra}}}A={\big\{{(x,u)\in{\ensuremath{{\mathcal H}}}\times{\ensuremath{{\mathcal H}}}}~\big |~{u\in Ax}\big\}}$, and ${\ensuremath{\text{\rm zer}\,}}A={\big\{{x\in{\ensuremath{{\mathcal H}}}}~\big |~{0\in Ax}\big\}}$. The inverse of $A$ is $A^{-1}\colon{\ensuremath{{\mathcal H}}}\mapsto 2^{{\ensuremath{{\mathcal H}}}}\colon u\mapsto
{\big\{{x\in{\ensuremath{{\mathcal H}}}}~\big |~{u\in Ax}\big\}}$, and the resolvent of $A$ is $$\label{e:resolvent}
J_A=({\ensuremath{\mathrm{Id}}\,}+A)^{-1}.$$ This operator is firmly nonexpansive if $A$ is monotone, i.e., $$(\forall(x,y)\in{\ensuremath{{\mathcal H}}}\times{\ensuremath{{\mathcal H}}})
(\forall(u,v)\in Ax\times Ay)\quad{{\left\langle{{x-y}\mid{u-v}}\right\rangle}}{\ensuremath{\geqslant}}0,$$ and ${\ensuremath{\text{\rm dom}\,}}J_A={\ensuremath{{\mathcal H}}}$ if, furthermore, $A$ is maximally monotone, i.e., there exists no monotone operator $B\colon{\ensuremath{{\mathcal H}}}\to2^{\ensuremath{{\mathcal H}}}$ such that ${\ensuremath{\text{\rm gra}}}A\subset{\ensuremath{\text{\rm gra}}}B$ and $A\neq B$. We denote by $\Gamma_0({\ensuremath{{\mathcal H}}})$ the class of proper lower semicontinuous convex functions $f\colon{\ensuremath{{\mathcal H}}}\to{\ensuremath{\left]-\infty,+\infty\right]}}$. Let $f\in\Gamma_0({\ensuremath{{\mathcal H}}})$. For every $x\in{\ensuremath{{\mathcal H}}}$, $f+\|x-\cdot\|^2/2$ possesses a unique minimizer, which is denoted by ${\ensuremath{\text{\rm prox}}}_fx$. We have $$\label{e:prox2}
{\ensuremath{\text{\rm prox}}}_f=J_{\partial f},\quad\text{where}\quad
\partial f\colon{\ensuremath{{\mathcal H}}}\to 2^{{\ensuremath{{\mathcal H}}}}\colon x\mapsto
{\big\{{u\in{\ensuremath{{\mathcal H}}}}~\big |~{(\forall y\in{\ensuremath{{\mathcal H}}})\;\:{{\left\langle{{y-x}\mid{u}}\right\rangle}}+f(x){\ensuremath{\leqslant}}f(y)}\big\}}$$ is the subdifferential of $f$.
We start with a specialization of Theorem \[cr7seGhn3243gd4\] to $m=2$.
\[koGhn843gd4\] Let $\varepsilon\in\left]0,1/2\right[$ and let $x_0\in{\ensuremath{{\mathcal H}}}$. For every every $n\in{\ensuremath{\mathbb N}}$, let $\alpha_{1,n}\in\left]0,1/(1+\varepsilon)\right]$, let $\alpha_{2,n}\in\left]0,1/(1+\varepsilon)\right]$, let $T_{1,n}\colon{\ensuremath{{\mathcal H}}}\to{\ensuremath{{\mathcal H}}}$ be $\alpha_{1,n}$-averaged, let $T_{2,n}\colon{\ensuremath{{\mathcal H}}}\to{\ensuremath{{\mathcal H}}}$ be $\alpha_{2,n}$-averaged, let $e_{1,n}\in{\ensuremath{{\mathcal H}}}$, and let $e_{2,n}\in{\ensuremath{{\mathcal H}}}$. In addition, for every every $n\in{\ensuremath{\mathbb N}}$, let $$\label{ewfw7reww23a}
\lambda_n\in\left[\varepsilon,
\frac{(1-\varepsilon)(1+\varepsilon\phi_n)}{\phi_n}\right],
\quad\text{where}\quad
\phi_n=\frac{\alpha_{1,n}+\alpha_{2,n}-2\alpha_{1,n}\alpha_{2,n}}
{1-\alpha_{1,n}\alpha_{2,n}},$$ and set $$\label{ooio9o0vd59}
x_{n+1}=x_n+\lambda_n\Big(T_{1,n}\big(T_{2,n}x_n+
e_{2,n}\big)+e_{1,n}-x_n\Big).$$ Suppose that $$\label{ooio9o0vd51}
S=\bigcap_{n\in{\ensuremath{\mathbb N}}}{\ensuremath{\text{\rm Fix}\,}}(T_{1,n} T_{2,n})\neq{\ensuremath{{\varnothing}}},\quad
{\ensuremath{\displaystyle\sum}}_{n\in{\ensuremath{\mathbb N}}}\lambda_n\|e_{1,n}\|<{\ensuremath{{+\infty}}},
\quad\text{and}\quad
{\ensuremath{\displaystyle\sum}}_{n\in{\ensuremath{\mathbb N}}}\lambda_n\|e_{2,n}\|<{\ensuremath{{+\infty}}}.$$ Then the following hold:
1. \[koGhn843gd4ii\] $(\forall x\in S)$ $\sum_{n\in{\ensuremath{\mathbb N}}}
\|T_{1,n}T_{2,n}x_n-T_{2,n}x_n+T_{2,n}x-x\|^2<{\ensuremath{{+\infty}}}$.
2. \[koGhn843gd4ii+\] $(\forall x\in S)$ $\sum_{n\in{\ensuremath{\mathbb N}}}\|T_{2,n}x_n-x_n-T_{2,n}x+x\|^2<{\ensuremath{{+\infty}}}$.
3. \[koGhn843gd4ii++\] $\sum_{n\in{\ensuremath{\mathbb N}}}\|T_{1,n}T_{2,n}x_n-x_n\|^2<{\ensuremath{{+\infty}}}$.
4. \[koGhn843gd4iii\] Suppose that every weak sequential cluster point of $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ is in $S$. Then $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ converges weakly to a point in $S$, and the convergence is strong if ${\ensuremath{\text{\rm int}\,}}S\neq{\ensuremath{{\varnothing}}}$.
5. \[koGhn843gd4iv\] Suppose that $\varliminf d_S(x_n)=0$. Then $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ converges strongly to a point in $S$.
For every $n\in{\ensuremath{\mathbb N}}$, since $\phi_n\in{\ensuremath{\left]0,1\right[}}$, $\varepsilon<1-\varepsilon<(1-\varepsilon)(1/\phi_n+\varepsilon)$ and $\lambda_n$ is therefore well defined in . Overall, the present setting is encompassed by that of Theorem \[cr7seGhn3243gd4\] with $m=2$.
\[koGhn843gd4ii\]–\[koGhn843gd4ii+\]: Let $x\in S$. We derive from Theorem \[cr7seGhn3243gd4\]\[cr7seGhn3243gd4ii\] with $m=2$ that $$\label{ewfw7reww23c}
\begin{cases}
{\ensuremath{\displaystyle\sum}}_{n\in{\ensuremath{\mathbb N}}}
{\displaystyle{\frac{\lambda_n(1-\alpha_{1,n})}{\alpha_{1,n}}}}
\left\|({\ensuremath{\mathrm{Id}}\,}-T_{1,n})T_{2,n}x_n-({\ensuremath{\mathrm{Id}}\,}-T_{1,n})T_{2,n}x
\right\|^2<{\ensuremath{{+\infty}}}\\
{\ensuremath{\displaystyle\sum}}_{n\in{\ensuremath{\mathbb N}}}{\displaystyle{\frac{\lambda_n(1-\alpha_{2,n})}{\alpha_{2,n}}}}
\left\|({\ensuremath{\mathrm{Id}}\,}-T_{2,n})x_n-({\ensuremath{\mathrm{Id}}\,}-T_{2,n})x
\right\|^2<{\ensuremath{{+\infty}}}.
\end{cases}$$ However, it follows from the assumptions that $$\label{ewfw7reww23b}
(\forall n\in{\ensuremath{\mathbb N}})\quad T_{1,n}T_{2,n}x=x,\quad
{\displaystyle{\frac{\lambda_n(1-\alpha_{1,n})}{\alpha_{1,n}}}}{\ensuremath{\geqslant}}\varepsilon^2,
\quad\text{and}\quad
{\displaystyle{\frac{\lambda_n(1-\alpha_{2,n})}{\alpha_{2,n}}}}{\ensuremath{\geqslant}}\varepsilon^2.$$ Combining and yields the claims.
\[koGhn843gd4ii++\]: Let $x\in S$. Then, for every $n\in{\ensuremath{\mathbb N}}$, $$\begin{aligned}
\label{ewfw7reww23d}
\|T_{1,n}T_{2,n}x_n-x_n\|^2
&=\|(T_{1,n}T_{2,n}x_n-T_{2,n}x_n+T_{2,n}x-x)
+(T_{2,n}x_n-x_n-T_{2,n}x+x)\|^2\nonumber\\
&{\ensuremath{\leqslant}}2\|T_{1,n}T_{2,n}x_n-T_{2,n}x_n+T_{2,n}x-x\|^2
+2\|T_{2,n}x_n-x_n-T_{2,n}x+x\|^2.\end{aligned}$$ Hence the claim follows from \[koGhn843gd4ii\]–\[koGhn843gd4ii+\].
\[koGhn843gd4iii\]–\[koGhn843gd4iv\]: These follow from Theorem \[cr7seGhn3243gd4\]\[cr7seGhn3243gd4iii\]–\[cr7seGhn3243gd4iv\].
[[@Sico10 Definition 2.3]]{} \[d:demir\] An operator $A\colon{\ensuremath{{\mathcal H}}}\to 2^{{\ensuremath{{\mathcal H}}}}$ is *demiregular* at $x\in{\ensuremath{\text{\rm dom}\,}}A$ if, for every sequence $((x_n,u_n))_{n\in{\ensuremath{\mathbb N}}}$ in ${\ensuremath{\text{\rm gra}}}A$ and every $u\in Ax$ such that $x_n{\ensuremath{\:\rightharpoonup\:}}x$ and $u_n\to u$, we have $x_n\to x$.
Here are some examples of demiregular monotone operators.
[[@Sico10 Proposition 2.4]]{} \[l:2009-09-20\] Let $A\colon{\ensuremath{{\mathcal H}}}\to 2^{{\ensuremath{{\mathcal H}}}}$ be monotone and suppose that $x\in{\ensuremath{\text{\rm dom}\,}}A$. Then $A$ is demiregular at $x$ in each of the following cases:
1. \[l:2009-09-20i\] $A$ is uniformly monotone at $x$, i.e., there exists an increasing function $\theta\colon{\ensuremath{\left[0,+\infty\right[}}\to{\ensuremath{\left[0,+\infty\right]}}$ that vanishes only at $0$ such that $(\forall u\in Ax)(\forall (y,v)\in{\ensuremath{\text{\rm gra}}}A)$ ${{\left\langle{{x-y}\mid{u-v}}\right\rangle}}{\ensuremath{\geqslant}}\theta(\|x-y\|)$.
2. \[l:2009-09-20ii\] $A$ is strongly monotone, i.e., there exists $\alpha\in{\ensuremath{\left]0,+\infty\right[}}$ such that $A-\alpha{\ensuremath{\mathrm{Id}}\,}$ is monotone.
3. \[l:2009-09-20iv-\] $J_A$ is compact, i.e., for every bounded set $C\subset{\ensuremath{{\mathcal H}}}$, the closure of $J_A(C)$ is compact. In particular, ${\ensuremath{\text{\rm dom}\,}}A$ is boundedly relatively compact, i.e., the intersection of its closure with every closed ball is compact.
4. \[l:2009-09-20vi\] $A\colon{\ensuremath{{\mathcal H}}}\to{\ensuremath{{\mathcal H}}}$ is single-valued with a single-valued continuous inverse.
5. \[l:2009-09-20vii\] $A$ is single-valued on ${\ensuremath{\text{\rm dom}\,}}A$ and ${\ensuremath{\mathrm{Id}}\,}-A$ is demicompact, i.e., for every bounded sequence $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ in ${\ensuremath{\text{\rm dom}\,}}A$ such that $(Ax_n)_{n\in{\ensuremath{\mathbb N}}}$ converges strongly, $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ admits a strong cluster point.
6. \[p:2009-09-20ii+\] $A=\partial f$, where $f\in\Gamma_0({\ensuremath{{\mathcal H}}})$ is uniformly convex at $x$, i.e., there exists an increasing function $\theta\colon{\ensuremath{\left[0,+\infty\right[}}\to{\ensuremath{\left[0,+\infty\right]}}$ that vanishes only at $0$ such that $$(\forall\alpha\in{\ensuremath{\left]0,1\right[}})(\forall y\in{\ensuremath{\text{\rm dom}\,}}f)\quad
f\big(\alpha x+(1-\alpha) y\big)+\alpha(1-\alpha)\theta(\|x-y\|)
{\ensuremath{\leqslant}}\alpha f(x)+(1-\alpha)f(y).$$
7. \[p:2009-09-20ii++++\] $A=\partial f$, where $f\in\Gamma_0({\ensuremath{{\mathcal H}}})$ and, for every $\xi\in{\ensuremath{\mathbb{R}}}$, ${\big\{{x\in{\ensuremath{{\mathcal H}}}}~\big |~{f(x){\ensuremath{\leqslant}}\xi}\big\}}$ is boundedly compact.
Our extended forward-backward splitting scheme can now be presented.
\[lI8uhT612\] Let $\beta\in{\ensuremath{\left]0,+\infty\right[}}$, let $\varepsilon\in\left]0,\min\{1/2,\beta\}\right[$, let $x_0\in{\ensuremath{{\mathcal H}}}$, let $A\colon{\ensuremath{{\mathcal H}}}\to 2^{{\ensuremath{{\mathcal H}}}}$ be maximally monotone, and let $B\colon{\ensuremath{{\mathcal H}}}\to{\ensuremath{{\mathcal H}}}$ be $\beta$-cocoercive, i.e., $$\label{e:cocoercive}
(\forall x\in{\ensuremath{{\mathcal H}}})(\forall y\in{\ensuremath{{\mathcal H}}})\quad
{{\left\langle{{x-y}\mid{Bx-By}}\right\rangle}}{\ensuremath{\geqslant}}\beta\|Bx-By\|^2.$$ Furthermore, let $(\gamma_n)_{n\in{\ensuremath{\mathbb N}}}$ be a sequence in $\left[\varepsilon,2\beta/(1+\varepsilon)\right]$, and let $(a_n)_{n\in{\ensuremath{\mathbb N}}}$ and $(b_n)_{n\in{\ensuremath{\mathbb N}}}$ be sequences in ${\ensuremath{{\mathcal H}}}$ such that $\sum_{n\in{\ensuremath{\mathbb N}}}\|a_n\|<{\ensuremath{{+\infty}}}$ and $\sum_{n\in{\ensuremath{\mathbb N}}}\|b_n\|<{\ensuremath{{+\infty}}}$. Suppose that ${\ensuremath{\text{\rm zer}\,}}(A+B)\neq{\ensuremath{{\varnothing}}}$ and, for every $n\in{\ensuremath{\mathbb N}}$, let $$\label{ewfw7reww25}
\lambda_n\in\left[\varepsilon,(1-\varepsilon)
\bigg(2+\varepsilon-{\displaystyle{\frac{\gamma_n}{2\beta}}}\bigg)\right]$$ and set $$\label{e:23juillet2007-1}
x_{n+1}=x_n+\lambda_n\Big(J_{\gamma_n A}
\big(x_n-\gamma_n(Bx_n+b_n)\big)+a_n-x_n\Big).$$ Then the following hold:
1. \[lI8uhT612i\] $\sum_{n\in{\ensuremath{\mathbb N}}}\|J_{\gamma_n A}(x_n-\gamma_nBx_n)-x_n\|^2<{\ensuremath{{+\infty}}}$.
2. \[lI8uhT612ii\] Let $x\in{\ensuremath{\text{\rm zer}\,}}(A+B)$. Then $\sum_{n\in{\ensuremath{\mathbb N}}}\|Bx_n-Bx\|^2<{\ensuremath{{+\infty}}}$.
3. \[lI8uhT612iii\] $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ converges weakly to a point in ${\ensuremath{\text{\rm zer}\,}}(A+B)$.
4. \[lI8uhT612iv\] Suppose that one of the following is satisfied:
1. \[lI8uhT612iva\] $A$ is demiregular at every point in ${\ensuremath{\text{\rm zer}\,}}(A+B)$.
2. \[lI8uhT612ivb\] $B$ is demiregular at every point in ${\ensuremath{\text{\rm zer}\,}}(A+B)$.
3. \[lI8uhT612ivc\] ${\ensuremath{\text{\rm int}\,}}S\neq{\ensuremath{{\varnothing}}}$.
Then $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ converges strongly to a point in ${\ensuremath{\text{\rm zer}\,}}(A+B)$.
We are going to establish the results as an application of Corollary \[koGhn843gd4\]. Set $$(\forall n\in{\ensuremath{\mathbb N}})\;\;
\label{ooio9o0vd54}
T_{1,n}=J_{\gamma_n A},\quad T_{2,n}={\ensuremath{\mathrm{Id}}\,}-\gamma_n B,\quad
e_{1,n}=a_n,\quad\text{and}\quad e_{2,n}=-\gamma_nb_n.$$ Then, for every $n\in{\ensuremath{\mathbb N}}$, $T_{1,n}$ is $\alpha_{1,n}$-averaged with $\alpha_{1,n}=1/2$ [@Livre1 Remark 4.24(iii) and Corollary 23.8] and $T_{2,n}$ is $\alpha_{2,n}$-averaged with $\alpha_{2,n}=\gamma_n/(2\beta)$ [@Livre1 Proposition 4.33]. Moreover, for every $n\in{\ensuremath{\mathbb N}}$, $$\label{ewfw7reww22s}
\phi_n=\frac{\alpha_{1,n}+\alpha_{2,n}-2\alpha_{1,n}\alpha_{2,n}}
{1-\alpha_{1,n}\alpha_{2,n}}=\frac{2\beta}{4\beta-\gamma_n}$$ and, therefore, $$\label{ewfw7reww22t}
\lambda_n\in\left[\varepsilon,(1-\varepsilon)
(1+\varepsilon\phi_n)/\phi_n\right],$$ in conformity with . In turn, Proposition \[paQg507-03”\]\[paQg507-03”i\] yields $$\label{ewfw7reww24z}
(\forall n\in{\ensuremath{\mathbb N}})\quad\lambda_n{\ensuremath{\leqslant}}\frac{1}{\phi_n}+\varepsilon
{\ensuremath{\leqslant}}\frac{1}{\alpha_{1,n}}+\varepsilon=2+\varepsilon.$$ Consequently, $$\label{ewfw7reww24t}
\sum_{n\in{\ensuremath{\mathbb N}}}\lambda_n\|e_{1,n}\|=(2+\varepsilon)
\sum_{n\in{\ensuremath{\mathbb N}}}\|a_n\|<{\ensuremath{{+\infty}}}\quad\text{and}\quad
\sum_{n\in{\ensuremath{\mathbb N}}}\lambda_n\|e_{2,n}\|{\ensuremath{\leqslant}}2(2+\varepsilon)\beta\sum_{n\in{\ensuremath{\mathbb N}}}\|b_n\|<{\ensuremath{{+\infty}}}.$$ On the other hand, [@Livre1 Proposition 25.1(iv)] yields $$\label{e:fz1}
(\forall n\in{\ensuremath{\mathbb N}})\quad{\ensuremath{\text{\rm zer}\,}}(A+B)={\ensuremath{\text{\rm Fix}\,}}(T_{1,n} T_{2,n}).$$ Altogether, $S={\ensuremath{\text{\rm zer}\,}}(A+B)\neq{\ensuremath{{\varnothing}}}$, is satisfied, and is an instance of .
\[lI8uhT612i\]: This is a consequence of Corollary \[koGhn843gd4\]\[koGhn843gd4ii++\] and .
\[lI8uhT612ii\]: Corollary \[koGhn843gd4\]\[koGhn843gd4ii+\] and yield $$\begin{aligned}
\sum_{n\in{\ensuremath{\mathbb N}}}\|Bx_n-Bx\|^2
&=\sum_{n\in{\ensuremath{\mathbb N}}}\gamma_n^{-2}\|T_{2,n}x_n-x_n-T_{2,n}x+x\|^2
\nonumber\\
&{\ensuremath{\leqslant}}\varepsilon^{-2}\sum_{n\in{\ensuremath{\mathbb N}}}\|T_{2,n}x_n-x_n-T_{2,n}x+x\|^2
\nonumber\\
&<{\ensuremath{{+\infty}}}.\end{aligned}$$
\[lI8uhT612iii\]: Let $(k_n)_{n\in{\ensuremath{\mathbb N}}}$ be a strictly increasing sequence in ${\ensuremath{\mathbb N}}$ and let $y\in{\ensuremath{{\mathcal H}}}$ be such that $x_{k_n}{\ensuremath{\:\rightharpoonup\:}}y$. In view of Corollary \[koGhn843gd4\]\[koGhn843gd4iii\], it remains to show that $y\in{\ensuremath{\text{\rm zer}\,}}(A+B)$. We set $$\label{e:burn}
(\forall n\in{\ensuremath{\mathbb N}})\quad y_n=J_{\gamma_n A}(x_n-\gamma_nBx_n)\quad
\text{and}\quad u_n=\frac{x_n-y_n}{\gamma_n}-Bx_n,$$ and note that $$\label{e:jhUy74d}
(\forall n\in{\ensuremath{\mathbb N}})\quad u_n\in Ay_n.$$ We derive from \[lI8uhT612i\] that $y_n-x_n\to 0$, hence $y_{k_n}{\ensuremath{\:\rightharpoonup\:}}y$. Now let $x\in{\ensuremath{\text{\rm zer}\,}}(A+B)$. Then \[lI8uhT612ii\] implies that $Bx_{n}\to Bx$, hence $u_n\to -Bx$. However, since implies that $B$ is maximally monotone [@Livre1 Example 20.28], it follows from the properties $x_{k_n}{\ensuremath{\:\rightharpoonup\:}}y$ and $Bx_{k_n}\to Bx$ that $By=Bx$ [@Livre1 Proposition 20.33(ii)]. Thus, $y_{k_n}{\ensuremath{\:\rightharpoonup\:}}y$ and $u_{k_n}\to -By$, and it therefore follows from and [@Livre1 Proposition 20.33(ii)] that $-By\in Ay$, i.e., $y\in{\ensuremath{\text{\rm zer}\,}}(A+B)$.
\[lI8uhT612iv\]: By \[lI8uhT612iii\], there exists $x\in{\ensuremath{\text{\rm zer}\,}}(A+B)$ such that $x_n{\ensuremath{\:\rightharpoonup\:}}x$. In addition, we derive from , \[lI8uhT612i\], and \[lI8uhT612ii\] that $y_n{\ensuremath{\:\rightharpoonup\:}}x$ and $u_n\to -B{x}\in A{x}$.
\[lI8uhT612iva\]: Suppose that $A$ is demiregular at $x$. Then yields $y_n\to{x}$ and \[lI8uhT612i\] implies that $x_n\to{x}$.
\[lI8uhT612ivb\]: Suppose that $B$ is demiregular at $x$. Since $x_n{\ensuremath{\:\rightharpoonup\:}}x$ and $Bx_n\to Bx$ by \[lI8uhT612ii\], we have $x_n\to x$.
\[lI8uhT612ivc\]: This follows from \[lI8uhT612iii\] and Corollary \[koGhn843gd4\]\[koGhn843gd4iii\].
Proposition \[lI8uhT612\] extends [@Opti04 Corollary 6.5] and [@Sico10 Theorem 2.8], which impose the additional assumption that the relaxation parameters $(\lambda_n)_{n\in{\ensuremath{\mathbb N}}}$ satisfy $(\forall n\in{\ensuremath{\mathbb N}})$ $\lambda_n{\ensuremath{\leqslant}}1$. By contrast, the relaxation range allowed in can be an arbitrarily large interval in $\left]0,2\right[$ and the maximum relaxation is always strictly greater than 1.
In Proposition \[lI8uhT612\], the parameters $(\gamma_n)_{n\in{\ensuremath{\mathbb N}}}$ are allowed to vary at each iteration. Now suppose that they are restricted to a fixed value $\gamma\in\left]0,2\beta\right[$. Then, as in , reduces to $x_{n+1}=x_n+\lambda_n(Tx_n+e_n-x_n)$, where $T=J_{\gamma A}({\ensuremath{\mathrm{Id}}\,}-\gamma B)$ is $\alpha$-averaged and $e_n$ is given by . In this special case, the weak convergence of $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ to a zero of $A+B$ can be derived from Proposition \[paQg507-04a\]\[paQg507-04a-iii\] applied with $T_n\equiv T$, $\alpha_n\equiv\alpha$, and $(\lambda_n)_{n\in{\ensuremath{\mathbb N}}}$ in $\left]0,1/\alpha\right[$ satisfying $\sum_{n\in{\ensuremath{\mathbb N}}}\lambda_n(1/\alpha-\lambda_n)={\ensuremath{{+\infty}}}$ (see also [@Livre1 Proposition 5.15(iii)]). This approach was proposed in [@Livre1 Theorem 25.8(i)] with the constant $\alpha=\widetilde{\phi}(1/2,\gamma/(2\beta))=1/(1/2+
\text{\rm min}\{1,\beta/\gamma\})$ of , and revisited in [@Cond13 Lemma 4.4] in the case of subdifferentials of convex functions with the sharper constant $\alpha={\phi}(1/2,\gamma/(2\beta))=2\beta/(4\beta-\gamma)$ of [@Ogur02 Theorem 3(b)] (see Remark \[KWghy5-f-09\]).
\[lI8uhT614\] Let $\beta\in{\ensuremath{\left]0,+\infty\right[}}$, let $\varepsilon\in\left]0,\min\{1/2,\beta\}\right[$, let $x_0\in{\ensuremath{{\mathcal H}}}$, let $f\in\Gamma_0({\ensuremath{{\mathcal H}}})$, let $g\colon{\ensuremath{{\mathcal H}}}\to{\ensuremath{\mathbb{R}}}$ be convex and differentiable with a $1/\beta$-Lipschitz gradient, and suppose that the set $S$ of solutions to the problem $$\label{ewfw7reww24u}
{\ensuremath{\underset{\substack{{x\in{\ensuremath{{\mathcal H}}}}}}{\text{\rm minimize}}\;\;f(x)+g(x) }}$$ is nonempty. Furthermore, let $(\gamma_n)_{n\in{\ensuremath{\mathbb N}}}$ be a sequence in $\left[\varepsilon,2\beta/(1+\varepsilon)\right]$, and let $(a_n)_{n\in{\ensuremath{\mathbb N}}}$ and $(b_n)_{n\in{\ensuremath{\mathbb N}}}$ be sequences in ${\ensuremath{{\mathcal H}}}$ such that $\sum_{n\in{\ensuremath{\mathbb N}}}\|a_n\|<{\ensuremath{{+\infty}}}$ and $\sum_{n\in{\ensuremath{\mathbb N}}}\|b_n\|<{\ensuremath{{+\infty}}}$. For every $n\in{\ensuremath{\mathbb N}}$, let $\lambda_n\in\left[\varepsilon,(1-\varepsilon)(2+\varepsilon-\gamma_n/(2\beta))\right]$ and set $$\label{ewfw7reww24v}
x_{n+1}=x_n+\lambda_n\Big({\ensuremath{\text{\rm prox}}}_{\gamma_n f}
\big(x_n-\gamma_n(\nabla g(x_n)+b_n)\big)+a_n-x_n\Big).$$ Then the following hold:
1. \[lI8uhT614i\] $\sum_{n\in{\ensuremath{\mathbb N}}}
\|{\ensuremath{\text{\rm prox}}}_{\gamma_n f}(x_n-\gamma_n\nabla g(x_n))-x_n\|^2<{\ensuremath{{+\infty}}}$.
2. \[lI8uhT614ii\] Let $x\in S$. Then $\sum_{n\in{\ensuremath{\mathbb N}}}\|\nabla g(x_n)-\nabla g(x)\|^2<{\ensuremath{{+\infty}}}$.
3. \[lI8uhT614iii\] $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ converges weakly to a point in $S$.
4. \[lI8uhT614iv\] Suppose that $\partial f$ or $\nabla g$ is demiregular at every point in $S$, or that ${\ensuremath{\text{\rm int}\,}}S\neq{\ensuremath{{\varnothing}}}$. Then $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ converges strongly to a point in $S$.
Using the same arguments as in [@Livre1 Section 27.3], one shows that this is the specialization of Proposition \[lI8uhT612\] to the case when $A=\partial f$ and $B=\nabla g$.
[99]{}
H. Attouch, L. M. Briceño-Arias, and P. L. Combettes, A parallel splitting method for coupled monotone inclusions, [*SIAM J. Control Optim.*]{}, vol. 48, pp. 3246–3270, 2010.
J.-B. Baillon and R. E. Bruck, Optimal rates of asymptotic regularity for averaged nonexpansive mappings, in: [*Proc. of the Second International Conference on Fixed Point Theory and Applications,*]{} (K. K. Tan, ed.), World Scientific Press, London, 1992, pp. 27–66.
J.-B. Baillon, R. E. Bruck, and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups, [*Houston J. Math.*]{}, vol. 4, pp. 1–9, 1978.
J.-B. Baillon, P. L. Combettes, and R. Cominetti, Asymptotic behavior of compositions of under-relaxed nonexpansive operators, [*J. Dynam. Games,*]{} vol. 1, pp. 331–346, 2014.
H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, [*SIAM Rev.,*]{} vol. 38, pp. 367–426, 1996.
H. H. Bauschke and P. L. Combettes, [*Convex Analysis and Monotone Operator Theory in Hilbert Spaces.*]{} Springer, New York, 2011.
C. L. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, [*Inverse Problems,*]{} vol. 20, pp. 103–120, 2004.
A. Cegielski, [*Iterative Methods for Fixed Point Problems in Hilbert Spaces,*]{} Lecture Notes in Mathematics, vol. 2057. Springer, Heidelberg, 2012.
Y. Censor, T. Elfving, and G. T. Herman, Averaging strings of sequential iterations for convex feasibility problems, in: [*Inherently Parallel Algorithms for Feasibility and Optimization*]{}, (D. Butnariu, Y. Censor, and S. Reich, eds.), pp. 101–113. Elsevier, New York, 2001.
P. L. Combettes, Quasi-Fejérian analysis of some optimization algorithms, in: [*Inherently Parallel Algorithms for Feasibility and Optimization*]{}, (D. Butnariu, Y. Censor, and S. Reich, eds.), pp. 115–152. Elsevier, New York, 2001.
P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, [*Optimization,*]{} vol. 53, pp. 475–504, 2004.
P. L. Combettes, Dinh Dũng, and B. C. Vũ, Dualization of signal recovery problems, [*Set-Valued Var. Anal.,*]{} vol. 18, pp. 373–404, 2010.
P. L. Combettes and B. C. Vũ, Variable metric forward-backward splitting with applications to monotone inclusions in duality, [*Optimization,*]{} vol. 63, pp. 1289–1318, 2014.
L. Condat, A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms, [*J. Optim. Theory Appl.,*]{} vol. 158, pp. 460–479, 2013.
G. Crombez, On averaged, firmly nonexpansive and quasi-nonexpansive operators, and more of that, [*J. Nonlinear Convex Anal.,*]{} vol. 8, pp. 277–288, 2007.
N. Ogura and I. Yamada, Non-strictly convex minimization over the fixed point set of an asymptotically shrinking nonexpansive mapping, [*Numer. Funct. Anal. Optim.,*]{} vol. 23, pp. 113–137, 2002.
B. T. Polyak, [*Introduction to Optimization.*]{} Optimization Software Inc., New York, 1987.
H. Raguet, J. Fadili, and G. Peyré, A generalized forward-backward splitting, [*SIAM J. Imaging Sci.,*]{} vol. 6, pp. 1199–1226, 2013.
Y.-W. Wen, M. K. Ng, and W.-K. Ching, Iterative algorithms based on decoupling of deblurring and denoising for image restoration, [*SIAM J. Sci. Comput.,*]{} vol. 30, pp. 2655–2674, 2008.
I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, in: [*Inherently Parallel Algorithms for Feasibility and Optimization*]{}, (D. Butnariu, Y. Censor, and S. Reich, eds.), pp. 473–504. Elsevier, New York, 2001.
I. Yamada and N. Ogura, Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, [*Numer. Funct. Anal. Optim.,*]{} vol. 25, pp. 619–655, 2004.
[^1]: Contact author: P. L. Combettes, [plc@ljll.math.upmc.fr]{}, phone: +33 1 4427 6319, fax: +33 1 4427 7200.
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---
abstract: 'Random matrix theory successfully connects the structure of interactions of large ecological communities to their ability to respond to perturbations. One of the most debated aspects of this approach is the missing role of population abundances. Despite being one of the most studied patterns in ecology, and one of the most empirically accessible quantities, population abundances are always neglected in random matrix approaches and their role in determining stability is still not understood. Here, we tackle this question by explicitly including population abundances in a random matrix framework. We obtain an analytical formula that describes the spectrum of a large community matrix for arbitrary feasible species abundance distributions. The emerging picture is remarkably simple: while population abundances affect the rate to return to equilibrium after a perturbation, the stability of large ecosystems is uniquely determined by the interaction matrix. We confirm this result by showing that the likelihood of having a feasible and unstable solution in the Lotka-Volterra system of equations decreases exponentially with the number of species for stable interaction matrices.'
author:
- Theo Gibbs
- Jacopo Grilli
- Tim Rogers
- Stefano Allesina
bibliography:
- 'abdrndmat.bib'
title: The effect of population abundances on the stability of large random ecosystems
---
Introduction
============
Since the work of Lotka and Volterra, ecologists have attempted to mathematize the interactions between populations to build predictive models of population dynamics. This is a complex problem – ecological communities are often composed of a large number of species [@May1988; @May1988], the equations describing their interactions have been debated for decades [@Arditi2012], and the estimation of parameters and initial conditions is often unfeasible from an empirical standpoint.
To circumvent this problem, Robert May [@May1972] introduced the idea of modeling complex ecological communities using random matrices. Consider the case in which the dynamics of the populations can be described by a system of ordinary differential equations:
$$\frac {dx_i(t)} {dt} = f_i({\boldsymbol{x}}(t)),
\label{eq:generic}$$
where ${\boldsymbol{x}}(t)$ is a vector containing the populations abundances at time $t$, and the function $f_i$ relates the abundance of all populations to the growth of population $i$. In general, $f_i$ is a nonlinear equation with several parameters.
Suppose that the system admits a feasible equilibrium point, i.e., a vector ${\boldsymbol{x}}^\ast$ such that $f_i({\boldsymbol{x}}^\ast) = 0$ and $x_i^\ast > 0$ for all $i$. If we start the system at this point, it will remain there indefinitely. We can therefore ask whether the system will go back to the equilibrium, or rather move away from it, following a perturbation. This type of stability analysis can be carried out by building the Jacobian matrix $J_{ij} = \partial
f_i({\boldsymbol{x}}(t)) / \partial x_j$ and evaluating it at the equilibrium point, yielding the so-called community matrix ${\boldsymbol{M}} = \left
. {\boldsymbol{J}} \right \rvert_{{\boldsymbol{x}}^\ast}$. If all the eigenvalues of ${\boldsymbol{M}}$ have negative real part, then the equilibrium is locally asymptotically stable, and the system will return to it after sufficiently small perturbations; if any of the eigenvalues have a positive real part, the system will move away from the equilibrium when perturbed.
Clearly, to build ${\boldsymbol{M}}$ one would need to precisely know the functions $f_i$, as well as their parameters, and solve for the equilibrium (or equilibria) ${\boldsymbol{x}}^\ast$. May took a radically different approach and analyzed the case in which ${\boldsymbol{M}}$ is a random matrix with independent, identically distributed off-diagonal elements, and constant diagonal elements [@May1972]. For this parameterization, he was able to show that the community matrices describing sufficiently large and complex ecological communities are always unstable. The random-matrix approach was recently extended and refined to include different types of interaction between the populations [@Allesina2012; @ReviewRMT], as well as to study the effect of more complex network structures, such as the hierarchical organization of food-webs [@Allesina2015a] and the modular pattern often displayed by biological networks [@Grilli2016].
By modeling directly the matrix ${\boldsymbol{M}}$ as a random matrix, one does not require a precise characterization of the functions $f_i$ and the equilibrium ${\boldsymbol{x}}^\ast$. While mathematically convenient, this approach does not explicitly take into account the abundance of the populations—a type of data that is empirically much more accessible than interaction coefficients or the elements of the community matrix.
The distribution of species abundances (SAD) has been shown to have remarkably similar features across different species rich communities [@Bell2001a] with a skewed shape and few highly abundant species. The log-series distribution [@Fisher1943a], discrete lognormal [@Preston1948] and negative binomial [@Volkov2007] have all been proposed to describe empirical SADs, and have been shown to emerge from either neutral [@Caswell1976; @Hubbell2001a; @Volkov2003; @Azaele2016] or niche mechanisms [@MacArthur1957a; @Vandermeer1966].
The role of species abundances in structuring the community matrix ${\boldsymbol{M}}$ can be easily seen by considering one of the simplest models of population dynamics, the Generalized Lotka-Volterra (GLV) model:
$$\frac {dx_i(t)} {dt} = x_i(t) \left(r_i + \sum _{j} A_{ij} x_j(t) \right) \ ,
\label{eq:LV}$$
where $r_i$ is the intrinsic growth rate of species $i$, and $A_{ij}$ is the per-capita effect of species $j$ on the growth of $i$. If a feasible equilibrium (i.e., one where all species have positive abundance) exists, then it can be found solving the system of equations
$$0 = r_i + \sum _{j} A_{ij} x^\ast_j \ ,$$
yielding the community matrix $M_{ij} = A_{ij} x_i^\ast$, which can be written in matrix form as
$${\boldsymbol{M}} = {\boldsymbol{X}} {\boldsymbol{A}} \ ,$$
where ${\boldsymbol{X}}$ is a diagonal matrix with $X_{ii} =
x_i^\ast$ and zeros elsewhere. Even if the elements of ${\boldsymbol{A}}$ were independent, identically distributed samples from a distribution, the elements of ${\boldsymbol{M}}$ would not be—the matrix of abundances ${\boldsymbol{X}}$ couples all the coefficients in the same row, such that the distribution of the elements in each row would in principle be different.
One of the main goals of this work is to extend the random matrix approach by considering a random matrix of abundances ${\boldsymbol{X}}$ and a random matrix of interactions ${\boldsymbol{A}}$, and determining the stability of ${\boldsymbol{M}}$ under these conditions. In this way, we address the effect of species abundances on stability, thereby lifting one of the main criticisms of the random matrix approach [@ReviewRMT; @amnatjames; @Jacquet2016].
As we stated above when analyzing coexistence, we need population abundances to be positive (*feasible*). Stability cannot, at least in principle, be disentangled from the constraint imposed by feasibility on interactions [@Roberts1974]. Diversity and interaction properties have important consequences for the range of parameters corresponding to feasible solutions [@Rohr2014; @Stone2016; @Grilli2017]. While the interest in feasibility has grown considerably in recent years, the relationship between feasibility and stability is still unclear. In fact, most of the studies on feasibility assume strong conditions on the interaction matrix (e.g., D-stability, diagonal stability) that guarantee stability of any feasible solution [@Rohr2014; @Grilli2017]. It is still unclear when these assumptions are justified and how likely it is for large random interaction matrices to meet these conditions.
In the second part of this work, we focus on the relationship between feasibility and stability. In particular, we study the relationship between the stability of ${\boldsymbol{A}}$ and that of ${\boldsymbol{M}}$ for the GLV model. Our results show that, given a stable random matrix ${\boldsymbol{A}}$, the probability that an arbitrary feasible equilibrium is unstable decreases exponentially with diversity. This result strongly suggests that, provided that the interaction matrix ${\boldsymbol{A}}$ is stable, feasible solutions are almost surely stable. We therefore provide a more robust justification to both May’s original paper—by showing that population abundances do not affect qualitatively stability—and the more recent work on feasibility that assumes stability—by predicting that this assumption is almost surely met for large random systems.
Constructing the community matrix with arbitrary population abundance
=====================================================================
We consider a system of $S$ interacting populations whose dynamics are described by the GLV model in equation \[eq:LV\], assume that a feasible equilibrium ${\boldsymbol{x}}^\ast$ exists, and define ${\boldsymbol{X}}$ as the diagonal matrix with diagonal entries $X_{ii} = x^\ast_i$. The feasible fixed point ${\boldsymbol{x}}^\ast$ is locally asymptotically stable if and only if all the eigenvalues of the community matrix ${\boldsymbol{M}} =
{\boldsymbol{X}} {\boldsymbol{A}}$, with components $M_{ij} = x^\ast_j A_{ij}$, have a negative real part. Here, we model ${\boldsymbol{A}}$ as a random matrix and ${\boldsymbol{x}}^\ast$ as a random vector with positive components, with the goal of studying the spectrum (distribution of the eigenvalues) of the community matrix ${\boldsymbol{M}} $ . From the GLV model, specifiying a feasible fixed point ${\boldsymbol{X}}^\ast$ is the same as specifying a vector of intrinsic growth rates ${\boldsymbol{r}}$ inside the feasibility domain [@Rohr2014; @Grilli2017].
More specifically, we assume that the diagonal entries of the diagonal matrix ${\boldsymbol{X}}$ are drawn from an arbitrary distribution with positive support, mean $\mu_X$, and variance $\sigma_X^2$. The diagonal entries of ${\boldsymbol{A}}$ are drawn from an arbitrary distribution with support in the negative axis, mean $\mu_d$, and variance $\sigma_d^2$. Finally, each off-diagonal pair $(A_{ij},A_{ji})$ in ${\boldsymbol{A}}$ is drawn independently from a bivariate distribution with identical marginal means $\mu$, variances $\sigma^2$, and correlation $\rho$. Unless otherwise specified, we focus on the case $\sigma_d =
0$, while we discuss in the [Supplementary Information]{} the effects of variability in self-regulation.
In the case of $\sigma_d = 0$ and in the limit of large $S$, the spectrum of ${\boldsymbol{A}}$ is known and is independent of the choice of the bivariate distribution (provided that mild conditions on the finiteness of the moments are satisfied [@nguyen2012elliptic]). In particular, ${\boldsymbol{A}}$ has one eigenvalue equal to $-\mu_d + S
\mu$ [@ORourke2014], while the others (the *bulk* of eigenvalues) are uniformly distributed in an ellipse in the complex plane centered in $-\mu_d-\mu$ with horizontal axis $\sqrt{S} \sigma
(1+\rho)$ and vertical axis $\sqrt{S} \sigma
(1-\rho)$ [@nguyen2012elliptic; @Allesina2012; @ORourke2014]. Figure \[fig:populationaffect\] shows an example of the spectrum of ${\boldsymbol{A}}$.
Figure \[fig:populationaffect\] also shows an example of the eigenvalues of the community matrix ${\boldsymbol{M}} = {\boldsymbol{X}} {\boldsymbol{A}}$ where the diagonal entries of ${\boldsymbol{X}}$ are independent random variables drawn from a uniform distribution. It is evident that the bulk of eigenvalues of ${\boldsymbol{M}}$ does not follow the elliptic law.
Disentangling the effect of the mean interaction strength {#sec:disentangle}
=========================================================
When the mean $\mu$ of the off-diagonal elements of the interaction matrix ${\boldsymbol{A}}$ does not equal zero, the spectra of ${\boldsymbol{A}}$ and ${\boldsymbol{M}}$ are characterized by the presence of an outlier. The value of this eigenvalue for the matrix ${\boldsymbol{A}}$ is known for the case $\sigma_d = 0$, and in the limit of large $S$ [@ORourke2014]. It can be obtained by decomposing the matrix ${\boldsymbol{A}}$ as a sum of three matrices
$${\boldsymbol{A}} = (\mu_d-\mu) {\boldsymbol{I}} + \mu {\boldsymbol{1}} + {\boldsymbol{B}} \ ,$$
where ${\boldsymbol{I}}$ is the identity matrix, ${\boldsymbol{1}}$ is a matrix of ones, and ${\boldsymbol{B}}$ is a random matrix with mean zero that follows the elliptic law. It has been proved [@ORourke2014] that the spectrum of ${\boldsymbol{A}}$ is characterized by a bulk of eigenvalues, determined by the spectrum of $(\mu_d-\mu) {\boldsymbol{I}} + {\boldsymbol{B}}$, and the presence of an outlier, whose value is (approximately) given by the largest eigenvalue of $(\mu_d-\mu) {\boldsymbol{I}} + \mu {\boldsymbol{1}}$, which has value $\mu_d + (S-1) \mu$.
Figure \[fig:observation\] shows that, if $\mu \neq 0$, the spectrum of ${\boldsymbol{M}}$ is also characterized by the presence of a bulk and of an outlying eigenvalue. By decomposing the matrix ${\boldsymbol{M}}$ as
$${\boldsymbol{M}} = {\boldsymbol{X}} \left( (\mu_d-\mu) {\boldsymbol{I}} + \mu {\boldsymbol{1}} + {\boldsymbol{B}} \right) \ ,$$
we show in the [Supplementary Information]{} that the bulk of the spectrum of ${\boldsymbol{M}}$ is determined by the eigenvalues of the matrix ${\boldsymbol{J}} =
{\boldsymbol{X}} \left( (\mu_d-\mu) {\boldsymbol{I}} + {\boldsymbol{B}} \right)$ and the outlier is given by largest eigenvalue of ${\boldsymbol{Q}} = {\boldsymbol{X}} \left(
(\mu_d-\mu) {\boldsymbol{I}} + \mu {\boldsymbol{1}} \right)$. Figure \[fig:observation\] shows an example of this decomposition, where it is evident that the bulks of eigenvalues of ${\boldsymbol{M}}$ and ${\boldsymbol{J}}$ are the same, and the outliers of ${\boldsymbol{M}}$ and ${\boldsymbol{Q}}$ match.
The trace of ${\boldsymbol{M}}$ is given by
$$\operatorname*{\text{tr}}\left({\boldsymbol{M}}\right) = \lambda_{\textrm{out}} + (S-1) \langle \lambda
\rangle_{\textrm{bulk}} \ ,$$
where $\lambda_{\textrm{out}}$ is the value of the outlier and $\langle \lambda \rangle_{\textrm{bulk}}$ is the average eigenvalue in the bulk. Since the bulks of the eigenvalues of ${\boldsymbol{M}}$ and ${\boldsymbol{J}}$ are the same, we have that
$$\langle \lambda \rangle_{\textrm{bulk}} = \frac{1}{S} \operatorname*{\text{tr}}\left({\boldsymbol{J}}\right) = \mu_X \left(\mu_d - \mu \right) \ .$$
Using the fact that
$$\operatorname*{\text{tr}}\left({\boldsymbol{M}}\right) = S \mu_d \mu_X \ ,$$
we see that the outlier is equal to
$$\lambda_{\textrm{out}} = \mu_X \left( \mu_d + (S-1) \mu \right) \ .
\label{eq:outlier}$$
Figure \[fig:outeigen\] shows that this analytical prediction closely matches the outlier of the spectrum of ${\boldsymbol{M}}$.
Analytical solution in the case $\rho = 0$
==========================================
In section \[sec:disentangle\] we showed that the spectrum of ${\boldsymbol{M}}$ is characterized by a bulk of eigenvalues and an outlier, which is determined by the mean of interaction matrix $\mu$. In the following, we focus on the bulk of eigenvalues, so we assume $\mu
= 0$.
Using the cavity method [@Rogers2008; @Rogers2009; @Grilli2016], we derive in the [Supplementary Information]{} a system of equations for the spectral density of the matrix ${\boldsymbol{M}}$. These equations cannot be explicitly solved in the most general case, but they take a particularly simple form in the case where the correlation $\rho = 0$. In this case, it is possible to write an implicit equation for the support of the spectrum, which takes the form
$$\int {\operatorname*{\text{d}}} x \ {\operatorname*{\text{d}}} s \ P_{XD}(x,s) \ \frac{ S x^2 \sigma^2} {| \lambda - s
x |^2 } = 1 \ ,
\label{eq:uncpred}$$
where $P_{XD}(x,s)$ is the joint distribution of the population abundances $x$, with mean $\mu_X$ and variance $\sigma_X^2$, and the self-regulation terms (i.e., the diagonal elements of the interaction matrix) with mean $\mu_d$ and variance $\sigma_d^2$. The complex solutions $\lambda$ of this equation define the support of the spectrum in the complex plane. In the [Supplementary Information]{} we explicitly solve the case of constant self-regulation terms (i.e., $\sigma_d = 0$) and population abundances drawn from a uniform distribution.
When the self-regulation terms are constant, equation \[eq:uncpred\] reduces to
$$\int {\operatorname*{\text{d}}}x \ P_{X}(x) \ \frac{S x^2 \sigma^2} {| \lambda - \mu_d x |^2 } = 1 \ ,
\label{eq:uncpred_diag}$$
where $P_{X}(x)$ is the species abundance distribution. Figure \[fig:uncorrelatedpredictions\] compares the analytical prediction with the bulk of eigenvalues of ${\boldsymbol{M}}$ for different distributions of ${\boldsymbol{X}}$, showing that the solutions of equation \[eq:uncpred\_diag\] closely match the support of the spectrum of ${\boldsymbol{M}}$.
Equation \[eq:uncpred\_diag\] also predicts that if ${\boldsymbol{A}}$ is stable, then ${\boldsymbol{M}}$ is stable. In fact, equation \[eq:uncpred\_diag\] predicts that the matrix ${\boldsymbol{A}}$ is stable iff $\mu_d + S \sigma^2 < 0$. If this condition is met, it is simple to observe that
$$\frac{S x^2 \sigma^2} {| \lambda - \mu_d x |^2 } < 1$$
for any complex $\lambda$ with positive real part and any positive real $x$. When this inequality is used in equation \[eq:uncpred\_diag\] one obtains that the points on the boundary of the support, and therefore all the eigenvalues, always have negative real part.
The stability of large community matrices does not depend on population abundance {#sec:dstab}
=================================================================================
In the previous section, we derived the spectrum in the case $\rho =
0$, finding that if the interaction matrix ${\boldsymbol{A}}$ is stable, then ${\boldsymbol{M}}$ is stable. The goal of this section is to study more deeply the relationship between the stability of ${\boldsymbol{A}}$ and that of ${\boldsymbol{M}}$. More specifically, given a stable random matrix ${\boldsymbol{A}}$, we ask what is the probability of finding a positive diagonal matrix ${\boldsymbol{X}}$, such that ${\boldsymbol{M}} = {\boldsymbol{X}} {\boldsymbol{A}}$ is stable.
A matrix ${\boldsymbol{A}}$ is *D-stable* if, for any positive diagonal matrix ${\boldsymbol{X}}$, ${\boldsymbol{X}} {\boldsymbol{A}}$ is stable [@Kaszkurewicz2000]. An explicit condition for D-stability that does not require checking all the possible choices of ${\boldsymbol{X}}$ is not known in dimension larger than four [@Redheffer1985]. Therefore, it is not known, in general, under which values of $\mu$, $\sigma$, $\rho$ and $\mu_d$ random matrices are expected to be *D-stable*.
A stronger condition for stability is *diagonal stability*. A matrix ${\boldsymbol{A}}$ is diagonally stable if there exists a positive diagonal matrix ${\boldsymbol{X}}$ such that ${\boldsymbol{X}}{\boldsymbol{A}}+{\boldsymbol{A}}^t{\boldsymbol{X}}$ is stable. Interestingly, diagonal stability implies D-stability [@Kaszkurewicz2000]. As for D-stability, a simple necessary and sufficient test for diagonal stability is not known. On the other hand, it is simple to observe that the stability of $({\boldsymbol{A}}+{\boldsymbol{A}}^t)/2$ is a sufficient condition for diagonal stability (corresponding to choosing a constant diagonal matrix ${\boldsymbol{X}}$), and therefore also implies D-stability.
All the eigenvalues of $({\boldsymbol{A}}+{\boldsymbol{A}}^t)/2$ are real and, if ${\boldsymbol{A}}$ is a symmetric random matrix of independently distributed entries with bounded higher moments, the bulk of eigenvalues of $({\boldsymbol{A}}+{\boldsymbol{A}}^t)/2$ follows Wigner’s semicircle distribution [@Wigner1958; @Tang2014]
$$\varrho_{\frac{{\boldsymbol{A}}+{\boldsymbol{A}}^t}{2}}(\lambda) = \frac{\sqrt{ 2 S \sigma^2 (1+\rho) - \left( \lambda - (\mu_d - \mu) \right)^2 }}{ \pi S \sigma^2 (1+\rho)} \ ,$$
with one outlying eigenvalue equal to $\mu_d + (S-1) \mu$.
For positive mean $\mu$, if $\mu > (1+\rho) \sigma / \sqrt{S}$, the rightmost eigenvalue is the outlier. In this case, the rightmost eigenvalue of ${\boldsymbol{A}}$ and of $({\boldsymbol{A}}+{\boldsymbol{A}}^t)/2$ are the same. Therefore, for non-negative $\mu$, stable random matrices are almost surely diagonally stable. Since diagonal stability implies D-stability, if ${\boldsymbol{A}}$ is stable, then ${\boldsymbol{M}}={\boldsymbol{X}}{\boldsymbol{A}}$ is stable. This argument is in agreement with our formula for the outlier of ${\boldsymbol{M}}$ in the case of non-vanishing mean $\mu$, obtained in equation \[eq:outlier\]. For positive mean $\mu$, the rightmost eigenvalue of ${\boldsymbol{M}}$ is equal to $\mu_X \lambda_{{\boldsymbol{A}}}$, where $\lambda_{{\boldsymbol{A}}}$ is the rightmost eigenvalue of ${\boldsymbol{A}}$ and $\mu_X$ is positive by definition. The sign of the rightmost eigenvalue of ${\boldsymbol{M}}$ is therefore the same as that of the rightmost eigenvalue of ${\boldsymbol{A}}$.
Since a negative $\mu$ only produces a equal shift in the rightmost eigenvalue of ${\boldsymbol{A}}$, $({\boldsymbol{A}}+{\boldsymbol{A}}^t)/2$ and ${\boldsymbol{M}}$, we can restrict our analysis to the case $\mu = 0$. For vanishing mean, the rightmost eigenvalue of $({\boldsymbol{A}}+{\boldsymbol{A}}^t)/2$ is equal to [@Tang2014]
$$\lambda_{\frac{{\boldsymbol{A}}+{\boldsymbol{A}}^t}{2}} = \mu_d + \sqrt{2 S \sigma^2 (1+\rho) } \ ,
\label{eq:lowerdiag}$$
which should be compared with the rightmost eigenvalue of ${\boldsymbol{A}}$
$$\lambda_{{\boldsymbol{A}}} = \mu_d + \sqrt{S \sigma^2} (1+\rho) \ .
\label{eq:stab}$$
As shown in [@Tang2014; @Grilli2017], $\lambda_{\frac{{\boldsymbol{A}}+{\boldsymbol{A}}^t}{2}} \geq \lambda_{{\boldsymbol{A}}}$ and they are equal in the case $\rho = 1$. Equation \[eq:lowerdiag\] imposes a sufficient condition on diagonal stability: if
$$\mu_d + \sqrt{2 S \sigma^2 (1+\rho) } < 0 \ ,
\label{eq:diagstabsuff}$$
${\boldsymbol{A}}$ is diagonally stable and, for any choice of positive diagonal matrix ${\boldsymbol{X}}$, ${\boldsymbol{M}}={\boldsymbol{X}}{\boldsymbol{A}}$ is stable. The non-trivial regime therefore corresponds to the values of parameters where $\mu_d + \sqrt{2 S \sigma^2 (1+\rho) } > 0$ and $\mu_d + \sqrt{S \sigma^2} (1+\rho) < 0$ [@Grilli2017].
Since an explicit condition for D-stability does not exist, we computed the probability that, given a stable random matrix ${\boldsymbol{A}}$, a positive diagonal matrix ${\boldsymbol{X}}$ would make ${\boldsymbol{M}} = {\boldsymbol{X}}
{\boldsymbol{A}}$ unstable. Note that, since any matrix has a non-null probability of being generated when entries are sampled from a bivariate distribution with infinite support, this probability is always non-zero. The relevant question in this context is therefore how this probability depends on the number of species $S$. Figure \[fig:unweightedprobabilities\] shows that the probability of finding a ${\boldsymbol{X}}$ with a destabilizing effect decreases exponentially with the number of species $S$, with a rate that depends on the rightmost eigenvalue $\lambda_{{\boldsymbol{A}}}$ and the correlation $\rho$. This implies that, for large values of $S$, ${\boldsymbol{M}}$ is almost surely stable if ${\boldsymbol{A}}$ is stable.
Fixed points are almost surely stable in large random Lotka-Volterra equations.
===============================================================================
If we consider the Lotka-Volterra equations (equation \[eq:LV\]), and we set the values of the intrinsic growth rates ${\boldsymbol{r}}$, the fixed point has components
$$x_i^\ast = \sum_j A^{-1}_{ij} r_j \ .
\label{eq:fixedpoint}$$
Let us also assume that all these components are positive (i.e., ${\boldsymbol{r}}$ is inside the feasibility domain). In section \[sec:dstab\] we showed that the matrix obtained by multiplying a stable random matrix ${\boldsymbol{A}}$ and a random positive diagonal matrix ${\boldsymbol{X}}$ is more and more likely to be stable as $S$ increases. It is evident (from equation \[eq:fixedpoint\]) that the components of ${\boldsymbol{x}}^\ast$ are not independent of the entries of the matrix ${\boldsymbol{A}}$. The presence of this correlation implies that, at least in principle, choosing a random vector ${\boldsymbol{r}}$ inside the feasibility domain to define ${\boldsymbol{X}}$ could produce different results from sampling independent entries from a specified species abundance distribution.
In this section we repeat the simulations detailed in section \[sec:dstab\], but instead of considering a random fixed point ${\boldsymbol{x}}^\ast$, we find the ${\boldsymbol{x}}^\ast$ determined by a random intrinsic growth rate vector ${\boldsymbol{r}}$ sampled uniformly from the feasibility domain. The most intuitive method for this simulation would consist of taking a random matrix ${\boldsymbol{A}}$, choosing a value ${\boldsymbol{r}}$ at random on the unit sphere, checking if it corresponds to a feasible fixed-point using equation \[eq:fixedpoint\], and finally computing the eigenvalue of ${\boldsymbol{M}} = {\boldsymbol{X}}{\boldsymbol{A}}$. However, as the number of species $S$ increases this method becomes practically unfeasible. In fact, the fraction of intrinsic growth rate vectors ${\boldsymbol{r}}$ corresponding to a feasible solution decreases exponentially with $S$ [@Grilli2017]. If this intuitive method was employed, most of the simulation time would be spent trying to find vectors ${\boldsymbol{r}}$ inside the feasibility domain.
On the other hand, since the relation between ${\boldsymbol{r}}$ and ${\boldsymbol{x}}^\ast$ (via equation \[eq:fixedpoint\]) is bijective, we can easily construct all the vectors ${\boldsymbol{r}}$ inside the feasibility domain by considering all the possible feasible solution ${\boldsymbol{x}}^\ast$. In section \[sec:dstab\] we specified a distribution on the ${\boldsymbol{x}}^\ast$. This distribution translates to a non-trivial distribution on the ${\boldsymbol{r}}$ (that can be obtained from equation \[eq:fixedpoint\]). In this section, we instead assume a distribution on the ${\boldsymbol{r}}$ and derive a corresponding distribution for the ${\boldsymbol{x}}^\ast$. For instance, if we assume that the vectors ${\boldsymbol{r}}$ are uniformly distributed on the unit sphere, the distribution of the ${\boldsymbol{x}}^\ast$ reads [@Grilli2017]
$$P({\boldsymbol{x}}^\ast | {\boldsymbol{A}} ) \propto | \det {\boldsymbol{A}} | \frac{ \delta
\left( \|{\boldsymbol{x}}^\ast\|^2 - 1 \right) }{ \| {\boldsymbol{A}} {\boldsymbol{x}}^\ast
\|^S } \ .
\label{eq:probx}$$
Sampling vectors ${\boldsymbol{x}}^\ast$ according to this distribution is equivalent to sampling vectors ${\boldsymbol{r}}$ uniformly from the feasibility domain. It is important to observe that when ${\boldsymbol{x}}^\ast$ is drawn according to this distribution, its entries are not independent and their densities depend on ${\boldsymbol{A}}$.
Figure \[fig:weightedprobabilitiess\] shows the stability of ${\boldsymbol{M}} = {\boldsymbol{X}} {\boldsymbol{A}}$ when the diagonal entries of ${\boldsymbol{X}}$ are sampled from the probability distribution defined in equation \[eq:probx\]. Despite the presence of a correlation between the entries of ${\boldsymbol{X}}$ and ${\boldsymbol{A}}$, the result obtained in section \[sec:dstab\] is confirmed: the probability of observing a stable ${\boldsymbol{A}}$ but an unstable ${\boldsymbol{M}}$ decreases exponentially with $S$. If the interaction matrix ${\boldsymbol{A}}$ is stable, in the limit of large $S$, the set of intrinsic growth rates corresponding to feasible unstable solutions has measure zero.
Discussion
==========
We explored the effect of population abundances on the stability of random interacting ecosystems. We derived an expression for the spectral density of a community matrix that explicitly includes the species abundance distribution. While the effect on the eigenvalues is highly heterogeneous and strongly depends on the specific choice of the abundance distribution, a remarkably simple message emerges for large randomly interacting ecosystems: the community matrix is stable if and only if the interaction matrix is stable. In other words, the abundances of species seem to not affect the sign of the eigenvalues. We further explored this intriguing result by explicitly estimating the probability of choosing a species abundance distribution leading to instability. While for finite systems this probability is always positive, it decreases exponentially with the number of species, confirming what was found studying the spectrum of the community matrix analytically.
Our results strongly suggest that large random matrices are D-stable *almost surely*: the set of destabilizing positive diagonal matrices has measure zero. This fact has important consequences on Lotka-Volterra systems of equations, implying that feasible unstable fixed-points are very unlikely. This result allows to disentangle the problem of feasibility (how often are fixed points feasible?) from the problem of stability (how often are fixed points stable?), justifying a-posteriori what assumed in many studies on feasibility [@Rohr2014; @Grilli2017] and expanding the validity of their results.
The generalized Lotka-Volterra equations display a rich dynamical behavior, leading to limit cycles when two or more species are considered and chaos with three or more species [@Smale1976; @Takeuchi1996]. Both limit cycles and chaos require the existence of an unstable fixed point in the interior of the feasibility domain [@Hofbauer1998]. Since the chance of observing a feasible unstable fixed point decays rapidly when the number of species increases, our results suggest that chaos and limit cycles are extremely rare in large random Lotka-Volterra systems.
A stronger notion than D-stability is diagonal stability. While for Lotka-Volterra systems, the former implies local asymptotic stability of any feasible solution, the latter implies global stability. We showed that large random stable matrices are always D-stable. Under which conditions they are also diagonally stable is an important open problem. A sufficient condition for diagonal stability is negative definiteness [@Grilli2017]. In the context of random matrices, negative definiteness is equivalent to the condition expressed in equation \[eq:diagstabsuff\]. The condition for negative definiteness should be compared to the condition for stability (see equation \[eq:stab\]). For large random matrices, two extreme scenarios are possible: negative definiteness is almost surely a necessary condition for diagonal stability, or stable random matrices are almost surely diagonally stable. It is also possible that the condition for diagonal stability is less trivial, corresponding to values of parameters between the conditions imposed by equations \[eq:diagstabsuff\] and \[eq:stab\]. Even more complicated, it is also possible that a sharp condition for diagonal stability does not exist for random matrices and, in the limit of large $S$, stable and non negative definite random matrices have a non-vanishing probability of being (or not being) diagonally stable.
Our results shed light on one of the most controversial aspects of the classic result of May [@May1972] and its extensions. Many authors [@Roberts1974; @Pimm1979; @King1983; @ReviewRMT; @amnatjames; @Jacquet2016] have argued that the unrealistic assumption of constant population abundances was a key choice in May’s paper, suggesting that more realistic abundance distribution would have produced drastically different results. We showed that the conditions obtained in the original paper and in its extension [@May1972; @Allesina2012] are in fact valid for any species abundance distribution. In other words, the stability of fixed points (i.e., the stability of the community matrix) is determined only by the stability of the interaction matrix.
We thank A. Maritan, S. Tang and G. Barabás for comments and discussions. T.G. and S.A. were supported by NSF grant DEB-1148867. J.G. was supported by the Human Frontier Science Program.
Notation and goals
==================
We aim to study the spectral density of a matrix ${\boldsymbol{M}}$ of the form ${\boldsymbol{M = XA}}$, where ${\boldsymbol{X}}$ is a positive diagonal matrix and ${\boldsymbol{A}}$ a random matrix with arbitrary distribution. The diagonal entries of ${\boldsymbol{X}}$ are drawn from an arbitrary distribution with positive support, mean $\mu_X$ and variance $\sigma_X^2$. The diagonal entries of ${\boldsymbol{A}}$ are drawn from an arbitrary distribution with negative support, mean $\mu_d$ and variance $\sigma_d^2$. Each pair of off-diagonal entries $(A_{ij}, A_{ji})$ is drawn from a bivariate distribution with identical marginal means $\mu$, variances $\sigma^2$ and correlation $\rho$.
Let $ {\boldsymbol{B}} $ be an $ S \times S $ random matrix with complex eigenvalues $ \lambda_i$ for $i = 1, ... , S$. Its spectral density is defined as $$\varrho(x,y) = \frac{1} {S} \sum_{i=1}^S\delta(x - \Re(\lambda_i))\delta(y - \Im(\lambda_i))$$ which, in the limit of large $S$, converges to $$\varrho(x,y) = \mathbb{E} [\delta(x - \Re(\lambda_i))\delta(y - \Im(\lambda_i)) ] \ ,$$ where $\mathbb{E}[\cdot]$ stands for the expectation over matrices in the ensemble.
We introduce the resolvent [@Rogers2010] $$\operatorname*{\mathcal{G}}({\boldsymbol{q}}; {\boldsymbol{B}}) = \frac{1} {S} \sum_{i=1}^S (\lambda_i - {\boldsymbol{q}})^{-1} = \frac{1}{S} \operatorname*{\text{tr}}\left( {\boldsymbol{B}} - {\boldsymbol{q}} {\boldsymbol{I}} \right)^{-1} \ .
\label{eq:resolvent_finite}$$ The variable ${\boldsymbol{q}} = \lambda + \epsilon j$ is a quaternion (see section \[sec:quaternions\] for definitions and notation) and the resolvent is a function $\operatorname*{\mathcal{G}}: \mathbb{H} \to \mathbb{H}$.
The resolvent and the spectral density are related by the following formulas [@Rogers2010]: $$\operatorname*{\mathcal{G}}({\boldsymbol{q}}; {\boldsymbol{B}}) = \int dx \ dy \ \varrho(x,y) (x+iy - {\boldsymbol{q}})^{-1}
\label{resolvent+spectrum}$$ and $$\varrho\left(x,y \right) = - \frac {1} {\pi} \lim_{\epsilon \to 0^+} \Re \left( \frac{\partial} {\partial \bar{ \lambda} } \operatorname*{\mathcal{G}}(\lambda + \epsilon j; {\boldsymbol{B}}) \right) \biggl|_{\lambda = x+iy} \ ,
\label{spectrum+resolvent}$$ where $\frac{\partial} {\partial \bar{ \lambda} }$ is the Wirtinger derivative $$\frac{\partial }{\partial \bar{\lambda}} := \frac{1}{2} \left( \frac{\partial }{\partial x} + i \frac{\partial }{\partial y} \right) \ .$$
Quaternions {#sec:quaternions}
===========
When constructing the complex numbers from the real numbers, one defines a variable $i$ to be a root of the equation $x^2 + 1 = 0$. The algebraic structure of $\mathbb{C}$ descends from the equation $i^2 = -1$ and the algebraic structure of $\mathbb{R}$. Similarly, the algebra of quaternions $\mathbb{H}$ can be defined by introducing the symbols $i$, $j$ and $k$ and the relations $$i^2 = j^2 = k^2 = ijk = -1 \ .$$ From these equations all the multiplication rules can be obtained. In particular, it follows that multiplication in $\mathbb{H}$ is not commutative (e.g. $ij = -ji$).
A quaternion ${\boldsymbol{q}}$ can be written as $$\textbf{q} = a + bi + cj + dk \ ,$$ where $a,b,c,d \in \mathbb{R}$. Equivalently, by introducing the two complex numbers $z = a + bi$ and $w = c + di$ and using $k = ij$, one can write $$\textbf{q} = z + w j \ .$$ Another equivalent way to represent quaternions is to write them in matrix form $$\textbf{q} = \left( \begin{matrix} z & w \\ \bar{w} & \bar{z} \\ \end{matrix} \right) \ .$$ It can be shown that, when written in this form, the multiplication rules of quaternions match the rules of matrix multiplication. In particular, one has that $$(z + w j)( u + v j ) = ( z u - w \bar{v} ) + (z v + w \bar{u} ) j \ .$$ We also introduce the operation $$(z + w j) \circ ( u + v j ) = z u - w v j \ ,$$ which, in matrix notation, corresponds to element-by-element multiplication.
The conjugate of a quaternion $\textbf{q} = z + wj$ is defined as $\bar{{\boldsymbol{q}}} = \bar{z} - wj$. From this definition, one obtains the norm of a quaternion $$| {\boldsymbol{q}} |^2 \equiv {\boldsymbol{q}} \bar{{\boldsymbol{q}}} = | z|^2 + |w|^2 \ ,$$ and the inverse $${\boldsymbol{q}}^{-1} \equiv \bar{{\boldsymbol{q}}} \frac{1}{|{\boldsymbol{q}}|^2} \ .$$ Moreover, the real part of a quaternion is defined as $$\Re({\boldsymbol{q}}) \equiv \bar{{\boldsymbol{q}}} + {\boldsymbol{q}} = \Re(z) = a \ .$$
Bulk and outliers of the spectrum of ${\boldsymbol{M}}$ {#sec:mudiff}
=======================================================
In this section, we want to show that the mean of ${\boldsymbol{A}}$ does not affect the bulk of eigenvalues of ${\boldsymbol{M}}$. We decompose the matrix $\bf{M}$ as $${\boldsymbol{M}} = {\boldsymbol{X}} ({\boldsymbol{D}} - \mu {\boldsymbol{I}} + {\boldsymbol{B}} + \mu {\boldsymbol{1}} )
\label{eq:matrix_decomposed}$$ where ${\boldsymbol{I}}$ is the identity matrix, ${\boldsymbol{1}}$ is a matrix of ones and ${\boldsymbol{D}} $ is the diagonal matrix consisting of the diagonal entries of $ {\boldsymbol{A}} $. Written in this way, ${\boldsymbol{B}}$ is a random matrix with mean zero and null diagonal. We will show that the bulk of the spectrum of ${\boldsymbol{M}}$ is equivalent to the bulk of eigenvalues of the matrix $ {\boldsymbol{J}} = {\boldsymbol{X}} ({\boldsymbol{D}} - \mu {\boldsymbol{I}}+ {\boldsymbol{B}}) $.
Using equation \[eq:matrix\_decomposed\], the resolvent of ${\boldsymbol{M}}$ can be written $$\operatorname*{\mathcal{G}}({\boldsymbol{q}}; {\boldsymbol{M}}) = \mathbb{E} [\frac{1} {S} \operatorname*{\text{tr}}\left( {\boldsymbol{q}} {\boldsymbol{I}} - {\boldsymbol{X}} {\boldsymbol{D}} - \mu {\boldsymbol{X}} - {\boldsymbol{X}} {\boldsymbol{B}} - \mu {\boldsymbol{X}} {\boldsymbol{1}} \right)^{-1}]
= \mathbb{E} [\frac{1} {S} \operatorname*{\text{tr}}\left( {\boldsymbol{q}} {\boldsymbol{I}} - {\boldsymbol{J}} - \mu {\boldsymbol{X}} {\boldsymbol{1}} \right)^{-1}].$$
Using the Sherman-Morrison formula, if ${\boldsymbol{Y}}$ and ${\boldsymbol{Y + Z}}$ are invertible matrices and ${\boldsymbol{Z}}$ has rank 1, then $$({\boldsymbol{Y}} + {\boldsymbol{Z}})^{-1} = {\boldsymbol{Y}}^{-1} + \frac{1}{1+ \operatorname*{\text{tr}}({\boldsymbol{Z}}{\boldsymbol{Y}}^{-1})} {\boldsymbol{Y}}^{-1} {\boldsymbol{Z}} {\boldsymbol{Y}}^{-1} \ .$$ Since $\mu {\boldsymbol{X}} {\boldsymbol{1}}$ has rank one, we have $$\left( {\boldsymbol{q}} {\boldsymbol{I}} - {\boldsymbol{J}} - \mu {\boldsymbol{X}}{\boldsymbol{1}} \right)^{-1} = (\textbf{q} {\boldsymbol{I}} - {\boldsymbol{J}})^{-1} +
\frac{1} {1+ \operatorname*{\text{tr}}(\mu {\boldsymbol{X}}{\boldsymbol{1}} (\textbf{q} {\boldsymbol{I}} - {\boldsymbol{J}})^{-1}) } (\textbf{q} {\boldsymbol{I}} - {\boldsymbol{J}})^{-1} \mu {\boldsymbol{X}}{\boldsymbol{1}} (\textbf{q} {\boldsymbol{I}} - {\boldsymbol{J}})^{-1} \ .
\label{inverse}$$ By introducing the linear operator $\langle \cdot \rangle$ defined by $\langle {\boldsymbol{C}} \rangle = \frac{1} {S} \operatorname*{\text{tr}}{{\boldsymbol{C}}} $ for an $S \times S $ matrix ${\boldsymbol{C}}$, we obtain $$\operatorname*{\mathcal{G}}({\boldsymbol{q}}; {\boldsymbol{M}}) = \operatorname*{\mathcal{G}}({\boldsymbol{q}}; {\boldsymbol{J}}) + \frac{\mu}{1+ S \mu \langle {\boldsymbol{X}}{\boldsymbol{1}} (\textbf{q} {\boldsymbol{I}} - {\boldsymbol{J}})^{-1}) \rangle }
\langle (\textbf{q} {\boldsymbol{I}} - {\boldsymbol{J}})^{-1} {\boldsymbol{X}} {\boldsymbol{1}} (\textbf{q} {\boldsymbol{I}} - {\boldsymbol{J}})^{-1} \rangle \ .
\label{eq:subleading}$$ In the limit of large $S$, the contribution from the second term in \[eq:subleading\] is subleading. Therefore, the resolvent of ${\boldsymbol{M}}$ converges to the resolvent of ${\boldsymbol{J}}$ when $S$ is large. In other words, as shown in Figure \[fig:M+J\], the bulks of the eigenvalues of ${\boldsymbol{M}}$ and ${\boldsymbol{J}}$ are the same — up to finite-size corrections.
The case $\sigma = 0$
=====================
When $\sigma = 0$, we derive the spectrum of a matrix ${\boldsymbol{Q}} = {\boldsymbol{X}} \left( {\boldsymbol{D}} + \mu {\boldsymbol{1}} \right)$. This case corresponds to setting ${\boldsymbol{B}} = 0$ in equation \[eq:matrix\_decomposed\]. As shown in the main text, the spectral density of the matrix ${\boldsymbol{Q}} $ is characterized by the presence of an outlier. In this section, we focus on the bulk of eigenvalues.
If we take $ {\boldsymbol{J}} = {\boldsymbol{X}} ( {\boldsymbol{D}} - \mu {\boldsymbol{I}} + {\boldsymbol{B}}) $ as before and set $\sigma = 0$, then ${\boldsymbol{B}} = 0 $, so that ${\boldsymbol{M}} = {\boldsymbol{Q}}$ and $ {\boldsymbol{J}} = {\boldsymbol{X}} ( {\boldsymbol{D}} - \mu {\boldsymbol{I}})$. The bulk of eigenvalues of ${\boldsymbol{Q}}$ and ${\boldsymbol{J}}$ will be the same. The resolvent of ${\boldsymbol{J}}$ in the case $\sigma = 0$ reads $$\begin{aligned}
G(\textbf{q}; {\boldsymbol{J}}) & = \frac{1} {S} \operatorname*{\text{tr}}\left( \textbf{q} - {\boldsymbol{J}} \right)^{-1} \\
& = \frac{1} {S} \operatorname*{\text{tr}}\left( \textbf{q} - {\boldsymbol{X}} {\boldsymbol{D}} \right)^{-1} \\
& = \frac{1} {S} \sum_{i = 1}^S \frac{1} {q - X_i D_i } \\
\label{resolventQ}
\end{aligned}$$ since $\textbf{q} - {\boldsymbol{X}} ( {\boldsymbol{D}} - \mu {\boldsymbol{I}})$ is symmetric. In the limit of large $S$, the sum in eq. \[resolventQ\] tends toward $ \mathbb{E} \left[(\textbf{q} - {\boldsymbol{X}} {\boldsymbol{D}} )^{-1} \right]$. If $P_{XD}(x,s)$ is the joint distribution of the entries of ${\boldsymbol{X}}$ and ${\boldsymbol{D}}$, we obtain $$\begin{aligned}
G(\textbf{q}; {\boldsymbol{J}}) & = \frac{1} {S} \sum_{i = 1}^S \frac{1} {q - X_iD_i + \mu X_i} \\
& = \mathbb{E} \left[(\textbf{q} - {\boldsymbol{X}} ( {\boldsymbol{D}} - \mu {\boldsymbol{I}}))^{-1} \right] \\
& = \int dx \ ds \ \frac{P_{XD}(x,s)} {q - x s} \ .
\label{spectrumQ}
\end{aligned}$$ In the case of a constant diagonal matrix ${\boldsymbol{D}} = d {\boldsymbol{I}}$, this equation simplifies to $$G(\textbf{q}; {\boldsymbol{J}}) = \int d x \ \frac{P_X(x)} {q - x d }
= \frac{1}{d}\int d y \ \frac{P_X\left(\frac{y}{d} \right)}{q - y} \\$$ with the change of variables $y = x d$. Using equation \[resolvent+spectrum\], we obtain that the spectral density $\varrho_{{\boldsymbol{J}}}(\lambda)$ will be $$\varrho_{{\boldsymbol{J}}}(\lambda) = \frac{1}{d} P_X \left(\frac{\lambda} { d} \right) \ .
\label{predictedQ}$$ In Figure \[fig:observationSI\], we plot the prediction from Equation \[predictedQ\] against the bulk of the spectrum of ${\boldsymbol{Q}} $ for two distributions of ${\boldsymbol{X}}$.
Derivation of the spectral density using the cavity method
==========================================================
In section \[sec:mudiff\], we showed that we can isolate the effect of $\mu \neq 0$. In this section, we use the cavity method [@Rogers2008; @Rogers2009] to derive the spectrum of the matrix $ {\boldsymbol{J}} = {\boldsymbol{X}} ( {\boldsymbol{D}} - \mu {\boldsymbol{I}} + {\boldsymbol{B}}) $, where ${\boldsymbol{D}}$ and ${\boldsymbol{X}}$ are two random diagonal matrices and ${\boldsymbol{B}}$ is a random matrix following the elliptic law.
We introduce the resolvent matrix $${\boldsymbol{G}} = ( {\boldsymbol{M}} - {\boldsymbol{q}} {\boldsymbol{I}} )^{-1} \ ,$$ The resolvent can be written as $$\operatorname*{\mathcal{G}}({\boldsymbol{q}}; {\boldsymbol{B}}) = \frac{1} {S} \operatorname*{\text{tr}}{\boldsymbol{G}} \ .$$ Note that each element of the resolvent matrix is a quaternion. In particular we will use the notation $${\boldsymbol{G}}_{ik} = \alpha_{ik} + \beta_{ik} j \equiv \left( \begin{matrix} \alpha_{ik} & \beta_{ik} \\ \bar{\beta_{ik}} & \bar{\alpha_{ik}} \\ \end{matrix} \right) \ ,$$ while $\operatorname*{\mathcal{G}}= \alpha + \beta j$, where $\alpha = \sum_i \alpha_{ii} / S$ and $\beta = \sum_i \beta_{ii} / S$. The cavity method allows us to compute the elements of ${\boldsymbol{G}}$ (and therefore the resolvent $\operatorname*{\mathcal{G}}$) if the matrix ${\boldsymbol{M}}$ has a tree structure [@Rogers2008; @Rogers2009]. It also allows to compute the spectral density for large, densely connected, random matrices [@Rogers2008; @Rogers2009; @Grilli2016]. In the limit of large $S$, for a densely connected matrix ${\boldsymbol{M}}$, the cavity equations read [@Rogers2009; @Grilli2016] $${\boldsymbol{G}}_{il} \equiv
\left( \begin{matrix} \alpha_{il} & \beta_{il} \\ \bar{\beta}_{il} & \bar{\alpha}_{il} \\ \end{matrix} \right) =
- \left(
\left( \begin{matrix} \lambda & \epsilon \\ \epsilon & \bar{\lambda} \\ \end{matrix} \right)
+ \sum_{jk}
\left( \begin{matrix} M_{ij} & 0 \\ 0 & M_{ji} \\ \end{matrix} \right)
\left( \begin{matrix} \alpha_{jk} & \beta_{jk} \\ \bar{\beta}_{jk} & \bar{\alpha}_{jk} \\ \end{matrix} \right)
\left( \begin{matrix} M_{kl} & 0 \\ 0 & M_{lk} \\ \end{matrix} \right)
\right)^{-1}
\ .$$ By introducing $${\tilde{\boldsymbol{M}}}_{ij} = \left( \begin{matrix} M_{ij} & 0 \\ 0 & M_{ji} \\ \end{matrix} \right)
\ ,$$ we obtain the more compact equation $${\boldsymbol{G}}_{il} = -\left(
{\boldsymbol{q}} + \sum_{jk} {\tilde{\boldsymbol{M}}}_{ij} {\boldsymbol{G}}_{jk} {\tilde{\boldsymbol{M}}}_{kl} \right)^{-1} \ .$$
Our goal is to find the resolvent for a random matrix of the form ${\boldsymbol{M}} = {\boldsymbol{X}} ({\boldsymbol{D}} + {\boldsymbol{B}})$, where ${\boldsymbol{X}}$ and ${\boldsymbol{D}}$ are diagonal matrices, while ${\boldsymbol{B}}$ is a random matrix following the elliptic law. We introduce the matrix $$\left( \begin{matrix}
-{\boldsymbol{q}} {\boldsymbol{I}} & {\tilde{\boldsymbol{X}}} \\
- {\tilde{\boldsymbol{D}}} - {\tilde{\boldsymbol{B}}} & {\boldsymbol{I}}
\end{matrix} \right) \ ,$$ which, when quaternions are represented as $2 \times 2$ matrices, is a $4S \times 4S$ matrix. In particular, this matrix is composed of $S^2$ $4 \times 4$ blocks with entries $$\left( \begin{matrix}
-\lambda \delta_{ij} & -\epsilon \delta_{ij} & X_{ii} \delta_{ij} & 0 \\
-\epsilon \delta_{ij} & -\bar{\lambda} \delta_{ij} & 0 & X_{ii} \delta_{ij} \\
-D_{ii} \delta_{ij} - B_{ij} & 0 & \delta_{ij} & 0 \\
0 & -D_{ii} \delta_{ij} - B_{ji} & 0 & \delta_{ij}
\end{matrix} \right) \ .$$
It is simple to observe that $${\boldsymbol{T}} = \left( \begin{matrix}
-\bf{q} & {\boldsymbol{X}} \\
{\boldsymbol{- A}} & {\boldsymbol{I}}
\end{matrix} \right)^{-1}
= \left( \begin{matrix}
\left( -{\boldsymbol{q}} {\boldsymbol{I}} + {\boldsymbol{X}} ({\boldsymbol{D}} + {\boldsymbol{B}}) \right)^{-1} & ... \\
... & ...
\end{matrix} \right)
= \left( \begin{matrix}
{\boldsymbol{G}} & ... \\
... & ...
\end{matrix} \right) \ .$$
If we write the cavity equation for ${\boldsymbol{T}}$, assuming dense matrices, we obtain $${\boldsymbol{T}}_{ii} = \left[ \left( \begin{matrix}
-{\boldsymbol{q}} & {\tilde{\boldsymbol{X}}}_{ii} \\
- {\tilde{\boldsymbol{D}}}_{ii} & 1
\end{matrix} \right)
- \sum_{j,k }
\left( \begin{matrix}
{\boldsymbol{0}} & {\boldsymbol{0}} \\
{\tilde{\boldsymbol{B}}}_{ij} & {\boldsymbol{0}}
\end{matrix} \right)
{\boldsymbol{T}}_{jk}
\left( \begin{matrix}
{\boldsymbol{0}} & {\boldsymbol{0}} \\
{\tilde{\boldsymbol{B}}}_{ki} & {\boldsymbol{0}}
\end{matrix} \right)
\right]^{-1} \ .$$
We can apply the law of large numbers to take the expectation of the matrices over ${\boldsymbol{B}}$. Using the following expectations over the elements of ${\boldsymbol{B}}$ $$\begin{split}
\mathbb{E}\left[ B_{ij} \right] &= 0 \\
\mathbb{E}\left[ (B_{ij} )^2 \right] & = \frac{\tilde{\sigma}^2}{S} \\
\mathbb{E}\left[ B_{ij} B_{ji} \right] & = \rho \frac{\tilde{\sigma}^2}{S} \ ,
\end{split}$$ we find that the non-diagonal terms of ${\boldsymbol{T}}$ go to zero in expectation. By introducing the notation $${\boldsymbol{T}}_{ii} \equiv
\left(
\begin{array} {cc}
{\boldsymbol{G}}_{ii} & {\boldsymbol{T}}^b_{ii} \\
... & ...
\end{array}
\right)$$ and $${\boldsymbol{T}}_{\star} \equiv \frac{1}{S} \sum_i {\boldsymbol{T}}_{ii} =
\left(
\begin{array} {cc}
\operatorname*{\mathcal{G}}& {\boldsymbol{T}}^b_{\star} \\
... & ...
\end{array}
\right) \ ,$$ the cavity equations for the diagonal terms read $${\boldsymbol{T}}_{ii}
= \left(
\begin{array} {cc}
-{\boldsymbol{q}} & X_{ii} \\
- D_{ii} - {\boldsymbol{t}} \circ {\boldsymbol{T}}^b_\star & {\boldsymbol{I}}
\end{array}
\right)^{-1}$$ where ${\boldsymbol{T}}_\star = \sum_i {\boldsymbol{T}}_{ii} / S $, while $\bf{t} = \tilde{\sigma}^2\rho + \tilde{\sigma}^2j$ with $\circ$ denoting the element-by-element matrix product (see section \[sec:quaternions\]).
We obtain a system of $2S$ quaternionic equations $$\begin{cases}
\displaystyle
{\boldsymbol{T}}^b_{ii} = - X_{ii} \left(-{\boldsymbol{q}} -X_{ii} \left( - D_{ii} - {\boldsymbol{t}} \circ {\boldsymbol{T}}^b_\star \right) \right)^{-1} \\
\displaystyle
{\boldsymbol{G}}_{ii} = \left( - {\boldsymbol{q}} - X_{ii} \left(- D_{ii} - {\boldsymbol{t}} \circ {\boldsymbol{T}}^b_\star \right) \right)^{-1} \ ,
\end{cases}$$ where $${\boldsymbol{T}}^b_\star = \frac{1}{S} \sum_i {\boldsymbol{T}}_{ii} \ .$$
Assuming that the elements of ${\boldsymbol{X}}$ and ${\boldsymbol{D}}$ are drawn from a given joint distirbution $P_{XD}$, we obtain the following two equations $$\begin{cases}
\displaystyle
{\boldsymbol{T}}^b_{\star} = - \int dx \ ds \ P_{XD}(x,s) x \left(-{\boldsymbol{q}} + x ( s + {\boldsymbol{t}} \circ {\boldsymbol{T}}^b_\star ) \right)^{-1} \\
\displaystyle
\operatorname*{\mathcal{G}}= \int dx \ ds \ P_{XD}(x,s) \left( - {\boldsymbol{q}} + x ( s + {\boldsymbol{t}} \circ {\boldsymbol{T}}^b_\star ) \right)^{-1} \ .
\end{cases}
\label{eq:tbstar}$$
At this point one can use ${\boldsymbol{T}}^b_{\star}= \alpha^b + \beta^b j $ and $\operatorname*{\mathcal{G}}= \alpha + \beta j$ to obtain $4$ equations of complex variables. It is simple to observe that $\beta^b = 0$ is always a solution and $\beta = 0$ if and only if $\beta^b = 0$. The solution $\beta = \beta^b = 0$ always corresponds to a null specral density [@Rogers2010]. The values of $\lambda$ for which a non-zero solution for $\beta$ exist correspond to the support of the spectral density. Setting $\beta^b \neq 0$ and $\epsilon = 0$, one obtains from equation \[eq:tbstar\] $$\begin{cases}
\displaystyle
\alpha^b = \int dx \ ds \ P_{XD}(x,s)
\frac{ x \left( \bar{\lambda} - s x - x \tilde{\sigma}^2 \rho \bar{\alpha}^b \right) }{
| \lambda + x \left( -s-\alpha^b \rho\tilde{\sigma}^2 \right) | ^2 + | x\tilde{\sigma}^2 \beta^b |^2 } \\
\displaystyle
1 = \int dx \ ds \ P_{XD}(x,s) \frac{ x^2 \tilde{\sigma}^2} {| \lambda + x \left( -s-\alpha^b \rho\tilde{\sigma}^2 \right) | ^2 + | x\tilde{\sigma}^2 \beta^b |^2 } \ .
\end{cases}\label{support:2}$$ By using the second equation, the system of equations further simplifies to $$\begin{cases}
\displaystyle
\alpha^b + \rho \bar{\alpha}^b = \int dx \ ds \ P_{XD}(x,s)
\frac{ x \left( \bar{\lambda} - s x \right) }{
| \lambda + x \left( -s-\alpha^b \rho\tilde{\sigma}^2 \right) | ^2 + | x\tilde{\sigma}^2 \beta^b |^2 } \\
\displaystyle
1 = \int dx \ ds \ P_{XD}(x,s) \frac{ x^2 \tilde{\sigma}^2} {| \lambda + x \left( -s-\alpha^b \rho\tilde{\sigma}^2 \right) | ^2 + | x\tilde{\sigma}^2 \beta^b |^2 } \ .
\end{cases}\label{support:3}$$
The values of $\lambda$ for which a solution of equations \[support:3\] exists are contained in the support of the spectral density. We assume that the solution $\beta^b$ of these equations vanishes at the boundaries of the support. In this case, the points at the boundaries of the support of the spectral density are the complex solutions $\lambda$ of $$\begin{cases}
\displaystyle
\alpha^b + \rho \bar{\alpha}^b = \int dx \ ds \ P_{XD}(x,s)
\frac{ x \left( \bar{\lambda} - s x \right) }{
| \lambda + x \left( -s-\alpha^b \rho\tilde{\sigma}^2 \right) | ^2 } \\
\displaystyle
1 = \int dx \ ds \ P_{XD}(x,s) \frac{ x^2 \tilde{\sigma}^2} {| \lambda + x \left( -s-\alpha^b \rho\tilde{\sigma}^2 \right) | ^2 } \ ,
\end{cases}
\label{eq:support}$$ where we used the second equation
We should note here that this method does not prove the convergence in any mode of the spectral bulk, but does yield a prediction for its support.
Support of the spectral density in the case $\rho = 0$ {#support-of-the-spectral-density-in-the-case-rho-0 .unnumbered}
======================================================
In the case $\rho = 0$, the two equations \[eq:support\] become independent, and the support is defined by the solutions of $$1 = \tilde{\sigma}^2 \int dx \ ds \ P_{XD}(x,s) \frac{ x^2 } {| \lambda - s x | ^2 } \ .$$ In the case $\tilde{\sigma}_d^2 = 0$, this equation further simplifies to $$1 = \tilde{\sigma}^2 \int dx \ P_{X}(x) \frac{ x^2 } {| \lambda - \mu_d x | ^2 } \ .$$ For instance, if $P_{X}$ is a uniform distribution on $[0,1]$, one can evaluate the integral, obtaining $$\begin{aligned}
1 & = \int_0^1 dx \ \frac{ x^2 \tilde{\sigma}^2} {| \lambda - \mu_d x | ^2 } \\
&= \frac{ \tilde{\sigma} ^ 2} { \mu_d^3} \left( 2 \lambda \mu_d - \mu_d ^ 2 + 2 \lambda (\mu_d - \lambda) \log \left| \frac{\lambda} {\lambda - \mu_d} \right| \right) \ .
\end{aligned}$$ The complex solutions $\lambda$ of this equation define the boundary of the support of the spectral density.
\[p\]
|
---
abstract: 'As an alternative to conventional magnetic field, the effective spin-orbit field in transition metals, derived from the Rashba field experienced by itinerant electrons confined in a spatial inversion asymmetric plane through the *s*-*d* exchange interaction, is proposed for the manipulation of magnetization. Magnetization switching in ferromagnetic thin films with perpendicular magnetocrystalline anisotropy can be achieved by current induced spin-orbit field, with small in-plane applied magnetic field. Spin-orbit field induced by current pulses as short as 10 ps can initiate ultrafast magnetization switching effectively, with experimentally achievable current densities. The whole switching process completes in about 100 ps.'
author:
- 'D. Wang'
title: 'Spin-orbit field switching of magnetization in ferromagnetic films with perpendicular anisotropy'
---
Ultrafast manipulation of magnetization is currently under intense investigation, partly driven by the ever increasing demand in information industry, partly inspired by the intriguing physics involved. Traditional methods use pulsed magnetic field to realize ultrafast switching of magnetization, through the spiral motion of magnetization in a magnetic field, applied in the inverse direction of the magnetization. However, due to domain wall instability [@kashuba06], ultrashort field pulses bring about stochastic behavior, thus imposing limitations on the ultimate switching speed [@tudosa04]. In practice, the limitation on this switching scheme is related to the difficulty in the generation of picosecond, strong magnetic field pulses, which entails the use of relativistic electron bunches nowadays. Precessional switching scheme, in which the magnetic field is applied perpendicular to the initial magnetization direction, circumvents this problem by maximizing the precession torque experienced by the magnetization [@back]. The deficit of precessional switching is manifested by the needed precise control of the pulse duration, on the time scale of the magnetization’s precession period. Instead of the conventional magnetic field, alternative means, such as light [@kirilyuk10], electric field [@ohno00] and electric current [@stt], can be used to manipulate magnetization. Recently, the effective spin-orbit field acting on the magnetization attracts much attention because of its potential applications. This spin-orbit field in transition metals results from the Rashba field [@bychkov84] experienced by itinerant electrons confined in a spatial inversion asymmetric potential through the *s*-*d* exchange interaction [@manchon]. Reversible switching of magnetization in perpendicularly magnetized Co nanodots was already demonstrated [@miron11], making the speculation of employing the spin-orbit field to control magnetization in ferromagnetic metals more than mere imagination, although the underlying mechanism responsible for the observed switching is still elusive. It is proposed, as will be shown in the following by macrospin simulation, that the pure spin-orbit field, in combination with the precessional motion induced by it, can explain qualitatively the observed experimental results . In addition, the feasibility of precessional switching utilizing the spin-orbit field will be addressed as well. It is found that, due to the large anisotropy and spin-orbit fields, both derived from the large spin-orbit coupling characteristic of systems with large perpendicular magnetocrystalline anisotropy (PMA), the switching time can be as short as 100 ps.
The prototype material system considered here is a trilayer Pt/Co 6 [Å]{}/AlO$_x$ nanodot, which is a representative of thin ferromagnetic metallic nanostructures with PMA. The strong perpendicular anisotropy results from the 3*d*-5*d* hybridization at the Pt/Co interface and the 3*d*-2*p* hybridization at the Co/AlO$_x$ interface [@rodmacq09]. The asymmetry of the top and bottom materials introduces a spin-orbit field for 3$d$ electrons confined in the thin Co layer. If current flows along the $x$ direction (c.f. Fig. 2 for the coordinate system used), and the trilayer structure lies in the $xy$ plane, then the spin-orbit field is **B**$_{so}$ = $-\alpha_{so}$ ($\hat{\textbf{z}}$ $\times$ **j**), where $\hat{\textbf{z}}$ is a unit vector along the $z$ axis, **j** is the current density, and $\alpha_{so}$ is the spin-orbit field constant, which is proportional to the spin-orbit coupling in Co. For the optimized thickness of Co considered here, $\alpha_{so}$ could be very large, $\alpha_{so}$ = 10$^{-12}$ T m$^2$/A [@miron10]. The induced large spin-orbit field by injecting high density current into the sample could have profound effects on the magnetization dynamics.
In the macrospin approximation, the uniform magnetization $\textbf{M}$ is treated as a macroscopic spin, whose dynamics is governed by the Landau-Lifshitz-Gilbert (LLG) equation [@llg] $$\label{llg} \frac{d\textbf{m}}{dt} = -\gamma \Big( (\textbf{m}
\times \textbf{B})+ \alpha\, \textbf{m} \times (\textbf{m} \times
\textbf{B})\Big),$$ where **m** = **M**/$M_s$ is the normalized magnetization vector ($M_s$ is the magnitude of **M**), $\gamma$ = 1.76 $\times$ 10$^{11}$ Hz/T is the free-electron gyromagnetic ratio, and $\alpha$ is the phenomenological Gilbert damping constant. The total magnetic field **B** = **B**$_a$ + **B**$_{app}$ + **B**$_{so}$ is a sum of the anisotropy (**B**$_a$), applied (**B**$_{app}$) and spin-orbit (**B**$_{so}$) fields. In the simulation, the current flow in Co is along the $x$ axis, while the magnetic field is applied in the $xz$ plane, 3$^\circ$ tilted away from the $x$ axis. The Gilbert damping is chosen to be $\alpha$ = 0.3 [@alpha]. The perpendicular anisotropy field has the form **B**$_a$ = $B_Km_z$$\hat{\textbf{z}}$, with $B_K$ = 0.92 T [@miron10]. To stabilize the perpendicular magnetization configuration, an external field $B_z$ = $\pm$ 5 mT is added to the total field, depending on the initial magnetization orientation.
{width="\linewidth"}
To investigate the effect of the spin-orbit field on the switching behavior, the time sequence for the applied field and current pulses, as shown in the insets of Fig. 1(a), is considered. Essentially, a hysteresis loop is simulated. But at each field value, current pulses with both polarities, positive and negative, are applied consecutively. The equilibrium magnetization direction after each pulse is then recorded. The current pulse is modelled by a 10 ns square wave with infinitely sharp rising and falling edges. The spin dynamics under the influence of current is dictated by the LLG equation, Eq. (\[llg\]). At the rising edge, due to the fact that the length of the current pulse is longer than the characteristic time scale of the magnetization dynamics triggered by the sudden application of current, the magnetization stops precessing far before the current pulse is terminated. Once the falling edge of the current pulse is reached, the magnetization vector will start precessing again, damping to a different equilibrium position, depending on the polarity of the current pulse. The $z$ component of the normalized magnetization, $m_z$, after positive and negative current pulses, as a function of the applied field, is shown in Figs. 1(a), 1(b) and 1(c). The current, or the corresponding spin-orbit field, effect can be clearly observed: When the current density is lower than 5 $\times$ 10$^{11}$ A/m$^2$ ($B_{so}$ = 0.5 T), only the coercivity is decreased (Fig. 1(a)). By increasing the current density to well above 1 $\times$ 10$^{12}$ A/m$^2$ ($B_{so}$ = 1 T), projection of the magnetization onto the $z$ axis is completely determined by the polarity of the current (Fig. 1(c)). Deterministic switching controlled by the polarity of current occurs. In the intermediate region, current controlled switching is effective only for a narrow field interval (Fig. 1(b)). Fig. 1(d) gives an overview of the different switching behavior of the magnetization, for current density ranging from 0 to 2 $\times$ 10$^{12}$ A/m$^2$. It can be seen that the coercivity decreases to zero with increasing current, while the maximum field for current induced switching remains almost constant, in agreement with experiment [@miron11].
{width="\linewidth"}
The physical mechanism responsible for the reversible, current induced switching can be understood by tracking the magnetization precession in time. In Fig. 2, two typical precession traces corresponding to $j$ = $\pm$1.5 $\times$ 10$^{12}$ A/m$^2$ and $B_{app}$ = 0.3 T are shown. The influence of the polarity of the current is obvious. It determines whether the magnetization will spiral upward or downward, initially. If the applied magnetic field is not too large, which means that the magnetization orientation pointing up or down is well separated, this initial discrepancy will lead to the difference in the final equilibrium position, i.e. whether the magnetization is pointing up or down. Effectively, the final orientation of the magnetization is defined by both the spin-orbit field and the applied field, through the cross product **B**$_{so}$ $\times$ **M** [@cross], which is nothing but the initial torque experienced by the magnetization when the current is turned off. This is consistent with the symmetry required by the perpendicular switching scheme [@miron11]. In the intermediate region (Fig. 1(b)), the spin-orbit torque is not large enough to induce switching by itself, applied field is required to overcome the action of anisotropy. If the applied field is not large enough, the equilibrium $m_z$ stays finite even in the presence of the spin-orbit field, because of the large PMA. When the current is removed, the magnetization never goes across the $xy$ plane, and switching could not occur. This explains why the spin-orbit torque induced switching is effective with large applied field, while there is no switching if the field is smaller than a critical value.
When the applied field is rotated away from the $x$ axis by an angle $\varphi>$ 0, the hysteretic behavior of the magnetization under the influence of current becomes asymmetric, because of the non-zero $y$ component of the applied magnetic field, $B_{app}
\sin \varphi$. For a positive current pulse to switch the magnetization, it has to overcome this positive $y$ field, making the current effect less efficient. The angular dependence of the maximum field (not shown) against which a positive current pulse can induce reversible switching supports this intuitive picture. But, in contrast to the experimental, linear relationship, theoretically, the dependence is determined by an almost quadratic relation, $B \propto \cos^2\varphi$. Nevertheless, the overall decrease in the switching efficiency when the applied field is rotated away from the current direction is observed unanimously.
{width="\linewidth"}
For the case of current flowing along the $x$ axis, the spin-orbit field is parallel to the $y$ axis. If the applied field is in the $xz$ plane, the magnetization is also in the $xz$ plane prior to the application of current pulses. This perpendicular configuration between the spin-orbit field and the magnetization maximizes the precession torque, thus facilitating precessional switching. Using a square-wave shaped current pulse, complete switching can be achieved in about 100 ps, as shown in Fig. 3. The length of the current pulse used in the simulation is 10 ps, which is about one half of the precession period corresponding to the spin-orbit field induced by the current pulse, with the current density $j$ = 1.5 $\times$ 10$^{12}$ A/m$^2$. The applied field is $B_{app}$ = 0.2 T. Due to the large spin-orbit field, the time needed to realize precessional switching is solely determined by the current density, whose direct consequence is the fact that a very short current pulse can effectively initiate the desired magnetization switching.
In the macrospin simulation, domain nucleation and the consequent domain wall motion, which is crucial for the actual determination of the coercivity, are neglected. Hence the simulated results are only of qualitative significance. However, as can be seen in Fig. 1, the qualitative agreement between the macrospin simulation and the experiment [@miron11] is satisfactory. Nevertheless, a detailed micromagnetic study, including finite temperature and finite size effects, is needed to gain further insight into the physics involved in the spin-orbit field induced reversible switching of magnetization in perpendicularly magnetized thin films. Experimentally, a thorough investigation of the magnetization dynamics following current excitation in such systems will prove to be important to clarify the role played by the spin-orbit field in manipulating the macroscopic state of magnetization. In Pt/Co/AlO$_x$ or similar systems, this can be achieved by time resolved magneto optical Kerr effect, which is already demonstrated to be a powerful technique for the study of magnetization dynamics in thin metallic magnetic films [@van; @Kampen02].
In summary, the spin-orbit field acting on the magnetization, mediated by the Rashba field experienced by itinerant electrons confined in a spatial inversion asymmetric plane, through the *s*-*d* exchange coupling, is proposed for the manipulation of magnetization. Perpendicular switching of magnetization in Pt/Co/AlO$_x$ nanodots, with in-plane applied field, can be realized using only the spin-orbit field, without the need of any extra fields. This simplifies the explanation for the experimental observation [@miron11]. Ultrafast switching, on the time scale of 100 ps, is made possible by the large magnitude of the spin-orbit field in systems with large PMA, such as Pt/Co/AlO$_x$. For perspectives, the spin-orbit field, properly tailored, can be used to coherently control spin oscillation and domain wall motion, in conjunction with the more familiar spin transfer torques, thus providing more freedom over the control of magnetization dynamics. The most recent experimental advance on this respect is the enhancement of domain wall velocity in perpendicularly magnetized Pt/Co/AlO$_x$ nanowires [@miron11dw]. Stimulated by the impetus from information technology, more advances are to be expected.
D.W. thanks group Physics of Nanostructures (FNA), Eindhoven University of Technology for hospitality. Enlightening discussions with Elena Mure and Sjors Schellekens are acknowledged. Constructive comments on the manuscript from Zengxiu Zhao and Jianmin Yuan are gratefully appreciated.
[99]{} A. Kashuba, Phys. Rev. Lett. **96**, 047601 (2006). I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann, J. Stöhr, G. Ju, B. Lu, and D. Weller, Nature **428**, 831 (2004). C. H. Back, D. Weller, J. Heidmann, D. Mauri, D. Guarisco, E. L. Garwin, and H. C. Siegmann, Phys. Rev. Lett. **81**, 3251 (1998); C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, D. Weller, E. L. Garwin, and H. C. Siegmann, Science **285**, 864 (1999). A. Kirilyuk, A. V. Kimel, and Th. Rasing, Rev. Mod. Phys. **82**, 2731 (2010). H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. Ohtani, Nature **408**, 944 (2000). L. Berger, Phys. Rev. B **54**, 9353 (1996); J. C. Slonczewski, J. Magn. Magn. Mater. **159**, L1 (1996). Yu. A. Bychkov and E. I. Rashba, J. Exp. Theor. Phys. Lett. **39**, 78 (1984). A. Manchon, and S. Zhang, Phys. Rev. B **78**, 212405 (2008); A. Manchon, and S. Zhang, *ibid* **79**, 094422 (2009); A. Matos-Abiague, and R. L. Rodriguez-Suarez, *ibid* **80**, 094424 (2009); I. Garate, and A. H. MacDonald, *ibid* **80**, 134403 (2009). I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Nature **476**, 189 (2011). B. Rodmacq, A. Manchon, C. Ducruet, S. Auffret, and B. Dieny, Phys. Rev. B **79**, 024423 (2009). I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella, Nature Mater. **9**, 230 (2010); P. Gambardella, and I. M. Miron, Phil. Trans. R. Soc. A **369**, 3175 (2011). L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, *Statistical Physics*, Part 2, 3rd ed. (Pergamon, Oxford), 1980; T. L. Gilbert, IEEE Trans. Mag. **40**, 3443 (2004). $\alpha$ = 0.3 is close to the experimenally determined Gilbert damping constant in a similar Pt/Co/AlO$_x$ sample. The detailed determination of the intrinsic $\alpha$ will be published elsewhere. Confusion can arise by simple reference to **M**. Miron *et al*. [@miron11] refer to the equilibrium magnetization without current in their expression, while **M** specifies the magnetization just prior to the removal of current here. For the sense of switching, the sign of the $z$ component of the cross product is of significance. Given that the two expressions, in which **M** has different meanings, produce the same sign for the $z$ component, we make no discrimination between them. They are both proportional to **B**$_{so}$ $\times$ **B**$_{app}$, if only the $z$ component is considered. M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W. J. M. de Jonge, and B. Koopmans, Phys. Rev. Lett. **88**, 227201 (2002). I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-Prejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, J. Vogel, M. Bonfim, A. Schuhl, and G. Gaudin, Nature Mater. **10**, 419 (2011).
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abstract: 'In this lecture I will introduce the concept of galactic chemical evolution, namely the study of how and where the chemical elements formed and how they were distributed in the stars and gas in galaxies. The main ingredients to build models of galactic chemical evolution will be described. They include: initial conditions, star formation history, stellar nucleosynthesis and gas flows in and out of galaxies. Then some simple analytical models and their solutions will be discussed together with the main criticisms associated to them. The yield per stellar generation will be defined and the hypothesis of instantaneous recycling approximation will be critically discussed. Detailed numerical models of chemical evolution of galaxies of different morphological type, able to follow the time evolution of the abundances of single elements, will be discussed and their predictions will be compared to observational data. The comparisons will include stellar abundances as well as interstellar medium ones, measured in galaxies. I will show how, from these comparisons, one can derive important constraints on stellar nucleosynthesis and galaxy formation mechanisms. Most of the concepts described in this lecture can be found in the monograph by Matteucci (2012).'
address: 'Department of Physics, Trieste University, Via G.B. Tiepolo 11, 34131 Triestwe, Italy'
author:
- Francesca Matteucci
title: Introduction to Galactic Chemical Evolution
---
Introduction
============
We call chemical evolution of galaxies the study of how the chemical elements formed in stars and were distributed in galaxies. During the Big Bang only the light elements (H, D, He, Li) were synthesized, while stars have been responsible for the formation and distribution of all the elements from carbon to uranium and beyond. Some light elements, such as Li, Be and B are formed during the spallation process, which is the interaction between the cosmic rays and C,N,O atoms present in the interstellar medium (ISM). Therefore, stars produce chemical elements in their interiors by means of nuclear reactions of fusion and then restore these elements into the ISM when they die. The fusion reactions occur up to $^{56}Fe$ which is the element with the maximum binding energy per nucleon. For elements heavier than $^{56}Fe$ the nuclear fusion is therefore inhibited and the nuclear fission is favored. The stars are born, live and die and they can die in a quiescent fashion like white dwarfs or violently as supernovae (SNe). The supernovae can be of various types: core-collapse SNe, namely the explosion of single massive stars ($M \ge 10 M_{\odot}$, Type II, Ib, Ic) , and Type Ia SNe, namely the explosion of a white dwarf occurring after accreting material from a companion in a binary system. In this lecture I will first describe the ingredients necessary to build galactic chemical evolution models. Such models can be analytical or numerical and they aim at following the evolution in time and space of the abundances of the chemical elements in the ISM. Then, I will focuse on some highlights in the galactic chemical evolution and on the comparison models-observations. In doing so, I will show how chemical evolution models can constrain stellar nucleosynthesis and galaxy formation timescales.
Basic ingredients for galactic chemical evolution
=================================================
The basic ingredients necessary to build a chemical evolution model are:
- [**Initial conditions**]{}
The initial conditions consist in deciding whether to assume an open or a closed model, in other words to decide if the gas, out of which the stars form, is present since the beginning or if it is accreted during the galactic lifetime; moreover, one should assume whether the initial gas has a primordial (only light elements from the Big Bang) or enriched chemical composition.
- [**Stellar birthrate function (SFRxIMF)**]{}
The stellar birthrate function is the history of star formation in a galaxy and it can be expressed as the product of the star formation rate (SFR) per the initial mass function (IMF), namely:
$$B(m,t)= \psi(t) \cdot \phi(m),$$
where the function $\psi(t)$ represents the SFR and it is generally assumed to be only a function of time, whereas $\phi(m)$ is the IMF which is assumed to be only a function of mass. The SFR represents how many solar masses go into stars per unit time, while the IMF is the initial stellar mass function describing the distribution of stars at birth as a function of the stellar mass. Clearly, these hypotheses are semplifications and we do not know whether the SFR is independent of mass and the IMF independent of time.
Parametrization of the SFR
--------------------------
The most common parametrization is the Schmidt (1959) law where the SFR is proportional to some power $k$ of the gas density. Kennicutt (1998) suggested $k=1.4 \pm 0.15$, as deduced by data relative to star forming galaxies. Other important parameters such as gas temperature, viscosity and magnetic field are usually ignored.
The Kennicutt SFR can be written as: $$\psi(t)= \nu \sigma_{gas}^{k},$$ where $\sigma_{gas}^{k}$ is the gas surface mass density and $\nu$ is the efficiency of star formation, namely the SFR per unit mass of gas.
{width="14pc"}
Parametrization of the IMF
--------------------------
The IMF, namely the number of stars born in the mass interval, m-m+dm, is generally expressed as a power law. It is possible to measure the IMF only in the solar vicinity since one needs to count the stars as functions of their magnitudes and it is still not possible to do it in external galaxies. Therefore, the only observational information we have is relative to the solar region of our Galaxy.
- [**Stellar nucleosynthesis**]{}
The stellar yields are defined as the masses of chemical elements produced by stars of different masses. These yields can represent both the newly formed elements or the elements already present in the star at its formation and restored into the ISM without being reprocessed. In particular, we can define the yield of a newly formed element, $p_{im}$, as the integral over the stellar lifetime of the mass loss rate multiplied by \[X(i)-Xo(i)\], where Xo(i) is the original abundance of the element $i$. To obtain the total stellar yield, it should be added to this term the mass ejected without processing, namely Xo(i) x mass lost, where with mass lost we intend the total mass ejected by a star into the ISM during its lifetime. Each stellar mass can produce and eject different chemical elements and the yields are therefore a function of the stellar mass but also of the original stellar metal content that we will indicate with $Z$. These yields are computed by means of detailed nucleosynthesis calculations taking into account all the main nuclear reactions in stars. Here we summarize briefly the element production in stars: i) low and intermediate mass stars (0.8-8 $M_{\odot}$): produce He, N, C and heavy s-process elements. They die as C-O white dwarfs, when single, and can die as Type Ia SNe when binaries. ii) Stars with $M<0.8M_{\odot}$ do not contribute to galactic chemical enrichment and have lifetimes longer than the Hubble time. iii) Massive stars ($M>8-10 M_{\odot}$): they produce mainly alpha-elements (O, Ne, Mg, S, Si, Ca), some Fe, light s-process elements and perhaps r-process elements (?) and explode as core-collapse SNe. However, r-process elements originating in neutron binary mergers seems to represent the most promising channel for r-process element production these days (see Matteucci et al, 2014 and references therein).
- [**Gas flows: infall, outflow, radial flows**]{}
In order to build a realistic galaxy one has to assume the presence of gas flows both in and out. The gas inflows are considered either as gas accretion or radial gas flows and they are influencing the chemical evolution of galaxies: in the case of accretion , usually assumed at a constant rate or exponentially decreasing in time, the main effect is to dilute the metal content. In the case of outflows or galactic winds also the effect is to decrease the metal concentration by decreasing the gas which is available for star formation. Galactic outflows are generally assumed to occur at a rate proportional to the star formation rate.
The yield per stellar generation
================================
The “yield per stellar generation” of a single chemical element, can be defined as (Tinsley 1980):
$$y_i={\int^{\infty}_{1}{mp_{im} \phi(m)dm} \over (1-R)}$$
where $p_{im}$ is the mass of the newly produced element $i$ ejected by a star of mass $m$, and $R$ is the returned fraction. The yield $y_i$ is therefore the mass fraction of the element $i$ newly produced by a generation of stars relative to the fraction of mass in remnants (white dwarfs, neutron stars and black holes) and never dying low mass stars ($M< 0.1M_{\odot}$).
We define “returned fraction” the fraction of mass ejected into the ISM by an entire stellar generation, namely: $$R= \int^{\infty}_{1}{(m-M_{rem}) \phi(m) dm}$$
The term fraction originates from the fact that both $y_i$ and $R$ are divided by the normalization condition of the IMF, namely: $$\int^{\infty}_{0.1}{m \phi(m)dm}=1.$$
In order to define $y_i$ and $R$ we have made a very specific assumption: the instantaneous recycling approximation (IRA), which states that [*all stars more massive than 1$M_{\odot}$ die instantaneously, while all stars less massive live forever*]{}. This assumption allows us to solve the chemical evolution equations analytically, as we will see in the following, but it is a very poor approximation for chemical elements produced on long timescales such as C, N and Fe. On the other hand, for oxygen, which is almost entirely produced by short lived core-collapse SNe, IRA is an acceptable approximation. In Figure 1 we show $y_O$ and $R$, computed for different initial metallicities of the stars and different IMFs. As one can see, the dependence of these quantities on Z is negligible whereas that on the IMF is strong. In that Figure it does not appear the Kroupa(2001) universal IMF, suggesting that the IMF in stellar clusters is an universal one. Kroupa (2001) IMF is a two-slope IMF with a slope for stars more massive than 0.5$M_{\odot}$ very similar to that of the Salpeter (1955) IMF, still widely used in model for external galaxies.
The Simple Model
================
The Simple Model of chemical evolution assumes that the system is evolving as a closed-box, without inflows or outflows, the IMF is constant in time, the chemical composition of the gas is primordial and the mixing between the chemical products ejected by stars and the ISM is instantaneous, plus IRA.
The initial gas mass is therefore, $M_{gas}(0)=M_{tot}=constant$, $\mu=M_{gas}/M_{tot}$ is the fractionary mass of gas, the metallicity is $Z=M_{Z}/M_{gas}$ is zero at the time t=0. The basic equations can be written as: $${dM_{gas} \over dt}= -\psi(t)\,\,+ \int^{\infty}_{m(t)}{(m-M_{R}) \psi(t-\tau_m) \varphi(m) dm}$$ which describes the evolution of gas and where the integral is the rate at which dying stars restore both the enriched and unenriched material into the ISM at the time $t$.
The equation for metals is:
$${d(ZM_{gas}) \over dt}=-Z\psi(t) \,\,+ \int^{\infty}_{m(t)}{[(m-M_{R})Z(t-\tau_m)+
mp_{Zm}] \psi(t-\tau_m) \varphi(m)dm},$$
where the first term in the square brackets represents the mass of pristine metals which are restored into the ISM without suffering any nuclear processing, whereas the second term contains the newly formed and ejected metals (Maeder 1992). When IRA is assumed, the SFR can be taken out of the integrals and the equation for metals be solved analytically. Its solution is: $$Z= y_{Z} ln({ 1 \over \mu}).$$
The metallicity yield per stellar generation $y_Z$ which appears in the above equation is known as [*effective yield*]{}, simply defined as the yield $y_{Z_{eff}}$ that would be deduced if the system were assumed to be described by the Simple Model.
Therefore, the effective yield is: $$y_{Z_{eff}}={Z \over ln(1/\mu)}.$$
Clearly, the true yield $y_Z$ will be always lower than the effective one in both cases of winds and infall of primordial gas. The only way to increase the effective yield is to assume an IMF more weighted towards massive stars than the canonical IMFs.
Primary and secondary elements
------------------------------
We define primary element an element produced directly from H and He A typical primary element is carbon or oxygen which originate from the 3-$\alpha$ reactions We define secondary element an element produced starting from metals already present in the star at birth (e.g. nitrogen produced in the CNO cycle). We recall that the solution of the chemical evolution equations for a secondary element, with abundance $X_S$, implies: $$X_S={ 1 \over 2}({y_S \over y_ZZ_{\odot}})Z^{2} ,$$ where $y_S$ is the yield per stellar generation of the secondary element. This means that, for a secondary element the simple closed-box model predicts that its abundance increases proportionally to the metallicity squared, namely:
$$X_S \propto Z^{2} .$$
Analytical solutions for outflow and inflow
===========================================
In the case of a model with outflow but no inflow occurring at a rate: $$W(t)=\lambda(1-R) \psi(t),$$ where $\lambda$ is a free parameter larger than or equal to zero, the solution for the metallicity of the system is:
$$Z={y_Z \over (1+ \lambda)}ln[(1 +\lambda) \mu^{-1}- \lambda].$$
It is clear that in the case of $\lambda=0$ the solution is the same as that of the Simple Model.
In the case of a model without outflow but inflow of primordial gas ($Z_{inf}=0$), occurring at a rate: $$A(t)=\Lambda (1-R) \psi(t)$$ with $\Lambda$ a positive constant different from zero and from 1, the solution is: $$Z= {y_Z \over \Lambda}[1-(\Lambda-(\Lambda-1)\mu^{-1})^{-\Lambda/(1-\Lambda)}].$$ Again, if $\Lambda=0$, the solution is the same as that of the Simple Model. The case $\Lambda=1$ is a particular one, called “extreme infall” and it has a different solution: $$Z = y_Z (1-e^{- \beta}),$$ where $\beta=\mu^{-1}-1$. For a more extensive discussion of these and other analytical solutions see Matteucci (2012).
Numerical models
================
When the stellar lifetimes are correctly taken into account the chemical evolution equations should be solved numerically. This allows us to follow in detail the temporal evolution of the abundances of single elements.
A complete chemical evolution model in the presence of both galactic wind, gas infall and radial flows can be described by a number of equations equal to the number of chemical species: in particular, if $M_i$ is the mass of the gas in the form of any chemical element $i$, we can write the following set of integer-differential equations which can be solved only numerically, if IRA is relaxed:
$$\begin{aligned}
\dot M_i(t) = -\psi(t)X_i(t)+
\int_{M_{L}}^{M_{Bm}}{\psi(t-\tau_m)
Q_{mi}(t-\tau_m)\varphi(m)dm}+\\ \nonumber
A_B\int_{M_{Bm}}^{M_{BM}}{\varphi(m)} \bigl[\int_{\mu_{B_{min}}}
^{0.5}{f(\mu_B)\psi(t-\tau_{m2}) Q_{mi}(t-\tau_{m2})d\mu_B \bigr]
dm}+\\ \nonumber (1-A_B)\int_{M_{Bm}}^
{M_{BM}}{\psi(t-\tau_{m})Q_{mi}(t-\tau_{m)}\varphi(m)dm}+\\
\nonumber \int_{M_{BM}}^{M_U}{\psi(t-\tau_m)Q_{mi}(t-\tau_m)
\varphi(m)dm} + X_{iA}(t) A(t)\\ \nonumber
- X_{i}(t) W(t) + X_{i}(t)I(t),\end{aligned}$$
where $M_i$ can be substituted by $\sigma_i$, namely the surface gas density of the element $i$. In several models of chemical evolution it is customarily to use normalized variables which should be substituted to $M_i(t)$ or to $\sigma_i(t)$, such as for example: $$G_i(t)= {\sigma_{i}(t) \over \sigma_{tot}(t_G)}= {M_i(t) \over M_{tot}(t_G)},$$ with $\sigma_{i}(t)=X_i(t) \sigma_{gas}(t)$, and $\sigma_{tot}(t_G)$ ($M_{tot}(t_G)$) being the total surface mass density (mass) at the present time $t_G$. The surface densities are more indicated for computing the chemical evolution of galactic disks, while for spheroids one can use the masses. The quantity $X_i(t) = {\sigma_i(t) \over \sigma_{gas}(t)}= {M_i(t) \over M_{gas}(t)}$ represents the abundance by mass of the element $i$ and by definition the summation over all the mass abundances of the elements present in the gas mixture is equal to unity. These equations include the products from Type Ia SNe (third term on the right) and the products of stars ending their lives as white dwarfs and core-collapse SNe (for a more extensive description of the equations see Matteucci, 2012).
Some highlights
===============
In this section we will illustrate some examples where predictions from chemical evolution models are compared to observations.
The chemical evolution of the Milky Way
---------------------------------------
We will start with the Milky Way, which is the galaxy for which we possess the majority of information. A good model for the chemical evolution of the Milky Way should reproduce several constraints, including the G-dwarf metallicity distribution, the \[X/Fe\] vs. \[Fe/H\][^1] relations (where X is a generic element from carbon up to the heaviest ones), abundance and gas gradients along the Galactic disk, among others.
In Figure 2 we show the predictions from Romano et al. (2010) concerning several chemical elements obtained by using different sets of stellar yields and compared to observation in stars. As one can see, the agreement is good for some chemical species whereas for others the agreement is still very poor. The reason for that probably resides in the uncertainties still existing in the theoretical stellar yields. The shapes of the \[X/Fe\] vs. \[Fe/H\] relation can be successfully interpreted as due to the [*time-delay model*]{}, namely the fact that elements such as C, N and Fe are mainly produced by long living stars, while others such as $\alpha$-elements (O, Mg, Si, Ca) are produced by short living stars. In particular, Fe is mainly produced by Type Ia SNe and only a small fraction of it originates in core-collapse (CC) SNe. Type I a SNe explode on longer timescales than core-collapse SNe and therefore ratios such as \[$\alpha$/Fe\] , where $\alpha$ indicates $\alpha$-elements which are mainly produced in CC SNe, can be used as cosmic clocks. The higher than solar value of the \[$\alpha$/Fe\] ratios for low \[Fe/H\] values, is then due to the CC SNe which restore the $\alpha$-elements on short timescales. When Type Ia SNe, originating from CO white dwarfs which have longer lifetimes (from 30 Myr up to a Hubble time), start restoring the bulk of Fe, then the \[$\alpha$/Fe\] ratios start decreasing. By means of the time-delay model we can interpret any abundace ratio.
{width="20pc"}
The model underlying the predictions of Figure 2 is an updated version of the two-infall model by Chiappini et al. (1997): this model assumes that the Milky Way formed by means of two main gas accretion episodes, one during which the halo and thick disk formed and another during which the thin disk formed on much longer timescales.
The Li-problem
--------------
In Figure 3 we show an illustration of the so-called Li-problem. In particular, we report the abundance of $^{7}Li$ (A(Li) = 12 +log(Li/H) versus \[Fe/H\]). The data sources are indicated inside the Figure. The upper envelope of the data should represent the evolution of the abundance of $^{7}Li$ in the ISM since the stars in the upper envelope should exhibit the Li abundance present in the gas when they formed. The stars below should instead have consumed the original Li which is very fragile and tends to react with protons to form $^{4}He$ when the temperature is $>2.8 \cdot 10^{6}$ K. According to this interpretation of the diagram, the younger stars with higher \[Fe/H\] should reflect the fact that the Li abundance has increased with galactic lifetime owing to Li stellar production by stars. Stars which can create, preserve and eject Li into the ISM are: low and intermediate mass stars during the asymptotic giant branch (AGB) phase, the CC SNe, cosmic rays and perhaps novae (see Izzo et al. 2015 for the possible detection of Li in a nova). The Li problem arises from the fact that both WMAP and Planck experiments have derived a primordial $^{7}Li$ abundance higher than the abundance found in the upper envelope of the Galactic halo stars (those with \[Fe/H\]$<$ -1.0 in Figure 3). This could mean that the oldest Galactic stars have consumed their primordial Li and roughly by the same amount for stars in the range -3.0 — -1.0 in \[Fe/H\] (the so-called Spite plateau). The lines in Figure 3 represent the predictions of a chemical evolution model (the two-infall model) including SNe, novae, cosmic rays and AGB stars as Li producers and starting from the primordial Li abundance deduced either by WMAP or by the value of the Spite plateau. There is not yet a solution for the Li-problem.
![A(Li) vs. \[Fe/H\] for solar vicinity stars and meteorites (symbols; see legend) compared to the predictions of chemical evolution models (lines and colored areas). The chemical evolution model is from Romano et al. (2010).The black line represents all the stellar Li producers, as described in the text. The red line is the same model but without novae. Note that the dotted theoretical curve starts with the primordial Li value from WMAP which lies well above the Li abundances of halo stars.Figure from Izzo et al. (2015).](litio_fine.eps){width="20pc"}
The Ultra-Faint Dwarfs (UFDs)
-----------------------------
In Figure 4 we show the model predictions and data for the UFD galaxy Hercules, a small (with stellar mass of $\sim 10^{7}M_{\odot}$) satellite of the Milky Way. The data refer to the \[Ca/Fe\] ratios measured in the stars of this galaxy and include also the data for Ca in the Milky Way and dwarf spheroidal stars. The data show that the Milky Way and Hercules evolved in a different way, since the \[Ca/Fe\] ratios are lower at the same \[Fe/H\] in the UFD galaxy, relative to the Milky Way. The models, computed with an efficiency of star formation one thousand times lower than assumed for the Milky Way can well reproduce the trend. The model for Hercules includes also a strong galactic wind proportional to the SFR. Because of the already mentioned time-delay model, the Fe delayed production coupled with a regime of slow star formation predict that the \[Ca/Fe\] ratios are lower. at the same \[Fe/H\], since when the Type Ia SNe start to be important in producing Fe, the Ca and the other metals produced by CC SNe have not yet attained the same abundances as in the Miilky Way which evolves with a much faster star formation.
![Theoretical and observed \[Ca/Fe\] ratios as functions of \[Fe/H\]. The red stars represent abundances in stars of Hercules, while the blue points represent dwarf spheroidals and black points the Milky Way. The red lines are models assuming a star formation efficiency ($\nu$) varying in the range 0.002-0.008 $Gyr^{-1}$. Figure from Koch et al. (2012).](Koch.eps){width="14pc"}
Therefore, in order to interpret the abundance patterns in galaxies we should consider the history of star formation, since it affects strongly the shape of the \[X/Fe\] vs. \[Fe/H\] relations. In this way, we can reconstruct the history of stars formation in galaxies just by looking at their abundances. This approach is known as [*astroarchaeological approach*]{}. From Figure 4 it apprears that the dwarf spheroidals have a higher star formation efficiency than the UFDs but a lower one than the Milky Way. From this Figure it is difficult to conclude that stars of the UFDs could have been the building blocks of the Galactic halo, but more data on UFDs are necessary before drawing firm conclusions on this point.
Acknowledgments
---------------
I acknowledge financial support from the PRIN2010-2011 project “The Chemical and Dynamical Evolution of the Milky Way and Local Group Galaxies”, prot.2010LY5N2T.
References {#references .unnumbered}
==========
[9]{}
Bonifacio, P. et al. 2007 [*A&A*]{} [**470**]{} 153
Chabrier, G. 2003 [*PASP*]{} [**115**]{} 763)
Charbonnel, C. & Primas, F. 2005 [*A&A*]{} [**442**]{} 961
Karakas, A.I. 2010 [*MNRAS*]{} [**403**]{} 1413
Kobayashi, C. et al. 2006 [*ApJ*]{} [**653**]{} 1145
Koch, A., Matteucci, F., & Feltzing, S. 2012 [*American Institute of Physics Conference Series*]{} [**1480**]{} 190
Kroupa, P: et al. 1993 [*MNRAS*]{} [**262**]{} 545
Kroupa, P. 2001 [*MNRAS*]{} [**322**]{} 231
Izzo et al. 2015 [*ApJL*]{} [**808**]{} L14
Lambert, D.L. & Reddy, B.E. 2004 [*MNRAS*]{} [**349**]{} 757
Lodders et al. 2009 [*Landolt B[ö]{}rnstein*]{} [**44**]{}
Maeder, A. 1992 [*A&A*]{} [**264**]{} 105
Matteucci, F. 2012 “Chemical Evolution of Galaxies” [*Astronomy and Astrophysics Library.Springer-Verlag Berlin Heidelberg*]{} 2012
Matteucci, F. et al. 2014 [*MNRAS*]{} [**438**]{} 3177
Ramirez, I. et al. 2012 [*ApJ*]{} [**756**]{} 46
Romano et al. 2010 [A&A]{} [**522**]{} 32
Salpeter, E.E. 1955 [*ApJ*]{} [**121**]{} 161
Sbordone et al. 2010 [*A&A*]{} [**522**]{} 26
Tinsley, B.M. 1980 [*Fund. Cosmic Physics*]{} [**5**]{} 287
van den Hoeck, L.B. & Groenewegen, M.A.T. 1997 [*A&ASuppl*]{} [**123**]{} 305
Vincenzo, F. et al. 2015 [*arXiv:1503.08300*]{}
Woosley, S.E. & Weaver, T.A. 1995 [*ApJS*]{} [**101**]{} 181
[^1]: \[Fe/H\]= $log(Fe/H)_* - log(Fe/H)_{\odot}$
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abstract: 'Shadow detection is a fundamental and challenging task, since it requires an understanding of global image semantics and there are various backgrounds around shadows. This paper presents a novel network for shadow detection by analyzing image context in a direction-aware manner. To achieve this, we first formulate the direction-aware attention mechanism in a spatial recurrent neural network (RNN) by introducing attention weights when aggregating spatial context features in the RNN. By learning these weights through training, we can recover direction-aware spatial context (DSC) for detecting shadows. This design is developed into the DSC module and embedded in a CNN to learn DSC features at different levels. Moreover, a weighted cross entropy loss is designed to make the training more effective. We employ two common shadow detection benchmark datasets and perform various experiments to evaluate our network. Experimental results show that our network outperforms state-of-the-art methods and achieves 97% accuracy and 38% reduction on balance error rate.'
author:
- |
Xiaowei Hu$^{1, \ast}$, Lei Zhu$^{2,}$[^1] , Chi-Wing Fu$^{1,3}$, Jing Qin$^{2}$, and Pheng-Ann Heng$^{1, 3}$\
$^1$ Department of Computer Science and Engineering, The Chinese University of Hong Kong\
$^2$ Centre for Smart Health, School of Nursing, The Hong Kong Polytechnic University\
$^3$ Guangdong Provincial Key Laboratory of Computer Vision and Virtual Reality Technology,\
Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, China
bibliography:
- 'egbib.bib'
title: 'Direction-aware Spatial Context Features for Shadow Detection'
---
Acknowledgments {#acknowledgments .unnumbered}
===============
The work is supported by the National Basic Program of China, 973 Program (Project no. 2015CB351706), the Research Grants Council of the Hong Kong Special Administrative Region (Project no. CUHK 14225616), the Shenzhen Science and Technology Program (No. JCYJ20170413162617606), the CUHK strategic recruitment fund, and the Innovation and Technology Fund of Hong Kong (Project no. ITS/304/16). We thank reviewers for their valuable comments, Michael S. Brown for his discussion, and Minh Hoai Nguyen for sharing their results. Xiaowei Hu is funded by the Hong Kong Ph.D. Fellowship.
[^1]: Joint first authors
|
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abstract: 'The combination of high optical nonlinearity in the electromagnetically induced transparency (EIT) effect and strong electric dipole-dipole interaction (DDI) among the Rydberg-state atoms can lead to important applications in quantum information processing and many-body physics. One can utilize the Rydberg-EIT system in the strongly-interacting regime to mediate photon-photon interaction or qubit-qubit operation. One can also employ the Rydberg-EIT system in the weakly-interacting regime to study the Bose-Einstein condensation of Rydberg polaritons. Most of the present theoretical models dealt with the strongly-interacting cases. Here, we consider the weakly-interacting regime and develop a mean field model based on the nearest-neighbor distribution. Using the mean field model, we further derive the analytical formulas for the attenuation coefficient and phase shift of the output probe field. The predictions from the formulas are consistent with the experimental data in the weakly-interacting regime, verifying the validity of our model. As the DDI-induced phase shift and attenuation can be seen as the consequences of elastic and inelastic collisions among particles, this work provides a very useful tool for conceiving ideas relevant to the EIT system of weakly-interacting Rydberg polaritons, and for evaluating experimental feasibility.'
author:
- 'Shih-Si Hsiao$^1$'
- 'Ko-Tang Chen$^1$'
- 'Ite A. Yu$^{1,2,}$'
title: ' Mean field theory of weakly-interacting Rydberg polaritons in the EIT system based on the nearest-neighbor distribution'
---
Introduction
============
The effect of electromagnetically induced transparency (EIT) involving Rydberg-state atoms is of great interest currently. The Rydberg-state atoms exhibit the strong electric dipole-dipole interaction (DDI) among each others [@blockade_Zoller2000; @blockade_Gould2004; @blockade_Pfau2007; @SaffmanRMP; @blockade_Pohl2013]. On the other hand, the EIT effect not only provides high optical nonlinearity for the atom-light interaction, but also gives rise to slow, stored, and stationary light for long interaction time [@EIT_Fleischhauer2005; @EIT_OurPRL2006; @SLP_OurPRL2012; @EIT_YFChen2016; @EIT1; @EIT2; @EIT3; @EIT4; @EIT5]. Thus, the combination of the strong DDI of Rydberg atoms and the high optical nonlinearity of EIT can efficiently mediate the interaction between photons via Rydberg polaritons in the dipole blockade regime, where the Rydberg polariton is the collective excitation involving the light and the atomic coherence between the ground and Rydberg states [@DSP_Fleischhauer2000; @DSP_Fleischhauer2002]. The Rydberg-EIT mechanism can lead to the applications of quantum optics and quantum information processing [@REIT_Adams2010; @REIT_Fleischhauer2011; @photon_interaction_Lukin2011; @REIT_Fleischhauer2015; @REIT_Lukin2012; @REIT_Hofferberth2016; @simulator_Lukin2018; @simulator_Lukin2017; @SP_transistor_Rempe2014; @SP_switch_; @Rempe2014; @SP_transistor_Hofferberth2014; @gate_Lukin2019; @gate_Rempe2019; @SP_Pfau2018; @XPM_Rempe2016; @Ruseckas2017].
To our knowledge, most of the present theoretical models dealt with the Rydberg-EIT system in the strongly-interacting regime, i.e., $r_B^3$ is comparable to $r_a^3$, where $r_B$ is the blockade radius and $r_a$ is the half mean distance between Rydberg polaritons. In Ref. [@REIT_Adams2010], J. D. Pritchard [*et al*]{}. utilized the $N$-atom model to analyze experiment phenomena of the optical nonlinearity and attenuation in the Rydberg-EIT system. In Ref. [@REIT_Fleischhauer2011], D. Petrosyan [*et al*]{}. modeled the propagation of light field in strongly-interacting Rydberg-EIT media by considering the superatoms with the volume of the blockade sphere. In Ref. [@photon_interaction_Lukin2011], A. V. Gorshkov [*et al*]{}. proposed a theory for the propagation of few-photon pulses in the system of strongly-interacting Rydberg polaritons. In Ref. [@REIT_Fleischhauer2015], M. Moos [*et al*]{}. utilized a one-dimensional model to describe the time evolution of Rydberg polaritons and analyze many-body phenomena in the strongly-interacting regime. In Ref. [@Ruseckas2017], J. Ruseckas [*et al*]{}. proposed a method to create two-photon states by making pairs of Rydberg atoms entangled during the storage.
In this article, we considered the weakly-interacting Rydberg-EIT system, and developed a mean field model to describe the attenuation and phase shift of the output probe field induced by the DDI effect. The Rydberg-EIT system is depicted in Fig. \[fig:Transition\](a), and the weakly-interacting condition requires $r_B^3 \ll r_a^3$ \[see Fig. \[fig:Transition\](b)\]. Under such condition, the system of Rydberg polaritons can be considered as nearly the ideal gas. Thus, the nearest-neighbor distribution (NND) shown by Ref. [@NNDistribution] is utilized in our model. The DDI-induced frequency shift between nonuniformly-distributed Rydberg excitations results in the effective phase shift and attenuation of light field. With the probability function of NND and the atom-light coupling equations of EIT system, we calculated the mean field results of transmission and phase shift spectra, and further derived the analytical formulas of the DDI-induced attenuation coefficient and phase shift. The theoretical predictions from the formulas are in good agreement with the experimental data in Ref. [@OurExp]. In the experiment of Ref. [@OurExp], we utilized the Rydberg state of a low principal number, the laser-cooled ensemble of a moderate atomic density, and the weak probe field of a low photon flux to make the mean number of Rydberg polaritons within the blockade sphere lower than 0.1. The good agreement verifies our model.
![(a) Transition diagram of the Rydberg-EIT system. $|1\rangle$, $|2\rangle$, and $|3\rangle$ represent the ground, Rydberg, and intermediate states. The weak probe and strong coupling fields form the ladder-type EIT configuration. (b) Top and bottom figures depict the strongly- and weakly-interacting systems. Red and blue balls represent the atoms with and without Rydberg excitations; dashed circles indicate the blockade spheres. As an example, let’s consider 8 photons in both systems. There are 8 Rydberg excitations or polaritons in the weakly-interacting system, but only 4 in the strongly-interacting system due to the dipole blockade effect.[]{data-label="fig:Transition"}](_Fig1){width="\columnwidth"}
Rydberg polaritons are regarded as bosonic quasi-particles, and the DDI-induced phase shift and attenuation coefficient can infer the elastic and inelastic collision rates in the ensemble of these particles [@DSP_BEC_Fleischhauer2008]. Weakly-interacting Rydberg polaritons assisted by a long interaction time of the EIT effect can be employed in the study of many-body physics such as the Bose-Einstein condensation of polaritons [@EPBEC_nature2006; @EPBEC_science2007; @DSP_BEC_Fleischhauer2008; @EPBEC_RMP2010]. The mean field theory developed in this work provides a useful tool to conceive ideas relevant to weakly-interacting EIT-based Rydberg polaritons and to evaluate feasibilities of experiments.
We organize the article as follows. In Sec. II, the theoretical model based on the probability function of NND, the atom-light coupling equations of the EIT system, and the ensemble average of the DDI-induced frequency shift are introduced. We obtain the mean field results of the real and imaginary parts of the steady-state absorption cross section of the probe field. In Sec. III, we numerically evaluate the integrals corresponding to the mean field results and present the spectra of transmission and phase shift of the output probe field. The DDI-induced phenomena observed from the spectra are discussed and explained. In Sec. IV, we derive the analytical formulas of the DDI-induced attenuation coefficient and phase shift. From the formulas, one can see how the DDI effects depend on the system parameters such as the optical depth, coupling and probe Rabi frequencies, coupling detuning, two-photon detuning, and decoherence rate. It is interesting to note that the DDI effects exhibit the asymmetric behavior with respect to the coupling detuning. In Sec. V, we briefly describe the experimental condition and data in Ref. [@OurExp], and calculate the predictions corresponding to the experimental condition from the analytical formulas. The predictions are in good agreement with the data. Finally, we give a summary in Sec. VI.
Theoretical Model
=================
In the system of Rydberg polaritons, the DDI induces a frequency shift of the Rydberg state. Since Rydberg excitations are nonuniformly distributed, the Rydberg-state frequency shift is not a constant in the medium. The system of low-density Rydberg excitations can be considered as the ideal gas, in which the nearest-neighbor distribution is given by [@NNDistribution] $$P(r)=\frac{3r^{2}}{r_{a}^{3}}e^{-r^{3}/r_{a}^{3}},
\label{eq:P(r)}$$ where $P(r)$ is the probability density, i.e., $P(r)dr$ is the probability of finding a particle’s nearest neighbor locating at the distant between $r$ and $r+dr$, and $r_{a}$ is the half mean distance between particles. The definition of $r_a$ is $$r_{a} \equiv \left( \frac{3}{4\pi n_R} \right)^{1/3},
\label{eq:Define_r_a}$$ where $n_R$ is the Rydberg-polariton density. Figure \[fig:Distribution\](a) shows $P(r)$ as a function of $r$.
The frequency shift of a Rydberg state induced by the DDI is $C_6/r^6$, where $C_6$ is the van der Waals coefficient [@C6_Saffman2008] and $r$ represents the distance between two particles. In the ensemble of Rydberg excitations, the Rydberg-state frequency shift consists of two parts. The first part is $C_6/r^6$ contributed from the nearest-neighbor Rydberg excitation at the distance $r$, and the second part is $n_R \int_{r}^{\infty} (C_6/r'^6) 4\pi r'^2 dr'$ contributed from all the other Rydberg excitations outside the sphere of the radius $r$. Here, we consider the particles in the second part are uniformly distributed. Thus, the Rydberg-state frequency shift is the following: $$\omega=\frac{C_6}{r^6}+\frac{C_6}{r_a^3 r^3}.
\label{eq:omega}$$ Using Eqs. (\[eq:P(r)\]) and (\[eq:omega\]), we can obtain frequency shift distribution $P(\omega)$, i.e., $P(\omega)d\omega$ is the probability of finding the Rydberg-state frequency shifted by the amount between $\omega$ and $\omega+d\omega$, given by $$\begin{aligned}
P(\omega) = &\frac{1}{\omega_a}
\frac{\left[ 1+\sqrt{1+4(\omega/\omega_a)} \right]^2}
{4 (\omega/\omega_a)^2 \sqrt{1+4(\omega/\omega_a)}} \\
&\times\exp{\left[ -\frac{1+\sqrt{1+4(\omega/\omega_a)}}{2(\omega/\omega_a)} \right]}, \\
\end{aligned}
\label{eq:P(omega)}$$ where $$\omega_a \equiv |C_6| /r_a^6.
\label{eq:Define_omega_a}$$ Since the value of distance, $r$, is always positive, only $\omega \geq 0$ is valid in $P(\omega)$. Figure \[fig:Distribution\](b) shows $P(\omega)$ as a function of $\omega$.
![(a) Probability density $P(r)$ as a function of distance $r$ in the nearest-neighbor distribution. Units of $r_a$ defined by Eq. (\[eq:Define\_r\_a\]) is the half mean distance between particles. (b) Probability density $P(\omega)$ as a function of frequency shift $\omega$. Units of $\omega_a$ defined by Eq. (\[eq:Define\_omega\_a\]) is the frequency shift corresponding to $r_a$.[]{data-label="fig:Distribution"}](_Fig2){width="\columnwidth"}
We utilize the optical Bloch equation (OBE) for the time evolution of atomic density matrix and the Maxwell-Schrödinger equation (MSE) for the propagation of probe field in the theoretical model. In the EIT system here, the weak probe drives the transition between the ground state $|1\rangle$ and the intermediate state $|3\rangle$, and the strong coupling field drives that between $|3\rangle$ and the Rydberg state $|2\rangle$. The two transitions form the ladder configuration. The OBE and MSE are shown below. $$\begin{aligned}
\label{eq:OBE_rho21}
\frac{\partial}{\partial t}\rho_{21} &=&
\frac{i}{2}\Omega_{c}\rho_{31} +i\delta\rho_{21}
-\left( \gamma_0+\frac{\Gamma_2}{2} \right) \rho_{21}, \\
\label{eq:OBE_rho31}
\frac{\partial}{\partial t}\rho_{31} &=&
\frac{i}{2}\Omega_{p} +\frac{i}{2}\Omega_{c}\rho_{21}
+i\Delta_{p}\rho_{31} -\frac{\Gamma}{2}\rho_{31}, \\
\label{eq:OBE_rho22}
\frac{\partial}{\partial t}\rho_{22} &=&
\frac{i}{2}\Omega_{c}\rho_{32}-\frac{i}{2}\Omega_{c}\rho_{32}^*
-\Gamma_2\rho_{22},\\
\label{eq:OBE_rho32}
\frac{\partial}{\partial t}\rho_{32} &=&
\frac{i}{2}\Omega_{p}\rho_{21}^* +\frac{i}{2}\Omega_{c}(\rho_{22}-\rho_{33})\nonumber\\
&\ &- \left( \frac{\Gamma_2+\Gamma}{2}+i \Delta_c \right) \rho_{32}, \\
\label{eq:OBE_rho33}
\frac{\partial}{\partial t}\rho_{33} &=&
-\frac{i}{2}\Omega_{p}^*\rho_{31} +\frac{i}{2}\Omega_{p}\rho_{31}^*
-\frac{i}{2}\Omega_{c}\rho_{32} \nonumber\\
&\ &+ \frac{i}{2}\Omega_{c}\rho_{32}^* -\Gamma\rho_{33}, \\
\label{eq:MSE}
\frac{1}{c}\frac{\partial}{\partial t}\Omega_p
&+& \frac{\partial}{\partial z}\Omega _p = i\frac{\alpha\Gamma}{2L}\rho_{31},\end{aligned}$$ where $\rho_{ij}$ is the density matrix element between states $|i\rangle$ and $|j\rangle$, $\Omega_{p}$ and $\Omega_{c}$ represent the probe and coupling Rabi frequencies, $\Delta_p$ and $\Delta_c$ are the one-photon detunings of the probe and coupling transitions, $\delta = \Delta_p + \Delta_c$ is the two-photon detuning, $\gamma_0$ is the decoherence or dephasing rate of the Rydberg coherence $\rho_{21}$, $\Gamma$ is the spontaneous decay rate of $|3\rangle$ which is $2\pi$$\times$6 MHz in our case of the state $|5P_{3/2}\rangle$ of $^{87}$Rb atoms, $\Gamma_2$ is the spontaneous decay rate of $|2\rangle$ which is $2\pi$$\times$5.4 kHz in our case of the state $|32D_{5/2}\rangle$, and $\alpha$ and $L$ are the optical depth (OD) and the length of the medium. Since $\Omega_p \ll \Omega_c$ and $\Omega_p \ll \Gamma$ in this work, we treat the probe field as a perturbation and keep only the terms of the lowest order of $\Omega_p$ in each equation. The value of $\Gamma_2$ is small and, thus, we set it to zero throughout this work.
{width="13.6cm"}
We will determine the optical coherence, $\rho_{31}$, which is responsible for the attenuation coefficient and phase shift of the probe field. The steady-state solution is considered here and, hence, all the time-derivative terms are dropped in Eqs. (\[eq:OBE\_rho21\])-(\[eq:MSE\]). Equations (\[eq:OBE\_rho21\]) and (\[eq:OBE\_rho31\]) are used to obtain the solution of $\rho_{31}$ given by $$\rho_{31}(\Delta_p , \Delta_c) =
\frac{\Delta_p + \Delta_c + i\gamma_0}
{\Omega_{c}^{2}/2- 2(\Delta_p+ i\Gamma/2)(\Delta_p + \Delta_c + i\gamma_0)}
\Omega_p.
\label{eq:rho31_omegap}$$ With above $\rho_{31}$, we solve Eq. (\[eq:MSE\]) and find the ratio of output to input probe Rabi frequencies as the following: $$\frac{\Omega_p(L)}{\Omega_p(0)} = \exp(i\phi-\beta/2),
\label{eq:out_Omegap}$$ where $\beta$ and $\phi$ represent the attenuation coefficient and phase shift of the probe field at the output, and the probe transmission is $\exp(-\beta)$. We use $\beta_0$ and $\phi_0$ to denote the attenuation coefficient and phase shift without the DDI effect. The optical coherence of the probe field determines $\beta_0$ and $\phi_0$ as the followings: $$\begin{aligned}
\label{eq:beta0}
\beta_0 (\Delta_p,\Delta_c) &=& \alpha \Gamma
\;{\rm Im}\! \left[ \frac{\rho_{31} (\Delta_p,\Delta_c)}{\Omega_p} \right], \\
\label{eq:phi0}
\phi_0 (\Delta_p,\Delta_c) &=& \frac{\alpha \Gamma}{2}
\,{\rm Re}\! \left[ \frac{\rho_{31} (\Delta_p,\Delta_c)}{\Omega_p} \right].\end{aligned}$$ The effect of DDI on the attenuation coefficient and phase shift of the probe field will be derived in the following. Due to the DDI-induced frequency shift of the Rydberg state, the one-photon detuning of the coupling field transition is shifted by the amount of $\omega$, i.e., $$\Delta_c \rightarrow \Delta_c\pm\omega. \nonumber$$ Because $\omega \geq 0$, the positive or negative sign in the above corresponds to negative or positive $C_6$, respectively, and we use $+\omega$ which corresponds to negative $C_6$ in the following. Under the DDI, the probe field propagates through the atoms with different DDI-induced frequency shifts, where the probability density $P(\omega)$ of the frequency shift distribution has been shown in Eq. (\[eq:P(omega)\]). We obtain the values of $\beta$ and $\phi$ by averaging $\rho_{31}$ over the frequency distribution as shown below. $$\beta (\Delta_p,\Delta_c) = \alpha \Gamma \int_{0}^{\infty} d\omega P(\omega)
\;{\rm Im}\!\left[ \frac{\rho_{31} (\Delta_p,\Delta_c+\omega)}{\Omega_p} \right],
\label{eq:beta_ddi}$$ $$\phi (\Delta_p,\Delta_c)=\frac{\alpha \Gamma}{2} \int_{0}^{\infty} d\omega P(\omega)
\;{\rm Re}\! \left[ \frac{\rho_{31} (\Delta_p,\Delta_c+\omega)}{\Omega_p} \right].
\label{eq:phi_ddi}$$ The dipole blockade effect is that an atom inside the blockade sphere cannot be excited to the Rydberg state, where the blockade sphere centering with a Rydberg excitation has the radius $r_B \equiv (2 C_6 \Gamma/\Omega_c^2)^{1/6}$ [@REIT_Lukin2012]. This effect has already been included in the above formulas. In the weakly-interacting system considered here, i.e., $r_{B}^3 \ll r_a^3$, the average number of Rydberg excitations per volume of the blockade sphere is far less than one, and thus the dipole blockade appears rarely.
To evaluate Eqs. (\[eq:beta\_ddi\]) and (\[eq:phi\_ddi\]), one needs to know the value of $\omega_a$ in $P(\omega)$. According to the definition of $\omega_a$ in Eq. (\[eq:Define\_omega\_a\]) and that of $r_a$ in Eq. (\[eq:Define\_r\_a\]), we can relate $\omega_a$ to the Rydberg-polariton density, $n_R$, as $\omega_a = |C_6| [(4\pi/3) n_R]^2$. The product of the atomic density, $n_{\rm atom}$, and the average Rydberg-state population, $\bar{\rho}_{22}$, gives $n_R$, and therefore $\omega_a = |C_6| [(4\pi/3) n_{\rm atom} \bar{\rho}_{22}]^2$. The DDI-induced nonlinear and many-body effects make $\bar{\rho}_{22}$ no longer be the steady-state solution of the OBE shown in Eqs. (\[eq:OBE\_rho21\])-(\[eq:OBE\_rho33\]). Nevertheless, one can phenomenologically associate $\bar{\rho}_{22}$ to the steady-state solution of Rydberg-state population at the input, $\rho_{22,{\rm in}}$, by introducing a parameter $\varepsilon$. Substituting $\varepsilon \rho_{22,{\rm in}}$ for $\bar{\rho}_{22}$, we obtain $$\omega_a = |C_6|
\left[ (4\pi/3) n_{\rm atom} \varepsilon \rho_{22,{\rm in}} \right]^2,
\label{eq:omega_a_eit}$$ where $\varepsilon$ is the phenomenological parameter representing the average value of entire ensemble.
Predictions of Transmission and Phase-Shift Spectra
===================================================
The spectra of probe transmission and phase shift under the DDI effect are obtained by numerically evaluating the integrals of Eqs. (\[eq:beta\_ddi\]) and (\[eq:phi\_ddi\]) with the value of $\omega_a$ given by Eq. (\[eq:omega\_a\_eit\]). Figures \[fig:Spectra\](a)-\[fig:Spectra\](c) show the probe transmission versus the probe detuning at the coupling detunings of $+1$$\Gamma$, 0, and $-1$$\Gamma$; similarly, Figs \[fig:Spectra\](d)-\[fig:Spectra\](e) show the probe phase shift. The spectra without and with the DDI are calculated with Eq. (\[eq:beta0\]) \[or Eq. (\[eq:phi0\])\] and Eq. (\[eq:beta\_ddi\]) \[or Eq. (\[eq:phi\_ddi\])\], respectively.
The DDI-induced phenomena exhibited in the transmission and phase shift spectra are summarized as follows: (1) A larger probe intensity results in a smaller transmission or larger attenuation. (2) A larger probe intensity results in a larger phase shift at $\delta = 0$. (3) With the same probe intensity, the attenuation at a positive coupling detuning (e.g., $\Delta_c = +1$$\Gamma$) is smaller than that at a negative coupling detuning (e.g., $\Delta_c = -1$$\Gamma$), where the positive and negative detunings have the same magnitude. (4) With the same probe intensity, the phase shift at $\delta = 0$ of a positive coupling detuning (e.g., $\Delta_c = +1$$\Gamma$) is larger than that of a negative coupling detuning (e.g., $\Delta_c = -1$$\Gamma$), where the positive and negative detunings have the same magnitude. (5) The position of the EIT peak transmission at $\Delta_c = +1$$\Gamma$ changes very little and locates around $\delta = 0$; that at $\Delta_c = -1$$\Gamma$ shifts away from $\delta = 0$ significantly and a larger probe intensity induces a greater shift. We will explain the first four phenomena in the next three paragraphs and the last one in the Appendix.
First of all, the peak transmission decreases against the probe Rabi frequency \[see Figs. \[fig:Spectra\](a), \[fig:Spectra\](b), and \[fig:Spectra\](c)\]. This is expected, because the Rydberg-state population is proportional to the probe intensity or Rabi frequency square. A larger Rydberg-state population or Rydberg-polariton density makes $\omega_a$ larger as shown by Eq. (\[eq:omega\_a\_eit\]). The probability density $P(\omega)$ with the larger $\omega_a$ has a broader width and a longer tail as demonstrated by Fig. \[fig:Distribution\](b). Under the broader $P(\omega)$, more atoms have the Rydberg-state frequency shifted away from the EIT resonance condition, reducing the peak transmission more. Secondly, the phase shift increases against the probe Rabi frequency \[see Figs. \[fig:Spectra\](d), \[fig:Spectra\](e), and \[fig:Spectra\](f)\]. The explanation is similar to that in the first phenomenon.
![ The imaginary and real parts of $\rho_{31}/\Omega_p$ as functions of the frequency shift $\omega$, calculated with Eq. (\[eq:rho31\_omegap\]) in which we make the substitution of $\Delta_c \rightarrow \Delta_c + \omega$ and $\Delta_p \rightarrow -\Delta_c$. In the calculation, $\Omega_c = 1.0\Gamma$, $\gamma_0 = 0$, $\delta = 0$, and $\Delta_c = 1.0$$\Gamma$ in (a,c) and $-1.0$$\Gamma$ in (b,d). []{data-label="fig:rho31"}](_Fig4){width="\columnwidth"}
The third phenomenon observed in the spectra is that the probe intensity or $\Omega_{p,\rm{in}}^2$ has a smaller effect on the reduction of peak transmission at $\Delta_c =$ $+1$$\Gamma$ as shown by Fig. \[fig:Spectra\](a) than that at $\Delta_c =$ $-1$$\Gamma$ as shown by Fig. \[fig:Spectra\](c). This can be explained with the help of Figs. \[fig:rho31\](a) and \[fig:rho31\](b), which show ${\rm Im}[\rho_{31}/\Omega_p]$ at $\Delta_c =$ 1$\Gamma$ and $-1$$\Gamma$, respectively. To obtain the probe transmission, the integration of Eq. (\[eq:beta\_ddi\]) is performed only for the region of $\omega >0$. In Fig. \[fig:rho31\](a), the value of ${\rm Im}[\rho_{31}/\Omega_p]$ is always small for $\omega > 0$, resulting in a smaller value of $\int_{0}^{\infty} d\omega P(\omega) {\rm Im}[\rho_{31}/\Omega_p]$, i.e., a higher probe transmission. On the other hand, in Fig. \[fig:rho31\](b), the value of ${\rm Im}[\rho_{31}/\Omega_p]$ has a large peak for $\omega > 0$, which corresponds to the absorption due to the two-photon transition. This large peak results in a larger value of $\int_{0}^{\infty} d\omega P(\omega) {\rm Im}[\rho_{31}/\Omega_p]$, i.e., a lower probe transmission. Therefore, with the same value of $\Omega_{p,\rm{in}}$, the peak transmission at $\Delta_c =$ 1$\Gamma$ shown by Fig. \[fig:Spectra\](a) is larger than that at $\Delta_c =$ $-1$$\Gamma$ shown by Fig. \[fig:Spectra\](c).
The fourth phenomenon observed in the spectra is that the probe intensity or $\Omega_{p,\rm{in}}^2$ has a much larger effect on the phase shift of $\delta = 0$ at $\Delta_c =$ 1$\Gamma$ as shown by Fig. \[fig:Spectra\](d) than that at $\Delta_c =$ $-1$$\Gamma$ as shown by Fig. \[fig:Spectra\](f). This can be explained with the help of Figs. \[fig:rho31\](c) and \[fig:rho31\](d), which show ${\rm Re}[\rho_{31}/\Omega_p]$ at $\Delta_c =$ 1$\Gamma$ and $-1$$\Gamma$, respectively. To obtain the phase shift, the integration of Eq. (\[eq:phi\_ddi\]) is performed only for the region of $\omega >0$. In Fig. \[fig:rho31\](c), the value of ${\rm Re}[\rho_{31}/\Omega_p]$ is always positive for $\omega > 0$, resulting in a larger value of $\int_{0}^{\infty} d\omega P(\omega) {\rm Re}[\rho_{31}/\Omega_p]$, i.e., a larger phase shift. On the other hand, in Fig. \[fig:rho31\](d), ${\rm Re}[\rho_{31}/\Omega_p]$ has both positive and negative values for $\omega > 0$, because the resonance of the two-photon transition locates at $\omega > 0$. The cancellation between positive and negative values of the integrand makes $\int_{0}^{\infty} d\omega P(\omega) {\rm Re}[\rho_{31}/\Omega_p]$ nearly zero, i.e., almost no phase shift. Therefore, with the same value of $\Omega_{p,\rm{in}}$, the phase shift at $\Delta_c =$ 1$\Gamma$ shown by Fig. \[fig:Spectra\](d) is significant, and that at $\Delta_c =$ $-1$$\Gamma$ shown by Fig. \[fig:Spectra\](f) is little.
Analytical Formulas of the DDI-Induced Attenuation Coefficient and Phase Shift
==============================================================================
We now derive the analytical formulas for the DDI-induced attenuation coefficient, $\Delta\beta$, and phase shift, $\Delta\phi$, at the condition of $\gamma_0 = 0$ and $\delta = 0$ (or $\Delta_p = -\Delta_c$). Here, $\Delta\beta$ (or $\Delta\phi$) is defined as the difference between the values of $\beta$ (or $\phi$) with and without the DDI effect, i.e., $\Delta\beta\equiv\beta -\beta_0$ and $\Delta\phi\equiv \phi -\phi_0$.
At $\gamma_0 = 0$ and $\delta = 0$, Eq. (\[eq:rho31\_omegap\]) gives $\beta_0 = 0$ and $\phi_0 = 0$, and thus $\Delta\beta$ = $\beta$ and $\Delta\phi$ = $\phi$. Replacing $\Delta_p$ by $-\Delta_c$ in $\beta$ of Eq. (\[eq:beta\_ddi\]) and in $\phi$ of Eq. (\[eq:phi\_ddi\]), we obtain $\Delta\beta$ and $\Delta\phi$ as follows: $$\begin{aligned}
\label{eq:Delta_beta_DDI_num}
\Delta\beta
&=& \alpha \Gamma \int_{0}^{\infty} d\omega P(\omega)
\frac{4\omega^2\Gamma}{4\omega^2 \Gamma^2+(4\omega\Delta_c+\Omega_c^2)^2}, \\
\label{eq:Delta_phi_DDI_num}
\Delta\phi
&=& \frac{\alpha \Gamma}{2} \int_{0}^{\infty} d\omega P(\omega)
\frac{8\omega^2 \Delta_c +2\omega \Omega_c^2}
{4\omega^2 \Gamma^2+(4\omega\Delta_c+\Omega_c^2)^2}.\end{aligned}$$ In the weakly-interacting or low-density system, the region of $\omega$ being the order of $\omega_a$ is very near the center of the EIT window, in which ${\rm Im}[\rho_{31}/\Omega_p]$ and ${\rm Re}[\rho_{31}/\Omega_p]$ are nearly zero and contribute to the above two integrals very little. On the other hand, the region of $\omega \gg \omega_a$ is away from the center of the EIT window, and contributes to the above two integrals predominately. Under $\omega \gg \omega_a$, in the integrands of Eqs. (\[eq:Delta\_beta\_DDI\_num\]) and (\[eq:Delta\_phi\_DDI\_num\]) we can make the approximation of $P(\omega)$ as $$P(\omega) \approx \frac{\sqrt{\omega_a}}{2 \omega^{3/2}} \equiv P'(\omega),
\label{eq:P(omega)2}$$ where $\omega_a$ is given by Eq. (\[eq:omega\_a\_eit\]). In Eq. (\[eq:omega\_a\_eit\]), the steady-state solution of $\rho_{22,{\rm in}}$ is $$\rho_{22,{\rm in}} = \frac{\Omega_{p,{\rm in}}^2 \Omega_c^2}
{4\delta^2\Gamma^2+(\Omega_c^2-4\delta\Delta_p)^2}
\approx \frac{\Omega_{p,{\rm in}}^2}{\Omega_c^2},$$ where $\delta\Gamma, \delta\Delta_p \ll \Omega_c^2$ is the typical condition in most of the EIT experiments. Without any other approximation, we use $P'(\omega)$ in Eqs. (\[eq:Delta\_beta\_DDI\_num\]) and (\[eq:Delta\_phi\_DDI\_num\]) and replace $\rho_{22,{\rm in}}$ in $\omega_a$ by $\Omega_{p,{\rm in}}^2/\Omega_c^2$ to obtain $$\begin{aligned}
\label{eq:Delta_beta_DDI_th_omegap}
\Delta\beta &=& 2 S_{\rm DDI} \sqrt{\frac{W_c-2\Delta_c}{W_c^2}}
\Omega_{p,{\rm in}}^2, \\
\label{eq:Delta_phi_DDI_th_omegap}
\Delta\phi &=& S_{\rm DDI} \sqrt{\frac{W_c+2\Delta_c}{W_c^2}} \Omega_{p,{\rm in}}^2,\end{aligned}$$ where $$\begin{aligned}
\label{eq:S_DDI}
S_{\rm DDI} &\equiv&
\frac{\pi^2\alpha\Gamma\sqrt{C_6} n_{\rm atom} \varepsilon}{3\Omega_c^3}, \\
W_c &\equiv& \sqrt{\Gamma^2+4\Delta_c^2}.\end{aligned}$$ The above results being good approximations imposes the condition that $\omega_a$ is much smaller than the EIT linewidth, $\Delta\omega_{\rm EIT}$, where $\Delta\omega_{\rm EIT} =$ $\Omega_c^2 \sqrt{\Gamma^2+8\Delta_c^2}/(\Gamma^2+4\Delta_c^2)$ derived from the spectrum of ${\rm Im}[\rho_{31}(\omega)]$ at $\delta = 0$. More precisely, the accuracy of the analytical formula of $\Delta\beta$ requires $(\omega_a/\Delta\omega_{\rm EIT})^{3/2} \ll 1$, and that of $\Delta\phi$ requires $(\omega_a/\Delta\omega_{\rm EIT})^{1/2} \ll 1$.
In Fig. \[fig:DDI\_Effect\], we compare the results of the above two analytical formulas with those of the numerical integrations of Eqs. (\[eq:Delta\_beta\_DDI\_num\]) and (\[eq:Delta\_phi\_DDI\_num\]) without the approximation of $P(\omega)$. The agreement between the results of the analytical formulas and numerical integrations is satisfactory except the line of $\Delta\phi$ at $\Omega_c = 1.0$$\Gamma$ in the region of $\Omega_{p,\rm{in}}^2 >$ 0.02$\Gamma^2$. In this region, $(\omega_a/\Delta\omega_{\rm EIT})^{1/2} \ll 1$ is no longer well satisfied, and the deviation between the analytical formula and the numerical integration becomes observable. Figures \[fig:DDI\_Effect\](a) and \[fig:DDI\_Effect\](c) demonstrate that both of $\Delta\beta$ and $\Delta\phi$ are proportional to $\Omega_{p,\rm{in}}^2/\Omega_c^3$. Figure \[fig:DDI\_Effect\](b) \[or \[fig:DDI\_Effect\](d)\] shows the asymmetric phenomenon that the value of $\Delta\beta$ (or $\Delta\phi$) at the coupling detuning of $|\Delta_c|$ is smaller (or larger) than that at the coupling detuning of $-|\Delta_c|$.
![(a,c) The DDI-induced attenuation coefficient $\Delta \beta$ and phase shift $\Delta \phi$ as functions of $\Omega_{p,\rm{in}}^2$ under $\Delta_p = \Delta_c =0$. (b,d) $\Delta \beta$ and $\Delta \phi$ as functions of $\Delta_c$ under $\delta = 0$ and $\Omega_{p,\rm{in}} = 0.1$$\Gamma$. The horizontal axes of the left two figures have the same scale, so do those of the right two figures. Red, cyan, and green solid lines represent the numerical evaluations of the integrals in Eqs. (\[eq:Delta\_beta\_DDI\_num\]) and (\[eq:Delta\_phi\_DDI\_num\]) at $\Omega_c =$ 1.0$\Gamma$, 1.4$\Gamma$, and 2.0$\Gamma$. Dashed lines are the results of the analytical formulas given by Eqs. (\[eq:Delta\_beta\_DDI\_th\_omegap\]) and (\[eq:Delta\_phi\_DDI\_th\_omegap\]). All the predictions are calculated with $\alpha$ = 81, $\gamma_0 = 0$, and $|C_6|[(4\pi/3)n_{\rm atom}\varepsilon]^2 = 0.35$$\Gamma$. []{data-label="fig:DDI_Effect"}](_Fig5){width="\columnwidth"}
In reality, there exist a nonzero decoherence rate $\gamma_0$ and the two-photon detuning $\delta$ in the system. We need to consider the corrections of $\gamma_0$ and $\delta$ to the analytical formulas. Under the condition of $\Omega_c^2 \gg \gamma_0 \Gamma,\delta\Gamma$, the attenuation coefficient and phase shift without the DDI effect, $\beta_0$ and $\phi_0$, are approximately given by $$\begin{aligned}
\label{eq:beta_0_correct}
\beta_0 &\approx &\frac{2\alpha\gamma_0\Gamma}{\Omega_c^2}
-\frac{16\alpha\gamma_0\delta\Delta_c\Gamma}{\Omega_c^4}, \\
\label{eq:phi_0_correct}
\phi_0&\approx &\frac{\alpha\Gamma\delta}{\Omega_c^2}
-\frac{4\alpha\gamma_0\delta\Gamma^2}{\Omega_c^4}
+\frac{4\alpha(\gamma_0^2-\delta^2)\Delta_c\Gamma}{\Omega_c^4}.\end{aligned}$$ To derive the DDI-induced attenuation coefficient, $\Delta\beta$, and phase shift, $\Delta\phi$, we first use the replacement of $\Delta_c \rightarrow \Delta_c + \omega$ and the relation of $\delta = \Delta_p + \Delta_c$ in $\rho_{31}/\Omega_p$ shown by Eq. (\[eq:rho31\_omegap\]). Then, we expand $\rho_{31}/\Omega_p$ with respect to $\gamma_0$ and $\delta$ under the assumption of $\Omega_c^2/\Gamma \gg \gamma_0, \delta$ to obtain $$\begin{aligned}
\label{eq:im_rho31_gamma_0}
{\rm Im}\! \left[\frac{\rho_{31}}{\Omega_p} \right]
&=& A_0 +A_1\gamma_0 +A_2\delta +\cdots, \\
\label{eq:re_rho31_gamma_0}
{\rm Re}\! \left[\frac{\rho_{31}}{\Omega_p} \right]
&=& B_0 +B_1\gamma_0 +B_2\delta +\cdots,\end{aligned}$$ where
$$\begin{aligned}
\label{eq:A0}
A_0 &=& \frac{4\omega^2\Gamma}
{4\omega^2\Gamma^2+(4\omega\Delta_c+\Omega_c^2)^2}, \\
\label{eq:A_gamma}
A_1 &=&
\frac{2\Omega_c^2[(4\omega^2\Delta_c+\Omega_c^2)^2-4\omega^2\Gamma^2]}
{[4\omega^2\Gamma^2+(4\omega\Delta_c+\Omega_c^2)^2]^2}, \\
\label{eq:A_delta}
A_2 &=& \frac{8\omega\Gamma\Omega_c^2(4\Delta_c\omega+\Omega_c^2)}
{[4\omega^2\Gamma^2+(4\omega\Delta_c+\Omega_c^2)^2]^2},\end{aligned}$$
and
$$\begin{aligned}
\label{eq:B0}
B_0 &=& \frac{8\Delta_c\omega^2+2\omega\Omega_c^2}
{4\omega^2\Gamma^2+(4\omega\Delta_c+\Omega_c^2)^2},\\
\label{eq:B_gamma}
B_1 &=& -\frac{8\omega\Gamma\Omega_c^2(4\Delta_c\omega+\Omega_c^2)}
{[4\omega^2\Gamma^2+(4\omega\Delta_c+\Omega_c^2)^2]^2}, \\
\label{eq:B_delta}
B_2 &=&
\frac{2\Omega_c^2[(4\omega^2\Delta_c+\Omega_c^2)^2-4\omega^2\Gamma^2]}
{[4\omega^2\Gamma^2+(4\omega\Delta_c+\Omega_c^2)^2]^2}.\end{aligned}$$
Next, we evaluate the two integrals of Eqs. (\[eq:beta\_ddi\]) and (\[eq:phi\_ddi\]) by substituting Eqs. (\[eq:im\_rho31\_gamma\_0\]) and (\[eq:re\_rho31\_gamma\_0\]) for ${\rm Im}[\rho_{31}/\Omega_p]$ and ${\rm Re}[\rho_{31}/\Omega_p]$ in the integrands. Since $\omega_a$ is much less than the EIT linewidth, $P'(\omega)$ shown in Eq. (\[eq:P(omega)2\]) can be employed in Eqs. (\[eq:beta\_ddi\]) and (\[eq:phi\_ddi\]) to replace $P(\omega)$. The results of the two integrals give $\beta$ and $\phi$. Finally, the analytical formulas of $\Delta\beta (= \beta-\beta_0)$ and $\Delta\phi (= \phi-\phi_0)$, including the corrections of $\gamma_0$ and $\delta$ are given by $$\begin{aligned}
\label{eq:beta_DDI_th_correct}
\Delta\beta &=& 2 S_{\rm DDI}\Bigg( \sqrt{\frac{W_c-2\Delta_c}{W_c^2}}
-\frac{3\gamma_0\sqrt{W_c +2\Delta_c}}{\Omega_c^2} \nonumber \\
&\ &
+\frac{3\delta\sqrt{W_c-2\Delta_c}}{\Omega_c^2} \Bigg) \Omega_{p,{\rm in}}^2, \\
\label{eq:phi_DDI_th_correct}
\Delta\phi &=& S_{\rm DDI}\Bigg( \sqrt{\frac{W_c+2\Delta_c}{W_c^2}}
-\frac{3\gamma_0\sqrt{W_c-2\Delta_c}}{\Omega_c^2} \nonumber \\
&\ &
-\frac{3\delta\sqrt{W_c+2\Delta_c}}{\Omega_c^2} \Bigg) \Omega_{p,{\rm in}}^2. \end{aligned}$$ Regarding $\Delta\beta$ as a function of $\Omega_{p,\rm{in}}^2$ in Fig. \[fig:DDI\_Effect\](a), the slope will decrease a little due to $\gamma_0$, and become a little larger (or smaller) due to positive (or negative) $\delta$. Regarding $\Delta\phi$ as a function of $\Omega_{p,\rm{in}}^2$ in Fig. \[fig:DDI\_Effect\](c), the slope will decrease a little due to $\gamma_0$, and become a little smaller (or larger) due to positive (or negative) $\delta$. When we consider $\beta$ and $\phi$ instead of $\Delta\beta$ and $\Delta\phi$ in Figs. \[fig:DDI\_Effect\](a) and \[fig:DDI\_Effect\](c), $\beta_0$ and $\phi_0$ make nonzero vertical-axis interceptions of those lines.
Simulation of the experimental data
===================================
To verify the mean field theory developed in this work, we systematically measured the attenuation coefficient, $\beta$, and phase shift, $\phi$, of the output probe field as shown in Fig. 2 of Ref. [@OurExp]. The experiment was carried out in cold $^{87}$Rb atoms with the temperature of 350 $\mu$K. The ground state $|1\rangle$, Rydberg state $|2\rangle$, and excited state $|3\rangle$ in the EIT system here correspond to $|5S_{1/2}, F=2, m_F=2\rangle$, $|32D_{5/2}, m_J=5/2\rangle$, and $|5P_{3/2}, F=3, m_F=3\rangle$ in the experiment. We set $\Omega_c = 1.0\Gamma$ and $\Omega_{p,{\rm in}} \leq 0.2\Gamma$, and the atomic density was about 0.05 $\mu$m$^{-3}$. Hence, the blockade radius $r_{B} =$ 2.1 $\mu$m, and the half mean distance between Rydberg polaritons $r_a \geq$ 4.9 $\mu$m. The weakly-interacting condition of $r_{B}^3 \ll r_a^3$ was satisfied in the measurement. Other experimental details can be found in Ref. [@OurExp].
![ Simulation of the experimental data shown in Fig. 2 of Ref. [@OurExp]. In the simulation, $\alpha$ = 81, $\Omega_c =$ 1.0$\Gamma$, $\delta = 0$, $\gamma_0 =$ 0.012$\Gamma$, and $|C_6|[(4\pi/3)n_{\rm atom}\varepsilon]^2 = 0.35$$\Gamma$. (a,c) Attenuation coefficient $\beta$ and phase shift $\phi$ as functions of $\Omega_{p,\rm{in}}^2$ at $\Delta_c = -2$$\Gamma$ (black), $-1$$\Gamma$ (red), 0 (blue), 1$\Gamma$ (magenta), and 2$\Gamma$ (olive). (b,d) Slope of $\beta$ versus $\Omega_{p,\rm{in}}^2$ and that of $\phi$ versus $\Omega_{p,\rm{in}}^2$ as functions of $\Delta_c$.[]{data-label="fig:DDI_exp_simulation"}](_Fig6){width="1\columnwidth"}
In Fig. \[fig:DDI\_exp\_simulation\], we made the predictions with Eqs. (\[eq:beta\_0\_correct\]), (\[eq:phi\_0\_correct\]), (\[eq:beta\_DDI\_th\_correct\]), and (\[eq:phi\_DDI\_th\_correct\]) for the comparison with the experimental data in Fig. 2 of Ref. [@OurExp]. The calculation parameters of OD, coupling Rabi frequency, two-photon detuning, and decoherence rate were determined experimentally. As for the value of $S_{\rm DDI}$, we used $C_6 = -2\pi$$\times$260 MHz$\cdot\mu$m$^6$ of the state $|32D_{5/2},m_J=5/2\rangle$, $n_{\rm atom} = 0.05$ $\mu$m$^{-3}$ estimated from the experimental condition, and $\varepsilon = 0.43$ determined by the experimental data. Figures \[fig:DDI\_exp\_simulation\](a) and \[fig:DDI\_exp\_simulation\](c) show the attenuation coefficient, $\beta$, and phase shift, $\phi$, of the output probe field as functions of $\Omega_{p,\rm{in}}^2$, where $\Omega_{p,\rm{in}}$ is the peak Rabi frequency of the Gaussian probe beam in the experiment. Figure \[fig:DDI\_exp\_simulation\](b) \[or \[fig:DDI\_exp\_simulation\](d)\] shows the slope of the straight line of $\beta$ (or $\phi$) versus $\Omega_{p,{\rm in}}^2$ as a function of $\Delta_c$. Note that the decoherence rate, $\gamma_0$, of 0.012$\Gamma$ makes the $y$-axis interception, i.e., $\beta_0$ or $\phi_0$, becomes nonzero according to Eqs. (\[eq:beta\_0\_correct\]) and (\[eq:phi\_0\_correct\]), and changes the slopes very little according to Eqs. (\[eq:beta\_DDI\_th\_correct\]) and (\[eq:phi\_DDI\_th\_correct\]).
In Fig. 2 of Ref. [@OurExp], the circles are the experimental data and the lines are their best fits. One can clearly observe the important characteristics of asymmetry in the data of slope versus $\Delta_c$. The consistency between the theoretical predictions in Fig. \[fig:DDI\_exp\_simulation\] here and the data or best fits in Fig. 2 of Ref. [@OurExp] is satisfactory. The discrepancies in the $y$-axis interceptions of straight lines between the predictions and best fits are minor, and can be explained by the uncertainties or fluctuations of $\delta$ and $\gamma_0$ in the experiment. Therefore, the mean field theory developed in this work is confirmed by the experimental data.
Conclusion
==========
In summary, a mean field theory based on the nearest-neighbor distribution is developed to describe the DDI effect in the system of weakly-interacting EIT-Rydberg polaritons. Employing the theory, we numerically calculate the spectra of probe transmission and phase shift as shown in Fig. \[fig:Spectra\]. We also explain the DDI-induced phenomena observed from the spectra. To make the theory convenient for predicting experimental outcomes and evaluating experimental feasibility, analytical formulas of the DDI-induced attenuation coefficient, $\Delta \beta$, and phase shift, $\Delta \phi$, are derived. As long as $\omega_a$ is much smaller than the EIT linewidth, the results of analytical formulas are in good agreement with those of numerical calculations. According to the formulas, $\Delta \beta$ and $\Delta \phi$ are linearly proportional to $\Omega_{p,\rm{in}}^2$ as demonstrated in Fig. \[fig:DDI\_Effect\](a) and \[fig:DDI\_Effect\](c), and $\Delta \beta$ and $\Delta \phi$ as functions of $\Delta_c$ are asymmetric with respect to $\Delta_c = 0$ as demonstrated in Fig. \[fig:DDI\_Effect\](b) and \[fig:DDI\_Effect\](d). We further consider the existences of nonzero but small decoherence rate and two-photon detuning in the system, and make corrections to the formulas of $\Delta \beta$ and $\Delta \phi$ as shown in Eqs. (\[eq:beta\_DDI\_th\_correct\]) and (\[eq:phi\_DDI\_th\_correct\]). Finally, we make the predictions with the parameters determined experimentally and compare them with the experimental data in Ref. [@OurExp]. The good agreement between the predictions and data demonstrates the validity of our theory. Rydberg polaritons are regarded as bosonic quasi-particles, and the DDI is the origin of the interaction between the particles. Thus, the DDI-induced phase shift and attenuation coefficient can infer the elastic and inelastic collision rates in the ensemble of these bosonic particles. Our mean field theory provides a useful tool for conceiving ideas relevant to the EIT system of weakly-interacting Rydberg polaritons, and for evaluating experimental feasibility.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This work was supported by Grant Nos. 107-2745-M-007-001 and 108-2639-M-007-001-ASP of the Ministry of Science and Technology, Taiwan.
APPENDIX {#appendix .unnumbered}
========
The fifth phenomenon observed in the spectra as shown by Fig. \[fig:Spectra\] is that the DDI effect shifts the position of EIT peak transmission very little at $\Delta_c =$ 1$\Gamma$ as shown by Fig. \[fig:Spectra\](a) and significantly at $\Delta_c =$ $-1$$\Gamma$ as shown by Fig. \[fig:Spectra\](c). We will explain the phenomenon in this Appendix.
There are two effects associating with the phenomenon. The first effect relates to the decoherence rate. According to Eq. (\[eq:rho31\_omegap\]), at $\Delta_c \neq 0$ the decoherence rate causes the EIT peak position to shift from $\delta = 0$ to $$\delta_{{\rm peak},\gamma} = \frac{2\gamma\Delta_c}{\Gamma}
\label{eq:omega_p_beta}$$ under the condition of $\Omega_c^2 \gg \gamma\Delta_c, \gamma\Gamma$, where $\gamma$ here can be the intrinsic decoherence rate $\gamma_0$, the DDI-induced decoherence rate $\gamma_{\rm eff}$, or sum of the two rates. Since the DDI-induced attenuation at $\Delta_c = +1$$\Gamma$ is less than that at $\Delta_c = -1$$\Gamma$, the value of $\gamma_{\rm eff}$ in the former is smaller than that in the latter. Thus, according to Eq. (\[eq:omega\_p\_beta\]) the shift of the EIT peak position due to $\gamma_{\rm eff}$ at $\Delta_c = +1$$\Gamma$ (denoted $\delta_{{\rm peak},\gamma,+}$) has a smaller magnitude than that at $\Delta_c = -1$$\Gamma$ (denoted $\delta_{{\rm peak},\gamma,-}$), i.e. $|\delta_{{\rm peak},\gamma,+}| < |\delta_{{\rm peak},\gamma,-}|$ where $\delta_{{\rm peak},\gamma,+} > 0$ and $\delta_{{\rm peak},\gamma,-} < 0$.
The second effect relates to the phase shift at $\delta = 0$ (denoted as $\phi_{\delta=0}$). This nonzero $\phi_{\delta=0}$ implies that the position of EIT peak transmission is shifted from $\delta = 0$ to $$\delta_{{\rm peak},\phi} = -\frac{\phi_{\delta=0}}{\tau_d},
\label{eq:omega_p_phi}$$ where $\tau_d=\alpha\Gamma/\Omega_c^2$ is the propagation delay time. The above equation shows the relation between the EIT peak position, $\delta_{{\rm peak},\phi}$, and the DDI-induced phase shift at $\delta = 0$. Figures \[fig:Spectra\](b) and \[fig:Spectra\](e) demonstrate this relation. At $\Delta_c = 0$, $\gamma_{\rm eff}$ cannot cause the shift of EIT peak position because of Eq. (\[eq:omega\_p\_beta\]), but $\phi_{\delta=0}$ can. Thus, the position of the EIT peak transmission shifted away from $\delta = 0$ as shown by Fig. \[fig:Spectra\](b) must be due to $\phi_{\delta=0}$. In Fig. \[fig:Spectra\](e), the phase shift spectra behave nearly straight lines. The peak positions in the transmission spectra, i.e., $\delta_{{\rm peak},\phi}$, of Fig. \[fig:Spectra\](b) approximately match the zero-crossing points of the corresponding straight lines in Fig. \[fig:Spectra\](e). In other words, the zero-crossing point is $\delta_{{\rm peak},\phi}$. The slopes of these straight lines are equal to the propagation delay time ($\tau_d$), i.e., $\tau_d = -\phi_{\delta=0}/\delta_{{\rm peak},\phi}$ where $\phi_{\delta=0} > 0$ and $\delta_{{\rm peak},\phi} <0$. One can clearly see the relation of Eq. (\[eq:omega\_p\_phi\]) from Figs. \[fig:Spectra\](b) and \[fig:Spectra\](e).
Now, we use Eq. (\[eq:omega\_p\_phi\]) to determine the frequency shift of the EIT peak caused by the second effect. Since the value of $\phi_{\delta=0}$ in Fig. \[fig:Spectra\](d) is positive and significant, the shift of the EIT peak position due to $\phi_{\delta=0}$ at $\Delta_c = +1$$\Gamma$ (denoted $\delta_{{\rm peak},\phi,+}$) is negative and non-negligible according to Eq. (\[eq:omega\_p\_phi\]). On the other hand, since the value of $\phi_{\delta=0}$ in Fig. \[fig:Spectra\](f) is very little, the shift of the EIT peak position due to $\phi_{\delta=0}$ at $\Delta_c = -1$$\Gamma$ (denoted $\delta_{{\rm peak},\phi,-}$) is nearly zero according to Eq. (\[eq:omega\_p\_phi\]).
The frequency shifts due to above two effects are summarized as $\delta_{{\rm peak},\gamma,+} > 0$, $\delta_{{\rm peak},\gamma,-} < 0$, $\delta_{{\rm peak},\phi,+} < 0$, $\delta_{{\rm peak},\phi,-} \approx 0$, and $|\delta_{{\rm peak},\gamma,-}| > |\delta_{{\rm peak},\gamma,+}| \approx |\delta_{{\rm peak},\phi,+}|$. Combining the two effects, we obtain the results of net frequency shift of the EIT peak in the followings: $\delta_{{\rm peak},\gamma,+} + \delta_{{\rm peak},\phi,+} \approx 0$ at $\Delta_c = +1$$\Gamma$, and $\delta_{{\rm peak},\gamma,-} + \delta_{{\rm peak},\phi,-} \approx \delta_{{\rm peak},\gamma,-} (< 0)$ at $\Delta_c = -1$$\Gamma$ . Hence, the shift of the EIT peak position at $\Delta_c = +1$$\Gamma$ is not observable in Fig. \[fig:Spectra\](a), and that at $\Delta_c = -1$$\Gamma$ is obvious and negative in Fig. \[fig:Spectra\](c).
[99]{} M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. [**87**]{}, 037901 (2001). D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Cote, E. E. Eyler, and P. L. Gould, Phys. Rev. Lett. [**93**]{}, 063001 (2004). R. Heidemann, U. Raitzsch, V. Bendkowsky, B. Butscher, R. Löw, L. Santos, and T. Pfau, Phys. Rev. Lett. [**99**]{}, 163601 (2007). M. Saffman, T. G. Walker, and K. M[ø]{}lmer, Rev. Mod. Phys., [**82**]{}, 2313 (2010). G. Bannasch, T. C. Killian, and T. Pohl, Phys. Rev. Lett. [**110**]{}, 253003 (2013). M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. [**77**]{}, 633 (2005). Y.-F. Chen, C.-Y. Wang, S.-H. Wang, and I. A. Yu, Phys. Rev. Lett. [**96**]{}, 043603 (2006). Z. B. Wang, K.-P. Marzlin, and B. C. Sanders, Phys. Rev. Lett. [**97**]{}, 063901 (2006). S. J. Li, X. D. Yang, X. M. Cao, C. H. Zhang, C. D. Xie, and H. Wang, Phys. Rev. Lett. [**101**]{}, 073602 (2008). B.-W. Shiau, M.-C. Wu, C.-C. Lin, and Y.-C. Chen, Phys. Rev. Lett. [**106**]{}, 193006 (2011). V. Venkataraman, K. Saha, and L. Gaeta, Nat. Photonics [**7**]{}, 138 (2012). Y.-H. Chen, M.-J. Lee, W. Hung, Y.-C. Chen, Y.-F. Chen, and I. A. Yu, Phys. Rev. Lett. [**108**]{}, 173603 (2012). A. Feizpour, M. Hallaji, G. Dmochowski, and A. M. Steinberg, Nat. Phys. [**11**]{}, 905 (2015). Z.-Y. Liu, Y- H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, Phys. Rev. Lett. [**117**]{}, 203601 (2016). M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett. [**84**]{}, 5094 (2000). M. Fleischhauer and M. D. Lukin, Phys. Rev. A [**65**]{}, 022314 (2002). J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adam, Phys. Rev. Lett. [**105**]{}, 193603 (2010). D. Petrosyan, J. Otterbach, and M. Fleischhauer, Phys. Rev. Lett. [**107**]{}, 213601 (2011). A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, Phys. Rev. Lett. [**107**]{}, 133602 (2011). T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, Nature (London) [**488**]{}, 57 (2012). S. Baur, D. Tiarks, G. Rempe, and S. Dürr, Phys. Rev. Lett. [**112**]{}, 073901 (2014). H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, Phys. Rev. Lett. [**113**]{}, 053601 (2014). D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, Phys. Rev. Lett. [**113**]{}, 053602 (2014). M. Moos, M. Höning, R. Unanyan, and M. Fleischhauer, Phys. Rev. A [**92**]{}, 053846 (2015). D. Tiarks, S. Schmidt, G. Rempe, and S. Dürr, Sci. Adv. [**2**]{}, e1600036 (2016). O. Firstenberg, C. S. Adams, and S. Hofferberth, J. Phys. B [**49**]{}, 152003 (2016). H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Nature (London) [**551**]{}, 579 (2017). J. Ruseckas, I. A. Yu, and G. Juzeliūnas, Phys. Rev. A [**95**]{}, 023807 (2017). H. Levine, A. Keesling, A. Omran, H. Bernien, S. Schwartz, A. S. Zibrov, M. Endres, M.G reiner, V. Vuletić, and M. D. Lukin, Phys. Rev. Lett. [**121**]{}, 123603 (2018). F. Ripka, H. Kübler, R. Löw, and T. Pfau, Science [**362**]{}, 446 (2018). H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bernien, M. Greiner, V. Vuletić, H. Pichler, and M. D. Lukin, Phys. Rev. Lett. [**123**]{}, 170503 (2019). D. Tiarks, S. Schmidt-Eberle, T. Stolz, G. Rempe, and S. Dürr, Nat. Phys. [**15**]{}, 124 (2019). S. Chandrasekhar, Rev. Mod. Phys. [**15**]{}, 1 (1943). B. Kim, K.-T. Chen, S.-S. Hsiao, S.-Y. Wang, K.-B. Li, J. Ruseckas, G. Juzeliūnas, T. Kirova, M. Auzinsh, Y.-C. Chen, Y.-F. Chen, and I. A. Yu, arXiv:2006.13526. M. Fleischhauer, J. Otterbach, and R. G. Unanyan, Phys. Rev. Lett. [**101**]{}, 163601 (2008). J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymańska, R. André, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and L. S. Dang, Nature (London) [**443**]{}, 409 (2006). R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, and K. West, Science [**316**]{}, 1007 (2007). H. Deng, H. Haug, and Y. Yamamoto, Rev. Mod. Phys. [**82**]{}, 1489 (2010). T. G. Walker and M. Saffman, Phys. Rev. A [**77**]{}, 032723 (2008).
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abstract: 'We study reheating processes and its cosmological consequences in the Starobinsky model embedded in the old-minimal supergravity. First, we consider minimal coupling between the gravity and matter sectors in the higher curvature theory, and transform it to the equivalent standard supergravity coupled to additional matter superfields. We then discuss characteristic decay modes of the inflaton and the reheating temperature $T_{\rm R}$. Considering a simple model of supersymmetry breaking sector, we estimate gravitino abundance from inflaton decay, and obtain limits on the masses of gravitino and supersymmetry breaking field. We find $T_{\rm R}\simeq 1.0\times10^9$ GeV and the allowed range of gravitino mass as $10^4$ GeV $\lesssim m_{3/2} \lesssim 10^5$ GeV, assuming anomaly-induced decay into the gauge sector as the dominant decay channel.'
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[ UT-14-46\
RESCEU-49/14\
WU-HEP-14-11]{}
.6in
[ **Reheating processes after Starobinsky inflation in old-minimal supergravity** ]{}
.5in
[Takahiro Terada$^{\spadesuit}$, Yuki Watanabe$^{\heartsuit}$, Yusuke Yamada$^{\diamondsuit}$, Jun’ichi Yokoyama$^{\heartsuit , \clubsuit}$ ]{}
0.25in
[*${}^\spadesuit$Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan\
${}^\heartsuit$Research Center for the Early Universe (RESCEU), Graduate School of Science,\
The University of Tokyo, Tokyo 113-0033, Japan\
${}^\diamondsuit$Department of Physics, Waseda University, Tokyo 169-8555, Japan\
${}^\clubsuit$Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU),\
WPI, TODIAS, The University of Tokyo, Chiba 277-8568, Japan* ]{}
.5in
Introduction
============
The recent observations of cosmic microwave background (CMB) by the WMAP [@WMAP] and Planck satellites [@Planck] indicate that nature is simple and minimal, providing increasing evidence to favor single field inflationary models.
Among the pioneering [@Starobinsky:1980te; @sato; @guth; @Linde:1981mu; @Albrecht:1982wi; @Linde:1983gd] and newer models of inflation [@review], Starobinsky model [@Starobinsky:1980te] occupies a unique position, since it does not require any new field to drive inflation which we call the inflaton. Its original version [@Starobinsky:1980te] is based on a higher curvature action which emerges by incorporating matter loops to the Einstein Hilbert action. Since all the second-order contributions of curvature tensors that affect the Einstein equation in a conformally flat geometry including the Robertson-Walker spacetime can be adequately described by the square scalar curvature term, currently popular version of Starobinsky model simply consists of linear and second-order terms of the Ricci scalar $R$ [@Mijic:1986iv].
In this model inflation is followed by an oscillatory behavior of the Hubble parameter, which results in gravitational particle production to reheat the Universe. Making use of a conformal transformation, one can also recast the system to the Einstein action containing a scalar field, dubbed as the scalaron acting as the inflaton, with a specific potential whose overall magnitude is determined by the coefficient of $R^2$ term in the original action [@Whitt:1984pd; @Maeda:1987xf]. In this picture reheating is described by the decay of the scalaron. Various aspects of reheating after $R^2$ inflation have been studied in Refs. [@Vilenkin:1985md; @Watanabe:2006ku; @Gorbunov:2010bn; @Arbuzova:2011fu; @Takeda:2014qma]. Ref. [@Gorbunov:2010bn] showed that dark matter and baryon asymmetry are produced at reheating by introducing Majorana neutrinos. Ref. [@Takeda:2014qma] showed that parametric resonance is not strong enough to form long-living localized objects, and thus reheating proceeds through perturbative decay of the inflaton.
The only adjustable parameter of Starobinsky model, namely the coefficient of the curvature square term, can be fixed by the amplitude of curvature perturbation [@Mukhanov:1981xt]. Its spectral index $n_s$ and the tensor-to-scalar ratio $r$ have also been confronted with observations, and interestingly, this oldest inflation model occupies just the central region of their error ellipses [@WMAP; @Planck]. This feature was challenged this March by BICEP2 collaboration [@Ade:2014xna], which claimed to have detected B-mode polarization of CMB corresponding to a value of $r$ much larger than favored by these satellite based observations. It turned out later, however, that the contamination of foreground dust may be so significant that one cannot rule out models with small $r$ yet at all [@Mortonson:2014bja; @Flauger:2014qra; @Adam:2014bub]. Thus the observational validity of Starobinsky model is still intact.
Needless to say, on the other hand, occupying the central region of the likelihood contour does not necessarily mean the model is the right one, and we should continue our efforts to further clarify features of Starobinsky model in the context of modern high energy theories, in particular, in supersymmetry (SUSY) which reduces the hierarchy problem significantly, naturally realizes gauge coupling unification, and provides cold dark matter candidates.
In the case of Starobinsky model, which is a theory of gravity, SUSY actually means supergravity (SUGRA) [@Ref:SUGRA]. There are two minimal choices for the SUGRA multiplet: the old-minimal [@Ferrara:1978em; @Stelle:1978ye; @Fradkin:1978jq] and new-minimal [@Sohnius:1981tp] formulations. These two formulations utilize different SUGRA auxiliary fields, but they coincide on-shell for the standard SUGRA action, whose bosonic part is General Relativity.[^1] This situation changes in the case of higher-derivative SUGRA, including SUGRA versions of Starobinsky model, because these auxiliary fields become propagating degrees of freedom. Embedding of the Starobinsky action into the old-minimal SUGRA was studied at the linearized level in Ref. [@Ferrara:1978rk] and the non-linear level in Ref. [@Cecotti:1987sa], where the duality to the standard SUGRA action with additional superfields was also established (in analogy with the bosonic case [@Whitt:1984pd; @Maeda:1987xf]). Tachyonic instability during the inflationary phase was cured in Ref. [@Kallosh:2013lkr]. In this SUGRA setup, the $R+R^2$ action (without higher order terms) emerges from generic $F$- and $D$-term action (without derivatives and superderivatives) [@Cecotti:1987sa; @Hindawi:1995qa] (see also Refs. [@Ketov:2013sfa; @Ketov:2013dfa]). The limited case, $F$-term generic action, was rediscovered in Ref. [@Gates:2009hu] and developed, *e.g.*, in Refs. [@Ketov:2012yz; @Ketov:2013wha] and its cosmological application was considered in Refs. [@Ketov:2012se; @Watanabe:2013lwa], but actually $D$-term action is required to realize Starobinsky inflation [@eno; @Ferrara:2013wka; @Ketov:2013dfa; @Ferrara:2013pla]. In the new-minimal SUGRA, embedding of Starobinsky model was studied in Ref. [@Cecotti:1987qe] and reconsidered in the inflationary context with higher order corrections in Ricci scalar in Ref. [@Farakos:2013cqa]. See also Ref. [@fklp].
Besides these pure SUGRA models without matter, there are many SUGRA models with matter that have Starobinsky-like scalar potentials (see, as an incomplete list, Refs. [@Cecotti:1987sa; @eno; @Kallosh:2013lkr; @SCDterm; @Farakos:2013cqa; @fklp; @Ferrara:2013wka; @Ketov:2013dfa; @Alexandre:2013nqa; @Pallis:2013yda]), so it is of prime importance to distinguish these models by studying cosmological scenarios after inflation. To discuss reheating of the universe, one has to couple the pure SUGRA inflation sector to matter sector. Ref. [@Hindawi:1996qi] studied soft SUSY breaking pattern in the old-minimal case, whereas Ref. [@Ferrara:2014cca] discussed some features of matter-coupling in the new-minimal setup.
In this paper, we consider generic old-minimal SUGRA models [@Cecotti:1987sa; @Hindawi:1995qa; @Ketov:2013dfa] that realize Starobinsky inflation focusing particularly on the model in Ref. [@Kallosh:2013lkr]. One of the reasons for the choice of the old-minimal formulation is that one has eventually to break R-symmetry to give gauginos their masses, but the new-minimal formulation has an exact R-symmetry. We assume the absence of even higher order terms in scalar curvature, corresponding to absence of superderivatives in the SUGRA action, because such terms may modify or hamper inflation [@Kamada:2014gma]. We introduce matter-coupling and study its cosmological consequences. In particular, we study various inflaton decay channels extensively. In contrast to the original non-SUSY version of reheating in the Starobinsky model, there is a long-lived particle, gravitino. The gravitino is the superpartner of the graviton and hence always present in SUGRA, and its abundance is a cosmologically important subject. We thus study the partial decay rate into gravitinos, and resultant constraints on parameters of the theory.
Before explaining our setup in section \[sec:setup\], we briefly emphasize the differences from the literature. In our setup, as we will see, the inflaton must have specific super- and Kähler potentials. For example, the exponential of Kähler potential *linearly* depends on the real part of the inflaton, while the gauge kinetic function never depends on the inflaton. Our setup, the old-minimal SUGRA realization of the Starobinsky model, is thus predictive. To the best of our knowledge, this is the first study of inflaton decay and gravitino production in the theory described by a modified action of supergravity. In section \[sec:decay\], we study various partial decay rates of the inflaton. In section \[sec:cosmology\], we discuss the cosmological constraints from gravitino abundance. We summarize and discuss differences from the original (non-SUSY) version of the Starobinsky model in section \[sec:conclusion\]. The duality transformation between a higher derivative SUGRA and the corresponding standard SUGRA is reviewed, and some generalization of it is discussed in Appendix \[app\]. We use the reduced Planck unit $c=\hbar=M_{\text{G}}=1$ with $M_{\text{G}}=M_{\text{Pl}}/\sqrt{8\pi} =1/\sqrt{8\pi G}$ unless otherwise stated, and basically use the notation and convention of Ref. [@Wess:1992cp].
Starobinsky model embedded in matter-coupled old-minimal supergravity {#sec:setup}
=====================================================================
The Starobinsky model is based on a pure gravity action with a second order term of scalar curvature. In the supergravity side, a generic (super)gravitational action up to matter and (super)derivatives is $$\begin{aligned}
S_{\rm grav}=& \int \mathrm{d}^4x\mathrm{d}^{4}\theta E N({\mathcal{R}},\bar{{\mathcal{R}}}) + \left[ \int \mathrm{d}^{4}x\mathrm{d}^{2}\Theta 2 {\mathscr{E}}
F({\mathcal{R}})+\text{H.c.}\right] ,\label{minimal}\end{aligned}$$ where ${\mathcal{R}}$ is the curvature chiral superfield, $E$ is the full density, ${\mathscr{E}}$ is the chiral density, $\Theta$ is the so-called new $\Theta$ variable [@Wess:1992cp], $N({\mathcal{R}},\bar{{\mathcal{R}}})$ is a Hermitian function, and $F({\mathcal{R}})$ is a holomorphic function.[^2]
To discuss inflaton decay and reheating of the universe, we consider a simple way of coupling the above action to matter sector. We take the minimal coupling between the SUGRA sector described by the curvature chiral superfield ${\mathcal{R}}$ and the matter sector described by chiral superfields $\phi^{i}$ and vector superfields $V^{A}$: $$\begin{aligned}
S=& \int \mathrm{d}^4x\mathrm{d}^{4}\theta E \left( N({\mathcal{R}},\bar{{\mathcal{R}}}) +J\left(\phi,\bar{\phi}e^{gV}\right) \right) \nonumber\\
&+ \left[ \int \mathrm{d}^{4}x\mathrm{d}^{2}\Theta 2 {\mathscr{E}}
\left( F({\mathcal{R}})+P(\phi) + \frac{1}{4}h_{AB}(\phi)W^{A}W^{B} \right)+\text{H.c.}\right] \nonumber \\
=&\int \mathrm{d}^4x\mathrm{d}^{4}\theta E N({\mathcal{R}},\bar{{\mathcal{R}}}) \nonumber \\
& + \left[ \int \mathrm{d}^{4}x\mathrm{d}^{2}\Theta 2 {\mathscr{E}}
\left( F({\mathcal{R}})+\frac{3}{8}\left( \bar{{\mathscr{D}}}\bar{{\mathscr{D}}}-8{\mathcal{R}}\right)e^{-K^{(\phi)}/3}+P(\phi)+ \frac{1}{4}h_{AB}(\phi)W^{A}W^{B}\right)+\text{H.c.}\right] ,\label{OriginalTheory}\end{aligned}$$ where $g$ is the gauge coupling constant, $\phi$ collectively denotes $\phi^i$’s, $J\left(\phi,\bar{\phi}e^{gV}\right)$ is a Hermitian function, $P(\phi)$ is a holomorphic function, and $K^{(\phi)}(\phi,\bar{\phi}e^{gV})=-3\ln \left( -\frac{J(\phi,\bar{\phi}e^{gV})}{3} \right)$ is the Kähler potential of the matter fields.
The above action can be recast into the following form [@Cecotti:1987sa; @Ferrara:2014cca]: $$\begin{aligned}
S=\int d^{4}xd^{2}\Theta 2{\mathscr{E}} \frac{3}{8}\left( \bar{{\mathscr{D}}}\bar{{\mathscr{D}}}-8{\mathcal{R}}\right)e^{-K/3}+W + \frac{1}{4}h_{AB}W^{A}W^{B}+\text{H.c.} ,\label{TransformedTheory}\end{aligned}$$ with the Kähler potential and superpotential specified as follows, $$\begin{aligned}
K &= -3\ln\left( \frac{T+\bar{T}-N(S,\bar{S})-J(\phi,\bar{\phi}e^{gV})}{3} \right) , \label{kahlerp} \\
W &=2TS+F(S)+P(\phi) \label{superp}.\end{aligned}$$ The derivation (in a more general setup) is reviewed in Appendix \[app\]. Note that the dependence of these potentials on the inflaton $T$ is completely determined by the structure of the theory: the origin of the inflaton $T$ is the Lagrange multiplier.[^3] This structure is not altered even if non-minimal couplings between ${\cal R}$ and matter superfields, which we do not discuss in this paper, are introduced because they become non-minimal couplings between $S$ (but not $T$) and matter superfields in the transformed theory.[^4] Therefore, in this sense, the couplings between $T$ and matters discussed in this paper are universal in old-minimal Starobinsky inflation.
The Kähler metric and its inverse are given by $$\begin{aligned}
g_{I\bar{J}}=&\frac{3}{\left( T+\bar{T}-N-J \right)^{2}}\begin{pmatrix} 1 & -N_{\bar{S}} & -J_{\bar{j}} \\
-N_{S} & N_{S\bar{S}}\left( T+\bar{T}-N-J \right)+N_{S}N_{\bar{S}} & N_{S} J_{\bar{j}} \\
-J_{i} & N_{\bar{S}}J_{i} & J_{i\bar{j}}\left( T+\bar{T}-N-J \right)+J_{i}J_{\bar{j}} \end{pmatrix}, \\
g^{\bar{I}J}=&\frac{T+\bar{T}-N-J}{3} \begin{pmatrix} \left(T+\bar{T}-N-J\right) +N_{S}N^{S}+J_{k}J^{k} & N^{S} & J^{j} \\
N^{\bar{S}} & N^{\bar{S}S} & 0 \\
J^{\bar{i}} & 0 & J^{\bar{i}j} \end{pmatrix},\end{aligned}$$ where $I,J,\dots=T, S, i,j,\dots \text{(or }\phi^{i},\phi^{j},\dots\text{)}$ are field indices, $N^{\bar{S}S}=(N_{S\bar{S}})^{-1}$, $J^{\bar{i}j}$ is the inverse matrix of $J_{i\bar{j}}$, and indices are uppered and lowered by these matrices, [*e.g.*]{} $N^{S}=N^{\bar{S}S}N_{\bar{S}}$ and $J^{\bar{i}}=J^{\bar{i}j}J_{j}$. The scalar potential is $$\begin{aligned}
V=&\left(\frac{3}{A}\right)^{2}\left( N^{\bar{S}S}\left| 2T+F_{S} \right|^{2}+\left| 2S \right|^{2}\left( A+N_{S}N^{S}+J_{i}J^{i} \right)+\bar{P}_{\bar{i}}J^{\bar{i}j}P_{j} \right. \nonumber \\
& \left. \phantom{N^{\bar{i}j} \bar{P}_{\bar{i}}P_{j} } +\left \{ 2\bar{S} \left [ \left(2T+F_{S} \right) N^{S}+P_{i}J^{i}-3W \right ] + \text{h.c.} \right \} \right) + \frac{g^{2}}{2}D^{A}D_{A} ,\end{aligned}$$ where we have defined a compact notation $A\equiv T+\bar{T}-N-J$.[^5] Indices of D-terms, $D^{A}\, (D_{A})$, are lowered (lifted) by (the inverse of) the real part of the gauge kinetic matrix function $h^{R}_{AB} \, (h_{R}^{AB})$.
The inflaton (or SUGRA) sector ($T$ and $S$) of this class of modified SUGRA models was studied in Ref. [@Ketov:2013dfa]. The Starobinsky model is realized in this setup essentially as the modified Cecotti model [@Kallosh:2013lkr]: $$\begin{aligned}
N(S,\bar{S})=&-3+\frac{12}{m_{\Phi}^{2}}S\bar{S}-\frac{\zeta}{m_{\Phi}^4} \left( S\bar{S} \right)^{2} \label{simpleN}, \\
F(S)=&0 \label{simpleF},\end{aligned}$$ where $m_{\Phi}$ is the inflaton mass at the vacuum, and $\zeta$ $(>0)$ gives a SUSY-breaking mass to $S$ and stabilizes its potential. The real part of $T$ becomes the inflaton, and the canonically normalized scalar potential is that of the Starobinsky model, $V=\frac{3m_{\Phi}^2}{4}\left(1-e^{-\sqrt{2/3}\widehat{\text{Re}T}}\right)^{2}$, where $\widehat{\text{Re}T}\equiv -K/\sqrt{6}$ is the canonically normalized inflaton field (during inflation). $S$ is the sGoldstino field that breaks SUSY during inflation. At the vacuum ($T=S=0$), SUSY is preserved.
Introduction of the linear term in $S/m_{\Phi}$ into eq. can make SUSY breaking vacua with an almost vanishing cosmological constant without spoiling inflation [@Dalianis:2014aya]. This is an interesting possibility because the higher derivative version of the purely supergravitational theory describes not only the inflation but also SUSY breaking. However, the SUSY breaking scale becomes the inflation scale ($m_{\Phi}\sim 10^{13}$ GeV), which typically makes the Higgs particle too heavy [@Giudice:2011cg]. Although the tree-level contributions to soft SUSY breaking parameters can be suppressed by assuming a minimal coupling between the MSSM sector and the SUGRA sector as in our setup, there are anomaly-mediated contributions to gaugino masses, which in turn give other particles their masses through renormalization group running.
Therefore, we concentrate on models that deviate (if any) only slightly from the simple model , . For definiteness, we assume $|N^{S}|$ and $|N^{\bar{S}S}F_{SS}|$ are at most of order the gravitino mass $m_{3/2}$, which is supposed to be much smaller than the inflaton mass, $m_{3/2}\ll m_{\Phi}$. Perturbation by higher order terms are negligible because VEV of $S$ is suppressed.[^6] Since the inflaton sector does not break SUSY at the vacuum, we introduce a hidden SUSY breaking sector. We treat the SUSY breaking sector as general as possible, but occasionally we assume a simple SUSY breaking sector described by $$\begin{aligned}
J(z, \bar{z})=& |z|^2 - \frac{|z|^4}{\Lambda^2} , \label{ZKahler} \\
P(z)=& \mu^2 z +W_0 , \label{Zsuper}\end{aligned}$$ where $J(z,\bar{z})$ and $P(z)$ are the Kähler potential and superpotential of the SUSY breaking field $z$ \[see equations and \]. We also assume that VEVs of $\phi^{i}$, $J(\phi^{i},\bar{\phi}^{\bar{j}})$, $P(\phi^{i})$, and their derivatives are negligibly small except for those of SUSY breaking field $z$, which is easily satisfied if $\phi^{i}$’s are charged under some unbroken symmetry.
All of the four scalar degrees of freedom and four fermionic degrees of freedom in the inflaton sector are degenerate in their masses ($=m_{\Phi}$) at the zeroth order of perturbation with respect to SUSY breaking ($m_{3/2}$). In the scalar sector, imaginary parts of $T$ and $S$ are still degenerate at the first order of gravitino mass, but the sum and difference of real parts of $T$ and $S$ have mass eigenvalues $m_{\Phi}\mp m_{3/2}$. Also, $S$ acquires its VEV, ${\left\langle S\right\rangle}={\left\langle W\right\rangle}/2$. Here we have neglected supersymmetric mass term of $S$ from its superpotential, $F_{SS}$. Fermionic mass eigenvalues depend on the detail of functions $N$ and $F$, but in the simplest case , , they are still degenerate at the first order in gravitino mass. For this kinematical reason, the decay of inflaton into particles in the inflaton sector (inflatino and gravitino), if possible, is extremely suppressed.
The mass eigenstates of the canonically normalized scalar linear fluctuations are approximately given by $$\begin{aligned}
\Phi_{R\pm}=&\frac{\sqrt{g_{T\bar{T}}}}{2} \left(T+\bar{T}\right) \pm \frac{\sqrt{g_{S\bar{S}}}}{2} \left( S+\bar{S} \right) \simeq \frac{1}{2\sqrt{3}}\left( T+ \bar{T}\right) \pm \frac{\sqrt{3}}{m_{\Phi}}\left( S+\bar{S}\right). \label{MassEigenstate}\end{aligned}$$ Because the $T$-$S$ oscillation time scale $\tau_{\text{osc}}\sim (2m_{3/2})^{-1}$ is much shorter than the lifetime $\tau_{\text{dec}}\sim (M_{\text{G}}^{2}/ m_{\Phi}^{3})$ for gravitino mass above GeV scale, decay rates from these mass eigenstates are appropriate quantities. However, the interactions are simply described in the basis of $T$ and $S$ but not of their linear combination, so for simplicity of presentation we describe partial decay rates of inflaton in the next section as if $T$ (or $S$) is the parent particle. The true rates are the averages of those for $T$ and $S$.
We take gravitino mass larger than TeV scale because we assume anomaly (or gravity) mediation of SUSY breaking, in which SUSY breaking is transmitted to the visible sector by the Planck suppressed coupling to the auxiliary field of the curvature superfield ${\mathcal{R}}$ in the transformed theory due to the trace anomaly (or by the Planck suppressed coupling to the hidden sector in the tree-level potential). We respect the philosophy of the Starobinsky model in this paper, that is, we exploit the (super-)gravitational sector as much as possible, and do not introduce an inflaton nor messenger fields by hand.
Inflaton decay {#sec:decay}
==============
Various modulus/inflaton decay modes and their cosmological consequences have been extensively studied in Ref. [@Endo:2007sz]. Inflaton decay in the case of no-scale supergravity has also been studied in Ref. [@Endo:2006xg], but in our case inflaton has supergravitational origin so that the form of inflaton Kähler potential is different from that in Ref. [@Endo:2006xg]. Moreover, these works suppose that the inflaton mass $m_{\Phi}$ comes mainly from the second derivative $W_{\Phi\Phi}$ of the superpotential with respect to the inflaton $\Phi$ itself. In our case, on the other hand, the origin of the inflaton mass $m_{T}(\equiv m_{\Phi})$ is from $W_{TS}$ rather than $W_{TT}$. We study inflaton decay in our setup taking these differences into account.
At the end of inflation, the inflaton oscillates around the minimum of the potential for a long time due to its Planck-suppressed decay rate. We have numerically checked that the energy stored in $\text{Re} T$ does not flow into $\text{Im} T$ or $S$ fields in this classical oscillation dynamics. In the following, we study various partial decay rates of the inflaton at the tree-level unless the one-loop process becomes leading. As stated at the end of the previous section, we first consider interactions involving $T$, followed by similar analyses for $S$.
Two-body decay of $T$ into scalars, spinors and gauge bosons
------------------------------------------------------------
### Decay into scalars
It is convenient to define the reduced scalar potential $\tilde{V}$ as $V=\left(\frac{3}{A}\right)^{2}\tilde{V}+\frac{g^{2}}{2}D^{A}D_{A}$, or equivalently, $$\begin{aligned}
\tilde{V}=& N^{\bar{S}S}\left| 2T+F_{S} \right|^{2}+\left| 2S \right|^{2}\left( A+N_{S}N^{S}+J_{i}J^{i} \right)+\bar{P}_{\bar{i}}J^{\bar{i}j}P_{j} \nonumber\\
& +\left \{ 2\bar{S} \left [ \left(2T+F_{S} \right) N^{S}+P_{i}J^{i}-3W \right ] + \text{h.c.} \right \}.\end{aligned}$$ Although $T$ and $S$ are singlets, derivatives of the $D$-term with respect to them are nonzero, $$\begin{aligned}
D_{AT}=&-ig_{T\bar{i}}\bar{X}_{A}^{\bar{i}}= \frac{3}{A^{2}}iJ_{\bar{i}}\bar{X}_{A}^{\bar{i}}=-\frac{1}{A}G_{\bar{i}}D_{A}{}^{\bar{i}}=-\frac{1}{A}D_{A}\simeq -\frac{1}{3}D_{A}, \\
D_{AS}=&-ig_{S\bar{i}}\bar{X}_{A}^{\bar{i}}= -\frac{3}{A^{2}}iN_{S}J_{\bar{i}}\bar{X}_{A}^{\bar{i}}=\frac{1}{A}N_{S}G_{\bar{i}}D_{A}{}^{\bar{i}}=\frac{1}{A}N_{A}D_{A}\simeq \frac{1}{3}N_{S}D_{A},\end{aligned}$$ where $G=K+\ln{|W|^2}$ is the total Kähler potential, $X_{A}$ is the Killing vector of the Kähler manifold, and we have used the gauge symmetry of the superpotential. With the aid of the condition of the vanishing cosmological constant, $V=0$, the stationary conditions for $T$ and $S$ at the vacuum, $V_{T}=V_{S}=0$, reduce to $\tilde{V}_{T}=\tilde{V}_{S}=0$.
Using the above formulas and the facts $\tilde{V}_{TT}=\tilde{V}_{Ti}=\tilde{V}_{T\bar{i}}=0$, the relevant vertex functions are derived as $$\begin{aligned}
V_{\tilde{T}\tilde{i}\tilde{j}}=-\frac{2}{A}V_{\tilde{i}\tilde{j}} \simeq -\frac{2}{3}V_{\tilde{i}\tilde{j}},\end{aligned}$$ where tilded indexes may take both of holomorphic and anti-holomorphic indexes like $\tilde{I}=I, \bar{I}$. This means that the interaction terms are proportional to the mass terms of scalars. There is a same order contribution from the kinetic term. Combining mass and kinetic term contributions, the rate is $$\begin{aligned}
\Gamma ( T \rightarrow \phi^{i}\bar{\phi}^{\bar{i}} ) = \frac{3m_{i}^{4}}{8\pi M_{\text{G}}^{2}m_{\Phi}},\end{aligned}$$ where $m_{i}$ is the mass of the daughter particle $\phi^{i}$. The kinetic term also provides the $\phi^{i}\phi^{j}$ production process with the rate $$\begin{aligned}
\Gamma (T \rightarrow \phi^{i}\phi^{j}) = \frac{m_{\Phi}^{3}}{96\pi M_{\text{G}}^{2}}|J_{ij}|^{2}.\end{aligned}$$
The partial decay rates of inflaton into $\text{Im}T$, $S$, or $\bar{S}$ and $\phi^{i}$ are suppressed by $J_{i}$ and phase space factors.
### Decay into spinors
It is convenient to define the reduced fermion mass matrix $\tilde{M}$ as $M_{IJ}=e^{G/2}\tilde{M}_{IJ}$, where $M_{IJ}$ is the fermion mass matrix, or equivalently, $$\begin{aligned}
\tilde{M}_{IJ}=\nabla_{I}G_{J}+G_{I}G_{J}-\frac{2}{3}\left( \langle G_{I} \rangle G_{J} + \langle G_{J} \rangle G_{I} \right)+\frac{2}{3}\langle G_{I}G_{J} \rangle.\end{aligned}$$ Terms with VEVs are induced by the redefinition of the gravitino field to absorb goldstino into gravitino. The inflaton-spinor-spinor vertex is obtained by differentiating the mass matrix, $M_{IJT}=G_{T}M_{IJ}/2 + e^{G/2}\tilde{M}_{IJT}\simeq m_{3/2}\tilde{M}_{IJT}$. Under the approximation like $A\simeq 3$ and $S\simeq W/2$, and neglecting $G_{i}, \, G_{T}$ and $G_{S}$, the reduced fermion matrix $\tilde{M}_{ij}$ is approximated as $\tilde{M}_{ij}\simeq P_{ij}/W +J_{ij}-J_{ij\bar{z}}G^{\bar{z}}$ where $z$ is the SUSY breaking field. Under the same approximation, $$\begin{aligned}
\tilde{M}_{ijT}\simeq -\tilde{M}_{ij}.\end{aligned}$$ On the other hand, $\tilde{M}_{ij\bar{T}}$ vanishes at the vacuum. The kinetic term gives a same order contribution. Combining the mass and kinetic term contributions, the partial decay rate is expressed as $$\begin{aligned}
\Gamma (T \rightarrow \chi^{i}\bar{\chi}^{\bar{i}} ) = \frac{m_{i}^{2}m_{\Phi}}{192\pi M_{\text{G}}^{2}},\end{aligned}$$ where $m_{i}$ is the mass of the spinor $\chi^{i}$. We have assumed here that the mixing terms between matter spinors and gauginos are smaller than the diagonal parts, $|M_{IA}|\ll |M_{JK}|$.
The partial decay rates of inflaton into inflatino or $S$-ino and $\chi^{i}$ are suppressed by $J_{i}$ and phase space factor.
### Anomaly-induced decay into gauge sector {#sec:anomaly}
The inflaton $T$ has the Lagrange multiplier origin so that it never appears in the gauge kinetic function. We have to consider decay into gauge sector via the anomaly-induced one loop process [@Endo:2007ih; @Endo:2007sz] unless we introduce a non-minimal term depending on $W^A$ in the $D$-term action (see Appendix \[app\]). The rate is [@Endo:2007ih; @Endo:2007sz] $$\begin{aligned}
\Gamma (T \rightarrow AA) + \Gamma (T \rightarrow \lambda\lambda )\simeq \frac{N_{\text{g}}\alpha^{2}}{256\pi^{3}}|X_{G}|^{2}m_{\Phi}^{3},\end{aligned}$$ where $N_{\text{g}}$ and $\alpha$ are the number of the generators and the fine structure constant of the gauge group, $X_{G}=\sqrt{6}\left [ (T_{G}-T_{R})K_{T}+\frac{2T_{R}}{d_{R}}\left( \log \det K|_{R}'' \right), _{T} \right ] $, $T_{G}$ and $T_{R}$ are the Dynkin indexes of the adjoint representation and representation $R$, $d_{R}$ is the dimension of the representation $R$, and $K|_{R}''$ is the Kähler metric restricted to the matter whose representation is $R$. In our case, the rate becomes (also see [@Watanabe:2010vy] for non-SUSY case) $$\begin{aligned}
\Gamma (T \rightarrow AA) + \Gamma (T \rightarrow \lambda\lambda )\simeq \frac{3N_{\text{g}}\alpha^{2}m_{\Phi}^{3}}{128\pi^{3}M_{\text{G}}^{2}}\left( T_{G}-\frac{1}{3}T_{R}\right)^{2}.\end{aligned}$$
Three-body decay of $T$
-----------------------
Let us first consider the decay channel into a scalar and two spinors involving Yukawa coupling. There are three diagrams at the tree level that are of the same order. The effective interaction term that reproduces the decay rate is found to be [@Endo:2007sz] $$\begin{aligned}
{\mathcal{L}}_{\text{eff}}\simeq -\frac{1}{2}e^{G/2}\left( G_{Tijk}-3\Gamma _{T(i}^{l}G_{jk)l} \right)T\phi^{i}\chi^{j}\chi^{k}+\text{h.c.}\end{aligned}$$ In our case, the leading terms, which could lead to the typical Planck-suppressed decay rate, cancel each other, and the remaining terms give at most $\Gamma\sim m_{3/2}^{2}m_{\Phi}^{3}/M_{\text{G}}^{4}$.
There are also scalar three-body decay. At the vacuum, the scalar four-point vertex is given by $$\begin{aligned}
V_{\tilde{T}\tilde{i}\tilde{j}\tilde{k}}\simeq & -\frac{2}{3}\left( \tilde{V}_{\tilde{i}\tilde{j}\tilde{k}}+V_{D\tilde{i}\tilde{j}\tilde{k}}\right) .\end{aligned}$$ The leading terms in $\tilde{V}_{ijk}$ cancel each other in the same way as for the above fermion case. $$\begin{aligned}
\tilde{V}_{ijk}\simeq & -3\bar{P}^{z}J_{z\bar{m}(i}J^{\bar{m}l}P_{jk)l} ,\\
\tilde{V}_{ij\bar{k}}\simeq & \bar{P}_{\bar{l}\bar{k}}J^{\bar{l}l}P_{ijl}-\bar{P}^{m}J_{m\bar{n}\bar{k}}J^{\bar{n}l}P_{ijl}+2\bar{S}P_{ijl}J^{\bar{l}l}\left( J_{\bar{l}\bar{k}}-J_{\bar{l}m\bar{k}}J^{m}\right) .\end{aligned}$$ The rates are suppressed by gravitino or matter mass squared, $\Gamma \sim m_{X}^{2}m_{\Phi}/M_{\text{G}}^{2}$ with $m_{X}=\max [ m_{\text{(matter)}}, \, m_{3/2} ]$ at most.
We also considered decay modes involving $\text{Im}T$, $S$, or their superpartners, but these rates are at most of order of $m_{\Phi}^{5}/M_{\text{G}}^{4}$ with additional phase space suppression. Four- or more- body decay rates are more suppressed by the phase space factor.
Decay of $S$
------------
In the same way as the previous subsections, we study decay channel of $S$ in this subsection. Although $S$ is basically conformally sequestered from the matter sector in our setup, it has unsupressed coupling with $T$ in the superpotential, which in turn couples to the matter sector universally. Consequently, $S$ has unsuppressed coupling to matter in some decay channels. Important partial decay rates are as follows, $$\begin{aligned}
\Gamma ( S \rightarrow \phi^{i}\phi^{i} ) \simeq & \frac{m_{i}^{2}m_{\Phi}}{48\pi M_{\text{G}}^{2}}, \\
\Gamma ( \bar{S} \rightarrow \chi^{i}\chi^{j} ) \simeq & \frac{m_{\Phi}^{3}}{48\pi M_{\text{G}}^{2}} \left| J_{ij} \right|^2 \end{aligned}$$ Beware $m_{S}= m_{\Phi}$. The above rates are calculated expanding mass terms. If there are no heavy matter particles, the following contribution from kinetic term becomes important, $$\begin{aligned}
\Gamma (S \rightarrow \phi^{i}\phi^{j} )\simeq \frac{m_{\Phi}m_{3/2}^{2}}{192\pi M_{\text{G}}^{2}}|J_{ij}|^{2},\end{aligned}$$ while other channels $\Gamma (S \rightarrow \phi^{i}\bar{\phi}^{\bar{j}})$ and $\Gamma (S \rightarrow \chi^{i}\chi^{j})$ from kinetic terms are suppressed by both of $N_{S}$ and matter masses. For decay modes of $S$ involving $T$, see the previous subsections, $\Gamma (S\rightarrow T X) = \Gamma (T\rightarrow S X)$. The anomaly-induced decay of $S$ involves an additional $-N_{S}$ factor compared to the case of $T$.
Gravitino production
--------------------
In this subsection we study gravitino production from the inflaton decay, which is one of the distinguishing features from the non-SUSY version of the Starobinsky model. Although we have treated $T$ or $S$ as the parent particle in the previous subsections, the mixing effect is essential in gravitino production [@Dine:2006ii; @Endo:2006tf]. We thus use the proper mass eigenstates in evaluating the inflaton decay rate into gravitinos.
### Single gravitino production
The partial decay rate of a scalar particle into its superpartner and a gravitino is calculated, [*e.g.*]{}, in Ref. [@Buchmuller:2004rq]. Because inflaton and inflatino are degenerate before SUSY breaking, their mass splitting is of the order of gravitino mass. We parametrize the mass difference as $ m_{\Phi}-m_{\tilde{\Phi}} = \Delta m_{3/2} $. The decay rate is approximately $$\begin{aligned}
\Gamma (\Phi \rightarrow \tilde{\Phi}\psi_{3/2} ) \simeq \frac{m_{\Phi}^{3}}{3\pi M_{\text{G}}^{2}}\left( \frac{m_{3/2}}{m_{\Phi}} \right)^{2}\Delta (\Delta^{2}-1)^{\frac{3}{2}},
\end{aligned}$$ where we explicitly wrote the reduced Planck mass $M_{\text{G}}$. Thus, the single gravitino production has the suppression factor $(m_{3/2}/m_{\Phi})^{2}$ and $(\Delta^{2}-1)^{3/2}$ compared to the typical Planck-suppressed decay rate $\mathcal{O}(m_{\Phi}^{3}/M_{\text{G}}^{2})$. In subsequent discussion, we neglect the single gravitino production rate because it is at most of order of gravitino pair production rate discussed below.
### Gravitino pair production
Gravitino pair production rate from modulus/inflaton decay has been extensively studied in the literature [@Endo:2006zj; @Nakamura:2006uc; @Dine:2006ii; @Endo:2006tf; @Endo:2012yg]. See also Refs. [@Kawasaki:2006gs; @Kawasaki:2006hm; @Endo:2006qk; @Asaka:2006bv; @Endo:2007ih; @Endo:2007sz] for other decay channels and cosmological consequences.
The gravitino pair production rate from a mass eigenstate $\Phi$ is given by [@Endo:2006zj; @Nakamura:2006uc; @Dine:2006ii; @Endo:2006tf; @Endo:2012yg] $$\begin{aligned}
\Gamma(\Phi \to \psi_{3/2}\psi_{3/2}) =
\frac{|\mathcal{G}^{\text{(eff)}}_{\Phi}|^{2}m_{\Phi}^{5}}{288 \pi m^{2}_{3/2 }},
\label{gravitinorate}\end{aligned}$$ where the mass hierarchy $m_{\Phi}\gg m_{3/2}$ is assumed, and the effective coupling is given by [@Endo:2012yg] $
\left | \mathcal{G}^{\text{(eff)}}_{\Phi} \right |^{2}
=2\left| G_{I} ({\mathcal{A}}^{-1}) ^{I}{}_{ \Phi} \right|^{2},
$ where ${\mathcal{A}}$ is the mixing matrix [@Endo:2012yg]. In our case, the inflaton is the real part of $T$, but the real parts of $T$ and $S$ mix almost maximally at the vacuum (see eq. ).
Because the SUSY breaking of $T$ and $S$ are small, $|G_{T}|, |G_{S}| \ll 1$, the effective coupling reduces to $$\begin{aligned}
\left | \mathcal{G}^{\text{(eff)}}_{\Phi_{\text{R}\pm}} \right |^{2}=2\left| \frac{\sqrt{3}}{2} G_{T} \pm \frac{m_{\Phi}}{4\sqrt{3}}G_{S} +({\mathcal{A}}^{-1})^{i}{}_{\Phi_{\text{R}\pm}}G_{i} \right|^{2}.\end{aligned}$$ We will first evaluate $G_T$ and $G_{S}$, and then proceed to $({\mathcal{A}}^{-1})^{i}{}_{\Phi_{\text{R}\pm}}$. We evaluate $G_{T}$ using the conditions $V=e^{G}(G_{I}G^{I}-3)+(g^{2}/2)D^{A}D_{A}=0$ for the vanishing cosmological constant and $V_{\bar{I}}=e^{G}(G_{\bar{I}}G^{J}G_{J}-2G_{\bar{I}}+G^{\bar{J}}\nabla_{\bar{I}}G_{\bar{J}})+g^{2}(-(h^{R}_{ABI}/2)D^{A}D^{B}+D^{A}D_{AI})=0$, where $\nabla_{I}G_{J}=G_{IJ}-G_{IJ\bar{K}}G^{\bar{K}}$, for the stationarity of the potential at the vacuum. The relevant equations are $$\begin{aligned}
G_{\bar{T}}+G^{\bar{I}}\nabla_{\bar{T}}G_{\bar{I}}=3\delta \left( G_{\bar{T}}+\frac{2}{3}\right), \label{VTbarEq} \\
G_{\bar{S}}+G^{\bar{I}}\nabla_{\bar{S}}G_{\bar{I}}=3\delta \left(G_{\bar{S}}-\frac{2}{3}N_{\bar{S}} \right) \label{VSbarEq},\end{aligned}$$ where $\delta=\frac{g^{2}}{6m_{3/2}^{2}}D^{A}D_{A}$ is the D-term SUSY breaking fraction. More explicitly, eq. is $$\begin{aligned}
3\delta G_{\bar{T}} +2\delta =G_{\bar{T}}+G_{T}g^{\bar{I}T}\nabla_{\bar{T}}G_{\bar{I}}+G_{i}g^{\bar{J}i}\nabla_{\bar{T}}G_{\bar{J}}+G_{S}g^{\bar{I}S}\nabla_{\bar{T}}G_{\bar{I}}.\end{aligned}$$ We concentrate on models that deviate only slightly from the simple model , , so we assume $|N_{S}|$ and $|F_{SS}|$ are at most of order $m_{3/2}/m_{\Phi}^{2}$. We also use $S\simeq W/2$. For example, $$\begin{aligned}
G_{Ti\bar{j}}=-\frac{3J_{i\bar{j}}}{(T+\bar{T}-N-J)^{2}}-\frac{6J_{i}J_{\bar{j}}}{(T+\bar{T}-N-J)^{3}}\simeq -\frac{1}{3}J_{i\bar{j}}.\end{aligned}$$ Similarly, $\nabla_{T}G_{T}\simeq -2/3$, $\nabla_{T}G_{S}\simeq 2/W$, and $\nabla_{T}G_{i}\simeq -2G_{i}/3 $. Equation becomes $$\begin{aligned}
0 \simeq (1-3\delta )G_{\bar{T}}+2\left(\frac{N^{\bar{S}}}{\bar{W}}-1\right)G_{T}-2+2G_{S}\frac{N^{\bar{S}S}}{\bar{W}}\simeq -2+2G_{S}\frac{N^{\bar{S}S}}{\bar{W}},\end{aligned}$$ so $G_S$ is approximately given by $$\begin{aligned}
G_{S}\simeq N_{S\bar{S}}\bar{W} \simeq \frac{12m_{3/2}}{m_{\Phi}^{2}}. \end{aligned}$$ This implies the tiny VEV of $T$: $$\begin{aligned}
T\simeq \frac{3m_{3/2}^{2}}{m_{\Phi}^{2}}-\frac{F_{S}}{2}. \label{tshift}\end{aligned}$$
In the same way, from eq. , $\nabla_{S}G_{S}\simeq F_{SS}/W$ and $\nabla_{S}G_{i}\simeq -N_{S}G_{i}/3$, we obtain $$\begin{aligned}
0 \simeq & \left(1-3\delta \right)G_{\bar{S}}+\left( \frac{2N^{S}+N^{\bar{S}S}\bar{F}_{\bar{S}\bar{S}}}{\bar{W}} \right)G_{S}+\left(\frac{6+N^{\bar{S}}\bar{F}_{\bar{S}\bar{S}}}{\bar{W}}\right)G_{T}+\frac{2J^{i}}{\bar{W}}G_{i}+2N_{\bar{S}}.\end{aligned}$$ To simplify the expression, let us assume $F_{SS}=\delta=0$, $N=-3+\frac{12}{m_{\Phi}^{2}}S\bar{S}$ with $S=W/2$ at the vacuum. Then, $G_{T}$ becomes $$\begin{aligned}
G_{T}\simeq -6\frac{m_{3/2}^{2}}{m_{\Phi}^{2}}-\frac{1}{3}J^{i}G_{i}.\end{aligned}$$ If we further assume for the SUSY breaking sector that $J(z,\bar{z})=|z|^{2}-\frac{|z|^{4}}{\Lambda^{2}}$ and $P(z)=\mu^{2}z+W_{0}$, $J^{i}$ is given by $J^{z}\simeq |z|\left( 1+ \frac{2}{\Lambda^{2}}|z|^{2}\right)\simeq |z| \simeq \sqrt{12}\left( \frac{m_{3/2}}{m_{z}} \right)^{2}$, where $m_{z}\simeq \sqrt{12}m_{3/2}/\Lambda$. It is implied that $$\begin{aligned}
S\simeq \frac{W}{2}\left(\frac{3}{A}+G_{T} \right)=\frac{W}{2}+{\mathcal{O}}\left( \frac{m_{3/2}^3}{m_{\Phi}^{3}} \, \text{or} \, \frac{m_{3/2}^{3}}{m_{\Phi}m_{z}^{2}} \right). \label{sshift}\end{aligned}$$ Equations and can be used to obtain shifts of quantities [*e.g.*]{} $A\simeq 3+ 3m_{3/2}^{2}/m_{\Phi}^{2}$ induced by SUSY breaking.
The mixing matrix ${\mathcal{A}}$ has two effects: canonicalization of kinetic terms and diagonalization of mass terms. We assume that there is a single SUSY breaking field $\phi^{z}=z$, and its kinetic term and mass term are dominated by the diagonal part (proportional not $zz$ nor $\bar{z}\bar{z}$ but to $z\bar{z}$) for simplicity, and then the matrix element is simplified [@Endo:2012yg] $$\begin{aligned}
({\mathcal{A}}^{-1})^{z}{}_{\Phi_{\text{R}\pm}}=&\frac{g^{\bar{z}z}}{m_{\Phi}^{2}-m_{z}^{2}} \left( \frac{\sqrt{3}}{2} \left( V_{T\bar{z}}+V_{\bar{T}\bar{z}}+J_{\bar{z}} \left(V_{T\bar{T}} +V_{\bar{T}\bar{T}}\right)\right)
\pm \frac{m_{\Phi}}{4\sqrt{3}} \left( V_{S\bar{z}}+V_{\bar{S}\bar{z}}+J_{\bar{z}} \left(V_{S\bar{T}} +V_{\bar{S}\bar{T}}\right)\right) \right) .\end{aligned}$$ For the former part regarding $T$, only the $V_{T\bar{T}}\simeq 4N^{\bar{S}S}$ term remains. If $m_{\Phi}^{2}\gg m_{z}^{2}$, this term cancels the term in $G_{T}$ proportional to $G_{z}$. For the latter part regarding $S$, all the four terms are nonzero: $$\begin{aligned}
\tilde{V}_{TS}=&-8N^{\bar{S}S}N_{SS\bar{S}}|S|^{2}-4\bar{S}, \\
\tilde{V}_{T\bar{S}}=&-8N^{\bar{S}S}N_{S\bar{S}\bar{S}}|S|^{2}+2N^{\bar{S}S}\left(\bar{F}_{\bar{S}\bar{S}} +2N_{\bar{S}}+2\bar{S}N_{\bar{S}\bar{S}}\right)-8S, \\
\tilde{V}_{S\bar{z}}+\tilde{V}_{\bar{S}\bar{z}}=&\left(4 \left(S+\bar{S}\right)J^{\bar{k}}+2P_{l}J^{l\bar{k}}\right)\left( J_{\bar{k}\bar{z}}-J_{\bar{z}k\bar{k}}J^{k}\right) -4\bar{P}_{\bar{z}} +2\bar{P}_{\bar{k}\bar{z}}J^{\bar{k}} -2\bar{P}_{\bar{k}}J^{\bar{k}l}J_{l\bar{m}\bar{z}}J^{\bar{m}},\end{aligned}$$ at the vacuum. Among these, $-4\bar{P}_{\bar{z}}$ cancels the leading term in $G_{S}=12m_{3/2}/m_{\Phi} + \cdots$ under the same condition $m_{\Phi}^2 \gg m_{z}^{2}$. Assuming $J(z,\bar{z})=|z|^{2}-\frac{|z|^{4}}{\Lambda^{2}}$ and $P(z)=\mu^{2}z+W_{0}$, subleading terms regarding this cancellation are still subdominant compared to terms in $G_{T}$.
In summary, the effective coupling is approximated as $$\begin{aligned}
\left | \mathcal{G}^{\text{(eff)}}_{\Phi_{\text{R}\pm}} \right |^{2}\simeq & 2 \left| \frac{\sqrt{3}}{2} \left( -6\frac{m_{3/2}^{2}}{m_{\Phi}^{2}}+\frac{1}{3}J^{z}G_{z}\frac{m_{z}^{2}}{m_{\Phi}^{2}-m_{z}^{2}} \right) \pm\frac{m_{\Phi}}{4\sqrt{3}}\left( \frac{12\bar{W}}{m_{\Phi}^{2}}-\frac{4G^{z}G_{z}\bar{W}}{m_{\Phi}^{2}-m_{z}^{2}} \right) \right|^{2} \nonumber \\
\simeq & 6 \left| 3\frac{m_{3/2}^{2}}{m_{\Phi}^{2}}+\frac{m_{z}^{2}}{m_{z}^{2}-m_{\Phi}^{2}} \left( \frac{1}{6}J^{z}G_{z}\mp \frac{\bar{W}}{m_{\Phi}} \right) \right|^{2}.\end{aligned}$$ Finally, the effective coupling is simplified when $m_{z}$ is in particular ranges: $$\begin{aligned}
\left | \mathcal{G}^{\text{(eff)}}_{\Phi_{\text{R}\pm}} \right |^{2}\simeq \begin{cases}
96\left(\frac{m_{3/2}}{m_{\Phi}}\right)^{4} & \left( m_{z}^{2}\ll m_{\Phi}m_{3/2} \right) \\
6\left( \frac{m_{z}^{2}m_{3/2}}{m_{\Phi}^{3}}\right)^{2} & \left( 3m_{\Phi}m_{3/2}\ll m_{z}^{2} \ll m_{\Phi}^{2} \right) \\
6 \left( \frac{m_{3/2}}{m_{\Phi}}\right)^{2} & \left( m_{\Phi}^{2}\ll m_{z}^{2}\right)
\end{cases},\end{aligned}$$ where we have assumed again $J(z,\bar{z})=|z|^{2}-\frac{|z|^{4}}{\Lambda^{2}}$ and $P(z)=\mu^{2}z+W_{0}$ to evaluate $J^{z}G_{z}$. Therefore, the gravitino pair production rate is $$\begin{aligned}
\Gamma(\Phi_{\text{R}\pm} \to \psi_{3/2}\psi_{3/2}) \simeq \frac{m_{\Phi}^{3}}{48\pi M_{\text{G}}^{2}} \times \begin{cases}
16 \left( \frac{m_{3/2}}{m_{\Phi}} \right)^{2} & \left( m_{z}^{2}\ll m_{\Phi}m_{3/2} \right) \\
\left( \frac{m_{z}}{m_{\Phi}}\right)^{4} & \left( 3m_{\Phi}m_{3/2}\ll m_{z}^{2} \ll m_{\Phi}^{2} \right) \\
1 & \left( m_{\Phi}^{2}\ll m_{z}^{2}\right)
\end{cases}.\end{aligned}$$
Constraints from gravitino abundance {#sec:cosmology}
====================================
We study gravitino abundance produced during and after reheating of the universe. Gravitino is generated by various processes, (i) direct decay of the inflaton, (ii) scattering in the thermal bath created by the inflaton decay, (iii) decay of particles such as $\chi^{S}$ and $z$ produced by inflaton decay, and (iv) decay of coherent oscillation of SUSY breaking field $z$. Similar analyses have been done in the literature, see Refs. [@Nakayama:2012hy; @Evans:2013nka; @Nakayama:2014xca] and references therein.
As for direct decay of inflaton (i), we have derived various partial decay rates in the previous sections. We assume no significant entropy dilution occurs after the reheating of the universe due to the inflaton decay. Note that the SUSY breaking field $z$ decays dominantly into a pair of gravitinos, so that it does not produce entropy when it decays. We parametrize the total decay rate of inflaton as $$\begin{aligned}
\Gamma _{\text{tot}}=X\frac{m_{\Phi}^3}{M_{\text{G}}^2} ,\label{total_decay_rate}\end{aligned}$$ where $X$ is defined by this equation. Among various decay channels, there is a generic decay channel via the anomaly-induced process. If we assume that this is the dominant mode, then $X$ is expressed as $X=N_{\text{g}}\alpha^2 b_0^2 /768\pi^3$ where $b_0=3T_{G}-T_{R}$. The branching ratio of the gravitino pair production is $$\begin{aligned}
\text{Br}(\Phi_{\text{R}\pm}\rightarrow \psi_{3/2}\psi_{3/2})\simeq \frac{1}{48\pi X} \times \begin{cases}
16 \left( \frac{m_{3/2}}{m_{\Phi}} \right)^{2} & \left( m_{z}^{2}\ll m_{\Phi}m_{3/2} \right) \\
\left( \frac{m_{z}}{m_{\Phi}}\right)^{4} & \left( 3m_{\Phi}m_{3/2}\ll m_{z}^{2} \ll m_{\Phi}^{2} \right) \\
1 & \left( m_{\Phi}^{2}\ll m_{z}^{2}\right)
\end{cases}.\end{aligned}$$
The gravitino yield $Y_{3/2}\equiv \frac{n_{3/2}}{s}$ where $n_{3/2}$ is gravitino number density and $s$ is entropy density, due to direct decay of inflaton is given by $$\begin{aligned}
Y_{3/2}^{\text{(direct)}}=\frac{3T_{\text{R}}\text{Br}_{3/2}}{2m_{\Phi}},\end{aligned}$$ where $\text{Br}_{3/2}$ is the branching ratio into a gravitino pair, and we define the reheating temperature $T_{\text{R}}$ as $$\begin{aligned}
T_{\text{R}}=\left( \frac{90}{\pi^2 g_{*}(T_{\text{R}})} \right)^{\frac{1}{4}} \sqrt{M_{\text{G}}\Gamma_{\text{tot}}}.\end{aligned}$$ The gravitino yield becomes $$\begin{aligned}
Y_{3/2}^{\text{(direct)}}=\left( \frac{90}{\pi^2 g_{*}(T_{\text{R}})} \right)^{\frac{1}{4}} \frac{1}{32\pi} \sqrt{\frac{m_{\Phi}}{XM_{\text{G}}}} \times \begin{cases}
16 \left( \frac{m_{3/2}}{m_{\Phi}} \right)^{2} & \left( m_{z}^{2}\ll m_{\Phi}m_{3/2} \right) \\
\left( \frac{m_{z}}{m_{\Phi}}\right)^{4} & \left( 3m_{\Phi}m_{3/2}\ll m_{z}^{2} \ll m_{\Phi}^{2} \right) \\
1 & \left( m_{\Phi}^{2}\ll m_{z}^{2}\right)
\end{cases}.\end{aligned}$$
The gravitino yield from thermal bath is known to be [@Bolz:2000fu; @Pradler:2006qh; @Pradler:2006hh; @Rychkov:2007uq; @Kohri:2005wn; @Nakayama:2014xca] $$\begin{aligned}
Y_{3/2}^{\text{(thermal)}}\simeq \begin{cases} \text{min}\left [ 2 \times 10^{-12} \left( 1+\frac{m_{\tilde{\text{g}}^{2}}}{3m_{3/2}^{2}} \right) \left( \frac{T_{\text{R}}}{10^{10}\text{GeV}} \right) , \frac{0.42}{g_{*s(T_{3/2})}} \right ] & (T_{\text{R}}\gtrsim m_{\text{SUSY}} ) \\ 0 & (T_{\text{R}}\lesssim m_{\text{SUSY}} ) \end{cases},\end{aligned}$$ where $m_{\tilde{\text{g}}}$ is the gaugino (gluino) mass at zero temperature, $m_{\text{SUSY}}$ is the typical soft SUSY breaking mass. We take them as $m_{\text{SUSY}}=m_{3/2}$, and $m_{\tilde{\text{g}}}=2.8\times 10^{-2}m_{3/2}$ (for $m_{3/2}\geq 10^{4.5}$ GeV; anomaly mediation) or $m_{\tilde{\text{g}}}=m_{3/2}$ (for $m_{3/2} < 10^{4.5}$ GeV; gravity mediation).
The inflaton decays into matter particles, gravitino, and SUSY breaking field. It also decays into other SUGRA sector particles ($T$, $S$, $\chi^T$, and $\chi^S$) if kinematically possible, but the rate should be highly suppressed by the phase space factor. Even if the decay is possible, these particles decay shortly after they are produced if there are Giudice-Masiero terms $|J_{ij}|\sim {\mathcal{O}}(1)$. Moreover, gravitino abundance from decay of these SUGRA sector particles $X$ will be multiply suppressed by tiny branching ratios of $\text{Br}(\Phi_{\text{R}\pm} \rightarrow X+\text{anything})$ and $\text{Br}(X \rightarrow \psi_{3/2}+\text{anything})$. Therefore we neglect effects of these SUGRA sector particles, and consider only the SUSY breaking field $z$ for the process of the type (iii).
The SUSY breaking field $z$ is produced as particles by the decay of inflaton, and it decays dominantly into a pair of gravitinos when $m_{\Phi}>2 m_{z}\gg m_{3/2}$ because the partial decay rate into them is enhanced by a factor $(m_{z}/m_{3/2})^2$, [@Endo:2006zj; @Nakamura:2006uc] $$\begin{aligned}
\Gamma ( z \rightarrow \psi_{3/2}\psi_{3/2} )= \frac{m_{z}^{5}}{96\pi m_{3/2}^{2} M_{\text{G}}^{2}}, \end{aligned}$$ while partial decay rates of other channels are of order $\Gamma = {\mathcal{O}}(m_{z}^3/4\pi M_{\text{G}}^{2})$.
The gravitino yield as a decay product of particle $z$, which in turn is created by decay of the inflaton, leads to $$\begin{aligned}
Y_{3/2}^{\text{(particle)}}=\frac{2n_{z}}{s}=\frac{3T_{\text{R}}}{m_{\Phi}}\text{Br}(\Phi_{\text{R}\pm} \rightarrow zz) = \frac{T_{\text{R}}m_{z}^{2}}{16\pi X m_{\Phi}^{3}}.\end{aligned}$$
Finally we consider the process of the type (iv). For matter fields, canonically normalized Hubble-induced mass is $\sqrt{2}H$. This value is close to that for critical damping $3H/2$, so matter fields rapidly moves to the instantaneous minimum, which can be regarded as zero for our purpose. The SUSY breaking field $z$ is also trapped near the origin until it decays at $H=H_{\text{D}}\simeq \Gamma ( z \rightarrow \psi_{3/2}\psi_{3/2} ) $ or until it starts coherent oscillation at $H=H_{\text{O}}\simeq m_{z}$. Here and hereafter the subscripts R, D, and O refer to the time of reheating, decay of $z$, and beginning of coherent oscillation of $z$, respectively. Assuming that the dominant channel is the model-independent anomaly-induced decay, $H_{\text{R}}\simeq 2.2$ GeV. For definiteness, we assume eqs. and for the SUSY breaking sector.
The VEV of $z$ is evaluated as ${\left\langle z\right\rangle}\simeq 2\sqrt{3}\left(\frac{m_{3/2}}{m_{z}}\right)^{2}$, and the energy density of coherent oscillation $z$ is $$\begin{aligned}
\rho_{z,\text{field}}=m_{z}^{2}{\left\langle z\right\rangle}^2=\frac{12m_{3/2}^4}{m_{z}^2} \times \begin{cases} 1 & (H>H_{\text{O}}) \\ \left( \frac{a}{a_{\text{O}}} \right)^{-3} & (H_{\text{O}}>H)
\end{cases},\end{aligned}$$ where $a$ is the cosmic scale factor. The entropy density is $$\begin{aligned}
s=\frac{4\rho}{3T}=\frac{4H^2}{T}.\end{aligned}$$ The gravitino yield from coherent oscillation of $z$ is thus $$\begin{aligned}
Y_{3/2}^{\text{(field)}}=\frac{2\rho_{z,\text{field}}}{m_{z}s} = \frac{ 6 m_{3/2}^4 T_{\text{R}} }{ m_{z}^5 } \hspace{20pt} (H_{\text{O}}>H_{\text{R}}>H_{\text{D}}) .\end{aligned}$$ If the mass scale of $z$ is larger than the inflation scale, $2m_{z}>m_{\Phi}$, $z$ goes close to its VEV during inflation, and the above quantity $Y_{3/2}^{\text{(field)}}$ is further suppressed by a factor $(m_{\Phi}^2/2m_{z}^2)$.
So far, we have implicitly assumed the decay of $z$ occurs at last. If the decay of $z$ occurs between $H_{\text{O}}$ and $H_{\text{R}}$, the energy density of gravitinos generated by the decay of coherent oscillation of $z$ at the time of reheating is $$\begin{aligned}
\rho_{3/2} =\frac{12m_{3/2}^4}{m_{z}^2}\left( \frac{a_{\text{D}}}{a_{\text{O}}} \right)^{-3} \left( \frac{a_{\text{NR}}}{a_{\text{D}}} \right)^{-4}\left( \frac{a_{\text{R}}}{a_{\text{NR}}} \right)^{-3},\end{aligned}$$ where NR stands for the time when gravitino becomes non-relativistic, $H_{\text{NR}}=(m_{3/2}/m_{z})^{3/2}H_{D}$. The gravitino yield is $$\begin{aligned}
Y_{3/2}^{\text{(field)}}=\frac{6m_{3/2}^{5}T_{\text{R}}}{m_{z}^6}.\end{aligned}$$ If $z$ decays even earlier than $H_{\text{O}}$, the gravitino energy density is given by $$\begin{aligned}
\rho_{3/2}=\frac{12m_{3/2}^4}{m_{z}^2} \left( \frac{a_{\text{NR}}}{a_{\text{D}}} \right)^{-4}\left( \frac{a_{\text{R}}}{a_{\text{NR}}} \right)^{-3},\end{aligned}$$ so the gravitino yield is $$\begin{aligned}
Y_{3/2}^{\text{(field)}}=\frac{6m_{3/2}^5 T_{\text{R}}}{m_{z}^4 H_{\text{D}}^2}.\end{aligned}$$
In summary, the gravitino yield from coherent $z$ field is given by $$\begin{aligned}
Y_{3/2}^{\text{(field)}}=\begin{cases}
\frac{ 6 m_{3/2}^4 T_{\text{R}} }{ m_{z}^5 } & (H_{\text{O}}>H_{\text{R}}>H_{\text{D}}) \\
\frac{6m_{3/2}^{5}T_{\text{R}}}{m_{z}^6} & (H_{\text{O}}>H_{\text{D}}>H_{\text{R}}) \\
\frac{6m_{3/2}^5 T_{\text{R}}}{m_{z}^4 H_{\text{D}}^2} & (H_{\text{D}}> H_{\text{O}}>H_{\text{R}})
\end{cases}.\end{aligned}$$
Because we assume no entropy production after inflaton decay until gravitino decay, the denominators of every $Y_{3/2}$ are common, so the cosmologically relevant gravitino yield is the sum of all four terms, $Y_{3/2}^{\text{(total)}}= Y_{3/2}^{\text{(direct)}} +Y_{3/2}^{\text{(thermal)}} +Y_{3/2}^{\text{(particle)}} +Y_{3/2}^{\text{(field)}}$.
![Constraint on masses of gravitino and SUSY breaking field from LSP overabundance from gravitino decay. Blue, red, yellow, and green shaded regions, corresponding to direct production, thermal production, $z$ particle decay, and $z$ coherent oscillation decay, are excluded.[]{data-label="fig:constraints"}](./constraints.eps){width="7cm"}
Now that we have derived generic expressions for gravitino abundance, let us discuss its cosmological consequences for a minimal setup. Gravitinos heavier than about $30$ TeV decay before big bang nucleosynthesis (BBN), but lightest supersymmetric particles (LSPs) produced by the gravitino decay chain may exceed the observed dark matter abundance. Such a constraint is shown in Fig. \[fig:constraints\] assuming wino LSP of anomaly mediation for $m_{3/2} \geq 10^{4.5} \text{GeV}$. For smaller gravitino mass, $m_{3/2} < 10^{4.5}\text{GeV}$, gravitino decay affects light element abundance. We assume gravity mediation for this mass region and impose the standard BBN constraints [@Kohri:2005wn] on the parameter space in Fig. \[fig:constraints\]. In this figure, the dominant decay mode of the inflaton is assumed to be a model-independent one, namely the anomaly-induced decay into gauge bosons and gauginos as discussed in subsection \[sec:anomaly\]. The inflaton mass is taken as $m_{\Phi}=3.2 \times 10^{13}$ GeV, and the reheating temperature after inflaton decay is $T_{\text{R}}\simeq1.0\times 10^9$ GeV. Instantaneous reheating occurs in spite of the Planck-suppressed interaction [@Harigaya:2013vwa]. As can be seen from the Figure, most of the parameter space are excluded. The lower unshaded region is also excluded by the standard constraint of the cosmological moduli problem [@Coughlan:1983ci; @Ref:Moduli] unless baryon asymmetry is regenerated *e.g.* by the Affleck-Dine mechanism [@Affleck:1984fy]. (In this case the modulus (Polonyi) field is the SUSY breaking field $z$.) Note that the range of gravitino mass $10^{6}\text{GeV} \lesssim m_{3/2} \lesssim 3\times10^{11}\text{GeV}$ (corresponding to $3 \text{TeV} \lesssim m_{\text{wino}} \lesssim T_{\text{R}}$; not shown in the Figure) is excluded by thermally produced wino abundance [@Hisano:2006nn] even without considering the wino LSP from gravitino decay. See also Ref. [@Moroi:2013sla] for non-thermal production of wino dark matter via the decay of long-lived particles. As usual, this problem is ameliorated or solved by assuming $R$-parity breaking so that LSP decays or thermal inflation [@Lyth:1995ka] so that it is diluted.
Summary and Discussion {#sec:conclusion}
======================
In this paper, we studied coupling of the SUSY Starobinsky model to matter sector in the old-minimal supergravity, inflaton decay and its cosmological consequences. To this end, we first transformed the supergravity theory of supercurvature ${\mathcal{R}}$ minimally coupled to matter to an equivalent one in the form of the standard no-scale type supergravity of inflaton $T$ plus another matter superfield $S$. The notable feature there is that the interactions of the inflaton $T$ to other superfields in the theory are completely determined by the fact that the origin of $T$ is a Lagrange multiplier. In particular, the inflaton $T$ does not enter in the gauge kinetic function. These are characteristic features of the SUSY Starobinsky model, unlike some other SUGRA models having Starobinsky-like scalar potentials.
On the other hand, interactions of $S$ have more freedom. In this paper, we assumed minimal coupling between SUGRA sector and matter sector in the first place, but it is not protected by any symmetries so more general coupling between $S$ and matter are possible. It may enhance decay rates of inflaton into matter through mixing between $T$ and $S$, which results in a suppressed branching ratio into gravitino.
We focused on model-independent decay channel of inflaton into gauge sector via the anomaly-induced decay in section \[sec:cosmology\], but presence of heavy matter, like right-handed (s)neutrinos, and large quadratic holomorphic term $J_{ij}$ in Kähler potential, which is used for the Giudice-Masiero mechanism [@Giudice:1988yz], are helpful to reheat the universe efficiently. These are simply because there are decay modes whose rates are proportional to matter mass or $J_{ij}$.
Taking anomaly-induced decay into the gauge sector as the dominant decay channel, the lower limit of the reheating temperature is a similar value, $T_{\text{R}}\simeq 1.0\times 10^{9}$ GeV, to that of the non-SUSY original Starobinsky model, and it is consistent with thermal leptogenesis [@Fukugita:1986hr]. The most striking difference to the non-SUSY case is presence of the built-in long lived particle in the theory, gravitino. We assumed gravity/anomaly mediation of SUSY breaking, and estimated the amount of LSPs produced from decay of gravitino, which is produced either by direct decay of inflaton, thermal scattering, decay of SUSY breaking particle or field $z$. The result is that most of the parameter space $(m_{3/2}, m_{z})$ is excluded unless $R$-parity is broken or thermal inflation occurs. Thus, our prediction of the mass of gravitino is $10^{4} \text{GeV} \lesssim m_{3/2} \lesssim 10^{5} \text{GeV}$. A way around this is considering more general coupling between SUGRA sector and matter sector in the original higher supercurvature SUGRA theory.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Iannis Dalianis for clarifying the model in Ref. [@Dalianis:2014aya]. TT thanks Motoi Endo, Kazunori Nakayama, Fuminobu Takahashi, and Masahiro Takimoto for valuable discussion. TT was supported partly by a grant of Advanced Leading Graduate Course for Photon Science in the University of Tokyo, and partly by a Grant-in-Aid for JSPS Fellows, and a Grant-in-Aid of the JSPS under No. 26$\cdot$10619. YW acknowledges supports from the JSPS Research Fellowship for Young Scientists No. 269337 and the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence “Origin and Structure of the Universe." YY acknowledges support from the JSPS Research Fellowship for Young Scientists No. 264236. JY acknowledges support form the JSPS Grant-in-Aid for Scientific Research (B) No. 23340058.
Duality transformation of higher derivative SUGRA models {#app}
=========================================================
In this Appendix, we briefly review the duality transformation between the higher derivative SUGRA system and the standard one described in Sec.
\[sec:setup\]. We explicitly show that the superpotential and the gauge kinetic function in the standard SUGRA are linear in and independent of $T$, respectively. We also discuss some generalizations.
The action including general couplings between ${\cal R}$ and matters [@Cecotti:1987sa] is given by $$\begin{aligned}
S=&\int d^4xd^4\theta EN\left({\cal R},\bar{\cal R},\phi,\bar{\phi}e^{gV}\right)\nonumber\\
&+\left[\int d^4x d^2\Theta 2{\mathscr{E}}\left(F\left({\cal R},\phi \right)+\frac{1}{4} h_{AB}\left({\cal R},\phi \right)W^AW^B\right)+{\rm H.c.}\right ],\label{genR2}\end{aligned}$$ where ${\cal R}$, $\phi$ and $W^A$ are the same as in Sec. \[sec:setup\], $N$ is a real function of ${\cal R}$, $\phi$ and their conjugates, and $F$ and $h_{AB}$ are holomorphic functions of ${\cal R}$ and $\phi$. By introducing the Lagrange multiplier chiral multiplet $T$ and a chiral multiplet $S$, the action (\[genR2\]) becomes the following form, $$\begin{aligned}
S=&\int d^4xd^4\theta EN\left(S,\bar{S},\phi,\bar{\phi}e^{gV}\right)\nonumber\\
&+\left[\int d^4x d^2\Theta 2{\mathscr{E}}\left(2T(S-{\cal R})+F(S,\phi )+\frac{1}{4}h_{AB}(S,\phi)W^AW^B \right)+{\rm H.c.}\right ].\label{multiplier}\end{aligned}$$ Varying it with respect to $T$ yields the equation ${\cal R}=S$, and we obtain the original action (\[genR2\]). We can also rewrite the action (\[multiplier\]) into the standard SUGRA form not containing higher curvature terms as $$\begin{aligned}
S=&\int d^4xd^4\theta E\left[ N\left(S,\bar{S},\phi,\bar{\phi}e^{gV} \right)-\left(T+\bar{T}\right)\right] \nonumber\\
&+\left[\int d^4x d^2\Theta 2{\mathscr{E}}\left( 2TS+ F(S,\phi )+\frac{1}{4}h_{AB}(S,\phi)W^AW^B\right)+{\rm H.c.}\right ]\nonumber\\
=&\int d^4xd^2\Theta 2{\mathscr{E}}\left[\frac{3}{8}\left(\bar{{\mathscr{D}}}\bar{{\mathscr{D}}}-8{\cal R}\right)e^{-K/3}+W+\frac{1}{4}h_{AB}(S,\phi)W^AW^B\right]+{\rm H.c.},\label{dual}\end{aligned}$$ where $$\begin{aligned}
K&=-3\ln\left(\frac{T+\bar{T}-N\left(S,\bar{S},\phi,\bar{\phi}e^{gV}\right)}{3}\right),\\
W&=2TS+F(S,\phi).\end{aligned}$$ Notice that in the dual action (\[dual\]), $T$ does not appear in the gauge kinetic function $h_{AB}$ even if there are couplings between ${\cal R}$ and $W^A_{\alpha}$ in the original action (\[genR2\]). The absence of $T$ in the gauge kinetic function $h_{AB}$ is a remarkable feature of the Starobinsky inflation in old-minimal SUGRA models, which restricts the main reheating processes to the anomaly induced decays into the gauge sector as discussed in Sec. \[sec:decay\].
We have shown that the naive generalization of the action (\[minimal\]) does not contain $T$-dependence in gauge kinetic function $h_{AB}$. One may wonder what happens if we introduce dependence of the gauge kinetic function on $T$ in eq. and transform it back to a higher derivative SUGRA. As a minimal extension of eq. , let us consider the following action in which the gauge kinetic function linearly depends on $T$, $$\begin{aligned}
S=&\int d^4xd^4\theta E\left[ N\left(S,\bar{S},\phi,\bar{\phi}e^{gV}\right)-\left(T+\bar{T}\right)\right]\nonumber\\
&+\left[\int d^4x d^2\Theta 2{\mathscr{E}}\left( 2TS+F(S,\phi )+\left(\frac{1}{4}h_{AB}(S,\phi)-2H_{AB}T\right)W^AW^B\right)+{\rm H.c.}\right ],\label{ext}\end{aligned}$$ where $H_{AB}$ is a constant. Here, to obtain the dual action of (\[ext\]), we follow the way discussed in Ref. [@Cecotti:2014ipa]. We can recast the action (\[ext\]) into the dual form as $$\begin{aligned}
S=&\int d^4xd^4\theta E N\left(S,\bar{S},\phi,\bar{\phi}e^{gV}\right)\nonumber\\
&+\left[\int d^4x d^2\Theta 2{\mathscr{E}}\left(2T \left(S-{\cal R}-H_{AB}W^AW^B \right)+F(S,\phi )+\frac{1}{4}h_{AB}(S,\phi)W^AW^B\right)+{\rm H.c.}\right ].\end{aligned}$$ Varying the above action with respect to $T$ yields $S={\cal R}+H_{AB}W^AW^B$. Substituting it into the action gives $$\begin{aligned}
S=&\int d^4xd^4\theta E N\left({\cal R}+H_{AB}W^A W^B,\bar{\cal R}+\bar{H}_{AB}\bar{W}^A \bar{W}^B,\phi,\bar{\phi}e^{gV}\right) \nonumber\\
&+\left[\int d^4x d^2\Theta 2{\mathscr{E}}\left(F\left({\cal R}+H_{AB}W^AW^B,\phi \right)+\frac{1}{4}h_{AB}\left({\cal R}+H_{CD}W^CW^D,\phi \right) W^AW^B \right)+{\rm H.c.}\right ].\label{exdual}\end{aligned}$$ Notice that the dual action (\[exdual\]) contains higher dimensional operators involving $H_{AB}W^AW^B$. It means that the theory contains the higher derivative terms of the gauge multiplets $V^A$. In such a case, the inflaton $T$ can decay into gauge bosons and gauginos through tree level couplings in the gauge kinetic function (or in the Kähler potential depending on the shift of $S$). Then, reheating processes can be different from the ones we discussed in Sec. \[sec:decay\].
For completeness, we finally discuss a possibility that the superpotential and the gauge kinetic function are non-linear in $T$. We generalize the $F$-term action as $$\begin{aligned}
S=&\int d^4xd^4\theta E\left[ N\left(S,\bar{S},\phi , \bar{\phi}e^{gV}\right)-\left(T+\bar{T}\right)\right]\nonumber\\
&+\left[\int d^4x d^2\Theta 2{\mathscr{E}}\left(F(T, S, \phi )+\frac{1}{4}h_{AB}(T, S, \phi) W^{A} W^{B}\right)+{\rm H.c.}\right ],\label{nl}\end{aligned}$$ where $F(T, S, \phi )$ and $h_{AB}(T, S, \phi )$ are holomorphic functions of $T$, $S$, and $\phi^i$. We can rewrite the action (\[nl\]) as $$\begin{aligned}
S=&\int d^4xd^4\theta EN\left(S,\bar{S},\phi , \bar{\phi}e^{gV}\right)\nonumber\\
&+\left[\int d^4x d^2\Theta 2{\mathscr{E}}\left(-2{\mathcal{R}}+F(T, S, \phi )+\frac{1}{4}h_{AB}(T, S, \phi) W^{A} W^{B}\right)+{\rm H.c.}\right ]. \label{nldual}\end{aligned}$$ Varying the above action with respect to $T$ yields $$\begin{aligned}
F_{T} (T, S, \phi )+ \frac{1}{4}h_{AB, T} (T, S, \phi ) W^{A}W^{B} - 2{\mathcal{R}}=0,\end{aligned}$$ and it can be implicitly solved as $S=S({\mathcal{R}}, T, \phi )$. Substituting it to eq. leads to a higher derivative SUGRA depending on ${\mathcal{R}}$, $\phi^i$, and an additional matter $T$. Notice that dependence on the additional matter $T$ vanishes if and only if $F(T, S, \phi )$ and $h_{AB}(T,S, \phi )$ are linear functions of $T$. Therefore, the non-linear dependence of $T$ in the superpotential or the gauge kinetic function requires a new chiral multiplet $T$ in the dual higher derivative SUGRA theory.
In this paper, we focus on the case that $T$ appears as the degree of freedom purely originated from the higher derivative SUGRA terms. In this case, the corresponding action of the standard SUGRA is given by eq. \[or more generally by eq. \].
[99]{} G. Hinshaw [*et al.*]{} \[WMAP Collaboration\], Astrophys. J. Suppl. [**208**]{} (2013) 19; arXiv:1212.5226 \[astro-ph.CO\].
P. A. R. Ade [*et al.*]{} \[Planck Collaboration\], arXiv:1303.5076 \[astro-ph.CO\]; arXiv:1303.5082 \[astro-ph.CO\]. A. A. Starobinsky, Phys. Lett. B [**91**]{} (1980) 99; in: *Quantum Gravity, Proc. of the Second Seminar “Quantum Theory of Gravity" (Moscow, 13-15 Oct. 1981)*, eds. M. A. Markov and P. C. West (Plenum Publ. Co., New York, 1984) pp. 103-128. K. Sato, Mon. Not. Roy. Astron. Soc. [**195**]{}, 467 (1981). A. H. Guth, Phys. Rev. D [**23**]{}, 347 (1981). A. D. Linde, Phys. Lett. B [**108**]{}, 389 (1982).
A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. [**48**]{}, 1220 (1982).
A. D. Linde, Phys. Lett. B [**129**]{}, 177 (1983).
For a review of inflation see [*e.g.*]{}, J. Yokoyama, PTEP [**2014**]{}, no. 6, 06B103 (2014); S. Tsujikawa, PTEP [**2014**]{}, no. 6, 06B104 (2014) M. B. Mijic, M. S. Morris and W. -M. Suen, Phys. Rev. D [**34**]{}, 2934 (1986).
B. Whitt, Phys. Lett. B [**145**]{} (1984) 176. K. Maeda, Phys. Rev. D [**37**]{}, 858 (1988). A. Vilenkin, Phys. Rev. D [**32**]{}, 2511 (1985). Y. Watanabe and E. Komatsu, Phys. Rev. D [**75**]{}, 061301(R) (2007) \[gr-qc/0612120\]; T. Faulkner, M. Tegmark, E. F. Bunn and Y. Mao, Phys. Rev. D [**76**]{}, 063505 (2007) \[astro-ph/0612569\].
D. S. Gorbunov and A. G. Panin, Phys. Lett. B [**700**]{} (2011) 157 \[arXiv:1009.2448 \[hep-ph\]\].
E. V. Arbuzova, A. D. Dolgov and L. Reverberi, JCAP [**1202**]{}, 049 (2012) \[arXiv:1112.4995 \[gr-qc\]\]; H. Motohashi and A. Nishizawa, Phys. Rev. D [**86**]{}, 083514 (2012) \[arXiv:1204.1472 \[astro-ph.CO\]\]; I. Rudenok, Y. Shtanov and S. Vilchinskii, Phys. Lett. B [**733**]{} (2014) 193 \[arXiv:1401.7298 \[astro-ph.CO\]\].
N. Takeda and Y. Watanabe, Phys. Rev. D [**90**]{}, 023519 (2014) \[arXiv:1405.3830 \[astro-ph.CO\]\].
V. F. Mukhanov and G. V. Chibisov, JETP Lett. [**33**]{}, 532 (1981) \[Pisma Zh. Eksp. Teor. Fiz. [**33**]{}, 549 (1981)\]; V. F. Mukhanov, astro-ph/0303077.
P. A. R. Ade [*et al.*]{} \[BICEP2 Collaboration\], Phys. Rev. Lett. [**112**]{} (2014) 241101 \[arXiv:1403.3985 \[astro-ph.CO\]\]. M. J. Mortonson and U. Seljak, JCAP [**1410**]{} (2014) 10, 035 \[arXiv:1405.5857 \[astro-ph.CO\]\]. R. Flauger, J. C. Hill and D. N. Spergel, JCAP [**1408**]{} (2014) 039 \[arXiv:1405.7351 \[astro-ph.CO\]\]. R. Adam [*et al.*]{} \[Planck Collaboration\], arXiv:1409.5738 \[astro-ph.CO\].
D. Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D [**13**]{} (1976) 3214;\
S. Deser and B. Zumino, Phys. Lett. B [**62**]{} (1976) 335. S. Ferrara and P. van Nieuwenhuizen, Phys. Lett. B [**74**]{} (1978) 333. K. S. Stelle and P. C. West, Phys. Lett. B [**74**]{} (1978) 330. E. S. Fradkin and M. A. Vasiliev, Lett. Nuovo Cim. [**22**]{} (1978) 651. M. F. Sohnius and P. C. West, Phys. Lett. B [**105**]{} (1981) 353. S. Ferrara, L. Girardello, T. Kugo and A. Van Proeyen, Nucl. Phys. B [**223**]{}, 191 (1983). S. Ferrara, M. T. Grisaru and P. van Nieuwenhuizen, Nucl. Phys. B [**138**]{} (1978) 430. S. Cecotti, Phys. Lett. B [**190**]{} (1987) 86. R. Kallosh and A. Linde, JCAP [**1306**]{} (2013) 028 \[arXiv:1306.3214 \[hep-th\]\]. A. Hindawi, B. A. Ovrut and D. Waldram, Nucl. Phys. B [**476**]{} (1996) 175 \[hep-th/9511223\]. S. V. Ketov, PTEP [**2013**]{} 123B04 \[arXiv:1309.0293 \[hep-th\]\]. S. V. Ketov and T. Terada, JHEP [**1312**]{} (2013) 040 \[arXiv:1309.7494 \[hep-th\]\]. S. J. Gates, Jr. and S. V. Ketov, Phys. Lett. B [**674**]{} (2009) 59 \[arXiv:0901.2467 \[hep-th\]\]. S. V. Ketov, Int. J. Mod. Phys. A [**28**]{} (2013) 1330021 \[arXiv:1201.2239 \[hep-th\]\]; Phys. Lett. B [**692**]{}, 272 (2010) \[arXiv:1005.3630 \[hep-th\]\]; S. V. Ketov and A. A. Starobinsky, JCAP [**1208**]{}, 022 (2012) \[arXiv:1203.0805 \[hep-th\]\].
S. V. Ketov and T. Terada, JHEP [**1307**]{} (2013) 127 \[arXiv:1304.4319 \[hep-th\]\]. S. V. Ketov and S. Tsujikawa, Phys. Rev. D [**86**]{}, 023529 (2012) \[arXiv:1205.2918 \[hep-th\]\]. Y. Watanabe and J. Yokoyama, Phys. Rev. D [**87**]{}, 103524 (2013) \[arXiv:1303.5191 \[hep-th\]\].
J. Ellis, D. V. Nanopoulos and K. A. Olive, Phys. Rev. Lett. [**111**]{} (2013) 111301 \[arXiv:1305.1247 \[hep-th\]\]; JCAP [**1310**]{} (2013) 009 \[arXiv:1307.3537\]. S. Ferrara, R. Kallosh and A. Van Proeyen, JHEP [**1311**]{} (2013) 134 \[arXiv:1309.4052 \[hep-th\]\]. S. Ferrara, A. Kehagias and M. Porrati, Phys. Lett. B [**727**]{} (2013) 314 \[arXiv:1310.0399 \[hep-th\]\]. S. Cecotti, S. Ferrara, M. Porrati and S. Sabharwal, Nucl. Phys. B [**306**]{} (1988) 160. F. Farakos, A. Kehagias and A. Riotto, Nucl. Phys. B [**876**]{} (2013) 187 \[arXiv:1307.1137\]. S. Ferrara, R. Kallosh, A. Linde and M. Porrati, Phys. Rev. D [**88**]{} (2013) 8, 085038 \[arXiv:1307.7696 \[hep-th\]\]; JCAP [**1311**]{} (2013) 046 \[arXiv:1309.1085 \[hep-th\]\]. W. Buchmüller, V. Domcke and K. Schmitz, JCAP [**1304**]{} (2013) 019 \[arXiv:1210.4105 \[hep-ph\]\]; W. Buchmuller, V. Domcke and K. Kamada, Phys. Lett. B [**726**]{} (2013) 467 \[arXiv:1306.3471 \[hep-th\]\].
J. Alexandre, N. Houston and N. E. Mavromatos, Phys. Rev. D [**89**]{} (2014) 2, 027703 \[arXiv:1312.5197 \[gr-qc\]\]. C. Pallis, JCAP [**1404**]{} (2014) 024 \[arXiv:1312.3623 \[hep-ph\]\]. A. Hindawi, B. A. Ovrut and D. Waldram, Phys. Lett. B [**381**]{} (1996) 154 \[hep-th/9602075\]. S. Ferrara and M. Porrati, Phys. Lett. B [**737**]{} (2014) 135 \[arXiv:1407.6164 \[hep-th\]\]. K. Kamada and J. Yokoyama, arXiv:1405.6732 \[hep-th\].
J. Wess and J. Bagger, “Supersymmetry and supergravity”, Princeton, USA: Univ. Pr. (1992) 259 p G. A. Diamandis, B. C. Georgalas, K. Kaskavelis, P. Kouroumalou, A. B. Lahanas and G. Pavlopoulos, arXiv:1411.5785 \[hep-th\]. S. Cecotti and R. Kallosh, JHEP [**1405**]{}, 114 (2014) \[arXiv:1403.2932 \[hep-th\]\]. T. Kugo and S. Uehara, Nucl. Phys. B [**222**]{}, 125 (1983). H. Abe, S. Aoki, F. Hasegawa and Y. Yamada, arXiv:1408.4875 \[hep-th\]. I. Dalianis, F. Farakos, A. Kehagias, A. Riotto and R. von Unge, arXiv:1409.8299 \[hep-th\].
G. F. Giudice and A. Strumia, Nucl. Phys. B [**858**]{} (2012) 63 \[arXiv:1108.6077 \[hep-ph\]\].
M. Endo, F. Takahashi and T. T. Yanagida, Phys. Rev. D [**76**]{} (2007) 083509 \[arXiv:0706.0986 \[hep-ph\]\]. M. Endo, K. Kadota, K. A. Olive, F. Takahashi and T. T. Yanagida, JCAP [**0702**]{} (2007) 018 \[hep-ph/0612263\]. M. Endo, F. Takahashi and T. T. Yanagida, Phys. Lett. B [**658**]{} (2008) 236 \[hep-ph/0701042\]. Y. Watanabe, Phys. Rev. D [**83**]{}, 043511 (2011) \[arXiv:1011.3348 \[hep-th\]\]; D. Gorbunov and A. Tokareva, JCAP [**1312**]{}, 021 (2013) \[arXiv:1212.4466 \[astro-ph.CO\]\].
M. Dine, R. Kitano, A. Morisse and Y. Shirman, Phys. Rev. D [**73**]{} (2006) 123518 \[hep-ph/0604140\]. M. Endo, K. Hamaguchi and F. Takahashi, Phys. Rev. D [**74**]{} (2006) 023531 \[hep-ph/0605091\]. W. Buchmuller, K. Hamaguchi, M. Ratz and T. Yanagida, Phys. Lett. B [**588**]{} (2004) 90 \[hep-ph/0402179\]. M. Endo, K. Hamaguchi and F. Takahashi, Phys. Rev. Lett. [**96**]{} (2006) 211301 \[hep-ph/0602061\]. S. Nakamura and M. Yamaguchi, Phys. Lett. B [**638**]{} (2006) 389 \[hep-ph/0602081\]. M. Endo, K. Hamaguchi and T. Terada, Phys. Rev. D [**86**]{} (2012) 083543 \[arXiv:1208.4432 \[hep-ph\]\]. M. Kawasaki, F. Takahashi and T. T. Yanagida, Phys. Lett. B [**638**]{} (2006) 8 \[hep-ph/0603265\]. M. Kawasaki, F. Takahashi and T. T. Yanagida, Phys. Rev. D [**74**]{} (2006) 043519 \[hep-ph/0605297\]. M. Endo, M. Kawasaki, F. Takahashi and T. T. Yanagida, Phys. Lett. B [**642**]{} (2006) 518 \[hep-ph/0607170\]. T. Asaka, S. Nakamura and M. Yamaguchi, Phys. Rev. D [**74**]{} (2006) 023520 \[hep-ph/0604132\]. K. Nakayama, F. Takahashi and T. T. Yanagida, Phys. Lett. B [**718**]{} (2012) 526 \[arXiv:1209.2583 \[hep-ph\]\]. J. L. Evans, M. A. G. Garcia and K. A. Olive, JCAP [**1403**]{} (2014) 022 \[arXiv:1311.0052 \[hep-ph\]\]. K. Nakayama, F. Takahashi and T. T. Yanagida, Phys. Lett. B [**734**]{} (2014) 358 \[arXiv:1404.2472 \[hep-ph\]\]. M. Bolz, A. Brandenburg and W. Buchmuller, Nucl. Phys. B [**606**]{} (2001) 518 \[Erratum-ibid. B [**790**]{} (2008) 336\] \[hep-ph/0012052\]. J. Pradler and F. D. Steffen, Phys. Rev. D [**75**]{} (2007) 023509 \[hep-ph/0608344\]. J. Pradler and F. D. Steffen, Phys. Lett. B [**648**]{} (2007) 224 \[hep-ph/0612291\]. V. S. Rychkov and A. Strumia, Phys. Rev. D [**75**]{} (2007) 075011 \[hep-ph/0701104\]. K. Kohri, T. Moroi and A. Yotsuyanagi, Phys. Rev. D [**73**]{} (2006) 123511 \[hep-ph/0507245\]. K. Harigaya and K. Mukaida, JHEP [**1405**]{} (2014) 006 \[arXiv:1312.3097 \[hep-ph\]\]. G. D. Coughlan, W. Fischler, E. W. Kolb, S. Raby and G. G. Ross, Phys. Lett. B [**131**]{} (1983) 59. T. Banks, D. B. Kaplan and A. E. Nelson, Phys. Rev. D [**49**]{} (1994) 779 \[hep-ph/9308292\];\
B. de Carlos, J. A. Casas, F. Quevedo and E. Roulet, Phys. Lett. B [**318**]{} (1993) 447 \[hep-ph/9308325\]. I. Affleck and M. Dine, Nucl. Phys. B [**249**]{} (1985) 361. J. Hisano, S. Matsumoto, M. Nagai, O. Saito and M. Senami, Phys. Lett. B [**646**]{} (2007) 34 \[hep-ph/0610249\]. T. Moroi, M. Nagai and M. Takimoto, JHEP [**1307**]{} (2013) 066 \[arXiv:1303.0948 \[hep-ph\]\]. D. H. Lyth and E. D. Stewart, Phys. Rev. D [**53**]{} (1996) 1784 \[hep-ph/9510204\]. G. F. Giudice and A. Masiero, Phys. Lett. B [**206**]{} (1988) 480. M. Fukugita and T. Yanagida, Phys. Lett. B [**174**]{} (1986) 45.
[^1]: In the case without any higher derivative terms, the equivalence between different formulations of SUGRA is shown from the conformal SUGRA viewpoint [@Ferrara:1983dh].
[^2]: The first, non-holomorphic term is called $D$-term action as it is from $D$-component of Kähler potential of ${\cal R}$, while the second, holomorphic term is called $F$-term action as it is from $F$-component of superpotential of ${\cal R}$.
[^3]: Recently, the work [@Diamandis:2014vxa] suggested a higher derivative SUGRA model in which a superpotential term of $S$ and $T$ is given by $W=g(T)S$. Such a superpotential can be realized if $T$ is not a Lagrange multiplier but a chiral multiplet coupled to $\cal R$ and $\bar{\cal R}$ (see Ref. [@Cecotti:2014ipa] for an earlier discussion). We briefly discuss similar extensions in Appendix \[app\]. In this work, we discuss the minimal case that the chiral multiplets $T$ and $S$ are purely originated from the gravitational multiplet and its higher derivative modes, and that the superpotential term of $T$ and $S$ is given by $W=2TS$ as in eq. (\[superp\]).
[^4]: We briefly discuss a possibility of $T$ dependent gauge kinetic functions in Appendix \[app\].
[^5]: It is often denoted as $\Omega=-3A$ in the standard notation [@Wess:1992cp], and $\tilde{\phi}=-3A$ in the conformal SUGRA notation [@Kugo:1982mr]. The functional form of $\Omega$ is important for the SUSY breaking effects on inflationary dynamics [@Abe:2014opa].
[^6]: Although it vanishes at the leading order, it has a value of the order of the gravitino mass after SUSY breaking. See the following discussion.
|
---
abstract: |
We consider models of inflation in supergravity with a shift symmetry. We focus on models with one moduli and one inflaton field. The presence of this symmetry guarantees the existence of a flat direction for the inflaton field. Mildly breaking the shift symmetry using a superpotential which depends not only on the moduli but also on the inflaton field allows one to lift the inflaton flat direction. Along the inflaton direction, the $\eta$-problem is alleviated. Combining the KKLT mechanism for moduli stabilization and a shift symmetry breaking superpotential of the chaotic inflation type, we find models reminiscent of “mutated hybrid inflation” where the inflationary trajectory is curved in the moduli–inflaton plane. We analyze the phenomenology of these models and stress their differences with both chaotic and hybrid inflation.
Keywords : Inflation, Supergravity, Superstring Theory, Cosmology
author:
- 'Philippe Brax [^1]'
- 'Jérôme Martin [^2]'
title: Shift Symmetry and Inflation in Supergravity
---
Introduction
============
The recent observations of the Cosmic Microwave Background (CMB) fluctuations [@wmap] give a strong hint in favor of an early period of inflation [@inflation] in the history of the Universe. Inflation is an attractive scenario as it allows us to avoid the difficulties which plague the standard hot big bang theory, [*e.g. *]{}the flatness and the horizon problems. Moreover, when combined with quantum mechanics, inflation can also give rise to a satisfactory model for structure formation, see Refs. [@pert]. As a matter of fact, inflation implies that the power spectrum of the cosmological perturbations should be (almost) scale invariant [@pert], a prediction which has been known for a long time to be in good agreement with the astrophysical data. In fact, the data are now so accurate that one can start probing the details of the inflationary scenario. For instance, the small deviation from scale invariance in the power spectrum predicted by inflation (for density perturbations and for gravitational waves) directly encodes the underlying high energy physics responsible for the phase of accelerated expansion [@slowroll] implying that the model building issue for inflation is meaningful [@LR].
The above-mentioned observations and the endeavor to better understand space–like singularities like the big-bang have sparked a renewed interest in cosmological models based on string theory and/or supergravity [@cosmological]. In this framework, alternatives to inflation have also been studied like, for instance, the pre big-bang scenario [@PBB] or the ekpyrotic model [@ekp], but so far no scenario has been as successful as inflation. Hence, finding a satisfactory inflationary scenario from the most recent ideas in string theory is an important challenge [@stringinflation]. In all these attempts, our universe is pictured as a moving brane embedded in a compactification space. The acceleration of the universe is caused by the motion of the brane [@Dvali]. At the level of the four dimensional effective description obtained after compactification, the theory results in particular supergravity models. Then, in general, two main issues must be addressed in order to obtain a satisfactory scenario.
The first problem is to obtain a sufficiently flat potential. One of the crucial stumbling blocks of F–term inflation in supergravity is the presence of large ${\cal O}\left(H \right)$ corrections (where $H$ is the Hubble parameter during inflation) to the inflaton mass, which spoil the flatness of the inflaton potential. In order to lead to realistic models, the string–inspired scenarios for inflation must overcome this problem. Another highly conspicuous problem is the stabilization issue. In string inflation, the 10–dimensional type IIB theory is compactified on a Calabi-Yau manifold. Calabi-Yau compactifications lead to two types of moduli fields describing the Kähler and complex structure deformations. There is also another scalar degree of freedom originating from the 10–dimensional dilaton. Now all these moduli fields need to be stabilized in order to guarantee the flatness of the inflaton potential, i.e. no runaway behavior in the moduli directions.
Different solutions to the above-mentioned questions have been proposed. In particular, it has been noted recently that one can obtain flat enough potentials by requiring that a shift symmetry $\phi \to \phi + c$, where $c$ is a real constant, is a symmetry of the Kähler potential, later broken mildly. This is particularly natural within the low energy description of brane dynamics emerging from string theory [@Tye; @shift; @Hsu; @D3; @koyama]. On the other hand, for the stabilization question, the complex structure moduli and the dilaton can be generically stabilized when fluxes are turned on. This leaves only the Kähler moduli as flat directions. The Kähler moduli can be also stabilized once strong coupling effects, such as gaugino condensation, take place on a stack of $D7$ branes wrapped around a four-cycle in the Calabi-Yau variety. This leads to an ${\rm AdS}_4$ supergravity background. Now de Sitter space can be achieved by incorporating an anti $D3$ brane whose energy density makes the total energy density positive, this is the KKLT stabilization mechanism [@KKLT]. It was then realized that the lifting of the vacuum energy can be performed in a supersymmetric way using Fayet–Iliopoulos terms [@Burgess2]. In supergravity, it has been recently argued that this mechanism is problematic as the Fayet–Iliopoulos terms of supergravity vanish in a supergravity vacuum [@nilles].
A first attempt to combine inflation with the KKLT mechanism was carried out in the KKLMMT paper, see Ref. [@KKLMMT]. In this model, string inflation is obtained when a pair of $D3$-anti $D3$ branes is added to the configuration needed for the KKLT mechanism and described above. The anti $D3$ is naturally sitting on top of the other anti $D3$ branes, the distance between the $D3$ and anti $D3$ branes playing the role of the inflaton. The shift symmetry is explicitly broken by the non-perturbative superpotential leading to the stabilization of the Kähler moduli. The flat directions corresponding the free motion of a $D3$ brane in the Calabi-Yau space are then lifted in a strong manner. As a consequence, since the shift symmetry is absent, the corresponding inflationary model suffers from the $\eta$ problem. More precisely, it has been shown that the squared mass of the inflaton is $2H^2$ spoiling the flatness of the potential. This example is typical of the general flatness problem of supergravity potentials when no shift symmetry is present.
Subsequent papers have tried to overcome this problem by taking into account the shift symmetry while still relying on the KKLT mechanism for stabilizing the moduli fields. The first model [@Hsu] to combine both aspects was based on an interesting type of configuration (different from the one envisaged in the KKLMMT paper) corresponding to $D7$ branes evolving in the background of a very heavy stack of $D3$ branes [@Hsu; @koyama; @D3]. In this case, one may consider the free motion of the $D7$ brane and the associated shift symmetry. The model realizes hybrid inflation in string inflation. The waterfall fields are represented by the charged open strings between the $D3$ and the $D7$ branes. When the distance between the branes is large enough, the configuration admits a flat direction which is lifted at one loop as in supersymmetric hybrid inflation hence giving a logarithmic slope to the potential in the inflaton direction. The presence of a flat direction is directly linked to the fact that the superpotential takes the form required to both preserve the shift symmetry and stabilize the Kähler moduli. For small distances between the branes, the waterfall fields condense and inflation ends.
Another possibility was studied in Refs. [@Tye]. It consists in implementing the shift symmetry in the context of the original KKLMMT scenario, where inflation is obtained when a pair of $D3$-anti $D3$ branes is present. In fact, the shift symmetry is present initially in the Kähler potential. Imposing the shift symmetry invariance of the superpotential leads to a flat potential for the inflation. Inflation is then due to the small interaction potential between the $D3$ and the anti $D3$ brane. As in usual brane inflation this requires an adjustment of the brane–antibrane potential. In this context, it has been noticed that threshold corrections in string theory give rise to an explicitly shift symmetry breaking superpotential. As a consequence, the model suffers from the $\eta $-problem unless the stabilization of the complex structure moduli is fine-tuned. We will discuss this model (and compare it to what is achieved in the present article) where the non-perturbative superpotential responsible for the stabilization of the Kähler moduli is multiplicatively corrected by loop effects [@mac; @berg]. Finally the shift symmetry is also instrumental in the race–track inflation model [@race].
Notice that combining the KKLT approach to stabilization and the shift symmetry in string theory requires the existence of isometries on Calabi–Yau threefolds as originally argued in Ref. [@shift] for compactifications on $\setK_3\times \setT^2$. In the following, we will concentrate on supergravity issues only and use string motivations as a guideline only.
Combining the flatness of the F–term inflation potential in supergravity using the shift symmetry and the stabilization of moduli (in particular using the KKLT mechanism) is the aim of this paper. We will extract the main ingredients from the D3/D7 system in string theory and deal with its supergravity description exclusively. The two main new aspects of the model presented here are the following. Firstly, we will consider a case where the shift symmetry is initially present in both the Kähler and the superpotential before being mildly broken by an explicit inflaton dependence of the superpotential. In our case, the fact that the superpotential can depend on the inflaton field allows us to give a (small) slope to the potential already at the tree level without having to compute the quantum corrections. Notice that the shift symmetry breaking superpotential has coefficients constrained by the COsmic Background Explorer (COBE) normalization and must therefore be small. We also insist on obtaining the inflationary potential from a strict $N=1$ supergravity context. Secondly, we find that the end of inflation is due to the presence of the moduli field playing the role of a waterfall field. Contrary to the hybrid inflation case, the end of inflation is not triggered by extra fields representing the charged open strings between the branes. In fact, the name waterfall field is not very appropriate in our case since we will see that the effective inflationary model is reminiscent of mutated inflation and that inflation stops due to the violation of the slow-roll conditions and not by instability.
The outline of the paper is the following. In section II, we present the flatness problem in F–term supergravity inflation and discuss the realization of the shift symmetry as a Kähler transformation. The shift symmetry alleviates the flatness problem and leads to restrictions on the type of possible superpotentials. In section III, we give examples of chaotic inflation models with a shift symmetry broken by the superpotential. The model has problems such as runaway potentials. In Section IV, we then introduce what we call “mutated chaotic inflation” based on a chaotic inflation superpotential and for which the inflationary trajectories become curved. This model does not suffer from the $\eta $-problem and the moduli is stabilized. We give a thorough analysis of the inflationary parameter space. In particular, we find that the model is different from hybrid inflation. In section V, we combine the KKLT stabilization mechanism and a chaotic inflation superpotential. This gives another realization of mutated chaotic inflation. Then, we discuss various aspects of our results. Finally, in section VI we present our conclusions.
Shift Symmetry in Supergravity
==============================
The $\eta$-problem
------------------
One of the stumbling blocks of F-term inflation in supergravity is the natural presence of ${\cal O}\left(H\right)$ corrections to the inflaton mass which would spoil the flatness of the potential (the so–called $\eta$-problem). Let us consider the Kähler potential for the inflaton of the form $K\left(\phi,\phi^\dagger \right) = \phi
\phi^\dagger$ and its role in the scalar potential of supergravity $$\exp \left[\kappa K\left(\phi,\phi^\dagger\right)\right]V_{\rm inf}\,
,$$ where $V_{\rm inf}$ is the inflationary potential when neglecting the supergravity corrections and where $\kappa $ is defined by $\kappa
\equiv 8\pi/{m_{_{\mathrm Pl}}}^2$. Expanding the exponential leads to $$\left[1+\kappa K\left(\phi,\phi^\dagger\right) +\cdots \right]V_{\rm
inf}\, .$$ The first term leads to the inflation potential, while the second one leads to a term in $H^2\phi\phi^\dagger$ which spoils the flatness of the potential, [*i.e. *]{}the quantity $$\frac{m_{\rm Pl}^2}{8\pi}\frac{1}{V}\frac{\partial ^2 V}{{\rm \partial
}\phi ^2}\, ,$$ becomes of order one. However, one should also remark the following. The above parameter (the so-called $\eta$-parameter) is not the parameter which controls whether inflation is taking place or not. Indeed, strictly speaking, the condition $\ddot{a}>0$, where $a(t)$ is the Friedmann-Lemaître-Robertson-Walker scale factor, is equivalent to $\epsilon\equiv -\dot{H}/H^2 <1$ or $m_{\rm
Pl}^2/(16\pi V^2)(\partial V/\partial \phi )^2 <1$. When the $\eta
$-parameter becomes of order one, we just have violation of the slow-roll conditions although, in principle, inflation could still proceed. Of course, one could argue, based on the well-known formula, $n_{_{\rm S}}-1\simeq 2\eta -6\epsilon$, that a parameter $\eta $ of order one would imply a scalar spectral index far from scale invariance, [*i.e. *]{}$\vert n_{_{\rm S}}-1\vert \gg 1$. Again, this conclusion is not rigorous because the previous formula is derived under the assumption that the slow-roll conditions are valid and, hence, not applicable when $\eta $ is large. In principle, one could imagine a situation where $\eta $ is large but where inflation proceeds and leads to an almost scale-invariant spectrum. Admittedly, this is probably not the most generic situation but this is possible in principle; for interesting recent remarks on the $\eta $ problem, see also Ref. [@kinney].
Shift symmetry
--------------
Let us now consider a typical ansatz, motivated by string inspired theories [@Hsu] given by $$\begin{aligned}
\label{deltanf}
K &=& -\frac{3}{\kappa }\ln \left[\kappa ^{1/2}\left(\rho
+\rho ^{\dagger}\right)-\sigma \kappa {\cal K}\left(\phi, \phi
^{\dagger}\right)\right]
\nonumber \\
& & +s{\cal G}\left(\phi ,\phi
^{\dagger}\right)\, , \\
W &=& W\left(\rho ,\phi \right)\, ,\end{aligned}$$ where $\sigma =0,1$ and/or $s=0,1$ according to the situation we want to describe. In the above expression, the field $\rho $ is a moduli while $\phi $ represents the inflaton. When $\sigma=1$ and $s=0$, the Kähler potential describes the motion of a $D3$ brane within a Calabi-Yau manifold (we have only retained one of the possible six directions). In that case ${\cal K}\left(\phi,\phi^\dagger\right)$ is the Kähler potential on the Calabi-Yau manifold. For small arguments, one can expand ${\cal
K}\left(\phi,\phi^\dagger\right)=\phi\phi^\dagger+\cdots$. Similarly when $\sigma=0$, ${\cal G}$ is also identified with the Kähler potential of the Calabi-Yau manifold and $\phi$ corresponds to the position of a $D7$ brane. These two cases will be exemplified later.
Let us come back to the issue of the shift symmetry $\phi \to \phi+c$ where $c$ is real which guarantees that the real part of the inflaton superfield is a flat direction. Following the above discussion, we focus on the Kähler potential $$\label{typicK}
K= -\frac{3}{\kappa }\ln \left[\kappa ^{1/2}(\rho+\rho^\dagger )
-\sigma \kappa \phi\phi^\dagger \right] +s\phi\phi^\dagger\, .$$ which follows from Eq. (\[deltanf\]) in the small argument limit. Let us also remind that a Kähler transformation is a transformation which leaves the Lagrangian of a supergravity theory invariant. It is of the form $$\begin{aligned}
K(\rho ,\phi ) &\to & K(\rho ,\phi ) +\xi\left (\rho , \phi \right)+\xi
^\dagger \left(\rho ^\dagger, \phi ^\dagger \right)\, ,\\
W(\rho ,\phi ) &\to &{\rm e}^{-\kappa \xi(\rho ,\phi )}W(\rho ,\phi )\, ,\end{aligned}$$ where $\xi $ is an arbitrary function. The shift symmetry, $\phi \to
\phi +c$, can be viewed as a Kähler transformation provided the moduli field $\rho $ transforms in a specific way and the superpotential possesses a given form, namely $$\begin{aligned}
\label{shiftsym}
\rho &\to & \rho + \sigma c \kappa ^{1/2}\phi +\sigma \kappa
^{1/2}\frac{c^2}{2}\, ,\\
\label{shiftsym2}
W(\rho,\phi) &=& {\rm e}^{-s\kappa
\phi^2/2}W\left(\rho -\sigma \kappa ^{1/2}\frac{\phi^2}{2}\right)\, .\end{aligned}$$ First of all, one can check that the quantities $\kappa
^{1/2}\left(\rho +\rho ^{\dagger }\right)-\sigma \kappa \phi \phi
^\dagger $ and $\rho -\sigma \kappa ^{1/2}\phi ^2/2$ are invariant under the above transformation of the fields $\phi $ and $\rho
$. Secondly, if the superpotential has the form given in Eq. (\[shiftsym2\]), then the corresponding Kähler transformation is described by the function $$\xi (\phi )=sc\phi +s\frac{c^2}{2}\, .$$ Therefore, if we want to implement the shift symmetry, one must restrict our considerations to models described by the Kähler potential given in Eq. (\[typicK\]) and superpotential given by Eq. (\[shiftsym2\]).
This class of models can be simplified further (or transform to another form). Two ingredients are necessary. The first one is a Kähler transformation described by the function $\xi =-s\phi ^2/2$ (this Kähler transformation has of course nothing to do with the other Kähler transformation considered before). In this case the Kähler potential and superpotential of the shift symmetry invariant model is equivalent to the one given by $$\begin{aligned}
K &=& -\frac{3}{\kappa }\ln \left[\kappa ^{1/2}(\rho+\rho^\dagger )
-\sigma \kappa \phi\phi^\dagger \right] \nonumber \\
& & -\frac{s}{2}\left(\phi-\phi^\dagger\right)^2\, , \\
W(\rho,\phi) &=& W\left(\rho -\sigma \kappa ^{1/2}
\frac{\phi^2}{2}\right)\, .\end{aligned}$$ The second ingredient consists in changing variables in a holomorphic way (as required by supersymmetry) $\rho \to \rho -\sigma \kappa
^{1/2}\phi^2/2$. In terms of the new variables, the Kähler potential becomes $$\begin{aligned}
\label{sigma}
K &=& -\frac{3}{\kappa }\ln \left[\kappa ^{1/2}\left(\rho+\rho ^\dagger
\right)+\frac{\sigma }{2}\kappa
\left(\phi-\phi^\dagger\right)^2\right]
\nonumber \\
& & -\frac{s}{2}\left(\phi-\phi^\dagger \right)^2 \, ,\\
\label{sigma2}
W(\rho,\phi) &=& W(\rho)\, .\end{aligned}$$ This representation guarantees that the inflaton field has a flat direction. This is the representation with ${\cal K}=-\sigma
\left(\phi-\phi^\dagger\right)^2/2$ and ${\cal
G}=-s\left(\phi-\phi^\dagger\right )^2/2$ in Eq. (\[deltanf\]). The previous model does not immediately lead to inflation in the $\phi$ direction as the scalar potential is exactly flat along the real $\Re
(\phi)$ direction due to the shift symmetry (but the potential could be lifted by quantum corrections). A large class of supergravity models with no $H^2$ corrections to the inflaton mass can be constructed by modifying the previous superpotential and including an explicit $\phi$ dependence which breaks the shift symmetry. Notice that the $H^2$ corrections will be absent as the Kähler potential is still shift symmetric. We now turn to the construction of such models.
Inflation and Shift Symmetry
============================
The scalar potential
--------------------
Following the discussion of the previous section, we now generalize the class of models invariant under the shift symmetry and consider the Kähler potential given by (we remind that, in the present context, the field $\phi $ will be viewed as the inflaton while the field $\rho $ will represent a moduli) $$K= -\frac{3}{\kappa }\ln \left[\kappa ^{1/2}\left(\rho+ \rho ^{\dagger
}\right)-\kappa {\cal K}\left(\phi-\phi^\dagger \right)\right]
+{\cal G}\left(\phi-\phi^\dagger\right)\,$$ in such a way that the shift symmetry is explicitly present. In the above expression, ${\cal K}$ and ${\cal G}$ are arbitrary functions, the form of which is not specified at this stage. In order to calculate the corresponding potential, one must first evaluate the matrix $G_{A\bar{B}}$ defined by $$G_{A\bar{B}}=\frac{\partial ^2}{\partial \varphi ^A
\partial \left(\varphi ^B\right)^{\dagger}}
\left[\kappa K+\ln \left(\kappa ^3\left\vert W
\right\vert^2\right)\right]\, ,$$ where $\varphi ^A=\{\rho ,\phi \}$ and where $W=W\left(\rho ,\phi
\right)$ is the superpotential which, as announced, explicitly depends on $\phi $. Explicitly, straightforward calculations lead to
$$\displaystyle
G_{A\bar{B}}=\left\{
\begin{matrix}
\displaystyle 3\frac{\kappa }{\Delta ^2} & \, \, \, \, \, &
\displaystyle 3\frac{\kappa
^{3/2}}{\Delta ^2} \frac{\partial {\cal K}}{\partial \left(\phi -\phi
^\dagger \right)} \cr \cr
\displaystyle -3\frac{\kappa ^{3/2}}{\Delta ^2}
\frac{\partial {\cal K}}{\partial \left(\phi -\phi ^\dagger \right)}
& \, \, \, \, \, &
\displaystyle -3\frac{\kappa }{\Delta } \frac{\partial ^2{\cal
K}}{\partial \left(\phi -\phi ^\dagger \right)^2} -3\frac{\kappa
^2}{\Delta ^2}\left[\frac{\partial {\cal K}}{\partial \left(\phi
-\phi ^\dagger \right)}\right]^2 -\kappa \frac{\partial ^2{\cal
G}}{\partial \left(\phi -\phi ^\dagger \right)^2}
\end{matrix}\right\}\, ,$$
where the quantity $\Delta $ is defined by $\Delta \equiv \kappa
^{1/2}\left(\rho +\rho ^{\dagger }\right)-\kappa {\cal K}\left(\phi
-\phi ^\dagger \right)$. These models are of the no-scale type with a cancellation of the $-3\vert W\vert^2$ term in the scalar potential $V={\rm e}^G\left(G^AG_A-3\right)/\kappa ^2$ \[we remind here that the function $G$ is given by $G\equiv \kappa K+\ln \left(\kappa ^3\vert
W\vert ^2\right)$\]. The potential reads $$\begin{aligned}
{\rm e}^{-\kappa {\cal G}}V (\rho ,\phi) &=& \frac{1}{3\Delta}\biggl
\vert \frac{\partial W}{\partial \rho }\biggr \vert^2 \times \left\{
1+\frac{\kappa }{\Delta {\cal D}} \left[\frac{\partial {\cal
K}}{\partial \left(\phi -\phi ^\dagger \right)}\right]^2\right\}
-\frac{1}{\Delta^2 {\cal D}} \left \vert\frac{\partial W}{\partial
\phi }\right \vert^2 \nonumber \\ & & -\frac{\kappa ^{1/2}}{\Delta^2}
\left(W\frac{\partial W^\dagger }{\partial \rho ^\dagger } +W^\dagger
\frac{\partial W }{\partial \rho }\right) \times \left[ 1-\frac{\kappa
}{{\cal D}} \frac{\partial {\cal K}}{\partial \left(\phi -\phi
^\dagger \right)} \frac{\partial {\cal G}}{\partial \left(\phi -\phi
^\dagger \right)}\right] \nonumber \\ & & +\frac{\kappa ^2}{\Delta
^2{\cal D}} \left[\frac{\partial {\cal G}}{\partial \left(\phi -\phi
^\dagger \right)}\right]^2\vert W\vert ^2 -\frac{\kappa ^{1/2}}{\Delta
^2{\cal D}} \frac{\partial {\cal K}}{\partial \left(\phi -\phi
^\dagger \right)} \left(\frac{\partial W}{\partial \rho }
\frac{\partial W^\dagger }{\partial \phi ^\dagger } -\frac{\partial
W^\dagger }{\partial \rho ^\dagger } \frac{\partial W }{\partial \phi
}\right) \nonumber \\ & & +\frac{\kappa }{\Delta^2{\cal D}}
\frac{\partial {\cal G}}{\partial \left(\phi -\phi ^\dagger \right)}
\left(W^\dagger \frac{\partial W }{\partial \phi } -W \frac{\partial W
^\dagger}{\partial \phi ^\dagger }\right)\, ,\end{aligned}$$ where the coefficient ${\cal D}$ is defined by the following expression $${\cal D}\equiv
3\frac{\partial ^2 {\cal K}}{\partial
\left(\phi -\phi ^\dagger \right)^2}+\Delta
\frac{\partial ^2 {\cal G}}
{\partial \left(\phi -\phi ^\dagger \right)^2}\, .$$ Let us notice that, if the fields $\rho $ and $\phi $ are real and if the superpotential $W$ is also real when $\rho $ and $\phi $ are real, then the above expression can be simplified further since the last two terms cancel out. Moreover, if ${\cal G}=0$ and if the superpotential does not depend on the field $\phi $ but only on the moduli $\rho$ , then one recovers the expression (5.12) found in Ref. [@KKLMMT] (which, therefore, appears to be a particular case of the most general formula established above), namely $$\begin{aligned}
V (\rho ) &=& \frac{1}{3\Delta}\biggl
\vert \frac{\partial W}{\partial \rho }\biggr \vert^2 \times \left\{
1+\frac{\kappa }{3\Delta }
\left[\frac{\partial ^2{\cal
K}}{\partial \left(\phi -\phi ^\dagger \right)^2}\right]^{-1}
\times \left[\frac{\partial {\cal
K}}{\partial \left(\phi -\phi ^\dagger \right)}\right]^2\right\}
-\frac{\kappa ^{1/2}}{\Delta^2}
\left(W\frac{\partial W^\dagger }{\partial \rho ^\dagger }
+W^\dagger \frac{\partial W }{\partial \rho }\right)\, .\end{aligned}$$ In this paper, we will consider a different situation. The functions ${\cal K}$ and ${\cal G}$ can always be Taylor expanded according to $${\cal K}=\sum _{n=0}^{\infty }\frac{a_n}{n!}\left(\phi -\phi
^\dagger \right)^n\, , \quad {\cal G}=\sum _{n=0}^{\infty
}\frac{b_n}{n!}\left(\phi -\phi ^\dagger \right)^n \, .$$ We will assume that the functions ${\cal K}$ and ${\cal G}$ satisfy the properties $${\cal K}\vert_{\phi=\phi^\dagger}={\cal G}\vert_{\phi=\phi^\dagger}=0
\, , \quad
\frac{\partial {\cal K}}{\partial\left(\phi -\phi ^\dagger \right)
}\biggl \vert_{\phi=\phi^\dagger}=
\frac{\partial {\cal G}}{\partial\left(\phi -\phi ^\dagger \right)
}\biggl \vert_{\phi=\phi^\dagger}=0\, ,$$ [*i.e. *]{}that $a_0=a_1=b_0=b_1=0$, the remaining coefficients being arbitrary. These simple assumptions are sufficient to render the matrix $G_{A\bar{B}}$ diagonal. Explicitly, the potential takes the form $$\label{potgeneral}
V (\rho ,\phi)= \frac{1}{3\Delta}\biggl \vert \frac{\partial
W}{\partial \rho }\biggr \vert^2 -\frac{1}{\Delta^2}
\left[3\frac{\partial ^2 {\cal K}}{\partial \left(\phi -\phi ^\dagger
\right)^2}+\Delta \frac{\partial ^2 {\cal G}}{\partial \left(\phi
-\phi ^\dagger \right)^2} \right]^{-1}\left \vert\frac{\partial
W}{\partial \phi }\right \vert^2 -\frac{\kappa ^{1/2}}{\Delta^2}
\left(W\frac{\partial W^\dagger }{\partial \rho ^\dagger } +W^\dagger
\frac{\partial W }{\partial \rho }\right)\, .$$ In the following, we will mainly focus on the ansatz (\[sigma\]) for which the potential becomes $$\label{potgeneral2} V (\rho ,\phi)= \frac{1}{3\Delta}\biggl \vert
\frac{\partial W}{\partial \rho }\biggr \vert^2 +\frac{1}{\Delta^2}
\left(3\sigma +s \Delta \right)^{-1}\left \vert\frac{\partial
W}{\partial \phi }\right \vert^2 -\frac{\kappa ^{1/2}}{\Delta^2}
\left(W\frac{\partial W^\dagger }{\partial \rho ^\dagger } +W^\dagger
\frac{\partial W }{\partial \rho }\right)\, .$$
This last equation is the main equation of this section. It gives the general form of the scalar potential in a theory where the shift symmetry is implemented in the Kähler potential and for a general superpotential which can depend both on the inflaton field $\phi $ but also on the moduli $\rho $. From the general form of the Kähler potential, we obtain $$\label{kinetic}
K_{\phi \phi ^\dagger }=-\frac{3}{\Delta } \frac{\partial ^2 {\cal
K}}{\partial \left(\phi -\phi ^\dagger \right)^2} -\frac{\partial ^2
{\cal G}}{\partial \left(\phi -\phi ^\dagger \right)^2}\, ,\quad
K_{\rho \rho ^\dagger }= \frac{3}{\Delta ^2}\, .$$ which will be used to normalize the fields. In the case of (\[sigma\]), this gives $$\label{kinetic2} K_{\phi \phi ^\dagger }=\frac{3\sigma }{\Delta } +s
\, ,\quad K_{\rho \rho ^\dagger }= \frac{3}{\Delta ^2}\, .$$
Let us stress once more why the shift symmetry is so crucial to alleviate the $\eta$-problem. To do so, we will compare with the results obtained in Ref. [@KKLMMT]. In this specific model the Kähler potential springs from ${\cal
K}(\phi,\phi^\dagger)=\phi\phi^\dagger$ and $\sigma=1$. As argued in section II, this combination is in fact explicitly shift symmetric (after using the transformations studied in that section). Now it is assumed that the non-perturbative superpotential depends only on the moduli $\rho$. Notice that this step is exactly where the shift symmetry is explicitly broken. Indeed the shift symmetry invariant combination is $\rho -\kappa^{1/2}\phi^2/2$. Now the scalar potential follows as before
$$\begin{aligned}
V (\rho ) &=& \frac{1}{3\Delta}\biggl \vert \frac{\partial W}{\partial
\rho }\biggr \vert^2 \left( 1+\frac{\kappa }{3\Delta }
\phi\phi^\dagger\right) -\frac{\kappa ^{1/2}}{\Delta^2}
\left(W\frac{\partial W^\dagger }{\partial \rho ^\dagger } +W^\dagger
\frac{\partial W }{\partial \rho }\right)\, ,\end{aligned}$$
where $\Delta\equiv \kappa^{1/2} (\rho+\rho^\dagger) -\kappa
\phi\phi^\dagger$. The potential is corrected by the presence of an anti D3-brane leading to a total potential $$\begin{aligned}
V (\rho ) &=& \frac{1}{3\Delta}\biggl \vert \frac{\partial
W}{\partial \rho }\biggr \vert^2 \left(1+\frac{\kappa }{3\Delta }
\phi\phi^\dagger\right) -\frac{\kappa ^{1/2}}{\Delta^2}
\left(W\frac{\partial W^\dagger }{\partial \rho ^\dagger }
+W^\dagger \frac{\partial W }{\partial \rho }\right)\
+\frac{E}{\Delta^2}\, ,\end{aligned}$$
for a given constant $E$. The extra potential is explicitly shift symmetric \[again using the transformation on the fields discussed at the beginning of this article in section II, the denominator of the new term becomes $\kappa^{1/2}(\rho +\rho ^{\dagger }) +\kappa (\phi
-\phi ^{\dagger })^2/2$\]. Now assume that $W(\rho)$ has been chosen in such a way that $\phi=0$ and $\rho=\rho_0$ is a minimum for a real $\rho_0$, the potential reads close to the minimum $$V=V(\rho_0)\left(1+\frac{2}{3}\kappa \Phi\Phi^\dagger\right)\, ,$$ for the canonically normalized inflaton $\Phi=\phi\sqrt{3/2\rho}$. We can see that the inflaton potential is not flat and runs into the $\eta$-problem.
Our goal is now to find inflationary models where the moduli is stabilized. We will consider a class of models where the superpotential breaks the shift symmetry mildly and does not jeopardize the flatness of the potential. Some examples are given in the next section and some stringy motivations for such a form are also presented. We find that the potentials lead to inflation along the $\phi$ direction. Notice that there is no cosmological constant and that the potential is a function of the inflaton field which is polynomial when the superpotential has a polynomial dependence on the inflaton field.
Chaotic Inflation
-----------------
As a warm up, let us now discuss a simple example of chaotic inflation as can be found in Ref. [@linde] where a similar case is treated. Explicitly, one assumes $$\begin{aligned}
{\cal K} &=& -\frac{1}{2}\left(\phi -\phi ^{\dagger }\right)^2\, , \quad
{\cal G}=+\frac{1}{2}\left(\phi -\phi ^{\dagger }\right)^2\, ,\\
W(\rho, \phi ) &=& \frac{1}{\sqrt{2}}m\phi^2 \, .\end{aligned}$$ The factor $1/\sqrt{2}$ in the inflationary part of the superpotential is chosen for future convenience. Notice that the shift symmetry is preserved by the Kähler potential while the superpotential breaks the shift symmetry explicitly. Then, using Eqs. (\[potgeneral\]) and (\[potgeneral2\]), straightforward calculations lead to $$V(\rho ,\phi)=\frac{1}{\Delta ^2} \frac{1}{3-\Delta}\left\vert
\frac{\partial W}{\partial \phi } \right\vert ^2\, .$$ The moduli $\rho$ can be stabilized for $\Delta =\kappa^{1/2}(\rho
+\rho ^\dagger )=2$ (since ${\cal K}=0$ for a real inflaton) and, therefore, one has $V(\phi, \rho )=\vert \partial W/\partial \phi
\vert ^2/4$. Notice the $1/4$ prefactor which comes from stabilizing the moduli and there is no $H^2$ contribution to the mass of the inflaton due to the partial shift symmetry preserved by the Kähler potential. In this model, we obtain chaotic inflation depending explicitly on the shift symmetry breaking superpotential. In particular for a quadratic superpotential, we obtain that $$V(\phi)=\frac{1}{2} m^2\phi^2 \, ,$$ which is nothing but the usual chaotic inflation potential and where one can check from Eq. (\[kinetic\]) that the real field $\phi $ is correctly normalized since $K_{\phi \phi ^\dagger }=1/2$. However, it happens that the present model is not the one favored in string theory as ${\cal G}$ has the wrong sign. In the following section we examine cases where the Kähler potential has a structure dictated by string theory.
Runaway potential
-----------------
In this subsection, we illustrate the stabilization problem on a simple example. As mentioned above, we now consider a model where the function ${\cal G}$ possesses an overall minus sign, [*i.e. *]{}where the Kähler potential has a form which can be justified in string theory. The superpotential is chosen to be the same as in the previous subsection. In this case, we show below that the moduli can no longer be stabilized. Therefore, when the Kähler potential is chosen according to the string-motivated considerations, it is necessary to consider more complicated forms for the superpotential (depending explicitly on the moduli). This will be done in the following section. Here we choose $${\cal K}=0\, , \quad {\cal G}=-\frac{1}{2}\left(\phi -\phi
^{\dagger }\right)^2\, ,\quad W(\rho, \phi )=2\sqrt{2}m\phi^2\, ,$$ where, again, the factor $2\sqrt{2}$ in front of the inflationary superpotential has been chosen for future convenience. This leads to the following scalar potential $$V(\rho ,\phi)=\frac{8}{\Delta
^3}m^2\phi ^2\, .$$ Redefine the fields in order to have properly normalized fields using $K_{\phi \phi ^\dagger}=1$ and $K_{\rho \rho ^\dagger }=3/\Delta ^2$ as given by Eq. (\[kinetic\]), we find that the normalized field $\bar{\phi }$ and $\bar{\rho }$ (assuming that $\rho =\rho ^{\dagger
}$) are given by $\bar{\phi }=\sqrt{2}\phi $ and $\kappa ^{1/2}\bar
{\rho}= \sqrt{3/2}\ln \left(\kappa ^{1/2}\rho \right)$. The potential for the redefined fields finally reads $$\label{potmod}
V(\bar{\rho },\bar{\phi }) =\frac{1}{2} m^2 \bar{\phi }^2 \exp
\left(-\sqrt{6}\kappa ^{1/2}\bar{\rho }\right)\, .$$ This potential is represented in Fig. \[potnonstab\]. Notice that $\rho$ (or $\bar{\rho }$) is not stabilized and is given by a runaway potential. This is the usual problem for moduli fields. In the following section, we give examples of potentials leading to inflation and a stabilization of the moduli. At this point, a last remark is in order. We will see that the KKLT procedure consists in adding a term $1/\rho ^2$ and/or $1/\rho ^3$ to the potential. It is clear here that this mechanism would not be enough to stabilize the moduli.
{width=".95\textwidth" height=".85\textwidth"}
Mutated chaotic inflation
=========================
Giving a mass to the inflaton
-----------------------------
In this section, we discuss a successful model of inflation combining moduli stabilization and chaotic inflation. To motivate the introduction of a mass term for the inflaton field, let us consider the D3/D7 system in string theory [@D3; @Hsu; @koyama]. This system can be modeled at low energy using three fields, the inflaton $\phi$ measuring the inter–brane distance and two charged fields $\phi^\pm$ representing the open strings between the two types of branes. The fields interact according to the superpotential $$\label{superpotpm}
W\left(\phi, \phi^+,\phi ^-\right)=\sqrt 2 g\left( \phi^+ \phi \phi^-
-\zeta ^2\phi \right)\, ,$$ where $g$ is the $U(1)$ gauge coupling and $\zeta $ is a constant term which is turned on when the compactification looks like a resolution of an orbifold singularity $\setC^2/\setZ_2$ locally. The term $\zeta
^2$ is stabilized at the same time as the complex structure moduli. This is not the case of the Fayet–Iliopoulos term which depends on the Kähler moduli. To simplify we consider the case where there is no Fayet-Iliopoulos term.
Let us also give the Kähler function of the model. As before, for the inflaton field $\phi $, we focus on the case where the function ${\cal K}$ vanishes and where ${\cal G}=-1/2 \left(\phi-\phi^{\dagger
}\right)^2$. On the other hand, the Kähler functions for the charged fields are standard and, therefore, the total Kähler function is given by $$K\left(\phi, \phi^+,\phi ^-\right)=-\frac12 \left(\phi-\phi^{\dagger
}\right)^2+\phi ^+\left(\phi ^+\right)^{\dagger}+\phi ^-\left(\phi
^-\right)^{\dagger}\, .$$
As we discussed in the introduction, stabilizing moduli can be achieved by considering non-perturbative superpotentials springing from gaugino condensation. In this case, the superpotential contains a part which explicitly depends on the moduli and reads $$W(\rho )=W_0-A\exp \left(-\beta \kappa ^{1/2}\rho \right)\, ,$$ where $W_0$, $A$ are free constants of dimension three and $\beta $ is dimensionless. The superpotential $W(\rho )$ has already been studied in Refs. [@KKLT] and [@Tye]. In string theory, the constant term springs from the stabilization of both the complex structure moduli and the dilaton. The exponential term is of non-perturbative origin. We take the total superpotential of the system to be the sum of the two previous expression, namely $$\label{Wtotal}
W\left(\phi, \phi^+,\phi ^-,\rho \right)=W\left(\phi, \phi^+,\phi
^-\right)+W(\rho )\, .$$ It is known from string considerations that the coupling constant $g$ and/or $\zeta $ can not depend on the moduli $\rho $ [@D3]. Therefore, this superpotential is the simplest way of coupling the moduli to inflation.
Finally, the total Kähler potential is also the sum of the Kähler potentials in the inflaton and moduli sector. It can be expressed as $$K\left(\phi, \phi^+,\phi ^-,\rho \right)=K\left(\phi, \phi^+,\phi
^-\right)-\frac{3}{\kappa }\ln \left[\kappa ^{1/2}\left(\rho +\rho
^{\dagger} \right)\right]\, .$$
Having specified what the Kähler and super potentials of the model are, one can now determine the corresponding scalar potential. It reads
$$\begin{aligned}
\label{pottotalpm} V &=& \frac{{\rm e}^{\kappa \left(\vert
\phi^+\vert^2 +\vert
\phi^-\vert^2\right)}}{\kappa^{3/2}\left(\rho+\rho^{\dagger}\right)^3}
\biggl\{\left\vert\sqrt{2}g\phi\phi^- +\kappa
\left(\phi^+\right)^{\dagger }W\right\vert ^2 +
\left\vert\sqrt{2}g\phi\phi^++\kappa \left(\phi^-\right)^{\dagger
}W\right\vert ^2 +2g^2\left\vert \phi^+\phi^--\zeta ^2\right\vert^2
\nonumber \\ & &-2\kappa^{3/2} \beta A \left(\rho+
\rho^\dagger\right)\Re \left[{\rm
e}^{-\beta\kappa^{1/2}\rho^\dagger}W\left(\phi,\phi^+,\phi^-\right)\right]\biggr\}
+ {\rm e}^{\kappa \left(\vert \phi^+\vert^2 +\vert
\phi^-\vert^2\right)} \tilde{\cal V}(\rho
)+\frac{g^2}{2}\left(\left\vert \phi^+\right \vert^2 - \left\vert
\phi^-\right \vert^2\right)^2\, ,\end{aligned}$$
where $y\equiv \beta \kappa^{1/2} \rho$ and where $W$ in the previous formula denotes the total superpotential defined by Eq. (\[Wtotal\]). The function $\tilde{\cal V}(\rho )$ is defined by $$\label{defV}
\tilde{\cal V}(\rho )\equiv \frac{\kappa A\beta ^2}{2y}{\rm e}^{-y}
\left[\frac{A\beta }{3}{\rm e}^{-y}-\frac{\beta }{y} \left(W_0-A{\rm
e}^{-y}\right)\right]\, .$$ The true vacuum of the above potential is obtained for $ \phi^+=
\phi^-= \zeta $, $\phi=-\kappa W(\rho _{{\rm min},\tilde{\cal
V}})/(\sqrt{2}g)$, $\rho= \rho_{{\rm min},\tilde{\cal V}}$ being the minimum of the function $\tilde{\cal V}$. This leads to an AdS vacuum with unbroken supersymmetry [@KKLT].
For large $\phi$ compared to $\zeta $, the potential has a flat inflationary valley where $\phi^\pm=0$ and the slope of the potential can be lifted by radiative corrections. When $\phi$ becomes small compared to $\zeta $, the field $\phi^\pm$ run towards the true minimum $\phi^+=\phi^-=\zeta $ and $\phi=-\kappa W/(\sqrt{2}g)$. This realizes a hybrid inflation scenario in string theory.
Now, let us consider the situation where $\phi$ is small enough and the charged fields are stuck at the minimum. In this regime, the field $\phi $ can be expanded according to $$\phi \simeq -\frac{\kappa W\left(\rho \right)}{\sqrt{2}g} +\delta \chi
\, .$$ Then the field $\delta \chi $ becomes massive with a potential $$\label{potmodelpm}
V=\tilde{\cal V}(\rho) +\frac{4g^2 \zeta ^2
}{\kappa^{3/2}(\rho+\rho^{\dagger })^3} \left \vert \delta \chi \right
\vert ^2\, .$$ We now notice that this potential can be also obtained using the much simpler superpotential $$W(\rho,\phi)= W(\rho) + g\zeta (\delta \chi) ^2\, ,$$ in the global supersymmetry limit where the Planck mass is taken to be very large compared to the inflation scale. This provides a motivation to use the above superpotential in a regime where supergravity corrections are taken into account and where one can be far from the true minimum. We show in the next subsection that the potential becomes very interesting and leads to “mutated chaotic inflation”.
The model
---------
Following the considerations presented before, we assume that the inflaton is a massive field like in chaotic inflation and discuss its coupling to the moduli stabilization sector. Since our model is a string inspired, we take ${\cal K}=0$ and ${\cal G}=-1/2\left(\phi
-\phi ^\dagger \right)^2$ for the Kähler potential. For the superpotential, we use the calculation of the previous subsection section and assume that $$\begin{aligned}
W(\rho ,\phi) &=& W(\rho )+W_{\rm inf}(\phi )\nonumber \\
&=& W_0-A\exp \left(-\beta \kappa ^{1/2}\rho \right)+\frac{\alpha
}{2}m\phi^2\, ,\end{aligned}$$ where the mass $m$ can be related to $\zeta $, namely $m=2g\zeta
/\alpha $. Then, we use the general formula established before, see Eqs. (\[potgeneral\]) and (\[potgeneral2\]), and we obtain $$\label{ourpot}
V\left(\rho, \phi \right)=\tilde{{\cal V}}(\rho ) +\tilde{{\cal
U}}(\rho )\phi ^2\, ,$$ where the function $\tilde{\cal V}$ has already been defined above in Eq. (\[defV\]) and $\tilde{\cal U}$ given by $$\tilde{{\cal U}}(\rho )=\frac{\alpha m\beta ^3}{4y^2}
\left(\frac{\alpha m}{2y}-\kappa A{\rm e}^{-y}\right)\, ,$$ and where we remind that $y\equiv \beta \kappa ^{1/2}\rho $. Let us notice that, if $\kappa \rightarrow 0$, then $\tilde{\cal U}\sim
1/\rho ^3$ in accordance with Eq. (\[potmodelpm\]). The function $\tilde{\cal U}(\rho )$, from the point of view of the field $\phi $, plays the role of an effective squared mass. The function $\tilde{\cal
V}(\rho )$ is not multiplied by a function of the field $\phi $ and, therefore, can be viewed as an “offset”. In order for the above potential to be relevant, it is necessary for the effective squared mass to be positive at the extremum of $\tilde{\cal U}$ where the moduli is stabilized.
{width=".95\textwidth" height=".65\textwidth"}
Let us also give the potential in terms of the normalized fields $\bar{\rho }$ and $\bar{\phi }$. These fields are given by the formulas in the text before Eq. (\[potmod\]) since the Kähler potential in the present section is the same as in the subsection of Eq. (\[potmod\]). Below, for convenience, we reproduce the relation between canonical and non-canonical fields $$\phi =\frac{\bar{\phi }}{\sqrt{2}}\, ,\quad y=\beta
\exp\left(\sqrt{\frac23}\kappa ^{1/2}\bar{\rho }\right)\, .$$ Therefore, the potential reads
$$\begin{aligned}
V\left(\bar{\rho} ,\bar{\phi }\right) &=& \frac{\kappa A\beta }{2}{\rm
e}^{-\sqrt{2/3}\kappa ^{1/2}\bar{\rho }} \exp \left(-\beta {\rm
e}^{\sqrt{2/3}\kappa ^{1/2}\bar{\rho }}\right) \biggl\{ \frac{A\beta
}{3}\exp \left(-\beta {\rm e}^{\sqrt{2/3}\kappa ^{1/2}\bar{\rho
}}\right) \nonumber \\ & & -{\rm e}^{-\sqrt{2/3}\kappa ^{1/2}\bar{\rho
}}\left[W_0 -A\exp \left(-\beta {\rm e}^{\sqrt{2/3}\kappa
^{1/2}\bar{\rho }}\right) \right]\biggr\} \nonumber \\ & &
+\frac{\alpha m\beta }{8} {\rm e}^{-2\sqrt{2/3}\kappa ^{1/2}\bar{\rho
}} \left[\frac{\alpha m}{2\beta } {\rm e}^{-\sqrt{2/3}\kappa
^{1/2}\bar{\rho }} -\kappa A \exp \left(-\beta {\rm
e}^{\sqrt{2/3}\kappa ^{1/2}\bar{\rho }}\right) \right]\bar{\phi }^2\, .\end{aligned}$$
Notice that, since the link between $(\rho, \phi )$, on one hand, and $(\bar{\rho }, \bar{\phi })$, on the other hand, is monotonic, the above change of variables does not modify the properties of the minima. In particular, in order to study how these properties depend on the free parameters, it is sufficient to work in terms of the non-canonically normalized fields which is simpler.
The squared mass function $\tilde{\cal U}(\rho )$
-------------------------------------------------
We now analyze whether the moduli can be stabilized to a value corresponding to a positive potential. For this purpose, we study the effective squared mass function $\tilde{\cal U}$ and requires that this function has a positive minimum. We will see that the valley where the moduli is stabilized is not exactly given by the minimum of the function $\tilde{\cal U}$ because, for small values of $\phi $, the offset function $\tilde{\cal V}$ also plays a role. In fact, strictly speaking, the minimum of $\tilde{\cal U}$ is the valley of stability for $\phi/{m_{_{\mathrm Pl}}}\rightarrow +\infty$ only. However, we emphasize that it is mandatory that the true minimum of $\tilde{\cal
U}$ be positive since this term is multiplied by $\phi ^2$. Otherwise the inflaton field becomes tachyonic. It is easy to calculate the derivative of the effective mass $\tilde{\cal U}$. This gives $$\frac{{\rm d}\tilde{\cal U}}{{\rm d}y}=
\frac{\alpha m\beta ^3}{4y^4}\left(-\frac32\alpha m+2\kappa Ay{\rm e}^{-y}
+\kappa Ay^2{\rm e}^{-y}\right)\, .$$ It vanishes at $y=y_{{\rm min},\tilde{\cal U}}$ where $y_{{\rm
min},\tilde{\cal U}}$ satisfies the following equation $$\label{deff}
f\left(y_{{\rm min},\tilde{\cal U}}\right)=y_{{\rm min},\tilde{\cal
U}}\left(y_{{\rm min},\tilde{\cal U}}+2\right){\rm e}^{-y_{{\rm
min},\tilde{\cal U}}}=\frac{3\alpha m}{2\kappa A}\, .$$ From this expression, one gets a constraint on the parameters $m$ and $A$, coming from the fact that $3\alpha m/(2\kappa A)$ must be smaller than the maximum value of the function $y_{{\rm
min},\tilde{\cal U}}(y_{{\rm min},\tilde{\cal U}}+2){\rm e}^{-y_{{\rm
min},\tilde{\cal U}}}$, otherwise the above equation has no solution, see Fig. \[studyu\]. This function vanishes at the origin, increases and reaches a maximum at $y_{{\rm min},\tilde{\cal U}}=\sqrt{2}$ and then exponentially decreases towards zero. Therefore, the function $\tilde{\cal U}$ possesses an extremum if and only if $0<3\alpha
m/(2\kappa A)<\sqrt{2}\left(2+\sqrt{2}\right){\rm e}^{-\sqrt{2}}\simeq
1.17387 $. The value of $\tilde{\cal U}$ at the extremum can be easily derived and one obtains $$\label{minu}
\tilde{\cal U}\left(y=y_{{\rm min},\tilde{\cal U}}\right)= \frac{\alpha
^2m^2\beta ^3}{8y_{{\rm min},\tilde{\cal U}}^3}\frac{y_{{\rm
min},\tilde{\cal U}}-1}{y_{{\rm min},\tilde{\cal U}}+2}\, .$$ Therefore, the extremum corresponds to a positive potential if $y_{{\rm min},\tilde{\cal U}}>1$ but, at this level, this does not require new constraints on $m$ and $A$.
{width="8.6cm" height="7.5cm"} {width="8.8cm" height="7.5cm"}
Now, let us check whether this is a maximum or a minimum. For this purpose, one calculate the second derivative of $\tilde{\cal U}$ at the extremum. One obtains $$\frac{{\rm d}^2\tilde{\cal U}}{{\rm d}y^2}\biggl\vert _{y=y_{{\rm
min},\tilde{\cal U}}}= -\frac{3\alpha ^2m^2\beta ^3}{8}\frac{y_{{\rm
min},\tilde{\cal U}}^2-2}{y_{{\rm min},\tilde{\cal U}}^5\left(y_{{\rm
min},\tilde{\cal U}}+2\right)}\, .$$ This is positive if $y_{{\rm min},\tilde{\cal
U}}<\sqrt{2}$. Therefore, we have a positive minimum if the parameters $m$ and $A$ are such that $1<y_{{\rm min},\tilde{\cal U}}<\sqrt{2}$ which in turn implies that one must have $3/{\rm e}\simeq
1.10364<3\alpha m/(2\kappa A)<\sqrt{2}\left(2+\sqrt{2}\right){\rm
e}^{-\sqrt{2}} \simeq 1.17387$ or $$\label{consparam}
1.27782 <\frac{\kappa A}{\alpha m}< 1.35914\, .$$ This interval is represented in Fig. \[studyu\] by the dashed region. It is quite clear from the above considerations that the ratio $m/(\kappa A)$ has to be adjusted precisely. However, this does not mean $m$ and/or $\kappa A$ must be tuned very accurately. As a matter of fact, they can a priori change over a large range of values, provided of course that their ratio satisfies the constraint derived above. The function $\tilde{\cal U}$ for various values of $m/(\kappa
A)$ is represented in Fig. \[plotuv\].
The offset function $\tilde{\cal V}(\rho )$
-------------------------------------------
Let us now study the effective offset function $\tilde{\cal V}(\rho )$ in more details. Firstly, let us evaluate $$\frac{{\rm d}\tilde{\cal V}}{{\rm d}y}=-\frac{\kappa A^2\beta
^3}{3y^3} {\rm e}^{-2y}\left[y^2+\frac72 y +3 -\frac{3W_0}{2A}{\rm
e}^y(y+2)\right]\, ,$$ and, therefore, there is a minimum if the following equation is satisfied $$\label{condder}
y_{{\rm min},\tilde{\cal V}}^2+\frac72y_{{\rm min},\tilde{\cal
V}}+3=\frac{3W_0}{2A}{\rm e}^{y_{{\rm min},\tilde{\cal V}}}
\left(y_{{\rm min},\tilde{\cal V}}+2\right)\, .$$ The existence of a solution to the above equation is controlled by the ratio $W_0/A$. If $W_0/A>1$, then there is no solution because the parabola in Eq. (\[condder\]) cannot intersect the function $3W_0/(2A){\rm e}^{y_{{\rm min},\tilde{\cal V}}} \left(y_{{\rm
min},\tilde{\cal V}}+2\right)$ (notice that for $W_0/A=1$ the two functions are equal at $y_{{\rm min},\tilde{\cal V}}=0$). Moreover, in this situation, we have $\lim _{y\rightarrow 0}\tilde{\cal V}=-\infty
$ and, therefore, the function $\tilde{\cal V}$ cannot be positive everywhere. On the other hand, if $W_0/A<1$, then $\lim _{y\rightarrow
0}\tilde{\cal V}=+\infty $ and, in addition, Eq. (\[condder\]) admits a solution, hence $\tilde{\cal V}$ possesses an extremum. Since we also have $$\begin{aligned}
\label{massV}
& & \frac{{\rm d}^2\tilde{\cal V}}{{\rm d}y^2}\biggl \vert _{y=y_{{\rm
min},\tilde{\cal V}}}=\frac{3\kappa W_0^2\beta^3\left(y_{{\rm
min},\tilde{\cal V}}+2\right)} {8y_{{\rm min},\tilde{\cal V}}^3\left(
y_{{\rm min},\tilde{\cal V}}^2+7y_{{\rm min},\tilde{\cal
V}}/2+3\right)^2} \nonumber \\ &\times & \left(2y_{{\rm
min},\tilde{\cal V}}^3+9 y_{{\rm min},\tilde{\cal V}}^2 +12y_{{\rm
min},\tilde{\cal V}}+4\right)>0 \, ,\end{aligned}$$ the function $\tilde{\cal V}$ possesses an extremum which is a minimum. However, the value of this function at this minimum is given by $$\begin{aligned}
\label{minV}
& & \tilde{\cal V}\left( y=y_{{\rm min},\tilde{\cal V}}
\right)=-\frac{\kappa AW_0\beta ^3}{4y_{{\rm min},\tilde{\cal
V}}}
\nonumber \\ &\times &
\frac{y_{{\rm min},\tilde{\cal V}}+2}{y_{{\rm min},\tilde{\cal
V}}^2+7y_{{\rm min},\tilde{\cal V}}/2+3} {\rm e}^{-y_{{\rm
min},\tilde{\cal V}}} <0 \, .\end{aligned}$$ As a consequence, the function $\tilde{\cal V}$ cannot be positive everywhere since it is negative at its extremum. However, this is not automatically a problem, as it would have been for the function $\tilde{\cal U}$, since the offset function can always be “renormalized” by adding a positive cosmological constant, namely $-\tilde{\cal V}\left( y=y_{{\rm min},\tilde{\cal V}} \right)$. We conclude from the above analysis that the function $\tilde{\cal V}$, regardless of the values of the free parameters $A$ and $W_0$, is necessarily negative somewhere. The regime of interest is given by $W_0/A<1$ since in this case $\tilde{\cal V}$ possesses a minimum which can be “renormalized” by adding a constant. The function $\tilde{\cal V}(\rho )$ is represented in Fig. \[plotuv\] for various values of the ratio $W_0/A$.
Another remark is in order in this subsection. In the following, we will study another mechanism of stabilization ([*i.e. *]{}the KKLT mechanism) which consists in adding a term $\propto 1/\rho ^3$ to the offset function. In this case one will show that one can always find a positive minimum regardless of $W_0/A$. Therefore, with this other mechanism, the above discussion of the shape of the offset function is modified and the condition $W_0/A<1$ can be relaxed.
Let us now discuss another class of models. In our models, inflation is only driven by the $F$ –terms originating from non–shift symmetric superpotentials. This is enough to lift the inflaton flat direction and leads to an inflation potential with no ${\cal O}(H)$ corrections to the inflaton mass. Of course we need to introduce a superpotential of chaotic inflation with a fine–tuning of the inflaton mass scale (see below where we apply the COBE normalization). This is the usual flatness problem in inflation model building. It has been argued in Ref. [@mac] that such a fine–tuning is also present in string compactification where threshold corrections are taken into account leading to a superpotential of the form $$W(\rho,\phi)=W(\rho)\left(1+ \delta \kappa \phi^2\right)\, ,$$ where $\delta$ has to be small to guarantee a small enough mass of the inflaton. This requires a tuning of the complex structure moduli. For this model the scalar potential becomes $$V\left(\rho ,\phi \right)=\tilde{\cal V} (\rho)\left(1+\delta \kappa
\phi ^2\right)^2 + \frac{4\kappa ^2\delta ^2}{3\Delta^2}\left\vert
W(\rho)\right\vert^2 \phi^2\, ,$$ where the Kähler potential corresponds to ${\cal G}=0$ and ${\cal
K}= -(\phi-\phi^{\dagger})^2/2$ and where $\tilde {\cal V}$ is the same function as studied above. For large $\phi$ the potential reduces to $$V\left(\rho ,\phi \right)\simeq \tilde{\cal V}(\rho)\delta^2 \kappa^4
\phi ^4 \, .$$ Now, when $\rho$ is at the minimum of ${\tilde {\cal V}}$, the coupling constant of the $\phi^4$ term becomes negative implying that this potential is not suitable to obtain inflation. This exemplifies how difficult it is to find a satisfactory model of inflation in supergravity.
Renormalizing the offset function by a constant
-----------------------------------------------
Having studied the functions $\tilde{\cal V}$ and $\tilde{\cal U}$ in some details, let us now turn to the properties of the full potential when shifted by a positive cosmological constant. It is represented in Fig. \[potrhophi2\].
{width=".95\textwidth" height=".75\textwidth"}
The derivatives of the potential in the two directions $\bar{\rho }$ and $\bar{\phi }$ are given by $$\begin{aligned}
\label{potder}
\frac{\partial V}{\partial \bar{\phi} } &=& \frac{{\rm d}\phi }{{\rm
d}\bar{\phi}}\frac{\partial V}{\partial \phi }=
\frac{1}{\sqrt{2}}\times 2\tilde{\cal U}\phi\, , \\
\label{potder2}
\frac{\partial V}{\partial \bar{\rho }} &=& \frac{{\rm d}y}{{\rm
d}\bar{\rho }}\frac{\partial V}{\partial y}=
\sqrt{\frac23}\kappa ^{1/2}y\left(\frac{{\rm d}\tilde{\cal V}}{{\rm
d}y} +\frac{{\rm d}\tilde{\cal U}}{{\rm d}y}\phi ^2\right)\, .\end{aligned}$$ From these expressions, we deduce that the potential possesses an absolute minimum located at $$\bar{\phi }=0\, , \quad
\bar{\rho }={\bar \rho}_{\rm min,\tilde{\cal V}}\, .$$ At this minimum, the potential vanishes exactly.
From the expressions of the derivatives of the potential, Eqs. (\[potder\]) and (\[potder2\]), one deduces that there also exists a valley of stability. This valley is also clearly seen in Fig. \[potrhophi2\] and is of course of utmost importance for us. It is given by the following trajectory in the $(\rho, \phi )$ plane
$$\label{trajecnoncanon}
\kappa \phi ^2_{\rm valley}\left(y\right)=\frac43\left(\frac{\kappa
A}{\alpha m}\right)^2 y{\rm
e}^{-2y}\left[y^2+\frac72y+3-\frac32\frac{W_0}{A}{\rm
e}^y\left(y+2\right)\right]\times \left[-\frac32+2\left(\frac{\kappa
A}{\alpha m}\right)y{\rm e}^{-y}+\left(\frac{\kappa A}{\alpha
m}\right)y^2{\rm e}^{-y} \right]^{-1}\, .$$
In terms of canonically normalized fields, the same trajectory reads $$\begin{aligned}
\label{inftrajec}
\kappa \bar{\phi }^2_{\rm valley}\left(\bar{\rho }\right)&=&\frac83
\left(\frac{\kappa A}{\alpha m}\right)^2 \beta
{\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}
\exp\left(-2\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}\right)
\biggl[\beta ^2{\rm e}^{2\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}
+\frac72\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}+3
\nonumber \\
& &
-\frac32\frac{W_0}{A}
\exp\left(\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}\right)
\left(\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}
+2\right)\biggr]\times
\biggl[-\frac32+2\left(\frac{\kappa
A}{\alpha m}\right)\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}
\exp\left(-\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho
}}\right)
\nonumber \\
& & +\left(\frac{\kappa A}{\alpha
m}\right)\beta ^2{\rm e}^{2\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}
\exp\left(-\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}\right)
\biggr]^{-1}\, .\end{aligned}$$
This trajectory is represented in Fig. \[courbmin\].
{width=".95\textwidth" height=".65\textwidth"}
For large values of the inflaton, $\bar{\phi }/{m_{_{\mathrm Pl}}}\gg 1$, the offset function is negligible and the potential is almost given by $V\simeq \tilde{\cal U}\phi ^2$. As a consequence, the valley of stability reduces to the simple equation $\bar{\rho }=\bar{\rho }_{\rm
min,\tilde{\cal U}}$ as can be directly checked in Fig. \[courbmin\]: in this regime (but in this regime only), as it is the case for hybrid inflation, the waterfall field is frozen. For small values of $\bar{\phi }/{m_{_{\mathrm Pl}}}$, the offset function becomes important, the trajectory bends and, as expected, joins the global minimum of the potential .
Let us now discuss how inflation proceeds in this model. Clearly, the potential plotted in Fig. \[potrhophi2\] is reminiscent of hybrid inflation where $\bar{\phi }$ is the inflaton and where the moduli $\bar{\rho }$ plays the role of the waterfall field [@hybrid]. In hybrid inflation, inflation proceeds along the valley either in the vacuum dominated regime, in which case the potential is almost a constant, or in the inflaton dominated regime, in which case $V\sim
\phi ^2$. There is also a version, well-motivated from the particle physics point of view, where the flat valley is lifted by quantum corrections [@LR]. In this case the potential along the valley is computed by means of the well-known Coleman-Weinberg formula. In our case, we have already noticed that, for $\bar{\phi }\gg {m_{_{\mathrm Pl}}}$, the potential in the valley takes the form $V\sim \tilde{\cal U}\bar{\phi
}^2$. Therefore, our case belongs to the inflaton dominated regime case.
The next question is to study how inflation ends. Typically, in hybrid inflation, inflation stops by instability. At some point along the inflationary valley, the effective squared mass in the direction perpendicular to the valley becomes negative and, as a consequence, the field can no longer be kept on tracks and quickly rolls down towards its minimum where reheating proceeds. We will show that this is not the case here. In our case, inflation stops while still in the valley due to a violation of the slow-roll conditions. The crucial ingredient here is that the valley is not a straight line but corresponds to a non-trivial path in the configuration space of the two fields. This aspect is reminiscent of mutated inflation [@mutated] (other interesting inflationary models where the trajectory is not trivial can be found in Refs. [@shifted]). In fact, the analogy can even been pushed further. Indeed, in the case of mutated inflation, the potential has typically the following form [@LR] $$\label{vmutated}
V\left(\psi, \phi \right)=V_0\left(1-\frac{\psi }{M}\right)
+\frac{\lambda }{4}\phi ^2\psi ^2 +\cdots \, ,$$ where $V_0$, $M$ and $\lambda $ are constants. In the above expression, $\phi $ is the inflaton and the second field $\psi $ plays the role of our moduli $\rho $. One of the main feature of mutated inflation is that an effective potential for the inflaton can be produced even if the original potential contains no term that depends only on $\phi $ as can be seen explicitly in the previous formula. In the configuration space, the trajectory reads $\psi \phi
^2=2V_0/(\lambda M)$ and after having inserted this expression into Eq. (\[vmutated\]) one gets $$V\left(\phi \right)=V_0\left(1-\frac{V_0}{\lambda ^2M^2\phi
^2}\right)\, ,$$ which is suitable for inflation. In the same manner, our potential given in Eq. (\[ourpot\]) does not contain any piece depending on the inflaton field only. However, inserting the trajectory $\rho (\phi
)$ in $V\left(\rho ,\phi \right)$ would lead to a potential $V(\phi )$ as above. The difficulty in our case is that Eq. (\[trajecnoncanon\]) gives rather $\phi (\rho )$ and that the expression is too complicated to be invertible. But, clearly, at the level of principles this is very similar. Therefore, the model presented here combines aspects from chaotic and mutated inflation hence the name “mutated chaotic inflation” given at the beginning of this section.
Phenomenological constraints
----------------------------
We now discuss the constraint on the parameter characterizing the inflaton sector, [*i.e. *]{}the mass $m$ (since the parameters $A$ and $W_0$ have already been discussed before). In order to simplify the discussion, we will assume that the initial conditions are such that the fields are, at the beginning of the evolution, in the valley of stability and more particularly in the straight line part of the valley (we will come back to the questions of the initial conditions later on and will discuss this assumption in some detail) where $V
\sim \tilde{\cal U}\left(\bar{\rho }_{\rm min, \tilde{\cal
U}}\right)\bar{\phi }^2/2$. Since the quantum fluctuations of the inflaton field $\bar{\phi }$ are at the origin of the CMB anisotropy observed today, the COBE and Wilkinson Microwave Anisotropy Probe (WMAP) normalizations fix the coupling constant of the inflaton potential, namely the mass function $\tilde{\cal U}\left(\bar{\rho
}_{\rm min, \tilde{\cal U}}\right)$ in the present context. Concretely, for small $\ell $, the multipole moments are given by $$C_{\ell }=\frac{2H^2}{25\epsilon {m_{_{\mathrm Pl}}}^2}\frac{1}{\ell (\ell +1)}\, ,$$ where $\epsilon $ is the first slow-roll parameter to be discussed below. What has been actually measured by the COBE and WMAP satellites is $Q^2_{\rm rms-PS}/T^2=5C_2/(4\pi )\simeq \left(18\times
10^{-6}/2.7\right)^2\simeq 36\times 10^{-12}$. The quantity $H$ is the Hubble parameter during inflation and is related to the potential by the slow-roll equation $H^2\simeq \kappa V/3$ evaluated at Hubble radius crossing. Putting everything together, we find that the inflaton mass is given by $$\left[\frac{\tilde{\cal U}\left(\bar{\rho }_{\rm min, \tilde{\cal
U}}\right)}{{m_{_{\mathrm Pl}}}}\right]^2\simeq 45 \pi \left(N_*+\frac12 \right)^{-2}
\frac{Q^2_{\rm rms-PS}}{T^2}\, ,$$ where $N_*\simeq 60$ ([*i.e. *]{}the number of e-folds between the time at which the modes of astrophysical interest today left the Hubble radius during inflation and the end of inflation, see Ref. [@number]), that is to say $$\sqrt{\tilde{\cal U}\left(\bar{\rho }_{\rm min, \tilde{\cal
U}}\right)} \simeq 1.3\times 10^{-6}\times {m_{_{\mathrm Pl}}}\, .$$ At this point, it is important to recall that the mass function has been defined by the expression $$\tilde{\cal U}\left(\bar{\rho }_{\rm min, \tilde{\cal U}}\right)
=\frac{(\alpha m)^2\beta ^3 }{4y_{\rm min, \tilde{\cal U}}}
\left(\frac{1}{2y_{\rm min, \tilde{\cal U}}}-\frac{\kappa A}{\alpha m}
{\rm e}^{-y_{\rm min, \tilde{\cal U}}}\right)\, .$$ In this expression, all the factors but $m$ are of order one, see the previous discussion about the constraints on the parameters $A$ and $W_0$. Therefore, this implies that $m\simeq {\cal
O}\left(10^{-6}\right)$ and this is the value that will be used in the following.
In order to compute the inflationary observables (as the spectral indices for instance), it is convenient to use the slow-roll approximation. The slow-roll approximation is controlled by two parameters (in fact, at leading order, there are three relevant slow-roll parameters but we will not need the third one) defined by [@slowroll] $$\begin{aligned}
\epsilon &\equiv & -\frac{\dot{H}}{H^2}\, ,\quad
\delta =-\frac{\dot{\epsilon}}{2H\epsilon }+\epsilon \, .\end{aligned}$$ The main advantage of these definitions is that they involve the background Hubble parameter $H$ only. Therefore, in some sense, they are independent from the matter content, in particular they do not require the knowledge of the number of scalar fields present in the underlying inflationary model. If we now assume that only one scalar field is present, then it is easy to obtain that $$\begin{aligned}
\label{srparam}
\epsilon &\simeq & \epsilon _{\bar{\phi }}\equiv \frac{{m_{_{\mathrm Pl}}}^2}{16\pi
}\left(\frac{V_{,{\bar{\phi}}}}{V}\right)^2\, ,\\ \delta &\simeq &
\delta _{\bar{\phi }}\equiv -\frac{{m_{_{\mathrm Pl}}}^2}{16\pi
}\left(\frac{V_{,\bar{\phi}}}{V}\right)^2 +\frac{{m_{_{\mathrm Pl}}}^2}{8\pi
}\frac{V_{,\bar{\phi}\bar{\phi}}}{V}\, .\end{aligned}$$ Two remarks are in order. Firstly, the condition $\epsilon <1$ is equivalent to $\ddot{a}>0$. Therefore, in order to have inflation, strictly speaking $\epsilon $ needs not to be small with respect to one, it only needs to be less than one. Secondly, the parameter $\delta $ is not positive definite contrary to the first slow-roll parameter.
In the case where two fields are present, they are different ways of generalizing the definition of the slow-roll parameters [@isopert]. A first method consists in following the trajectory in configuration space. The trajectory is given by $\bar{\phi }=\bar{\phi }(N)$ and $\bar{\rho }= \bar{\rho }(N)$, where $N$ is the total number of e-folds counted from the beginning of inflation (not to be confused with $N_*$). As a consequence the vector tangent to this trajectory, ${\bf e}_{\parallel}=(e_{\bar{\phi
}},e_{\bar{\rho }})$, can be expressed as $$\begin{aligned}
e_{\bar{\phi }} &=& \frac{\displaystyle \frac{{\rm d}\bar{\phi }}{{\rm
d}N}} {\sqrt{\displaystyle\left(\frac{{\rm d}\bar{\phi }}{{\rm
d}N}\right)^2 +\left(\frac{{\rm d}\bar{\rho }}{{\rm
d}N}\right)^2}} =\cos \theta \, , \\ e_{\bar{\rho }}&=&
\frac{\displaystyle \frac{{\rm d}\bar{\rho }}{{\rm
d}N}} {\sqrt{\displaystyle\left(\frac{{\rm d}\bar{\phi }}{{\rm
d}N}\right)^2 +\left(\frac{{\rm d}\bar{\rho }}{{\rm
d}N}\right)^2}}=\sin \theta \, .\end{aligned}$$ We can then define “directional slow-roll parameters” by replacing the first and second derivatives in Eqs. (\[srparam\]) by directional derivatives of the potential, namely $$\begin{aligned}
\label{directionalsr}
\epsilon _{\parallel} &=& \frac{{m_{_{\mathrm Pl}}}^2}{16\pi V^2}\left(\cos \theta
V_{,{\bar{\phi}}}+\sin \theta V_{,{\bar{\rho }}}\right)^2\, ,\\
\label{directionalsr2}
\delta_{\parallel} &=& -\epsilon _{\parallel}+\frac{{m_{_{\mathrm Pl}}}^2}{8\pi V}(\cos
^2\theta V_{,\bar{\phi}\bar{\phi}} +2\cos \theta \sin \theta
V_{,\bar{\phi}\bar{\rho }} \nonumber \\
& & +\sin ^2 \theta V_{,\bar{\rho }\bar{\rho
}})\, .\end{aligned}$$ However, it is clear that we no longer have the equivalence between $\epsilon _{\parallel}<1$ and $\ddot{a}>0$. For this reason, it is also interesting to keep the original definition of $\epsilon$ (only in terms of the “geometry”), [*i.e. *]{}$\epsilon =-\dot{H}/H^2$, and express it in terms of the derivatives of the potential. This leads to $$\epsilon= \epsilon _{\bar{\phi }}+\epsilon _{\bar{\rho }} =
\frac{{m_{_{\mathrm Pl}}}^2}{16\pi }\left(\frac{V_{,{\bar{\phi}}}}{V}\right)^2 +
\frac{{m_{_{\mathrm Pl}}}^2}{16\pi }\left(\frac{V_{,{\bar{\rho }}}}{V}\right)^2 \, .$$ It is interesting to establish the link between the two types of slow-roll parameters. For this purpose, let us introduce the vector ${\bf e}_{\perp}\equiv (e_{\bar{\rho }},-e_{\bar{\phi }})$, which is perpendicular to ${\bf e}_{\parallel}$. Then, one can define a directional slow-roll parameter in the direction perpendicular to the trajectory by means of the expression $$\epsilon _{\perp}=\frac{{m_{_{\mathrm Pl}}}^2}{16\pi V^2}\left(\sin \theta
V_{,{\bar{\phi}}}-\cos \theta V_{,{\bar{\rho }}}\right)^2\, .$$ From this expression, it is not difficult to show that $$\epsilon _{\parallel}+\epsilon _{\perp} = \epsilon _{\bar{\phi
}}+\epsilon _{\bar{\rho }} = \epsilon \, .$$ Obviously, if the inflationary valley is a straight line then one has $\epsilon \simeq \epsilon _{\parallel} \simeq \epsilon _{\bar{\phi}}$ and this is the case for the model under consideration in this article provided that $\bar{\phi }\gg {m_{_{\mathrm Pl}}}$. In this situation, the model is equivalent to chaotic inflation and therefore the slow-roll parameters are given by $$\label{chaoticsr}
\epsilon = \frac{1}{2N_*+1}, \quad
\delta = 0\, ,$$ where $N_*\simeq 60$ (there is some freedom in the choice of this number, see Ref. [@number]) has already been defined before. In the situation where these parameters are small, namely $\epsilon \ll
1$ and $\delta \ll 1$, the equation of motion of the inflaton field can be easily integrated. One finds $$\label{srinflaton}
\frac{\bar{\phi }}{{m_{_{\mathrm Pl}}}}=\sqrt{\left(\frac{\bar{\phi }_{\rm
ini}}{{m_{_{\mathrm Pl}}}}\right)^2 -\frac{N}{2\pi }}\, ,$$ where $\bar{\phi }_{\rm ini}$ is the initial value of the field. Let us emphasize again that this is valid only if the scales of astrophysical interest leave the Hubble radius in the straight line part of the potential. If this happens in the curved part of the potential the above result is no longer valid.
Finally, let us introduce the squared mass in the direction perpendicular to the inflationary trajectory. This is nothing but the second order directional derivative of the potential along ${\bf
e}_{\perp}$ given by $$\label{massperp} m_{\perp}^2\equiv (\cos ^2\theta
V_{,\bar{\phi}\bar{\phi}} -2\cos \theta \sin \theta
V_{,\bar{\phi}\bar{\rho }} +\sin ^2 \theta V_{,\bar{\rho }\bar{\rho
}})\, .$$ This quantity allows us to distinguish whether inflation ends by instability or not. If, as it is the case for hybrid inflation, an instability occurs, $m_{\perp}^2$ becomes negative.
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Numerical results
-----------------
To prove the above claims that inflation ends by violation of the slow-roll conditions and not by instability, it is necessary to determine the inflationary trajectory exactly. Clearly, the potential is too complicated to permit an analytical integration of the exact motion and, therefore, we will perform a numerical integration of the two Klein-Gordon equations and of the Friedmann equation. For convenience, as already mentioned, the total number of e-folds $N\equiv \ln \left(a/a_{\rm ini}\right)$ will be used as time variable. In this case, the Klein-Gordon equation reads (here for the inflaton field; this is of course also the case for the moduli) $$\begin{aligned}
& & \label{kgefold} \frac{{\rm d}^2}{{\rm d}N^2}\left(\frac{\bar{\phi
}}{{m_{_{\mathrm Pl}}}}\right) +\left(3+\frac{1}{H}\frac{{\rm d}H}{{\rm d}N}\right)
\frac{{\rm d}}{{\rm d}N}\left(\frac{\bar{\phi }}{{m_{_{\mathrm Pl}}}}\right)
\nonumber \\
& & +\left(\frac{{m_{_{\mathrm Pl}}}}{H}\right)^2 \frac{\partial
\left(V/{m_{_{\mathrm Pl}}}^4\right)}{\partial (\bar{\phi }/{m_{_{\mathrm Pl}}})}=0\, ,\end{aligned}$$ where $H$ is the Hubble parameter during inflation.
The result of our numerical integration is displayed in Figs. \[case3\] and \[case10\]. In Fig. \[case3\], we have chosen the parameters such that $\alpha =\sqrt{2}$, $\beta =1$, $m=10^{-6}{m_{_{\mathrm Pl}}}$ (in accordance with the COBE and WMAP normalizations, see the discussion above), $\kappa A/(\alpha m)=1.35135$ and $W_0/A=0.41111$. This means that the minimum of the inflationary valley is located at $y_{{\rm min},\tilde{\cal U}}\simeq 1.067$ or, in terms of the canonically normalized fields, at $\bar{\rho }_{{\rm
min},\tilde{\cal U}}\simeq 0.004 \times {m_{_{\mathrm Pl}}}$. The absolute minimum of the potential is at $\phi =0$, $y=y_{{\rm min},\tilde{\cal V}}\simeq
1.622$ or, in terms of the canonically normalized fields, at $\bar{\phi }=0$, $\bar{\rho }=\bar{\rho }_{{\rm min},\tilde{\cal
V}}\simeq 0.118 \times {m_{_{\mathrm Pl}}}$. The initial conditions have been chosen to be $\phi _{\rm ini}=3\times {m_{_{\mathrm Pl}}}$ or $\bar{\phi }_{\rm ini}\simeq
4.243 \times {m_{_{\mathrm Pl}}}$ and $y_{\rm ini}=y_{{\rm min},\tilde{\cal U}}\simeq
1.067$ or $\bar{\rho }_{\rm ini}=\bar{\rho }_{{\rm min},\tilde{\cal
U}}\simeq 0.004 \times {m_{_{\mathrm Pl}}}$. This means that the evolution starts at the bottom of the valley. The first plot (first line, on the left) shows the evolution of the field $\bar{\phi }$ versus the total number of e-folds (solid black line). The red dotted line represents the slow-roll approximation given by Eq. (\[srinflaton\]), valid in the case where there is only one field. At the beginning of the evolution, the field $\bar{\phi }$ closely follows the slow-roll equation Eq. (\[srinflaton\]). Clearly, this is because the valley is a straight line and, therefore, everything is as if there were only one field. Then, the valley bends and the black dotted curve separates from the red dotted line. Interestingly enough, this is not associated with a rapid evolution of the inflaton which is already an indication that, although the valley is now curved, the slow-roll conditions are probably not violated. Then, the field joins its minimum at $\bar{\phi
}=0$ and there are small oscillations around that minimum (which are difficult to distinguish with the scales used in this particular plot). The second figure (first line, on the right) shows the evolution of the moduli $\bar{\rho }$ with the total number of e-folds. At the beginning, $\bar{\rho }$ is frozen at the bottom of the inflationary valley. When the valley bends, $\bar{\rho }$ also joins the absolute minimum. The third plot (second line, on the left) displays the trajectory $\bar{\phi }(\bar{\rho })$. The most striking feature of the plot is that the trajectory exactly follows the inflationary valley (shown as the red dotted curve). Of course, this is maybe not so surprising given the fact that, initially, the moduli field is at the bottom of the valley and that the initial velocities of both fields vanish. Nevertheless, in this case, Eq. (\[inftrajec\]) is an analytical expression of the non trivial inflationary trajectory. The next plots show the directional slow-roll parameters $\epsilon _{\parallel}$ (second line, on the right) and $\delta _{\parallel}$ (third line, one the right). One can explicitly check that these slow-roll parameters remain small even, and this is crucial here, when the trajectory bends. It is only at the very end, close to the absolute minimum of the potential, that the slow-roll conditions are violated. Therefore, we have proven that, contrary to the case of hybrid inflation, a variation of the waterfall field is not associated with a violation of the slow-roll conditions. In other words, the slow-roll conditions continue to hold even when the trajectory is curved, except, of course, when the absolute minimum is approached. The cusp present in the plot of the second slow-roll parameter $\delta _{\parallel}$ (around $N\simeq 130$) is due to the fact that $\delta _{\parallel}$ becomes negative (recall that, contrary to $\epsilon _{\parallel}$, $\delta _{\parallel}$ is not positive-definite). One also notices the presence of quite large oscillations at the very beginning of the evolution and at the very end. The oscillations at the very end are clearly the oscillations occurring when the absolute minimum is joined and when reheating proceeds. The oscillations at the beginning of the evolution are worth interpreting. They do not seem to be associated with some numerical problems since it can be checked that they remain even if the parameter controlling the accuracy of the code is modified. Our interpretation is the following. As discussed above, the initial value of $\bar{\rho }$ has been chosen such that it corresponds to the bottom of the valley in the regime $\bar{\phi }\gg {m_{_{\mathrm Pl}}}$, in fact strictly speaking, in the limit $\bar{\phi }/{m_{_{\mathrm Pl}}}\rightarrow
+\infty$. On the other hand, the initial condition on the inflaton is $\bar{\phi} \simeq 4.2 \times {m_{_{\mathrm Pl}}}$ and, for this value of the field, the bottom of the valley is not exactly located at the value obtained before, in the limit $\bar{\phi }/{m_{_{\mathrm Pl}}}\rightarrow +\infty$. The oscillations are nothing but a transient regime during which the moduli field is settling at the bottom of the valley.
As noticed before, the directional slow-roll parameter $\epsilon
_{\parallel }$ does not control the end of inflation. However, we have checked that, during almost all the evolution, the difference between the parameter $\epsilon =-\dot{H}/H^2$ and $\epsilon _{\parallel}$ is small. As a consequence, we see that the total number of e-folds (during which we have slow-roll inflation) is $N_{_{\rm T}}\simeq
160$. This has to be compared with the single field expression for $N_{_{\rm T}}$, $$N_{_{\rm T}}=2\pi \left(\frac{\bar{\phi }_{\rm
ini}}{{m_{_{\mathrm Pl}}}}\right)^2-\frac12\, ,$$ evaluated for the same initial conditions. In this case, this gives $N_{_{\rm T}}\simeq 113$. This means that, for the same initial conditions, the model under investigation in this article leads to a larger number of total e-folds, probably because the inflationary path is, in some sense, “longer”. Finally, the last plot (third line, on the right) represents $m_{\perp}^2$, see Eq. (\[massperp\]), versus the total number of e-folds. This figure is important because it proves that inflation does not end by instability, as it is the case for hybrid inflation. This is because $m_{\perp}^2$ always remains positive, although it is decreasing as the fields are rolling down the valley, meaning that this valley opens up as one is approaching the absolute minimum. Therefore, in chaotic mutated inflation, inflation stops by violation of the slow-roll conditions in the inflationary valley (and after this valley has bent), [*i.e. *]{}close to the absolute minimum of the potential.
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Our next step is to study whether the previous conclusions are robust and can be modified if we change either the initial conditions and/or the parameters of the model. In Fig. \[case10\], we have considered another initial condition for the inflaton field, namely $\phi _{\rm
ini}=10\times {m_{_{\mathrm Pl}}}$ or $\bar{\phi }_{\rm ini}\simeq 14.142 \times {m_{_{\mathrm Pl}}}$, the other parameters being the same as in Fig. \[case3\] \[ i.e. $\alpha =\sqrt{2}$, $\beta =1$, $m=10^{-6}{m_{_{\mathrm Pl}}}$, $\kappa A/(\alpha
m)=1.35135$ and $W_0/A=0.41111$. The initial condition of the moduli waterfall field is $y_{\rm ini}=y_{{\rm min},\tilde{\cal U}}\simeq
1.067$ or $\bar{\rho }_{\rm ini}=\bar{\rho }_{{\rm min},\tilde{\cal
U}}\simeq 0.004 \times {m_{_{\mathrm Pl}}}$, [*i.e. *]{}still at the bottom of the valley\]. As can be seen all the remarks and conclusions obtained before remain valid for this case. Another remark is in order at this point. In the valley, since we have $V\sim \tilde{\cal U}\bar{\phi
}^2$, the parameter $\delta _{\parallel}$ should vanish, see Eqs. (\[directionalsr2\]) and (\[chaoticsr\]). We see in Fig. \[case3\] that, on the contrary, $\epsilon _{\parallel }$ and $\delta _{\parallel}$ are initially of the same order of magnitude, [*i.e. *]{}$10^{-3}$. The reason is that, initially, the two fields are not sufficiently “deep”in the valley. On the contrary, with the new initial condition $\bar{\phi }_{\rm ini}\simeq 14.142\times {m_{_{\mathrm Pl}}}$, the fields are really in the straight part of the valley. As a consequence, one can check that the slow-roll parameter $\delta
_{\parallel}$ ($\simeq 10^{-5}$) is now two orders of magnitude smaller than $\epsilon _{\parallel}$ ($\simeq 10^{-3}$), in full agreement with the arguments presented above. One can even check that the numerical values are consistent with the above interpretation. Indeed, as a time-dependent function, the slow-roll parameter $\epsilon $ is given by $\epsilon={m_{_{\mathrm Pl}}}^2/(4\pi \phi^2
)$. Therefore, initially one has $\epsilon={m_{_{\mathrm Pl}}}^2/(4\pi \phi _{\rm
ini}^2 )\simeq 0.39\times 10^{-3}$ which is the value seen in Fig. \[case10\] for small $N$.
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There exists another way of modifying the initial conditions. Instead of changing the initial value of the inflaton $\bar{\phi }$, we can also study what happens if the moduli field $\bar{\rho }$ is initially displaced from the bottom of the valley. This is especially relevant because it is known that hybrid inflation is very sensitive to the initial conditions and that only a very small fraction of possible initial conditions leads to successful inflation, see Refs. [@hybridini]. In particular, it has been shown in these articles that the waterfall field must be precisely tuned at the bottom of the inflationary valley in order to obtain a satisfactory subsequent evolution. In Fig. \[caseoff\] (left panel), we have used the same set of parameters as in the previous figures but the initial conditions are now $\bar{\phi }_{\rm ini}=14.142\times {m_{_{\mathrm Pl}}}$ and $\bar{\rho }_{\rm ini}=0.0479 \times {m_{_{\mathrm Pl}}}$. The initial value of the moduli field is approximatively one order of magnitude larger than in Figs. \[case3\] and \[case10\]. We see that successful inflation is still obtained. After very rapid oscillations, the moduli stabilizes at the bottom of the valley and then the evolution proceeds as before. Therefore, it seems that the present model is more stable to modifications of the initial conditions than standard hybrid inflation. Of course, what should really be done, as in Refs. [@hybridini], is a systematic scan of the space of initial conditions but this is beyond the scope of the present article.
Finally, we also need to study what happens if we change the parameters of the model, [*i.e. *]{}$A$ and $W_0$ (since $m$ is fixed by the CMB normalization). In Fig. \[caseoff\] (right panel), the trajectory in the space $(\bar{\phi },\bar{\rho })$ is displayed for the following choice of parameters: $\alpha =\sqrt{2}$, $\beta =1$, $m=10^{-6}{m_{_{\mathrm Pl}}}$, $\kappa A/(\alpha m)=1.30$ and $W_0/A=0.41111$. There is a new value for the parameter $A$ (and in fact a new value for $W_0$ but such that the ratio $W_0/A$ is left unchanged), of course still compatible with the constraints derived above. The position of the absolute minimum is unaffected because it only depends on $W_0/A$. On the other hand, the location of the valley ([*i.e. *]{}the location of the minimum of the mass function) is changed, since it depends on $\kappa A/(\alpha m)$ and is now located at $\bar{\rho
}=\bar{\rho }_{{\rm min},\tilde{\cal U}}\simeq 0.0415 \times {m_{_{\mathrm Pl}}}$. We have chosen the initial conditions such that $\phi _{\rm ini}=10\times
{m_{_{\mathrm Pl}}}$ or $\bar{\phi }_{\rm ini}\simeq 14.142 \times {m_{_{\mathrm Pl}}}$ and such that the evolution starts from the (new) bottom of the valley exactly, namely $y_{\rm ini}=y_{{\rm min},\tilde{\cal U}}\simeq 1.165$ or $\bar{\rho }_{\rm ini}=\bar{\rho }_{{\rm min},\tilde{\cal U}}\simeq
0.0415 \times {m_{_{\mathrm Pl}}}$. We notice that the above conclusions remain unchanged: the fields follow the inflationary valley and join the absolute minimum of the potential as it was the case in the previous examples.
Our conclusion is that the main features of the mutated chaotic inflationary scenario seem to be robust either to modifications of the initial conditions or to changes of the parameters of the model, namely $A$ and/or $W_0$.
Quantum stability
-----------------
Let us finish this section by a discussion on the quantum stability of inflation along the valley where the inflaton rolls slowly. As can be seen in Fig. \[potrhophi2\], the valley is bordered on each side by potential barriers. There is the infinite barrier at $\rho=0$ and a finite height barrier located at $y_{{\rm max},\tilde{\cal U}}$ defined as the second root of Eq. (\[deff\]). Now this barrier separates the inflation valley and the vacuum at infinity with vanishing potential. There are two sources of instability of the valley. The first one is the tunneling of the moduli field through the barrier followed by the down roll towards $\rho=\infty$. The second one is stochastic evolution of the moduli field when its mass in the valley is much less than the Hubble rate. In our case, the mass of the moduli field is of the order of the Hubble rate implying that the moduli field is not light and does not fluctuate like a stochastic field with root mean square excursion $H/2\pi$. The only possibility for the moduli field to go through the barrier is tunneling. The tunneling time is approximated by the Coleman-De Luccia instanton [@coleman]. In the thin-wall approximation where the height of the barrier is large, [*i.e. *]{}$\tilde U(y_{{\rm min},\tilde{\cal
U}})/\tilde U(y_{{\rm max},\tilde{\cal U}})\ll 1$, and the width of the potential barrier is large in Planck units (see Fig. \[plotuv\]), the tunneling time is given by $$t_{\rm Decay}\simeq t_{_{\rm Pl}}{\rm e}^{24{m_{_{\mathrm Pl}}}^4\pi^2/V_0}\,
,$$ where $V_0$ is the potential in the valley and $t_{_{\rm Pl}}$ the Planck time. Using $V_0\simeq 3{m_{_{\mathrm Pl}}}^2 H^2$ and $H\approx 10^{-6} {m_{_{\mathrm Pl}}}$, one finds that the decay time is exponentially longer than the Planck time. For all practical purposes the valley is quantum stable. At the end of the evolution along the valley, the moduli becomes sensitive to the existence of the global minimum of the potential and rolls down the potential towards the global minimum.
Mutated chaotic inflation and KKLT stabilization
================================================
In the previous section we have introduced a model of inflation with moduli stabilization and mutated chaotic inflation. Unfortunately, the vacuum energy at the end of inflation becomes negative. We have compensated this negative energy by introducing a constant and positive energy of unknown origin. Here, we will combine a string inspired stabilization mechanism (the KKLT stabilization mechanism) with mutated chaotic inflation. This is obtained by introducing an explicit moduli dependent potential which lifts the vacuum energy towards positive values.
Lifting AdS to dS
-----------------
There are two equivalent ways of lifting the potential energy for the moduli. The first one comes from the D3/D7 system that we have already studied at the beginning of the previous section. Instead of studying the potential (\[pottotalpm\]) in the vicinity of the minimum, which leads to Eq. (\[potmodelpm\]), one can focus on the regime where the waterfall fields vanish $\phi^\pm=0$. Then, Eq. (\[pottotalpm\]) leads to $$V=\frac{2g^2\zeta ^4}{\kappa^{3/2}\left(\rho+\rho ^{\dagger
}\right)^3}+ \tilde{\cal V}(\rho)\, .$$ This is a KKLT potential with a correction term $\propto 1/\rho ^3$, with a positive minimum provided $\zeta $ is chosen appropriately. However, we have already used the potential (\[pottotalpm\]) to give a mass to the inflaton and, as discussed before, this was in another regime. Therefore, if we want to preserve this mechanism and stabilize the moduli by the KKLT method, the term $\propto 1/\rho ^3$ must have another origin that we now discuss.
Assume that the model possesses a $U(1)$ gauge field with a Fayet–Iliopoulos term. The Kähler potential of the moduli is modified and becomes $$K=-\frac{3}{\kappa } \ln \left[\kappa^{1/2}\left(\rho+\rho ^{\dagger
}\right) + \xi V \right]\, .$$ where $V$ is the vector superfield and $\xi $ is the Fayet–Iliopoulos term. This could be due to an anomalous symmetry cured by the Green-Schwarz mechanism. Moreover we assume that the gauge coupling function reads $$f(\rho )= \frac{\kappa^{1/2}\rho}{\tilde g^2}\, ,$$ where $\tilde g$ is a constant. Assuming that no field is charged under this $U(1)$ symmetry, the D-term associated to this gauge symmetry is $$V_{_{\rm D}}= \frac{D}{\rho^3}\, ,$$ where $D=\tilde{g}^2\xi ^2/16$ [@Burgess2]. The total potential that we obtain is the potential of Eq. (\[pottotalpm\]) plus $V_{_{\rm D}}$. Then, using our usual mechanism to give a mass to the inflaton and following the same step as before we find that the new potential reads $$\label{VKKLT}
V\left(\rho ,\phi \right)=\tilde{{\cal V}}(\rho )+\frac{D}{\rho ^3}
+\tilde{{\cal U}}(\rho )\phi ^2\, .$$ We see that this only amounts to have a new offset function $\hat{{\cal V}}(\rho ) \equiv \tilde{{\cal V}}(\rho )+D/\rho ^3$. Hence, all the results obtained before on the mass function $\tilde{\cal U}$ are still valid.
KKLT stabilization and mutated chaotic inflation
------------------------------------------------
Let us now discuss how the parameter $D$ can be fixed. We will not present a complete analysis of the parameter space as such an analysis is complicated. However, we will demonstrate that the KKLT mechanism also works in the case under consideration. The parameter $D$ must be chosen such that, at the absolute minimum of the potential, the potential exactly vanishes (or is equal to the value of the vacuum energy today. Since this one is tiny and since we only study the inflationary era, we will just assume that the minimum is zero). This means that we have to solve simultaneously the equations $$\begin{aligned}
& & y^2_{{\rm min},\hat{\cal V}}+\frac72 y_{{\rm min},\hat{\cal V}}+3
= \frac{3W_0}{2A}{\rm e}^{y_{{\rm min},\hat{\cal V}}} \left(y_{{\rm
min},\hat{\cal V}} +2\right)
\nonumber \\
& & -\frac{9D\kappa ^{1/2}}{A^2}\frac{{\rm
e}^{2y_{{\rm min},\hat{\cal V}}}}{y_{{\rm min},\hat{\cal V}} }\, ,
\\ & & \frac{{\rm e}^{-y_{{\rm min},\hat{\cal V}} }}{2y_{{\rm
min},\hat{\cal V}} }\left[\frac{{\rm e}^{-y_{{\rm min},\hat{\cal V}}
}}{3}-\frac{1}{y_{{\rm min},\hat{\cal V}}} \left(\frac{W_0}{A}-{\rm
e}^{-y_{{\rm min},\hat{\cal V}}}\right)\right]
\nonumber \\
& & +\frac{\kappa
^{1/2}D}{A^2y_{{\rm min},\hat{\cal V}} ^3} =0\, .\end{aligned}$$ The first condition is a condition on the derivative of the new offset function and is similar to Eq. (\[condder\]) while the second equation is nothing but the condition that the potential is zero at the minimum. We see that the relevant new parameter is in fact the dimensionless quantity $\kappa ^{1/2}D/A^2$.
Let us first consider the case where the parameters are $\alpha
=\sqrt{2}$, $\beta =1$, $m=10^{-6}{m_{_{\mathrm Pl}}}$, $W_0/A=0.4111$ and $\kappa
A/(\alpha m)=1.35135$, [*i.e. *]{}the case envisaged before. Then, a solution to the two above equations can be found and reads: $\kappa
^{1/2}D/A^2\simeq 0.027$ and $y_{{\rm min},\hat{\cal V}}\simeq
2.349$. The corresponding offset function is represented in Fig. \[plotvKKL\] (solid line). One can check that there is indeed a minimum and that, at the minimum, the offset function vanishes. Let us now consider another case, namely $\alpha =\sqrt{2}$, $\beta =1$, $m=10^{-6}{m_{_{\mathrm Pl}}}$, $W_0/A=1.2$ and $\kappa A/(\alpha m)=1.35135$, [*i.e. *]{}the value of the ratio $W_0/A$ is now different and, most importantly, greater than one. A solution can also be obtained and is: $\kappa
^{1/2}D/A^2\simeq 0.145$ and $y_{{\rm min},\hat{\cal V}}\simeq
1.458$. The corresponding offset function is plotted in Fig. \[plotvKKL\] (dotted line) and we check again that there is a vanishing minimum where the moduli can be stabilized. This case is in fact more interesting than the first one. Indeed, one sees that the ratio $W_0/A>1$ violates the bound established previously. Therefore, the KKLT mechanism allows us to find a vanishing minimum to the potential even if $W_0/A>1$. In other words, the constraint $W_0/A<1$ is relaxed.
{width=".95\textwidth" height=".65\textwidth"}
{width=".95\textwidth" height=".65\textwidth"}
{width=".95\textwidth" height=".65\textwidth"}
The above property is illustrated in Fig. \[potnonKKL\] where we have plotted the potential $V(\bar{\rho} ,\bar{\phi} )$ for the following set of parameters: $\alpha =\sqrt{2}$, $\beta =1$, $m=10^{-6}{m_{_{\mathrm Pl}}}$, $W_0/A=1.2$, $\kappa A/(\alpha m)=1.35135$ and $\kappa
^{1/2}D/A^2=0$. As already discussed above, the moduli cannot be stabilized in this case because $W_0/A>1$. The “hole” that can be seen in this figure represents the region of instability, [*i.e. *]{}the region where the offset function goes to $-\infty$. In Fig. \[potKKL\], we have added the KKLT term $D/\rho ^3$. The value of the parameters are $\alpha =\sqrt{2}$, $\beta =1$, $m=10^{-6}{m_{_{\mathrm Pl}}}$, $W_0/A=1.2$, $\kappa A/(\alpha m)=1.35135$ and $\kappa
^{1/2}D/A^2\simeq 0.145$, [*i.e. *]{}as for the dotted line in Fig. \[plotvKKL\]. The “hole” has now disappeared and the shape of the potential is very reminiscent to that of the potential studied in the previous subsection, see Fig. \[potrhophi2\]. In particular, it is clear that there is a new valley of stability.
It is now interesting to study the valley in more details. Since adding the term $D/\rho ^3$ just amounts to modifying the form of the offset function, the analytical calculations which lead to the equation of the valley are very similar to those which resulted in Eq. (\[inftrajec\]). Straightforward manipulations yields
$$\begin{aligned}
\kappa \phi ^2_{\rm valley}\left(y\right) &=& \frac43\left(\frac{\kappa
A}{\alpha m}\right)^2 y{\rm
e}^{-2y}\left[y^2+\frac72y+3-\frac32\frac{W_0}{A}{\rm
e}^y\left(y+2\right)+\frac{9\kappa ^{1/2}D}{A^2}\frac{{\rm e}^{2y}}{y}
\right]
\\ \nonumber
& & \times \left[-\frac32+2\left(\frac{\kappa A}{\alpha
m}\right)y{\rm e}^{-y}+\left(\frac{\kappa A}{\alpha m}\right)y^2{\rm
e}^{-y} \right]^{-1}\, .\end{aligned}$$
As expected the only difference is the presence of the term proportional to $D$ at the numerator of the above equation. It is also interesting to give the trajectory expressed in terms of canonically normalized fields, [*i.e. *]{}the equivalent of Eq. (\[inftrajec\]). It can be expressed as $$\begin{aligned}
\label{inftrajecKKL}
\kappa \bar{\phi }^2_{\rm valley}\left(\bar{\rho }\right)&=&\frac83
\left(\frac{\kappa A}{\alpha m}\right)^2 \beta
{\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}
\exp\left(-2\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}\right)
\biggl[\beta ^2{\rm e}^{2\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}
+\frac72\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}+3
\nonumber \\
& -& \frac32\frac{W_0}{A}
\exp\left(\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}\right)
\left(\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}
+2\right)+\frac{9\kappa ^{1/2}D}{A^2\beta }{\rm
e}^{-\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }} \exp\left(2\beta {\rm
e}^{\sqrt{\frac23}
\kappa ^{1/2}\bar{\rho }}\right) \biggr]
\nonumber \\
& \times & \biggl[-\frac32+2\left(\frac{\kappa
A}{\alpha m}\right)\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}
\exp\left(-\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho
}}\right)+\left(\frac{\kappa A}{\alpha
m}\right)\beta ^2{\rm e}^{2\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}
\exp\left(-\beta {\rm e}^{\sqrt{\frac23} \kappa ^{1/2}\bar{\rho }}\right)
\biggr]^{-1}\, .\end{aligned}$$
The valley is represented in Fig. \[courbminKKL\] and compared to the valley obtained previously without the term $D/\rho ^3$. As is clear form the figure, the two trajectories have the same features.
{width=".95\textwidth" height=".65\textwidth"}
Numerical Results with KKLT stabilization
-----------------------------------------
Our next step is to study the potential given by Eq. (\[VKKLT\]) numerically. The results are displayed in Fig. \[case10KKLT\]. This figure should be compared with Figs. \[case3\] and \[case10\] where the same quantities, [*i.e. *]{}$\bar{\phi }(N)$, $\bar{\rho }(N)$, $\bar{\phi }(\bar{\rho })$, $\epsilon _{\parallel}$, $\delta
_{\parallel}$ and $m^2_{\perp}$, have also been displayed. The main conclusion that can be drawn from these plots is that the inflationary trajectory obtained in the case where the KKLT mechanism is responsible for the stabilization of the moduli is very similar to the trajectory obtained before (simply by adding a cosmological constant to renormalize the true vacuum). Therefore, all the properties that were discussed in the previous section are still valid in the present context.
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{width="8.8cm" height="7.5cm"} {width="8.8cm" height="7.5cm"}\
{width="8.8cm" height="7.5cm"} {width="8.8cm" height="7.5cm"}
Let us now discuss in more details the inflationary scenario proposed in this article. Previously, we have studied the properties of the inflationary background only. However, it is clear that, if one wants to investigate all the consequences of the model, one must compare its predictions to the high accuracy cosmological data presently available in cosmology, in particular to the CMB data which are likely to carry an imprint from inflation. This requires the calculation of the cosmological perturbations and, in the present context, this is not a trivial task. The difficulty comes from the fact that we have more than one scalar field and hence the standard slow-roll single field result is not applicable here. In particular, we now have non-adiabatic perturbations. The non-adiabatic component is expected to be particularly important if the scales of astrophysical interest today left the Hubble radius in the “curved part” of the inflationary trajectory [@wr]. This is the case, for instance, in Fig. \[case10KKLT\] where $60$ e-folds before the end of inflation corresponds to a regime where the trajectory has already bent. To be more precise, it is quite easy to determine the spectral indices at Hubble crossing during inflation. They are simply given by the well-known equations [@wr] $$\begin{aligned}
n_{_{\rm S}} &=& -4\epsilon _{\parallel}+2\delta _{\parallel}\, ,
\\
n_{_{\rm T}} &=& -2\epsilon _{\parallel}\, .\end{aligned}$$ In these equations, $n_{_{\rm S}}$ is the spectral index of the adiabatic part of the density perturbations while $n_{_{\rm T}}$ is the tensor spectral index. One could have also given the spectral index of the isocurvature power spectrum but we do not need it here and it can be found in Ref. [@wr]. We see that these formulae are a simple generalization of the one-field equations, $\epsilon
_{\bar{\phi}}$ and $\delta _{\bar{\phi}}$ just being replaced by $\epsilon _{\parallel}$ and $\delta _{\parallel}$. We have computed $n_{_{\rm S}}$ and $n_{_{\rm T}}$ for some cases envisaged before, see Tab. \[indices\].
[c c c c c c c c c c c]{} & $\displaystyle\frac{W_0}{A}$ & $\displaystyle \frac{\kappa A}{\alpha
m}$ & $\displaystyle \frac{\kappa ^{1/2}D}{A^2}$ & $\bar{\rho
}_{{\rm min},\tilde{\cal U}}$ & $\bar{\rho }_{{\rm
min},\tilde{\cal V}}$ & $\bar{\rho }_{\rm ini}$ & $\bar{\phi
}_{\rm ini}$ & $N_{_{\rm T}}$ & $n_{_{\rm S}}$ & $n_{_{\rm T}}$\
\
\
& $0.411$ & $1.3513$ & $0$ & $0.004\times {m_{_{\mathrm Pl}}}$ & $0.118\times {m_{_{\mathrm Pl}}}$ & $0.004\times {m_{_{\mathrm Pl}}}$ & $14.142\times {m_{_{\mathrm Pl}}}$ & $1356$ & $0.979$ & $-0.012$\
\
& $0.411$ & $1.3513$ & $0$ & $0.004\times {m_{_{\mathrm Pl}}}$ & $0.118\times {m_{_{\mathrm Pl}}}$ & $0.004\times {m_{_{\mathrm Pl}}}$ & $4.242\times {m_{_{\mathrm Pl}}}$ & $161$ & $0.979$ & $-0.012$\
\
& $1.2$ & $1.3513$ & $0.145$ & $0.004\times {m_{_{\mathrm Pl}}}$ & $0.092\times {m_{_{\mathrm Pl}}}$ & $0.004\times {m_{_{\mathrm Pl}}}$ & $14.142\times {m_{_{\mathrm Pl}}}$ & $1331$ & $0.974$ & $-0.0139$\
\
& $0.6$ & $1.3513$ & $0$ & $0.004\times {m_{_{\mathrm Pl}}}$ & $0.008\times {m_{_{\mathrm Pl}}}$ & $0.004\times {m_{_{\mathrm Pl}}}$ & $14.142\times {m_{_{\mathrm Pl}}}$ & $1258$ & $0.966$ & $-0.0167$\
\
& $0.411$ & $1.330$ & $0$ & $0.017\times {m_{_{\mathrm Pl}}}$ & $0.118\times {m_{_{\mathrm Pl}}}$ & $0.017\times {m_{_{\mathrm Pl}}}$ & $14.142\times {m_{_{\mathrm Pl}}}$ & $1278$ & $0.968$ & $-0.0165$\
\
These numbers must be compared with those obtained in the case of single field chaotic inflation (for a potential quadratic in the field), $$n_{_{\rm S}} \simeq 0.967\, ,\quad n_{_{\rm T}} \simeq -0.0165\, .$$ We see they are quite similar although not identical. In fact, the values of the spectral indices depend on the details of the inflationary trajectory. More precisely, what is important is how the trajectory is curved $N_*$ e-folds before the end of inflation. For instance, let us compare the cases corresponding to the two first columns in Tab. \[indices\] to the cases corresponding to the two last columns. One can check that, for the two first cases, the inflationary trajectory is “more curved”, [*i.e. *]{}deviates from a straight line more strongly, than for the two last cases. As a consequence, the spectral indices obtained for the two first cases differ more from those obtained in the standard chaotic model than the spectral indices calculated in the two last cases do. Let us also remark that one could have expected a smaller difference in the case where the initial conditions are such that the fields start deep in the valley ([*i.e. *]{}, for instance, spectral indices closer to chaotic inflation in the case where $\bar{\phi }_{\rm ini}=14.142 \times {m_{_{\mathrm Pl}}}$ than for the case $\bar{\phi }_{\rm ini}=4.242 \times {m_{_{\mathrm Pl}}}$). However, what really matters is the value of the slow-roll parameters $N_*$ e-folds before the end of inflation and not at the beginning of inflation. Therefore, the previous reasoning does not work in our case, as confirmed by the numbers in Tab. \[indices\]. Finally, one notices that the differences observed are quite small and, although it seems easy to interpret them as we have just done above, we have not been able to find a simple criterion which would allow us to predict, from the values of the free parameters $W_0$, $A$ and $D$, how far from the fiducial model the spectral indices will be. It seems that this really depends on the fine structure of the valley near the Hubble scale exit.
As discussed in Ref. [@wr], the point is that the previous indices are not those that are observable. The reason is that the evolution of the perturbations after the Hubble radius exit during inflation is non trivial in presence of isocurvature perturbations. Technically, this is because the standard conserved quantity (on super-Hubble scales) $\zeta =-{\cal R}$ is sourced by the non-adiabatic pressure $\delta
p_{\rm nad}$. The curvature and entropy perturbations evolve according to the equation $$\begin{pmatrix}
{\cal R} \cr {\cal S}
\end{pmatrix}=
\begin{pmatrix}
1 & T_{{\cal R}{\cal S}} \cr
0 & T_{{\cal S}{\cal S}}
\end{pmatrix}
\begin{pmatrix}
{\cal R} \cr {\cal S}
\end{pmatrix}
_{\rm exit}\, ,$$ where the subscript “exit” means the corresponding quantities evaluated at the exit of the Hubble radius during inflation. They correspond to the quantities given in Tab. \[indices\]. Then, the next step consists in defining the correlation angle by [@wr] $$\cos \Delta \simeq \frac{T_{{\cal R}{\cal S}}}{\sqrt{1+T_{{\cal
R}{\cal S}}^2}}\, ,$$ which appears in the final expressions of the observable spectral indices (these expressions can be found in Ref. [@wr]). As shown in Ref. [@wr], the correlation angle is the only quantity needed in order to calculate the observable spectral indices from the directional slow-roll parameters introduced before. Unfortunately, this quantity is not easy to obtain. In the present context, this would require to numerically integrate the equations governing the evolution of the cosmological perturbations (and not only the equations governing the evolution of the background as done before). This is clearly beyond the scope of the present article. However, if $\Delta $ is not too far from $\pi /2$, then the estimates given in Tab. \[indices\] are sufficient to demonstrate that the model seems to be presently compatible with the CMB data. We hope to address the question of determining the spectral indices exactly elsewhere. Let us finally notice that the value of the correlation angle has to be in agreement with the CMB constraints on the contribution originating from isocurvature perturbations obtained from the WMAP data, see for instance Ref. [@isowmap].
Another point worth discussing is the production of topological defects at the end of inflation. As was discussed recently in Ref. [@mairiJ], there exists quite tight constraints on the amount of cosmic string produced at the end of hybrid inflation. In the present context, this problem does not exist because the models studied here have only a single true vacuum. Hence, the production of topological defects at the end of inflation is simply not possible.
Conclusions
===========
We now quickly summarize our main results. Firstly, we have emphasized the role that the shift symmetry plays in order to generate flat enough potentials in F–term inflation supergravity. Secondly, we have treated the issue of moduli stabilization and considered two different possibilities, namely a simple renormalization of the potential by a constant and the stringy motivated KKLT mechanism. Thirdly, we have combined the two above mentioned ingredients in order to construct inflationary models. We have shown that, quite generically, this gives rise to models that are reminiscent of mutated inflation where the inflationary path in the configuration space is non trivial. We have also demonstrated that, in these models, inflation ends by violation of the slow-roll conditions and not by instability as it is the case in standard hybrid inflation. Finally, we have pointed out that the calculations of cosmological perturbations may be non trivial due to the possible presence of non-adiabatic perturbations.
**Acknowledgments**
We wish to thank F. Quevedo for several interesting comments.
C. L. Bennet [*et al. *]{}, Astrophys. J. Suppl. **148**, 1 (2003), [astro-ph/0302207]{}; G. Hinshaw [*et al. *]{}, Astrophys. J. Suppl. **148**, 135 (2003), [astro-ph/0302217]{}; L. Verde [*et al. *]{}, Astrophys. J. Suppl. **148**, 195 (2003), [astro-ph/0302218]{}; H. V. Peiris [*et al. *]{}Astrophys. J. Suppl. **148**, 213 (2003), [astro-ph/0302225]{}; A. Kogut [*et al. *]{}, Astrophys. J. Suppl. **148**, 161 (2003), [astro-ph/0302213]{}.
A. H. Guth, Phys. Rev. D [**23**]{}, 347 (1981); A. D. Linde, Phys. Lett. [**B108**]{}, 389 (1982); A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. [**48**]{}, 1220 (1982); A. Linde, Phys. Lett. B [**129**]{}, 177 (1983).
V. Mukhanov and G. Chibisov, JETP Lett. [**33**]{}, 532 (1981); Sov. Phys. JETP [**56**]{}, 258 (1982); S. Hawking, Phys. Lett. [**115B**]{}, 295 (1982); A. Starobinsky, Phys. Lett. [**117B**]{}, 175 (1982); A. Guth and S.Y. Pi, Phys. Rev. Lett. [**49**]{}, 1110 (1982); J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D [**28**]{}, 679 (1983).
V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, Phys. Rep. [**215**]{}, 203 (1992); J. E. Lidsey [*et al. *]{}, Rev. Mod. Phys. [**69**]{}, 373 (1997), [astro-ph/9508078]{}; J. Martin and D. J. Schwarz, , 3302 (1998), [gr-qc/9704049]{}; J. Martin and D. J. Schwarz, , 103520 (2000), [astro-ph/9912005]{}; J. Martin, A. Riazuelo and D. J. Schwarz, Astrophys. J. [**543**]{}, L99 (2000), [astro-ph/0006392]{}; D. J. Schwarz, C. A. Terrero-Escalante, and A. A. García, Phys. Lett. B [**517**]{}, 243 (2001), [astro-ph/0106020]{}; S. M. Leach, A. R. Liddle, J. Martin and D. J. Schwarz, , 023515 (2002), [astro-ph/0202094]{}; J. Martin, Proceedings of the XXIV Brazilian National Meeting on Particles and Fields, Caxambu, Brazil, (2004), [astro-ph/0312492]{}; J. Martin, Lecture notes of the 40th Karpacz Winter School on Theoretical Physics, Poland, (2004), [hep-th/0406011]{}.
D. Lyth and A. Riotto, Phys. Rept. [**314**]{}, 1 (1999), [hep-ph/9807278]{}.
N. Turok, M. Perry and P. J. Steinhardt, , 029901 (2005), [hep-th/0408083]{}; L. Cornalba and M. S. Costa, Fortsch. Phys. [**52**]{}, 145 (2004), [hep-th/0310099]{}; B. Craps and B. A. Ovrut , 066001 (2004), [hep-th/0308057]{}; M. Berkooz and B. Pioline, JCAP [**0311**]{}, 007 (2003), [hep-th/0307280]{}; L. Cornalba and M. S. Costa, Class. Quant. Grav. [**20**]{}, 3969 (2003), [hep-th/0302137]{}; E. Dudas, J. Mourad and C. Timirgaziu, Nucl. Phys. [**B660**]{}, 3 (2003), [hep-th/0209176]{}; M. Fabinger and J. Mc Greevy, JHEP [**0306**]{}, 042 (2003), [hep-th/0206196]{}; H. Liu, G. Moore and N. Seiberg, JHEP [**0210**]{}, 031 (2002), [hep-th/0206182]{}; H. Liu, G. Moore and N. Seiberg, JHEP [**0206**]{}, 045 (2002), [hep-th/0204168]{}; L. Cornalba and M. S. Costa, , 066001 (2002), [hep-th/0203031]{}; N. Ohta, Int. J. Mod. Phys. [**A20**]{}, 1 (2005), [hep-th/0411230]{}; U. H. Danielsson, [hep-th/0409274]{}.
M. Gasperini and G. Veneziano, Astropart. Phys. [**1**]{}, 317 (1993); G. Veneziano, in [*The primordial Universe*]{}, Les Houches, session LXXI, edited by P. Binétruy [*et al.*]{}, (EDP Science & Springer, Paris, 2000); M. Gasperini and G. Veneziano, Phys. Rep. [**373**]{}, 1 (2003), [hep-th/0207130]{}.
J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, , 123522 (2001), [hep-th/0103239]{}; J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, [hep-th/0105212]{}; J. Khoury, B. A. Ovrut, N. Seiberg, P. J. Steinhardt and N. Turok, , 086007 (2002), [hep-th/0108187]{}; J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, , 046005 (2002), [hep-th/0109050]{}; R. Durrer, [hep-th/0112026]{}; R. Kallosh, L. Kofman and A. Linde, , 123523 (2001), [hep-th/0104073]{}; R. Kallosh, L. Kofman, A. Linde and A. Tseytlin, , 123524 (2001), [hep-th/0106241]{}; D. H. Lyth, Phys. Lett. B [**524**]{}, 1 (2002), [hep-ph/0106153]{}; R. H. Brandenberger and F. Finelli, JHEP [**0111**]{}, 056 (2001), [hep-th/0109004]{}; J. Hwang, , 063514 (2002), [astro-ph/0109045]{}; D. H. Lyth, Phys. Lett. B [**526**]{}, 173 (2002), [hep-ph/0110007]{}; J. Martin, P. Peter, N. Pinto-Neto and D. J. Schwarz, , 123513 (2002), [hep-th/0112128]{}; J. Martin and P. Peter, , 103517 (2003), [hep-th/0307077]{}. J. Martin and P. Peter, , 061301 (2004), [astro-ph/0312488]{}. J. Martin and P. Peter, , 107301 (2004), [hep-th/0403173]{}.
J. M. Cline, [hep-th/0501179]{}; C. P. Burgess, Pramana [**63**]{}, 1269 (2004), [hep-th/0408037]{}; F. Quevedo, Class. Quant. Grav. [**19**]{}, 5721 (2002), [ hep-th/0210292]{}; C. P. Burgess, J. M. Cline, H. Stoica and F. Quevedo, JHEP [**0409**]{}, 033 (2004), [hep-th/0403119]{}; C. P. Burgess, P. Martineau, F. Quevedo, G. Rajesh and R.-J. Zhang, JHEP [**0203**]{}, 052 (2002), [hep-th/0111025]{}; C. P. Burgess, M. Majumdar, D. Nolte, F. Quevedo, G. Rajesh and R.-J. Zhang, JHEP [**0107**]{}, 047 (2001), [hep-th/0105204]{}; N. Jones, H. Stoica and S. H. H. Tye, JHEP [**0207**]{}, 051 (2002) [hep-th/0203163]{}, G. Shiu and S-H. H. Tye Phys. Lett. [**B513**]{}, 251 (2001), [ hep-th/0105307]{}.
G. R. Dvali and S.-H. H. Tye, Phys. Lett. [**B450**]{}, 72 (1999), [hep-ph/9812483]{}.
H. Firouzjahi and S.-H. H. Tye. Phys. Lett. [**B584**]{}, 147 (2004), [hep-th/0312020]{}.
J. P. Hsu and R. Kallosh, JHEP [**0404**]{}, 042 (2004), [hep-th/0402047]{}.
J. P. Hsu, R. Kallosh and S. Prokushkin, JCAP [**0312**]{}, 009 (2003), [hep-th/0311077]{}.
K. Dasgupta, J. P. Hsu, R. Kallosh, A. Linde and M. Zagermann, JHEP [**0408**]{}, 030 (2004), [hep-th/0405247]{}.
F. Koyama, Y. Tachikawa and T. Watari, 106001 (2004), [hep-th/0311191]{}.
S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, , 046005 (2003), [hep-th/0301240]{}.
C. P. Burgess, R. Kallosh and F. Quevedo, JHEP [**0310**]{}, 056 (2003) [hep-th/0309187]{}.
K. Choi, A. Falkowski, H. P. Nilles and M. Olechowski, [hep-th/0503216]{}.
S. Kachru, R. Kallosh, A. Linde, J. Maldacena, L. Mc Allister and S. P. Trivedi, JCAP [**0310**]{}, 013 (2003), [hep-th/0308055]{}.
L. Mc Allister [hep-th/0502001]{}.
M. Berg, M. Haack and B. Kors, , 026005 (2005), [hep-th/0404087]{}; M. Berg, M. Haack and B. Kors, contribution ot proceedings of “PASCOS’04”, [hep-th/0409282]{}.
J. J. Blanco–Pillado, C. P. Burgess, J. M. Cline, C. Escoda, M. Gomez–Reino, R. Kallosh, A. Linde and F. Quevedo, JHEP 0411 (2004) 063, [hep-th/0406230]{}.
W. H. Kinney, [gr-qc/0503017]{}.
A. Linde, [*Particle Physics and Inflationary Cosmology*]{} (Harwood Academic Publishers, Chur, Switzerland, 1990).
A. Linde, , 38 (1991); A. Linde, , 748 (1994), [astro-ph/9307002]{}; E. J. Copeland, A. R. Liddle, D. H. Lyth, E. D. Stewart and D. Wands, , 6410 (1994), [astro-ph/9401011]{}.
E. D. Stewart, , 414 (1995), [astro-ph/9407040]{}.
R. Jeannerot, S. Khalil, G. Lazarides and Q. Shafi, JHEP [**0010**]{}, 012 (2000), [hep-ph/0002151]{}; R. Jeannerot, S. Khalil and G. Lazarides, JHEP [**0207**]{}, 069 (2002), [hep-ph/0207244]{}.
S. Leach and A. R. Liddle, , 103503 (2003), [astro-ph/0305263]{}.
C. Gordon, D. Wands, B. A. Bassett and R. Maartens, , 023506 (2001), [astro-ph/0009131]{}; S. Groot Nibbelink and B. J. van Tent, Class. Quant. Grav. [**19**]{}, 613 (2002), [hep-ph/0107272]{}; B. J. W. van Tent, Class. Quant. Grav. [**21**]{}, 349 (2004), [astro-ph/0307048]{}.
G. Lazarides, C. Panagiotakopoulos and N. D. Vlachos, , 1369 (1996), [hep-ph/9606297]{}; G. Lazarides and N. D. Vlachos, , 4562 (1997), [ hep-ph/9707296]{}; N. Tetradis, , 5997 (1998), [ astro-ph/9707214]{}; L. E. Mendes and A. R. Liddle, , 103511 (2000), [astro-ph/0006020]{}.
S. Coleman and F. de Lucia, , 3305 (1980).
D. Wands, N. Bartolo, S. Matarrese and A. Riotto, , 043520 (2002), [astro-ph/0205253]{}.
K. Moodley, M. Bucher, J. Dunkley, P. G. Ferreira and C. Skordis, , 103520 (2004), [ astro-ph/0407304]{}.
J. Rocher and M. Sakellariadou, [hep-ph/0405133]{}; J. Rocher and M. Sakellariadou, , 011303 (2005), [hep-ph/0412143]{}.
[^1]: Electronic mail: pbrax@cea.fr\
Also at Institut d’Astrophysique de Paris, [${\cal G}\setR\varepsilon\setC{\cal O}$]{}, 98bis boulevard Arago, 75014 Paris, France
[^2]: Electronic mail: jmartin@iap.fr
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---
abstract: 'A right Engel sink of an element $g$ of a group $G$ is a set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) It is proved that if every element of a compact (Hausdorff) group $G$ has a countable (or finite) right Engel sink, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent.'
address:
- 'Charlotte Scott Research Centre for Algebra, University of Lincoln, U.K., and Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia'
- 'Department of Mathematics, University of Brasilia, DF 70910-900, Brazil'
author:
- 'E. I. Khukhro'
- 'P. Shumyatsky'
title: Compact groups in which all elements have countable right Engel sinks
---
Introduction
============
A group $G$ is called an Engel group if for every $x,g\in G$ the equation $[x,\,{}_{n} g]=1$ holds for some $n=n(x,g)$ depending on $x$ and $g$. Henceforth, we use the left-normed simple commutator notation $[a_1,a_2,a_3,\dots ,a_r]:=[...[[a_1,a_2],a_3],\dots ,a_r]$ and the abbreviation $[a,\,{}_kb]:=[a,b,b,\dots, b]$ where $b$ is repeated $k$ times. A group is said to be locally nilpotent if every finite subset generates a nilpotent subgroup. Clearly, any locally nilpotent group is an Engel group. Wilson and Zelmanov [@wi-ze] proved the converse for profinite groups: any Engel profinite group is locally nilpotent. Later Medvedev [@med] extended this result to Engel compact groups. (Henceforth by compact groups we mean compact Hausdorff groups.)
Generalizations of Engel groups can be defined in terms of Engel sinks.
\[d\] A *left Engel sink* of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[x,g,g,\dots ,g]$ belong to ${\mathscr E}(g)$, that is, for every $x\in G$ there is a positive integer $l=l(x,g)$ such that $[x,\,{}_{l}g]\in {\mathscr E}(g)$ for all $l{\geqslant}l(x,g).
$
A *right Engel sink* of an element $g$ of a group $G$ is a set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[g,x,x,\dots ,x]$ belong to ${\mathscr R}(g)$, that is, for every $x\in G$ there is a positive integer $r(x,g)$ such that $[g,\,{}_{r}x]\in {\mathscr R}(g)$ for all $r{\geqslant}r(x,g).
$
(Thus, $g$ is a left Engel element precisely when we can choose ${\mathscr E}(g)=\{ 1\}$, and a right Engel element when we can choose ${\mathscr R}(g)=\{ 1\}$.)
Earlier we considered in [@khu-shu; @khu-shu191] compact groups $G$ in which every element has a countable or finite left Engel sink and proved the following theorem.
\[t-left\] If every element of a compact group $G$ has a countable left Engel sink, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent.
(Henceforth by “countable” we mean “finite or denumerable”.)
For right Engel sinks we proved earlier in [@khu-shu172] that if every element of a compact group has a finite right Engel sink, then the group is finite-by-(locally nilpotent). In the present paper we extend this result to countable right Engel sinks.
\[t-main\] Suppose that $G$ is a compact group in which every element has a countable right Engel sink. Then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent.
In Theorem \[t-main\] it also follows that there is a locally nilpotent subgroup of finite index — just consider $C_G( N)$.
While it is well-known that the inverse of a right Engel element is a left Engel element, it is unclear if the existence of a countable (or finite) right Engel sink of a given element implies the existence of a countable (or finite) left Engel sink of this element or its inverse. It is only by virtue of our Theorem \[t-main\] that if all elements of a compact group have countable right Engel sinks, then in fact all elements have finite right and left Engel sinks contained in the same finite normal subgroup.
The proof uses the aforementioned Wilson–Zelmanov theorem for profinite groups. First the case of pro-$p$ groups is considered, where Lie ring methods are applied including Zelmanov’s theorem on Lie algebras satisfying a polynomial identity and generated by elements all of whose products are ad-nilpotent [@ze92; @ze95; @ze17]. As we noted in [@khu-shu172], it is easy to see that if every element of a pro-$p$ group has a finite right Engel sink, then the group is locally nilpotent. But in the present paper, with countable right Engel sinks, the case of pro-$p$ groups requires substantial efforts. Then the case of prosoluble groups is settled by using properties of coprime actions including a profinite analogue of a theorem of Thompson [@th]. The general case of profinite groups is dealt with by bounding the nonsoluble length of the group, which enables induction on this length. (We introduced the nonsoluble length in [@khu-shu131], although bounds for nonsoluble length had been implicitly used in various earlier papers, for example, in the celebrated Hall–Higman paper [@ha-hi], or in Wilson’s paper [@wil83]; more recently, bounds for the nonsoluble length were used in the study of verbal subgroups in finite and profinite groups [@dms1; @68; @austral; @khu-shu132].) Finally, the result for compact groups is derived with the use of the structure theorems for compact groups.
Preliminaries
=============
In this section we recall some notation and terminology and establish some general properties of left and right Engel sinks in compact and profinite groups.
Our notation and terminology for profinite and compact groups is standard; see, for example, [@rib-zal], [@wil], and [@hof-mor]. A subgroup (topologically) generated by a subset $S$ is denoted by $\langle S\rangle$. Recall that centralizers are closed subgroups, while commutator subgroups $[B,A]=\langle [b,a]\mid b\in B,\;a\in A\rangle$ are the closures of the corresponding abstract commutator subgroups.
For a group $A$ acting by automorphisms on a group $B$ we use the usual notation for commutators $[b,a]=b^{-1}b^a$ and commutator subgroups $[B,A]=\langle [b,a]\mid b\in B,\;a\in A\rangle$, as well as for centralizers $C_B(A)=\{b\in B\mid b^a=b \text{ for all }a\in A\}$ and $C_A(B)=\{a\in A\mid b^a=b\text{ for all }b\in B\}$.
We record for convenience the following simple lemma.
\[l-fng\] Suppose that $\varphi$ is a continuous automorphism of a compact group $G$ such that $G=[G,\varphi ]$. If $N$ is a normal subgroup of $G$ contained in $C_G(\varphi )$, then $N{\leqslant}Z(G)$.
We denote by $\pi (k)$ the set of prime divisors of $k$, where $k$ may be a positive integer or a Steinitz number, and by $\pi (G)$ the set of prime divisors of the orders of elements of a (profinite) group $G$. Let $\sigma$ be a set of primes. An element $g$ of a group is a $\sigma$-element if $\pi(|g|)\subseteq \sigma$, and a group $G$ is a $\sigma$-group if all of its elements are $\sigma$-elements. We denote by $\sigma'$ the complement of $\sigma$ in the set of all primes. When $\sigma=\{p\}$, we write $p$-element, $p'$-element, etc.
Recall that a pro-$p$ group is an inverse limit of finite $p$-groups, a pro-$\sigma $ group is an inverse limit of finite $\sigma$-groups, a pronilpotent group is an inverse limit of finite nilpotent groups, a prosoluble group is an inverse limit of finite soluble groups.
We denote by $\gamma _{\infty}(G)=\bigcap _i\gamma _i(G)$ the intersection of the lower central series of a group $G$. A profinite group $G$ is pronilpotent if and only if $\gamma _{\infty}(G)=1$.
Profinite groups have Sylow $p$-subgroups and satisfy analogues of the Sylow theorems. Prosoluble groups satisfy analogues of the theorems on Hall $\pi$-subgroups. We refer the reader to the corresponding chapters in [@rib-zal Ch. 2] and [@wil Ch. 2]. We add a simple folklore lemma.
\[l-prosol-by-prosol\] A profinite group $G$ that is an extension of a prosoluble group $N$ by a prosoluble group $G/N$ is prosoluble.
We shall use several times the following well-known fact, which is straightforward from the Baire Category Theorem (see [@kel Theorem 34]).
\[bct\] If a compact Hausdorff group is a countable union of closed subsets, then one of these subsets has non-empty interior.
We now establish some general properties of Engel sinks. Clearly, the intersection of two left Engel sinks of a given element $g$ of a group $G$ is again a left Engel sink of $g$, with the corresponding function $l(x,g)$ being the maximum of the two functions. Therefore, if $g$ has a *finite* left Engel sink, then $g$ has a unique smallest left Engel sink, which has the following characterization.
\[l-min\] If an element $g$ of a group $G$ has a finite left Engel sink, then $g$ has a smallest left Engel sink $\mathscr E (g)$ and for every $s\in \mathscr E (g)$ there is an integer $k{\geqslant}1$ such that $s=[s,\,{}_kg]$.
There are similar observations about right Engel sinks. The intersection of two right Engel sinks of a given element $g$ of a group $G$ is again a right Engel sink of $g$, with the corresponding function $r(x,g)$ being the maximum of the two functions. Therefore, if $g$ has a *finite* right Engel sink, then $g$ has a unique smallest right Engel sink, which has the following characterization.
\[l-min-r\] If an element $g$ of a group $G$ has a finite right Engel sink, then $g$ has a smallest right Engel sink $\mathscr R(g)$ and for every $z\in \mathscr R(g)$ there are integers $n{\geqslant}1$ and $m{\geqslant}1$ and an element $x\in G$ such that $z=[g,{}_nx]=[g,{}_{n+m}x]$.
(Of course, the elements $x$ and numbers $m,n$ in the above lemma vary for different $z$ and are not unique.)
The following lemma was proved by Heineken [@hei].
\[l-hei\] If $g$ is a right Engel element of a group $G$, then $g^{-1}$ is a left Engel element.
Furthermore, for metabelian groups we have the following.
\[l-metab\] If $G$ is a metabelian group, then a right Engel sink of the inverse $g^{-1}$ of an element $g\in G$ is a left Engel sink of $g$.
Thus, if $G$ is a metabelian group in which all elements have finite right Engel sinks, then all elements of $G$ also have finite left Engel sinks, and if all elements of $G$ have countable right Engel sinks, then all elements of $G$ also have countable left Engel sinks.
If every element of a group has a countable right Engel sink, then this condition is inherited by every section of the group, and we shall use this property without special references. The same applies to a group in which every element has a finite right Engel sink. Similar properties hold for left Engel sinks.
Pronilpotent groups
===================
When $G$ is a pro-$p$ group, or more generally a pronilpotent group, the conclusion of the main Theorem \[t-main\] is equivalent to $G$ being locally nilpotent, and this is what we prove in this section.
\[t2\] Suppose that $G$ is a pronilpotent group in which every element has a countable right Engel sink. Then $G$ is locally nilpotent.
First we establish an Engel-like property.
\[l-ksm\] Suppose that $G$ is a profinite group in which every element has a countable right Engel sink. For any elements $a,b\in G$ there exist positive integers $k,s,m$ (depending on $a,b$) such that $$[[b,\,{}_ka^{s}],a^{t}]=1.$$
Let $\{s_1,s_2,\dots \}$ be a countable right Engel sink of $b$. Consider the subsets $$T_{i,j,k}=\{x\in \langle a\rangle \mid [b,\,{}_kx]=s_i\text{ and } [s_i,x]=s_j\}$$ (where $\langle a\rangle$ is the procyclic subgroup generated by $a$). Note that each $T_{i,j,k}$ is a closed subset of $\langle a\rangle$.
By the definition of a right Engel sink, we have $$\langle a\rangle=\bigcup _{i,j,k}T_{i,j,k}.$$ By Theorem \[bct\] some $T_{i,j,k}$ contains an open subset of $\langle a\rangle$, so $Nd\subset T_{i,j,k}$ for some open subgroup $N$ of $\langle a\rangle$ and some $d\in \langle a\rangle$. Since $\langle a\rangle/N$ is finite, we can assume that $d=a^{s}$ for some positive integer $s$.
Since $[s_i,nd]=s_j$ for all $n\in N$, it follows that $[s_i,N]=1$. Since $\langle a\rangle/N$ is finite, we have $a^{t}\in N$ for some positive integer $t$, so that $[s_i,a^{t}]=1$. As a result, $$\begin{aligned}
[[b,\,{}_ka^{s}],a^{t}]= [[b,\,{}_kd],a^{t}]
&=[s_i,a^{t}]=1.\qedhere
\end{aligned}$$
The bulk of the proof of Theorem \[t2\] is about the case where $G$ is a pro-$p$ group. First we remind the reader of important Lie ring methods in the theory of pro-$p$ groups.
For a prime number $p $, the *Zassenhaus $p $-filtration* of a group $G$ (also called the *$p $-dimension series*) is defined by $$G_i=\langle g^{p ^k}\mid g\in \gamma _j(G),\;\, jp ^k\geqslant i\rangle \qquad\text{for}\quad i\in {\Bbb N}.$$ This is indeed a *filtration* (or an *$N$-series*, or a *strongly central series*) in the sense that $$\label{e-fil}
[G_i,G_j] \leqslant G_{i+j}\qquad \text{for all}\quad i, j.$$
Then the Lie ring $D_p (G)$ is defined with the additive group $$D_p(G)=\bigoplus _{i}G_i/G_{i+1},$$ where the factors $Q_i=G_i/G_{i+1}$ are additively written. The Lie product is defined on homogeneous elements $xG_{i+1}\in Q_i$, $yG_{j+1}\in Q_j$ via the group commutators by $$[xG_{i+1},\, yG_{j+1}] = [x, y]G_{i+j+1}\in Q_{i+j}$$ and extended to arbitrary elements of $D_p(G)$ by linearity. Condition ensures that this product is well-defined, and group commutator identities imply that $D_p(G)$ with these operations is a Lie ring. Since all the factors $G_i/G_{i+1}$ have prime exponent $p $, we can view $D_p(G)$ as a Lie algebra over the field of $p $ elements $\mathbb{F}_p $. We denote by $L_p (G)$ the subalgebra generated by the first factor $G/G_2$. (Sometimes, the notation $L_p (G)$ is used for $D_p (G)$.) If $u\in G_i\setminus G_{i+1}$, then we define $\delta (u)=i$ to be the *degree* of $u$ with respect to the Zassenhaus filtration.
A group $G$ is said to satisfy a *coset identity* if there is a group word $w(x_1,\dots ,x_m)$, elements $a_1,\dots,a_m$, and a subgroup $H {\leqslant}G$ such that $w(a_1h_1,\dots,a_mh_m) = 1$ for any $h_1,\dots,h_m\in H$. We shall use the following result of Wilson and Zelmanov [@wi-ze] about coset identities.
\[t-coset\] If a group $G$ satisfies a coset identity on cosets of a subgroup of finite index, then for every prime $p$ the Lie algebra $L_p (G)$ constructed with respect to the Zassenhaus $p $-filtration satisfies a polynomial identity.
Theorem \[t-coset\] was used in the proof of the above-mentioned theorem on profinite Engel groups, which we state here for convenience.
\[t-wz\] Every profinite Engel group is locally nilpotent.
The proof of Theorem \[t-wz\] was based on the following deep result of Zelmanov [@ze92; @ze95; @ze17], which is also used in our paper.
\[tz\] Let $L$ be a Lie algebra over a field and suppose that $L$ satisfies a polynomial identity. If $L$ can be generated by a finite set $X$ such that every commutator in elements of $X$ is ad-nilpotent, then $L$ is nilpotent.
We now consider pro-$p$ groups with countable right Engel sinks.
\[pr-pro-p\] Suppose that $P$ is a finitely generated pro-$p$ group in which every element has a countable right Engel sink. Then $P$ is nilpotent.
Our immediate aim is an application of Theorem \[tz\] to the Lie algebra $L_p(P)$ of $P$, which will show that $L_p(P)$ is nilpotent. We need to verify that the conditions of that theorem are satisfied.
\[l-ad\] The Lie algebra $L_p(P)$ is generated by finitely many elements all commutators in which are ad-nilpotent.
The image of the finite generating set of $P$ in the first homogeneous component of the Lie algebra $L_p(P)$ is a finite set of generators of $L_p(P)$. We claim that all commutators in these generators are ad-nilpotent. In fact, we prove that every homogeneous element $\bar a$ of $L_p(P)$ is ad-nilpotent. We may assume that $\bar a$ is the image of an element $a\in P$ in the corresponding factor $P_{\delta (a)}/P_{\delta (a)+1}$ of the Zassenhaus filtration, where $\delta (a)$ is the degree of $a$. We fix the notation $a$ and $\bar a$ for the rest of the proof of this lemma.
For our $a\in P$, we consider the sets $$U_{k,s,t}=\{x\in P\mid [[x,\,{}_ka^{s}],a^{t}]=1\}, \qquad k,s,t\in {\Bbb N}.$$ Each set $U_{k,s,t}$ is closed, and $$P=\bigcup_{k,s,t}U_{k,s,t}$$ by Lemma \[l-ksm\]. Therefore by Theorem \[bct\] some $U_{k,s,t}$ contains a coset $Nd$ of an open normal subgroup $N$ of $P$. We obtain that $$\label{e-ndpsm}
[[nd,\,{}_ka^{s}],a^{t}]=1\qquad \text{for all}\quad n\in N.$$ We are going to derive from this equation the desired ad-nilpotency of $\bar a$ in $L_p(P)$.
Let $s=s_1p^l$ and $t=t_1p^m$ for $s_1, t_1$ coprime to $p$. Since $\langle a^t\rangle=\langle a^{s_1p^m}\rangle$, we can replace $t$ with $s_1p^m$ in , so that $$\label{e-ndpsm2}
[[nd,\,{}_ka^{s_1p^l}],a^{s_1p^m}]=1\qquad \text{for all}\quad n\in N,$$ where $(s_1,p)=1$. Since the image $\overline{a^{s_1}}$ of $a^{s_1}$ in $P_{\delta (a)}/P_{\delta (a)+1}$ is equal to $s_1\bar a$ and $s_1$ is coprime to the characteristic $p$ of the ground field of $L_p$, it is sufficient to prove that $\overline{a^{s_1}}$ is ad-nilpotent. Replacing $a$ with $a^{s_1}$ we change notation in , so that we have $$\label{e-ndpsm3}
[[nd,\,{}_ka^{p^l}],a^{p^m}]=1\qquad \text{for all}\quad n\in N.$$
For generators $x,y,z,h$ of a free group we write $$[[xy,\,{}_k z],h]=[[x,\,{}_k z],h]\cdot [[y,\,{}_k z],h]\cdot v(x,y,z,h),$$ where the word $v(x,y,z,h)$ is a product of commutators of weight at least $k + 3$, each of which involves $x$, $y$, $h$ and involves $z$ at least $k$ times. Substituting $x=n$, $y=d$, $z=a^{p^l}$, and $h=a^{p^m}$ and using we obtain that $$[[n,\,{}_ka^{p^l}],a^{p^m}]=v(n,d,a^{p^l},a^{p^m})^{-1}
\qquad \text{for all}\;\, n\in N.$$ If $|P/N|=p^r$, then for any $g\in P$ we have $[g,\,{}_ra^{p^l}]\in N$, so that we also have $$\label{eq-ad3}
[[g,\,{}_{k+r}a^{p^l}],a^{p^m}]=v([g,\,{}_ra^{p^l}],d,a^{p^l},a^{p^m})^{-1}.$$ We claim that $\bar a$ is ad-nilpotent in $L_p(P)$ of index $(k+r)p^l+p^m$.
Recall that $\delta (u)$ denotes the degree of an element $u \in P$ with respect to the Zassenhaus filtration. It is well known that $$\label{e-pd}
u^p\in P_{p\delta (u)}.$$ Furthermore, in $L_p(P)$ for the images of $u$ and $u^p$ in $P_{\delta (u)}/P_{\delta (u)+1}$ and $P_{p\delta (u)}/P_{p\delta (u)+1}$, respectively, we have $$\label{e-pd2}
[x, \bar{u^p}]=[x,\,{}_p \bar{u}]$$ (see, for example, [@bou Ch. II, §5, Exercise 10]).
By the degree of $v([g,\,{}_ra^{p^l}],d,a^{p^l},a^{p^m})$ on the right of is at least $\delta (d)+\delta (g)+((k+r)p^l+p^m)\delta (a)$, which is strictly greater than $w=\delta (g)+((k+r)p^l+p^m)\delta (a)$. This means that the image of the right-hand side of in $P_w/P_{w+1}$ is trivial. At the same time, by the image of the left-hand side of in $P_w/P_{w+1}$ is equal to the image of $[g,\,{}_{(k+r)p^l+p^m} a]$ in $P_w/P_{w+1}$, which is in turn equal to the element $[\bar g,\,{}_{(k+r)p^l+p^m}\bar a]$ in $L_p(P)$. Thus, for the corresponding homogeneous elements of $L_p(P)$ we have $$[\bar g,\,{}_{(k+r)p^l+p^m}\bar a]=0.$$ Since here $\bar g$ can be any homogeneous element, this means that $\bar a$ is ad-nilpotent of index $(k+r)p^l+p^m$, as claimed.
\[l-pi\] The Lie algebra $L_p(P)$ satisfies a polynomial identity.
Consider the subsets of the direct product $P\times P$ $$V_{k,s,t}=\{(x,y)\in P\times P\mid [[x,\,{}_ky^{s}],y^{t}]=1\}, \qquad k,s,t\in{\Bbb N}.$$ Note that each subset $V_{k,s,t}$ is closed in the product topology of $P\times P$. By Lemma \[l-ksm\] we have $$P\times P=\bigcup_{k,s,t}V_{k,s,t}.$$ By Theorem \[bct\] one of the sets $V_{k,s,t}$ contains an open subset of $P\times P$. This means that there are cosets $aN$ and $bN$ of an open normal subgroup $N$ of $P$ and positive integers $k,s,t$ such that $$[[x,\,{}_ky^{s}],y^{t}]=1\qquad \text{ for any }x\in aN,\; y\in bN.$$ Thus, $P$ satisfies a coset identity on cosets of a subgroup of finite index and therefore the Lie algebra $L_p(P)$ satisfies a polynomial identity by Theorem \[t-coset\].
We can now finish the proof of Proposition \[pr-pro-p\]. By Lemmas \[l-ad\] and \[l-pi\] the Lie algebra $L_p(P)$ satisfies the hypotheses of Theorem \[tz\], by which $L_p(P)$ is nilpotent. The nilpotency of the Lie algebra $L_p (P)$ of the finitely generated pro-$p $ group $P$ implies that $P$ is a $p $-adic analytic group. This result goes back to Lazard [@laz]; see also [@sha Corollary D]. Furthermore, by a theorem of Breuillard and Gelander [@br-ge Theorem 8.3], a $p $-adic analytic group satisfying a coset identity on cosets of a subgroup of finite index is soluble.
Thus, $P$ is soluble, and we prove that $P$ is nilpotent by induction on the derived length of $P$. By induction hypothesis, $P$ has an abelian normal subgroup $U$ such that $P/U$ is nilpotent. We aim to show that $P$ is an Engel group. Since $P/U$ is nilpotent, it is sufficient to show that every element $a\in P$ is an Engel element in the product $U\langle a\rangle$. Since this product is a metabelian group, all of its elements also have countable left Engel sinks by Lemma \[l-metab\] and then $U\langle a\rangle$ is nilpotent by Theorem \[t-left\].
Thus, $P$ is an Engel group and therefore, being a finitely generated pro-$p$ group, $P$ is nilpotent by Theorem \[t-wz\].
We now consider the general case of a pronilpotent group.
Let $G$ be a pronilpotent group in which every element has a countable right Engel sink; we need to prove that $G$ is locally nilpotent. By Theorem \[t-wz\], it is sufficient to prove that $G$ is an Engel group.
For each prime $p$, let $G_p$ denote the Sylow $p$-subgroup of $G$, so that $G$ is a Cartesian product of the $G_p$, since $G$ is pronilpotent. Given any two elements $a,g\in G$, we write $g=\prod _pg_p$ and $a=\prod _pa_p$, where $a_p,g_p\in G_p$. Clearly, $[g_q,a_p]=1$ for $q\ne p$. We need to show that $[g,\,{}_ma]=1$ for some positive integer $m=m(a,g)$.
Let $\{s_1,s_2,\dots \}$ be a countable right Engel sink of $g$. Consider the subsets $$T_{i,j,k}=\{x\in \langle a\rangle \mid [g,\,{}_kx]=s_i\text{ and } [s_i,x]=s_j\}$$ (where $\langle a\rangle$ is the procyclic subgroup generated by $a$). Note that each $T_{i,j,k}$ is a closed subset of $\langle a\rangle$. By the definition of a right Engel sink, we have $$\langle a\rangle=\bigcup _{i,j,k}T_{i,j,k}.$$ By Theorem \[bct\] some $T_{i,j,k}$ contains an open subset of $\langle a\rangle$, so $Nd\subset T_{i,j,k}$ for some open subgroup $N$ of $\langle a\rangle$ and some $d\in \langle a\rangle$, so that $$[g,\,{}_kx]=s_i\quad\text{and}\quad [s_i,x]=s_j \qquad\text{for any } x\in Nd.$$ Since $\langle a\rangle/N$ is finite, we can assume that $d=a^{s}$ for some positive integer $s$. Since $[s_i,nd]=s_j$ for all $n\in N$, it follows that $[s_i,N]=1$. Thus, $$\label{e-nn}
[[g,\,{}_k(n_1a^s)],n_2]=1\qquad\text{for any}\quad n_1,n_2\in N.$$
Let $\sigma=\pi ( |\langle a\rangle/N|)$ be the (finite) set of prime divisors of the order of $\langle a\rangle/N$. Then $a_q\in N$ for any $q\not\in \sigma$. Choosing $n_1=a_q^{1-s}$ and $n_2=a_q$ in we obtain for the $q$-components $$\label{e-qk}
[g_q,\,{}_{k+1}a_q]= [[g_q,\,{}_ka_q],a_q]=1\qquad\text{for any}\quad q\not\in \sigma.$$
For every prime $p$ the group $G_p$ is locally nilpotent by Proposition \[pr-pro-p\], so there is a positive integer $k_p$ such that $[g_p,\,{}_{k_p}a_p]=1$. Now for $$m=\max\{k+1, \,\max_{p\in \sigma } \{k_p\}\}$$ in view of we have $[g_p,\,{}_{m}a_p]=1$ for all $p$, which means that $[g,\,{}_{m}a]=1$. Thus, $G$ is an Engel group and therefore it is locally nilpotent by Theorem \[t-wz\].
Prosoluble groups
=================
Any profinite group $G$ has the largest normal pronilpotent subgroup $F(G)$, called the Fitting subgroup of $G$. Further terms of the Fitting series are defined by induction: $F_1(G)=F(G)$ and $F_{i+1}(G)$ is the inverse image of $F(G/F_i(G))$. By definition a group has finite Fitting height if $F_k(G)=G$ for some $k\in {\mathbb N}$.
Recall that by Theorem \[t2\] any pronilpotent group with countable right Engel sinks is locally nilpotent. Therefore, if $G$ is a profinite group with countable right Engel sinks, then the Fitting subgroup $F(G)$ is locally nilpotent.
The proof of the main Theorem \[t-main\] for a prosoluble group $G$ with countable right Engel sinks will follow from a key proposition stating that $F(G)\ne 1$. We approach the proof of this proposition in a number of steps. First we list several profinite analogues of the properties of coprime automorphisms of finite groups, which are used in this section in relation to Engel sinks.
If $\varphi$ is an automorphism of a finite group $H$ of coprime order, that is, such that $(|\varphi |,|H|)=1$, then we say for brevity that $\varphi$ is a coprime automorphism of $H$. This definition is extended to profinite groups as follows. We say that $\varphi$ is a *coprime automorphism* of a profinite group $H$ meaning that a procyclic group $\langle\varphi\rangle$ faithfully acts on $H$ by continuous automorphisms and $\pi (\langle \varphi\rangle)\cap \pi (H)=\varnothing$. Since the semidirect product $H\langle \varphi\rangle$ is also a profinite group, $\varphi$ is a coprime automorphism of $H$ if and only if for every open normal $\varphi$-invariant subgroup $N$ of $H$ the automorphism (of finite order) induced by $\varphi$ on $H/N$ is a coprime automorphism. The following lemma is derived from an analogue of the Schur–Zassenhaus theorem for profinite groups; we shall freely use this fact without special references.
\[l-inv\] If $\varphi$ is a coprime automorphism of a profinite group $G$, then for every prime $q\in \pi (G)$ there is a $\varphi$-invariant Sylow $q$-subgroup of $G$. If $G$ is in addition prosoluble, then for every subset $\sigma\subseteq \pi (G)$ there is a $\varphi$-invariant Hall $\sigma$-subgroup of $G$.
The following lemma is a special case of [@rib-zal Proposition 2.3.16].
\[l-cover\] If $\varphi $ is a coprime automorphism of a profinite group $G$ and $N$ is a $\varphi$-invariant closed normal subgroup of $G$, then every fixed point of $\varphi$ in $G/N$ is an image of a fixed point of $\varphi$ in $G$, that is, $C_{G/N}(\varphi)=C(\varphi )N/N$.
As a consequence, we have the following.
\[l-gff\] If $\varphi $ is a coprime automorphism of a profinite group $G$, then $[[G,\varphi],\varphi]=[G,\varphi]$.
We will be applying the following profinite version of a theorem of Thompson. Namely, Thompson [@th] proved that if $G$ is a finite soluble group on which a finite soluble group $A$ of coprime order acts by automorphisms, then the Fitting height $h(G)$ is bounded in terms of $h(C_G(A))$ and the number of prime divisors of $|A|$ counting multiplicities. (Further results in this direction were devoted to improving the corresponding bounds, with best possible one obtained by Turull, see his survey [@turull].) The profinite version of Thompson’s theorem can be deduced by standard arguments in the spirit of [@wil83 Lemma 2], since the hypotheses are inherited by quotients by closed normal subgroups by Lemma \[l-cover\].
\[thompson\] Let $G$ be a prosoluble group on which a finite soluble group $A$ acts by continuous automorphisms and suppose that $\pi(G)\cap \pi(A)=\varnothing$. If $C_G(A)$ has finite Fitting height, then $G$ also has finite Fitting height.
We begin our step-by-step approach to proving that $F(G)\ne 1$ in any nontrivial prosoluble group with countable right Engel sinks. The first step is considering the case where all Sylow subgroups are finite.
\[l-finsyl\] Suppose that $G$ is a nontrivial prosoluble group in which Sylow $p$-subgroups are finite for all primes $p$. If every element of $G$ has a countable right Engel sink, then $F(G)\ne 1$.
For a given arbitrary element $g\in G$, let $\{s_1,s_2,\dots \}$ be a countable right Engel sink of $g$. Consider the subsets $$T_{i,j,k}=\{x\in G \mid [g,\,{}_kx]=s_i\text{ and } [s_i,x]=s_j\}.$$ Note that each $T_{i,j,k}$ is a closed subset of $G$. By the definition of a right Engel sink, we have $$G=\bigcup _{i,j,k}T_{i,j,k}.$$ By Theorem \[bct\] some $T_{i,j,k}$ contains a coset $Nd$ of an open normal subgroup $N$ of $G$, so that $$\label{e-gss}
[g,\,{}_kx]=s_i\quad\text{and}\quad [s_i,x]=s_j \quad\text{for any } x\in Nd.$$ Since $[s_i,nd]=s_j$ for all $n\in N$, it follows that $[s_i,N]=1$. Then $s_i$ has centralizer of finite index in $G$, and $s_i$ has finitely many conjugates in $G$. Hence the normal closure $\langle s_i^G\rangle$ is central-by-finite. Given that $G$ is a prosoluble group, it follows that if $s_1\neq1$, then $G$ has a normal abelian subgroup and therefore also a closed normal abelian subgroup, so that $F(G)\ne 1 $.
Thus, the lemma is proved unless for every $g\in G$ the element $s_i$ given by the above argument in is trivial. In other words, it remains to consider the case where for every $g\in G$ there is a positive integer $k=k(g)$ and a coset $Nd$ of an open subgroup $N$ of $G$ such that $$\label{e-gss2}
[g,\,{}_kx]=1 \quad\text{for any } x\in Nd.$$
We now observe that the group $G$ has only countably many open normal subgroups. Indeed, any normal subgroup of finite index $n$ must contain the normal subgroup $H({\pi (n))}$ generated by all Sylow $q$-subgroups for all primes $q\not\in\pi(n)$, and each quotient by $G/H({\pi (n))}$ is finite, since all Sylow subgroups of $G$ are finite. Therefore $G$ has only finitely many normal subgroups of any given finite index and hence only countably many open normal subgroups. Consequently, $G$ has countably many cosets of such subgroups, say, $\{D_1,D_2,\dots \}$. Consider the subsets $$U_{j,k}=\{x\in G \mid [x,\,{}_ky]=1\text{ for any } y\in D_j\}.$$ Note that each $U_{j,k}$ is a closed subset of $G$. By our assumption involving , we have $$G=\bigcup _{j,k}U_{j,k}.$$ By Theorem \[bct\] some $U_{j,k}$ contains a coset $Kb$ of an open normal subgroup $K$ of $G$, so that $$[x,\,{}_ky]=1\quad\text{ for any } x\in Kb\text{ and any }y\in D_j,$$ where $D_j=Nd$ for some $d\in G$ and an open normal subgroup $N$ of $G$. Setting $L=K\cap N$ we obtain $$\label{e-gss3}
[ub,\,{}_kvd]=1\quad\text{ for any } u,v\in L.$$ Let ${\sigma }=\pi (G/L)$, which is a finite set of primes. Then $G=LH$ for a Hall ${\sigma }$-subgroup $H$, which is finite, since all Sylow subgroups are finite. We can choose the coset representatives $b,d\in H$ satisfying . Then $B=\langle b , d \rangle$ is a finite subgroup of $H$.
By standard commutator formulae equation implies that $$1=[ub,\,{}_kvd]=[b,\,{}_k d][u,\,{}_kv]=[u,\,{}_kv]\quad\text{ for any } u,v\in C_L(B).$$ Hence $C_L(B)$ is a $k$-Engel group. Therefore $C_L(B)$ is locally nilpotent by Theorem \[t-wz\].
Recall that ${\sigma }=\pi (G/L)$ and $H$ is a Hall ${\sigma }$-subgroup. Since $H$ is finite, there is an open normal subgroup $M$ of $G$ that intersects $H$ trivially; replacing $L$ with $L\cap M$, we can assume that $L$ is a ${\sigma }'$-subgroup.
By Theorem \[thompson\] applied to the action of $B$ on $L$ we obtain that $L$ has a finite series of characteristic closed subgroups with pronilpotent factors. This implies that $F(G)\ne 1$ (even if $L=1$, since $G$ is prosoluble).
We now consider coprime automorphisms in relation to Engel sinks. In the proof of the following lemma we use the well-known fact that if $G$ is nilpotent and $G/G'$ is finite, then $G$ is finite (see, for example, [@rob 5.2.6]).
\[l-copr3\] Let $\varphi$ be a coprime automorphism of a pronilpotent group $G$. If all elements of the semidirect product $G\langle \varphi\rangle$ have countable right Engel sinks, then $\gamma _{\infty} (G\langle \varphi\rangle)$ is finite and $\gamma _{\infty}(G\langle \varphi\rangle)= [G,\varphi]$.
The group $G$ is locally nilpotent by Theorem \[t2\]. The quotient $G\langle \varphi\rangle/[G,\varphi]$ is obviously the direct product of the images of $G$ and $\langle \varphi\rangle$ and therefore is pronilpotent. Hence, $\gamma _{\infty}(G\langle \varphi\rangle){\leqslant}[G,\varphi]$. By Lemma \[l-gff\], $$[[G,\varphi],\varphi]=[G,\varphi].$$ Therefore also $\gamma _{\infty}(G\langle \varphi\rangle){\geqslant}[G,\varphi]$, so that $\gamma _{\infty}(G\langle \varphi\rangle)= [G,\varphi]$.
Let $V$ be the quotient of $[G,\varphi ]$ by its derived subgroup. The semidirect product $V\langle \varphi\rangle$ is metabelian and therefore all elements of it also have countable left Engel sinks. By Theorem \[t-left\] then $\gamma _{\infty}(V\langle \varphi\rangle)= [V,\varphi]=V$ is finite. It follows that the locally nilpotent pronilpotent group $[G,\varphi ]$ is finitely generated and therefore nilpotent and finite.
We shall further need the following simple lemma about finite groups.
\[l-fq\] Suppose that $G$ is a finite $q$-soluble group admitting a coprime automorphism $\alpha$ such that $G=[G,\alpha ]$. If $Q$ is a nontrivial $\alpha$-invariant Sylow $q$-subgroup of $G$, then $[Q,\alpha ]\ne 1$.
We can assume from the outset that $O_{q'}(G)=1$; then $O_q(G)\ne 1$ and $C_G(O_q(G)){\leqslant}O_q(G)$. Suppose the opposite: $[Q,\alpha ]=1$. Then $[O_q(G),\alpha ]=1$, whence $[O_q(G),[\alpha ,G] ]=[O_q(G),G]=1$ by Lemma \[l-fng\], so that $G=O_q(G)$ and $G=[G,\alpha ]=[Q,\alpha]=1$, a contradiction.
We now consider the action of a coprime automorphism of a prosoluble group.
\[l-copr4\] Let $\varphi$ be a nontrivial coprime automorphism of a prosoluble group $G$. If all elements of the semidirect product $G\langle \varphi\rangle$ have countable right Engel sinks, then $F([G, {\varphi}])\ne\nobreak 1$.
By Lemma \[l-gff\] we can assume from the outset that $G=[G,{\varphi}]$.
For a prime $q$, let $Q$ be a ${\varphi}$-invariant Sylow $q$-subgroup of $G$. Then $[Q,{\varphi}]$ is finite by Lemma \[l-copr3\]. Since $G\langle{\varphi}\rangle$ is a profinite group, it has an open normal subgroup $U$ such that $U\cap [Q,{\varphi}]=1$; then $N=G\cap U$ is a ${\varphi}$-invariant open normal subgroup of $G$ such that $N\cap [Q,{\varphi}]=1$.
We claim that $[N,{\varphi}]$ is a $q'$-group. Indeed, $Q\cap N$ is a Sylow $q$-subgroup of $N$, and then $Q_1=Q\cap [N,{\varphi}]$ is a Sylow $q$-subgroup of $[N,{\varphi}]$. We have $[Q_1,{\varphi}]{\leqslant}[Q,{\varphi}]\cap N=1$. Applying Lemma \[l-fq\] to every finite quotient of $[N,{\varphi}]\langle{\varphi}\rangle$, we obtain that $[N,{\varphi}]$ is a $q'$-group.
Then $[N,{\varphi}]{\leqslant}O_{q'}(N){\leqslant}O_{q'}(G)$. In the quotient $\bar G=G/O_{q'}(G)$ we have $[\bar N,{\varphi}]=1$, so $\bar N{\leqslant}Z(\bar G)$ by Lemma \[l-fng\]. This means that $\bar G$ is central-by-finite, whence $\bar G'$ is finite by Schur’s theorem [@rob 10.1.4]. Since $G/G'=[G/G',{\varphi}]$ is also finite by Lemma \[l-copr3\], we obtain that $G/O_{q'}(G)$ is finite. Therefore a Sylow $q$-subgroup of $G$ is finite for every prime $q$. The result now follows from Lemma \[l-finsyl\].
We now consider the case of prosoluble groups of finite Fitting height.
\[l-height\] Let $G$ be a prosoluble group of finite Fitting height. If every element of $G$ has a countable right Engel sink, then $\gamma _{\infty} (G)$ is finite.
It is sufficient to prove the result for the case of Fitting height 2. Then the general case will follow by induction on the Fitting height $k$ of $G$. Indeed, then $\gamma _{\infty}(G/ \gamma _{\infty}(F_{k-1}(G)))$ is finite, while $\gamma _{\infty}(F_{k-1}(G))$ is finite by the induction hypothesis, and as a result, $\gamma _{\infty}(G)$ is finite.
Thus, we assume that $G=F_2(G)$. By Theorem \[t-left\], it is sufficient to show that every element $a\in G$ has a finite left Engel sink. Since $G/F(G)$ is locally nilpotent, a left Engel sink of $a$ in $F(G)\langle a\rangle$ is also a left Engel sink of $a$ in $G$. Therefore it is sufficient to prove that $\gamma _{\infty}(F(G)\langle a\rangle)$ is finite.
For a prime $p$, let $F_p$ be a Sylow $p$-subgroup of $F(G)$, and write $a=a_pa_{p'}$, where $a_p$ is a $p$-element, $a_{p'}$ is a $p'$-element, and $[a_p,a_{p'}]=1$. Then $F_p\langle a_p\rangle$ is a normal Sylow $p$-subgroup of $F_p\langle a\rangle$, on which $a_{p'}$ induces by conjugation a coprime automorphism. By Lemma \[l-copr3\] the subgroup $\gamma _{\infty}(F_p\langle a\rangle)=[F_p,a_{p'}]$ is finite. Since $$\gamma _{\infty}(F(G)\langle a\rangle)=\prod_p\gamma _{\infty}(F_p\langle a\rangle)=\prod_p [F_p,a_{p'}],$$ it remains to prove that $\gamma _{\infty}(F_p\langle a\rangle)=[F_p,a_{p'}]=1$ for all but finitely many primes $p$.
Let $V=F(G)/[F(G),F(G)]$. In the metabelian group $V\langle a\rangle$ all elements also have countable left Engel sinks by Lemma \[l-metab\]. By Theorem \[t-left\] the subgroup $\gamma _{\infty}(V\langle a\rangle)$ is finite. Since $$[F(G),F(G)] =\prod_p [F_p,F_p],$$ it follows that $[F_p,a_{p'}]{\leqslant}[F_p,F_p]$ for all but finitely many primes $p$. But if $[F_p,a_{p'}]{\leqslant}[F_p,F_p]$, then $C_{F_p}(a_{p'})[F_p,F_p]=F_p$ by Lemma \[l-cover\], whence $C_{F_p}(a_{p'})=F_p$, that is, $[F_p,a_{p'}]=\nobreak 1$. Hence the result.
\[l-copr5\] Let $\varphi$ be a coprime automorphism of a prosoluble group $G$. If all elements of the semidirect product $G\langle \varphi\rangle$ have countable right Engel sinks, then $[G,{\varphi}]$ is finite.
By Lemma \[l-gff\] we can assume from the outset that $G=[G,{\varphi}]$. Since ${\gamma }_{\infty}(F_2(G))$ is finite by Lemma \[l-height\], there is a ${\varphi}$-invariant open normal subgroup $N$ of $G$ such that $N\cap {\gamma }_{\infty}(F_2(G))=1$. It follows that ${\gamma }_{\infty}(F_2(N))=1$, which means that $F_2(N)=F(N)$. Then ${\varphi}$ must act trivially on $N/F(N)$, since otherwise $F_2(N)\ne F(N)$ by Lemma \[l-copr4\] applied to $N/F(N)$. Thus, $[N,{\varphi}]{\leqslant}F(N){\leqslant}F(G)$. Then $NF(G)/F(G){\leqslant}Z(G/F(G))$ by Lemma \[l-fng\], since $G=[G,{\varphi}]$ by our assumption. In particular, the Fitting height of $G$ is finite, and we obtain that $[G,{\varphi}]$ is finite by Lemma \[l-height\].
We now prove the key proposition of this section.
\[pr-f\] If every element of a nontrivial prosoluble group $G$ has a countable right Engel sink, then $F(G)\ne 1$.
Since $G$ is prosoluble, we have $G\ne
\gamma _{\infty}(G)$. Choose a prime $p\in \pi (G/\gamma _{\infty}(G))$. If $\gamma _{\infty}(G)$ is a pro-$p$ group, then $F(G)\ne 1$ and we are done. Otherwise, let $H$ be a Hall $p'$-subgroup of $\gamma _{\infty}(G)$. By an analogue of the Frattini argument we have $G=N_G(H)\gamma _{\infty}(G)$. Indeed, for any $x\in G$ the Hall $p'$-subgroups $H$ and $H^x$ of $\gamma _{\infty}(G)$ are conjugate in $\gamma _{\infty}(G)$, so that $H=H^{xy}$ for $y\in \gamma _{\infty}(G)$ and then $x\in N_G(H)y^{-1}$.
We can now choose a nontrivial $p$-element $a\in N_G(H)$ (so that $|a|=p^k$, where $k\in {\Bbb N}\cup \{\infty\}$). By Lemma \[l-copr5\] the subgroup $[H,a]$ is finite. Therefore there is an open normal $a$-invariant subgroup $N$ of $\gamma _{\infty}(G)$ such that $N\cap [H,a]=1$. Then $[N\cap H,a]=1$. Since $N=(H\cap N)P$ for an $a$-invariant Sylow $p$-subgroup $P$ of ${\gamma }_{\infty}(G)$, we have $[N,a]=[(H\cap N)P,a]=[P,a]$. Hence the subgroup $[N,a]$ is pronilpotent, and therefore, $$[N,a]{\leqslant}F(N){\leqslant}F(\gamma _{\infty}(G)){\leqslant}F(G).$$ Thus, the proposition is proved if $[N,a]\ne 1$.
If $[N,a]=1$, then also $[N,[\gamma _{\infty}(G),a]]=1$. Then $[\gamma _{\infty}(G),a]$ has a central subgroup of finite index and therefore has finite Fitting height. By Lemma \[l-height\], $\gamma _{\infty}([\gamma _{\infty}(G),a])$ is finite, and therefore $F([\gamma _{\infty}(G),a])\ne 1$ unless $[\gamma _{\infty}(G),a]=1$. Since $$F([\gamma _{\infty}(G),a]){\leqslant}F(\gamma _{\infty}(G)){\leqslant}F(G),$$ the proof is complete if $[\gamma _{\infty}(G),a]\ne 1$. Finally, if $[\gamma _{\infty}(G),a]=1$, then $a$ is an Engel element since $G/\gamma _{\infty}(G)$ is locally nilpotent by Theorem \[t2\]. Then the normal subgroup $[G,a]\langle a\rangle$ is pronilpotent by Baer’s theorem [@hup Satz III.6.15], and $F(G)\ne 1$.
We are now ready to prove the main result of this section.
\[t3\] Suppose that $G$ is a prosoluble group in which every element has a countable right Engel sink. Then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent.
By Theorem \[t2\] it is sufficient to prove that $\gamma _{\infty}(G)$ is finite. Since ${\gamma }_{\infty}(F_2(G))$ is finite by Lemma \[l-height\], there is an open normal subgroup $N$ of $G$ such that $N\cap {\gamma }_{\infty}(F_2(G))=1$. It follows that ${\gamma }_{\infty}(F_2(N))=1$, which means that $F_2(N)=F(N)$. It follows from Proposition \[pr-f\] that $F(N)=N$ is locally nilpotent, so that $N{\leqslant}F(G)$. Hence the quotient group $G/F(G)$ is finite, and therefore the Fitting height of $G$ is finite. We obtain that ${\gamma }_{\infty}(G)$ is finite by Lemma \[l-height\].
Here we also derive the following corollary for a virtually prosoluble group (that is, a group with a prosoluble open normal subgroup), which will be needed in the sequel.
\[c-virt\] Suppose that $G$ is a virtually prosoluble group in which every element has a countable right Engel sink. Then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent.
By Theorem \[t2\] it is sufficient to show that $\gamma _{\infty}(G)$ is finite. By hypothesis, $G$ has an open normal prosoluble subgroup $H$. By Theorem \[t3\], $\gamma _{\infty}(H)$ is finite. Therefore, passing to the quotient group, we can assume that $\gamma _{\infty}(H)=1$ and the Fitting subgroup $F(G)$ is open.
Since $G/F(G)$ is finite, we can use induction on $|G/F(G)|$. The basis of this induction includes the trivial case $G/F(G)=1$ when $\gamma _{\infty}(G)=1$. But the bulk of the proof deals with the case where $G/F(G)$ is a finite simple group. If $G/F(G)$ is abelian, then $G$ has Fitting height 2 and $\gamma _{\infty }( G)$ is finite by Lemma \[l-height\] and the proof is complete.
Thus, suppose that $G/F(G)$ is a non-abelian finite simple group. Let $p$ be a prime divisor of $|G/F(G)|$, and $g\in G\setminus F(G)$ an element of order $p^n$, where $n$ is either a positive integer or $\infty$ (so $p^n$ is a Steinitz number). Let $T$ be the Hall $p'$-subgroup of $F(G)$. By Lemma \[l-copr3\] the subgroup $[T , g]$ is finite.
Since $[T, g]$ is normal in $F(G)$, its normal closure $R=\langle [T, g]^G\rangle $ in $G$ is a product of finitely many conjugates and is therefore also finite. Therefore it is sufficient to prove that $\gamma _{\infty }(G/R)$ is finite. Thus, we can assume that $R=1$. Note that then $[T, g^a]=1$ for any conjugate $g^a$ of $g$.
Choose a transversal $\{u_1,\dots, u_k\}$ of $G$ modulo $F(G)$. Let $G_1=\langle g^{u_1}, \dots ,g^{u_k}\rangle$. Clearly, $G_1F(G)/F(G)$ is generated by the conjugacy class of the image of $g$. Since $G/F(G)$ is simple, we have $G_1F(G)=G$. By our assumption, the Hall $p'$-subgroup $T$ of $F(G)$ is centralized by all elements $g^{u_i}$. Hence, $[G_1, T]=1$. Let $P$ be the Sylow $p$-subgroup of $F(G)$ (possibly, trivial). Then also $[PG_1, T]=1$, and therefore $$\gamma _{\infty }(G)=\gamma _{\infty }(G_1F(G))= \gamma _{\infty }(PG_1).$$
Let the bar denote images in $\bar G=G/P$. Note that $\gamma _{\infty}(\bar G)=\gamma _{\infty}(\bar G_1)$, while $F(\bar G)=\bar T$ and $\bar G/\bar T=\bar G_1\bar T/\bar T\cong F/F(G)$ is a non-abelian finite simple group. Hence, $\bar G=\gamma _{\infty}(\bar G_1)\bar T$. Therefore, since $[\gamma _{\infty}(\bar G_1), \bar T]=1$, $$\gamma _{\infty}(\bar G_1)=[\gamma _{\infty}(\bar G_1), \bar G_1]=[\gamma _{\infty}(\bar G_1),\gamma _{\infty}(\bar G_1)\bar T]=[\gamma _{\infty}(\bar G_1),\gamma _{\infty}(\bar G_1)].$$ As a result, $\gamma _{\infty}(\bar G_1)\cap \bar{T}$ is contained both in the centre and the derived subgroup of $\gamma _{\infty}(\bar G_1)$, and therefore is isomorphic to a subgroup of the Schur multiplier of the finite group $\gamma _{\infty}(\bar G_1)/ (\gamma _{\infty}(\bar G_1)\cap \bar{T})\cong G/F(G)$. Since the Schur multiplier of a finite group is finite [@hup Hauptsatz V.23.5], we obtain that $\gamma _{\infty}(\bar G_1)\cap \bar{T}$ is finite. Since $\bar T$ is canonically isomorphic to $T$, it follows that $$\gamma _{\infty }(G)\cap T\cong\gamma _{\infty }(\bar G)\cap \bar T=\gamma _{\infty }(\bar G_1)\cap \bar T$$ is also finite. Therefore we can assume that $T=1$, in other words, that $F(G)$ is a $p$-group.
Since $G/F(G)$ is a non-abelian simple group, we can choose another prime $r\ne p$ dividing $|G/F(G)|$ and repeat the same arguments as above with $r$ in place of $p$. As a result, we reduce the proof to the case $F(G)=1$, where the result is obvious.
We now finish the proof of Corollary \[c-virt\] by induction on $|G/F(G)|$. The basis of this induction where $G/F(G)$ is a simple group was proved above. Now suppose that $G/F(G)$ has a nontrivial proper normal subgroup with full inverse image $N$, so that $F(G)<N\lhd G$. Since $F(N)=F(G)$, by induction applied to $N$ the group $\gamma _{\infty }(N)$ is finite. Since $N/\gamma _{\infty }(N){\leqslant}F( G/\gamma _{\infty }(N))$, by induction applied to $G/\gamma _{\infty }(N)$ the group $ \gamma _{\infty }(G/\gamma _{\infty }(N) )$ is also finite. As a result, $\gamma _{\infty }(G) $ is finite, as required.
Profinite groups
================
We approach the general case of profinite groups by obtaining bounds for the so-called nonprosoluble length. These bounds follow from the bounds for nonsoluble length of the corresponding finite quotients. We begin with the relevant definitions.
The *nonsoluble length* $\lambda (H)$ of a finite group $H$ is defined as the minimum number of nonsoluble factors in a normal series in which every factor either is soluble or is a direct product of non-abelian simple groups. (In particular, the group is soluble if and only if its nonsoluble length is $0$.) Clearly, every finite group has a normal series with these properties, and therefore its nonsoluble length is well defined. It is easy to see that the nonsoluble length $\lambda (H)$ is equal to the least positive integer $l$ such that there is a series of characteristic subgroups $$1=L_0\leqslant R_0 < L_1\leqslant R_1< \dots \leqslant R_{l}=H$$ in which each quotient $L_i/R_{i-1}$ is a (nontrivial) direct product of non-abelian simple groups, and each quotient $R_i/L_{i}$ is soluble (possibly trivial).
We shall use the following result of Wilson [@wil83], which we state in the special case of $p=2$ using the terminology of nonsoluble length.
\[t-wil83\] Let $K$ be a normal subgroup of a finite group $G$. If a Sylow $2$-subgroup $Q$ of $K$ has a coset $tQ$ consisting of elements of order dividing $2^k$, then the nonsoluble length of $K$ is at most $k$.
We now turn to profinite groups. It is natural to say that a profinite group $G$ has finite *nonprosoluble length* at most $l$ if $G$ has a normal series $$1=L_0\leqslant R_0 < L_1\leqslant R_1< \dots \leqslant R_{l}=G$$ in which each quotient $L_i/R_{i-1}$ is a (nontrivial) Cartesian product of non-abelian finite simple groups, and each quotient $R_i/L_{i}$ is prosoluble (possibly trivial). As a special case of a general result in Wilson’s paper [@wil83] we have the following.
\[l-nsl\] If, for some positive integer $m$, all continuous finite quotients of a profinite group $G$ have nonsoluble length at most $m$, then $G$ has finite nonprosoluble length at most $m$.
We now prove a key proposition on bounds for the nonprosoluble length.
\[pr-fnl\] Suppose that $G$ is a profinite group in which every element has a countable right Engel sink. Then $G$ has finite nonprosoluble length.
Let $ H=\bigcap G^{(i)} $ be the intersection of the derived series of $G$ (where $G^{(1)}=[G,G]$ and by induction $G^{(i+1)}=[G^{(i)},G^{(i)}]$). Then $H=[H,H]$. Indeed, if $H\ne [H,H]$, then the quotient $G/[H,H]$ is a prosoluble group by Lemma \[l-prosol-by-prosol\], whence $\bigcap G^{(i)}=H{\leqslant}[H,H]$, a contradiction. Since the quotient $G/H$ is prosoluble, it is sufficient to prove the proposition for $H$. Thus, we can assume from the outset that $G=[G,G]$.
Let $T$ be a Sylow $2$-subgroup of $G$. By Theorem \[t2\] the group $T$ is locally nilpotent. Consider the subsets of the direct product $T\times T$ $$S_{i}=\{(x,y)\in T\times T\mid \text{the subgroup }\langle x,y\rangle\text{ is nilpotent of class at most }i\}.$$ Note that each subset $S_{i}$ is closed in the product topology of $T\times T$, because the condition defining $S_i$ means that all commutators of weight $i+1$ in $x,y$ are trivial. Since every $2$-generator subgroup of $T$ is nilpotent, we have $$\bigcup _iS_{i}=T\times T.$$ By Theorem \[bct\] one of the sets $S_i$ contains an open subset of $T\times T$. This means that there are cosets $aN$ and $bN$ of an open normal subgroup $N$ of $T$ and a positive integer $c$ such that $$\label{e-2nilp}
\langle x,y\rangle\text{ is nilpotent of class }c\text{ for any }x\in aN,\; y\in bN.$$ (The subsequent arguments include the case where $N=T$, even with certain simplifications.)
Let $K$ be an open normal subgroup of $G$ such that $K\cap T{\leqslant}N$. If we replace $N$ by $K\cap T$, then still holds with the same $a,b$. Hence we can assume that $N$ is a Sylow $2$-subgroup of $K$.
We now apply the following general fact (which, for example, immediately follows from [@rib-zal Lemma 2.8.15]).
\[l-dop\] Let $G$ be a profinite group and $K$ a normal open subgroup of $G$. There exists a subgroup $H$ of $G$ such that $G=KH$ and $K\cap H$ is pronilpotent.
Let $H$ be the subgroup given by this lemma for our group $G$ and subgroup $K$. Since $H$ is virtually pronilpotent and every element has a countable right Engel sink, by Corollary \[c-virt\] the subgroup $\gamma _\infty(H)$ is finite. Recalling our assumption that $G=[G,G]$, we obtain $$G=[G,G]=\gamma _{\infty}(G){\leqslant}\gamma _{\infty}(HK){\leqslant}\gamma _{\infty}(H)K.$$ Thus, $G=\gamma _{\infty}(H)K$, where $\gamma _{\infty}(H)$ is a finite subgroup.
Hence we can choose the coset representative $a$ satisfying in a conjugate of a Sylow $2$-subgroup of $\gamma _{\infty}(H)$, and therefore having finite order, say, $|a|=2^n$.
For any $y\in bN$ the $2$-subgroup $\langle a,y\rangle$ is nilpotent of class at most $c$, while $a^{2^n}=1$. Then $$\label{e-ay2nc}
[a,y^{2^{n(c-1)}}]=1.$$ This follows from well-known commutator formulae (and for any $p$-group); see, for example, [@shu00 Lemma 4.1].
In particular, for any $z\in N$ by using we obtain $$\label{e-2eng2}
[z,\,{}_c y^{2^{n(c-1)}}]=[az,\,{}_c y^{2^{n(c-1)}}]=1,$$ since $\langle az,y^{2^{n(c-1)}}\rangle$ is a subgroup of $\langle az,y\rangle$, which is nilpotent of class $c$ by . (Note that in the case $N=T$ we could have put $K=G$, $H=1$, $a=b=1$, and $n=0$.)
Our aim is to show that there is a uniform bound, in terms of $|G:K|$, $c$, and $n$, for the nonsoluble length of all continuous finite quotients of $G$. Let $M$ be an open normal subgroup of $G$ and let the bar denote the images in $\bar G=G/M$. It is clearly sufficient to obtain a required bound for the nonsoluble length of $\bar K$.
Let $R_0$ be the soluble radical of $\bar K$, and $L_1$ the inverse image of the generalized Fitting subgroup of $\bar K/R_0$, so that $$\label{e-soc}
L_1/R_0=S_1\times S_2\times \dots\times S_k$$ is a direct product of non-abelian finite simple groups. Note that $R_0$ and $L_1$ are normal subgroups of $\bar G$. The group $\bar G$ acting by conjugation induces a permutational action on the set $\{S_1,S_2,\dots ,S_k\}$. The kernel of the restriction of this permutational action to $\bar K$ is contained in the inverse image $R_1$ of the soluble radical of $\bar K/L_1$: $$\label{e-soc2}
\bigcap _iN_{\bar K}(S_i){\leqslant}R_1.$$ This follows from the validity of Schreier’s conjecture on the solubility of the outer automorphism groups of non-abelian finite simple groups, confirmed by the classification of the latter, because $L_1/R_0$ contains its centralizer in $\bar K/R_0 $.
Let $e$ be the least positive integer such that $2^{e}{\geqslant}c$, and let $t= 2^{n(c-1)+e}$. We claim that for any $y\in \bar b\bar N$ the element $y^{2^t}$ normalizes each factor $S_i$ in . Arguing by contradiction, suppose that the element $y^{2^t}$ has a nontrivial orbit on the set of the $S_i$. Then the element $y^{2^{n(c-1)}}$ has an orbit of length $2^s{\geqslant}2^{e+1}$ on this set; let $\{T_1,T_2,\dots , T_{2^s}\}$ be such an orbit cyclically permuted by $y^{2^{n(c-1)}}$. Since non-abelian finite simple groups have even order (by the Feit–Thompson theorem [@fei-tho]) and the subgroups $S_i$ are subnormal in $\bar K/R_0$, each subgroup $S_i$ contains a nontrivial element of $\bar NR_0/R_0$. If $x$ is a nontrivial element of $T_1\cap \bar NR_0/R_0$, then the commutator $$[x,\,{}_c \bar y^{2^{n(c-1)}}],$$ written as an element of $T_1\times T_2\times \dots\times T_{2^s}$, has a nontrivial component in $T_{c+1}$ since $2^s{\geqslant}2^{e+1}>c$. This, however, contradicts .
Thus, for any element $y\in \bar b\bar N$ the power $y^{2^t}$ normalizes each factor $S_i$ in . Let $2^{d}$ be the highest power of $2$ dividing $|G:K|$, and let $u=\max\{t,d\}$. Then $y^{2^u}\in R_1$ by , since $y^{2^u}\in \bar K$ and $y^{2^u}$ normalizes each $S_i$ in by the choice of $u$.
As a result, in the quotient $\bar G/R_1$ all elements of the coset $\bar b\bar NR_1/R_1$ of the Sylow $2$-subgroup $\bar NR_1/R_1$ of $\bar K/R_1$ have order dividing $2^u$. We can now apply Theorem \[t-wil83\], by which the nonsoluble length of $\bar K/R_1$ is at most $u$. Then the nonsoluble length of $\bar K$ is at most $u+1$. Clearly, the nonsoluble length of $\bar G/\bar K$ is bounded in terms of $|G:K|$. As a result, since the number $u$ depends only on $|G:K|$, $n$, and $c$, the nonsoluble length of $\bar G$ is bounded in terms of these parameters only. Since this holds for any continuous finite quotient of the profinite group $G$, the group $G$ has finite nonprosoluble length by Lemma \[l-nsl\]. This completes the proof of Proposition \[pr-fnl\].
We are now ready to handle the general case of profinite groups using Corollary \[c-virt\] on virtually prosoluble groups and induction on the nonprosoluble length. First we eliminate infinite Cartesian products of non-abelian finite simple groups.
\[l-cart\] Suppose that $G$ is a profinite group that is a Cartesian product of non-abelian finite simple groups. If every element of $G$ has a countable right Engel sink, then $G$ is finite.
Suppose the opposite: then $G$ is a Cartesian product of infinitely many non-abelian finite simple groups $G_i$ over an infinite set of indices $i\in I$.
Every non-abelian finite simple groups $S$ contains an element $s\in S$ with a nontrivial smallest right Engel sink $\mathscr R(s)\ne \{1\}$. Actually, any nontrivial element $s\in S\setminus\{1\}$ has nontrivial minimal right Engel sink. Indeed, otherwise $s$ is a right Engel element of $S$, and right Engel elements of a finite group belong to its hypercentre by Baer’s theorem [@rob 12.3.7]. By Lemma \[l-min-r\], for any $z\in \mathscr R(s)$ we have $$z=[s,{}_nx]=[s,{}_{n+m}x]$$ for some $x\in S$ and some $n{\geqslant}1$ and $m{\geqslant}1$, and then also $$z=[s,{}_nx]=[s,{}_{n+ml}x]\quad \text{for any}\;\, l\in {\mathbb N}.$$
For every $i$, we choose a nontrivial element $g_i\in G_i$, a nontrivial element $z_{i}\in \mathscr R(g_i)\subseteq G_i$, and the corresponding $x_i\in G_i$ such that for some $n_i{\geqslant}1$ and $m_i{\geqslant}1$ $$\label{e-cycl2}
z_i=[g_i,{}_{n_i}x_i]=[g_i,{}_{n_i+m_il}x_i]\quad \text{for any}\;\, l\in {\mathbb N}.$$
Consider the element $$g=\prod _{i\in I} g_{i}.$$ For any subset $J\subseteq I$, consider the element $$x_{J}=\prod _{j\in J} x_{j}.$$ If $\mathscr R(g)$ is any right Engel sink of $g$ in $G$, then for some $k\in {\mathbb N}$ the commutator $[g, \,{}_kx_{J}]$ belongs to $\mathscr R(a)$. Because of the properties , all the components of $[g, \,{}_kx_{J}]$ in the factors $G_j$ for $j\in J$ are nontrivial, while all the other components in $G_i$ for $i\not\in J$ are trivial by construction. Therefore for different subsets $J\subseteq I$ we thus obtain different elements of $\mathscr R(g)$. The infinite set $I$ has at least continuum of different subsets, whence $\mathscr R(g)$ is uncountable, contrary to $g$ having a countable Engel sink by the hypothesis.
\[t4\] Suppose that $G$ is a profinite group in which every element has a countable Engel sink. Then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent.
By Proposition \[pr-fnl\] the group $G$ has finite nonprosoluble length $l$. This means that $G$ has a normal series $$1=L_0\leqslant R_0 < L_1\leqslant R_1< L_1 \leqslant \dots \leqslant R_{l}=G$$ in which each quotient $L_i/R_{i-1}$ is a (nontrivial) Cartesian product of non-abelian finite simple groups, and each quotient $R_i/L_{i}$ is prosoluble (possibly trivial). We argue by induction on $l$. When $l=0$, the group $G$ is prosoluble, and the result follows by Theorem \[t3\].
Now let $l{\geqslant}1$. By Lemma \[l-cart\] each of the nonprosoluble factors $L_i/R_{i-1}$ is finite. In particular, the subgroup $L_1$ is virtually prosoluble, and therefore $\gamma _{\infty}(L_1)$ is finite by Corollary \[c-virt\]. The quotient $R_1/ \gamma _{\infty}(L_1)$ is prosoluble by Lemma \[l-prosol-by-prosol\]. Hence the nonprosoluble length of $G/\gamma _{\infty}(L_1)$ is $l-1$. By the induction hypothesis we obtain that $\gamma _{\infty}(G/\gamma _{\infty}(L_1))$ is finite, and therefore $\gamma _{\infty}(G)$ is finite. By Theorem \[t2\] the quotient $G/\gamma _{\infty}(G)$ is locally nilpotent, and the proof is complete.
Compact groups {#s-comp}
==============
In this section we prove the main Theorem \[t-main\] about compact groups with countable right Engel sinks. We use the structure theorems for compact groups and the results of the preceding section on profinite groups. Parts of the proof are similar to the proof of the main results of [@khu-shu; @khu-shu191] about finite and countable left Engel sinks. In the end we reduce the proof to the situation where every element has a finite left Engel sink and then apply Theorem \[t-left\].
By the well-known structure theorems (see, for example, [@hof-mor Theorems 9.24 and 9.35]), the connected component of the identity $G_0$ of a compact (Hausdorff) group $G$ is a divisible normal subgroup such that $G_0/Z(G_0)$ is a Cartesian product of (non-abelian) simple compact Lie groups, while the quotient $G/G_0$ is a profinite group. (Recall that a group $H$ is said to be *divisible* if for every $h\in H$ and every positive integer $k$ there is an element $x\in H$ such that $x^k=h$.)
We shall be using the following lemma from [@khu-shu].
\[l-eng\] Suppose that $G$ is a compact group in which every element has a finite left Engel sink and the connected component of the identity $G_0$ is abelian. Then for every $g\in G$ and for any $x\in G_0$ we have $$[x,\,{}_kg]=1\quad \text{for some}\;\,k=k(x,g)\in {\mathbb N}.$$
For compact groups with countable right Engel sinks, we begin with eliminating simple Lie groups.
\[l-lie\] A non-abelian simple compact Lie group contains an element all of whose right Engel sinks are uncountable.
It is well known that any non-abelian compact Lie group $G$ contains a subgroup isomorphic either to $SO_3 (\mathbb{R})$ or $SU_2( \mathbb{C})$ (see, for example, [@hof-mor Proposition 6.46]), and therefore in any case, a section isomorphic to $SO_3 (\mathbb{R})$. Since the property that every element has a countable right Engel sink is inherited by sections, it is sufficient to consider the case $G=SO_3 (\mathbb{R})$.
Consider the following elements of $SO_3 (\mathbb{R})$: $$a_\vartheta =\begin{pmatrix} \cos \vartheta &\sin \vartheta &0\\-\sin \vartheta &\cos \vartheta &0\\0&0&1\end{pmatrix},\quad \vartheta\in \mathbb{R},$$ and $$g=\begin{pmatrix} -1&0&0\\0&1&0\\0&0&-1\end{pmatrix}.$$ We have $$\begin{aligned}
[a_\vartheta ,g]=a_\vartheta ^{-1}a^g&=\begin{pmatrix} \cos (-\vartheta )&\sin (-\vartheta )&0\\-\sin (-\vartheta ) &\cos (-\vartheta )&0\\0&0&1\end{pmatrix}\cdot \begin{pmatrix} \cos \vartheta &-\sin \vartheta &0\\\sin \vartheta &\cos \vartheta &0\\0&0&1\end{pmatrix}\\ &=\begin{pmatrix} \cos (-2\vartheta )&\sin (-2\vartheta )&0\\-\sin(-2\vartheta ) &\cos (-2\vartheta )&0\\0&0&1\end{pmatrix}=a_{-2\vartheta },\end{aligned}$$ and then by induction, $$[a_\vartheta ,\,{}_ng]=a_{(-2)^n\vartheta }.$$ Therefore any left Engel sink of $g$ must contain, for every $\vartheta \in \mathbb{R}$, an element of the form $a_{(-2)^{n(\vartheta )}\vartheta }$ for some $n(\vartheta )\in {\mathbb N}$. Since for $\vartheta $ we can choose continuum elements of $\mathbb{R}$ that are linearly independent over $\mathbb{Q}$, any left Engel sink of $g$ must be uncountable.
All the elements $a_{\vartheta}$ form an abelian subgroup $A$, which is normalized by $g$. The above arguments actually show that any left Engel sink of $g$ in $A\langle g\rangle$ must be uncountable.
Since the group $A\langle g\rangle$ is metabelian, by Lemma \[l-metab\] a right Engel sink of $g=g^{-1}$ in $A\langle g\rangle$ is a left Engel sink of $g$ in $A\langle g\rangle$. Therefore any right Engel sink of $g$ must be uncountable.
The next lemma is a step towards proving that every element has a finite Engel sink.
\[l-ab\] Suppose that $G$ is a compact group in which every element has a countable right Engel sink. If $G$ has an abelian subgroup $A$ with locally nilpotent quotient $G/A$, then every element of $G$ has a finite left Engel sink.
Since $G/A$ is locally nilpotent, for showing that an element $g\in G$ has a finite left Engel sink we can obviously assume that $G=A\langle g\rangle$. Since the group $A\langle g\rangle$ is metabelian, by Lemma \[l-metab\] every element of it also has a countable left Engel sink. Then $g$ has a finite left Engel sink by Theorem \[t-left\].
We are now ready to prove the main result.
\[t5\] Suppose that $G$ is a compact group in which every element has a countable right Engel sink. Then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent.
In view of Lemma \[l-lie\], the connected component of the identity $G_0$ is an abelian divisible normal subgroup.
\[l-eng2\] For every $g\in G$ and for any $x\in G_0$ we have $$[x,\,{}_kg]=1\quad \text{for some}\;\,k=k(x,g)\in {\mathbb N}.$$
We can obviously assume that $G=G_0\langle g\rangle$. The group $G_0\langle g\rangle$ satisfies the hypothesis of Lemma \[l-ab\] and therefore every element in it has a finite left Engel sink. Then for any $x\in G_0$ we have $[x,\,{}_kg]=1$ for some $k=k(x,g)\in {\mathbb N}$ by Lemma \[l-eng\].
We proceed with the proof of Theorem \[t5\]. Applying Theorem \[t4\] to the profinite group $\bar G=G/G_0$ we obtain a finite normal subgroup $D$ with locally nilpotent quotient. Then every element $g\in\bar G$ has a finite smallest left Engel sink $\bar{\mathscr E}(g)$ contained in $D$. Consider the subgroup generated by all such sinks: $$E=\langle \bar{\mathscr E}(g)\mid g\in \bar G\rangle{\leqslant}D.$$ Clearly, $\bar{\mathscr E}(g)^h=\bar{\mathscr E}(g^h)$ for any $h\in\bar G$; hence $E$ is a normal finite subgroup of $\bar G$. Note that $\bar G/E$ is also locally nilpotent by Theorem \[t-wz\] as an Engel profinite group.
We now consider the action of $\bar G$ by automorphisms on $G_0$ induced by conjugation.
\[l-central\] The subgroup $E$ acts trivially on $G_0$.
In the proof of this lemma, we consider $G_0$ as an abstract abelian divisible group. Thus, $G_0$ is a direct product $A_0\times\bigoplus _pA_p$ of a torsion-free divisible group $A_0$ and divisible Sylow $p$-subgroups $A_p$ over various primes $p$. Clearly, every Sylow subgroup $A_p$ is normal in $G$.
First we show that $E$ acts trivially on each $A_p$. It is sufficient to show that for every $g\in \bar G$ every element $z\in \bar{\mathscr E}(g)$ acts trivially on $A_p$. Consider the action of $\langle z, g\rangle$ on $A_p$. Note that $\langle z, g\rangle=\langle z^{\langle g\rangle}\rangle\langle g\rangle$, where $\langle z^{\langle g\rangle}\rangle$ is a finite $g$-invariant subgroup, since it is contained in the finite subgroup $E$. For any $a\in A_p$ we have $[a,\,{}_kg]=1$ for some $k=k(a,g)\in {\mathbb N}$ by Lemma \[l-eng2\]. Hence the subgroup $$\langle a^{\langle g\rangle}\rangle=\langle a,[a,g], [a,g,g],\dots \rangle$$ is a finite $p$-group; note that this subgroup is $g$-invariant. The images of $\langle a^{\langle g\rangle}\rangle$ under the action of elements of the finite group $\langle z^{\langle g\rangle}\rangle$ generate a finite $p$-group $B$, which is $\langle z, g\rangle$-invariant. It follows from Lemma \[l-eng2\] that $\langle z, g\rangle/C_{\langle z, g\rangle}(B)$ must be a $p$-group. Indeed, otherwise there is a $p'$-element $y\in \langle z, g\rangle/C_{\langle z, g\rangle}(B)$ that acts non-trivially on the Frattini quotient $V=B/\Phi (B)$. Then $[[V,y],y]=[V,y]\ne 1$ and $C_{[V,y]}(y)=1$, whence $[V,y]=\{[v,y]\mid v\in [V,y]\}$ and therefore also $[V,y]=\{[v,\,{}_ny]\mid v\in [V,y]\} $ for any $n$, contrary to Lemma \[l-eng2\]. Thus, $\langle z, g\rangle/C_{\langle z, g\rangle}(B)$ is a finite $p$-group. But since $z\in \bar{\mathscr E}(g)$, by Lemma \[l-min\] we have $z=[z,{}_mg]$ for some $m\in {\mathbb N}$. Since a finite $p$-group is nilpotent, this implies that $z\in C_{\langle z, g\rangle}(B)$. In particular, $z$ centralizes $a$. Thus, $E$ acts trivially on $A_p$, for every prime $p$.
We now show that $E$ also acts trivially on the quotient $W=G_0/\bigoplus _pA_p$ of $G_0$ by its torsion part. Note that $W$ can be regarded as a vector space over ${\mathbb Q}$. Every element $y\in E$ has finite order and therefore by Maschke’s theorem $W=[W,y]\oplus C_W(y)$ and $[W,y]=\{[w,\,{}_ny]\mid w\in [W,y]\} $ for any $n$. If $[W,y]\ne 0$, then this contradicts Lemma \[l-eng2\].
Thus, $E$ acts trivially both on $W$ and on $\bigoplus _pA_p$. Then any automorphism $\eta$ of $G_0$ induced by conjugation by $h\in E$ acts on every element $a\in A_0$ as $a^{\eta}=a^h=at$, where $t=t(a,h)$ is an element of finite order in $G_0$. Then $a^{\eta ^i}=at^i$, and therefore the order of $t$ must divide the order of $\eta$.
Assuming the action of $E$ on $G_0$ to be non-trivial, choose an element $h\in E$ acting on $G_0$ as an automorphism $\eta$ of some prime order $p$. Then there is $a\in A_0$ such that $a^h=as$, where $s\in A_p$ has order $p$. There is an element $a_1\in A_0$ such that $a_1^{p}=a$. Then $a_1^h=a_1s_1$, where $s_1^{p}=s$. Thus, $|s_1|=p^{2}$, and therefore $p^{2}$ divides the order of $\eta $. We arrived at a contradiction with $|\eta |=p$.
We now finish the proof of Theorem \[t5\]. Let $F$ be the full inverse image of $E$ in $G$. Then we have normal subgroups $G_0{\leqslant}F{\leqslant}G$ such that $G/F$ is locally nilpotent, $F/G_0$ is finite, and $G_0$ is contained in the centre of $F$ by Lemma \[l-central\]. Since $F$ has centre of finite index, the derived subgroup $F'$ is finite by Schur’s theorem [@hup Satz IV.2.3]. The quotient $G/F'$ is an extension of an abelian subgroup by a locally nilpotent group. Hence every element of $G/F'$ has a finite left Engel sink by Lemma \[l-ab\]. By Theorem \[t-left\] the group $G/F'$ has a finite normal subgroup with locally nilpotent quotient. The full inverse image of this subgroup is a required finite normal subgroup $N$ such that $G/N$ is locally nilpotent. The proof of Theorem \[t5\] is complete.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank John Wilson for stimulating discussions.
The first author was supported by Mathematical Center in Akademgorodok, the agreement with Ministry of Science and High Education of the Russian Federation no. 075-15-2019-1613. The second author was supported by FAPDF and CNPq-Brazil.
[99]{} N. Bourbaki, Elements of Mathematics. Lie Groups and Lie Algebras. Part I: Chapters 1–3, Hermann, Paris, Addison-Wesley, Reading, MA, 1975.
E. Breuillard and T. Gelander, A topological Tits alternative, *Ann. Math. (2)* [**166**]{}, no. 2 (2007), 427–474.
E. Detomi, M. Morigi, P. Shumyatsky, [ Bounding the exponent of a verbal subgroup]{}, *Annali Mat.* **193** (2014), 1431–1441.
W. Feit and J. G. Thompson, Solvability of groups of odd order, *Pac. J. Math.* **13** (1963), 775–1029.
P. Hall and G. Higman, The $p$-length of a $p$-soluble group and reduction theorems for Burnside’s problem, *Proc. London Math. Soc. (3)* [ **6**]{} (1956), 1–42.
H. Heineken, Eine Bemerkung Über engelsche Elemente, *Arch. Math. (Basel)* [**11**]{} (1960), 321.
K. H. Hofmann and S. A. Morris, *The Structure of Compact Groups*, De Gruyter, Berlin, 2006.
B. Huppert, *Endliche Gruppen*. I, Springer, Berlin, 1967.
J. L. Kelley, *General topology*, Grad. Texts in Math., vol. 27, Springer, New York, 1975.
E. I. Khukhro and P. Shumyatsky, Nonsoluble and non-$p$-soluble length of finite groups, *Israel J. Math.*, **207** (2015), 507–525.
E. I. Khukhro and P. Shumyatsky, Words and pronilpotent subgroups in profinite groups, *J. Austral. Math. Soc.* [**97**]{}, no. 3 (2014), 343–364.
E. I. Khukhro and P. Shumyatsky, Almost Engel compact groups, *J. Algebra* [**500**]{} (2018), 439–456.
E. I. Khukhro and P. Shumyatsky, Compact groups all elements of which are almost right Engel, [*Quart. J. Math*]{} **70** (2019), 879–893.
E. I. Khukhro and P. Shumyatsky, Compact groups with countable Engel sinks, *to appear in Bull. Math. Sci.*, `https://arXiv:1908.11637`.
M. Lazard, Groupes analytiques $p$-adiques, *Publ. Math. Inst. Hautes Études Sci.* **26** (1965), 389–603.
Yu. Medvedev, On compact Engel groups, Israel J. Math. [**185**]{} (2003), 147–156.
L. Ribes and P. Zalesskii, *Profinite groups*, Springer, Berlin, 2010.
D. J. S. Robinson, A course in the theory of groups, Springer, New York, 1996.
A. Shalev, Polynomial identities in graded group rings, restricted Lie algebras and $p$-adic analytic groups, *Trans. Amer. Math. Soc.* **337**, no. 1 (1993), 451–462.
P. Shumyatsky, On pro-$p$ groups admitting a fixed-point-free automorphism, *J. Algebra* **228**, no. 1 (2000), 357–366.
P. Shumyatsky, Commutators in residually finite groups, *Israel J. Math.* [**182**]{} (2011), 149–156.
P. Shumyatsky, On the exponent of a verbal subgroup in a finite group, *J. Austral. Math. Soc.*, [**93**]{} (2012), 325–332.
J. G. Thompson, Automorphisms of solvable groups, *J. Algebra* [**1**]{} (1964), 259–267.
A. Turull, Character theory and length problems, in: *Finite and Locally Finite Groups*, NATO ASI Series, [**471**]{}, Kluwer, Dordrecht, 1995, 377–400.
J. S. Wilson, On the structure of compact torsion groups, [*Monatsh. Math.*]{} [**96**]{} (1983), 57–66.
J. S. Wilson, *Profinite groups* (Clarendon Press, Oxford, 1998).
J. S. Wilson and E. I. Zelmanov, Identities for Lie algebras of pro-$p$ groups, *J. Pure Appl. Algebra* [**81**]{}, no. 1 (1992), 103–109.
E. Zelmanov, Nil rings and periodic groups, *Korean Math. Soc. Lecture Notes in Math.*, Seoul, 1992.
E. Zelmanov, Lie methods in the theory of nilpotent groups, in: *Groups’93 Galaway/St Andrews*, Cambridge Univ. Press, Cambridge, 1995, 567–585.
E. Zelmanov, Lie algebras and torsion groups with identity, *J. Comb. Algebra* **1**, no. 3 (2017), 289–340.
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abstract: 'Language generation tasks that seek to mimic human ability to use language creatively are difficult to evaluate, since one must consider creativity, style, and other non-trivial aspects of the generated text. The goal of this paper is to develop evaluation methods for one such task, ghostwriting of rap lyrics, and to provide an explicit, quantifiable foundation for the goals and future directions of this task. Ghostwriting must produce text that is similar in style to the emulated artist, yet distinct in content. We develop a novel evaluation methodology that addresses several complementary aspects of this task, and illustrate how such evaluation can be used to meaningfully analyze system performance. We provide a corpus of lyrics for 13 rap artists, annotated for stylistic similarity, which allows us to assess the feasibility of manual evaluation for generated verse.'
author:
- |
Peter Potash, Alexey Romanov, Anna Rumshisky\
University of Massachusetts Lowell\
Department of Computer Science\
[{ppotash,aromanov,arum}@cs.uml.edu]{}\
bibliography:
- 'tacl.bib'
title: 'Evaluating Creative Language Generation: The Case of Rap Lyric Ghostwriting'
---
Introduction
============
Language generation tasks are often among the most difficult to evaluate. Evaluating machine translation, image captioning, summarization, and other similar tasks is typically done via comparison with existing human-generated “references”. However, human beings also use language creatively, and for the language generation tasks that seek to mimic this ability, determining how accurately the generated text represents its target is insufficient, as one also needs to evaluate creativity and style. We believe that one of the reasons such tasks receive little attention is the lack of sound evaluation methodology, without which no task is well-defined, and no progress can be made. The goal of this paper is to develop an evaluation methodology for one such task, ghostwriting, or more specifically, ghostwriting of rap lyrics.
Ghostwriting is ubiquitous in politics, literature, and music. As such, it introduces a distinction between the performer/presenter of text, lyrics, etc, and the creator of text/lyrics. The goal of ghostwriting is to present something in a style that is believable enough to be credited to the performer. In the domain of rap specifically, rappers sometimes function as ghostwriters early on before embarking on their own public careers, and there are even businesses that provide written lyrics as a service[^1]. The goal of automatic ghostwriting is therefore to create a system that can take as input a given artist’s work and generate **similar** yet **unique** lyrics. Our objective in this work is to provide a quantifiable direction and foundation for the task of rap lyric generation and similar tasks through (1) developing an evaluation methodology for such models, and (2) illustrating how such evaluation can be used to analyze system performance, including advantages and limitations of a specific language model developed for this task. As an illustration case, we use the ghostwriter model previously proposed in exploratory work by Potash et al. , which uses a recurrent neural network (RNN) with Long Short-Term Memory (LSTM) for rap lyric generation.
The following are the main contributions of this paper. We present a comprehensive manual evaluation methodology of the generated verses along three key aspects: fluency, coherence, and style matching. We introduce an improvement to the semi-automatic methodology used by Potash et al. that automatically penalizes repetitive text, which removes the need for manual intervention and enables a large-scale analysis. Finally, we build a corpus of lyrics for 13 rap artists, each with his own unique style, and conduct a comprehensive evaluation of the LSTM model performance using the new evaluation methodology. The corpus includes style matching annotation for select verses in dataset, which can form a gold standard for future work on automatic representation of similarity between artists’ styles. The resulting rap lyric dataset is publicly available from the authors’ website. Additionally, we believe that the annotation method we propose for manual style evaluation can be used for other similar generation tasks. One example is ’Deep Art’ work in the computer vision community that seeks to apply the style of a particular painting to other images [@gatys2015neural; @li2016combining]. One of the drawbacks of such work is a lack of systematic evaluation. For example, compared the results of the model with previous work by doing a manual inspection during an informal user study. The presence of a systematic formal evaluation process would lead to a clearer comparison between models and facilitate progress in this area of research. With this in mind, we make the interface used for style evaluation in this work available for public use.
Our evaluation results highlight the truly multi-faceted nature of the ghostwriting task. While having a single measure of success is clearly desirable, our analysis shows the need for complementary metrics that evaluate different components of the overall task. Indeed, despite the fact that our test-case LSTM model outperforms a baseline model across numerous artists based on automated evaluation, the full set of evaluation metrics is able to showcase the LSTM model’s strengths and weakness. The coherence evaluation demonstrates the difficulty of incorporating large amounts of training data into the LSTM model, which intuitively would be desirable to create a flexible ghostwriting model. The style matching experiments suggest that the LSTM is effective at capturing an artist’s general style. However, this may indicate that it tends to form ‘average’ verses, which are then more likely to be matched with existing verses from an artist rather than another random verse from the same artist. Overall, the evaluation methodology we present provides an explicit, quantifiable foundation for the ghostwriting task, allowing for a deeper understanding of the task’s goals and future research directions.
Related Work
============
In the past few years there has been a significant amount of work dedicated to the evaluation of natural language generation [@hastie2014comparative], dealing with different aspects of evaluation methodology. However, most of this work focuses on simple tasks, such as referring expressions generation. For example, Belz and Kow investigated the impact of continuous and discrete scales for generated weather descriptions, as well as and simple image descriptions that typically consist of a few words (e.g., “`the small blue fan`”).
Previous work that explores text generation for artistic purposes, such as poetry and lyrics, generally uses either automated or manual evaluation. In terms of manual evaluation, Barbieri et al. have a set of annotators evaluate generated lyrics along two separate dimensions: grammar and semantic relatedness to song title. The annotators rated the dimensions with scores 1-3. A similar strategy was used by Gervás , where the author had annotators evaluate generated verses with regard to syntactic correctness and overall aesthetic value, providing scores in the range 1-5. Wu et al. had annotators determine the effectiveness of various systems based on fluency as well as rhyming.
Some heuristic-based automated approaches have also been used. For example, Oliveira et al. use a simple automatic heuristic that awards lines for ending in a termination previously used in the generated stanza. Malmi et al. evaluate their generated lyrics based on the verses’ rhyme density, on the assumption that a higher rhyme density means better lyrics. Note that none of the work cited above provide a comprehensive evaluation methodology, but rather focus on certain specific aspects of generated verses, such as rhyme density or syntactic correctness. Moreover, the methodology for generating lyrics, proposed by the various authors, influences the evaluation process. For instance, did not evaluate the presence of rhymes because the model was constrained to produce only rhyming verses. Furthermore, none of the aforementioned works implement models that generate complete verses at the token level(including verse structure), which is the goal of the models we aim to evaluate. In contrast to previous approaches that evaluate whole verses, our evaluation methodology uses a more fine-grained, line-by-line scheme, which makes it easier for human annotators, as they no longer need to evaluate the whole verse at once. In addition, despite the fact the each line is annotated using a discrete scale, our methodology produces a continuous numeric score for the whole verse, enabling better comparison.
Dataset
=======
For our evaluation experiments, we selected the following list of artists in four different categories:
- Three top-selling rap artists according to Wikipedia[^2]: Eminem, Jay Z, Tupac
- Artists with the largest vocabulary according to Pop Chart Lab[^3]: Aesop Rock, GZA, Sage Francis
- Artists with the smallest vocabulary according to Pop Chart Lab: DMX, Drake
- Best classified artists from Hirjee and Brown using rhyme detection features: Fabolous, Nototious B.I.G., Lil’ Wayne
We collected all available songs from the above artists from the site *The Original Hip-Hop (Rap) Lyrics Archive - OHHLA.com - Hip-Hop Since 1992*[^4]. We removed the metadata, line repetiton markup, and chorus lines, and tokenized the lyrics using the NLTK library [@BirdKleinLoper09]. Since the preprocessing was done heuristically, the resulting dataset may still contain some text that is not actual verse, but rather dialogue or chorus lines. We therefore filter out all verses that are shorter than 20 tokens. Statistics of our dataset are shown in Table \[table:dataset-stat\].
Evaluation Methodology
======================
We believe that adequate evaluation for the ghostwriting task requires both manual and automatic approaches. The automated evaluation methodology enables large-scale analysis of the generated verse. However, given the nature of the task, the automated evaluation is not able to assess certain critical aspects of fluency and style, such as the vocabulary, tone, and themes preferred by a particular artist. In this section, we present a manual methodology for evaluating these aspects of the generated verse, as well as an improvement to the automatic methodology proposed by Potash et al. .
Manual Evaluation
-----------------
We have designed two annotation tasks for manual evaluation. The first task is to determine how fluent and coherent the generated verses are. The second task is to evaluate manually how well the generated verses match the style of the target artist.
#### Fluency/Coherence Evaluation {#sec:Fluency/Coherence Evaluation .unnumbered}
Given a generated verse, we ask annotators to determine the fluency and coherence of the lyrics. Even though our evaluation is for systems that produce entire verses, we follow the work of Wu and annotate fluency, as well as coherence, at the line level. To assess fluency, we ask to what extent a given line can be considered a valid English utterance. Since a language model may produce highly disjointed verses as it progresses through the training process, we offer the annotator three options for grading fluency: strongly fluent, weakly fluent, and not fluent. If a line is disjointed, i.e., it is only fluent in specific segments of the line, the annotators are instructed to mark it as weakly fluent. The grade of not fluent is reserved for highly incoherent text.
To assess coherence, we ask the annotator how well a given line matches the preceding line. That is, how believable is it that these two lines would follow each other in a rap verse. We offer the annotators the same choices as in the fluency evaluation: strongly coherent, weakly coherent, and not coherent. During the training process, a language model may output the same line repeatedly. We account for this in our coherence evaluation by defining the consecutive repetition of a line as not coherent. This is important to define because the line on its own may be strongly fluent, however, a coherent verse cannot consist of a single fluent line repeated indefinitely.
#### Style Matching {#style-matching .unnumbered}
The goal of the style matching annotation is to determine how well a given verse captures the style of the target artist. In this annotation task, a user is presented with an evaluation verse and asked to compare it against four other verses. The goal is to pick the verse that is written in a similar style. One of the four choices is always a verse from the same artist that was used to generate the verse being evaluated. The other three verses are chosen from the remaining artists in our dataset. Each verse is evaluated in this manner four times, each time against different verses, so that it has the chance to get matched with a verse from each of the remaining twelve artists. The generated verse is considered stylistically consistent if the annotators tend to select the verse that belongs to the target artist. To evaluate the difficulty of this task, we also perform style matching annotation for authentic verse, in which the evaluated verse is not generated, but rather is an actual existing verse from the target artist. [^5]
Automated Evaluation {#subsection:auto_eval}
--------------------
The automated evaluation we describe below attempts to capture computationally the dual aspects of “unique yet similar” in a manner originally proposed by .
#### Uniqueness of Generated Lyrics {#sec:eval-methods-sim .unnumbered}
We use a modified tf-idf representation for verses, and calculate cosine similarity between generated verses and the verses from the training data to determine novelty (or lack thereof). In order to evaluate the novelty of generated lyrics, we compare the similarity of the generated lyrics to the lyrics in our training set. We used an algorithm proposed by [@mahedero2005natural] for calculating the similarity between produced lyrics and all verses from the same artist. This algorithm is based on the well-known Inverse Document Frequency and cosine distance between documents. First, we build the Term-Document Matrix with weights for each token in each song: $$\label{eq:sim_vectors}
w_{ij} = f_{ij} \log\big( \frac{N}{n_{j}} \big)$$ where $N$ is the total number of documents (verses, in our case), $n_{j}$ is the number of verses that contains term $j$ and $f_{ij}$ is the frequency of term $j$ in the $ith$ verse. Using this matrix, we can calculate the cosine distance between verses and use it as measure of similarity. Verses that have a similar distribution of word usage will have a high cosine similarity. In order to more directly penalize generated verses that are primarily the reproduction of a single verse from the training set, we calculate the maximum similarity score across all training verses. That is, we do not want generated verses that contain text from a single training verse, which in turn rewards generated verses that draw from numerous training verses.
#### Stylistic Similarity via Rhyme Density of Lyrics {#stylistic-similarity-via-rhyme-density-of-lyrics .unnumbered}
We use the rhyme density method proposed by to evaluate how well the generated verse models an artist’s style. The point of an effective system is not to produce arbitrary rhymes: it is to produce rhyme types and rhyme frequency similar to the target artist. We note that the ghostwriter methodology we implement trains exclusively on the verses of a given artist, causing the vocabulary of the generated verse to be closed with respect to the training data. In this case, assessing how similar the generated vocabulary is to the target artist is not important. Instead, we focus on rhyme density, which is defined as the number of rhymed syllables divided by the total number of syllables [@hirjee2010using]. Certain artists distinguish themselves by having more complicated rhyme schemes, such as the use of internal[^6] or polysyllabic rhymes[^7]. Rhyme density is able to capture this in a single metric, since the tool we use is able to detect these various forms of rhymes. Moreover, as was shown in Wu , the inter-annotator agreement (IAA) for manual rhyme detection is low (the highest IAA was only 0.283 using a two-scale annotation scheme), which is expected due to the subjective nature of the task. Therefore, an objective automatic methodology is desirable. Since this tool is trained on a distinct corpus of lyrics, it can provide a “uniform” experience and give an impartial and objective score.
However, the rhyme detection method is not designed to deal with highly repetitive text, which the LSTM model produces often in the early stages of training. Since the same phoneme is repeated (because the same word is repeated), the rhyme detection tool generates a false positive. deal with this by manually inspecting the rhyme densities of verses generated in the early stages of training to determine if a generated verse should be kept for the evaluation procedure. This approach is suitable for their work since they processed only one artist, but it is clearly not scalable, and therefore not applicable to our case. In order to fully automate this method, we propose to handle highly repetitive textby weighting the rhyme density of a given verse by its entropy. More specifically, for a given verse, we calculate entropy at the token level and divide by total number of tokens in that verse. Verses with highly repetitive text will have a low entropy, which results in down-weighting the rhyme density of verses that produce false positive rhymes due to their repetitive text.
To evaluate our method, we applied it to the artist used by Potash et al. and obtained exactly the same average rhyme density without any manual inspection of the generated verses; this despite the presence of false positive rhymes automatically detected in the beginning of training.
#### Merging Uniqueness and Similarity {#merging-uniqueness-and-similarity .unnumbered}
Since ghostwriting is a balancing act of the two opposing forces of textual uniqueness and stylistic similarity, we want a low correlation between rhyme density (stylistic similarity) and maximum verse similarity (lack of textual uniqueness). However, our goal is not to have a high rhyme density, but rather to have a rhyme density similar to the target artist, while simultaneously keeping the maximum similarity score low. As the model overfits the training data, both the value of maximum similarity and the rhyme density will increase, until the model generates the original verse directly. Therefore, our goal is to evaluate the value of the maximum similarity at the point where the rhyme density has the value of the target artist. In order to accomplish this, we follow and plot the values of rhyme density and maximum similarity obtained at different points during model training. We use regression lines for these points to identify the value of the maximum similarity line at the point where the rhyme density line has the value of the target artist. We give more detail below.
Lyric Generation Experiments {#sec:lyric-generation-experimnents}
============================
Baseline
--------
To compare with the results of an LSTM model [@potashghostwriter], we followed the work of [@barbieri2012markov] and created a Markov model for lyric generation. Since the goal of our work is to make an unsupervised system, we do not use any constraints or templates to produce the lyrics. Thus, our baseline simplifies to a basic n-gram model. Given a history of $w_{k+n-1}$,...,$w_{k}$, the system generates a new token $t$ as follows: $$\label{eq:ngram_generation}
\begin{split}
& P(w_{k+n}=t|w_{k+n-1},...,w_{k}) = \\
&\frac{|w_{k}...w_{k+n-1}t|}{|w_{k}...w_{k+n-1}\bullet|}
\end{split}$$ where $|w_{k}...w_{k+n-1}t|$ is the number of times the context $w_{k+n-1}$,...,$w_{1}$ is followed by $t$ in the training data, and $|w_{k}...w_{k+n-1}\bullet|$ is the number of times the context appears followed by any token. There is the possibility, particularly when $n$ is large, that the context has never been encountered in the training data. When this occurs, we back off to a smaller n-gram model: $$\label{eq:ngram_backoff}
\begin{split}
&P(w_{k+n}=t|w_{k+n-2},...,w_{k}) =\\
&\frac{|w_{k}...w_{k+n-2}\bullet t|}{|w_{k}...w_{k+n-2}\bullet\bullet|}
\end{split}$$ The model may have to backoff multiple times before it encounters context it has seen in the training data. Once we back off to the point where we compute $P(w_{n+k}=t|w_{k})$ we are guaranteed to have at least one non-zero probability, because $w_{k}$ must have appeared in the vocabulary for it to have been generated previously (see section 4.3 on model initialization). Because of this, we do not need to implement smoothing into the model. Note that the model to which we backoff is not necessarily a lower order n-gram model, but rather a lower order skip-gram model. Also, given the initial context of just the “$<$startVerse$>$” token, the model initializes as a unigram model, then becomes a bigram model, and so on until there is enough context to use the full n-gram model.
The main generative model we use in our evaluation experiments is an LSTM. Similar to , we use an n-gram model as a baseline system for automated evaluation. We refer the reader to the original work for a detailed description. After every 100 iterations of training[^8] the LSTM model generates a verse. For the baseline model, we generate five verses at values 1-9 for $n$. We see a correspondence between higher $n$ and higher iteration: as both increase, the models become more ‘fit’ to the training data.
For the baseline model, we use the verses generated at different n-gram lengths ($n\in \{1,...,9\}$) to obtain the values for regression. At every value of $n$, we take the average rhyme density and maximum similarity score of the five verses that we generate to create a single data point for rhyme density and maximum similarity score, respectively.
To enable comparison, we also create nine data points from the verses generated by the LSTM as follows. A separate model for each artist is trained for a minimum of 16,400 iterations. We take the verses generated every 2,000 iterations, from 0 to 16,000 iterations, giving us nine points. The averages for each point are obtained by using the verses generated in iterations $\pm x, x \in \{100,200,300,400\}$ for each interval of 2,000.
Results
=======
We present the results of our evaluation experiments using both manual and automated evaluations.
Fluency/Coherence
-----------------
In order to fairly compare the fluency/coherence of verses across artists, we use the verses generated by each artist’s model at 16,000 iterations. We apply the fluency/coherence annotation methodology from Section \[sec:Fluency/Coherence Evaluation\]. Each line is annotated by two annotators. Annotation results are shown in Figure \[fig:fluency\_comparasion\] and Figure \[fig:coherence\_comparasion\]. For each annotated verse, we report the percentage of lines annotated as strongly fluent, weakly fluent, and not fluent, as well as the corresponding percentages for coherence. We convert the raw annotation results into a single score for each verse by treating the labels “strongly fluent”, “weakly fluent”, and “not fluent” as numeric values 1, 0.5, and 0, respectively. Treating each annotation on a given line separately, we calculate the average numeric rating for a given verse: $$\mbox{\textit{Fluency}} = \frac{\#\mbox{\textit{sf}} + 0.5 \#\mbox{\textit{wf}}}{\#a}$$ where $\#\mbox{\textit{sf}}$ is the number of times any line is labeled strongly fluent, $\#\mbox{\textit{wf}}$ is the number of times any line is labeled weakly fluent, and $\#a$ is the total annotations provided for a verse, which is equal to the number of lines $\times$ 2. *Coherence* is calculated in a similar manner. Raw inter-annotator agreement (IAA) for fluency annotation was 0.67. For coherence annotation, the IAA was 0.43. We believe coherence has a lower agreement because it is more semantic, as opposed to syntactic, in nature, causing it to be more subjective. Note that while the agreement is relatively low, it is expected, given the subjective nature of the task. For example, report similar agreement values for the fluency annotation they perform.
![Percentage of lines annotated as strongly fluent, weakly fluent, and not fluent. The numbers above the bars reflect the total score of the artist (higher is better). The resulting mean is 0.723 and the standard deviation is 0.178.[]{data-label="fig:fluency_comparasion"}](fluency_comparasion){width="\linewidth"}
![Percentage of lines annotated as strongly coherent, weakly coherent, and not coherent. The numbers above the bars reflect the total score of the artist (higher is better). The resulting mean is 0.713 and the standard deviation is 0.256.[]{data-label="fig:coherence_comparasion"}](coherence_comparasion){width="\linewidth"}
Style Matching
--------------
We performed style-matching annotation for the verses generated at iterations 16,000–16,400 for each artist. For the experiment with authentic verses, we randomly chose five verses from each artist, with a verse length of at least 40 tokens. Each page was annotated twice, by native English-speaking rap fans. The results of our style matching annotations are shown in Table \[tbl:style\_matches\]. We present two different views of the results. First, each annotation for a page is considered separately and we calculate: $$Match \% = \frac{\#m}{\#a}$$ where $\#m$ is the number of times, on a given page, the chosen verse actually came from the target artist, and $\#a$ is the total number of annotations done. For a given artist, five verses were evaluated, each verse appeared on four separate pages, and each page is annotated twice, so $\#a$ is equal to 40. Since in each case (i.e., page) the classes are different, we cannot use Fleiss’s kappa directly. Raw agreement for style annotation, which corresponds to the percentage of times annotators picked the same verse (whether or not they are correct) is shown in the column ’Raw agreement %’ in Table \[tbl:style\_matches\]. We also report annotators’ joint ability to guess the target artist correctly, which we compute as follows: $$Match_A\% = \frac{\#m_A}{\#s_A}$$ where $\#s_A$ is the number of times the annotators agreed on a verse on the same page, and $\#m_A$ is the number of times that the agreed upon verse is from the target artist.
### Artist Confusion
![Fraction of confusions between artists[]{data-label="fig:confusions_baseline"}](confusions_baseline){width="\linewidth"}
The results of style-matching annotation also provides us with an interesting insight into the similarity between two artists’ styles. This is captured by the *confusion* between two artists during the annotation of the pages with authentic verses, which is computed as follows: $$Confusion(a,b) = \frac{\#c(a,b)+\#c(b,a)}{\#p(a,b)+\#p(b,a)}$$
where $\#p(a,b)$ is the number of times a verse from artist $a$ is presented for evaluation and a verse from artist $b$ is shown as one of four choices; $\#c(a,b)$ is the number of times the verse from artist $b$ was chosen as the matching verse. The resulting confusion matrix is presented in Figure \[fig:confusions\_baseline\]. We intend for this data to provide a gold standard for future experiments that would attempt to encode the similarity of artists’ styles.
Automated Evaluation {#sec:automated_evaluation}
--------------------
****
The results of our automated evaluation are shown in Table \[tbl:regression\_results\]. For each artist, we calculate their average rhyme density across all verses. We then use this value to determine at which iteration this rhyme density is achieved during generation (using the regression line for rhyme density). Next, we use the maximum similarity regression line to determine the maximum similarity score at that iteration. Low maximum similarity score indicates that we have maintained stylistic similarity while producing new, previously unseen lyrics.
Note the presence of negative numbers in Table \[tbl:regression\_results\]. The reason is that in the beginning of training (in the LSTM’s case) and at a low n-gram length (for the baseline model), the models actually achieved a rhyme density that exceeded the artist’s average rhyme density. As a result, the rhyme density regression line hits the average rhyme density on a negative iteration.
Discussion
==========
In order to better understand the interaction between the four metrics we have introduced in this paper, we examined correlation coefficients between different measures of quality for generated verse (see Table \[tbl:coherence\_fluency\_sim\_math\_correlation\]). The lack of strong correlation supports the notion that different aspects of verse quality should be addressed separately. Moreover, the metrics are in fact complementary. Even the measures of *fluency* and *coherence*, despite sharing a similar goal, have a relatively low correlation of 0.4. Such low correlations emphasize our contribution, since other works [@barbieri2012markov; @wuevaluating; @malmi2015dopelearning] do not provide a comprehensive evaluation methodology, and evaluate just one or two particular aspects. For example, evaluated only fluency and rhyming, and evaluated only syntactic correctness and semantic relatedness to the title, whereas we present complementary approaches for evaluating different aspects of the generated verses.
[0.45]{}
[0.45]{}
Interestingly, the number of verses a rapper has in our dataset has a strong negative correlation with coherence score (cf. Table \[tbl:coherency\_fluency\_corr\]). This can be explained by the following consideration: on iteration 16,000, the model for the authors with the smaller number of verses has seen the same verses more times than the model trained on a larger number of verses. Therefore, it is easier for the former to produce more coherent lyrics since it saw more of the same patterns. As a result, models trained on a larger number of verses have a lower coherence score. For example, Lil’ Wayne has the most verses in our data, and correspondingly, the model for his verse has the worst coherence score. Note that the fluency score does not have this negative correlation with the number of verses. Based on our evaluation, 16,000 iterations is enough to learn a language model for the given artist that produces fluent lines. However, these lines will not necessarily form a coherent verse if the artist has a large number of verses. As can be seen from Table \[tbl:style\_matches\], the $Match\%$ score suggests that the LSTM-generated verses are able to capture the style of the artist as well as the original verses. Furthermore, $Match_A\%$ is significantly higher for the LSTM model, which means that the annotators agreed on matching verses more frequently. We believe this means that the LSTM model, trained on all verses from a given artist, is able to capture the artist’s “average” style, whereas authentic verses represent a random selection that are less likely, statistically speaking, to be similar to another random verse. Note that, as we expect, there is a strong correlation between the number of tokens in the artist’s data and the frequency of agreed-upon correct style matches (cf. Table \[tbl:coherency\_fluency\_corr\]). Since verses vary in length, this correlation is not observed for verses. Finally, the lack of strong correlation with vocabulary richness suggests that token uniqueness is not as important as the sheer volume.
One aspect of the generated verse we have not discussed so far is the structure of the generated verse. For example, the length of the generated verses should be evaluated, since the models we examined do generate line breaks and also decide when to end the verse. Table \[tbl:max\_len\_epoch\] shows the longest verse generated for each artist, and also the point at which it was achieved during the training. We note that although 10 of the 11 models are able to generate long verses (up to a full standard deviation above the average verse length for that author), it takes a substantial amount of time, and the correlation between the average verse length for a given an artist and the verse length achieved by the model is weak (0.258). This suggests that modeling the specific verse structure, including length, is one aspect that requires improvement.
Lastly, we note that the fully automated methodology we propose is able to replicate the results of the previously available semi-automatic method for the rapper Fabolous, which was the only artist evaluated by . Furthermore, the results of automated evaluation for the 11 artists confirm that the LSTM model generalizes better than the baseline model.
Conclusions and Future Work
===========================
In this paper, we have presented a comprehensive evaluation methodology for the task of ghostwriting rap lyrics, which captures complementary aspects of this task and its goals. We developed a manual evaluation method that assesses several key properties of generated verse, and created a data set of authentic verse, manually annotated for style matching. A previously proposed semi-automatic evaluation method has now been fully automated, and shown to replicate results of the original method. We have shown how the proposed evaluation methodology can be used to evaluate an LSTM-based ghostwriter model. We believe our evaluation experiments also clearly demonstrate that complementary evaluation methods are required to capture different aspects of the ghostwriting task.
Lastly, our evaluation provides key insights into future directions for generative models. For example, the automated evaluation shows how the LSTM’s inability to integrate new vocabulary makes it difficult to achieve truly desirable similarity scores; future generative models can draw on the work of and in an attempt to leverage other artists’ lyrics.
Acknowledgments {#acknowledgments .unnumbered}
===============
Do not number the acknowledgment section. Do not include this section when submitting your paper for review.
[^1]: <http://www.rap-rebirth.com/>,\
<http://www.precisionwrittens.com/rap-ghostwriters-for-hire/>
[^2]: <http://en.wikipedia.org/wiki/List_of_best-selling_music_artists>
[^3]: <http://popchartlab.com/products/the-hip-hop-flow-chart>
[^4]: <http://www.ohhla.com/>
[^5]: We have made the annotation interface available on (<https://github.com/placeholder>).
[^6]: e.g. “New York City gritty committee pity the fool” and “How I made it you salivated over my calibrated”
[^7]: e.g. “But it was your op to shop stolen art/Catch a swollen heart form not rolling smart”.
[^8]: Training is done in batches with two verses per batch.
|
---
abstract: 'We present a dissipative protocol to engineer two $^{87}Rb$ atoms into a form of three-dimensional entangled state via spontaneous emission. The combination of coupling between ground states via microwave fields and dissipation induced by spontaneous emission make the current scheme deterministic and a stationary entangled state can always be achieved without state initialization. Moreover, this scheme can be straightforwardly generalized to preparation of an $N$-dimensional entangled state in principle.'
author:
- 'Xiao-Qiang Shao[^1]'
- 'Tai-Yu Zheng'
- 'C. H. Oh'
- Shou Zhang
title: 'Dissipative creation of three-dimensional entangled state in optical cavity via spontaneous emission'
---
introduction
============
For an open quantum system, the dissipation process must be accompanied by entanglement generation, i.e. the populations of quantum states are altered due to entanglement with an external environment. Thus researchers are dedicating themselves to find efficient ways avoiding decoherence during quantum information process. Currently, the feasible methods include an active error-correction approach based on the assumption that the most probable errors occur independently to a few qubits, which can be corrected via subsequent quantum operation [@bennett; @chi; @kosut; @moussa; @reed], and alternative passive error-prevention scheme, where the logical qubits are encoded into subspaces which do not decohere because of symmetry [@lidar; @beige; @kempe; @chen; @pushin]. Recently, the function of dissipation is reexamined in Ref. [@diehl; @verstraete; @vacanti; @kastoryano; @busch; @shen; @torre; @rao; @lin], where the environment along can be used as a resource to preparing entanglement and implementing universal quantum computing. In particular, Kastoryano [*et al.*]{} consider a dissipative scheme for preparing a maximally entangled state of two $\Lambda$-atoms in a high finesse optical cavity [@kastoryano], in which a pure steady singlet state is achieved with no need of state initialization.
Compared with other kinds of entanglement, high-dimensional entangled states have attracted much interest, since it can enhance the violations of local realism and the security of quantum cryptography. In the fields of linear optics, two experiments utilize the spatial modes of the electromagnetic field carrying orbital angular momentum to create high-dimensional entanglement. In the context of cavity quantum electrodynamics (QED), three-dimensional entanglement has also been realized in the unitary evolutionary dynamics based on resonant, and dispersive atom-cavity interactions [@mair; @vaziri; @zou; @ye; @lu; @li]. In this paper, we put forward a dissipative method for preparing a stationary three-dimensional entangled state. The motivation of our proposal is mainly based on the following truth: The typical decoherence factors in cavity QED system consist of atomic spontaneous emission and cavity decay, which have detrimental effects on schemes based on unitary dynamics. However, the loss of cavity can used to stabilize a pure maximally entangled state when a suitable feedback control is applied [@1; @2; @3; @shao]. Thus the spontaneous emission of atom becomes the only one detrimental factor. The result of our work shows that atomic spontaneous emission is able to be a useful resource in respect of entanglement preparation, especially the fidelity of target state can even be better than the unitary evolution based schemes.
The structure of the manuscript is as follows. We derive the Lindblad master equation for preparation of three-dimensional entangled state with effective operator method in section II. We then generalize the scheme to realization of an $N$-dimensional entangled state via introducing multi-level atoms and multi-mode cavity and discuss the effect of cavity decay on the fidelity in section III. This paper ends up with a conclusion in section IV.
Effective master equation for open quantum systems
==================================================
We take into account a system composed by two $^{87}Rb$ atoms trapped in a bi-mode optical field, as shown in Fig. \[p0\]. The quantum states $|g_L\rangle$, $|g_0\rangle$, $|g_R\rangle$, and $|g_a\rangle$ correspond to atomic levels $|F=1,m_f=-1\rangle,$ $|F=1,m_f=0\rangle,$ $|F=1,m_f=1\rangle,$ and $|F=2,m_f=0\rangle$ of $5S_{1/2}$, and $|e_L\rangle$, $|e_0\rangle$, $|e_R\rangle$ correspond to $|F=1,m_f=-1\rangle,$ $|F=1,m_f=0\rangle,$ and $|F=2,m_f=1\rangle$ of $5P_{3/2}$. Without loss of generality, we apply two off-resonance $\pi$-polarized optical lasers, with Rabi frequencies $\Omega_{1(2)}$, detuning $\Delta$ to drive the transitions $|e_0\rangle\leftrightarrow|g_a\rangle$ for the first atom and $|e_L\rangle\leftrightarrow|g_L\rangle$ and $|e_R\rangle\leftrightarrow|g_R\rangle$ for the second atom. The transition $|e_{0(R)}\rangle\leftrightarrow|g_{L(0)}\rangle$ and $|e_{0(L)}\rangle\leftrightarrow|g_{R(0)}\rangle$ are coupled to the cavity modes $a_L$ and $a_R$ with coupling strength $g_L$ and $g_R$, detuning $\Delta-\delta$, respectively. In addition, two microwave fields with Rabi frequencies $\omega_1$ and $\omega_2$ are introduced to resonantly couple ground states as to acquire a steady-state entanglement during the dissipative process.
In a rotating frame, the master equation describing the interaction between quantum systems and external environment is in the Lindblad form $$\label{H1}
\dot{\hat{\rho}}=i[\hat{\rho},\hat{H}]+\sum_j\hat{L}_j\hat{\rho} \hat{L}_j^{\dag}-\frac{1}{2}(\hat{L}_j^{\dag}\hat{L}_j\hat{\rho}+\hat{\rho} \hat{L}_j^{\dag}\hat{L}_j).$$ In connection with cavity QED, the Lindblad operator $L_j$ is closly related to two typical decoherence factors, i.e. the spontaneous emission rate $\gamma$ from the excited state of $^{87}Rb$ atom and the leaky rate $\kappa$ of photon from the optical cavity. Thus the nine dissipation channels are denoted by $L^{\gamma_1,g_{L(a,R)}}=\sqrt{\gamma/3}|g_{L,(a,R)}\rangle\langle e_0|$, $L^{\gamma_2,g_{L(0)}}=\sqrt{\gamma/2}|g_{L,(0)}\rangle\langle e_L|$, $L^{\gamma_2,g_{R(0)}}=\sqrt{\gamma/2}|g_{R,(0)}\rangle\langle e_R|$, $L^{a_L}=\sqrt{\kappa}a_L$ and $L^{a_R}=\sqrt{\kappa}a_R$, respectively. For the sake of convenience, we have assumed a uniform dissipation rate for atoms and cavity modes. The Hamiltonian of the total system reads $$\begin{aligned}
\label{111}
\hat{H}&=&\hat{H}_0+\hat{H}_g+\hat{V}_++\hat{V}_-,\\
\hat{H}_0&=&\delta {\hat{a}}_L^{\dag}{\hat{a}}_L+\big[g_L(|g_L\rangle_{11}\langle e_0|+|g_0\rangle_{22}\langle e_R|){\hat{a}}_L^{\dag}+{\rm H.c.}\big]\nonumber\\
&&+\delta {\hat{a}}_R^{\dag}{\hat{a}}_R+\big[g_R(|g_R\rangle_{11}\langle e_0|+|g_0\rangle_{22}\langle e_L|){\hat{a}}_R^{\dag}+{\rm H.c.}\big]\nonumber\\
&&+\Delta(|e_0\rangle_{11}\langle e_0|+|e_L\rangle_{22}\langle e_L|+|e_R\rangle_{22}\langle e_R|),\\
\hat{H}_g&=&\omega_1(|g_L\rangle_{11}\langle g_a|+|g_R\rangle_{11}\langle g_a|)\nonumber\\&&+\omega_2(|g_L\rangle_{22}\langle g_0|+|g_R\rangle_{22}\langle g_0|)+{\rm H.c.},\\
\hat{V}_+&=&\hat{V}_-^{\dag}=\Omega_1(|e_0\rangle_{11}\langle g_a|)\nonumber\\&&+\Omega_2(|e_L\rangle_{22}\langle g_L|+|e_R\rangle_{22}\langle g_R|),\end{aligned}$$ where $\hat{H}_0$ characterizes the strong interaction between atoms and quantized cavity fields, and $\hat{H}_g$ and $\hat{V}_{\pm}$ correspond to the weakly driven fields of microwave and optical lasers, respectively. For simplicity, we set $g_L=g_R=g$, $\Omega_1=\Omega_2=\Omega$, and $\omega_1=-\omega_2=\omega$ in the following. To gain a better insight into the effect of spontaneous emission on the preparation of entanglement state, we first consider a perfect cavity without decay. According to the effective operator method [@reiter], the excited states of the atoms and the cavity field modes can be adiabatically eliminated, provided that the Rabi frequency $\Omega$ of the optical pumping laser is sufficiently weak enough compared with $g$, $\delta$ and $\Delta$, and the excited states are not initially populated. Then we obtain the effective master equation as $$\begin{aligned}
\label{HHH}
\dot{\hat{\rho}}&=&i[\hat{\rho},\hat{H}_{\rm eff}]+\sum_j\hat{L}_{{\rm eff},j}\hat{\rho} \hat{L}_{{\rm eff},j}^{\dag}-\frac{1}{2}(\hat{L}_{{\rm eff},j}^{\dag}\hat{L}_{{\rm eff},j}\hat{\rho}\nonumber\\&&+\hat{\rho} \hat{L}_{{\rm eff},j}^{\dag}\hat{L}_{{\rm eff},j}),\end{aligned}$$ where $$\begin{aligned}
\label{H7}
&&\hat{H}_{\rm eff}=-\frac{1}{2}[\hat{V}_-\hat{H}_{NH}^{-1}\hat{V}_++\hat{V}_-(\hat{H}_{NH}^{-1})^{\dag}\hat{V}_+]+\hat{H}_g,\nonumber\\
&&\hat{L}_{{\rm eff},j}=\hat{L}_j\hat{H}_{NH}^{-1}\hat{V}_+.\end{aligned}$$ In the above expression, $\hat{H}_{NH}=\hat{H}_0-\frac{i}{2}\sum_j\hat{L}_j^{\dag}\hat{L}_j$ is a non-Hermitian Hamiltonian, and its inverted matrix can be written as $\hat{H}_{ NH}^{-1}=\hat{H}_{ NH_1}^{-1}+\hat{H}_{ NH_2}^{-1}+\hat{H}_{ NH_3}^{-1}$, explicitly $$\begin{aligned}
\label{H8}
\hat{H}_{ NH_1}^{-1}&=&\frac{g^2-3\delta\Delta^{'}}{9g^2\Delta^{'}-3\delta\Delta^{'2}}|X_1\rangle\langle X_1|
+\frac{1}{9}\bigg[\frac{8}{\Delta^{'}}-\frac{\delta}{3g^2-\delta\Delta^{'}}\bigg]\nonumber\\
&&|X_2\rangle\langle X_2|-\frac{1}{g^2-\delta\Delta^{'}}(\delta|X_3\rangle\langle X_3|+\Delta^{'}|-\rangle\langle -|)\nonumber\\
&&-\frac{\Delta^{'}}{3g^2-\delta\Delta^{'}}|+\rangle\langle+|+\bigg[\frac{2\sqrt{2}g^2}{9g^2\Delta^{'}-3\delta\Delta^{'2}}|X_2\rangle\langle X_1|\nonumber\\
&&+\frac{2\sqrt{2}g}{\sqrt{3}(3g^2-\delta\Delta^{'})}|+\rangle\langle X_1|-\frac{g}{\sqrt{3}(3g^2-\delta\Delta^{'})}\nonumber\\
&&|+\rangle\langle X_2|+\frac{g}{g^2-\delta\Delta^{'}}|-\rangle\langle X_3|+{\rm H.c.}\bigg],\end{aligned}$$ $$\begin{aligned}
\label{HHH}
\hat{H}_{ NH_2}^{-1}&=&-\frac{\delta}{2g^2-\delta\Delta^{'}}(|e_0g_L\rangle\langle e_0g_L|+|e_0g_R\rangle\langle e_0g_R|)\nonumber\\&&+\bigg[\frac{1}{\delta}-\frac{g^2}{2g^2\delta-\delta^{2}\Delta^{'}}\bigg]
(|g_Lg_L\rangle|1_L\rangle\langle g_Lg_L|\langle 1_L|\nonumber\\&&
+|g_Rg_L\rangle|1_R\rangle\langle g_Rg_L|\langle 1_R|+|g_Lg_R\rangle|1_L\rangle\langle g_Lg_R|\nonumber\\
&&\langle 1_L|+|g_Rg_R\rangle|1_R\rangle\langle g_Rg_R|\langle 1_R|)\nonumber\\&&
+\bigg\{\frac{g}{2g^2-\delta\Delta^{'}}[(|g_Lg_L\rangle|1_L\rangle+|g_Rg_L\rangle|1_R\rangle)\nonumber\\
&&\langle e_0g_L|+(|g_Lg_R\rangle|1_L\rangle+|g_Rg_R\rangle|1_R\rangle)\langle e_0g_R|]\nonumber\\
&&-\frac{g^2}{2g^2\delta-\delta^{2}\Delta^{'}}(|g_Rg_L\rangle|1_R\rangle\langle g_Lg_L|\langle 1_L|\nonumber\\
&&+|g_Rg_R\rangle|1_R\rangle\langle g_Lg_R|\langle 1_L|)+{\rm H.c.}\bigg\},\end{aligned}$$ $$\begin{aligned}
\label{HHH}
\hat{H}_{ NH_3}^{-1}&=&-\frac{\delta}{g^2-\delta\Delta^{'}}(|g_ae_L\rangle\langle g_ae_L|+
|g_ae_R\rangle\langle g_ae_R|\nonumber\\
&&+|g_Le_L\rangle\langle g_Le_L|+|g_Re_R\rangle\langle g_Re_R|)
\nonumber\\&&-\frac{\Delta^{'}}{g^2-\delta\Delta^{'}}(|g_ag_0\rangle|1_R\rangle\langle g_ag_0|\langle 1_R|\nonumber\\
&&+|g_ag_0\rangle|1_L\rangle\langle g_ag_0|\langle 1_L|+|g_Lg_0\rangle|1_R\rangle\langle g_Lg_0|\langle 1_R|\nonumber\\
&&+|g_Rg_0\rangle|1_L\rangle\langle g_Rg_0|\langle 1_L|)\nonumber\\
&&+\frac{g}{g^2-\delta\Delta^{'}}(
|g_ag_0\rangle|1_R\rangle\langle g_ae_L|\nonumber\\
&&+|g_ag_0\rangle|1_L\rangle\langle g_ae_R|+|g_Lg_0\rangle|1_R\rangle\langle g_Le_L|\nonumber\\&&+
|g_Rg_0\rangle|1_L\rangle\langle g_Re_R|+{\rm H.c.}),\end{aligned}$$ where $\Delta^{'}=\Delta-\frac{i\gamma}{2}$ and the vacuum states of cavity modes are discarded and we have adopted the notation $|X_1\rangle=\frac{1}{\sqrt{3}}(|g_Le_R\rangle+|g_Re_L\rangle+|e_0 g_a\rangle)$, $|X_2\rangle=\frac{1}{\sqrt{6}}(|g_Le_R\rangle+|g_Re_L\rangle-2|e_0 g_a\rangle)$, $|X_3\rangle=\frac{1}{\sqrt{2}}(|g_Le_R\rangle-|g_Re_L\rangle)$, and $|+\rangle=\frac{1}{\sqrt{2}}(|g_Lg_0\rangle|1_L\rangle+|g_Rg_0\rangle|1_R\rangle)$, $|-\rangle=\frac{1}{\sqrt{2}}(|g_Lg_0\rangle|1_L\rangle-|g_Rg_0\rangle|1_R\rangle)$. On the basis of Eq. (\[H7\]), we have the effective Hamiltonian as $$\begin{aligned}
\label{HHH}
\hat{H}_{\rm eff}&=&\Omega^2 {\rm Re}\bigg[\frac{\delta}{g^2-\delta^{'}\Delta^{'}}+\frac{\delta}{2g^2-\delta^{'}\Delta^{'}}\bigg](|g_ag_L\rangle\langle g_ag_L|\nonumber\\
&&+|g_ag_R\rangle\langle g_ag_R|)+\Omega^2 {\rm Re}\bigg[\frac{\delta}{g^2-\delta\Delta^{'}}\bigg](|g_Lg_L\rangle\langle g_Lg_L|\nonumber\\&&+|g_Rg_R\rangle\langle g_Rg_R|+|T_3\rangle\langle T_3|)\nonumber\\&&-\Omega^2 {\rm Re}\bigg[\frac{g^2-3\delta\Delta^{'}}{9g^2\Delta^{'}-3\delta\Delta^{'2}}\bigg]|T_1\rangle\langle T_1|\nonumber\\&&-\Omega^2 {\rm Re}\bigg[\frac{2\sqrt{2}g^2}{9g^2\Delta^{'}-3\delta\Delta^{'2}}\bigg](|T_1\rangle\langle T_2|+{\rm H.c.})\nonumber\\
&&-\Omega^2 {\rm Re}\bigg\{\frac{1}{9}\bigg[\frac{8}{\Delta^{'}}-\frac{\delta}{3g^2-\delta\Delta^{'}}\bigg]\bigg\}|T_2\rangle\langle T_2|+\hat{H}_g,\end{aligned}$$ where $
|T_1\rangle=\frac{1}{\sqrt{3}}(|g_Lg_R\rangle+|g_Rg_L\rangle+|g_ag_0\rangle)
$ is the desired three-dimensional entangled state and $|T_2\rangle=\frac{1}{\sqrt{6}}(|g_Lg_R\rangle+|g_Rg_L\rangle-2|g_ag_0\rangle)$, $|T_3\rangle=\frac{1}{\sqrt{2}}(|g_Lg_R\rangle-|g_Rg_L\rangle)$. The effective Lindblad operators induced by the spontaneous emission take the form of $$\begin{aligned}
\label{HHH}
\hat{L}_{\rm eff}^{\gamma_{1,g_{L(a,R)}}}&=&\frac{\Omega\sqrt{\gamma}}{\sqrt{3}}\bigg\{\frac{1}{\sqrt{3}}|g_{L(a,R)}
g_0\rangle\bigg[\bigg(\frac{g^2-3\delta\Delta^{'}}{9g^2\Delta^{'}-3\delta\Delta^{'2}}
\nonumber\\
&&-\frac{4g^2}{9g^2\Delta^{'}-3\delta\Delta^{'2}}\bigg)\langle T_1|+\frac{\sqrt{2}}{\sqrt{3}}\bigg(\frac{2g^2}{9g^2\Delta^{'}-3\delta\Delta^{'2}}
\nonumber\\
&&-\frac{8}{9\Delta^{'}}+\frac{\delta}{3g^2-\delta\Delta^{'}}\bigg)\langle T_2|\bigg]-\frac{\delta}{2g^2-\delta\Delta^{'}}\nonumber\\
&&(|g_{L(a,R)}g_L\rangle\langle g_ag_L|+|g_{L(a,R)}g_R\rangle\langle g_ag_R|)\bigg\},\end{aligned}$$ $$\begin{aligned}
\label{HHH}
\hat{L}_{\rm eff}^{\gamma_{2,g_{L(R)}}}&=&\frac{\Omega\sqrt{\gamma}}{\sqrt{2}}
\bigg\{\frac{1}{\sqrt{3}}|g_{R(L)}g_{L(R)}\rangle\bigg[\bigg(\frac{g^2-3\delta\Delta^{'}}{9g^2\Delta^{'}-3\delta\Delta^{'2}}
\nonumber\\
&&+\frac{2g^2}{9g^2\Delta^{'}-3\delta\Delta^{'2}}\bigg)\langle T_1|+\frac{\sqrt{2}}{\sqrt{3}}\bigg(\frac{2g^2}{9g^2\Delta^{'}-3\delta\Delta^{'2}}
\nonumber\\
&&+\frac{8}{9\Delta^{'}}-\frac{\delta}{3g^2-\delta\Delta^{'}}\bigg)\langle T_2|\bigg]-\frac{\delta}{g^2-\delta\Delta^{'}}\nonumber\\
&&(|g_ag_{L(R)}\rangle\langle g_ag_{L(R)}|+|g_{L(R)}g_{L(R)}\rangle\langle g_{L(R)}g_{L(R)}|\nonumber\\&&\mp\frac{1}{\sqrt{2}}|g_{R(L)}g_{L(R)}\rangle\langle T_3|)\bigg\},\end{aligned}$$ $$\begin{aligned}
\label{HHH}
\hat{L}_{\rm eff}^{\gamma_{2,g_{0}}}&=&\frac{\Omega\sqrt{\gamma}}{\sqrt{2}}\bigg\{\frac{1}{\sqrt{3}}(|g_Rg_0\rangle+|g_Lg_0\rangle)
\bigg[\bigg(\frac{g^2-3\delta\Delta^{'}}{9g^2\Delta^{'}-3\delta\Delta^{'2}}
\nonumber\\
&&+\frac{2g^2}{9g^2\Delta^{'}-3\delta\Delta^{'2}}\bigg)\langle T_1|+\frac{\sqrt{2}}{\sqrt{3}}\bigg(\frac{2g^2}{9g^2\Delta^{'}-3\delta\Delta^{'2}}
\nonumber\\
&&+\frac{8}{9\Delta^{'}}-\frac{\delta}{3g^2-\delta\Delta^{'}}\bigg)\langle T_2|\bigg]-\frac{\delta}{g^2-\delta\Delta^{'}}\big[|g_ag_0\rangle\nonumber\\
&&(\langle g_ag_L|+\langle g_ag_R|)+|g_Lg_0\rangle\langle g_Lg_L|\nonumber\\&&+|g_Rg_0\rangle\langle g_Rg_R|-\frac{1}{\sqrt{2}}(|g_Rg_0\rangle\nonumber\\
&&-|g_Lg_0\rangle)\langle T_3|\big]\bigg\}.\end{aligned}$$ In order to have a compact form for the above expression, we have employed $|g_Lg_R\rangle$, $|g_Rg_L\rangle$, and $|g_ag_0\rangle$ to represent the [*ket*]{}s instead of $|T_{1(2,3)}\rangle$. It is worth noting that if we set the cavity detuning $\delta$ from two photon resonance meeting the requirements $\delta=g^2/\Delta$, $\Delta\gg \gamma$, the other decay rates approximately equal to zero except the following dominant parts
$$\begin{aligned}
\label{H15}
\hat{L}_{\rm eff}^{\gamma_{2,g_{L(R)}}}&=&\frac{\sqrt{\gamma}}{\sqrt{2}}\frac{g_{\rm eff}}{\delta\gamma
/(2\Delta)}\bigg[\bigg(\frac{1}{2}|T_3\rangle\mp\frac{1}{\sqrt{6}}
|T_1\rangle\mp\frac{1}{2\sqrt{3}}
|T_2\rangle\bigg)\nonumber\\
&&\langle T_3|+|g_{L(R)}g_{L(R)}\rangle\langle g_{L(R)}g_{L(R)}|\nonumber\\&&+|g_ag_{L(R)}\rangle\langle g_ag_{L(R)}|\bigg],\end{aligned}$$
$$\begin{aligned}
\label{H16}
\hat{L}_{\rm eff}^{\gamma_{2,g_{0}}}&=&\frac{\sqrt{\gamma}}{\sqrt{2}}\frac{g_{\rm eff}}{\delta\gamma
/(2\Delta)}\bigg[\bigg(\frac{1}{\sqrt{3}}
|T_1\rangle-\frac{2}{\sqrt{6}}|T_2\rangle\bigg)(\langle g_ag_L|\nonumber\\
&&+\langle g_ag_R|)+|g_Lg_0\rangle\langle g_Lg_L|+|g_Rg_0\rangle\langle g_Rg_R|\nonumber\\&&-\frac{1}{\sqrt{2}}(|g_Rg_0\rangle-|g_Lg_0\rangle)\langle T_3|\bigg].\end{aligned}$$
where $g_{\rm eff}=g\Omega/\Delta$. The application of microwave fields is crucial to our scheme, because it guarantees $|T_1\rangle$ remains the dark state while other ground states are coupled to each other. Therefore, the three-dimensional entangled state $|T_1\rangle$ is able to be achieved from an arbitrary initial state via the effective dissipation induced by spontaneous emission. In the left panel of Fig. \[p1\], we plot the fidelities $F(|T_1\rangle,\hat{\rho})=\langle T_1|\hat{\rho}|T_1\rangle$ for creation of $|T_1\rangle$ with the full and the effective master equations, from which we see that under the given parameters the full and the effective dynamics of the system are in excellent agreement. In the right panel, we further optimize the parameters to make the entangled state reach stable in a shorter time.
Generalization to high-dimensional entangled state
==================================================
The successful use of dissipation to deterministic creation of three-dimensional entangled state mainly relies on the effective level structure of atoms, i.e. we require transitions from a common excited (ground) state of first (second) atom to two ground (excited) states coupled by two orthogonal cavity modes, while other transitions are driven by off-resonance optical lasers. Thus it is possible to generalize our model to prepare high-dimensional entangled state if we design the atomic energy-level diagram following the similar rules. In Fig. \[px\], we suppose two potential multi-level atoms strongly interact with a multi-mode optical cavity, which is a direct extension of Fig. \[p0\]. By introducing microwave fields that drive the transitions $|g_0\rangle\leftrightarrow|g_{i}\rangle$ where $i=1,\cdots,N-1$, an $N$-dimensional entangled state $1/\sqrt{N}(|g_ag_a\rangle+|g_1g_1\rangle+|g_2g_2\rangle+\cdots+|g_{N-1}g_{N-1}\rangle)$ will be carried out via spontaneous emission. In confirmation of our assumption, we numerically simulation of the fidelity for generating the four-dimensional entangled state with the full master equation in the left panel of Fig. \[py\]. Compared with the case of three-dimensional entangled state, a longer time is needed to stabilize the target state above the fidelity $90\%$. Hence it is not difficult to conclude that the increase of dimension is at the cost of convergence time.
Now we briefly discuss the effect of cavity decay on the performance for entanglement preparation. In the right panel of Fig. \[py\], we plot the fidelity by numerically solving the full master equation of Eq. (\[H1\]) incorporating $\kappa$, three curves correspond to different parameters of dissipation, i.e. $\kappa=\gamma=0.05g$, $\kappa=\gamma=0.1g$ and $\kappa=\gamma/2=0.1g$. The decrease of population for $|T_1\rangle$ undoubtedly accompanied by a increase of population for other state. As the system approach to equilibrium, we will obtain a steady-mixed entanglement state. For certain cavity setup, the coupling strength between atom and cavity $g$, the cavity leakage rate $\kappa$, and the spontaneous emission rate $\gamma$ are fixed, thus we are allowed to modulate other parameters to achieve a three-dimensional entangled state with a relatively high fidelity. Fig \[pz\] illustrates the evolution of fidelity versus time with cavity parameters extracted from a recent experiment $(g, \kappa, \gamma)\sim 2\pi\times(750,2.62,3.5)$MHz [@spi]. A selection of $\Omega=0.02g,\omega=0.4\Omega,\Delta=g$ will lead to a fidelity about $98\%$, which overwhelms with the value based on the unitary dynamics [@ye; @lu; @li].
Conclusion
==========
In conclusion, we have achieved a stationary three-dimensional entangled state via using the dissipation caused by spontaneous emission of atoms. The numerical simulation reveals the theory for effective operator agrees well with the full master equation under given parameters. This proposal is then extended to realize the $N$-dimensional entangled state in theory by considering two multi-level atoms interacting with a multi-mode cavity, which is confirmed by the simulation of implementing a four-dimensional entangled state. The cavity decay plays a negative role on the state preparation, thus corresponding to different experimental situations, we need to regulate the Rabi frequencies of both optical and microwave fields accurately so as to obtain a relatively high fidelity. We believe that our work will be useful for the experimental realization of quantum information in the near future.
[**[ACKNOWLEDGMENT]{}**]{}
This work is supported by Fundamental Research Funds for the Central Universities under Grant No. 12SSXM001, National Natural Science Foundation of China under Grant Nos. 11204028 and 11175044, and National Research Foundation and Ministry of Education, Singapore (Grant No. WBS: R-710-000-008-271). X. Q. Shao was also supported in part by the Government of China through CSC.
[999]{} C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A [**54**]{}, 3824-3851 (1996). J. Chiaverini, D. Leibfried, T. Schaetz, M. D. Barrett, R. B. Blakestad, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, R. Ozeri , and D. J. Wineland, Nature [**432**]{}, 602-605 (2004). R. L. Kosut, A. Shabani, and D. A. Lidar, Phys. Rev. Lett. [**100**]{}, 020502 (2008). O. Moussa, J. Baugh, C. A. Ryan, and R. Laflamme, Phys. Rev. Lett. [**107**]{}, 160501 (2011). M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Nature [**482**]{}, 382-385 (2013).
D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett. [**81**]{}, 2594-2597 (1998). A. Beige, D. Braun, B. Tregenna, and Peter L. Knight, Phys. Rev. Lett. [**85**]{}, 1762-1765 (2000). J. Kempe, D. Bacon, D. A. Lidar, and K. B. Whaley, Phys. Rev. A [**63**]{}, 042307 (2001). Qiong Chen and Mang Feng, Phys. Rev. A [**82**]{}, 052329 (2010). D. A. Pushin, M. G. Huber, M. Arif, and D. G. Cory, Phys. Rev. Lett. [**107**]{}, 150401 (2011).
S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and P. Zoller, Nature Physics [**4**]{}, 878-883 (2008). F. Verstraete, M. M. Wolf and J. I. Cirac, Nature Physics [**5**]{}, 633-636 (2009). G. Vacanti and A. Beige, New J. Phys. [**11**]{}, 083008 (2009). M. J. Kastoryano, F. Reiter, and A. S. S[ø]{}rensen, Phys. Rev. Lett. [**106**]{}, 090502 (2011). J. Busch, S. De, S. S. Ivanov, B. T. Torosov, T. P. Spiller, and A. Beige, Phys. Rev. A [**84**]{}, 022316 (2011). L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, Phys. Rev. A [**84**]{}, 064302 (2011).
E. G. D. Torre, J. Otterbach, E. Demler, V. Vuletic, and M. D. Lukin, Phys. Rev. Lett. [**110**]{}, 120402 (2013). D. D. B. Rao and K. M[ø]{}lmer, Phys. Rev. Lett. [**111**]{}, 033606 (2013). Y. Lin, J. P. Gaebler, F. Reiter, T. R. Tan, R. Bowler, A. S. S[ø]{}ensen, D. Leibfried, and D. J. Wineland, Nature, doi:10.1038/nature12801.
A. Mair, A. Vaziri, G.Weihs, and A. Zeilinger, Nature [**412**]{}, 313 (2001). A. Vaziri, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. [**89**]{}, 240401 (2002). X. B. Zou, K. Pahlke, and W. Mathis, Phys. Rev. A [**67**]{}, 044301 (2003).
S. Y. Ye, Z. R. Zhong,and S. B. Zheng, Phys. Rev. A [**77**]{}, 014303 (2008). X. Y. Lü, J. B. Liu, C. L. Ding, and J. H. Li, Phys. Rev. A [**78**]{}, 032305 (2008). W. A. Li and G. Y. Huang, Phys. Rev. A [**83**]{}, 022322 (2011).
A. R. R. Carvalho and J. J. Hope, Phys. Rev. A [**76**]{}, 010301(R) (2007). A. R. R. Carvalho, A. J. S. Reid, and J. J. Hope, Phys. Rev. A [**78**]{}, 012334 (2008). R. N. Stevenson, J. J. Hope, and A. R. R. Carvalho, Phys. Rev. A [**84**]{}, 022332 (2011). X. Q. Shao, T. Y. Zheng, and Shou Zhang, Phys. Rev. A [**85**]{}, 042308 (2012).
F. Reiter and A. S. S[ø]{}rensen, Phys. Rev. A [**85**]{}, 032111 (2012). S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, Phys. Rev. A [**71**]{}, 013817 (2005).
[^1]: E-mail: xqshao@yahoo.com
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---
abstract: 'We conjecture a [*universal upper bound*]{} to the entropy of a [*rotating*]{} system. The entropy bound follows from application of the generalized second law of thermodynamics to an idealized gedanken experiment in which an entropy-bearing rotating system falls into a black hole. This bound is [*stronger*]{} than the Bekenstein entropy bound for non-rotating systems.'
address: 'The Racah Institute for Physics, The Hebrew University, Jerusalem 91904, Israel'
author:
- Shahar Hod
title: Universal Entropy Bound for Rotating Systems
---
One of the most intriguing features of both the classical and quantum theory of black-holes is the striking analogy between the laws of black-hole physics and the universal laws of thermodynamics [@Chris; @ChrisRuf; @Haw1; @Haw2; @Car; @BarCarHaw]. In particular, Hawking’s (classical) theorem [@Haw1], “The surface area of a black hole never decreases,” is a property reminiscent of the entropy of a closed system. This striking analogy had led Bekenstein [@Beken1; @Beken2; @Beken3] to conjecture that the area of a black hole (in suitable units) may be regarded as the black-hole entropy – entropy in the sense of information about the black-hole interior inaccessible to observers outside the black hole. This conjecture is logically related to a second conjecture, known as [*the generalized second law of thermodynamics*]{} (GSL): “[*The sum of the black-hole entropy*]{} (now known to be $1 \over 4$ of the horizon’s surface area) [*and the common (ordinary) entropy in the black-hole exterior never decreases*]{}”.
The general belief in the validity of the ordinary second law of thermodynamics rests mainly on the repeated failure over the years of attempts to violate it. There currently exists no general proof of the law based on the known microscopic laws of physics. In the analog case of the GSL considerably less is known since the fundamental microscopic laws of physics, namely, the laws of quantum gravity are not yet known. Hence, one is forced to consider gedanken experiments in order to test the validity of the GSL. Such experiments are important since the validity of the GSL underlies the relationship between black-hole physics and thermodynamics. If the GSL is valid, then it is very plausible that the laws of black-hole physics are simply the ordinary laws of thermodynamics applied to a self-gravitating quantum system. This conclusion, if true, would provide a striking demonstration of the [*unity*]{} of physics. Thus, it is of considerable interest to test the validity of the GSL in various gedanken experiments.
In a [*classical*]{} context, a basic physical mechanism is known by which a violation of the GSL can be achieved: Consider a box filled with matter of proper energy $E$ and entropy $S$ which is dropped into a black hole. The energy delivered to the black hole can be arbitrarily [*red-shifted*]{} by letting the assimilation point approach the black-hole horizon. As shown by Bekenstein [@Beken3; @Beken4], if the box is deposited with no radial momentum a proper distance $R$ above the horizon, and then allowed to fall in such that
$$\label{Eq1}
R < \hbar S/2 \pi E\ ,$$
then the black-hole area increase (or equivalently, the increase in black-hole entropy) is not large enough to compensate for the decrease of $S$ in common (ordinary) entropy. Bekenstein has proposed a resolution of this apparent violation of the GSL which is based on the [*quantum*]{} nature of the matter dropped into the black hole. He has proposed the existence of a universal upper bound on the entropy $S$ of any system of total energy $E$ and maximal radius $R$ [@Beken5]:
$$\label{Eq2}
S \leq 2\pi RE/\hbar\ .$$
It has been argued [@Beken5; @Beken6; @Beken7], and disputed [@UnWal; @PelWal] that this restriction is [*necessary*]{} for enforcement of the GSL; the box’s entropy disappears but an increase in black-hole entropy occurs which ensures that the GSL is respected provided $S$ is bounded as in Eq. (\[Eq2\]). Other derivations of the universal bound Eq. (\[Eq2\]) which are based on black-hole physics have been given by Zaslavskii[@Zas1; @Zas2; @Zas3] and by Li and Liu [@LiLiu]. Few pieces of evidence exist concerning the validity of the bound for self-gravitating systems [@Zas1; @Sorkin; @Zurek]. However, the universal bound Eq. (\[Eq2\]) is known to be true independently of black-hole physics for a variety of systems in which gravity is negligible [@Beken8; @BekenSchi; @SchiBeken; @BekenGuen]. In particular, Schiffer and Bekenstein [@SchiBeken] had provided an analytic proof of the bound for free scalar, electromagnetic and massless spinor fields enclosed in boxes of arbitrary shape and topology.
In this paper we test the validity of the GSL in an idealized gedanken experiment in which an entropy-bearing [*rotating*]{} system falls into a stationary black hole. We argue that while the bound Eq. (\[Eq2\]) may be a necessary condition for the fulfillment of the GSL, it may [*not*]{} be a sufficient one.
It is not difficult to see why a [*stronger*]{} upper bound must exist for the entropy $S$ of an arbitrary system with energy $E$, intrinsic angular momentum $s$ and (maximal) radius $R$: The [*gravitational spin-orbit interaction*]{} [@Wald] (the analog of the more familiar electromagnetic spin-orbit interaction) experienced by the spinning body (which, of course, was not relevant in the above mentioned gedanken experiment) can decrease the energy delivered to the black hole. This would [*decrease*]{} the change in black-hole entropy (area). Hence, the GSL will be violated unless the spinning-system entropy (what disappears from the black-hole exterior) is restricted by a bound [*stronger*]{} than Eq. (\[Eq2\]).
Furthermore, there is one disturbing feature of the universal bound Eq. (\[Eq2\]). As was pointed out by Bekenstein [@Beken5], Kerr black holes conform to the bound; however, only the Schwarzschild hole actually [*attains*]{} the bound. This uniqueness of the Schwarzschild black hole (in the sense that it is the [*only*]{} black hole which have the maximum entropy allowed by quantum theory and general relativity) among the electrically neutral Kerr-family solutions is somewhat disturbing. Clearly, the unity of physics demands a stronger bound for [*rotating*]{} systems in general, and for black holes in particular (see also [@HBML]).
In fact, the plausible existence of an upper bound stronger than Eq. (\[Eq2\]) on the entropy of a rotating system has nothing to do with black-hole physics. Classically, entropy is a measure of the phase space available to the system in question. Consider a system whose energy is no more than $E$. The limitation imposed on $E$ amounts to a limitation on the momentum space available to the system’s components (provided the potential energy is bounded from below). Now, if part of the system’s energy is in the form of a coherent (global) kinetic energy (in contrast to random motion of its constituents), then the momentum space available to the system’s components is [*further*]{} limited (part of the energy of the system is irrelevant for the system’s statistical properties). If the system has a finite dimension in space, then its phase space is limited. This amounts to an upper bound on its entropy. This bound evidently [*decreases*]{} with the absolute value of the intrinsic angular momentum of the system. However, our simple argument cannot yield the exact dependence of the entropy bound on the system’s parameters: its energy, intrinsic angular momentum (spin), and proper radius.
In fact, black-hole physics (more precisely, the GSL) provides a concrete universal upper bound for rotating systems. We consider a spinning body of rest mass $\mu$, (intrinsic) spin $s$ and proper cylindrical radius $R$, which is descending into a black hole. We consider plane (equatorial) motions of the body in a Kerr-Newman background [@MTW], with the (intrinsic) spin orthogonal to the plane (the general motion of a spinning particle in a Kerr-Newman background is very complicated, and has not been analyzed so far). The black-hole (event and inner) horizons are located at
$$\label{Eq3}
r \pm =M \pm (M^2-Q^2-a^2)^{1/2}\ ,$$
where $M$, $Q$ and $a$ are the mass, charge and angular-momentum per unit mass of the hole, respectively (we use gravitational units in which $G=c=1$). The test particle approximation implies $|s| / (\mu r_+) \ll 1$.
The equation of motion of a spinning body in the equatorial plane of a Kerr-Newman background is a quadratic equation for the conserved energy (energy-at-infinity) $E$ of the body [@Hojman]
$$\label{Eq4}
\tilde \alpha E^2-2 \tilde \beta E + \tilde \gamma=0\ ,$$
where the expression for $\tilde \alpha, \tilde \beta$ and $\tilde \gamma$ are given in [@Hojman].
The actual role of buoyancy forces in the context of the GSL is controversial (see e.g., [@Beken6; @Beken7; @UnWal; @PelWal]). Bekenstein [@Beken7] has recently shown that buoyancy protects the GSL, provided the floating point (see [@UnWal; @Beken6; @Beken7]) is close to the black-hole horizon. In addition, Bekenstein [@Beken7] has proved that one can derive the universal entropy bound Eq. (\[Eq2\]) from the GSL when the floating point is near the horizon (this is the relevant physical situation for macroscopic and mesoscopic objects with a moderate number of species in the radiation, which seems to be the case in our world). The entropy bound Eq. (\[Eq2\]) is also a [*sufficient*]{} condition for the validity of the GSL. For simplicity, and in the spirit of the original analysis of Bekenstein [@Beken5], we neglect buoyancy contribution to the energy bookkeeping of the body. As in the case of non rotating systems [@Beken7] we expect this not to effect the final entropy bound.
The gradual approach to the black hole must stop when the proper distance from the body’s center of mass to the black-hole horizon equals $R$, the body’s radius. Thus, in order to find the change in black-hole surface area caused by an assimilation of the spinning body, one should first solve Eq. (\[Eq4\]) for $E$ and then evaluate it at the point of capture $r=r_{+}+ \delta (R)$, where $\delta(R)$ is determined by
$$\label{Eq5}
\int_{r_{+}}^{r_{+}+ \delta (R)} (g_{rr})^{1/2} dr = R\ ,$$
with $g_{rr}=(r^2+a^2 cos^2 \theta) \Delta^{-1}$, and $\Delta =(r-r_-)(r-r_+)$. Integrating Eq. (\[Eq5\]) one finds (for $\theta = \pi / 2$ and $R \ll r_{+}$)
$$\label{Eq6}
\delta (R)=(r_{+}-r_{-}) {R^2 \over {4{r^2_+}}}\ .$$
Thus, the conserved energy $E$ of a body having a radial turning point at $r=r_{+}+ \delta (R)$ [@note1] is
$$\label{Eq7}
E = {{aJ} \over \alpha} - {{Js(r_+ - r_-)r_+}
\over {2 \mu \alpha^2}} + {{R (r_+-r_-)} \over {2 \alpha}} \sqrt{\mu^2
+J^2 {r_+^2 \over \alpha^2}}\ ,$$
where the “rationalized area” $\alpha$ is related to the black hole surface area $A$ by $\alpha = A/4 \pi$, and $J$ is the body’s total angular momentum. The second term on the r.h.s. of Eq. (\[Eq7\]) represents the above mentioned gravitational [*spin-orbit*]{} interaction between the orbital angular momentum of the body and its intrinsic angular momentum (spin).
An assimilation of the spinning body by the black hole results in a change $dM=E$ in the black-hole mass and a change $dL=J$ in its angular momentum. Using the first-law of black hole thermodynamics [@BarCarHaw]
$$\label{Eq8}
dM={\kappa \over {8\pi}} dA + \Omega dL\ ,$$
where $\kappa=(r_{+}-r_{-})/2\alpha$ and $\Omega=a/ \alpha$ are the surface gravity ($2\pi$ times the Hawking temperature [@Haw3]) and rotational angular frequency of the black hole, respectively, we find
$$\label{Eq9}
d\alpha =-{{2Jsr_+} \over {\mu \alpha}} +2R\sqrt{\mu^2
+J^2 {r_+^2 \over \alpha^2}}\ .$$
The increase in black-hole surface area Eq. (\[Eq9\]) can be [*minimized*]{} if the total angular momentum of the body is given by
$$\label{Eq10}
J=J^* \equiv {{s \alpha} \over {{R r_+}
\sqrt {1 -\left({s \over {\mu R}} \right)^2}}}\ .$$
For this value of $J$ the area increase is
$$\label{Eq11}
(\Delta A)_{min}=8 \pi \mu R \sqrt {1 -\left ({s
\over {\mu R}} \right)^2}\ ,$$
which is the minimal increase in black-hole surface area caused by an assimilation of a spinning body with given parameters $\mu, s$ and $R$. Obviously, a minimum exists only for $s \leq \mu R$. Otherwise, $\Delta A$ can be made (arbitrarily) negative, violating the GSL. Moller’s well-known theorem [@Moller] therefore protects the GSL.
Arguing from the GSL, we derive an [*upper bound*]{} to the entropy $S$ of an arbitrary system of proper energy $E$, intrinsic angular momentum $s$ and proper radius $R$:
$$\label{Eq12}
S \leq 2 \pi \sqrt {(RE)^2 -s^2}/\hbar \ .$$
It is evident from this suggestive argument that in order for the GSL to be satisfied $[(\Delta S)_{tot} \equiv (\Delta S)_{bh} -S \geq 0]$, the entropy $S$ of the rotating system should be bounded as in Eq. (\[Eq12\]). This upper bound is [*universal*]{} in the sense that it depends only on the [*system’s*]{} parameters (it is [*independent*]{} of the black-hole parameters which were used to suggest it).
It is in order to emphasize an important assumption made in obtaining the upper bound Eq. (\[Eq12\]); We have not taken into account [*second*]{}-order interactions between the particle’s angular momentum and the black hole, which are expected to be of order $O(J^2/M^3)$. Taking cognizance of Eq. (\[Eq10\]) we learn that this approximation is justified for rotating systems with negligible self gravity, i.e., rotating systems with $\mu \ll R$.
Although our derivation of the entropy bound is valid only for rotating systems with negligible self-gravity, we [*conjecture*]{} that it might be applicable also for strongly gravitating systems; A positive evidence for the validity of the bound is the fact that any Kerr black hole saturates it, provided the effective radius $R$ is properly defined for the black hole: consider an electrically neutral Kerr black hole. Let its energy and angular momentum be $E=M$ and $s=Ma$, respectively. The black-hole entropy $S_{BH}=A/4\hbar=\pi(r^2_++a^2)/\hbar$ exactly saturates the entropy bound provided one identifies the [*effective*]{} radius $R$ with $(r^2_++a^2)^{1/2}$, where $r_+=M+(M^2-a^2)^{1/2}$ is the radial Boyer-Lindquist coordinate for the Kerr black-hole horizon. The identification may be reasonable because $4\pi (r^2_++a^2)$ is exactly the black-hole surface area.
Evidently, systems with negligible self-gravity (the rotating system in our gedanken experiment) and systems with maximal gravitational effects (i.e., rotating black holes) both satisfy the upper bound Eq. (\[Eq12\]). Thus, this bound appears to be of universal validity. The intriguing feature of our derivation is that it uses a law whose very meaning stems from gravitation (the GSL, or equivalently the area-entropy relation for black holes) to derive a universal bound which has [*nothing*]{} to do with gravitation \[written out fully, the bound Eq. (\[Eq12\]) would involve $\hbar$ and $c$, but [*not*]{} $G$\]. This provides a striking illustration of the [*unity*]{} of physics.
In summary, an application the generalized second law of thermodynamics to an idealized gedanken experiment in which an entropy-bearing rotating system falls into a black hole, enables us to conjecture an [*improved upper bound*]{} to the entropy of a [*rotating*]{} system. The bound is stronger than Bekenstein’s bound for non-rotating systems. Moreover, this bound seems to be remarkable from a black-hole physics point of view: provided the effective radius $R$ is properly defined, [*all*]{} Kerr black holes [*saturate*]{} it (although we emphasize again that our specific derivation of the bound is consistent only for systems with [*negligible*]{} self gravity). This suggests that the Schwarzschild black hole is [*not*]{} unique from a black-hole entropy point of view, removing the disturbing feature of the entropy bound Eq. (\[Eq2\]). Thus, [*all*]{} electrically neutral black holes seem to have the [*maximum*]{} entropy allowed by quantum theory and general relativity. This provides a striking illustration of the [*extreme*]{} character displayed by (all) black holes, which is, however, still [*within*]{} the boundaries of more mundane physics.
[**ACKNOWLEDGMENTS**]{}
I thank Jacob D. Bekenstein for helpful discussions. This research was supported by a grant from the Israel Science Foundation.
[99]{}
D. Christodoulou, Phys. Rev. Lett. [**25**]{}, 1596 (1970).
D. Christodoulou and R. Ruffini, Phys. Rev. D [**4**]{}, 3552 (1971).
S. W. Hawking, Phys. Rev. Lett. [**26**]{}, 1344 (1971).
S. W. Hawking, Commun. Math. Phys. [**25**]{}, 152 (1972).
B. Carter, Nat. Phys. Sci. [**238**]{}, 71 (1972).
J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. [**31**]{}, 161 (1973).
J. D. Bekenstein, Ph.D. thesis, Princeton University,1972 (unpublished).
J. D. Bekenstein, Lett. Nuov. Cim. [**4**]{}, 737 (1972).
J. D. Bekenstein, Phys. Rev. D [**7**]{}, 2333 (1973).
J. D. Bekenstein, Phys. Rev. D [**9**]{}, 3292 (1974).
J. D. Bekenstein, Phys. Rev. D [**23**]{}, 287 (1981).
J. D. Bekenstein, Phys. Rev. D [**49**]{}, 1912 (1994).
J. D. Bekenstein, e-print gr-qc/9906058.
W. G. Unruh and R. M. Wald, Phys. Rev. D [**25**]{}, 942 (1982); W. G. Unruh and R. M. Wald, Phys. Rev. D [**27**]{}, 2271 (1983).
M. A. Pelath and R. M. Wald, e-print, gr-qc/9901032.
O. B. Zaslavskii, Phys. Lett. [**A 160**]{}, 339 (1991).
O. B. Zaslavskii, Gen. Rel. Grav. [**24**]{}, 973 (1992).
O. B. Zaslavskii, “Bekenstein entropy upper bound and laws of thermodynamics”, Kharkov University preprint, 1993.
L. X. Li and L. Liu, Phys. Rev. D [**46**]{}, 3296 (1992).
R. D. Sorkin, R. M. Wald and Z. Z. Jiu, Gen. Rel. Grav. [**13**]{}, 1127 (1981).
W. H. Zurek, and D. N. Page, Phys. Rev. D [**29**]{}, 628 (1984).
J. D. Bekenstein, Phys. Rev. D [**30**]{}, 1669 (1984).
J. D. Bekenstein and M. Schiffer, Int. J. Mod. Phys. C [**1**]{}, 355 (1990).
M. Schiffer and J. D. Bekenstein, Phys. Rev. D [**39**]{}, 1109 (1989).
J. D. Bekenstein and E. I. Guendelman, Phys. Rev. D [**35**]{}, 716 (1987).
R. Wald, Phys. Rev. D [**6**]{}, 406 (1972).
S. Hod, e-print gr-qc/9901035; S. Hod, e-print gr-qc/9903010; S. Hod, Phys. Rev. D (1999), to be published; J. D. Bekenstein and A. E. Mayo, e-print gr-qc/9903002, Phys. Rev. D (1999), to be published; B. Linet, e-print gr-qc/9903064; B. Linet, e-print gr-qc/9904044.
C. W. Misner, K. S. Thorne and J. A. Wheeler, [*Gravitation*]{} (Freeman, San Francisco, 1973).
R. Hojman and S. Hojman, Phys. Rev. D [**15**]{}, 2724 (1977).
As our task is to test the validity of the GSL in the most ’dangerous’ situation \[with $(\Delta S)_{bh}$ as small as possible\] we consider the case of a spinning body which is captured from a radial turning point of its motion. This minimize the increase in black hole surface area, and thus, allows one to derive the strongest bounds on the various physical quantities.
S. W. Hawking, Nature [**248**]{}, 30 (1974); Commun. Math. Phys. [**43**]{}, 199 (1975).
C. Moller, Commun. Dublin Inst. Advan. Stud. A [**5**]{}, 1 (1949).
|
SLAC-PUB-11380\
hep-ph/0508047\
August 2005\
**A precise sum rule among four $B\to K\pi$ CP asymmetries [^1]**
Michael Gronau
*Stanford Linear Accelerator Center*
*Stanford University, Stanford, CA 94309*
and
*Physics Department, Technion*
*Haifa 32000, Israel*
> A sum rule relation is proposed for direct CP asymmetries in $B\to K\pi$ decays. Leading terms are identical in the isospin symmetry limit, while subleading terms are equal in the flavor SU(3) and heavy quark limits. The sum rule predicts $A_{\rm CP}(B^0\to K^0\pi^0) = -0.17
> \pm 0.06$ using current asymmetry measurements for the other three $B\to K\pi$ decays. A violation of the sum rule would be evidence for New Physics in $b\to s\bar q q$ transitions.
CP asymmetry measurements in neutral $B$ decays involving an interference between $B^0$–$\bar B^0$ mixing and $b\to c\bar c s$ or $b\to u\bar ud$ transitions improve our knowledge of the Cabibbo-Kobayashi-Maskawa (CKM) phases $\beta\equiv
{\rm Arg}(-V^*_{cb}V_{cd}/V^*_{tb}V_{td})$ and $\alpha\equiv
{\rm Arg}(-V^*_{tb}V_{td}/V^*_{ub}V_{ud})$ beyond information obtained from all other CKM constraints [@MGSak]. While time-dependent asymmetries in $b\to s\bar q q$ transitions ($q=u,d,s$) indicate a potential deviation from $\sin 2\beta$ [@beta], the current statistical significance of the discrepancy is insufficient for claiming a serious anomaly.
Extraction of the weak phase $\gamma\equiv {\rm Arg}
(-V^*_{ub}V_{ud}/V^*_{cb}V_{cd})$ from the direct CP asymmetry measured recently in $B^0\to K^+\pi^-$ [@K+pi-Ba; @K+pi-Be] is obstructed by large theoretical uncertainties in strong interaction phases. Direct CP asymmetries can provide evidence for New Physics in $B^+\to
\pi^+\pi^0$, where the Standard Model predicts a vanishing asymmetry, including tiny electroweak penguin contributions [@GPY]. Other tests based on direct asymmetries, which require studying carefully U-spin symmetry breaking effects, are provided by pairs of processes, e.g. $B^+\to K^0\pi^+$ and $B^+\to \bar{K^0}K^+$ or $B^0\to K^+\pi^-$ and $B_s\to \pi^+K^-$, in which CP asymmetries are related by U-spin symmetry interchanging $d$ and $s$ quarks [@Uspin]. Precision tests would be provided by CP asymmetry relations, in which isospin relates dominant terms in the asymmetries while flavor SU(3) relates subdominant terms. The motivation for this study is proposing such a relation among $B\to K\pi$ asymmetries, a violation of which could serve as an alternative clue for physics beyond the Standard Model in $b\to s \bar q q$ transitions.
Direct CP asymmetries in all four $B\to K\pi$ decay processes, measured by the Babar [@Babar] and Belle [@Belle] collaborations, are quoted in Table I together with their averages. (We do not quote earlier CLEO measurements which involve considerably larger errors.) One defines by convention A\_[CP]{}(B f) . A nonzero asymmetry was measured in $B^0\to K^+\pi^-$, $A_{\rm CP}=
(-12.0 \pm 1.9)\%$, where the experimental error is smallest among the four $B\to K\pi$ decays. The other three asymmetries, of which that in $B\to K^0\pi^0$ involves the largest experimental error, are consistent with zero. Table I quotes also for later use corresponding CP-averaged branching ratios in units of $10^{-6}$ [@HFAG], including a very recent Babar measurement of $\b(B^0\to K^+\pi^-)$ [@Ford].
--------------------------------------------------------------------------------------------------------
Decay mode Babar [@Babar] Belle [@Belle] Average
-------------------- ----------------------------- -------------------------------- --------------------
$B^0\to K^+ \pi^-$ $-0.133\pm 0.030 \pm 0.009$ $-0.113\pm $-0.120 \pm 0.019$
0.022\pm 0.008$
$(19.2 \pm 0.6 \pm 0.6)$ $(18.5 \pm 1.0 \pm 0.7)$ ($18.9 \pm 0.7$)
$B^+\to K^+ \pi^0$ $0.06\pm 0.06\pm 0.01$ $0.04 \pm 0.04\pm 0.02$ $0.05 \pm 0.04$
$(12.0 \pm 0.7 \pm 0.6)$ $(12.0 \pm 1.3^{+1.3}_{-0.9})$ $(12.1 \pm 0.8)$
$B^0\to K^0 \pi^0$ $-0.06\pm 0.18\pm 0.03$ $0.11 \pm 0.18\pm 0.08$ $0.02 \pm 0.13$
$(11.4 \pm 0.9 \pm 0.6)$ $(11.7 \pm 2.3^{+1.2}_{-1.3})$ $(11.5 \pm 1.0)$
$B^+\to K^0 \pi^+$ $-0.09\pm 0.05 \pm 0.01$ $0.05\pm 0.05\pm 0.01$ $-0.02\pm0.04$
$(26.0 \pm 1.3 \pm 1.0)$ $(22.0 \pm 1.9 \pm 1.1)$ $(24.1 \pm 1.3)$
--------------------------------------------------------------------------------------------------------
: CP asymmetries $A_{CP}$ for $B \to K \pi$ decays. In parentheses are corresponding branching ratios in units of $10^{-6}$. \[tab:Kpiasym\]
The purpose of this Letter is to prove a sum rule among the four $B\to K\pi$ CP rate differences, \[DSR\] (K\^+\^-) + (K\^0\^+) 2(K\^+\^0) + 2(K\^0\^0) , where we define \[Deltadef\] (Bf) (|B|f) - (Bf) . This sum rule, reminiscent of a similar sum rule among partial decay rates [@GR99; @Lipkin], is expected to hold within an accuracy of several percent. Using the approximation (see branching ratios in Table I and the discussion in the paragraph below Eq. (\[C\])), \[Gamma\] (K\^+\^-) (K\^0\^+) 2(K\^+\^0) 2(K\^0\^0) , this implies at a somewhat lower precision a sum rule among CP asymmetries, \[SR\] A\_[CP]{}(K\^+\^-) + A\_[CP]{}(K\^0\^+) A\_[CP]{}(K\^+\^0) + A\_[CP]{}(K\^0\^0) . The equality of leading terms in the sum rule (\[DSR\]) will be shown to follow from isospin symmetry, while subleading terms are equal in the flavor SU(3) and heavy quark limits. For the most part, we will make no assumption relating $B\to K\pi$ decays to $B\to\pi\pi$ decays.
A somewhat less precise relation excluding $A_{\rm CP}(K^0\pi^+)$ in (\[SR\]), which holds under more restricted conditions, was proposed recently in a broader context [@bspeng]. An oversimplified but too crude relation, \[naive\] A\_[CP]{}(K\^+\^-) \~A\_[CP]{}(K\^+\^0) , was suggested several years ago [@GR99] making too strong an assumption about color suppressed tree amplitudes. The latter relation, which is quite far from what is being measured (see Table I), has recently provoked discussions about an anomalously large color-suppressed amplitude, an enhanced electroweak penguin amplitude and possible New Physics effects [@NP].
Let us recapitulate for completeness the structure of hadronic amplitudes in charmless $B$ decays, specifying carefully our assumptions. The effective Hamiltonian governing $B\to K\pi$ decays is given by [@BBL] \[H\] [H]{}\_[eff]{} = -. where $\lambda_U = V^*_{Ub}V_{Us}, \lambda_t\equiv V^*_{tb}V_{ts}$. The dozen operators $O^U_j$ and $O_i$ are four-quark operators, with given flavor and chiral structure, including current-current operator $O^U_{1,2}$, QCD-penguin operators $O_i,
i=3-6$, and electroweak penguin (EWP) operators $O_i, i=7-10$. The real Wilson coefficients, which were calculated beyond the leading logarithmic approximation, are $c_1 \approx 1.10,~c_2 \approx -0.20,~c_{3-6}\sim
{\rm few}\times10^{-2},~c_{7,8}\sim {\rm few}\times10^{-4},
c_9 \approx -0.010$ and $c_{10} \approx 0.0020$. Contributions of $O_7$ and $O_8$ can be safely neglected, as one does not expect a huge enhancement of their hadronic matrix elements relative to those of $O_9$ and $O_{10}$. The latter operators, involving larger Wilson coefficients, have a (V-A)(V-A) structure similar to the current-current operators.
All four quark operators can be written as a sum of SU(3) representations, ${\overline {\bf 15}}$, ${\bf 6}$ and ${\bar {\bf 3}}$, into which the product ${\bar {\bf 3}}\otimes {\bf 3}\otimes {\bar {\bf 3}}$ can be decomposed [@GHLR; @GrLe]. Current-current and EWP operators which involve the same (V-A)(V-A) structure consist of identical SU(3) operators. Thus one finds simple proportionality relations between current-current (here denoted by a subscript $T$ for “tree”) and EWP operators belonging to $\overline {\bf 15}$ and ${\bf 6}$ representations [@GPY; @NR; @VP], \[15\] [H]{}\_[EWP]{}() &=& -32 \_T() ,\
\[6\] [H]{}\_[EWP]{}([**6**]{}) &=& 32 \_T([**6**]{}) . These operator relations have useful consequences in $B\to K\pi$ decay amplitudes. The first relation was applied in [@NR] and both relations were used in [@GPY]. In the following discussion we will apply SU(3) to the subdominant EWP amplitudes by using Eqs. (\[15\])–(\[6\]). Dominant terms in $B\to K\pi$ asymmetries will be shown to be related by isospin symmetry alone.
The effective Hamiltonian (\[H\]) permits a general decomposition of the four $B\to K\pi$ amplitudes into terms of distinct topologies representing hadronic matrix elements of corresponding operators in (\[H\]). Using the unitarity of the CKM matrix, $\lambda_u +\lambda_c + \lambda_t =0$, and defining $P_{tc}\equiv
P_t - P_c, P_{uc}\equiv P_u - P_c$, one has [@GHLR]: \[+-\] -A(K\^+\^-) & = & \_u(P\_[uc]{} + T) + \_t(P\_[tc]{} + P\^C\_[EW]{}) ,\
-A(K\^+\^0) & = & \_u(P\_[uc]{} + T + C + A) + \_t(P\_[tc]{} + P\_[EW]{} + P\^C\_[EW]{}) ,\
A(K\^0\^0) & = & \_u(P\_[uc]{} - C) + \_t(P\_[tc]{} - P\_[EW]{} - P\^C\_[EW]{}) ,\
\[0+\] A(K\^0\^+) & = & \_u(P\_[uc]{} + A) + \_t(P\_[tc]{} - P\^C\_[EW]{}) . The amplitudes $P_u, T, C, A$ and $P_c$ are contributions from the first sum in (\[H\]), corresponding to $U=u$ and $U=c$, respectively, while $P_t, P_{EW}$ and $P_{EW}^c$ originate from the second sum. The terms $P, T, C$ and $A$ represent penguin, color-allowed tree, color-suppressed tree and annihilation topologies, respectively. Specific EWP contributions were expressed in terms of color-allowed and color suppressed amplitudes, $P_{EW}$ and $P^C_{EW}$, using a simple substitution [@GHLR], \_u C \_u C + \_t P\_[EW]{} , \_u T \_u T + \_t P\^C\_[EW]{} , \_u P\_[uc]{} \_u P\_[uc]{} - \_t P\^C\_[EW]{} .
The four physical amplitudes can also be decomposed into three isospin amplitudes [@isospin], a contribution $B_{1/2}$ with $I(K\pi)=1/2$ from the isosinglet part of ${\cal H}_{\rm eff}$, and two amplitudes $A_{1/2,3/2}$ with $I(K\pi)
=1/2,3/2$ from the isotriplet part of ${\cal H}_{\rm eff}$: \[I-1\] -A(K\^+\^-) & = & B\_[1/2]{} - A\_[1/2]{} - A\_[3/2]{} ,\
-A(K\^+\^0) & = & B\_[1/2]{} + A\_[1/2]{} - 2A\_[3/2]{} ,\
A(K\^0\^0) & = & B\_[1/2]{} - A\_[1/2]{} + 2A\_[3/2]{} ,\
\[I-4\] A(K\^0\^+) & = & B\_[1/2]{} + A\_[1/2]{} + A\_[3/2]{} . One has \[B\] B\_ & = & \_u+\_t ,\
A\_ & = & -\_u + \_t ,\
\[A3/2\] A\_ & = & -\_u - \_t .
Eqs. (\[+-\])-(\[0+\]) are quite general, providing a common basis for QCD calculations of $B\to K\pi$ amplitudes [@BBNS; @KLS]. The terms in parentheses involve magnitudes of hadronic amplitudes and strong interaction phases, which are hard to calculate without making further assumptions. For instance, the term $P_c$ may involve sizable long distance “charming penguin” contributions which must be fitted to the data [@Charmpeng]. Our following arguments will be independent of specific hadronic calculations, relying mainly on isospin and flavor SU(3) symmetry properties of certain terms. SU(3) breaking corrections will be estimated using generalized factorization [@MN].
We will use Eqs. (\[15\]) and (\[6\]), which imply approximate SU(3) relations between $P_{EW}, P^C_{EW}$, on the one hand, and $T$ and $C$, on the other [@GPY; @NR], \[T+C\] P\_[EW]{} + P\^C\_[EW]{} & & - (T+C) ,\
\[C\] P\^C\_[EW]{} & & - C . In the second equation we used $(c_9 - c_{10})/(c_1 - c_2) \approx
(c_9 + c_{10})/(c_1 + c_2)$ [@BBL], neglecting a small exchange contribution [@E] which vanishes at leading order in $1/m_b$ and $\alpha_s$ [@SCET]. SU(3) breaking effects on (\[T+C\]), calculated by using generalized factorization [@MN], were found to be about 10$\%$ in the magnitude of ratio $(P_{EW} + P^C_{EW})/(T+C)$ and less than $5^\circ$ in its phase. Similar effects will be assumed in (\[C\]), as estimated by similar considerations.
The terms in the amplitudes (\[+-\])-(\[0+\]) multiplying $\lambda_t$ dominate the decay amplitudes because $|\lambda_u/\lambda_t|\approx
0.02$. The penguin amplitude $P_{tc}$ is pure isosinglet, thus contributing equally to the two decay amplitudes involving a charged pion and contributing a term smaller by factor $\s$ to the two amplitudes involving $\pi^0$. Dominance by $P_{tc}$ is exhibited clearly by the four $K\pi$ branching ratios in Table I which obey Eq. (\[Gamma\]) to a reasonable approximation. (The effect of a lifetime difference between $B^+$ and $B^0$ will be discussed later.) All other terms in (\[+-\])-(\[0+\]) are smaller than $\lambda_tP_{tc}$ and may be considered subdominant.
Using Eq. (\[T+C\]) and noting that $T+C$ dominates the amplitude of $B^+\to\pi^+\pi^0$ [@GHLR], the measured ratio of branching ratios $\b(\pi^+\pi^0)/\b(K^0\pi^+)$ shows that the higher order electroweak amplitude $P_{EW} + P^C_{EW}$ is indeed much smaller than $P_{tc}$ [@NR; @GLR], 0.11 , where $f_\pi$ and $f_K$ are meson decay constants, and a value $\b(\pi^+\pi^0) = (5.5\pm 0.6)\times 10^{-6}$ [@HFAG] was used.
We will not assume color suppression for $C$ and $P^C_{EW}$, nor will we assume that $P_{uc}$ is smaller than $T$ or $C$. That is, the triplet of amplitudes $(T, C, P_{uc})$ and the doublet $(P_{EW}, P^C_{EW})$ could each consist of amplitudes with comparable magnitudes. Several questions have been raised recently concerning these relative magnitudes [@Yoshi; @KpiSR; @CGRS; @BHLDS; @BF] in view of an apparent disagreement with a hierarchy assumption [@GHLR] $|C| \sim 0.2|T|,~|P^C_{EW}| \sim 0.2|P_{EW}|$ and with calculations in QCD [@BBNS; @KLS]. We will make use of the fact that the amplitude $A$ and the strong phase of $C/T$ vanish to leading order in $1/m_b$ and $\alpha_s$ [@SCET]. We note that a small value of ${\rm Arg}(C/T)$ is not favored by a global SU(3) fit to all $B$ meson decays into two charmless pseudoscalars [@CGRS], although the error on the output value of this phase is still very large. While the fit assumes common magnitudes and strong phases for SU(3) amplitudes in $B\to K\pi$ and $B\to \pi\pi$ decays, our assumption about SU(3) in Eqs. (\[T+C\])-(\[C\]) is restricted to $B\to K\pi$. As mentioned, these SU(3) breaking effects have been calculated to be very small implying $|{\rm Arg}[(P_{EW}+P^C_{EW})/(T+C)]| < 5^\circ$.
Direct CP asymmetries in $B\to K\pi$ processes occur through the interference of two terms in the amplitudes involving different CKM factors, $\lambda_t$ and $\lambda_u$, and different strong phases. Using the definition (\[Deltadef\]) we find (K\^+\^-) & = & [Im]{}I ,\
2(K\^+\^0) & = & [Im]{}I ,\
2(K\^0\^0) & = & [Im]{}I ,\
(K\^0\^+) & = & [Im]{}I , where $I=4\,{\rm Im}(\lambda_t\lambda^*_u)$ is a common CKM factor.
Combining the four CP rate differences by defining a difference $\delta_{K\pi}$ between pairs involving charged and neutral pions, \_[K]{}(K\^+\^-) + (K\^0\^+) - 2(K\^+\^0) - 2(K\^0\^0) , we find \[delta\] \_[K]{} = -[Im]{}I . All terms involving the dominant $P_{tc}$ term cancel in $\delta_{K\pi}$. This follows from isospin symmetry [@AS], as these terms describe an interference of $P_{tc}$ with a term multiplying $\lambda_u$ in a combination which vanishes by Eqs. (\[I-1\])-(\[I-4\]), \[comb\] -A(K\^+\^-) + A(K\^0\^+) + A(K\^+\^0) - A(K\^0\^0) = 0 .
All the terms on the right-hand-side of (\[delta\]) involve EWP amplitudes and are thus suppressed relative to corresponding terms involving $P_{tc}$ by about an order of magnitude. The first term vanishes in the SU(3) limit because it involves an interference of two contributions which carry a common strong phase by (\[T+C\]). A potential SU(3) breaking strong phase difference, $|{\rm Arg}[(P_{EW}+P^C_{EW})/(T+C)]| <
5^\circ$ [@MN], suppresses this term by at least an order of magnitude. The second term vanishes in the SU(3) limit at leading order in $1/m_b$ and $\alpha_s$, as can be seen by using (\[T+C\])-(\[C\]) and ${\rm Arg}(C/T)
\approx 0$. This implies a suppression of this term either by an order of magnitude from SU(3) breaking, or by $1/m_b$ and $\alpha_s$. The last term on the right-hand-side involves an interference between two subdominant amplitudes, $P_{EW} + P^C_{EW}$ and $A$, each of which is suppressed relative to corresponding leading terms, $P_{tc}$ and $T$, respectively. Since all three terms on the right-hand-side of (\[delta\]) are doubly suppressed relative to $\Delta(K^+\pi^-)$ by two factors each about an order of magnitude, we expect the ratio $\delta_{K\pi}/\Delta(K^+\pi^-)$ to be at most several percent. Therefore, one may safely take $\delta_{K\pi} =0$ which proves (\[DSR\]).
The proposed sum rule may be written in terms of CP asymmetries, taking into account differences among the four $B\to K\pi$ CP-averaged branching ratios and the $B^+$ to $B^0$ lifetime ratio $\tau_+/\tau_0 = 1.076 \pm
0.008$ [@HFAG]: \[exact\] A\_[CP]{}(K\^+\^-) & + & A\_[CP]{}(K\^0\^+)\
& = & A\_[CP]{}(K\^+\^0) + A\_[CP]{}(K\^0\^0) . Using branching ratios from Table I, we predict a negative CP asymmetry in $B^0\to K^0\pi^0$ in terms of the other asymmetries, A\_[CP]{}(K\^0\^0) = -0.17 0.06 . This value is not inconsistent with the average measured value in Table I. Alternatively, the approximate sum rule (\[SR\]) among CP asymmetries reads in terms of corresponding current measurements, (-0.120 0.019) + (-0.02 0.04) (0.05 0.04) + (0.02 0.13) . While central values on the two sides have opposite signs, errors in the asymmetries (in particular that in $B^0\to K^0\pi^0$) must be reduced before claiming a discrepancy.
The proposed sum rule (\[DSR\]) makes no assumption about the smallness of the amplitudes $P_{uc}$ and $C$ relative to $T$, or about their strong phases relative to that of the dominant $P_{tc}$ amplitude. The contribution of $P_{uc}$ to the asymmetries has been neglected in a sum rule suggested recently when studying $b\to s$ penguin amplitudes in $B$ meson decays into two pseudoscalars [@bspeng]. A $P_{uc}$ contribution comparable to $T$ would be observed by a nonzero $A_{\rm CP}(K^0\pi^+)$, unless the strong phase of $P_{uc}$ relative to $P_{tc}$ is very small. A sizable $P_{uc}$ comparable to $T$ is an output of a global SU(3) fit to $B\to K\pi$ and $B\to\pi\pi$ decays [@CGRS]. Bounds on $A_{\rm CP}(K^0\pi^+)$ derived from Table I favor a small relative phase between $P_{tc}$ and a sizable $P_{uc}$. Another output of the fit, a large amplitude $C$ comparable to $T$, also obtained in separate analyses of $B\to K\pi$ [@BHLDS] and $B\to\pi\pi$ [@BF], provides a simple interpretation for the failure of the oversimplified relation (\[naive\]) which had assumed $C$ to be color-suppressed.
To conclude, we have shown that direct CP asymmetries in the four $B\to K\pi$ decay processes obey the sum rule (\[DSR\]) within several percent, or the sum rule (\[SR\]) in the approximation of equal rates in (\[Gamma\]). Isospin and flavor SU(3) symmetries have been used to relate leading QCD penguin and subleading electroweak penguin terms in the sum rule, respectively. While we assumed a suppression of an annihilation amplitude relative to a color-allowed tree amplitude and a suppression of ${\rm Arg}(C/T)$, no assumption was made about the magnitudes of color-suppressed tree, electroweak penguin amplitudes and a term $P_{uc}$ associated with intermediate $u$ and $c$ quarks. A violation of the sum rule would provide evidence for New Physics in $b\to s\bar qq$ transitions. The most likely interpretation of the origin of a potential violation would be an anomalous $\Delta I=1$ operator in the effective Hamiltonian. A generalization of our argument using (\[comb\]) implies that in the isospin symmetry limit contributions to CP asymmetries from any $\Delta I=0$ operator cancel in the sum rule.
I wish to thank the SLAC theory group for its very kind hospitality. I am grateful to Helen Quinn, Jonathan Rosner and Denis Suprun for very useful discussions. This work was supported in part by the Department of Energy contract DE-AC02-76SF00515, by the Israel Science Foundation founded by the Israel Academy of Science and Humanities, Grant No. 1052/04, and by the German-Israeli Foundation for Scientific Research and Development, Grant No. I-781-55.14/2003.
\#1\#2\#3[Am. J. Phys. [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[Ann. Phys. (N.Y.) [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[Acta Phys. Polonica [**\#1**]{} (\#3) \#2]{} \#1\#2\#3[Ann. Rev. Nucl. Part. Sci. [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[Comments on Nucl. Part. Phys. [**\#1**]{}, \#2 (\#3)]{} 89[[*CP Violation,*]{} edited by C. Jarlskog (World Scientific, Singapore, 1989)]{} \#1\#2\#3[Electronic Conference Proceedings [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[Eur. Phys. J. C [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[ [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[Int. J. Mod. Phys. A [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[JHEP [**\#1**]{} (\#3) \#2]{} \#1\#2\#3[J. Phys. B [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[Mod. Phys. Lett. A [**\#1**]{} (\#3) \#2]{} \#1\#2\#3[Nature [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[Nuovo Cim. [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[Nucl. Instr. Meth. A [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[Nucl. Phys. B [**\#1**]{} (\#3) \#2]{} \#1\#2\#3[Nucl. Phys. Proc. Suppl. [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3\#4[Pis’ma Zh. Eksp. Teor. Fiz. [**\#1**]{}, \#2 (\#3) \[JETP Lett. [**\#1**]{}, \#4 (\#3)\]]{} \#1\#2\#3[Phys. Lett. [**\#1**]{} (\#3) \#2]{} \#1\#2\#3[Phys. Lett. A [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[Phys. Lett. B [**\#1**]{} (\#3) \#2]{} \#1\#2\#3[Phys. Rev. Lett. [**\#1**]{} (\#3) \#2]{} \#1\#2\#3[Phys. Rev. D [**\#1**]{} (\#3) \#2]{} \#1\#2\#3[Phys. Rep. [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[Prog. Theor. Phys. [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[Rev. Mod. Phys. [**\#1**]{} (\#3) \#2]{} \#1\#2\#3\#4[Yad. Fiz. [**\#1**]{}, \#2 (\#3) \[Sov. J. Nucl. Phys., \#4 (\#3)\]]{} \#1\#2\#3\#4\#5\#6[Zh. Eksp. Teor. Fiz. [**\#1**]{}, \#2 (\#3) \[Sov.Phys. - JETP [**\#4**]{}, \#5 (\#6)\]]{} \#1\#2\#3[Zeit. Phys. C [**\#1**]{}, \#2 (\#3)]{} \#1\#2\#3[Zeit. Phys. D [**\#1**]{}, \#2 (\#3)]{}
[99]{}
For two very recent reviews, see M. Gronau, invited talk presented at the Tenth International Conference on $B$ Physics at Hadron Machines (Beauty 2005), 20–24 June 2005, Assisi, Perugia, Italy, http://www.pg.infn.it/beauty2005; Y. Sakai, invited talk presented at the 25th International Symposium on Physics in Collision (PIC 2005), 6–9 July 2005, Prague, Czech Republic, http://www.particle.cz/conferences/pic2005.
For two very recent reviews, see K. Abe, invited talk presented at the 22nd International Symposium on Lepton-Photon Interactions at High Energy (LP 2005), 30 June – 7 July 2005, Uppsala, Sweden, http://lp2005.tsl.uu.se/\~lp2005; R. Bartoldus, invited talk presented at the 25th International Symposium on Physics in Collision (PIC 2005), 6–9 July 2005, Prague, Czech Republic, http://www.particle.cz/conferences/pic2005.
Babar , B. Aubert , .
Belle , Y. Chao , ; K. Abe , hep-ex/0507045.
M. Gronau, D. Pirjol and T. M. Yan, .
M. Gronau, .
Babar , B. Aubert , Ref. [@K+pi-Ba]; ; hep-ex/0503011; hep-ex/0507023.
Belle , Ref. [@K+pi-Be]; Y. Chao , ; K. Abe , hep-ex/0507037.
J. Alexander [*et al.*]{}, Heavy Flavor Averaging Group, hep-ex/0412073. Updated results and references are tabulated periodically by this group: http://www.slac.stanford.edu/xorg/hfag/rare.
Babar , reported by W. T. Ford at the 25th International Symposium on Physics in Collision (PIC 05), 6–9 July 2005, Prague, Czech Republic, http://www.particle.cz/conferences/pic2005.
M. Gronau and J. L. Rosner, .
H. J. Lipkin, .
M. Gronau and J. L. Rosner, .
M. Neubert, talk presented at the Third Workshop on the Unitarity Triangle: CKM 2005, 15–18 March 2005, San Diego, CA, http://ckm2005.ucsd.edu/; W. S. Hou, M. Nagashima and A. Soddu, hep-ph/050307; M. Beneke, talk presented at the International Conference on QCD and Hadronic Physics, 16–20 June 2005, Beijing, China, http://www.phy.pku.edu.cn/qcd/.
G. Buchalla, A. J. Buras and M. E. Lautenbacher, .
M. Gronau, O. Hernández, D. London, and J. L. Rosner, ; .
B. Grinstein and R. F. Lebed, .
M. Neubert and J. L. Rosner, .
M. Gronau, .
H. J. Lipkin, Y. Nir, H. R. Quinn and A. Snyder, ; M. Gronau, .
M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, ; M. Beneke and M. Neubert, .
Y. Y. Keum, H. N. Li and A. I. Sanda, ; Y. Y. Keum and A. I. Sanda, .
M. Ciuchini, E. Franco, G. Martinelli and L. Silvestrini, ; M. Ciuchini, R. Contino, E. Franco, G. Martinelli and L. Silvestrini, ; \[Erratum-ibid. B [**531**]{} (1998) 656\].
M. Neubert, .
Evidence for suppression by an order of magnitude of an exchange amplitude relative to a tree amplitude exists in $B^0\to D^-_s
K^+$ [@PDG]. The measured suppression factor in the charmless decay process $B^0 \to K^+ K^-$, for which a strict upper limit has been recently obtained [@Ford; @K+K-], has not yet reached this level.
S. Eidelman (Particle Data Group Collaboration), .
Belle , K. Abe , hep-ex/0506080.
C. W. Bauer and D. Pirjol, ; C. W. Bauer, D. Pirjol, I. Z. Rothstein and I. W. Stewart, .
M. Gronau, J. L. Rosner and D. London, .
T. Yoshikawa, .
M Gronau and J. L. Rosner, .
C. W. Chiang, M. Gronau, J. L. Rosner and D. Suprun, . This study uses instead of $P_{uc}$ a quantity $P_{tu} = V^*_{ub}V_{ud}(P_{tc} - P_{uc})$.
S. Baek, P. Hamel, D. London, A. Datta and D. Suprun, .
A. J. Buras, R. Fleischer, S. Recksiegel and F. Schwab, ; ; .
D. Atwood and A. Soni, . This paper mentioned a similar sum rule among CP differences of branching ratios rather than partial decay rates, using isospin symmetry and neglecting EWP contributions. I thank David Atwood and Amarjit Soni for pointing this out.
[^1]: To be published in Physics Letters B.
|
=1200 = =cmmib10 9==“0920 =”091E to 1,5truecm
**J/$\bfpsi$ AND $\bfpsi$’ SUPPRESSION IN HEAVY ION COLLISIONS**
by
[**A. Capella, A. Kaidalov**]{}[^1][Permanent address : ITEP, B. Cheremushkinskaya 25, 117259 Moscow, Russia]{}[**, A. Kouider Akil**]{}
Laboratoire de Physique Théorique et Hautes Energies [^2][Laboratoire associé au Centre National de la Recherche Scientifique - URA D0063]{}
Université de Paris XI, b timent 211, 91405 Orsay cedex, France
**C. Gerschel**
Institut de Physique Nucléaire
Université de Paris XI, b timent 100, 91406 Orsay cedex, France
${\bf Abstract}$
We study the combined effect of nuclear absorption and final state interaction with co-moving hadrons on the $J/\psi$ and $\psi '$ suppression in proton-nucleus and nucleus-nucleus collisions. We show that a reasonable description of the experimental data can be achieved with theoretically meaningful values of the cross-sections involved and without introducing any discontinuity in the $J/\psi$ or $\psi '$ survival probabilities.
to 4 truecm
LPTHE Orsay 96-55
July 1996
In 1986 Matsui and Satz \[1\] proposed $J/\psi$ suppression in heavy ion collisions as a signal of quark gluon plasma (QGP) formation. This suppression results from Debye screening in a medium of deconfined quarks and gluons. Shortly afterwards, the NA38 collaboration tested this idea and found that in $O$-$U$ and $S$-$U$ collisions the ratio $J/\psi$ over di-muon continuum decreases with increasing centrality \[2\]. However, very soon, two alternative explanations involving conventional physics - i.e. no phase transition - were proposed. In the first one, known as nuclear absorption \[3\] \[4\], the $c\bar{c}$ pair wave packet produced inside the nucleus, is modified by nuclear collisions in such a way that it does not project into $J/\psi$ but into open charm. It was shown that the absorptive cross-section $\sigma_{abs}$ needed to explain the $A$-dependence of $J/\psi$ production in $pA$ collisions also does explain the suppression found in nucleus-nucleus. This cross-section turns out to be about 6 mb. In the second explanation, known as interaction with co-movers, the $J/\psi$ produced outside the nucleus is surrounded by a dense system of hadrons (mainly pions) and converts into open charm due to interactions in the medium \[5\]. This interaction takes place at low energies - which, however, have to be large enough to overcome threshold effects. Reliable theoretical calculations of the corresponding cross-section $\sigma_{co}^{\psi}$ show that it increases very slowly with energy from threshold \[6\]. In view of that, the second interpretation has been progressively abandoned in favor of nuclear absorption \[7\].
Very recent data for $Pb$ $Pb$ collisions obtained by the NA50 collaboration show an anomalous $J/\psi$ suppression \[8\]. The ratio $J/\psi$ over Drell-Yan (DY) is two times smaller than the extrapolation of the $O$-$U$ and $SU$ data based on nuclear absorption - the statistical significance of this discrepancy being of nine standard deviations. These data provide a most exciting hint of QGP formation and some interpretations in this context have been presented at the QM’96 conference \[9\]. In them a discontinuity in the $J/\psi$ survival probability is assumed when some threshold of local energy density is reached.
In this note we present an attempt to describe the observed $J/\psi$ suppression using the combined effect of nuclear absorption and interaction with co-movers - without introducing any discontinuity in the survival probability \[10\]. For the nuclear absorption we adopt the formalism of refs. \[3, 4\]. For simplicity, we use the exponential form of the $J/\psi$ suppression given in ref. \[4\] and used in the experimental papers, namely $$S_1 = \exp (- \rho \ L \ \sigma_{abs} ) \eqno(1)$$ where $\rho = 0.138$ nucleon/fm$^3$ is the nuclear density and $L$ is the length of nuclear matter crossed by the $c\bar{c}$ pair. The more involved formalism of ref. \[3\] gives results very similar to those obtained from (1).
We turn next to the final state interaction with produced hadrons (co-movers). A rigourous treatment of this interaction is very complicated and is not available in the literature. We use the simple treatment proposed in refs. \[11\]. The decrease in the spatial density of $J/\psi$ at point $x$ due to the interaction $\psi$-$h$ is given by \[12\] $$\Delta {dN^{\psi} \over d^4x} = \rho^{\psi} (x) \ \rho^h(x)
\sigma_{co}^{\psi}
\eqno(2)$$ with $d^4x = \tau d\tau dy d^2s$, where $\tau$ is the proper time, $y$ the space-time rapidity (to be later on identified with the usual rapidity) and $d^2s$ an element of transverse area. $\sigma_{co}^{\psi}$ is the part of the $\psi$-$h$ cross-section which does not contain the $J/\psi$ in the final state, averaged over the momentum distribution of the colliding particles. The effect of thresholds will be taken into account in an effective way through the value of $\sigma_{co}^{\psi}$. Assuming longitudinal boost invariance and a dilution of the densities $\rho^{\psi}$ and $\rho^h$ of the type $1/\tau$ (i.e. neglecting transverse expansion), we get from (2) $$\left . {dN^{\psi} \over dy} \right |_{\tau_0 + \Delta \tau} = \int d^2s
{dN^{\psi}
\over dy d^2s} {dN^h \over dy d^2s} \sigma_{co}^{\psi} \ell n \left | {\tau_0 +
\tau
\over \tau_0} \right | \eqno(3)$$ where $\tau_0$ is the formation time and $\tau$ is the duration of the hadronic phase.
We want to express the densities $dN/dyd^2s$ in terms of the observables $dN/dy$. We have \[11\] $$\int d^2s {dN^{\psi} \over dy d^2s}(b) {dN^h \over dy d^2s}(b) = G(b)
{dN^{\psi}
\over dy}(b) {dN^h \over dy}(b) \eqno(4)$$ where $b$ is the impact parameter of the collision and the geometrical factor $G(b)$ is given by $$G(b) = {\int d^2s \ T_A^2(s) \ T_B^2(b -s) \over T_{AB}^2(b)} \ \ \ ,
\eqno(5)$$ which has an obvious geometrical interpretation. For the nuclear profile $T_A(b)$ we use standard Saxon-Woods. For the proton we use a Gaussian profile with $R_p = 0.6$ fm. Using (3)-(5) we have : $$\left . {dN^{\psi} \over dy} \right |_{\tau_0 + \Delta
\tau}(b) =
\left . {dN^{\psi} \over dy} \right |_{\tau_0} (b) \left [ 1 -
\sigma_{co}^{\psi} G(b)
\ell n \left | {\tau_0 + \Delta \tau \over \tau_0} \right | {dN^h \over dy}(b)
\right ]
\eqno(6)$$ and, for a finite time interval, $${dN^{\psi} \over dy}(b) = \left . {dN^{\psi} \over dy} \right |_{\tau_0}(b)
\exp
\left [ - \sigma_{co}^{\psi} G(b) \ell n \left | {\tau_0 + \tau (b) \over
\tau_0}
\right | {dN^h \over dy} (b) \right ] \equiv \left . {dN^{\psi} \over dy}
\right
|_{\tau_0}(b) \ S_2(b) \ . \eqno(7)$$ We have now to specify the duration time of the interaction $\tau (b)$. This time is not well known. We use the following ansatz \[11\]. In the case of Gaussian profiles one has $$G = {1 \over 2 \pi} \left [ {2 \over 3} {R_A^2 \cdot R_B^2 \over R_A^2 +
R_B^2}
\right ] \eqno(8)$$ where $R$ are the rms radii. In ref. \[13\] it was found that the quantity in brackets in (8) is precisely the geometrical HBT squared transverse radius. Following the arguments of \[14\] we take it as a measure of the duration of interaction and therefore use $$\tau (b) = [2 \pi G(b) ]^{-1/2} \eqno(9)$$ For the formation time $\tau_0$ we take $\tau_0$ = 1 fm \[14\]. Of course our results depend on the value of $\tau_0$. However, this dependence can, to a large extent, be compensated by a small change of $\sigma_{co}^{\psi}$. Finally $dN^h(b)/dy$ is a measurable quantity. In order to avoid model estimates we use the experimental value of $E_{T}$ as a measure of the hadronic activity. More precisely, for $SU$ collisions, where the NA38 calorimeter covers a range $- 1.3 < \eta_{cm} < 1.1$ not far from the one of the dimuon ($0 < \eta < 1$), we take $${dN^h \over dy} (b) = {3 E_T(b) \over \Delta \eta
<p_T>} \ \ \ , \eqno(10)$$ where $E_T$ is the average energy of neutrals measured by the calorimeter in each centrality bin, $\Delta \eta = 2.4$, and $<p_T>$ = 0.35 GeV. Note, however, that our results depend only on the product $\sigma_{co}^{\psi} dN^h/dy$. The value of $b$ in each bin is determined from the NA38 code \[15\]. Unfortunately, in $Pb$ $Pb$ collisions, the calorimeter does not have the same acceptance as in $SU$ and, moreover, is not located at mid-rapidities. Due to these differences the $E_T$ measured in $Pb$ $Pb$ has to be multiplied by a factor $2.35 \pm 0.15$ \[16\] in order to be comparable to the $E_T$ measured in $SU$. Finally for a $pA$ collision we take $${dN^{pA \to h} \over dy}(b) = {3 \over 2} (\bar{\nu}
+ 1)
{dN^{NN \to h^-} \over dy} \ \ \ , \eqno(11)$$ where $\bar{\nu}$ is the average number of collisions. From (1) and (7) we obtain the combined result of nuclear absorption and destruction of the $J/\psi$ via interactions with co-moving hadrons, as $${dN^{\psi} \over dy} (b) = \left . {dN^{\psi} \over dy} \right |_{\tau_0} (b)
\
S_1(b) \ S_2(b) \eqno(12)$$ Note that $dN^{\psi}/dy$ at $\tau_0$ is close but not identical to $AB$ times the corresponding value in $pp$ collisions. This is due to the fact that, contrary to $S_1$, $S_2 \not= 1$ for $pp$. Therefore it has to be determined from $$\left . {dN \over dy}\right |_{\tau_0}^{AB \to \psi} (b) = AB
{dN^{pp
\to \psi} \over dy} S_2^{pp} (b) \ \ \ . \eqno(13)$$ We can now compute the absolute yield of $J/\psi$ in any reaction - or the ratio $J/\psi$ over DY since, in the latter case, $S_1 = S_2 = 1$. The results, which depend on two parameters $\sigma_{abs}$ and $\sigma_{co}^{\psi}$, are presented in Fig. 1 and compared with the NA38 and NA50 data. The agreement with experiment is reasonably good. In particular the strong suppression between $SU$ and $Pb$ $Pb$ is obtained with no discontinuity in the parameters. However, our $L$-dependence is somewhat too weak in $pA$ collisions and too strong in $SU$. It is important that the values of the parameters $$\sigma_{abs} = 4.1 \ {\rm mb} \qquad , \qquad \sigma_{co}^{\psi} =
0.46 \
{\rm mb} \eqno(14)$$ are very reasonable. Had we needed a much larger value of $\sigma_{co}^{\psi}$ our interpretation of the $J/\psi$ suppression should be dismissed on theoretical grounds \[6\].
So far we have considered, besides nuclear absorption, all the destruction channels $h +
\psi \to D + \bar{D} + X$, ... , with cross-section $\sigma_{co}^{\psi}$. Likewise, in order to study $\psi '$ suppression we have to consider the channels $h + \psi ' \to D + \bar{D} + X$, ... , which do not involve $\psi '$ in the final state. The corresponding cross-section will be denoted $\sigma_{co}^{\psi '}$. Due to the different geometrical sizes, $\sigma_{co}^{\psi '}$ is larger than $\sigma_{co}^{\psi}$ at very high energies. Their difference is even bigger at low energies due to the dramatic differences in the energy behaviour of these two cross-sections near threshold \[6\]. With this sole extra parameter at our disposal, it is not possible to reproduce the $\psi '/\psi$ ratio in both $SU$ and $Pb$ $Pb$ systems. If we choose $\sigma_{co}^{\psi '}$ such as to reproduce the $SU$ data, the result for central $Pb$ $Pb$ is an order of magnitude too low. However, in this case the above destruction channels are not the only relevant ones. One has also to consider the exchange channels $$\psi + \pi \longrightarrow \psi ' + X \quad ,
\quad \psi ' + \pi \longrightarrow \psi + X \eqno(15)$$ with cross-sections $\sigma_{ex}^{\psi}$ and $\sigma_{ex}^{\psi '}$ respectively. Asymptotically, $\sigma_{ex}^{\psi} = \sigma_{ex}^{\psi '}$. However, at low energies $\sigma_{ex}^{\psi '}$ is expected to be much larger than $\sigma_{ex}^{\psi}$ due to the different thresholds. The presence of these channels has little effect on the $J/\psi$ over DY ratio but it changes considerably the $\psi
'/\psi$ one and allows to cure the problem mentioned above. Indeed, for central $Pb$ $Pb$ collisions, when the $\psi '/\psi$ ratio becomes very small, channels (15) produce a feeding of $\psi '$ at the expense of $\psi$, thereby increasing the ratio $\psi
'/\psi$.
Let us now discuss the combined effect of all destruction and exchange channels. The destruction channel for the $\psi '$ is treated in the same way as for the $\psi$ - with $\sigma_{co}^{\psi}$ replaced by $\sigma_{co}^{\psi '}$ ($\sigma_{co}^{\psi
'} > \sigma_{co}^{\psi}$). For the exchange channels (15), there is a gain of $\psi '$ due to $\psi \to \psi '$ conversion and a loss of $\psi$ due to the inverse reaction. The net gain of $\psi '$ is $$\Delta (b) = \left . {dN^{\psi} \over dy} \right |_{\tau_0} (b) \left [
\sigma_{ex}^{\psi} - \sigma_{ex}^{\psi '} R(b) \right ] G(b) \ \ell n \left |
{\tau_0 +
\Delta \tau \over \tau_0} \right | {dN^h \over dy}(b) \eqno(16)$$ where $R(b)$ is the ratio of $\psi '$ over $\psi$ rapidity densities at time $\tau_0$. The net gain of $\psi$ is obviously given by the same eq. (16) with opposite sign.
Combining (1), (6) and (16) we have $$\left . {dN^{\psi} \over dy} \right |_{\tau_0 + \Delta \tau} (b) = \left [
\left .
{dN^{\psi} \over dy} \right |_{\tau_0}(b) \exp \left [ - \sigma_{co}^{\psi}
G(b) \ell n
\left | {\tau_0 + \Delta \tau \over \tau_0} \right | {dN^h \over dy}(b) \right
] -
\Delta (b) \right ] S_1(b) \eqno(17)$$ and $$\left . {dN^{\psi '} \over dy} \right |_{\tau_0 + \Delta \tau} (b) = \left [
\left .
{dN^{\psi '} \over dy} \right |_{\tau_0}(b) \exp \left [ - \sigma_{co}^{\psi '}
G(b) \ell n \left | {\tau_0 + \Delta \tau \over \tau_0} \right | {dN^h \over
dy}(b)
\right ] + \Delta (b) \right ] S_1(b) \ \ \ . \eqno(18)$$ Contrarily to (6), eqs. (17) (18), have to be solved numerically, because, not only ${dN^{\psi} \over dy}$ changes with increasing $\tau$, but also $R(b)$. Therefore, it is not possible to get a close formula at freeze-out time $\tau$ - but only the variation during an infinitesimal interval $\Delta \tau$. One has to solve the problem numerically, dividing the total $\ell n \tau$ interval into a very large number of subintervals, and using as initial condition in each subinterval the result obtained at the end of the previous one.
The results for the ratios $J/\psi$ over DY and $\psi '/\psi$ are given in Figs. 1 and 2. We have used the following values of the parameters $$\sigma_{abs} = 4.1 \ {\rm mb} \ , \ \sigma_{co}^{\psi} = 0.40 \ {\rm mb}
\ , \ \sigma_{co}^{\psi '} = 2.6 \ {\rm mb} \ , \ \sigma_{ex}^{\psi} =
0.1 \ {\rm mb} \ , \ \sigma^{\psi '}_{ex} = 0.65 \ {\rm mb} \eqno(19)$$ The result presented in Fig. 1 for the $J/\psi$ over $DY$ ratio is not changed by the introduction of the exchange channels (15) (within 1 $\%$). More precisely, a small change in $\sigma_{co}^{\psi}$ from 0.46 (14) to 0.40 mb (19) has compensated for their effect. The value of $\sigma_{co}^{\psi '}$ is basically determined from the data on $\psi '/\psi$ for SU. Finally, the value of $\sigma_{ex}^{\psi}$ is determined in such a way to get enough feeding of $\psi
'$ from $\psi$ in $Pb$ $Pb$. Due to the smallness of $R(b)$, our results are rather unsensitive to the ratio $\sigma_{ex}^{\psi '}/\sigma_{ex}^{\psi}$ and, in order to decrease the number of parameters, we have taken it equal to $\sigma_{co}^{\psi
'}/\sigma_{co}^{\psi} = 6.5$. (A ratio $\sigma_{ex}^{\psi '}/\sigma_{ex}^{\psi}
= 1$ with $\sigma_{ex}^{\psi} = 0.06$ also gives acceptable results). Although we have not attempted a best fit of the data we describe the $\psi '/\psi$ ratio reasonably well. In particular, we have a mild decrease of this ratio both in $pA$ and $Pb$ $Pb$ collisions and a faster decrease in $SU$. This striking feature is also present in the experimental data. However, our $\psi '/\psi$ ratio in $pA$ collisions decreases somewhat faster than the experimental one.
Before concluding it should be noted that the values of $E_T$ measured in $Pb$ $Pb$ collisions, relative to those measured in $SU$, are 20 to 30 $\%$ larger than expected from scaling in the number of participant nucleons and from Monte Carlo codes. At present this point is not well understood either theoretically or experimentally. If the $E_T$ values in $Pb$ $Pb$ were to be decreased by such an amount, the values of the ratio $\psi '/\psi$ in $Pb$ $Pb$ would increase without spoiling the agreement with experiment. However, the ratio $J/\psi$ over DY for $Pb$ $Pb$ collisions would increase (by as much as 20 $\%$ in the most central bin of $Pb$ $Pb$) as shown in Fig. 1. In this case, the mechanism described above would not reproduce entirely the NA50 data.
Note also that an important part of the effect of the co-movers comes from the region of $\tau$ near $\tau_0$ where the densities are very high and one can wonder whether such a dense system can be regarded as a hadronic one. In any case our mechanism of $J/\psi$ suppression is different from Debye screening.
In conclusion, combining nuclear absorption and final state interaction with co-moving hadrons, we have obtained a reasonable description of the $J/\psi$ and $\psi '$ data. This description is better for $SU$ and $Pb$ $Pb$ collisions than for $pA$. It has been achieved with theoretically meaningful values of the cross-sections involved and without introducing any discontinuity in the $J/\psi$ or $\psi '$ survival probabilities.
3 truemm
[**Acknowledgments**]{}
It is a pleasure to thank J. P. Blaizot, M. Braun, B. Chaurand, S. Gavin, M. Gonin, R. Hwa, D. Kharzeev, L. Kluberg, A. Krzywicki, A. Mueller, J. Y. Ollitrault, C. Pajares, H. Satz, Y . Shabelski, D. Schiff and J. Tran Thanh Van for discussions.
**References**
3 truemm =16 pt
[\[1\]]{} T. Matsui and H. Satz, Phys. Lett. [**B178**]{} (1986) 416.
[\[2\]]{} NA38 collaboration : C. Baglin et al, Phys. Lett. [**B201**]{} (1989) 471 ; Phys. Lett. [**B255**]{} (1991) 459.
[\[3\]]{} A. Capella, J. A. Casado, C. Pajares, A.V. Ramallo and J. Tran Thanh Van, Phys. Lett. [**B206**]{} (1988) 354.
A. Capella, C. Merino, C. Pajares, A. V. Ramallo and J. Tran Thanh Van, Phys. Lett. [**B230**]{} (1989) 149.
[\[4\]]{} C. Gerschel and J. H fner, Phys. Lett. [**B207**]{} (1988) 253 ; Z. f r Phys. [**C56**]{} (1992) 71.
[\[5\]]{} J.P. Blaizot and J.Y. Ollitrault, Phys. Rev. [**D39**]{} (1989) 232.
S. Gavin, M. Gyulassy and A. Jackson, Phys. Lett. [**207**]{} (1988) 194.
J. Ftacnik, P. Lichard and J. Pitsut, Phys. Lett. [**B207**]{} (1988) 194.
R. Vogt, M. Parakash, P. Koch and T. H. Hansson, Phys. Lett. [**B207**]{} (1988) 263.
S. Gavin and R. Vogt, Nucl. Phys. [**B345**]{} (1990) 1104.
S. Gavin, H. Satz, R. Thews and R. Vogt, Z. Phys. [**C61**]{} (1994) 351.
[\[6\]]{} G. Bhanot and M. E. Peskin, Nucl. Phys. [**B156**]{} (1979) 365.
A. Kaidalov and P. Volkovitsky, Phys. Rev. Lett. [**69**]{} (1992) 3155.
A. Kaidalov, Proceedings XXVIII Rencontres de Moriond (1993), ed. J. Tran Thanh Van.
JM. Luke et al, Phys. Lett. [**B288**]{} (1992) 355.
D. Kharzeev and H. Satz, Phys. Lett. [**B306**]{} (1994) 155.
[\[7\]]{} D. Kharzeev and H. Satz, Phys. Lett. [**B366**]{} (1996) 316.
B. Kopeliovich and J. H fner, Phys. Rev. Lett. [**76**]{} (1996) 192.
[\[8\]]{} NA50 collaboration : P. Bordaló et al, Proceedings XXI Rencontres de Moriond (1996), ibid ; M. Gonin et al, Proceedings Quark Matter 96, to be published in Nucl. Phys. A.
[\[9\]]{} J.P. Blaizot and J.Y. Ollitrault, Proceedings Quark Matter 96, ibid and Saclay preprint 1996.
C.Y. Wong, Proceedings Quark Matter 96, ibid.
D. Kharzeev and H. Satz, Proceedings QM’96, ibid.
[\[10\]]{} Preliminary results of this work were presented by A. Capella during the discussion session in the QM’96 conference. For a contribution based on a similar approach, see S. Gavin and R. Vogt, Proceedings of QM’96 ibid.
[\[11\]]{} A. Capella, Phys. Lett. [**B364**]{} (1995) 175.
A. Capella, A. Kaidalov, A. Kouider Akil, C. Merino and J. Tran Thanh Van, Z. Phys. [**C70**]{} (1996) 507.
[\[12\]]{} P. Koch, U. Heinz, J. Pitsut, Phys. Lett. [**B243**]{} (1990) 149.
[\[13\]]{} A. Capella and A. Krzywicki, Z. Phys. [**C41**]{} (1989) 659.
[\[14\]]{} NA35 collaboration : G. Roland, Nucl. Phys. [**A566**]{} (1994) 527c.
D. Ferenc, Proceedings XXIX Rencontres de Moriond (1994) ibid.
[\[15\]]{} NA38 collaboration : C. Baglin et al, Phys. Lett. [**B251**]{} (1990) 472.
[\[16\]]{} NA50 collaboration : private communication. 2 truecm
**Figure Captions**
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=1 cm
[**Fig. 1**]{} The ratio $B_{\mu \mu} \sigma (J/\psi )/\sigma (DY)$ versus the interaction length $L$ in the final state for $pp$, $pA$, $SU$ and $Pb$ $Pb$ collisions. The data are from ref. \[8\]. The theoretical values are obtained from eq. (17) with the values of the parameters in (19). The same result (within 1 $\%$) is obtained from eq. (12) with the values of the parameters in (14). The straight line corresponds to nuclear absorption alone (eq. (1)), with $\sigma_{abs} = 6.2$ mb.
3 mm
[**Fig. 2**]{} The ratio $B_{\mu \mu} \sigma (\psi ')/B_{\mu \mu} \sigma
(J/\psi )$ versus $L$ in $pp$, $pA$, $SU$ and $Pb$ $Pb$ collisions. The data are from ref. \[8\]. The theoretical values are obtained from eq. (18), with the values of the parameters in (19). 3 mm
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abstract: 'We consider the influence of the perturbative bulk viscosity on the evolution of the Hubble parameter in the QCD era of the early Universe. For the geometry of the Universe we assume the homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker metric, while the background matter is assumed to be characterized by barotropic equations of state, obtained from recent lattice QCD simulations, and heavy–ion collisions, respectively. Taking into account a perturbative form for the bulk viscosity coefficient, we obtain the evolution of the Hubble parameter, and we compare it with its evolution for an ideal (non–viscous) cosmological matter. A numerical solution for the viscous QCD plasma in the framework of the causal Israel-Stewart thermodynamics is also obtained. Both the perturbative approach and the numerical solution qualitatively agree in reproducing the viscous corrections to the Hubble parameter, which in the viscous case turns out to be slightly different as compared to the non–viscous case. Our results are strictly limited within a very narrow temperature– or time–interval in the QCD era, where the quark-gluon plasma is likely dominant.'
author:
- 'A. Tawfik'
- 'M. Wahba'
- 'H. Mansour'
- 'T. Harko'
title: Hubble Parameter in QCD Universe for finite Bulk Viscosity
---
Introduction {#sec:intro}
============
The bulk viscosity is assumed to play an essential role in various eras of the early Universe [@tawTD]. Causal bulk viscous thermodynamics has been extensively used for describing the dynamics of the early Universe, and in astrophysical applications. Due to the complicated nature of the cosmological evolution equations in the early Universe, very few exact solutions are known. Isotropic homogeneous Universes filled with causal viscous fluid obeying the relation $\xi \sim \rho^{s}$, where $\xi $ is the bulk viscosity coefficient, and $\rho$ is the energy density, have been studied for special cases of the exponent $s$, $s=1$ and $s=1/2$, respectively, in [@earlyQGP] and [@ChJa97; @MaHa99b; @MaTr97], respectively. Arbitrary values of the exponent $s$ are also considered in Refs. [@MaHa99a; @earlyQGP]. It has been proposed that causal bulk viscous thermodynamics can give a model for the phenomenological matter creation in the early Universe [@earlyQGP; @ChJa97].
Recent RHIC results apparently indicate that hot and dense matter has been formed in heavy-ion collisions [@reff1], which likely agrees with the [*new state of matter*]{} predicted in lattice QCD simulations [@reff5; @mueller2] and clearly points out that the bulk viscosity $\xi$ is non-negligible in regions close to the QCD critical temperature, $T_c$. Nevertheless, there has been a debate on how to modify the cosmological standard model [@harrison], and the possibility of applying the QCD barotropic equations of state (EoS) [@barotrp] in the QCD era of the early Universe [@earlyQGP; @TawCosmos]. Reasons for such a [*”categorical resistance”*]{} might be the mathematical difficulties associated with the second Abel type non-homogeneous and non-linear differential equations describing the dynamics of causal bulk viscous models [@earlyQGP; @TawCosmos; @maartensR]. The long history of cosmological studies in which background matter has been modeled as an ideal (non-viscous) fluid would represent another reason. Obviously, the viscous characteristics of cosmological matter likely leads to important consequences in cosmology and astronomy [@taw08].
It is the purpose of the present work to investigate the QCD cosmological era of the evolution of the Universe. We assume that the Universe, described by the standard Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmic background geometry, is filled with a relativistic viscous QCD plasma, whose bulk viscosity is supposed to be finite. The equation of state of the plasma is obtained from recent heavy-ion collisions and lattice QCD simulations, respectively [@karsch07; @Cheng:2007jq]. The QCD plasma is assumed to be formed at temperatures in the range $0.2\leq T\leq10\;$GeV, or in the time–period $18.35\leq t \leq 0.0073\;$GeV$^{-1}$, respectively.
To fully describe the effects of bulk viscosity on the cosmological evolution we analyze several scenarios. Firstly, we assume a perturbative bulk viscosity, $\xi=\alpha$, where $\alpha$ is a constant independent of $T$ and/or $\rho$. However, this assumption has an insufficient thermodynamical motivation. Nevertheless, in the absence of a better alternative, we can consider this approach in order to obtain some indications for the deviation from the zero bulk viscosity case $\xi=0$. On the other hand the assumption of the constant bulk viscosity coefficient is physically motivated, if perturbative methods are used to describe bulk viscous processes. Thus, the $T$– and/or the $\rho$–dependence of $\xi$ or $\alpha$ can be ignored. A better approach could be the use of ${\cal N}=4$ SYM [@baier], which is apparently valid at very high $T$.
The main concern about considering a constant $\xi$ stems from the corresponding $T$– and/or $\rho$–dependence of the relaxation time $\tau$, and particle decay width $\Gamma$, which has to ensure that the speed of viscous pulses does not exceed the speed of light. For example, a $T$–dependence of $\tau$ of the form $\tau=(2-\ln2)/(2\pi T)$, guarantees a positive heat capacity. A constant relaxation time can be interpreted as indicating that the relaxation towards the thermal equilibrium is very slow. On the other hand, both the $1/T$ and the $\tau$ times scale become arbitrarily small with decreasing time $t$. At $T>T_c$, the quark masses are much smaller than $T$. Then the only way to construct a characteristic time scale for $t$ is $1/T$. Also, powers of dimensionless coupling $\alpha_s$ can be multiplied by $1/T$. Once again, the perturbative methods are consistent with such an approximation.
Secondly, we consider numerical solutions for the evolution equation in the case of finite viscous background matter characterized by the Israel-Stewart causal thermodynamics. Barotropic EoS for both $\xi$ and $\tau$ are adopted from recent lattice QCD simulations and quasi–particle (QP) effective models [@qpm], respectively. The numerical methods are very much sensitive to the boundary conditions. To keep this sensitivity as small as possible, we use the results obtained from the perturbative treatment.
The present paper is organized as follows. The QCD plasma model is described in Section \[sec:model\]. The perturbative treatment and the numerical solution are considered in Section \[sec:new\]. We discuss and conclude our results on the evolution of the Hubble parameter in the QCD era in Section \[sec:concl\].
Geometry, field equations, and consequences {#sec:model}
===========================================
In spherical coordinates $\left(t,r,\theta , \phi \right)$, the line element of a homogeneous and isotropic flat Universe can be represented in the standard FLRW form as $$\label{1}
ds^{2}=dt^{2}-a^{2}(t) \left[dr^{2}+r^{2}\left( d\theta
^{2}+\sin ^{2}\theta d\phi^{2}\right) \right],$$ where $a$ is the scale factor. The Hubble parameter is defined as $H=\dot a/a$. For a vanishing cosmological constant $\Lambda $, the dynamics of the Universe is described by the Einstein gravitational field equations, given by $$\label{ein}
R_{ik}-\frac{1}{2}g_{ik}R=8\pi\; G\; T_{ik}.$$ The energy-momentum tensor of the bulk viscous cosmological fluid filling the very early Universe is given by $$T_{i}^{k}=\left( \rho +p+\Pi\right) u_{i}u^{k}-\left( p+\Pi\right)
\delta_{i}^{k},\label{1_a}$$ where $i,k$ takes the values $0,1,2,3$, $\rho$ is the mass density, $p$ the thermodynamic pressure, $\Pi $ the bulk viscous pressure, and $u_{i}$ is the four velocity, satisfying the condition $u_{i}u^{i}=1$. The particle and entropy fluxes are defined according to $N^{i}=nu^{i}$ and $S^{i}=sN^{i}-\left( \tau\Pi^{2}/2\xi T\right)
u^{i}$, where $n$ is the number density, $s$ the specific entropy, $T\geq0$ the temperature, $\xi$ the bulk viscosity coefficient, and $\tau\geq0$ the relaxation coefficient for transient bulk viscous effect (i.e. the relaxation time), respectively. In the following we shall also suppose that the energy-momentum tensor of the cosmological fluid is conserved, that is $T_{i;k}^{k}=0$.
The bulk viscous effects can be generally described by means of an effective pressure $\Pi $, formally included in the effective thermodynamic pressure $p_{eff}=p+\Pi $ [@maartensR]. Then in the comoving frame the energy momentum tensor has the components $T_{0}^{0}=\rho ,T_{1}^{1}=T_{2}^{2}=T_{3}^{3}=-p_{eff}$. For the line element given by Eq. (\[1\]), the Einstein field equations read $$\begin{aligned}
\label{2n}
\left( \frac{\dot{a}}{a}\right)^{2} &=& \frac{8\pi}{3}G \;\rho, \\
\frac{\ddot{a}}{a} &=& -\frac{4\pi}{3}G \; \left( 3p_{eff}+\rho \right),
\label{3}\end{aligned}$$ where the dot denotes the derivative with respect to the time $t$, and $G$ is the gravitational constant. The energy density of the cosmic matter fulfills the conservation law: $$\label{5n}
\dot{\rho}+3H\left( p_{eff}+\rho \right) =0.$$
In presence of bulk viscous stress $\Pi $, the effective thermodynamic pressure term becomes $p_{eff}=p+\Pi $. Then Eq. (\[5n\]) can be written as $$\label{6n}
\dot{\rho}+3H\left( p+\rho \right) =-3\Pi H.$$
For the evolution of the bulk viscous pressure we adopt the causal evolution equation [@maartensR], obtained in the simplest way (linear in $\Pi)$ to satisfy the $H$-theorem (i.e., for the entropy production to be non-negative, $S_{;i}^{i}=\Pi^{2}/\xi T\geq0$ [@Is76; @IsSt76]). According to the causal relativistic Israel-Stewart theory, the evolution equation of the bulk viscous pressure reads [@maartensR] $$\label{8n}
\tau \dot{\Pi}+\Pi =-3\xi H-\frac{1}{2}\tau \Pi \left( 3H+\frac{\dot{\tau}}{\tau }-\frac{\dot{\xi}}{\xi }-\frac{\dot{T}}{T}\right).$$ In order to close the system of equations (\[2n\]), (\[6n\]) and (\[8n\]) we have to add the equations of state for $p$ and $T$. The equation of state, the temperature and the bulk viscosity of the quark-gluon plasma (QGP), can be determined approximately at high temperatures from recent lattice QCD calculations as [@Cheng:2007jq; @karsch07] $$\label{13an}
P = \omega \rho,\hspace*{1cm}T = \beta \rho^r,\hspace*{1cm}\xi = \alpha \rho + \frac{9}{\omega_0} T_c^4,$$ $$\label{13bn}
\alpha = \frac{1}{9\omega_0} \frac{9\gamma^2-24\gamma+16}{\gamma-1},$$ where $\omega = (\gamma-1)$, $\gamma \simeq 1.183$, $r\simeq 0.213$, $\beta\simeq 0.718$, and $\omega_0 \simeq 0.5-1.5$ GeV, respectively. In the following we assume that $\alpha \rho >> 9/\omega_0 T_c^4$, and therefore we take $\xi \simeq \alpha \rho$. In order to close the system of the cosmological equations, we have also to give the expression of the relaxation time $\tau $, for which we adopt the expression [@maartensR], $$\label{tau}
\tau=\xi\rho^{-1}\simeq\alpha .$$
Eqs. (\[13an\]) are standard in the study of the viscous cosmological models, whereas the equation for $\tau$ is a simple procedure to ensure that the speed of viscous pulses does not exceed the speed of light. Eq. (\[tau\]) implies that the relaxation time in our treatment is constant, but strongly depends on the EoS. These equations are without sufficient thermodynamical motivation, but in the absence of better alternatives, we shall follow the practice of adopting them in the hope that they will at least provide some indication of the range of bulk viscous effects. The temperature law is the simplest law guaranteeing positive heat capacity.
Perturbative and numerical solutions of the field equations {#sec:new}
===========================================================
For a viscous FLRW Universe filled with a viscous quark–gluon plasma (QGP), the evolution equation of the Hubble parameter is given by $$\begin{aligned}
\label{eq:1}
\alpha H\ddot H+\frac{3}{2}[1+(1-r)\gamma]\alpha H^2\dot H+H\dot H-(1+r)\alpha\dot H^2+\frac{9}{4}(\gamma-2)\alpha H^4+\frac{3}{2}\gamma H^3 = 0.\end{aligned}$$ The parameters, $\alpha$, $r$ and $\gamma$ are the coefficients and the exponents of the barotropic QGP EoS, discussed later in Eqs. (\[13an\])–(\[13bn\]) [@earlyQGP]. In the limit of a vanishing $\alpha$, a non–viscous evolution equation can be obtained from Eq. (\[eq:1\]). Then, $$\begin{aligned}
\label{eq:2}
H_0(t)\dot H_0(t)+\frac{3}{2}\gamma H^3_0(t)=0,\end{aligned}$$ where $H_0(t)$ is the Hubble parameter corresponding to the non–viscous background matter. Eq. (\[eq:2\]) can easily be solved by $$\label{eq:3}
H_0(t)= \frac{2}{3\gamma}\,t^{-1},$$ The mathematical solution $H_0=0$ of Eq. (\[eq:2\]), which can be interpreted physically as describing a static Universe with a vacuum background geometry, has been ignored, as irrelevant to the present case. The second solution refers to a dynamic state. The acceleration (deceleration) of the Universe reads $$\begin{aligned}
\dot H_0(t) &=& \frac{\ddot a}{a}-H_0^2(t),\\
\ddot a &=& a\left(-\frac{2}{3\gamma}\frac{1}{t^2}+H_0^2(t)\right)\label{eq:4},\end{aligned}$$ implying that the type of the expansion of the Universe is exclusively determined by $\gamma$. For $\gamma>2/3$, the Universe decelerates. As shown below, in a quark-gluon plasma $\gamma$ could be as large as two times this threshold. For the viscous QGP model, the expressions for $H$ will be obtained by means of a perturbative analysis and by considering numerical methods.
Perturbative Bulk Viscosity
---------------------------
In the following we consider the effects of the inclusion of the bulk viscosity in the cosmological standard model [@harrison] by means of a perturbative approach. It is essential to highlight that the validity of this treatment is restricted to the QCD era. It is considered that the QCD era lasted a much shorter time interval than the era corresponding to a finite scale and a strong running coupling $\alpha_s$ [@alfasA; @alfasB]. As a first step in our analysis we obtain perturbatively the deviations of the Hubble parameter from the non–viscous cosmological picture.
### A First–Order Perturbative Correction
By using a first–order perturbative correction, $f(t)$, the solution of Eq. (\[eq:1\]) can be written as $$\label{eq:6}
H(t)=H_0(t)+\alpha f(t),$$ where $H_0(t)$ is the Hubble parameter corresponding to a vanishing bulk viscosity. Substituting this into Eq. (\[eq:1\]) leads to $$\begin{aligned}
\label{eq:7}
&& \alpha H_0(t)\ddot H_0(t)+\frac{3}{2}[1+(1-r)\gamma]\alpha H^2_0(t)\dot H_0(t)+H_0(t)\dot H_0(t)+\alpha H_0(t)\dot f(t)\nonumber\\
&& +\alpha f(t)\dot H_0(t)-(1+r)\alpha\dot H^2_0(t)+\frac{9}{4}(\gamma-2)\alpha H^4_0(t)+\frac{3}{2}\gamma H^3_0(t)+\frac{9}{2}\alpha\gamma H^2_0(t)f(t)=0.\end{aligned}$$ This can be written in the form \[eq:8\] f(t)+G\_1(t)f(t)+G\_0(t)&=&0, where $$\begin{aligned}
G_0(t) &=& -\ddot H_0(t)-\frac{3}{2}[1+(1-r)\gamma]H_0(t)\dot H_0(t)-\frac{\dot H_0(t)}{\alpha}+(1+r)\frac{\dot H^2_0(t)}{H_0(t)}
-\frac{9}{4}(\gamma-2)H^3_0(t)-\frac{3}{2}\gamma\frac{H^2_0(t)}{\alpha}, \\
G_1(t) &=& \frac{\dot H_0(t)}{H_0(t)}+\frac{9}{2}\gamma H_0(t).\end{aligned}$$ To eliminate $H_0(t)$ and its derivatives, we can use Eqs. (\[eq:3\]), (\[eq:4\]) and H\_0(t) &=& -3H\_0(t) + 2H\^3\_0(t) = ,respectively, thus closing the coupled set of equations needed to determine the non–homogeneous coefficients $G_0(t)$ and $G_1(t)$. Then $$\begin{aligned}
G_0(t) &=& A\,t^{-3},\label{eq:9}\\
G_1(t) &=& 2\,t^{-1}\label{eq:10},\end{aligned}$$ where the coefficient $A$ is given by $$A=\frac{2}{3\gamma^2}[1+(1-r)\gamma]-\frac{4}{3\gamma}-\frac{2}{3\gamma}(1+r)
-\frac{2}{3}\frac{\gamma-2}{\gamma^3}.$$
Equation (\[eq:8\]) is a first–order linear differential equation and has the general solution given by $$\label{eq:11}
f(t)= e^{-F}\left(\int e^F G_0(t) dt+C\right),$$ where $F=\int G_1(t) dt$, and $C$ is an arbitrary integration constant. Substituting Eqs. (\[eq:9\]) and (\[eq:10\]) into Eq. (\[eq:11\]) leads to, $$\label{eq:1ft}
f(t)=t^{-2}[A \ln(t)-C].$$ Therefore, the solution of Eq. (\[eq:1\]) reads $$\label{eq:12}
H(t)=\frac{2}{3\gamma}\frac{1}{t}+\alpha t^{-2}[A \ln(t) - C].$$ Since $C=A\,\ln(t_0)$ does not depend on $t$, the acceleration/deceleration is obtained as \[eq:hdot1\] (t) &=& - + .
### A Second–Order Perturbative Correction
A second–order perturbative correction to the Hubble parameter can be represented as $$\label{eq:13}
H(t)=H_0(t)+\alpha f(t)+\alpha^2 h(t).$$ In order to solve the resulting evolution equation for the new perturbation function $h(t)$, we can follow the same procedure used in the previous Section. Substituting Eq. (\[eq:13\]) and its derivatives into Eq. (\[eq:1\]) reduces the problem to a second–order non–homogeneous differential equation given by \[eq:2nonh\] f\_2(t)h(t)+f\_1(t)h(t)+f\_0(t)h(t)&=&k(t), where the coefficients of the non–homogeneous equations are f\_2(t) &=& \^4A ,\
f\_1(t) &=& (1+r) f\_2(t) + ,\
f\_0(t) &=& - (f\_1(t)-(1+r)f\_2(t)),\
k(t) &=& -f\_2(t)-\
&& -\
&& -\
&& -\
&& -\
&& -\
&& +\
&& f\_2(t) + f\_2(t) + (-2A+c(4+))f\_2(t) +\
&& f\_2(t)We assume that $f(t)$, given by Eq. (\[eq:1ft\]), represents a particular solution of the following second–order homogeneous differential equation: \[eq:2h\] f\_2(t)g(t)+f\_1(t)g(t)+f\_0(t)g(t)&=&0. The solution $g(t)$ of Eq. (\[eq:2h\]) reads g(t) &=& f(t)\_[t\_0]{}\^t dt, where $F=\int_{t_0}^t f_2(t)/f_1(t)dt$. Then g(t) &=& \_[t\_0]{}\^t dt,\
where we have denoted \_1=\^4 A, && \_2=\^4 A (1+r), \_3= (1+r-r).A second particular solution for the homogeneous differential equation is given by \[eq\_2ps\] g(t) &=& {-(2r-3)\^[-(1+B)]{} }, where $\Gamma$ is the incomplete gamma function, and B&=& 2. The numerical values of this second–order perturbative correction are negligibly small. Fig. \[fig:1a2\] shows the real part of Eqs. (\[eq\_2ps\]). Obviously, it is valid up to $t=1$.
The general solution of the non–homogeneous differential equation, Eq. (\[eq:2nonh\]), can be obtained with the use of the Wronskian determinant, $W=\exp(-F)$, h(t) &=& c\_1f(t)+c\_2g(t) + g(t)f(t) - f(t)g(t), where $c_1$ and $c_2$ are arbitrary constants of integration.
Hence, we conclude that the first–order perturbative correction $h(t)$ seems to give the dominant term, at least according to the quantitative comparison with the numerical solution given in the next Section. The function $g(t)$ likely makes a very little contribution to the final results. Taking the non–viscous case as a particular solution leads to a similar conclusion. Therefore, there is no need to include the second order correction in Fig. \[fig:1a1\].
![Left panel: second particular solution $g(t)$ as a function of time $t$. Only its real part is taken into account, which obviously exists only in $t\in[t_0,1]\,$GeV$^{-1}$. Right panel shows a comparison between first (dashed curve) and second (solid curve) particular solutions. []{data-label="fig:1a2"}](gOFt.eps "fig:"){width="8.5cm"} ![Left panel: second particular solution $g(t)$ as a function of time $t$. Only its real part is taken into account, which obviously exists only in $t\in[t_0,1]\,$GeV$^{-1}$. Right panel shows a comparison between first (dashed curve) and second (solid curve) particular solutions. []{data-label="fig:1a2"}](g_fOFt.eps "fig:"){width="8.65cm"}
Numerical solution for the causal cosmological quark-gluon plasma
-----------------------------------------------------------------
The inclusion of the bulk viscous effects can generally be done through an effective pressure $\Pi$, which is formally included in the effective thermodynamic pressure $p_{eff}$, where $p_{eff}=p+\Pi $, and $p$ is the thermodynamic pressure. EoS’s for $p$ and $T$ are necessary in order to close the system of equations Eq. (\[2n\]), (\[6n\]) and (\[8n\]). $\tau$ and $\xi$ can be determined by using some phenomenological approaches. For instance, $\xi$ in QGP matter at high $T$ can be estimated from recent lattice QCD simulations [@karsch07; @Cheng:2007jq]. The relaxation time can also be derived from the quasi–particle (QP) model, which is effectively used to reproduce the lattice QCD thermodynamics [@qpm]. p&=&, \[13a\]\
T&=&\^r, \[13b\] where $\omega=(\gamma-1)$, $\omega_0 \simeq 0.5-1.5$ $GeV$, $r=0.39$ and $\beta=0.718$. According to recent lattice QCD simulations [@karsch07], it is found that $s=1$. In the quasi–particle model, it has been found that the bulk viscosity $\xi$ and the relaxation time $\tau$ have the following barotropic forms, &=& (0.9590.006) \^[(0.8630.001)]{},\
&=& (64.1891.667) \^[(-0.0750.003)]{} -(26.2761.774), which are graphically shown in Fig. \[fig:tau\]. In the left panel, the dependence of the bulk viscosity (in units of GeV/fm$^{2}$) on $\rho$ (in units of GeV/fm$^{3}$) is presented. The right panel shows the decay of the relaxation time $\tau$ in QGP matter with increasing $\rho$. The parameters of the quasi–particle model are first adjusted to reproduce the thermodynamics of lattice QCD simulations, and then they are used to calculate both $\xi$ and $\tau$. The thermodynamic consistency of the barotropic relations of $\xi$ and $\tau$ is guaranteed [@tawuro]. This can be partly seen from the fact that the ratio $\xi/\tau$ strongly depends on $\rho$.
Substituting Eq. (\[8n\]) into Eq. (\[6n\]) leads to Eq. (\[eq:1\]). When assuming that the last two terms in the right hand side of this equation are vanishing, then we are left with a Bernoulli type differential equation, with the solutions $H=0$ and $H\approx -2\gamma/[3\alpha(2-\gamma)]$. The case $H=0$ corresponds to a vacuum Universe, and hence this solution can be ignored. If it would be possible to express $\xi$ as a function of $t$, we would obtain $2\dot H+3\gamma H^2-3\xi(t)H=0$, which is a Bernoulli type equation, and has the solution $$H(t)=\frac{\exp{\left[\left(3/2\right)\int\xi(t)dt\right]}}{C+\exp{\left[\left(3/2\right)\gamma \int\xi(t)dt\right]}},$$ where $C$ is an arbitrary constant of integration. A numerical solution of Eq. (\[eq:1\]) for $H$ is presented in Fig. \[fig:3a1\]. It strongly depends on the initial conditions for $H$, $\dot H$ and $C$. As mentioned previously, the initial time is defined as $t_0=0.734\;$GeV$^{-1}$, at which $T=1.0\;$GeV.
Discussion and final remarks {#sec:concl}
============================
According to the standard cosmological standard the QCD era is characterized by a temperature range of– $0.2<T<10\;$GeV, or a time–interval of $18.35\leq t \leq 0.0073\;$GeV$^{-1}$. In Fig. \[fig:1a1\], the boundaries of the QCD era are drawn by vertical lines. The dashed one refers to $T=0.5\;$GeV, or $t=2.937\;$GeV$^{-1}$, respectively. The right and the left vertical lines give the boundaries, $T=0.2\;$GeV or $t=18.35\;$GeV$^{-1}$ and $T=1.0\;$GeV or $t=0.734\;$GeV$^{-1}$, respectively.
![Various measurements of strong running coupling $\alpha_s(Q)$ as a function of the energy scale $Q$ in GeV units. The graph is taken from Ref. [@alfasA]. The curves represent the QCD predictions, and their systematic certainty.[]{data-label="fig:alfas"}](asq-2006.eps){width="7.cm"}
In principle, the QCD era is defined according to the energy scale available in the early Universe, and according to the asymptotic behavior of the partonic matter. Although QCD does not predict the absolute size of the strong running coupling $\alpha_s$, it makes a precise prediction for its energy–dependence. Since $\alpha_s$ is the most important degree of freedom of QCD, we define the region for which our treatment is consistent as the region where $\alpha_s$ remains finite. It is important to mention that this degree of freedom appears even if the quarks are entirely excluded. The scale–dependence of the coupling constant $\alpha_s$ is governed by the function $\beta$, which encodes the running of the coupling, and whose perturbative expansion in the four–loop approximation is [@betaalpha]. (\_s(Q\^2)) &=& - \_0\_s\^2(Q\^2) - \_1\_s\^3(Q\^2) - \_2\_s\^4(Q\^2) - \_3\_s\^5(Q\^2) + , where the coefficients $\beta_0$, $\beta_1$, $\beta_3$, and $\beta_3$ have been determined by perturbative methods, and found to be dependent on the number of active quark flavors $n_f$ at the corresponding energy scale $Q$. For simplicity, let us consider the one–loop approximation. Then \[eq:alfaQCD\] \_s(Q\^2) && - , where $\Lambda$ is the QCD energy scale, depending on $n_f$, and on the renormalization scheme. If $Q=\Lambda$, then $\alpha_s$ diverges. On the other hand, $\alpha_s \rightarrow 0$ at very large energy scale. A summary of $\alpha_s$–measurements is graphically displayed in Fig. \[fig:alfas\]. It is obvious that the measurements unambiguously confirm the QCD predictions for $\alpha_s$, i.e, Eq. (\[eq:alfaQCD\]), as well as the approach towards asymptotic freedom. The figure is taken from Ref. [@alfasA]. The QCD era is conjectured to start from a very high energy scale, i.e, very small $\alpha_s$, and ends up when $Q\rightarrow \Lambda$, i.e, at divergent $\alpha_s$. As a matter of precaution, we assume that the QGP matter exists in a much narrower energy scale, up to just $3-5$ times $\Lambda$. According to the recent lattice QCD simulations [@karsch07], $\Lambda$ can be localized at $T_c\simeq0.2\,$GeV, which sets the [*latest*]{} end of the QCD era. Also, we know that the bulk viscosity is about to diverge at $T_c$, and decays with increasing temperature. Based on these two ingredients, we set some [*narrow*]{} limits to the validity of including finite bulk viscosity in the QGP era.
![Numerical solution for the Hubble parameter $H(t)$ for a bulk viscous fluid with finite bulk viscosity in the full causal Israel-Stewart theory (dashed–dotted bottom curve). The dashed curve (top) represents $H(t)$ in the QCD era with non–viscous background matter. The physical units in both axis are GeV. The solid curve (left) gives the evolution in the pre–QCD eras, where vanishing viscosity characterizes the background matter. []{data-label="fig:3a1"}](QP-IS_EvolutionEq.eps){width="10.cm"}
In the left panel of Fig. \[fig:1a1\], the evolution of the Hubble parameter $H$ is given for different values of the first–order bulk viscosity coefficient $\alpha$. Obviously, the deviation from the non–viscous evolution increases with increasing $\alpha$. The vertical lines determine when or where QGP matter is becoming dominant in the early Universe. Pre– and post–cosmological eras are probably characterized by EoS’s that might be different from the ones used in the present work. Under these assumptions, the viscosity effects are assumed to be localized within the QGP era. In the right panel, the evolution of the time derivative of $H$ is presented for different values of $\alpha$. As shown in Eq. (\[eq:3\]), $\dot H$ is the sum of two terms. The first term gives a ratio of acceleration, $\ddot a$, to a scale parameter, $a$. The second term is the square of $H$ itself. The sum of these two terms is negative at small $t$. At large $t$, it the two terms are identical not only qualitatively, but also quantitatively. That the Universe is obviously accelerating, $\ddot a>0$, is also shown by the values of the scale parameter $a$, which increase with increasing $t$. Therefore, we conclude that including finite bulk viscosity seems to affect the expansion of the Universe in a significant way. The evolution of the Hubble parameter likely decays with a fast rate, that increases with increasing $t$. Furthermore, we point out that increasing the values of the bulk viscosity coefficient $\alpha$ accelerates the decay of $H$.
A numerical solution for the evolution equation, Eq. (\[eq:1\]), at finite bulk viscosity, is presented in Fig. \[fig:3a1\]. Here, we take into consideration the barotropic dependence of $\xi$, as it has been obtained in the lattice QCD simulations. We do not make any further assumptions on it. Also, the dependence of the relaxation time $\tau$ on the energy density has not been modified. These two ingredients have been strictly implemented. The consistency of these barotropic EoS’s with the laws of thermodynamics has been discussed in Ref. [@tawuro]. It seems that all barotropic EoS’s used in the present (and also in the previous work [@earlyQGP; @TawCosmos]) are thermodynamically consistent. Therefore, we numerically solve the evolution equation for the Hubble parameter. The analytic solution has been obtained in Ref. [@earlyQGP]. This solution is sensitive to the particular assumptions made in order to solve the second Abel–type non–linear non–homogeneous differential equation. To avoid their effects, we consider the numerical solution to the evolution equation. The boundary conditions required for the numerical methods are defined at the boundaries of QGP era in the early Universe.
A comparison between the time evolution of the Hubble parameter $H$ in the non–viscous and viscous background matter is presented in Fig. \[fig:3a1\]. The top dashed curve represents the evolution of the non–viscous cosmological matter, given by Eq. (\[eq:3\]). The dotted–dashed curve gives the evolution of the viscous cosmological matter. In order to compare with the results presented in Fig. \[fig:1a1\], the [*earliest*]{} (left) time boundary has been zoomed in. By comparing the numerical and analytical results leads us to the conclusion that the two methods (perturbative and numerical) agree in reproducing the evolution of $H$. Also, we conclude that the time evolution of $H$ for viscous cosmological matter is faster than the evolution for non–viscous cosmological matter. Such a difference can be estimated, quantitatively. Therefore, essential cosmological consequences are to be expected from the inclusion of bulk viscous effects in the description of the cosmological expansion of the Universe.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank to the two anonymous referees, whose comments and suggestions helped us to significantly improve the manuscript. TH is partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 701808P).
[99]{}
A. Tawfik, to appear in Can. J. Phys., arXiv:1002.0296 \[gr-qc\].
A. Tawfik, M. Wahba, H. Mansour and T. Harko, arXiv:1001.2814 \[gr-qc\] to appear in Ann. Phys.
L. P. Chimento and A. S. Jakubi, Class. Quantum Grav. [**14**]{}, 1811 (1997); L. P. Chimento and A. S. Jakubi, Int. J. Mod. Phys. D [**7**]{}, 177 (1998); M. K. Mak and T. Harko, Gen. Rel. Grav. [**30**]{}, 1171 (1998); Gen. Rel. Grav. [**31**]{}, 273 (1999); J. Math. Phys. [**39**]{}, 5458 (1998).
M. K. Mak and T. Harko, Aust. J. Phys. [**52**]{}, 659 (1999); Int. J. Mod. Phys. D [**9**]{}, 97 (2000); Aust. J. Phys. [**53**]{}, 241 (2000); Int. J. Mod. Phys. D [**9**]{}, 475 (2000).
R. Maartens and J. Triginer, Phys. Rev. D [**56**]{}, 4640 (1997).
T. Harko and M. K. Mak, Int. J. Theor. Phys. [**38**]{}, 1561 (1999).
BRAHMS Collaboration, I. Arsene [*et al.*]{}, Nucl. Phys. A [**757**]{}, 1 (2005); PHENIX Collaboration, K. Adcox [*et al.*]{}, Nucl. Phys. A [**757**]{}, 28 (2005); STAR Collaboration, J. Adams [*et al.*]{}, Nucl. Phys. A [**757**]{}, 102 (2005); PHOBOS Collaboration, B. B. Back [*et al.*]{}, Nucl. Phys. A [**757**]{}, 295 (2005).
E. R. Harrison, Ann. Phys. [**11**]{}, 155-186 (1973).
L. P. Chimento and A. S. Jakubi, Class. Quantum Grav. [**14**]{}, 1811-1820 (1997); B. Cheng, Phys. Lett. A [**160**]{}, 329-338 (1991); G. Papini, M. Weiss, Lett. Nuovo Cim. [**44**]{}, 612 (1985).
A. Tawfik, H. Mansour and M. Wahba, Invited talk given at the 7th international conference on “Modern Problems of Nuclear Physics”, 22-25 September 2009, Tashkent-Uzbekistan, arXiv:0911.4105 \[gr-qc\]; Talk given at 12th Marcel Grossmann Meeting on “General Relativity”, Paris-France, 12-18 July 2009, arXiv:0912.0115 \[gr-qc\].
R. Maartens, Class. Quantum Grav. [**12**]{}, 1455 (1995); R. Maartens, Causal thermodynamics in relativity, astro-ph/9609119 (1996).
F. Karsch, E. Laermann, P. Petreczky, S. Stickan and I. Wetzorke, 2001 [*Proccedings of NIC Symposium*]{}, Ed. H. Rollnik and D. Wolf, John von Neumann Institute for Computing, Jülich, NIC Series, [**9**]{}, (2002).
D. Kharzeev and K. Tuchin, JHEP [**0809**]{}, 093 (2008).
A. Tawfik, AIP Conf. Proc. [**1115**]{}, 239 (2009), arXiv:0809.3825 \[hep-ph\].
F. Karsch, D. Kharzeev, K. Tuchin, Phys. Lett. B [**663**]{}, 217 (2008).
M. Cheng [*et al.*]{}, Phys. Rev. D [**77**]{} 014511 (2008), arXiv:0710.0354 \[hep-lat\]. R. Baier, P. Romatschke, D. T. Son, A. O. Starinets and M. A. Stephanov, JHEP [**0804**]{}, 100 (2008).
M. Bluhm, B. Kampfer and K. Redlich, Nucl. Phys. A [**830**]{}, 737C-740C (2009).
S. Bethke, Prog. Part. Nucl. Phys. [**58**]{}, 351-386 (2007); A. I. Alekseev, Few Body Syst. [**32**]{}, 193-217 (2003).
P. A. M. Fernandez, [*Studien zur Quantenchromodynamik und Messung der starken Kopplungskonstanten $\alpha_s$ bei $\sqrt{s}=14-44\;$GeV mit dem JADE–Detektor*]{}, Ph. D. Thesis, Aachen University (2002); M. Schmelling, invited talk at 15th Int. Conf. on Physics in Collision - PHYSCOLL 95, Cracow, Poland, 8 - 10 Jun 1995.
W. Israel, Ann. Phys. [**100**]{}, 310 (1976).
W. Israel, J. M. Stewart, Phys. Lett. A [**58**]{}, 213 (1976).
A. A. Huerta and A. Tawfik, in preparation
T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, Phys. Lett. B [**400**]{}, 379 (1997).
|
---
abstract: |
The flux penetration near a semicircular indentation at the edge of a thin superconducting strip placed in a transverse magnetic field is investigated. The flux front distortion due to the indentation is calculated numerically by solving the Maxwell equations with a highly nonlinear $E(j)$ law. We find that the excess penetration, $\Delta$, can be significantly ($\sim$ 50 %) larger than the indentation radius $r_0$, in contrast to a bulk supercondutor in the critical state where $\Delta=r_0$. It is also shown that the flux creep tends to smoothen the flux front, i.e. reduce $\Delta$. The results are in very good agreement with magneto-optical studies of flux penetration into an YBa$_2$Cu$_3$O$_x$ film having an edge defect.
author:
- 'J. I. Vestgården, D. V. Shantsev, Y. M. Galperin and T. H. Johansen'
title: 'Flux Penetration in Superconducting Strip with Edge-Indentation'
---
Introduction
============
Magnetic field penetrates type-II superconductors as a set of quantized flux lines – vortices. Macroscopically, the vortex matter can be considered as a “flux liquid”. An important feature of this matter is pinning of vortices leading to zero electrical resistance at zero temperature. The pinning results in a non-uniform distribution of magnetic flux forming a *critical state*. The critical state determines the macroscopic properties, e.g. the maximum current density and magnetic susceptibility, that are important for applications. According to the critical state model,[@bean64] at any point of the sample the local value of the electrical current density is equal to its critical value, $j_c$, for a given magnetic field and temperature.
An interesting property of the critical state is that [*local*]{} material defects affect the field and current distributions on a [*global*]{} scale. For example, even a small non-superconducting cavity or an edge indentation create sample-spanning discontinuity lines where the current flow direction changes abruptly.[@CCS] At a non-zero temperature, the critical state is relaxed due to flux creep that is conventionally described by a highly nonlinear current-voltage curve, $E \propto j^n$ where $n\gg 1$ and $E$ is electric field. Nevertheless, the same tendency persists: a small cavity of size $\ell$ in a bulk superconductor perturbs the field distributions on a much larger scale of $\sim n\ell$.[@gurevich00] Many applications of superconductors are based on thin films where this tendency must be even stronger since the relation between the magnetic field and current is [*nonlocal*]{}.[@brandt93] Usually this leads to poorer performance of superconducting devices whose global properties are deteriorated by numerous natural defects blocking the current flow. However, the same tendency can help control the flux motion on a global scale by patterning the superconductor with arrays of small holes designed, e.g., to guide the flux in a particular direction.[@wordenweber04; @yurchenko06]
Surprisingly, a quantitative understanding on how a single local defect affects the flux penetration into a superconducting film is still rather poor. Even the simple case of an infinitely long thin strip with a semicircular edge-indentation is not solved. For a bulk superconductor in the critical state such an indentation creates an excess flux penetration exactly equal to the indentation radius.[@CCS] However the nonlocal electrodynamics in thin films and hence the presence of Meissner currents in the flux-free regions make the picture much more complicated.[@eisenmenger01] It has been observed using magneto-optic imaging that the excess flux penetration in films can significantly exceed the size of the indentation it originates from.[@schuster96] The physical mechanism behind this enhancement is however not yet understood. It could be related to the effect of thin-film geometry, to the flux creep or to thermal instabilities nucleated at the indentation.
This work aims to clarify this question by presenting a detailed study of flux penetration into a strip with a semicircular edge-indentation in the flux creep regime. We determine how the excess penetration depends on the size of the indentation, the applied magnetic field and the creep exponent $n$.
Model
=====
Consider a thin superconducting strip of thickness $d$ placed in a transverse magnetic field. The strip is infinite in the $y$-direction, has the width $2w\gg d$ in the $x$-direction and a semicircular indentation with radius $r_0$ at the edge, see Fig. \[fig:sample\]. The flux dynamics in the creep regime is conventionally described using a local relation between electric field $\mathbf E$ and current density $\mathbf j$,[@brandt95-prl; @zeldov90; @brandt95] $$\mathbf E = \rho \mathbf j\, ,$$ with a highly nonlinear resistivity $$\rho=\rho_0\left(j/j_c\right)^{n-1}\, ,
\label{material-law}$$ which does not explicitly depend on the magnetic induction $B$. Here $\rho_0$ is a constant, $j_c$ is the critical current density, while $n$ is the creep exponent, $n\gg 1$. This exponent can be related to the activation energy $U$ for thermal depinning as $n\sim U/kT$. Hence, large $n$ means small creep, and the Bean critical state model[@bean64] is regained in the limit $n\to\infty$.
For numerical simulations of flux penetration into the strip we use the formalism developed by Brandt[@brandt92; @brandt93; @schuster95; @brandt95-prl; @brandt95; @mints96; @brandt97; @brandt01; @brandt05] that can be applied to thin type-II superconductors of various shapes. For a thin superconductor, it is appropriate to look at length scales larger than the thickness $d$, and introduce a sheet current $$\mathbf{J}(\mathbf r)=\int_{-d/2}^{d/2} dz~\mathbf{j}(\mathbf r,z)\, ,$$ where $\mathbf r=(x,y)$ are in-plane coordinates. Due to the current conservation, $\nabla\cdot\mathbf J=0$, the sheet current can be expressed through a scalar function $g(\mathbf r)$ as $$\label{eq:cd001}
\mathbf J = \nabla \times \hat z g$$ where $g$ has the interpretation of the local magnetization.[@brandt95-prl] Substituting the current from Eq. (\[eq:cd001\]) into the Biot-Savart law one arrives at a non-local relation between $B_z$ and $g$, $$B_z(\mathbf r,z)
= \mu_0H_a+
\int_{A} d^2 r'\, Q(\mathbf r, \mathbf r',z)\, g(\mathbf r') \, .
\label{hfromg}$$ Here $H_a$ is the applied field and $A$ is the sample area. The integral kernel is equal to the field of a dipole of unit strength, $$Q(\mathbf r,\mathbf r', z) = \frac{\mu_0}{4\pi}
\frac{2z^2-(\mathbf r - \mathbf r')^2}
{\left[z^2+(\mathbf r-\mathbf r')^2\right]^{5/2}}\, .
\label{kernel1}$$ The integral Eq. with kernel Eq. is divergent at $\mathbf r \to \mathbf r'$ and $z\to 0$. In a numerical procedure, the divergence can be handled in three ways: (i) by keeping a finite $z$ during the calculation;[@loerincz04] (ii) by working in the Fourier space;[@brandt95] (iii) by converting the integral to a matrix form and using the flux conservation to determine the diagonal elements.[@brandt05; @brandt92; @brandt01] Here we use the third method. Since, for $H_a=0$, the total flux through the $z = 0$ plane is zero, the kernel should have the property $\int d^2r\, Q(\mathbf r,\mathbf r',0)=0$. This yields $$\frac{1}{\mu_0}B_z(\mathbf r)
= H_a+
g(\mathbf r) C(\mathbf r)
-
\int_{A} \frac{d^2r'}{4\pi}
\frac{g(\mathbf r')- g(\mathbf r)}
{|\mathbf r- \mathbf r'|^3}
,
\label{biot-savart-2}$$ where the scalar function $C$ is an integral over the area *outside* the superconductor $$\label{C}
C(\mathbf r)
=
\int_{\text{outside}}
\frac{dr'^2}
{4\pi |\mathbf r - \mathbf r'|^3}\, .$$ For a uniform strip of width $2w$ it yields $$C_\text{strip}(x)=\frac{1}{\pi}\frac{w}{w^2-x^2}.$$ In addition, the indentation gives a contribution from the semicircle, which is calculated numerically from Eq. .
\[fig:H\]
In the following we use an equidistant square grid and ascribe the same area $s$ to each grid point. The discrete version of the kernel then acquires the form[@brandt05] $$\frac{Q_{ij}}{\mu_0}
=
\delta_{ij}\left(\frac{C_i}{s}+\sum_l q_{il}\right)
- q_{ij} \, ,
\label{kernel-discrete}$$ where $q_{ij}= 1/4\pi|\mathbf r_i - \mathbf r_j|^3$ for $i\neq j$ and $q_{ii}=0$. All elements of the discrete kernel Eq. are nondivergent and the flux conservation, $\int d^2r \, B(\mathbf r)=0$, is guaranteed. Relating the magnetic field and the current by the Faraday’s law, and using the inverted Biot-Savart law one obtains the dynamic equation for the local magnetization: $$\dot g(\mathbf r,t)
=
\int_A d^2 r'\, Q^{-1}(\mathbf r,\mathbf r')\, \left[\hat{f}
g(\mathbf{r}',t)
-\dot H_a(t)\right]
,
\label{dynamics}$$ where $$\hat{f} g \equiv \nabla\cdot(\rho\nabla g) / d\mu_0\, .$$ For discrete formulation of the problem the inverse kernel $Q^{-1}$ is just the inverse of the matrix Eq. , hence the matrix must be calculated and inverted only once.
$\begin{array}{c}
\epsfig{file=fluxfront-n.eps,width=9cm} \\
\end{array}$
$\begin{array}{c}
\epsfig{file=distortion_r0_10.eps,width=8cm} \\
\epsfig{file=distortion_r0_20.eps,width=8cm} \\
\end{array}$
Results and Discussion
======================
#### Magnetic field and current
The simulations were performed by ramping the applied field at a constant rate $\mu_0\dot H_a=\rho_0J_c/wd$, starting at zero field and a flux-free strip. The flux penetrates from the edges forming well-defined flux fronts that move towards the strip center as the applied field increases. Shown in Fig. \[fig:H\] is a typical result of the flux density distribution presented as seen in a magneto-optical image, i.e., the image brightness represents the magnitude of the perpendicular magnetic field. The sample edge is seen as a bright line, i.e., the flux density is highest at the edge.
Far from the indentation the flux penetration front is straight, and leaves a fraction $a/w$ of the strip in the flux-free Meissner state, seen here as a black region. The penetration of this straight front versus applied field is shown in Fig. \[fig:fluxfront\] for different values of the creep exponent. For large $n$ the simulations approach the Bean-model result,[@brandt93] $a_\text{Bean}=w/\cosh(\pi H_a/J_c)$, while for smaller $n$, i.e., stronger flux creep, the penetration is deeper, all as expected for a strip with straight edges.
Near the indentation the flux penetration largely follows the circular shape. At both sides of the indentation there are dark regions of reduced flux density. As penetration gets deeper these will become narrow $d$-lines, where the current stream lines make sharp turns.[@CCS] In the Bean limit $n\to\infty$, the $d$-lines of semicircular indentations have parabolic shape. With finite $n$ the parabolic shape is only approximated. However, the main effect of the indentation is that it pushes magnetic field deeper into the sample. In order to quantify this we define the excess penetration $\Delta$ as the difference between the deepest penetration and the penetration far away from the indentation. Fig. \[fig:distortion\] shows how $\Delta$ evolves with increasing $H_a$. Evidently, the excess penetration is not equal to the indentation radius, $r_0$, as in the case of the bulk Bean model.[@CCS; @schuster94; @gurevich00] Moreover, $\Delta$ turns out to be field-dependent. Initially, $\Delta$ increases, then reaches a maximum followed by a decrease at larger $H_a$. This surprising non-monotonous behaviour is supported by magneto-optical measurements of the flux penetration in a uniform YBa$_2$Cu$_3$O$_x$ film containing an edge defect, see Fig. \[fig:ybco\]. The film was shaped as a strip of half-width $w=0.4$ mm, and the figure shows the flux distribution at 25 K for 3 different applied fields. In (a) the field was very small, $\mu_0 H_a = 3 $ mT, creating negligible penetration so that the actual shape of the defect appears in the image as the bright “bay area” inside the strip. In this state the excess penetration is equal to the depth of the defect, and measures $\Delta = 80~\mu$m. In (b) and (c) the applied field is 17 mT and 36 mT, respectively, and the corresponding excess penetration is $\Delta = 115~\mu$m and 100 $\mu$m. This gives for $\Delta/w = 0.20, 0.29$ and 0.25, demonstrating an excess penetration that exceeds the depth of the indentation by nearly 40 %, in very good agreement with the Bean model results plotted in Fig. \[fig:distortion\].
The Fig. \[fig:distortion\] includes the behaviour of $\Delta/w$ for two different $r_0/w$. Comparing the two panels we see that larger indentations produce a larger $\Delta$. However, the relative excess penetration, $\Delta/r_0$ is larger for the [*smaller*]{} indentation. The excess penetration can exceed the indentation depth by almost 50 % for $r_0=0.1w$ and large values of $n$. For smaller values of the creep exponent one always finds smaller $\Delta$, implying that creep tends to smoothen perturbations in the flux front.[@gurevich00]
Our results demonstrate that an indentation in a thin film affects the flux distribution in a stronger and more complex way than it does in bulk superconductors. This must be due to the non-local electrodynamics of thin films, and in particular due to the presence of Meissner currents in the flux free regions. These Meissner currents do not make the same sharp turns as the critical currents in the flux penetrated region, see Fig. \[fig:H+J\] and also Refs. . As a result, the Meissner currents concentrate in front of the indentation where their density reaches $j_c$ and hence leads to even deeper flux penetration. This is why the flux front near the indentation advances faster than in the rest of the film. This accelerated advancement eventually terminates when the penetration depth becomes comparable to the strip halfwidth. The reason is simply that in the limit of full penetration all flux fronts reach the middle of the strip and hence $\Delta \to 0$.
$
\begin{array}{l}
\begin{array}{cc}
\epsfig{file=detail-1.eps,height=2.4cm} &
\epsfig{file=detail-2.eps,height=2.4cm}
\end{array} \\
\epsfig{file=detail-3.eps,height=2.4cm}
\end{array}
$
#### Electric field
The Lorentz force pushing magnetic flux is directed perpendicular to the local current density. Even a small indentation distorts the current stream lines over a large area, and hence significantly modifies the trajectories of flux motion. In particular, all the flux arriving to the fan-shaped region rooted at the indentation must have entered the sample through this indentation, see Fig. \[fig:ybco\]. It creates a dramatic local enhancement of electric field since $E$ is a direct measure of the intensity of flux traffic.
Analytical solution for the electric field distribution around an indentation in thin films is not available. Therefore the results obtained for the case of a slab are often utilized as approximations also for films.[@mints96; @gurevich00; @gurevich01] We will now analyze to what extent such estimates are valid by comparing them with our simulation results for a strip.
In the fan-shaped region that originates from the semicircular indentation, the electric field can be found by solving the Maxwell equation $\nabla\times \mathbf E=-\dot {\mathbf B}$ in cylindrical coordinates. Since the evolution of $B$-distribution is usually not very far from the Bean model, one can assume $\dot B=\mu_0\dot H_a$, which leads to the solution[@mints96] $$E_1(x) = \frac{\mu_0\dot
H_a}{2}\left[\frac{\left(w-a+r_0\right)^2}{w-|x|}-(w-|x|)\right] \label{eq:bulk001}$$ for $|x| > a - r_0$ and zero for $|x| < a-r_0$. Far away from the indentation the solution of the same equation in cartesian coordinates, $\partial_x E=-\mu_0\dot H_a$, is $$E_0(x) = \mu_0\dot H_a\left(|x|-a\right) \label{eq:bulk0012}$$ for $|x|>a$ and zero for $|x|<a$. Note that the width $w$ enters Eq. (\[eq:bulk001\]) only because of the specific choice of the $x$-coordinate, where the edge is located at $x=w$. Replacement $x \to x+w$ removes the $w$-dependence.
$\begin{array}{ccc}
\epsfig{file=H1.eps,width=2.4cm} &
\epsfig{file=H2.eps,width=2.4cm} &
\epsfig{file=H3.eps,width=2.4cm} \\
\epsfig{file=j1.eps,width=2.4cm} &
\epsfig{file=j2.eps,width=2.4cm} &
\epsfig{file=j3.eps,width=2.4cm}
\end{array}$
Figure \[fig:E-profile\] compares $E_0(x)$ and $E_1(x)$ with the simulated electric field profiles. The quantitative agreement is poor, though the shape of profiles (both across the indentation and away from it) is fairly well reproduced, in agreement with Ref. . The expected enhancement of $E$ due to indentation is also obvious. The formulas above predict the relative enhancement for the peak values $E_1^{(\max)} / E_0^{(\max)} = (w-a)/2r_0+1$ for a bulk sample. One can see from the plot that the effect of indentation is even stronger for thin films: the ratio $E_1^{(\max)} / E_0^{(\max)}$ is slightly higher and the excess penetration is larger (the flux front here corresponds to the point where $E(x)=0$).
A locally enhanced electric field near edge indentations and hence enhanced Joule heating is predicted to facilitate nucleation of a thermal instability.[@mints96; @gurevich01] The instability in thin superconductors is usually observed in form of macroscopic dendritic flux avalanches[@denisov05] or macroscopic uniform flux jumps[@prozorov]. However, a third scenario is also possible when a series of [*microscopic*]{} flux avalanches repeatedly take place in the same region, each leading to a small advancement of the flux front.[@shantsev05] It creates an additional front distortion since the avalanches are expected to be larger and occur more frequently at the indentation, where the local $E$ is maximal. Experimentally the individual avalanches can be very small, and hence it is not easy to determine whether the thermal effects contribute to an observed front distortion. To identify the penetration mechanism one can compare the observed flux profiles with the simulations. The maximal excess penetration due to non-thermal effects is found to be 150 % of the indentation radius for our parameters. Consequently, when the observed excess penetration is larger, the flux penetration probably occurs via thermal micro-avalanches.
Conclusions
===========
We have numerically solved the Maxwell equations to describe flux penetration into a thin superconducting strip with an edge-indentation and analyzed the time evolution of flux front in an increasing applied field, $H_a$. The excess penetration, $\Delta$, due to the indentation is not equal to the indentation radius, $r_0$, in contrast to the well-known case of a bulk superconductor in the Bean model. Three different mechanisms that influence the excess penetration were analyzed. (i) The nonlocal electrodynamics in films leads to a characteristic $\Delta(H_a)$ dependence with a smooth peak. The ratio $\Delta/r_0$ at the peak equals 1.5 when $r_0$ is 0.1 of the strip half-width and becomes even larger for smaller $r_0$. (ii) The flux creep always tends to smoothen the flux front and decrease the excess penetration. (iii) Thermal flux avalanches are more likely to occur at the indentation, which can increase the apparent front distortion. Our results can be very helpful in order to identify which of these three mechanisms is the dominant one in a concrete experiment.\
We thank C. Romero and Ch. Jooss for fruitful discussions. This work was supported financially by The Norwegian Research Council, Grant No. 158518/431 (NANOMAT) and by FUNMAT@UIO.
Numerical details
=================
The simulations are carried out on an equidistant square grid with $N\times N$ points, $x_m=w(2m+1)/N-w$ and $y_n=w(2n+1)/N-w$, for $0\leq m,n<N$. The system has two symmetries that must be incorporated in the kernel: first, the periodic boundary, which means that we must add a mirror strips at $x<-w$ and $x>w$. Second, the symmetry around $x=0$. The latter means that we can work with half the kernel.[@brandt95] The simulations use a grid size of $N=100$, which means that a $5000\times 5000$ matrix must be put in memory and inverted. The memory consumption is the main limiting factor of the simulations. The kernel is stable, so there is no need for additional smoothening. For most exponents a pure power law is used, but for the Bean limit, $n=101$, a cutoff on the resistivity $\rho<\rho_{\text{max}}$ was necessary to ensure stability. The flux front position was determined at every time step and then smoothened as a function of time. It allows the front position to be determined with an accuracy much better than the distance between two grid points.
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abstract: 'Interaction between a central outflow and a surrounding wind is common in astrophysical sources powered by accretion. Understanding how the interaction might help to collimate the inner central outflow is of interest for assessing astrophysical jet formation paradigms. In this context, we studied the interaction between two nested supersonic plasma flows generated by focusing a long pulse high-energy laser beam onto a solid target. A nested geometry was created by shaping the energy distribution at the focal spot with a dedicated phase plate. Optical and X-ray diagnostics were used to study the interacting flows. Experimental results and numerical hydrodynamic simulations indeed show the formation of strongly collimated jets. Our work experimentally confirms the “shock-focused inertial confinement" mechanism proposed in previous theoretical astrophysics investigations.'
author:
- 'R. Yurchak'
- 'A. Ravasio'
- 'A. Pelka'
- 'S. Pikuz Jr.'
- 'E. Falize'
- 'T. Vinci'
- 'M. Koenig'
- 'B. Loupias'
- 'A. Benuzzi-Mounaix'
- 'M. Fatenejad'
- 'P. Tzeferacos'
- 'D. Q. Lamb'
- 'E.G. Blackman'
title: Experimental demonstration of an inertial collimation mechanism in nested outflows
---
[^1]
[*Introduction*]{}-Supersonic jets are common in astrophysics, emanate from such sources as newly forming young stellar objects (YSOs) [@reipurth01], active galactic nuclei (AGN) [@Ferrari1998; @Marscher2006], planetary and pre-planetary nebulae (PPN and PPN) [@Balick] and micro-quasars [@mirabel]. Their sustained collimation over large distances is not yet completely understood. Both magnetohydrodynamic (MHD) and hydrodynamic (HD) processes may be important. Often the jets propagate within a surrounding wind or envelope, as observed in YSOs [@arce02], AGN [@Tombesi2012], and in PN, where fast collimated winds sweep into a slower denser wind ejected most strongly during the PPN phase [@bujarrabal; @rizzo2013]. For YSOs and AGN a direct connection between disks and jets has been established and there is emerging consensus for such in the PPN/PN context as well [@soker98; @witt09; @blackmanlucchini]. The question of how different time-dependent ambient thermal and ram pressures affect jet collimation arises quite generally [@icke93; @Frank1994; @Frank1996; @BlackmanApJ2004; @DennisApJ2009].\
The role of the ambient medium can be important even if the inner outflows are magnetically driven [@Fendt; @lb03]. Recent 3D MHD simulations of laser driven plasma experiments have looked at the possible magnetic field collimation of wide-angle winds into HD jets [@Ciardi] and interpreted this as analogous to hydrodynamic collimation of an inner flow by a torus. Astrophysical jet launch regions are generally not observationally resolved, being obscured by high opacities. It is there- fore valuable to distill the distinct physics of MHD and HD effects via alternative methods.\
Combined with numerical simulations and theory, experiments bring new contributions to the subject. Some jet propagation and collimation mechanisms within steady ambient backgrounds have been studied experimentally [@Loupias2007; @Farley1999; @Lebedev2002]. Crosswinds were also used to study jet deflection and C-shape structures [@Lebedev2004]. Here we present results from a new experimental approach aimed at investigating the time dependent HD collimation of an inner isotropically supersonic expanding plasma by a surrounding time-evolving supersonic ambient flow. [*Experimental Setup-*]{}The experiment was performed on the LULI2000 laser facility at the LULI Laboratory, in France. The set-up is shown schematically in Fig. \[fig: setup\]. A long pulse ($\tau_L\sim$ 1.5 ns) high-energy ($E_L\sim$ 400 J at $\lambda_L=$ 527 nm) laser beam was used to produce supersonic plasma flows via interaction with solid targets. To create the nested configuration, we have designed a phase plate able to generate a laser energy distribution with a (100 $\mu$m) central circular spot and a thinner (75 $\mu$m) outer ring. Targets manufactured to match this pattern were made of a central iron disk (15 $\mu$m thick Fe) and a peripheral plastic ring (CH, also 15 $\mu$m thick), sitting on a CHAl pusher. Upon laser impact, a shock wave is launched in the pusher and transmitted to the Fe and CH layers. Once this shock reaches the rear side of the target, supersonic plasmas are formed from the outer plastic ring and from the central iron disk.\
To probe the interacting flows, we used rear-side and transverse optical diagnostics, in addition to transverse X-ray radiography. Optical probes were applied to the low density CH plasma and X-rays were used to characterize the inner iron flow which is opaque to optical radiation. Transverse optical diagnostics included time-resolved self-emission, shadowography and interferometry, while time-resolved 1D self-emission and 2D self-emission snapshots were implemented at the rear side of the target. This ensemble allowed us to measure the plastic flow velocity, morphology and electron density.\
[*Results-*]{} Typical data showing time resolved 1-D self-emission are shown in Fig. \[fig: SOP1D\], In Fig. \[fig: SOP1D\]a, the ring part of the laser spot was blocked and the data show a typical plasma release, with the iron expanding and cooling into vacuum. Fig. \[fig: SOP1D\]b shows the case of a complete target, wtih the plastic ring added. Since the plastic is transparent to visible light, we can follow the shock front in the plastic layer, and measure the shock velocity *D*. Typical *D* values are $\sim$30 km/s. When the shock breaks out, the CH unloads into vacuum at $\sim$70 km/s, as measured from the transverse time-resolved self-emission. The shock wave propagates more slowly in iron than plastic because of different impedances. Therefore the Fe flow unloads into vacuum after the CH and then swiftly collides with the radially expanding CH. Bright emission is observed from this collision and it is associated to a shock generation, as we will discuss below. The time evolution of the collision emission shows that the iron is confined by the surrounding CH flow. Later, the CH flows themselves collapse on axis, generating highly collimated emission.
X-ray radiography confirms the iron collimation by the plastic flows (Fig. \[radiography:time\_series\]). The X-ray source was generated by driving a copper back-lighter with the short pulse beam ($\sim$1 ps) of the pico2000 laser system. The incident X-ray spectrum accounted for intense K-$\alpha$ emission line at $\sim$8 keV, superimposed on a weaker bremsstrahlung continuum. At these X-ray energies, the outer plastic plasma is nearly transparent, while the iron flow is highly absorbing. The central (iron) jet morphology can therefore be identified without being disturbed by the surrounding flow. Again we have compared shots taken with and without the surrounding plastic. Typical results are presented in figure \[radiography:time\_series\]a and \[radiography:time\_series\]e for a probing delay of 35 ns. As in Fig. \[fig: SOP1D\]a, the case of iron alone (Fig. \[radiography:time\_series\]a) exhibits quasi-spherical adiabatic expansion. As soon as the plastic flow is added, the expansion is strongly reduced and the flow is collimated. Fig. \[radiography:time\_series\]e thus confirms what the optical data suggested more indirectly. By varying the delay of the back-lighter to the main beam from 8ns to 100ns, we monitored the time evolution of the iron flow. The results are presented in Fig.\[radiography:time\_series\]b-h, confirming the emergence of a thin jet from an initially uncollimated plasma.
Overall, the data of Fig. \[radiography:time\_series\] reveal different phases of jet evolution. First, the iron expands. A high density layer is formed at the interface between the iron and plastic plasma, detected as a reduced transmission at the Fe boundary in the radiographs of figure \[radiography:time\_series\]c. The plasma flow is subsequently focused on the axis, with a convergence point clearly observable at 35 ns (Fig. \[radiography:time\_series\]e). At longer times, a narrow collimated feature is observed remaining stable for 80 ns (figure \[radiography:time\_series\]g).\
The jet longitudinal extension (i.e. along the propagation axis) linearly increases from a few 100 $\mu$m at earlier times (Fig. \[radiography:time\_series\]b-e) to mm scales at later times (Fig. \[radiography:time\_series\]f-g). Its radius shrinks in time and can be fit by the expression $r(\mu$m)$\sim$56$\cdot$e$^{-t (ns)/13}$+53. The iron confinement sustains high surface densities. From transmission data we measure 100 g$\cdot$ cm$^{-3}\cdot \mu$m at earlier times (8 ns) and 12g$\cdot$ cm$^{-3}\cdot \mu$m at 80 ns. The corresponding densities at the mid length of the jet are obtained by Abel inversion resulting in 0.6 g/cc and 0.1 g/cc respectively. At these densities, typical aspect ratios (AR, length-to-width) up to $\sim$5 are obtained.
[*Numerical Simulations and Physical Interpretation-*]{} Using the FLASH multi-physics AMR (Adaptive Mesh Refinement) code [@Fryxell2000] (recently extended to include high energy density physics capabilities [@Orban2013]), we have simulated the interaction of the iron in with the surrounding plastic in 2-D. We used the un-split 3-temperature HD solver with the energy deposition module and a radiation transfer modeled by multi-group diffusion with 32 groups. Iron and plastic shock breakout times measured by the rear-side self-emission diagnostics, are used to calibrate the laser intensity in the simulations. The reliability of the simulations up to 55 ns is also verified by interferometry and shadowgraphy data. At later times, the cumulative incertitude associated with equations of state, opacities, conduction models, species mixing, etc, limits the accuracy of the numerical results. Detailed modeling of the plasma parameters is beyond the present scope, but the present simulations do very much help to convey the global flow dynamics and physical processes\
Fig. \[Simulations\]a shows density and pressure maps at different times, with the corresponding synthetic radiographs (Fig. \[Simulations\]b-f). These panels can be directly compared to the experimental results. Generally, there is substantial agreement with the data shown in Fig. \[radiography:time\_series\]b-h with respect to the presence and time evolution of the iron jet. All of the different evolutionary phases are seen in the simulations: confinement; focusing with a convergence point (Fig. \[Simulations\]c-d); and long lasting collimation (Fig. \[Simulations\]e-f). Simulations also show that the iron collimation initiates at a shock wave from the collision between supersonic CH and Fe plasmas. Signatures of the shock are seen in the simulated density and pressure maps, which indicate three density discontinuities and two pressure jumps, corresponding to the inner and outer density features (Fig. \[fig: streamlines\]). These features correspond to a transmitted shock in Fe, a reflected shock in CH and a contact discontinuity between them. The shock is also seen in the simulated X-ray radiographs as a stronger absorption layer and corresponds to the high density shocked iron, already observed in the experimental data. Together with the increase in the emitted radiation recorded at the iron/plastic boundary from the rear side self-emission, the experimental measurements are consistent with the picture revealed by simulations.
The presence of the shock and its shape reveals the dynamics and collimation of the iron flow. The shock shape is determined by the relative expansion of the CH and Fe plasmas. In our case, a converging conical shock is generated in Fe, since the CH plasma forms earlier and expands farther than the Fe. As the Fe expands and strikes the shock surface obliquely, only the normal component of the post-shock velocity is reduced, so the shock marks the locus at which the Fe flow vector focuses toward the axis. (Fig. \[fig: streamlines\]). This mechanism was previously identified analytically in astrophysical studies [@Sanders] and later in HD simulations [@icke93; @Frank1994; @Frank1996b]. It is called “shock-focused inertial confinement" (SFIC) and may help collimate flows from Young Stellar Objects (YSO) and PN, particularly when cooling is added [@Mellema1997]. Our results give the first experimental confirmation of this scenario, without cooling. [*Astrophysical Relevance-*]{} The importance of the experiment is bolstered when experimental parameters correspond to those of specific astrophysical systems [@Ryutov99]. The characteristic experimental parameters for the iron flow are shown in the first column of table \[tab:table1\]. They indicate a highly collimated (AR$\sim5$), supersonic flow (M$\sim$10) in a pure HD regime where radiative ($\chi\gg1$) and microphysical conductive ($Pe\gg1$) effects are negligible. Table \[tab:table1\] also shows representative parameter regimes for YSOs [@Hartigan1993; @reipurth01; @arce02], AGN [@Ferrari1998; @Tombesi2012], and PPN [@bujarrabal; @Balick; @blackmanlucchini; @witt09; @rizzo2013], near the inner collimation scales of these jets. Note that $Pe>>1$ in all cases so that microphysics of thermal conduction does no affect the bulk dynamics. YSOs are the most similar to the experiments, except for their cooling.\
For PPN, the young jets of low density seem to interact with the denser wind of the post-AGB star [@witt09], resulting in a density ratio $<1$. AGN jets are also of lower density than their surrounding wide angle winds and they are relativistic, differing in those respects from the experiments. Nevertheless, jet collimation in the experiment arises because the momentum of the outer outflow can redirect that of the inner outflow, and this requirement would be the same regardless of the density ratio of whether the flows are relativistic. So the nested wind structure and basic principles of inertial collimation still apply to PPN and AGN, but the specific predictions for shock location and geometry could be different. *Conclusions-* Motivated by astrophysical contexts where jets are associated with accretion engines, we have established an experimental platform to study the collimating interaction between high Mach outflows. We have experimentally confirmed the efficacy of inertial mechanisms in producing highly collimated outflows in an HD regime, similar to jets from YSOs, but also relevant for systems such as AGN and PPN. Most importantly, we have experimentally verified the SFIC mechanism suggested in previous astrophysical studies, exemplifying the contribution that such laboratory experiments can bring to the enterprise of astrophysics. Further insights on the jet collimation paradigm can be accessed by combining inertial with magnetic and radiative effects in future experiments.
We gratefully acknowledge support of the technical staff at the LULI 2000 laser facility and H. Nakatsutsumi for target fabrication. The FLASH code was developed in part by the U.S. DOE and NSF-funded Flash Center for Computational Science at the Univ. of Chicago. This work was performed using HPC resources from GENCI-IDRIS (Grant 2013-i2013057066). EB acknowledges NSF grant AST-1109285.
[40]{}
B. Reipurth, J. Bally, ARA&A, 39, **403** (2001) A. Ferrari, ARA&A **36** 539 (1998). A.P. Marscher AIP Conf. Proc. **856**, 1 (2006). B. Balick, A. Frank, ARA&A **40**, 439 (2002). I.F. Mirabel, L.F. Rodr[í]{}guez ARA&A, 37, **409**(1999). H. Arce, A.A. Goodman, ApJ, 575, **928** (2002). F. Tombesi *et al.* Mon. Not. R. Astron. Soc, **424**, 754 (2012). V. Bujarrabal , A. Castro-Carrizo, J. Alcolea, C. S[á]{}nchez Contreras, A&A, 377, **868** (2001) J.R. Rizzo et al. A&A, 560, **A82** (2013) N. Soker, ApJ, 496, **833** (1998) E.G. Blackman, S. Lucchini , in press MNRAS, arXiv:1312.5372 (2014) A.N. Witt et al. ApJ, 693, **1946** (2009) V. Icke, G. Mellema, B. Balick, F. Eulderink, A. Frank, Nature, 355 **524**, (1993) A. Frank, A. Noriega-Crespo, Astron. Astrophys. **290**, 643 (1994). A. Frank *et al.* **471**, **L53** (1996). E.G. Blackman, R. Perna, , **601**, L71 (2004). T.J. Dennis *et al.*, **707**, 1485 (2009). C. Fendt, , **692**, 346 (2009) D. Lynden-Bell, MNRAS, 341, **1360** (2003). A. Ciardi *et al.*, **110**, 025002 (2013). S.V. Lebedev *et al.*, **564**, 113 (2002). B. Loupias *et al.*, **99**, 265001 (2007). D.R. Farley *et al.*, **83**, 1982 (1999). S.V. Lebedev *et al.*, **616**, 988 (2004). B. Fryxell, *et al.*., **131**, 273 (2000). C. Orban, *et al.* eprint arXiv:1306.1584 (2013). R.H. Sanders, , **266**, 73 (1983). A. Frank, G. Mellema **472**, 684 (1996). G. Mellema,A. Frank, Mon. Not. R. Astr. Soc. **292**, 795 (1997). D.D. Ryutov *et al* **518**, 821 (1999). P. Hartigan *et al.*, , **414**, L121 (1993).
FIG. 1: Schematic of the experimental setup, focal spot (measured) and target geometry. The laser energy distribution is modeled by means of a hybrid phase plate (HPP) resulting in a central disc and an outer ring. The nested target fits this geometry, consisting in a central iron (Fe 15$\mu$m) disc and and outer plastic (CH 15$\mu$m) ring on a common CHAl pusher. The laser hits the front side of the target and the plasma emanates from the rear..\
FIG. 2: Typical time resolved rear side self-emission data for a) the iron disc only and b) the complete nested target.\
FIG. 3: X-ray radiography of the iron flows from a) the central disk alone, where the outer ring was blocked on the laser side and b)-h) from the complete nested target. In this case, the temporal evolution shows the formation of a collimated flow being stable up to very long delays (80 ns) and reaching mm sizes. In the insertion at 100 ns we have enhanced the contrast.\
FIG. 4: Upper panel: simulated density (upper) and pressure (lower) maps at different time delays. Lower panel: corresponding synthetic radiographs obtained using the simulated density map and the experimental X-ray spectrum.\
FIG. 5: Streamlines on density and pressure maps for simulations at 8 ns (a) and 20 ns (b) showing the “focusing" of the iron on the axis as it strikes the shock. The transmitted shock in Fe, the reflected shock in CH and the contact discontinuity in-between are clearly distinguishable in the density map at 20 ns. Color scheme is same as in Fig. 4\
TABLE 1: Parameter values for the experiment and typical astrophysical cases on scales where collimation occurs. The index j stands for “jet" while the index a for “ambient" , represented in the experiment by the Fe and CH respectively . V is the velocity, $c_s$ the sound velocity, l the longitudinal length, r the radial extension, $\rho$ the density, $t_{rad}$ is the cooling time, while $t_{hydro}$ is the hydrodynamic one. $\chi$ is the thermal diffusivity, calculated as in [@Ryutov99]. c is the speed of light
![Schematic of the experimental setup, focal spot (measured) and target geometry. The laser energy distribution is modeled with a hybrid phase plate (HPP) resulting in a central disc and an outer ring. The nested target fits this geometry, consisting in a central iron (Fe 15$\mu$m) disc and and outer plastic (CH 15$\mu$m) ring on a common CH-Al pusher. []{data-label="fig: setup"}](SetUP_b.pdf)
![Typical time resolved rear side self-emission data for a) the iron disc only and b) the complete nested target.[]{data-label="fig: SOP1D"}](SOP1D_b.pdf)
![X-ray radiography of the iron flows from a) the central disk alone, where the outer ring was blocked on the laser side and b)-h) from the complete nested target In this case, we see the formation of a collimated flow being stable up to very long delays (80 ns) and reaching mm sizes. In the insertion at 100 ns we have enhanced the contrast. \[radiography:time\_series\]](testRadio_e){width="100.00000%"}
![Upper panel: simulated density (upper) and pressure (lower) maps at different time delays. Lower panel: corresponding synthetic radiographs obtained using the simulated density map and the experimental X-ray spectrum. \[Simulations\]](Sim_e.pdf){width="100.00000%"}
![Streamlines on density and pressure maps for simulations at 8 ns (a) and 20 ns (b) showing the “focusing" of the iron on the axis as it strikes the shock. The transmitted shock in Fe, the reflected shock in CH and the contact discontinuity in-between are clearly distinguishable in the density map at 20 ns. Color scheme is same as in Fig. \[Simulations\][]{data-label="fig: streamlines"}](Test_streamline-e.pdf){width="50.00000%"}
Parameter Lab. YSO PPN AGN
------------------------------------ ----------- -------------- ------------ ----------
collimation scale $ 1$mm $10^{-3} pc$ $<0.01$pc $$0.1pc
Int. Mach $M_{int}$=$V_j/c_{s,j}$ 5-10 $>10$ $>10 $ $>10$
Ext. Mach $M_{ext}$=$V_j/c_{s,a}$ 5-10 $>10$ $>10$ $> 10$
aspect ratio $AR=l_j/r_j$ 5 10 10 $>10 $
density ratio $\eta=\rho_j/\rho_a$ 5-10 10 $< 1$ $<<1$
Cooling $\chi=t_{rad}/t_{hydro}$ $100$ $<1$ $<1$ $>>1$
Peclet $Pe=\rho r V_j/\chi$ $10^4$ $>>1$ $>>1$ $>>1$
$\beta=V_j/c$ $10^{-4}$ 10$^{-3}$ $ 10^{-3}$ 0.9-0.99
: Experimental vs. astrophysical parameters on scales where collimation occurs. The indices j and a refer to “jet” and “ambient" , represented in the experiment by the Fe and CH respectively . V is velocity, $c_s$ is sound speed, l is the longitudinal length, $r$ the radial extension, $\rho$ the density, $t_{rad}$ is the cooling time, $t_{hydro}$ is the hydrodynamic time. $\chi$ is the thermal diffusivity, as in [@Ryutov99]. $c$ is the speed of light[]{data-label="tab:table1"}
[^1]: alessandra.ravasio@polytechnique.fr
|
---
author:
- '[**Bhaskar Bagchi**]{}$^{\rm a}$ and [**Basudeb Datta**]{}$^{\rm b}$'
date:
-
- 'To appear in ‘Journal of Combinatorial Theory, Ser. A’'
title: '**Minimal triangulations of sphere bundles over the circle**'
---
9.14in
-0.6in -0.6in
\[section\] \[section\] \[section\] \[theo\][[**Corollary**]{}]{} \[section\] \[section\]
=msbm10 scaled1
=msbm10 scaled2
$^{\rm a}$Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore 560059, India
$^{\rm b}$Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
------------------------------------------------------------------------
[**Abstract**]{}
[For integers $d \geq 2$ and $\varepsilon = 0$ or $1$, let $S^{\hspace{.1mm}1, d - 1}(\varepsilon)$ denote the sphere product $S^{\hspace{.1mm}1} \times S^{\hspace{.1mm}d - 1}$ if $\varepsilon = 0$ and the twisted sphere product ${S^{\hspace{.1mm}1}\! \times \hspace{-3.3mm}_{-} \, S^{\hspace{.2mm}d-1}}$ if $\varepsilon = 1$. The main results of this paper are: $(a)$ if $d \equiv \varepsilon$ (mod 2) then $S^{\hspace{.1mm}1, d -
1}(\varepsilon)$ has a unique minimal triangulation using $2d+3$ vertices, and $(b)$ if $d \equiv 1 - \varepsilon$ (mod 2) then $S^{\hspace{.1mm}1, d - 1}(\varepsilon)$ has minimal triangulations (not unique) using $2d+4$ vertices. In this context, a minimal triangulation of a manifold is a triangulation using the least possible number of vertices. The second result confirms a recent conjecture of Lutz. The first result provides the first known infinite family of closed manifolds (other than spheres) for which the minimal triangulation is unique. Actually, we show that while $S^{\hspace{.1mm}1, d - 1}(\varepsilon)$ has at most one $(2d + 3)$-vertex triangulation (one if $d \equiv
\varepsilon$ (mod 2), zero otherwise), in sharp contrast, the number of non-isomorphic $(2d + 4)$-vertex triangulations of these $d$-manifolds grows exponentially with $d$ for either choice of $\varepsilon$. The result in $(a)$, as well as the minimality part in $(b)$, is a consequence of the following result: $(c)$ for $d\geq 3$, there is a unique $(2d +
3)$-vertex simplicial complex which triangulates a non-simply connected closed manifold of dimension $d$. This amazing simplicial complex was first constructed by Kühnel in 1986. Generalizing a 1987 result of Brehm and Kühnel, we prove that $(d)$ any triangulation of a non-simply connected closed $d$-manifold requires at least $2d + 3$ vertices. The result $(c)$ completely describes the case of equality in $(d)$. The proofs rest on the Lower Bound Theorem for normal pseudomanifolds and on a combinatorial version of Alexander duality.]{}
[*Mathematics Subject Classification (2000):*]{} 57Q15, 57R05.
[*Keywords:*]{} Triangulated manifolds; Stacked spheres; Non-simply connected manifolds.
------------------------------------------------------------------------
Preliminaries
=============
With a single exception in Section 3, all simplicial complexes considered here are finite. For a simplicial complex $X$, $V(X)$ will denote the set of all the vertices of $X$ and $|X|$ will denote the geometric carrier of $X$. One says that $X$ is a [*triangulation*]{} of the topological space $|X|$. If $|X|$ is a manifold then we say that $X$ is a [*triangulated manifold*]{}. The unique $(d + 2)$-vertex triangulation of the $d$-sphere $S^{\hspace{.1mm}d}$ is denoted by $S^{\hspace{.1mm}d}_{d + 2}$ and is called the [*standard $d$-sphere*]{}. The unique $(d+1)$-vertex triangulation of the $d$-ball is denoted by $B^{\hspace{.1mm}d}_{d + 1}$ and is called the [*standard $d$-ball*]{}. For $n\geq 3$, the unique $n$-vertex triangulation of the circle $S^1$ is denoted by $S^1_n$ and is called the [*$n$-cycle*]{}.
For $i = 1, 2$, the $i$-faces of a simplicial complex $K$ are also called the [*edges*]{} and [*triangles*]{} of $K$, respectively. For a simplicial complex $K$, the graph whose vertices and edges are the vertices and edges of $K$ is called the [*edge graph*]{} (or [*$1$-skeleton*]{}) of $K$. Recall that a [*graph*]{} is nothing but a simplicial complex of dimension at most $1$. A set of vertices in a graph is called a [*clique*]{} if these vertices are mutually adjacent (i.e., any two of them form an edge). Note that any simplex in a simplicial complex is a clique in its edge graph.
For a simplex $\sigma$ in a simplicial complex $K$, the number of vertices in ${\rm lk}_K(\sigma)$ is called the [*degree*]{} of $\sigma$ in $K$ and is denoted by $\deg_K(\sigma)$ (or by $\deg(\sigma)$). So, the degree of a vertex $v$ in $K$ is the same as the degree of $v$ in the edge graph of $K$.
Recall that for any face $\alpha$ of a complex $X$, its [*link*]{} ${\rm lk}_X(\alpha)$ is the simplicial complex whose faces are the faces $\beta$ of $X$ such that $\alpha\cap \beta =
\emptyset$ and $\alpha\cup \beta\in X$. Likewise, the [*star*]{} ${\rm st}_X(\alpha)$ of the face $\alpha$ has all the maximal faces $\gamma \supseteq \alpha$ of $X$ as its maximal faces.
A simplicial complex $X$ is called a [*combinatorial $d$-sphere*]{} (respectively, [*combinatorial $d$-ball*]{}) if $|X|$ (with the induced pl structure from $X$) is pl homeomorphic to $|S^{\hspace{.2mm}d}_{d + 2}|$ (respectively, $|B^{\hspace{.1mm}d}_{d + 1}|$). A simplicial complex $X$ is said to be a [*combinatorial $d$-manifold*]{} if $|X|$ (with the induced pl structure) is a pl $d$-manifold. Equivalently, $X$ is a combinatorial $d$-manifold if all its vertex links are combinatorial spheres or combinatorial balls. In this case, we also say that $X$ is a [*combinatorial triangulation*]{} of $|X|$. A simplicial complex $X$ is a combinatorial manifold without boundary if all its vertex links are combinatorial spheres. A combinatorial manifold will usually mean one without boundary.
A simplicial complex $K$ is called [*pure*]{} if all the maximal faces ([*facets*]{}) of $K$ have the same dimension. For $d \geq 1$, a $d$-dimensional pure simplicial complex is said to be a [*weak pseudomanifold*]{} if each $(d - 1)$-simplex is in exactly two facets. Clearly, any $d$-dimensional weak pseudomanifold has at least $d + 2$ vertices, with equality only for $S^{\hspace{.2mm}d}_{d+2}$.
For a pure $d$-dimensional simplicial complex $K$, let ${\Lambda}(K)$ be the graph whose vertices are the facets of $K$, two such vertices being adjacent in ${\Lambda}(K)$ if the corresponding facets intersect in a $(d - 1)$-face. If $\Lambda(K)$ is connected then $K$ is called [*strongly connected*]{}. A strongly connected weak pseudomanifold is called a [*pseudomanifold*]{}. Thus, for a $d$-pseudomanifold $K$, $\Lambda(K)$ is a connected $(d +
1)$-regular graph. This implies that $K$ has no proper subcomplex which is also a $d$-pseudomanifold. (Or else, the facets of such a subcomplex would provide a disconnection of $\Lambda(X)$.) By convention, $S^{\hspace{.2mm}0}_2$ is the only $0$-pseudomanifold.
A connected $d$-dimensional weak pseudomanifold is said to be a [*normal pseudomanifold*]{} if the links of all the simplices of dimension up to $d - 2$ are connected. Clearly, the normal $2$-pseudomanifolds are just the connected combinatorial $2$-manifolds. But, normal $d$-pseudomanifolds form a broader class than connected combinatorial $d$-manifolds for $d \geq 3$. In fact, any triangulation of a connected closed manifold is a normal pseudomanifold. Observe that if $X$ is a normal pseudomanifold then $X$ is a pseudomanifold. (If ${\Lambda}(X)$ is not connected then, since $X$ is connected, ${\Lambda}(X)$ has two components $G_1$ and $G_2$ and two intersecting facets $\sigma_1$, $\sigma_2$ such that $\sigma_i
\in G_i$, $i = 1, 2$. Choose $\sigma_1$, $\sigma_2$ among all such pairs such that $\dim(\sigma_1 \cap \sigma_2)$ is maximum. Then $\dim(\sigma_1 \cap \sigma_2) \leq d-2$ and ${\rm
lk}_X(\sigma_1 \cap \sigma_2)$ is not connected, a contradiction.) Notice that all the links of simplices of dimensions up to $d-2$ in a normal $d$-pseudomanifold are normal pseudomanifolds.
Let $X$, $Y$ be two simplicial complexes with disjoint vertex sets. (Since we identify isomorphic complexes, this is no real restriction on $X$, $Y$.) Then their [*join*]{} $X \ast Y$ is the simplicial complex whose simplices are those of $X$ and of $Y$, and the (disjoint) unions of simplices of $X$ with simplices of $Y$. It is easy to see that if $X$ and $Y$ are combinatorial spheres (respectively normal pseudomanifolds) then their join $X
\ast Y$ is a combinatorial sphere (respectively normal pseudomanifold).
By a [*subdivision*]{} of a simplicial complex $K$ we mean a simplicial complex $K^{\hspace{.1mm}\prime}$ together with a homeomorphism from $|K^{\hspace{.1mm}\prime}|$ onto $|K|$ which is facewise linear. Two complexes $K$, $L$ have isomorphic subdivisions if and only if $|K|$ and $|L|$ are pl homeomorphic. Let $X$ be a pure $d$-dimensional simplicial complex and $\sigma$ be a facet of $X$, then take a symbol $v$ outside $V(X)$ and consider the pure $d$-dimensional simplicial complex $Y$ with vertex set $V(X) \cup \{v\}$ whose facets are facets of $X$ other than $\sigma$ and the $(d + 1)$-sets $\tau \cup \{v\}$ where $\tau$ runs over the $(d - 1)$-simplices in $\sigma$. Clearly, $Y$ is a subdivision of $X$. The complex $Y$ is called the subdivision obtained from $X$ by [*starring a new vertex $v$ in the facet $\sigma$*]{}.
If $U$ is a non-empty subset of the vertex set $V(X)$ of a simplicial complex $X$ then the simplices of $X$ which are subsets of $U$ form a simplicial complex. This simplicial complex is called the [*induced subcomplex*]{} of $X$ on the vertex set $U$ and is denoted by $X[U]$.
$\!\!\!$[**.**]{} \[SC\] [If $Y$ is an induced subcomplex of a simplicial complex $X$ then the [*simplicial complement*]{} $C(Y, X)$ of $Y$ in $X$ is the induced subcomplex of $X$ with vertex set $V(X) \setminus
V(Y)$. By abuse of notation, for any face $\sigma$ of $X$, the induced subcomplex of $X$ on the complement of $\sigma$ will be denoted by $C(\sigma, X)$.]{}
$\!\!\!$[**.**]{} \[DAB\] [Let $\sigma_1$, $\sigma_2$ be two facets in a pure simplicial complex $X$. Let $\psi : \sigma_1 \to \sigma_2$ be a bijection. We shall say that $\psi$ is [*admissible*]{} if ($\psi$ is a bijection and) the distance between $x$ and $\psi(x)$ in the edge graph of $X$ is $\geq 3$ for each $x\in
\sigma_1$. Notice that if $\sigma_1$, $\sigma_2$ are from different connected components of $X$ then any bijection between them is admissible. Also note that, in general, for the existence of an admissible map $\psi : \sigma_1 \to \sigma_2$, the facets $\sigma_1$ and $\sigma_2$ must be disjoint.]{}
$\!\!\!$[**.**]{} \[DEHA\] [Let $X$ be a weak pseudomanifold with disjoint facets $\sigma_1$, $\sigma_2$. Let $\psi \colon \sigma_1\to \sigma_2$ be an admissible bijection. Let $X^{\psi}$ denote the weak pseudomanifold obtained from $X \setminus \{\sigma_1, \sigma_2\}$ by identifying $x$ with $\psi(x)$ for each $x\in \sigma_1$. Then $X^{\psi}$ is said to be obtained from $X$ by an [*elementary handle addition*]{}. If $X_1$, $X_2$ are two $d$-dimensional weak pseudomanifolds with disjoint vertex-sets, $\sigma_i$ a facet of $X_i$ ($i=1, 2$) and $\psi \colon \sigma_1 \to \sigma_2$ any bijection, then $(X_1\sqcup X_2)^{\psi}$ is called an [*elementary connected sum*]{} of $X_1$ and $X_2$, and is denoted by $X_1 \#_{\psi} X_2$ (or simply by $X_1\# X_2$). Note that the combinatorial type of $X_1 \#_{\psi} X_2$ depends on the choice of the bijection $\psi$. However, when $X_1$, $X_2$ are connected triangulated $d$-manifolds, $|X_1 \#_{\psi} X_2|$ is the topological connected sum of $|X_1|$ and $|X_2|$ (taken with appropriate orientations). Thus, $X_1 \#_{\psi} X_2$ is a triangulated manifold whenever $X_1$, $X_2$ are triangulated $d$-manifolds. ]{}
$\!\!\!$[**.**]{} \[sd-s\] Let $N$ be a $(d- 1)$-dimensional induced subcomplex of a $d$-dimensional simplicial complex $M$. If both $M$ and $N$ are normal pseudomanifolds then
1. for any vertex $u$ of $N$ and any vertex $v$ of the simplicial complement $C(N, M)$, there is a path $P$ $($in $M)$ joining $u$ to $v$ such that $u$ is the only vertex in $P\cap N$, and
2. the simplicial complement $C(N, M)$ has at most two connected components.
[**Proof.**]{} Part $(a)$ is trivial if $d=1$ (in which case, $N = S^{\hspace{.2mm}0}_2$ and $M = S^{\hspace{.2mm}1}_n$). So, assume $d > 1$ and we have the result for smaller dimensions. Clearly, there is a path $P$ (in the edge graph of $M$) joining $u$ to $v$ such that $P = x_1x_2 \cdots x_ky_1 \cdots y_l$ where $x_1 = u$, $y_l = v$ and $x_i$’s are the only vertices of $P$ from $N$. Choose $k$ to be the smallest possible. We claim that $k
=1$, so that the result follows. If not, then $x_{k-1} \in {\rm
lk}_N(x_k) \subset {\rm lk}_M(x_k)$ and $y_1 \in C({\rm
lk}_N(x_k), {\rm lk}_M(x_k))$. Then, by induction hypothesis, there is a path $Q$ in ${\rm lk}_M(x_k)$ joining $x_{k-1}$ and $y_1$ in which $x_{k-1}$ is the only vertex from ${\rm
lk}_N(x_k)$. Replacing the part $x_{k-1}x_ky_1$ of $P$ by the path $Q$, we get a path $P^{\hspace{.2mm} \prime}$ from $u$ to $v$ where only the first $k-1$ vertices of $P^{\hspace{.2mm}
\prime}$ are from $N$. This contradicts the choice of $k$.
The proof of Part $(b)$ is also by induction on the dimension $d$. The result is trivial for $d=1$. For $d > 1$, fix a vertex $u$ of $N$. By induction hypothesis, $C({\rm lk}_N(u), {\rm lk}_M(u))$ has at most two connected components. By Part $(a)$ of this lemma, every vertex $v$ of $C(N, M)$ is joined by a path in $C(N,
M)$ to a vertex in one of these components. Hence the result. $\Box$
Let $N$ be an induced subcomplex of a simplicial complex $M$. One says that $N$ is [*two-sided*]{} in $M$ if $|N|$ has a (tubular) neighbourhood in $|M|$ homeomorphic to $|N| \times [-1, 1]$ such that the image of $|N|$ (under this homeomorphism) is $|N|\times
\{0\}$.
$\!\!\!$[**.**]{} \[m-sts\] Let $M$ be a normal pseudomanifold of dimension $d \geq 2$ and $A$ be a set of vertices of $M$ such that the induced subcomplex $M[A]$ of $M$ on $A$ is a $(d - 1)$-dimensional normal pseudomanifold. Let $G$ be the graph whose vertices are the edges of $M$ with exactly one end in $A$, two such vertices being adjacent in $G$ if the union of the corresponding edges is a $2$-simplex of $M$. Then $G$ has at most two connected components. If, further, $M[A]$ is two-sided in $M$ then $G$ has exactly two connected components.
[**Proof.**]{} Let $E = V(G)$ be the set of edges of $M$ with exactly one end in $A$. For $x\in A$, set $E_x = \{e\in E :
x\in e\}$, and let $G_x = G[E_x]$ be the induced subgraph of $G$ on $E_x$. Note that $G_x$ is isomorphic to the edge graph of $C({\rm lk}_{M[A]}(x), {\rm lk}_M(x))$. Therefore, by Lemma \[sd-s\] $(b)$, $G_x$ has at most two components for each $x\in
A$. Also, for an edge $xy$ in $M[A]$, there is a $d$-simplex $\sigma$ of $M$ such that $xy$ is in $\sigma$. Since the induced complex $M[A]$ is $(d - 1)$-dimensional, there is a vertex $u \in
\sigma \setminus A$. Then $e_1 = xu \in E_x$ and $e_2 = yu \in
E_y$ are adjacent in $G$. Thus, if $x$, $y$ are adjacent vertices in $M[A]$ then there is an edge of $G$ between $E_x$ and $E_y$. Since $M[A]$ is connected and $V(G) = \cup_{x \in A} E_x$, it follows that $G$ has at most two connected components.
Now suppose $S = M[A]$ is two-sided in $M$. Let $U$ be a tubular neighbourhood of $|S|$ in $|M|$ such that $U \setminus |S|$ has two components, say $U^{+}$ and $U^{-}$. Since $|S|$ is compact, we can choose $U$ sufficiently small so that $U$ does not contain any vertex from $V(M) \setminus A$. Then, for $e \in E$, $|e|$ meets either $U^{+}$ or $U^{-}$ but not both. Put $E^{\pm} = \{e
\in E : |e| \cap U^{\pm} \neq \emptyset\}$. Then no element of $E^{+}$ is adjacent in $G$ with any element of $E^{-}$. From the previous argument, one sees that each $x\in A$ is in an edge from $E^{+}$ and in an edge from $E^{-}$. Thus, both $E^{+}$ and $E^{-}$ are non-empty. So, $G$ is disconnected. $\Box$
$\!\!\!$[**.**]{} \[LEHD\] Let $X$ be a normal $d$-pseudomanifold with an induced two-sided standard $(d - 1)$-sphere $S$. Then there is a $d$-dimensional weak pseudomanifold $\widetilde{X}$ such that $X$ is obtained from $\widetilde{X}$ by elementary handle addition. Further,
1. the connected components of $\widetilde{X}$ are normal $d$-pseudomanifolds,
2. $\widetilde{X}$ has at most two connected components,
3. if $\widetilde{X}$ is not connected, then $X = Y_1 \#
Y_2$, where $Y_1$, $Y_2$ are the connected components of $\widetilde{X}$, and
4. if $C(S, X)$ is connected then $\widetilde{X}$ is connected.
[**Proof.**]{} As above, let $E$ be the set of all edges of $X$ with exactly one end in $S$. Let $E^{+}$ and $E^{-}$ be the connected components of the graph $G$ (with vertex-set $E$) defined above (cf. Lemma \[m-sts\]). Notice that if a facet $\sigma$ intersects $V(S)$ then $\sigma$ contains edges from $E$, and the graph $G$ induces a connected subgraph on the set $E_{\sigma} = \{e\in E : e\subseteq \sigma\}$. (Indeed, this subgraph is the line graph of a complete bipartite graph.) Consequently, either $E_{\sigma} \subseteq E^{+}$ or $E_{\sigma}
\subseteq E^{-}$. Accordingly, we say that the facet $\sigma$ is positive or negative (relative to $S$). If a facet $\sigma$ of $X$ does not intersect $V(S)$ then we shall say that $\sigma$ is a neutral facet.
Let $V(S) = W$ and $V(X) \setminus V(S) = U$. Take two disjoint sets $W^{+}$ and $W^{-}$, both disjoint from $U$, together with two bijections $f_{\pm} \colon W \to W^{\pm}$. We define a pure simplicial complex $\widetilde{X}$ as follows. The vertex-set of $\widetilde{X}$ is $U \sqcup W^{+} \sqcup W^{-}$. The facets of $\widetilde{X}$ are: (i) $W^{+}$, $W^{-}$, (ii) all the neutral facets of $X$, (iii) for each positive facet $\sigma$ of $X$, the set $\widetilde{\sigma} := (\sigma \cap U) \sqcup f_{+}(\sigma
\cap W)$, and (iv) for each negative facet $\tau$ of $X$, the set $\widetilde{\tau} := (\tau \cap U) \sqcup f_{\!-}(\tau \cap W)$. Clearly, $\widetilde{X}$ is a weak pseudomanifold. Let $\psi =
f_{-} \circ f_{+}^{-1} \colon W^{+} \to W^{-}$. It is easy to see that $\psi$ is admissible and $X = (\widetilde{X})^{\psi}$.
Since the links of faces of dimension up to $d - 2$ in $X$ are connected, it follows that the links of faces of dimension up to $d - 2$ in $\widetilde{X}$ are connected. This proves $(a)$.
As $X$ is connected, choosing two vertices $f_{\pm}(x_0) \in
W^{\pm}$ of $\widetilde{X}$, one sees that each vertex of $\widetilde{X}$ is joined by a path in the edge graph of $\widetilde{X}$ to either $f_{+}(x_0)$ or $f_{-}(x_0)$. Hence $\widetilde{X}$ has at most two components. This proves $(b)$. This arguments also shows that when $\widetilde{X}$ is disconnected, $W^{+}$ and $W^{-}$ are facets in different components of $\widetilde{X}$. Hence $(c)$ follows.
Observe that $C(S, X) = C(W^{+} \sqcup W^{-}, \widetilde{X})$. Assume that $C(S, X)$ is connected. Now, for any $(d-1)$-simplex $\tau \subseteq W^{+}$, there is a vertex $x$ in $C(S, X)$ such that $\tau\cup\{x\}$ is a facet of $\widetilde{X}$. So, $C(S, X)$ and $W^{+}$ are in the same connected component of $\widetilde{X}$. Similarly, $C(S, X)$ and $W^{-}$ are in the same connected component of $\widetilde{X}$. This proves $(d)$. $\Box$
$\!\!\!$[**.**]{} \[DEHD\] [If $S$ is an induced two-sided $S^{\hspace{.35mm}d - 1}_{d +
1}$ in a normal $d$-pseudomanifold $X$, then the pure simplicial complex $\widetilde{X}$ constructed above is said to be obtained from $X$ by an]{} elementary handle deletion [over $S$.]{}
$\!\!\!$[**.**]{} \[REHD\] [In Lemma \[LEHD\], if $X$ is a triangulated manifold then it is easy to see that $\widetilde{X}$ is also a triangulated manifold.]{}
Stacked spheres
===============
Let $X$ be a pure $d$-dimensional simplicial complex and $Y$ be obtained from $X$ by starring a new vertex $v$ in a facet $\sigma$. Clearly, $Y$ is a normal pseudomanifold if and only if $X$ is so. Since $Y$ is a subdivision of $X$, it follows that $X$ is a combinatorial manifold (respectively, combinatorial sphere) if and only if $Y$ is a combinatorial manifold (respectively, combinatorial sphere). Notice that the new vertex $v$ is of degree $d+1$ in $Y$, and when $d > 1$ the edge graph of $X$ is the induced subgraph of the edge graph of $Y$ on the vertex set $V(Y) \setminus \{v\}$.
Now, if $Y$ is a normal $d$-pseudomanifold, then note that for any vertex $u$ of $Y$, ${\rm lk}_Y(u)$ is a normal $(d -
1)$-pseudomanifold, hence has at least $d + 1$ vertices. Thus, each vertex of $Y$ has degree $\geq d + 1$. If $u$ is a vertex of $Y$ of (minimal) degree $d+1$ and the number of vertices in $Y$ is $> d+2$, then consider the pure simplicial complex $X$ with vertex set $V(Y) \setminus \{u\}$, whose facets are the facets of $Y$ not passing through $u$, and the set of all $d+1$ neighbours of $u$. We say that $X$ is obtained from $Y$ by [*collapsing*]{} the vertex $u$. Clearly, this is the reverse of the operation of starring a vertex $u$ in a facet of $X$.
$\!\!\!$[**.**]{} \[DSS1\] [A simplicial complex $X$ is said to be a]{} stacked $d$-sphere [if there is a finite sequence $X_0$, $X_1, \dots, X_m$ of simplicial complexes such that $X_0 = S^{\hspace{.2mm}d}_{d +
2}$, the standard $d$-sphere, $X_m = X$ and $X_{i}$ is obtained from $X_{i - 1}$ by starring a new vertex in a facet of $X_{i -
1}$ for $1 \leq i \leq m$. Thus an $n$-vertex stacked $d$-sphere is obtained from the standard $d$-sphere by $(n - d - 2)$-fold starring. This implies that every stacked sphere is a combinatorial sphere. Since, for $d > 1$, each starring increases the number of edges by $d+1$, it follows that any $n$-vertex stacked $d$-sphere has exactly ${d + 2 \choose 2} + (n - d - 2)(d
+ 1) = n(d + 1) - {d + 2 \choose 2}$ edges. ]{}
In [@ba2], Barnette proved that any $n$-vertex polytopal $d$-sphere has at least $n(d + 1) - {d + 2 \choose 2}$ edges. In [@ka], Kalai proved this result for triangulated manifolds and also proved that, for $d \geq 3$, equality holds in this inequality only for stacked spheres. In [@ta], Tay generalized these results to normal pseudomanifolds to prove:
$\!\!\!$[**.**]{} $($[Lower Bound Theorem for Normal Pseudomanifolds]{}$)$ \[LBT-NPM\] For $d \geq 2$, any $n$-vertex normal $d$-pseudomanifold has at least $n(d + 1) - {d +
2 \choose 2}$ edges. For $d \geq 3$, equality holds only for stacked spheres.
In [@bd7], we have presented a self-contained combinatorial proof of Theorem \[LBT-NPM\]. Using induction, it is not difficult to prove the next four lemmas (see [@bd7] for complete proofs).
$\!\!\!$[**.**]{} \[LSS1\] Let $X$ be a normal pseudomanifold of dimension $d \geq 2$.
1. If $X$ is not the standard $d$-sphere then any two vertices of degree $d + 1$ in $X$ are non-adjacent.
2. If $X$ is a stacked sphere then $X$ has at least two vertices of degree $d + 1$.
$\!\!\!$[**.**]{} \[LSS2\] Let $X$, $Y$ be normal $d$-pseudomanifolds. Suppose $Y$ is obtained from $X$ by starring a new vertex in a facet of $X$. Then $Y$ is a stacked sphere if and only if $X$ is a stacked sphere.
$\!\!\!$[**.**]{} \[LSS3\] The link of a vertex in a stacked sphere is a stacked sphere.
$\!\!\!$[**.**]{} \[LSS4\] Any stacked sphere is uniquely determined by its edge graph.
$\!\!\!$[**.**]{} \[LSS5\] Let $X_1$, $X_2$ be normal $d$-pseudomanifolds. Then $X_1 \# X_2$ is a stacked $d$-sphere if and only if both $X_1$, $X_2$ are stacked $d$-spheres.
[**Proof.**]{} Induction on the number $n \geq d + 3$ of vertices in $X_1 \# X_2$. If $n = d + 3$ then both $X_1$, $X_2$ must be standard $d$-spheres (hence stacked spheres) and then $X_1 \# X_2 = S^{\hspace{.2mm}0}_{2} \ast S^{\hspace{.2mm}d -
1}_{d + 1}$ is easily seen to be a stacked sphere. So, assume $n
> d + 3$, so that at least one of $X_1$, $X_2$ is not the standard $d$-sphere. Without loss of generality, say $X_1$ is not the standard $d$-sphere. Of course, $X = X_1 \# X_2$ is not a standard $d$-sphere. Let $X$ be obtained from $X_1 \sqcup X_2
\setminus \{\sigma_1, \sigma_2\}$ by identifying a facet $\sigma_1$ of $X_1$ with a facet $\sigma_2$ of $X_2$ by some bijection. Then, $\sigma_1 = \sigma_2$ is a clique in the edge graph of $X$, though it is not a facet of $X$. Notice that a vertex $x \in V(X_1) \setminus \sigma_1$ is of degree $d + 1$ in $X_1$ if and only if it is of degree $d + 1$ in $X$. If either $X_1$ is a stacked sphere or $X$ is a stacked sphere then, by Lemma \[LSS1\], such a vertex $x$ exists. Let $\widetilde{X}_1$ (respectively, $\widetilde{X}$) be obtained from $X_1$ (respectively, $X$) by collapsing this vertex $x$. Notice that $\widetilde{X} = \widetilde{X}_1 \# X_2$. Therefore, by induction hypothesis and Lemma \[LSS2\], we have: $X$ is a stacked sphere $\Longleftrightarrow$ $\widetilde{X}$ is a stacked sphere $\Longleftrightarrow$ both $\widetilde{X}_1$ and $X_2$ are stacked spheres $\Longleftrightarrow$ both $X_1$ and $X_2$ are stacked spheres. $\Box$
$\!\!\!$[**.**]{} \[DSS2\] [For $d \geq 2$, ${\cal K}(d)$ will denote the family of all normal $d$-pseudomanifolds $X$ such that the link of each vertex of $X$ is a stacked $(d - 1)$-sphere. Since all stacked spheres are combinatorial spheres, it follows that the members of ${\cal
K}(d)$ are combinatorial $d$-manifolds. Notice that, Lemma \[LSS3\] says that all stacked $d$-spheres belong to the class ${\cal K}(d)$. Also, for $d \geq 2$, $K^{\hspace{.1mm}d}_{2d +
3}$ and all the simplicial complexes $K^{\hspace{.1mm}d}_{2d +
4}(p)$ constructed in Section 3 are in the class ${\cal K}(d)$ (cf. Proof of Lemma \[LSE2\]).]{}
$\!\!\!$[**.**]{} $($Walkup$)$ \[LSS6\] Let $X$ be a normal $d$-pseudomanifold and $\psi \colon \sigma_1 \to
\sigma_2$ be an admissible bijection, where $\sigma_1, \sigma_2$ are facets of $X$. Then $X^{\psi} \in {\cal K}(d)$ if and only if $X \in {\cal K}(d)$.
[**Proof.**]{} For a vertex $v$ of $X$, let $\bar{v}$ denote the corresponding vertex of $X^{\psi}$. Observe that ${\rm
lk}_{X^{\psi}}(\bar{v})$ is isomorphic to ${\rm lk}_{X}(v)$ if $v
\in V(X) \setminus (\sigma_1 \cup \sigma_2)$ and ${\rm
lk}_{X^{\psi}}(\bar{v}) = {\rm lk}_{X}(v) \# {\rm
lk}_{X}(\psi(v))$ if $v \in \sigma_1$. The result now follows from Lemma \[LSS5\]. $\Box$
$\!\!\!$[**.**]{} \[TSS\] For $d \geq 2$, there is a unique $(3d + 4)$-vertex stacked $d$-sphere ${\cal S} = {\cal S}^d_{3d + 4}$ which has a pair of facets with an admissible bijection between them. Further, this pair of facets and the admissible bijection between them is unique up to automorphisms of ${\cal S}$.
[**Proof.**]{} [Uniqueness:]{} Let $V^{+}$ and $V^{-}$ be two (disjoint) facets in a $(3d + 4)$-vertex stacked $d$-sphere ${\cal S}$, and $\psi \colon V^{+} \to V^{-}$ be an admissible bijection. Put $V({\cal S}) = U \sqcup V^{+} \sqcup
V^{-}$. Thus, $\#(U) = d + 2$. Since $\psi$ is admissible, for each $x \in V^{+}$, none of the $3d + 2$ vertices of ${\cal S}$ other than $x$ and $\psi(x)$ is adjacent (in the edge graph of ${\cal S}$) with both $x$ and $\psi(x)$. Further, $x$ and $\psi(x)$ are non-adjacent. Therefore, $$\label{ESS1}
\deg(x) + \deg(\psi(x)) \leq 3d + 2, ~ x \in V^{+}.$$ Also, for $y\in U$, $y$ is adjacent to at most one vertex in the pair $\{x, \psi(x)\}$ for each $x \in V^{+}$, and these $d + 1$ pairs partition $V({\cal S}) \setminus U$. So, each $y \in U$ has at most $d + 1$ neighbours outside $U$. Since $y$ can have at most $d + 1 = \#(U \setminus \{y\})$ neighbours in $U$, it follows that $$\label{ESS2}
\deg(y) \leq 2d + 2, ~ y \in U.$$ From (\[ESS1\]) and (\[ESS2\]), we get by addition, $$\sum_{x \in V^{+}} \deg(x) + \sum_{x \in V^{+}} \deg(\psi(x)) +
\sum_{y \in U} \deg(y) \leq (d+1)(3d+2) + (d+2)(2d+2) =
(d+1)(5d+6).$$ Now, the left hand side in this inequality is the sum of the degrees of all the vertices of ${\cal S}$, which equals twice the number of edges of ${\cal S}$. Thus ${\cal S}$ has at most $(d+1)(5d+6)/2$ edges. But, as ${\cal S}$ is a $(3d+4)$-vertex stacked $d$-sphere and $d\geq 2$, it has exactly $(3d+4)(d+1) -
{d+2 \choose 2} = (d+1)(5d+6)/2$ edges. Hence we must have equality in (\[ESS1\]) and (\[ESS2\]). Thus we have equality throughout the arguments leading to (\[ESS1\]) and (\[ESS2\]). Therefore we have: $(a)$ $U$ is a $(d+2)$-clique in the edge graph $G$ of ${\cal S}$, and $(b)$ for each $y\in U$ and $x\in V^{+}$, $y$ is adjacent to exactly one of the vertices $x$ and $\psi(x)$. Notice that, since $U$, $V^{+}$ and $V^{-}$ are cliques and there is no edge between $V^{+}$ and $V^{-}$, it follows that $G$ is completely determined by its (bipartite) subgraph $H$ whose edges are the edges of $G$ between $U$ and $V^{+}$.
Let $0 \leq m \leq d + 1$.
[Claim.]{} There exist $x_i^{+}$, $1\leq i\leq m$, in $V^{+}$ and $y_i$, $1\leq i\leq m$, in $U$ such that for each $i$ ($1\leq i\leq m$), the $i$ vertices $y_1, \dots, y_i$ are the only vertices from $U$ adjacent to $x_i^{+}$. Further, there is a stacked $d$-sphere $X(m)$ with vertex-set $V({\cal S}) \setminus
\{x_i^{+} : 1\leq i\leq m\}$ whose edge graph is the induced subgraph $G_{m}$ of $G$ on this vertex set.
We prove the claim by finite induction on $m$. The claim is trivially correct for $m = 0$ (take $X(0) = {\cal S}$, $G_0 =
G$). So, assume $1 \leq m \leq d + 1$ and the claim is valid for all smaller values of $m$. By Lemma \[LSS1\], $X(m - 1)$ has at least two vertices of degree $d+1$ and they are non-adjacent in $G_{m-1}$. Since each vertex of $U$ has degree $2d + 2$ in $G$, it has degree $\geq 2d + 2 - (m - 1) > d + 1$ in $G_{m-1}$. Since $V^{-}$ is a clique of $G_{m-1}$, at least one of the degree $d +
1$ vertices of $G_{m - 1}$ is in $V^{+} \setminus \{x^{+}_i : 1
\leq i < m\}$. Let $x^{+}_m$ be a vertex of degree $d + 1$ in $G_{m - 1}$ from $V^{+} \setminus \{x^{+}_i : 1 \leq i < m\}$. Notice that $x^{+}_{m - 1}$ is a vertex of degree $d + 1$ in $X(m
- 2)$; its set of neighbours in $G_{m - 2}$ is $\{y_j : 1 \leq j
\leq m - 1\} \sqcup (V^{+} \setminus \{x^{+}_i : 1 \leq i \leq m
- 1\})$. Since ${\rm lk}_{X(m - 2)}(x^{+}_{m - 1})$ is an $S^{\hspace{.2mm}d - 1}_{d + 1}$, all the neighbours of $x^{+}_{m-1}$ are mutually adjacent (in $G_{m-2}$ and hence) in $G$. Thus, the vertices $y_j$, $1\leq j\leq m-1$, are adjacent in $G$ with each vertex in $V^{+} \setminus \{x^{+}_i : 1\leq i \leq
m-1\}$. In particular, $x^{+}_m$ is adjacent (in $G$ and hence) in $G_{m-1}$ to the $m-1$ vertices $y_j$, $1\leq j\leq m-1$ in $U$. It is also adjacent to the $d+1-m$ vertices in $V^{+}
\setminus\{x^{+}_i : 1\leq i\leq m\}$ and to no vertex in $V^{-}$. Since $x^{+}_m$ is of degree $d+1$ in $G_{m-1}$, it follows that there is a unique vertex $y_m\in U\setminus \{y_i :
1\leq i\leq m-1\}$ which is adjacent to $x^{+}_m$ (in $G_{m-1}$ and hence) in $G$. By construction, $y_1, \dots, y_m$ are the only vertices in $U$ adjacent to $x^{+}_m$. Now, let $X(m)$ be obtained from $X(m -1)$ by collapsing the vertex $x^{+}_m$ of degree $d+1$. By Lemma \[LSS2\], $X(m)$ is a stacked sphere. Its edge graph is the induced subgraph $G_m$ of $G$ on the vertex-set $V({\cal S}) \setminus \{x^{+}_i : 1\leq i\leq m\}$. This completes the induction step and hence proves the claim.
Now, by the final step $m = d+1$, we have named the vertices in $V^{+}$ as $x^{+}_i$, $1\leq i\leq d+1$. We have also named $d+1$ of the vertices in $U$ as $y_i$, $1\leq i\leq d+1$. Let $y_{d+2}$ be the unique vertex in $U \setminus \{y_i : 1\leq i \leq d+1\}$. Also, put $x_i^{-} = \psi(x^{+}_i) \in V^{-}$, $1\leq i\leq d+1$. Thus, $x^{-}_i$ is adjacent to $y_j$ if and only if $x^{+}_i$ is non-adjacent with $y_j$. This completes the description of the edge graph $G$ of ${\cal S}$. The vertices of $G$ are $x^{+}_i$, $x^{-}_i$ ($1\leq i\leq d+1$) and $y_j$, $1\leq j\leq d+2$. $x^{+}_i$ and $x^{+}_j$ (as well as $x^{-}_i$ and $x^{-}_j$) are adjacent in $G$ for $i\neq j$. $y_i$ and $y_j$ are adjacent in $G$ for $i\neq j$. $x^{+}_i$ and $x^{-}_j$ are non-adjacent in $G$ for all $i, j$. $x_i^{+}$ and $y_j$ are adjacent in $G$ if and only if $j \leq i$. $x_i^{-}$ and $y_j$ are adjacent in $G$ if and only if $j > i$.
Since the edge graph $G$ is thus completely determined by the given datum, Lemma \[LSS4\] implies that ${\cal S}$ is uniquely determined. Notice that the graph $G$ has maximum vertex degree $2d +2$, and the set $U$ is uniquely determined by $G$ as the set of its vertices of maximum degree. Also, the facets $V^{+}$, $V^{-}$ are determined by $G$ as the connected components of the induced subgraph of $G$ on the complement of $U$. Finally, the above argument shows that the admissible bijection $\psi \colon
V^{+} \to V^{-}$ is also determined by $G$ since it must map the unique vertex of degree $d+i$ in $V^{+}$ to the unique vertex of degree $2d+2-i$ in $V^{-}$ ($1\leq i \leq d+1$). Notice that $S$ has an automorphism of order two which interchanges $x^{+}_i$ and $x^{-}_{d+2-i}$ for each $i$ and interchanges $y_j$ and $y_{d+3-j}$ for each $j$. This automorphism interchanges $V^{+}$ and $V^{-}$ and replaces $\psi$ by $\psi^{-1}$. This completes the uniqueness proof.
[Existence of ${\cal S}^d_{3d + 4}$:]{} The simplicial complex $\partial N^{d+1}_{3d+4}$ constructed in the next section is a $(3d+4)$-vertex stacked $d$-sphere (cf. proof of Lemma \[LSE2\]) with an admissible bijection $\psi_0 \colon
B_{2d+3} \to A_{2d+3}$ (cf. the paragraph before Lemma \[LSE3\]). This proves the existence. $\Box$
$\!\!\!$[**.**]{} \[RSS\] [$(a)$ The proof of Theorem \[TSS\], in conjunction with the Lower Bound Theorem, actually shows the following. If $X$ is an $n$-vertex normal $d$-pseudomanifold with an admissible bijection, then $n \geq 3d + 4$, and equality holds only for $X =
{\cal S}^{d}_{3d + 4}$. $(b)$ If $\psi$ is the admissible bijection on ${\cal S}^{d}_{3d + 4}$, then it is possible to verify directly that $({\cal S}^{d}_{3d + 4})^{\psi} =
K^{\hspace{.1mm}d}_{2d + 3}$. This is also immediate from the proof of Theorem \[TUK\] below. ]{}
Some Examples
=============
Recall that for any positive integer $n$, a [*partition*]{} of $n$ is a finite weakly increasing sequence of positive integers adding to $n$. The terms of the sequence are called the [*parts*]{} of the partition. Let’s say that a partition of $n$ is [*even*]{} (respectively, [*odd*]{}) if it has an even (respectively, odd) number of even parts. Let $P(n)$ (respectively $P_0(n)$, respectively $P_1(n)$) denote the total number of partitions (respectively even partitions, respectively odd partitions) of $n$.
To appreciate the construction given below, it is important to understand the growth rate of these number theoretic functions $P_{\varepsilon}$, $\varepsilon = 0, 1$. Recall that if $f$, $g$ are two real valued functions on the set of positive integers, then one says that $f$, $g$ are [*asymptotically equal*]{} (in symbols, $f(n) \sim g(n)$) if ${\displaystyle \lim_{n\to\infty}}
\frac{f(n)}{g(n)} =1$. A famous theorem of Hardy and Ramanujan (cf. [@ra]) says that $$\label{Pn}
P(n) \sim \frac{c_1}{n}e^{c_2\sqrt{n}} ~ \mbox{ as } ~ n\to\infty,$$ where the absolute constants $c_1$, $c_2$ are given by $$c_1 = \frac{1}{4\sqrt{3}}, ~~ c_2 = \pi\sqrt{\frac{2}{3}}.$$ We observe that:
$\!\!$[**.**]{} \[LSE1\] $P_0(n) \sim \frac{c_1}{2n} e^{c_2 \sqrt{n}}$, $P_1(n) \sim
\frac{c_1}{2n} e^{c_2 \sqrt{n}}$ as $n \to \infty$.
[**Proof.**]{} In view of (\[Pn\]), it suffices to show that $P_0(n) \sim \frac{1}{2} P(n)$, $P_1(n) \sim \frac{1}{2}
P(n)$ as $n \to \infty$. Now, $(p_1, \dots, p_k) \mapsto (1, p_1,
\dots, p_k)$ is a one to one function from the set of even (respectively, odd) partitions of $n - 1$ to the set of even (respectively, odd) partitions of $n$. Also, $(p_1, \dots, p_k)
\mapsto (p_1, \dots, p_{k - 1}, p_k + 1)$ is a one to one function from the set of even (respectively, odd) partitions of $n-1$ to the set of odd (respectively, even) partitions of $n$. Therefore, $\min(P_0(n), P_1(n)) \geq \max(P_0(n - 1), P_1(n - 1))$. Since $P_0(n - 1)+ P_1(n - 1) = P(n - 1)$, it follows that $$P_0(n) \geq \frac{1}{2} P(n - 1) ~ \mbox{ and } ~ P_1(n) \geq
\frac{1}{2} P(n - 1).$$ But, from (\[Pn\]) it follows that $P(n - 1) \sim P(n)$. Therefore, ${\displaystyle \liminf_{n \to \infty}}
\frac{P_0(n)}{P(n)} \geq \frac{1}{2}$, ${\displaystyle \liminf_{n
\to \infty}} \frac{P_1(n)}{P(n)} \geq \frac{1}{2}$. But, $P_0(n)+
P_1(n) = P(n)$. Therefore, ${\displaystyle \lim_{n \to \infty}}
\frac{P_0(n)}{P(n)} = \frac{1}{2} = {\displaystyle \lim_{n \to
\infty}} \frac{P_1(n)}{P(n)}$. $\Box$
[**The Construction:**]{} For $d \geq 2$, let $N^{\,d +
1}$ denote the pure $(d + 1)$-dimensional simplicial complex with vertex-set ${\mbox{\bbb Z}}$ (the set of all integers) such that the facets of $N^{d + 1}$ are the sets of $d + 2$ consecutive integers. Then $N^{d + 1}$ is a combinatorial $(d + 1)$-manifold with boundary $M^{d} = \partial N^{d + 1}$. Now, $M^{d}$ is a combinatorial $d$-manifold ($\in {\cal K}(d)$) and triangulates ${\mbox{\bbb R}}\times
S^{\hspace{.2mm}d - 1}$ (cf. [@ku1]). Clearly, the facets of $M^{d}$ are of the form $\sigma_{n, i} := \{n, n + 1, \dots, n + d
+ 1\}\setminus\{n + i\}$, $1 \leq i \leq d$, $n \in {\mbox{\bbb Z}}$ (intervals of length $d + 2$ minus an interior point).
For $m \geq 1$, let $N^{d + 1}_{m + d + 1}$ (respectively, $M^{d}_{m + d +1}$) denote the induced subcomplex of $N^{\,d +
1}$ (respectively, $M^{d}$) on $m + d + 1$ consecutive vertices (without loss of generality we may take $V(N^{d + 1}_{m + d + 1})
= V(M^{d}_{m + d + 1}) = \{1, 2, \dots, m+d+1\}$). Clearly, $M^{d}_{m + d + 1}$ triangulates $[0, 1] \times S^{\hspace{.2mm}d
- 1}$ and $\partial M^{d}_{m + d + 1} = S^{\hspace{.2mm}d - 1}_{d
+ 1}(A_m) \sqcup S^{\hspace{.2mm}d - 1}_{d + 1}(B_m)$, where $A_m
= \{1, \dots, d + 1\}$ and $B_m = \{m + 1, \dots, m + d + 1\}$.
$\!\!\!$[**.**]{} \[LSE2\] $(a)$ $\partial N^{d + 1}_{m + d + 1}$ is a stacked $d$-sphere and $A_m$, $B_m$ are two of its facets. $(b)$ If $\psi \colon B_m
\to A_m$ is an admissible bijection then $X^{\hspace{.1mm}d}_m(\psi) := (\partial N^{d + 1}_{m + d +
1})^{\psi}$ is a combinatorial $d$-manifold and triangulates $S^{\hspace{.2mm}1, d - 1}(\varepsilon)$, where $\varepsilon = 0$ if $X^{\hspace{.1mm}d}_m(\psi)$ is orientable and $\varepsilon =
1$ otherwise.
[**Proof.**]{} Observe that $\partial N^{d + 1}_{d + 2}$ is the standard $d$-sphere and for $m \geq 2$, $\partial N^{d +
1}_{m + d + 1}$ is obtained from $\partial N^{d + 1}_{m + d}$ by starring the new vertex $m + d + 1$ in the facet $B_{m - 1} =
\{m, \dots, m + d\}$ of $\partial N^{d + 1}_{m + d}$. Thus, $\partial N^{d + 1}_{m + d + 1}$ is a stacked $d$-sphere. $A_m$ is a facet of $\partial N^{d + 1}_{i + d + 1}$ for all $i\geq 1$ and from construction, $B_m$ is a facet of $\partial N^{d + 1}_{m
+ d + 1}$. This proves $(a)$.
Thus, by Lemma \[LSS3\], $\partial N^{d + 1}_{m + d + 1}$ is in ${\cal K}(d)$. Then, by Lemma \[LSS6\], $X^{\hspace{.1mm}d}_m(\psi)$ is in the class ${\cal K}(d)$. In consequence, $X^{\hspace{.1mm}d}_m(\psi)$ is a combinatorial $d$-manifold. Since $M^{d}_{m + d + 1}$ triangulates $[0, 1]
\times S^{\hspace{.2mm}d - 1}$ and $M^{d}_{m + d + 1} = \partial
N^{d + 1}_{m + d + 1}\setminus\{A_m, B_m\}$, it follows that $X^{\hspace{.1mm}d}_m(\psi)$ ($= (\partial N^{d + 1}_{m + d +
1})^{\psi}$) triangulates an $S^{\hspace{.2mm}d - 1}$-bundle over $S^{\hspace{.2mm}1}$. But, there are only two such bundles: $S^{\hspace{.2mm}1, d - 1}(\varepsilon)$, $\varepsilon = 0, 1$ (cf. [@st pages 134–135]). This is orientable for $\varepsilon = 0$ and non-orientable for $\varepsilon = 1$. Hence the result. $\Box$
Notice that $x\in B_m$ is at a distance $\geq 3$ from $y\in A_m$ (in the edge graph of $\partial N^{d + 1}_{m + d + 1}$) if and only if $x - y \geq 2d + 3$. Therefore, if $m \leq 2d + 2$, it is easy to see that there is no admissible bijection $\psi \colon
B_m \to A_m$. For $m \geq 2d + 3$ the map $\psi_0 \colon B_m \to
A_m$ given by $\psi_0(m + i) = i$ is admissible. When $m = 2d +
3$, it is the only admissible map and the resulting combinatorial manifold $X^{\,d}_{2d + 3}(\psi_0)$ is Kühnel’s $K^{\,d}_{2d
+ 3}$, triangulating $S^{\hspace{.2mm}1, d - 1}(\varepsilon)$, $d
\equiv \varepsilon$ (mod 2), whose uniqueness we prove in Section 4 below. For $m \geq 2d + 3$, Kühnel and Lassmann constructed $X^{\,d}_{m}(\psi_0)$ and proved that for $m$ odd $X^{\,d}_{m}(\psi_0)$ is orientable if and only if $d$ is even (cf. [@kl]). Here we have:
$\!\!\!$[**.**]{} \[LSE3\] Let $m \geq 2d + 3$. If $md$ is even then for any admissible $\psi
\colon B_m \to A_m$, the combinatorial $d$-manifold $X^{d}_{m}(\psi)$ is orientable if and only if $\psi \circ
\psi_0^{-1}$ is an even permutation. In other words, if $\psi
\circ \psi_0^{-1}$ is an even $($respectively, odd$)$ permutation then $X^{d}_{m}(\psi)$ is a combinatorial triangulation of $S^{\hspace{.2mm}1, d - 1}(0)$ $($respectively, $S^{\hspace{.2mm}1, d - 1}(1))$.
[**Proof.**]{} For $1 \leq k \leq m$, $1\leq i \leq d$, let $\sigma_{k, i}$ denote the facet $\{k, k + 1, \dots, k + d +
1\} \setminus \{k + i\}$ and for $0 \leq i < j \leq d + 1$, $(i,
j) \neq (0, d+1)$, let $\sigma_{k, i, j}$ denote the $(d -
1)$-simplex $\{k, k + 1, \dots, k + d + 1\} \setminus \{k + i, k
+ j\}$ of $M^{d}_{m + d + 1}$. Consider the orientation on $M^{d}_{m + d + 1}$ given by: $$\begin{aligned}
\label{ESE}
+ \sigma_{k, i, j} & = & (-1)^{kd + i + j} \langle k, \dots, k +
i - 1, k + i + 1, \dots, k + j - 1, k + j + 1, \dots, k + d +
1\rangle, \nonumber \\
+ \sigma_{k, i} & = & (-1)^{kd + i} \langle k, k + 1, \dots, k + i
- 1, k + i + 1, \dots, k + d + 1\rangle.\end{aligned}$$
By an easy computation one sees that the incidence numbers satisfy the following: $[\sigma_{k, i}, \sigma_{k, i, j}] = -
1$, $[\sigma_{k, j}, \sigma_{k, i, j}] = 1$ for $1\leq i < j \leq
d$, $1 \leq k \leq m$ and $[\sigma_{k, i}, \sigma_{k, 0, i}] =
1$, $[\sigma_{k + 1, i - 1}, \sigma_{k, 0, i}] = [\sigma_{k + 1,
i - 1}, \sigma_{k + 1, i - 1, d + 1}] = (- 1)^{2d -1} = -1$ for $1
\leq i \leq d$, $1 \leq k < m$. Thus, (\[ESE\]) gives an orientation on $M^{d}_{m + d + 1}$.
Let $\bar{\sigma}_{k, i}$ and $\bar{\sigma}_{k, i, j}$ denote the corresponding simplices in $X^{d}_{m}(\psi_0)$. Observe that $\bar{\sigma}_{k, 0, j} = \bar{\sigma}_{k + 1, j - 1, d + 1}$ for $1 \leq k < m$ and $\bar{\sigma}_{m, 0, j} = \bar{\sigma}_{1,
j-1, d + 1}$. (The vertex-set of $X^{d}_{m}(\psi_0)$ is the set of integers modulo $m$.) Then the above orientation induces an orientation on $X^{d}_{m}(\psi_0)$. (This is well defined since $+ \sigma_{m, 0, j} = (-1)^{md + j} \langle m + 1, \dots, m + j -
1, m + j + 1, \dots, m + d + 1 \rangle = (- 1)^{j} \langle 1,
\dots, j - 1, j + 1, \dots, d + 1 \rangle = (- 1)^{d + (j - 1) +
(d + 1)} \langle 1, \dots, j - 1, j + 1, \dots, d + 1 \rangle = +
\sigma_{1, j - 1, d + 1}$.) Now, $[\bar{\sigma}_{m, j},
\bar{\sigma}_{m, 0, j}] = 1$, $[\bar{\sigma}_{1, j - 1},
\bar{\sigma}_{m, 0, j}] = [\bar{\sigma}_{1, j - 1},
\bar{\sigma}_{1, j - 1, d + 1}] = - 1$. Thus, $[\bar{\sigma}_{m,
j}, \bar{\sigma}_{m, 0, j}] = - [\bar{\sigma}_{1, j-1},
\bar{\sigma}_{m, 0, j}]$. Therefore, the induced orientation on $X^{d}_{m}(\psi_0)$ is coherent. So, $X^{d}_{m}(\psi_0)$ is orientable. This implies that $X^{d}_{m}(\psi_0)$ triangulates $S^1\times S^{\hspace{.2mm}d - 1} = S^{\hspace{.2mm}1, d-1}(0)$.
Since $|M^{d}_{m + d + 1}|$ is homeomorphic to $|S^{\hspace{.2mm}d
- 1}_{d + 1}(B_m)| \times [0, 1]$, we can choose an orientation on $|S^{\hspace{.2mm}d - 1}_{d + 1}(B_m)|$ so that the orientation on $|M^{d}_{m + d + 1}|$ as the product $|S^{\hspace{.2mm}d -
1}_{d + 1}(B_m)| \times [0, 1]$ is the same as the orientation given in (\[ESE\]). This also induces an orientation on $|S^{\hspace{.2mm}d - 1}_{d + 1}(A_m)|$. Let $S_B$ (respectively, $S_A$) denote the oriented sphere $|S^{\hspace{.2mm}d - 1}_{d +
1}(B_m)|$ (respectively $|S^{\hspace{.2mm}d - 1}_{d + 1}(A_m)|$) with this orientation. Then, as the boundary of an oriented manifold, $\partial (|M^d_{d + m + 1}|) = S_A \cup (- S_B)$. \[In fact, it is not difficult to see that the orientation defined in (\[ESE\]) on $S^{\hspace{.2mm}d - 1}_{d + 1}(A_m)$ (respectively $S^{\hspace{.2mm}d - 1}_{d + 1}(B_m)$) is the same as the orientation in $S_A$ (respectively $S_B$).\]
Let $|\psi_0| \colon S_B \to S_A$ be the homeomorphism induced by $\psi_0$. Since $|X^{d}_{m}(\psi_0)|$ is orientable, it follows that $|\psi_0| \colon S_B \to S_A$ is orientation preserving (cf. [@st pages 134–135]).
Therefore, $\psi \circ \psi_0^{- 1}$ is an even (respectively odd) permutation $\Longrightarrow$ $|\psi \circ \psi_0^{- 1}|
\colon S_A \to S_A$ is orientation preserving (respectively reversing) $\Longrightarrow$ $|\psi| = |\psi \circ \psi_0^{- 1}|
\circ |\psi_0| \colon S_B \to S_A$ is orientation preserving (respectively reversing) $\Longrightarrow$ $|X^{d}_{m}(\psi)|$ is orientable (respectively non-orientable). Hence, the result follows from Lemma \[LSE2\]. $\Box$
Now take $m = 2d + 4$. A bijection $\psi \colon \{2d + 5, \dots,
3d + 5\} \to \{1, \dots, d + 1\}$ is admissible for $\partial
N^{d+1}_{3d+5}$ if and only if $x - \psi(x) \geq 2d + 3$ for $2d+5 \leq x \leq 3d+5$. It turns out that there are $2^d$ distinct admissible choices for $\psi$. But it seems difficult to decide when two admissible choices for $\psi$ yield isomorphic complexes $X^{\,d}_{2d + 4}(\psi)$. So, we specialize as follows:
Let $p = (p_1, p_2, \dots, p_k)$ be a partition of $d + 1$. Put $s_0 = 0$ and $s_j = \sum_{i=1}^{j} p_i$ for $1 \leq j \leq k$. (Thus, in particular, $s_1 = p_1$ and $s_k = d + 1$.) Let $\pi_p$ be the permutation of $\{1, 2, \dots, d+1\}$ which is the product of $k$ disjoint cycles $(s_{j - 1} + 1, s_{j - 1} +2, \dots,
s_{j})$, $1\leq j\leq k$. Notice that $\pi_p$ is an even (respectively, odd) permutation if $p$ is an even (respectively, odd) partition of $d + 1$. Now, define the bijection $\psi_p
\colon \{2d + 5, 2d + 6, \dots, 3d + 5\} \to \{1, 2, \dots, d +
1\}$ by $\psi_p(2d + 4 + i) = \pi_p(i)$, $1 \leq i \leq d + 1$. Since $\pi_p(i) \leq i + 1$ for $1 \leq i \leq d + 1$, it follows that $\psi_p$ is an admissible bijection. Clearly, the corresponding complex $X^{d}_{2d + 4}(\psi_p)$ depends only on the partition $p$ of $d+1$. We denote it by $K^{d}_{2d + 4}(p)$. Note that $\pi_p = \psi_p\circ \psi_0^{-1}$. Therefore, by Lemma \[LSE3\], $K^{d}_{2d + 4}(p)$ triangulates $S^{\hspace{.2mm}1,
d - 1}(0)$ (respectively, $S^{\hspace{.2mm}1, d - 1}(1)$) if $p$ is an even (respectively odd) partition of $d + 1$.
Let $G_p$ denote the non-edge graph of $K^{d}_{2d + 4}(p)$. Its vertex-set is $V(K^{d}_{2d + 4}(p))$, and two distinct vertices $x$, $y$ are adjacent in $G_p$ if $xy$ is not an edge of $K^{d}_{2d + 4}(p)$. It turns out that $G_p$ has a clear description in terms of the partition $p$. For $b\geq 1$, let $K_{1, b}$ denote the unique graph with one vertex of degree $b$ and $b$ vertices of degree one. Also, let $p = (p_1, p_2, \dots,
p_k)$, and put $p_0 = 1$. Then a computation shows that $G_p$ is the disjoint union of $K_{1, p_i}$, $0\leq i \leq k$. Thus, if $p$ and $q$ are distinct partitions of $d+1$ then $G_p$ and $G_q$ are non-isomorphic (this is where our assumption that $p$, $q$ are weakly increasing sequences comes into play!) and hence $K^{d}_{2d + 4}(p)$ and $K^{d}_{2d + 4}(q)$ are non-isomorphic complexes. Thus we have proved:
$\!\!\!$[**.**]{} \[th1.1\] For any partition $p$ of $d + 1 \geq 3$, let $\varepsilon =
\varepsilon(p) = 0$ if $p$ is even and $= 1$ if $p$ is odd. Then $K^{d}_{2d + 4}(p)$ is a $(2d + 4)$-vertex triangulation of $S^{\hspace{.2mm}1, d - 1}(\varepsilon)$. Further, distinct partitions $p$ of $d + 1$ correspond to non-isomorphic triangulations of $S^{\hspace{.2mm}1, d - 1}(\varepsilon)$. In consequence, for $\varepsilon = 0, 1$, there are $(2d+4)$-vertex combinatorial triangulations of $S^{\hspace{.2mm}1, d -
1}(\varepsilon)$ and the number of non-isomorphic triangulations is at least $P_{\varepsilon}(d+1) \sim
\frac{c_1}{2d}e^{c_2\sqrt{d}}$.
This theorem provides an affirmative solution of the conjecture (made by Lutz in [@lu1]) that $S^{\hspace{.2mm}1, d - 1}(1)$ can be triangulated by $2d + 4$ vertices for $d$ even. Notice that each $(2d+4)$-vertex triangulation of $S^{\hspace{.2mm}1, d -
1}(\varepsilon)$ constructed here has $d+2$ non-edges. We conjecture that this is the maximum possible number of non-edges. If this is true then, for $d \equiv 1- \varepsilon$ (mod 2), our construction yields triangulations of $S^{\hspace{.2mm}1, d -
1}(\varepsilon)$ with the minimum number of vertices and edges.
Uniqueness of $K^{d}_{2d + 3}$
==============================
Recall from Section 3 that for $d \geq 2$, $K^{d}_{2d+3}$ is the $(2d+3)$-vertex combinatorial $d$-manifold constructed by Kühnel in [@ku1]. It triangulates $S^{\hspace{.2mm}1,
d-1}(\varepsilon)$, where $\varepsilon \in \{0, 1\}$ is given by $\varepsilon \equiv d$ (mod 2). One description of $K^{\hspace{.1mm}d}_{2d + 3}$ is implicit in Section 3. An equivalent (and somewhat simpler) description is as follows. It is the boundary complex of the combinatorial $(d +1)$-manifold with boundary whose vertices are the vertices of a cycle $S^{\hspace{.2mm}1}_{2d + 3}$ of length $2d + 3$, and facets are the sets of $d + 2$ vertices spanning a path in the cycle. From this picture, it is clear that the dihedral group of order $4d +
6$ ($= {\rm Aut}(S^{\hspace{.2mm}1}_{2d + 3})$) is the full automorphism group of $K^{d}_{2d + 3}$. Here we prove that for $d
\geq 3$, up to simplicial isomorphism, $K^{d}_{2d + 3}$ is the unique $(2d + 3)$-vertex non-simply connected triangulated $d$-manifold.
$\!\!\!$[**.**]{} $($Simplicial Alexander duality$)$ \[LUK1\] Let $L \subset L^{\prime}$ be induced subcomplexes of a triangulated $d$-manifold $X$. Let $R\supset R^{\hspace{.2mm}
\prime}$ be the simplicial complements in $X$ of $L$ and $L^{\prime}$ respectively. Then $H_{d - j}(L^{\prime}, L; {\mbox{\bbb Z}}_2)
\cong H_{j}(R, R^{\hspace{.2mm}\prime}; {\mbox{\bbb Z}}_2)$ for $0 \leq j \leq
d$.
[**Proof.**]{} Fix a piecewise linear map $f \colon |X|
\to {\mbox{\bbb R}}$ such that for all vertices $u$ of $L$, $v$ of $R$ we have $f(u) < f(v)$, and for all vertices $u^{\hspace{.2mm} \prime}$ of $L^{\prime}$, $v^{\hspace{.2mm}\prime}$ of $R^{\hspace{.2mm}
\prime}$ we have $f(u^{\hspace{.2mm} \prime}) < f(v^{\hspace{.2mm}
\prime})$. Choose $c < c^{\hspace{.2mm} \prime}$ in ${\mbox{\bbb R}}$ such that $f(u) < c < f(v)$ and $f(u^{\hspace{.2mm} \prime}) <
c^{\hspace{.2mm} \prime} < f(v^{\hspace{.2mm} \prime})$ for all such $u, v, u^{\hspace{.2mm} \prime}, v^{\hspace{.2mm} \prime}$. Define ${\cal L} = \{x \in |X| : f(x) \leq c\}$, ${\cal R} = \{x
\in |X| : f(x) > c\}$, ${\cal L}^{\prime} = \{x \in |X| : f(x)
\leq c^{\hspace{.2mm} \prime}\}$, ${\cal R}^{\hspace{.2mm}
\prime} = \{x \in |X| : f(x) > c^{\hspace{.2mm} \prime}\}$. Since $f$ is piecewise linear, it follows that ${\cal L}, {\cal
L}^{\prime}$ are compact polyhedra (i.e., geometric carriers of finite simplicial complexes). Also, $(|L^{\prime}|, |L|)$ (respectively $(|R|, |R^{\hspace{.2mm} \prime}|)$) is a strong deformation retract of $({\cal L}^{\prime}, {\cal L})$ (respectively $({\cal R}, {\cal R}^{\hspace{.2mm} \prime})$). Hence we have $$\begin{aligned}
&& H_{d-j}(L^{\prime}, L; {\mbox{\bbb Z}}_2) \cong H_{d-j}(|L^{\prime}|, |L|;
{\mbox{\bbb Z}}_2) \cong H_{d-j}({\cal L}^{\prime}, {\cal L}; {\mbox{\bbb Z}}_2) \cong
H^{\hspace{.2mm}d-j}({\cal L}^{\prime}, {\cal L}; {\mbox{\bbb Z}}_2) \\
&& \cong H_{j}({\cal R}, {\cal R}^{\hspace{.2mm} \prime}; {\mbox{\bbb Z}}_2)
\cong H_{j}(|R|, |R^{\hspace{.2mm} \prime}|; {\mbox{\bbb Z}}_2) \cong H_{j}(R,
R^{\hspace{.2mm} \prime}; {\mbox{\bbb Z}}_2) ~ \mbox { for } 0 \leq j \leq d.\end{aligned}$$ Here, the fourth isomorphism is because of Alexander duality (cf. [@sp Theorem 17, Page 296]). The usual statement of this duality refers to Alexander cohomology, but this agrees with singular cohomology for polyhedral pairs (cf. [@sp Corollary 11, Page 291]). Also, Alexander duality applies to orientable closed manifolds, but any closed manifold (such as $|X|$ in our application) is orientable over ${\mbox{\bbb Z}}_2$. The third isomorphism holds since over a field, homology and cohomology are isomorphic. $\Box$
$\!\!\!$[**.**]{} \[LUK2\] Let $X$ be a non-simply connected $n$-vertex triangulated manifold of dimension $d \geq 3$. Then $n \geq 2d + 3$. If further, $n = 2d + 3$, then for any facet $\sigma$ of $X$ and any vertex $x$ outside $\sigma$, either the induced subcomplex of $X$ on $V(X) \setminus (\sigma \cup \{x\})$ is an $S^{\hspace{.2mm}d
-1}_{d + 1}$ or the induced subcomplex ${\rm lk}_X(x)[\sigma]$ of ${\rm lk}_X(x)$ on the vertex set $\sigma$ is disconnected.
[**Proof.**]{} Let $\sigma$ be a facet and $C = C(\sigma,
X)$ be its simplicial complement. Choose a small (simply connected) neighbourhood $U$ of $|\sigma|$ in $|X|$ such that $U
\cap (|X| \setminus |\sigma|)$ is homeomorphic to $S^{\hspace{.2mm}d -1} \times (0, 1)$. Now, $|X|$ is non-simply connected, $|X| = U \cup (|X| \setminus |\sigma|)$ and $d \geq
3$. So, by Van Kampen’s theorem, $|X| \setminus |\sigma|$ is non-simply connected. But $|C|$ is a strong deformation retract of $|X| \setminus |\sigma|$. Therefore, $C$ is non-simply connected.
Now fix a facet $\sigma$ of $X$. Choose an ordering $x_1, x_2,
\dots, x_n$ of $V(X)$ so that $\sigma = \{x_1, \dots, x_{d +
1}\}$. For $1 \leq i \leq n$, let $L_i$ (respectively $R_i$) be the induced subcomplex of $X$ on the vertex-set $\{x_1, \dots,
x_i\}$ (respectively $\{x_{i + 1}, \dots, x_n\}$). Then, by Lemma \[LUK1\], $$\label{dual1}
H_{j}(R_{i}, R_{i+1}) \cong H_{d-j}(L_{i+1}, L_{i}), ~ \mbox{ for
} 0 \leq j \leq d ~ \mbox{ and } 1 \leq i < n.$$ Here the homologies are taken with coefficients in ${\mbox{\bbb Z}}_2$.
Since $L_1 = \{x_1\}$ is simply connected but $L_{n} = X$ is not, it follows that there is a (smallest) index $i$ such that $L_i$ is simply connected but $L_{i + 1}$ is not. Note that $i \geq d +
1$. Choose this $i$. Since $L_{i + 1} = L_{i} \cup {\rm st}_{L_{i
+ 1}}(x_{i + 1})$ and $L_{i} \cap {\rm st}_{L_{i + 1}}(x_{i + 1})
= {\rm lk}_{L_{i + 1}}(x_{i + 1})$, Van Kampen’s theorem implies that ${\rm lk}_{L_{i + 1}}(x_{i + 1})$ is not connected. Hence $H_1(L_{i + 1}, L_i) \cong H_1({\rm st}_{L_{i + 1}}(x_{i + 1}),
{\rm lk}_{L_{i + 1}}(x_{i + 1})) \cong \widetilde{H}_0({\rm
lk}_{L_{i + 1}}(x_{i + 1})) \neq \{0\}$. Thus, there is an index $i \geq d + 1$ such that $H_1(L_{i + 1}, L_{i}) \neq \{0\}$. Hence, from (\[dual1\]), it follows that $$\label{dual2}
H_{d-2}({\rm lk}_{R_i}(x_{i+1})) \cong H_{d - 1}(R_{i},
R_{i+1})\neq \{0\} ~ \mbox{ for some } i \geq d + 1.$$ Notice that we have $R_{i+1} \subset R_{i} \subseteq C =
C(\sigma, X)$. Since $H_{d - 1}(R_{i}, R_{i+1})\neq \{0\}$, $R_i$ contains at least two $(d-1)$-faces. Hence the number of vertices in $R_i$ is $\geq d+1$.
First suppose $R_{i}$ has exactly $d + 1$ vertices. Since $H_{d -
2}({\rm lk}_{R_i}(x_{i + 1})) \neq \{0\}$ and ${\rm
lk}_{R_i}(x_{i + 1})$ has at most $d$ vertices, it follows that ${\rm lk}_{R_i}(x_{i + 1}) = S^{\hspace{.2mm}d - 2}_d$. Since $d
\geq 3$, it follows that $R_i$ is simply connected. As $C$ is not simply connected, we have $R_i \subset C$ (proper inclusion). Thus $n \geq (d + 1) + 1 + (d + 1) = 2d + 3$. Also, if the number $n - i$ of vertices in $R_i$ is $\geq d + 2$. Then $n \geq i + d
+ 2 \geq 2d + 3$. This proves the inequality.
Now assume that $n = 2d + 3$. Let $x \not\in \sigma$ be a vertex such that ${\rm lk}_X(x) \cap L_{d+1}$ ($= {\rm st}_X(x) \cap
L_{d+1}$) is connected. Choosing the vertex order so that $x_{d +
2} = x$, we get that $L_{d + 2}$ is simply connected (by Van Kampen theorem). Therefore $i \geq d + 2$. Hence $R_i$ has $\leq
n - d - 2 = d + 1$ vertices. But, $H_{d - 1}(R_{i}, R_{i + 1})
\neq \{0\}$, so that $R_{i}$ has $\geq d + 1$ vertices. Therefore $R_i$ has exactly $d + 1$ vertices and hence $i = d + 2$. Thus, $H_{d - 2}({\rm lk}_{R_{d + 2}}(x_{d + 3})) \cong H_{d - 1}(R_{d
+ 2}, R_{d + 3}) \neq \{0\}$. Since ${\rm lk}_{R_{d + 2}}(x_{d +
3})$ has at most $d$ vertices, it follows that ${\rm lk}_{R_{d +
2}}(x_{d + 3}) = S^{\hspace{.2mm}d - 2}_d$. Since any vertex of $R_{d + 2}$ may be chosen to be $x_{d + 3}$ in this argument, we get that all the vertex links of $R_{d + 2}$ are isomorphic to $S^{\hspace{.2mm}d - 2}_d$. Hence the induced subcomplex $R_{d+2}$ of $C$ on the vertex set $V(X) \setminus (\sigma \cup
\{x\})$ is an $S^{\hspace{.2mm}d - 1}_{d + 1}$. This proves the lemma. $\Box$
$\!\!\!$[**.**]{} \[RBK\] [For combinatorial manifolds, the inequality in Lemma \[LUK2\] is a theorem due to Brehm and Kühnel [@bk].]{}
$\!\!\!$[**.**]{} \[LUK3\] Let $X$ be a $(2d + 3)$-vertex non-simply connected triangulated manifold of dimension $d \geq 3$. Then, there is a facet $\sigma$ of $X$ such that its simplicial complement $C(\sigma, X)$ contains an induced $S^{\hspace{.2mm}d - 1}_{d + 1}$.
[**Proof.**]{} Suppose the contrary. Then, by Lemma \[LUK2\], for each facet $\sigma$ of $X$ and each vertex $x
\not \in \sigma$, the induced subcomplex ${\rm lk}_X(x)[\sigma]$ of ${\rm lk}_X(x)$ on $\sigma$ is disconnected. If $\tau$ were a $(d-2)$-face of $X$ of degree 3, say with ${\rm lk}_X(\tau) =
S^{\hspace{.2mm}1}_3(\{x_1, x_2, x_3\})$, then the induced subcomplex of ${\rm lk}_X(x_3)$ on the facet $\tau \cup \{x_1,
x_2\}$ would be connected - a contradiction. So, $X$ has no $(d -
2)$-face of degree 3. Now, no face $\gamma$ of $X$ of dimension $e \leq d-2$ can have (minimal) degree $d - e + 1$. (In other words, the link of $\gamma$ can not be a standard sphere.) Or else, any $(d-2)$-face $\tau \supseteq \gamma$ of $X$ would have degree 3. So, no standard sphere of positive dimension occurs as a link in $X$.
Now fix a facet $\sigma$ of $X$. For each $x \in \sigma$, there is a unique vertex $x^{\hspace{.15mm}\prime} \not \in\sigma$ such that $(\sigma \setminus \{x\}) \cup \{x^{\hspace{.15mm}\prime}\}$ is a facet. This defines a map $x \mapsto x^{\prime}$ from $\sigma$ to its complement. This map is injective: if we had $x^{\hspace{.15mm}\prime}_1 = y = x^{\hspace{.15mm} \prime}_2$ for $x_1 \neq x_2$ then the induced subcomplex of ${\rm lk}_X(y)$ on $\sigma$ would be connected. Also, since ${\rm
lk}_X(x^{\hspace{.15mm}\prime})[\sigma]$ is disconnected, it follows that $x$ must be an isolated vertex in ${\rm
lk}_X(x^{\hspace{.15mm}\prime})[\sigma]$. This implies that $x
x^{\hspace{.15mm}\prime}$ is an edge of $X$, and $V({\rm lk}_X(x
x^{\hspace{.15mm}\prime})) \subseteq V(X) \setminus (\sigma
\cup\{x^{\hspace{.15mm}\prime}\})$. Hence $x x^{\hspace{.15mm}
\prime}$ is an edge of degree $\leq d + 1$. Therefore, by the observation in the previous paragraph (with $e = 1$), $\deg_X(x
x^{\hspace{.15mm} \prime}) = d + 1$. In consequence, ${\rm lk}_X(x
x^{\hspace{.15mm} \prime})$ is a $(d + 1)$-vertex normal $(d -
2)$-pseudomanifold. But all such normal pseudomanifolds are known: we must have ${\rm lk}_X(x x^{\hspace{.15mm}\prime}) =
S^{\hspace{.2mm}m}_{m + 2} \ast S^{\hspace{.2mm}n}_{n + 2}$ for some $m, n \geq 0$ with $m + n = d-3$ (cf. [@bd2]). If $m >
0$ or $n > 0$ then $S^{\hspace{.2mm}1}_{3}$ occurs as a link (of some $(d - 4)$-simplex) in this sphere and hence it occurs as the link of a $(d - 2)$-simplex (containing $x x^{\hspace{.15mm}
\prime}$) in $X$. Hence, we must have $m = n = 0$. Thus $d = 3$ and each of the four edges $x x^{\hspace{.15mm} \prime}$ ($x\in
\sigma$) is of degree 4.
Then ${\rm lk}_X(x x^{\hspace{.15mm}\prime})$ is an $S^{\hspace{.2mm}1}_{4} = S^{\hspace{.3mm}0}_{2} \ast
S^{\hspace{.3mm}0}_{2}$ with vertex set $V(X) \setminus
(\sigma\cup\{x^{\hspace{.15mm}\prime}\})$. In consequence, putting $C = C(\sigma, X)$, one sees that $C$ is a 5-vertex non-simply connected simplicial complex (by the proof of Lemma \[LUK2\]) such that for at least four of the vertices $x^{\hspace{.15mm}\prime}$ in $C$, ${\rm lk}_C(x^{\hspace{.15mm}
\prime}) \supseteq S^{\hspace{.2mm}1}_{4}$. In consequence, all ${5 \choose 2} = 10$ edges occur in $C$. Since $C$ is non-simply connected, it follows that $C$ has at least one missing triangle (induced $S^{\hspace{.2mm}1}_{3}$), say with vertices $y_1, y_2,
y_3$. At least two of these vertices (say $y_1, y_2$) have $S^{\hspace{.2mm}1}_{4}$ in their links. It follows that ${\rm
lk}_C(y_1) \supseteq S^{\hspace{.3mm}0}_{2}(\{y_2, y_3\}) \ast
S^{\hspace{.3mm}0}_{2}(\{y_4, y_5\})$ and ${\rm lk}_C(y_2)
\supseteq S^{\hspace{.3mm}0}_{2}(\{y_1, y_3\}) \ast
S^{\hspace{.3mm}0}_{2}(\{y_4, y_5\})$ where $y_4$, $y_5$ are the two other vertices of $C$. Hence $C \supseteq C_0 =
(S^{\hspace{.2mm}1}_{3}(\{y_1, y_2, y_3\}) \ast
S^{\hspace{.3mm}0}_{2}(\{y_4, y_5\})) \cup \{y_4y_5\}$. But all 5-vertex simplicial complexes properly containing $C_0$ and not containing the 2-simplex $y_1y_2y_3$ are simply connected. So, $C
= C_0$. But, then two of the vertices of $C$ (viz. $y_4$, $y_5$) have no $S^{\hspace{.2mm}1}_{4}$ in their links, a contradiction. This completes the proof. $\Box$
$\!\!\!$[**.**]{} \[TUK\] For $d \geq 3$, Kühnel’s complex $K^{d}_{2d+3}$ is the only non-simply connected $(2d+3)$-vertex triangulated manifold of dimension $d$.
[**Proof.**]{} Let $X$ be a non-simply connected $(2d +
3)$-vertex triangulated manifold of dimension $d \geq 3$. By Lemma \[LUK3\], $X$ must have a facet $\sigma$ such that $C(\sigma, X)$ contains an induced subcomplex $S$ which is an $S^{\hspace{.2mm}d - 1}_{d + 1}$. Let $x$ be the unique vertex in $C(\sigma, X) \setminus S$. If $xy$ is a non-edge for each $y\in
\sigma$ then the normal $(d-1)$-pseudomanifold ${\rm lk}_X(x)$ is a subcomplex of the $(d - 1)$-sphere $S$ and hence ${\rm lk}_X(x)
= S$. This implies that $C(\sigma, X)$ is the combinatorial $d$-ball $\{x\} \ast S$. This is not possible since $C(\sigma,
X)$ is non-simply connected. Thus, $x$ forms an edge with a vertex in $\sigma$. This implies that $C(S, X)$ is connected.
Thus, $S$ is an induced $S^{\hspace{.21mm}d - 1}_{d + 1}$ in $X$, and $C(S, X)$ is connected. Since $d\geq 3$, $S$ is two-sided in $X$. By Lemma \[LEHD\], we may delete the handle over $S$ to get a $(3d + 4)$-vertex normal $d$-pseudomanifold $\widetilde{X}$. Since $X$ has at most ${2d + 3 \choose 2}$ edges, it follows that $\widetilde{X}$ has at most ${2d + 3 \choose 2} + {d + 1 \choose
2}$ edges. But ${2d + 3 \choose 2} + {d + 1 \choose 2} = (3d +
4)(d + 1) - {d + 2 \choose 2}$ is the lower bound on the number of edges of a $(3d + 4)$-vertex normal $d$-pseudomanifold given by the Lower Bound Theorem (cf. Theorem \[LBT-NPM\]). Therefore, $\widetilde{X}$ attains the lower bound, and hence, by Theorem \[LBT-NPM\], $\widetilde{X}$ is a stacked sphere. Now, Lemma \[LEHD\] implies that $X = \widetilde{X}^{\psi}$ where $\psi \colon \sigma_1 \to \sigma_2$ is an admissible bijection between two facets of $\widetilde{X}$. Thus, $\widetilde{X}$ is a $(3d + 4)$-vertex stacked $d$-sphere with an admissible bijection $\psi$. Therefore, by Theorem \[TSS\], $\widetilde{X} = {\cal
S}^{\hspace{.2mm}d}_{3d + 4}$ and $\psi$ are uniquely determined, hence so is $X = \widetilde{X}^{\psi}$. Since $K^{\hspace{.1mm}d}_{2d + 3}$ satisfies the hypothesis, it follows that $X = K^{\hspace{.1mm}d}_{2d + 3}$. $\Box$
$\!\!\!$[**.**]{} \[CUK1\] Let $X$ be an $n$-vertex triangulation of an $S^{\hspace{.2mm}d -
1}$-bundle over $S^{\hspace{.2mm}1}$. If $d \geq 2$ then $n \geq
2d+3$. Further, if $n = 2d + 3$, then $X$ is isomorphic to $K^{d}_{2d + 3}$.
[**Proof.**]{} Since an $S^{\hspace{.2mm}d - 1}$-bundle over $S^{\hspace{.2mm}1}$ is non-simply connected, the result is immediate from Lemma \[LUK2\] and Theorem \[TUK\] for $d \geq
3$. For $d = 2$, this result is classical. $\Box$
$\!\!\!$[**.**]{} \[CUK2\] If $d \geq 2$, $\varepsilon \equiv d$ $($mod $2)$ then $S^{\hspace{.2mm}1, d - 1}(\varepsilon)$ has a unique $(2d +
3)$-vertex triangulation, namely $K^{d}_{2d + 3}$.
[**Proof.**]{} Since $S^{\hspace{.2mm}1, d -
1}(\varepsilon)$ (with $\varepsilon \equiv d$ (mod 2)) is non-simply connected and is the geometric carrier of $K^{\hspace{.1mm}d}_{2d + 3}$, the result is immediate from Theorem \[TUK\] for $d \geq 3$. For $d = 2$, this result is classical. $\Box$
$\!\!\!$[**.**]{} \[CUK3\] If $d \geq 2$, $\varepsilon \not \equiv d$ $($mod $2)$ then any triangulation of $S^{\hspace{.2mm}1, d - 1}(\varepsilon)$ requires at least $2d + 4$ vertices. Thus, for this manifold, the $(2d + 4)$-vertex triangulations in Section $3$ are vertex minimal.
[**Proof.**]{} Since $S^{\hspace{.2mm}1, d -
1}(\varepsilon)$ (with $\varepsilon \not\equiv d$ (mod 2)) is non-simply connected and $K^{\hspace{.1mm}d}_{2d + 3}$ does not triangulate this space, the result is immediate from Theorem \[TUK\] for $d \geq 3$. For $d = 2$, this result is classical. $\Box$
$\!\!\!$[**.**]{} $($Walkup, Altshuler and Steinberg$)$ \[CUK4\] $K^{3}_{9}$ is the unique $9$-vertex triangulated $3$-manifold which is not a combinatorial $3$-sphere. In consequence, every closed $3$-manifold other than $S^{\hspace{.2mm}3}$ and ${S^{\hspace{.1mm}1}\! \times \hspace{-3.3mm}_{-} \,
S^{\hspace{.2mm}2}}$ requires at least $10$ vertices for a triangulation.
[**Proof.**]{} Note that any triangulated 3-manifold is a combinatorial 3-manifold. The result is immediate from Theorem \[TUK\], since by the Poincaré-Perelman theorem, the $3$-sphere is the only simply connected closed $3$-manifold. However, it is not necessary to invoke such a powerful result. Since a simply connected 3-manifold is clearly a homology 3-sphere, and by a result of [@bd5] any homology 3-sphere (other that $S^{\hspace{.2mm}3}$) requires at least 12 vertices, the corollary follows from Theorem \[TUK\]. $\Box$
A few days after we posted the first two versions of this paper in the arXiv (arXiv:math. GT/0610829) a similar paper [@css] was posted in the arXiv (arXiv:math.CO/0611039) by Chestnut, Sapir and Swartz. In that paper, the authors prove the uniqueness of $K^{\hspace{.1mm}d}_{2d + 3}$ in the broader class of homology $d$-manifolds (compared to the class of triangulated $d$-manifolds considered here) but with a much more restrictive topological condition (viz., $\beta_1 \neq 0$ and $\beta_2 = 0$, compared to our hypothesis of non-simply connectedness).
[**Acknowledgement:**]{} The authors thank the anonymous referees for many useful comments which led to substantial improvements in the presentation of this paper. The authors are thankful to Siddhartha Gadgil for useful conversations. The second author was partially supported by DST (Grant: SR/S4/MS-272/05) and by UGC-SAP/DSA-IV.
[99]{} A. Altshuler, L. Steinberg, An enumeration of combinatorial 3-manifolds with nine vertices, [*Discrete Math.*]{} [**16**]{} (1976) 91–108. B. Bagchi, B. Datta, A structure theorem for pseudomanifolds, [*Discrete Math.*]{} [**168**]{} (1998) 41–60. B. Bagchi, B. Datta, Combinatorial triangulations of homology spheres, [*Discrete Math.*]{} [**305**]{} (2005) 1–17. B. Bagchi, B. Datta, Lower bound theorems for pseudomanifolds (preprint). D. Barnette, A proof of the lower bound conjecture for convex polytopes, [*Pacific J. Math.*]{} [**46**]{} (1973), 349–354. U. Brehm, W. Kühnel, Combinatorial manifolds with few vertices, [*Topology*]{} [**26**]{} (1987) 465–473. J. Chestnut, J. Sapir, E. Swartz, Enumerative properties of triangulations of spherical boundles over $S^1$, [*European J. Combin.*]{} (to appear). G. Kalai, Rigidity and the lower bound theorem 1, [*Invent. math.*]{} [**88**]{} (1987) 125–151. W. Kühnel, Higher dimensional analogues of Császár’s torus, [*Results in Mathematics*]{} [**9**]{} (1986) 95–106. W. Kühnel, G. Lassmann, Permuted difference cycles and triangulated sphere bundles, [*Discrete Math.*]{} [**162**]{} (1996) 215–227. F. H. Lutz, Triangulated manifolds with few vertices: Combinatorial manifolds, v1 (2005) arXiv:math.CO/0506372. H. Rademacher, [*Topics in Analytic Number Theory*]{}, Springer-Verlag, New York Heidelberg, 1973. E. H. Spanier, [*Algebraic Topology*]{}, Springer-Verlag, New York Berlin Heidelberg, 1966. N. Steenrod, [*The topology of fibre bundles*]{}, Princeton Univ. Press, Princeton, 1951. T. S. Tay, Lower-bound theorems for pseudomanifolds, [*Discrete Comput Geom.*]{} [**13**]{} (1995) 203–216. D. W. Walkup, The lower bound conjecture for 3- and 4-manifolds, [*Acta Math.*]{} [**125**]{} (1970) 75–107.
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abstract: 'Dynamical systems which are invariant under $\mathcal{N}=1$ supersymmetric extension of the $l$-conformal Galilei algebra are constructed. These include a free $\mathcal{N}=1$ superparticle which is governed by higher derivative equations of motion and an $\mathcal{N}=1$ supersymmetric Pais-Uhlenbeck oscillator for a particular choice of its frequencies. A Niederer-like transformation which links the models is proposed.'
---
LMP-TPU– 3/14\
[**Dynamical realizations of $\mathcal{N}=1$** ]{}\
0.5cm [**$l$-conformal Galilei superalgebra**]{}\
$
\textrm{\Large Ivan Masterov\ }
$ 0.7cm [*Laboratory of Mathematical Physics, Tomsk Polytechnic University,\
634050 Tomsk, Lenin Ave. 30, Russian Federation*]{} 0.7cm [*Department of Physics, Tomsk State University, 634050 Tomsk,\
Lenin Ave. 36, Russian Federation*]{} 0.7cm [E-mail: masterov@tpu.ru]{}
PACS numbers: 11.30.-j, 11.25.Hf, 02.20.Sv
0.5cm
Keywords: conformal Galilei algebra, Pais-Uhlenbeck oscillator, supersymmetry
[**1. Introduction**]{} 0.5cm
In recent years nonrelativistic (super)conformal algebras have attracted considerable attention [@Henkel_2]-[@Hosseiny]. On the one hand, this interest originates from the current investigation of the nonrelativistic version of the AdS/CFT-correspondence [@Son; @McGreevy]. On the other hand, this research is motivated by the desire to construct new integrable models and explore novel correlations.
As is well-known, the Galilei algebra is relevant for physics in flat nonrelativistic spacetime. Conformal extension of the Galilei algebra is feasible and, moreover, it is not unique [@Henkel]-[@Negro_2]. In general, conformally extended Galilei algebra involves $(2l+1)$ vector generators where $l$ is a positive integer or half-integer. Such an extension is called the $l$-conformal Galilei algebra [@Negro_1]. The first two options $l=\frac{1}{2}$ and $l=1$, which are known in the literature as the Schrödinger algebra [@Niederer_1] and the conformal Galilei algebra [@Lukierski_1; @Havas], have been the focus of most studies [@Henkel_2]-[@Galajinsky_7], [@Duval_2]-[@Fedoruk_2]. More recently, various aspects of $l>1$ conformal Galilei symmetry have been extensively investigated. These include the construction of dynamical realizations [@Gomis]-[@Gonera_1], [@Aizawa_3], the study of admissible central and infinite dimensional extensions [@Galajinsky_3; @Masterov; @Hosseiny], the analysis of supersymmetric generalizations [@Masterov; @Aizawa_1; @Aizawa_2], the investigation of irreducible representations [@Aizawa_4; @Lu] and the possible twist deformations [@Dasz].
The Galilei algebra can be obtained from the Poincaré algebra by the nonrelativistic contraction [@Inonu]. Likewise, the so-called Newton-Hooke algebra [@Bacry; @Gibbons] can be derived from the (anti) de Sitter algebra. A specific feature of the Newton-Hooke algebra is that its structure relations involve the nonrelativistic cosmological constant $\Lambda=\mp\frac{1}{R^2}$, where $R$ is the characteristic time which is proportional to the radius of the parent (anti) de Sitter spacetime [@Gibbons]. The Newton-Hooke algebra also admits the $l$-conformal extension, which, however, is isomorphic to the $l$-conformal Galilei algebra [@Galajinsky_3; @Negro_1; @Negro_2]. The change of the basis in the $l$-conformal Galilei algebra \[change1\] K\_[-1]{}K\_[-1]{}K\_[1]{}, where $K_{-1}$ is the generator of time translations and $K_{1}$ is the generator of special conformal transformations, leads to the structure relations of the $l$-conformal Newton-Hooke algebra with negative (upper sign) or positive (lower sign) cosmological constant. By this reason it is customary to speak about realizations of one and the same algebra in a flat spacetime and in the Newton-Hooke spacetime [@Gibbons].
Supersymmetric extensions of the $l=\frac{1}{2}$-conformal Galilei algebra have been studied in [@Henkel_1; @Galajinsky_7; @Galajinsky_4; @Gomis_1; @Leblanc]. In particular, such supersymmetry was revealed in a nonrelativistic spin-$\frac{1}{2}$ particle, the nonrelativistic limit of the Chern-Simons matter systems, and quantum many-body mechanics. $\mathcal{N}$-extended version was systematically studied in [@Duval_3]. More recently, in [@Masterov; @Aizawa_1; @Aizawa_2] various supersymmetric extensions of the $l$-conformal Galilei algebra were constructed for the case of arbitrary $l$, but their dynamical realizations remain completely unexplored. The purpose of this work is to construct new dynamical realizations of $\mathcal{N}=1$ supersymmetric extension of the $l$-conformal Galilei algebra in the basis chosen in [@Aizawa_1].
The paper is organized as follows. In Section 2, we recall the basic facts about the $l$-conformal Galilei algebra. In Section 3 and Section 4, we construct dynamical realizations of $\mathcal{N}=1$ $l$-conformal Galilei superalgebra in flat superspace and in Newton-Hooke superspace, respectively. In Section 5, we summarize our results and discuss possible further developments. An infinite-dimensional generalization of the $\mathcal{N}=1$ $l$-conformal Galilei algebra in $d=1$ is discussed in Appendix A.
0.5cm [**2. The $l$-conformal Galilei algebra**]{} 0.5cm
Let us recall the structure of the $l$-conformal Galilei algebra. Besides the generators $K_{-1}$ and $K_1$ mentioned above, it involves the generator of dilatations $K_{0}$, the chain of vector generators $C_i^{(n)}$ with $n=0,1,..,2l$, and the generators of spatial rotations $M_{ij}$. The structure relations read $$\begin{aligned}
\label{algebraG}
&
[K_{p},K_{m}]=(m-p)K_{p+m}, && [K_{p},C^{(n)}_i]=(n-l(p+1))C_i^{(p+n)},
\nonumber\\[2pt]
&
[M_{ij},C^{(n)}_k]=\delta_{ik} C^{(n)}_j-\delta_{jk} C^{(n)}_i, && [M_{ij},M_{kl}]=\delta_{ik} M_{jl}+\delta_{jl} M_{ik}-
\delta_{il} M_{jk}-\delta_{jk} M_{il}.\end{aligned}$$
The algebra admits the central extension whose form depends on whether $l$ is even or odd [@Galajinsky_3] \[MCE\] \[C\_i\^[(n)]{},C\_j\^[(m)]{}\]=(-1)\^n n!m!\_[ij]{}\_[n+m,2l]{}M, where \_[ij]{}={
\_[ij]{},&i,j=1,2,..,d,&;\
\_[ij]{},&i,j=1,2,&,\
. $\epsilon_{12}=1$ and $M$ is the central charge. In dynamical realizations the central charges correspond to physical parameters of a systems. As we shall see below in the Sections 3 and 4, the central charges play the crucial role in constructing the dynamical realizations.
0.5cm [**3. Dynamical realization of $\mathcal{N}=1$ $l$-conformal Galilei superalgebra**]{} 0.5cm
Let us consider $\mathcal{N}=1$ supersymmetric extension of the $l$-conformal Galilei algebra presented in [@Aizawa_1]. In addition to the generators considered in the preceding section, it involves the supersymmetry generator $G_{-\frac{1}{2}}$, the generator of superconformal transformations $G_{\frac{1}{2}}$ and the fermionic partners $L_i^{(n)}$, with $n=0,1,..,2l-1$, of the vector generators. Along with (\[algebraG\]) the nonvanishing (anti)commutation relations of the superalgebra include $$\begin{aligned}
\label{algebraGN1}
&
\{G_r,G_s\}=2iK_{r+s}, && [K_p,L_i^{(m)}]=(m-(l-1/2)(p+1))L_i^{(p+m)},
\nonumber\\[2pt]
&
\{G_r,L_i^{(n)}\}=iC_i^{(n+r+1/2)}, && [G_r,C_i^{(n)}]=\left(n-2l\left(r+1/2\right)\right)L_i^{(r+n-1/2)},
\nonumber\\[2pt]
&
[K_p,G_r]=\left(r-\frac{p}{2}\right)G_{n+r}, && [M_{ij},L_k^{(n)}]=\d_{ik}L_j^{(n)}-\d_{jk}L_i^{(n)}.\end{aligned}$$
The structure relations (\[algebraG\]) and (\[algebraGN1\]) are compatible with (\[MCE\]) only if the anticommutators of the fermionic vector generators are modified as follows [@Aizawa_1]: \[MCEN1\] {L\_i\^[(n)]{},L\_j\^[(m)]{}}=i(-1)\^[n]{}n! m!\_[ij]{}\_[n+m,2l-1]{}M.
As the first step, let us check that all the scalar generators in the superalgebra, as well as space rotations, can be realized as quadratic combinations of the bosonic and fermionic vector generators. Indeed, choosing the ansatz for $K_n$ K\_n=\_[k,m=0]{}\^[2l]{}(k,m;n)\_[ij]{}C\_i\^[(k)]{}C\_j\^[(m)]{}+\_[k,m=0]{}\^[2l-1]{}(k,m;n)\_[ij]{}L\_i\^[(k)]{}L\_j\^[(m)]{}, one can unambiguously fix the constants $\alpha(k,m;n)$ and $\beta(k,m;n)$ by imposing the structure relations of the superalgebra. The explicit form of the generators obtained in this way is $$\begin{aligned}
\label{biprod}
&
K_n=\sum_{k=0}^{2l}\frac{\alpha_k}{2} (k-l(n+1))\lambda_{ij}C_i^{(2l-k)}C_j^{(k+n)}+\sum_{k=0}^{2l-1}\frac{\beta_k}{2}(k-(l-1/2)(n+1))\lambda_{ij}L_i^{(2l-1-k)}L_j^{(k+n)},
\nonumber\\
&
G_r=\sum_{k=1}^{2l}\alpha_k k\lambda_{ij} C_i^{(2l-k+r+1/2)}L_j^{(k-1)},\quad\;
M_{ij}=\sum_{k=0}^{2l}\alpha_k C_i^{(2l-k)}C_j^{(k)}+\sum_{k=0}^{2l-1}\beta_k L_i^{(2l-k-1)}L_j^{(k)}.\end{aligned}$$ Note that for integer $l$ the generator of rotation reads \[12\] M\_[12]{}=-\_[k=0]{}\^[2l]{} C\_i\^[(2l-k)]{}C\_i\^[(k)]{}-\_[k=0]{}\^[2l-1]{} L\_i\^[(2l-k-1)]{}L\_i\^[(k)]{}, where we denoted \_k=,\_k=. For $\mathcal{N}=2$ supersymmetric extensions of the $l$-conformal Galilei algebra the analogs of (\[biprod\]) and (\[12\]) are presented in [@Aizawa_1], [@Aizawa_2]. Throughout the work the summation over repeated spatial indices is understood.
Thus if one succeeds in constructing a system with conserved vector charges obeying (\[MCE\]) and (\[MCEN1\]) with respect to some graded Poisson bracket, one can automatically produce additional integrals of motion by making use of (\[biprod\]) and (\[12\]). Together with $C_i^{(n)}$, $L_i^{(n)}$ they will obey the structure relations (\[algebraG\]), (\[algebraGN1\]) with respect to the same bracket.
Let us construct such a system by applying the method of nonlinear realizations [@Coleman_1; @Coleman_2] to the subalgebra formed by $K_{-1}$, $C_i^{(n)}$, $L_i^{(n)}$, and $M$. To this end, one starts with a generic subgroup element $e^{aK_{-1}}e^{\zeta_i^{(n)}C_i^{(n)}}e^{i\xi_i^{(n)}L_i^{(n)}}e^{\chi M}$, where $a,\zeta_i^{(n)},\xi_i^{(n)},\chi$ are parameters, and considers the transformation of the space \[spaceN1\] G=e\^[tK\_[-1]{}]{}e\^[x\_i\^[(n)]{}C\_i\^[(n)]{}]{}e\^[i\_i\^[(n)]{} L\_i\^[(n)]{}]{}e\^[M]{}, parametrized by the coordinates $t,x_i^{(n)},\psi_i^{(n)},\varphi$, which is generated by the left multiplication with the subgroup element. It is assumed that $\xi_i^{(n)}$ and $\psi_i^{(n)}$ anticommute with $L_i^{(n)}$ as well as with each other. The resulting infinitesimal coordinate transformations read && t=a,x\_i\^[(n)]{}=\_[k=n]{}\^[2l]{}t\^[k-n]{}\_i\^[(k)]{},\_i\^[(n)]{}=\_[k=n]{}\^[2l-1]{}t\^[k-n]{}\_i\^[(k)]{},\
&&\[trN1\] =+\_[n=0]{}\^[2l]{}\_[k=n]{}\^[2l]{}t\^[k-n]{}\_i\^[(k)]{}\_[ij]{}x\_j\^[(2l-n)]{}+\
&& +i\_[n=0]{}\^[2l-1]{}\_[k=n]{}\^[2l-1]{}t\^[k-n]{}\_i\^[(k)]{}\_[ij]{}\_j\^[(2l-n-1)]{}.
Then one constructs the Maurer-Cartan one-forms G\^[-1]{}dG=\_K K\_[-1]{}+\_[i]{}\^[(n)]{}C\_i\^[(n)]{}+i\_i\^[(n)]{}L\_i\^[(n)]{}+\_M M, where $$\begin{aligned}
\label{MC}
&
\omega_K=dt,\quad\qquad\omega_i^{(n)}=dx_i^{(n)}+(n+1)x_i^{(n+1)}dt,\quad\qquad \tilde{\omega}_i^{(n)}=d\psi_i^{(n)}+(n+1)\psi_i^{(n+1)}dt,
\\[2pt]
&
\omega_M=d\varphi+\frac{\la_{ij}}{2}\left(\sum_{n=0}^{2l}(-1)^{n}\omega_i^{(n)}x_j^{(2l-n)}n!(2l-n)!+i\sum_{n=0}^{2l-1}(-1)^n\tilde{\omega}_i^{(n)}\psi_j^{(2l-n-1)}n!(2l-n-1)!\right),
\nonumber\end{aligned}$$ which hold invariant under all the transformations (\[trN1\]). By definition, $x_i^{(2l+1)}=\psi_i^{(2l)}=0$.
In general, one can either reduce the number of degrees of freedom or obtain the dynamical equations of motion by setting some of the Maurer-Cartan one-forms to vanish [@Ivanov]. Let us choose the restrictions \[restr\] \_i\^[(n)]{}=0,\_i\^[(n)]{}=0, which allow us to exclude all the vector variables except for $x_i^{(0)}\equiv x_i$ and $\psi_i^{(0)}\equiv\psi_i$. Then taking $t$ to be a temporal coordinate, from (\[restr\]) one obtains the constraints \[constr\] x\_i\^[(n)]{}=,\_i\^[(n)]{}=, as well as the dynamical equations of motion \[EOM\] =0,=0.
Note that the equations (\[EOM\]) can be derived from the action functional \[action\] S=(-1)\^[2l+1]{} \_M=dt\_[ij]{}(x\_i-i\_i), which is derived from $\omega_M$ by taking into account the constraints (\[constr\]). The model (\[action\]) is an $\,\mathcal{N}=1$ supersymmetric generalization of the free higher derivative particle studied in [@Gomis; @Gonera_5; @Gonera_4].
In accord with (\[trN1\]), the action (\[action\]) is invariant under the transformations \[tr\] t=a,x\_i=\_[n=0]{}\^[2l]{}\_i\^[(n)]{}t\^n,\_i=\_[n=0]{}\^[2l-1]{}\_i\^[(n)]{}t\^n, where $\tilde{\zeta}_i^{(n)}=(-1)^n\zeta_i^{(n)}$, $\tilde{\xi}^{(n)}=(-1)^n\xi_i^{(n)}$. Then the Noether theorem yields the vector constants of the motion \[vect\] C\_i\^[(n)]{}=\_[ij]{}\_[k=0]{}\^[n]{}t\^[n-k]{} x\_j\^[(2l-k)]{},L\_i\^[(n)]{}=i\_[ij]{}\_[k=0]{}\^[n]{}t\^[n-k]{} \_j\^[(2l-k-1)]{}, as well as $K_{-1}$ \[K1\] K\_[-1]{}=\_[ij]{}(\_[k=1]{}\^[2l]{}(-1)\^[k+1]{}x\_i\^[(k)]{}x\_j\^[(2l-k+1)]{}+i\_[k=1]{}\^[2l-1]{}(-1)\^k \_i\^[(k)]{}\_j\^[(2l-k)]{}). Here and in what follows the upper superscript in braces, which is attached to coordinates, denotes the number of time derivatives. Note that the latter is related to (\[vect\]) via (\[biprod\]).
Introducing the graded Poisson bracket \[bracketG\] \[A,B}=\_[ij]{}\_[n=0]{}\^[2l]{}(-1)\^n +i\_[ij]{}\_[n=0]{}\^[2l-1]{}(-1)\^[n+1]{}, it is straightforward to check that the integrals of motion (\[biprod\]) and (\[vect\]) do obey the structure relations of the centrally extended $\mathcal{N}=1$ $l$-conformal Galilei superalgebra (\[algebraG\]), (\[MCE\]), (\[algebraGN1\]), (\[MCEN1\]) with $M=1$. When verifying the algebra, the following relations \[111\] \[x\_i\^[(n)]{},x\_j\^[(m)]{}}=(-1)\^[n]{}\_[n+m,2l]{}\_[ij]{},\[\_i\^[(n)]{},\_j\^[(m)]{}}=i(-1)\^[n+1]{}\_[n+m,2l-1]{}\_[ij]{} prove to be helpful. It should be noted that the equations $[x_i^{(n)},K_{-1}\}=x_i^{(n+1)},\, [\psi_i^{(n)},K_{-1}\}=\psi_i^{(n+1)}$ hold.
Concluding this section, we display the infinitesimal symmetry transformations of the action (\[action\]) which correspond to the integrals of motion constructed above \[transform\] && K\_n:t=t\^[n+1]{}a\_n,x\_i=l(n+1)t\^n x\_i a\_n,\_i=(l-1/2)(n+1)t\^n\_i a\_n,\
&& G\_r:x\_i=it\^[r+1/2]{}\_i\_r,\_i=(t\^[r+1/2]{}\_i-2l(r+1/2)x\_i)\_r,\
&& M\_[ij]{}: x\_i=w\_[ij]{}x\_j, \_i=w\_[ij]{}\_j,(w\_[ij]{}=-w\_[ji]{}).
To summarize, a free superparticle obeying the higher derivative equations of motion (\[EOM\]) provides the simplest dynamical realization of $\mathcal{N}=1$ $l$-conformal Galilei superalgebra. Note that this result is in agreement with the previous studies in [@Gomis]-[@Gonera_5] and [@Gonera_4]-[@Gonera_2].
0.5cm [**4. Dynamical realization of $\mathcal{N}=1$ $l$-conformal Newton-Hooke superalgebra**]{} 0.5cm
As is known, realizations of the $l$-conformal Galilei algebra in a flat spacetime and in the Newton-Hooke spacetime [@Negro_1; @Negro_2] are related by the coordinate transformations which, for the case of a negative cosmological constant, reads [@Duval_2; @Galajinsky_3; @Niederer] \[nied\] t’=R(t/R),x\_i’(t’)=x\_i(t)/\^[2l]{}(t/R). Here the prime designates coordinates parametrizing flat spacetime.
Let us construct an analogous transformation which links the model (\[action\]) to its Newton-Hooke counterpart. To this end, we first note that the equations of motion for $\psi_i$ in (\[EOM\]) can be formally obtained from the equations of motion for $x_i$ by the substitution $x_i\rightarrow\psi_i$, $l\rightarrow l-1/2$. Using this observation, one obtains the transformation for the odd variables \[niedN1\] \_i’(t’)=\_i(t)/\^[2l-1]{}(t/R).
Implementing the transformations (\[nied\]) and (\[niedN1\]) to (\[action\]), one derives the action functional \[PU1\] S=dt(x\_i\_[k=1]{}\^[l+]{}(+)x\_i- i\_i\_[k=1]{}\^[l-]{}(+)\_i), which is valid for half-integer $l$. For integer $l$, one gets \[PU2\] S=dt \_[ij]{}(x\_i\_[k=1]{}\^[l]{}(+)\_j- i\_i\_[k=1]{}\^[l]{}(+)\_j). These actions describe $\mathcal{N}=1$ supersymmetric extensions of the Pais-Uhlenbeck oscillator [@Pais] (for a review see [@Smilga]) for a particular choice of its frequencies. The bosonic limit of (\[PU1\]) has been considered in detail in a recent work [@PU] (see also [@Galajinsky_1]).
The form of the symmetry transformations which leave the actions (\[PU1\]) and (\[PU2\]) invariant, as well as the form of the associated integrals of motion, is readily obtained by applying (\[nied\]), (\[nied1\]) to (\[tr\]), (\[transform\]), (\[vect\]) and (\[biprod\]), respectively. In particular, from (\[vect\]) one derives the transformations corresponding to the vector generators \[transf1\] x\_i=\_[n=0]{}\^[2l]{}\_i\^[(n)]{}R\^n \^[n]{}\^[2l-n]{}, \_i=\_[n=0]{}\^[2l-1]{}\_i\^[(n)]{}R\^n \^[n]{}\^[2l-n-1]{}, and the integrals of motion associated with them \[vect1\] C\_i\^[(n)]{}=\_[ij]{}\_[k=0]{}\^[n]{}\^[n-k]{} ([x’]{}\_j)\^[(2l-k)]{}, L\_i\^[(n)]{}=i\_[ij]{}\_[k=0]{}\^[n]{}\^[n-k]{} ([’]{}\_j)\^[(2l-k-1)]{}. With respect to the graded bracket \[bracketG1\] \[A,B}=\_[ij]{}\_[n=0]{}\^[2l]{}(-1)\^n +\_[ij]{}\_[n=0]{}\^[2l-1]{}(-1)\^n, they obey the structure relations (\[MCE\]) and (\[MCEN1\]). The existence of the relations similar to (\[biprod\]) for the bosonic limit of (\[PU1\]) was anticipated in [@Andrzejewski]. Eqs. (\[vect1\]) and (\[bracketG1\]) involve the derivatives of ${x'}_i$ and ${\psi'}_i$ with respect to $t'$, i.e., $\frac{d}{dt'}=\cos^{2}{\frac{t}{R}}\frac{d}{dt}$.
The remaining symmetries of the actions (\[PU1\]) and (\[PU2\]) read \[transformPU\] && K\_[0]{}:t=a\_[0]{},x\_i=lx\_i a\_[0]{}, \_i=(l-)\_i a\_[0]{};\
&& K\_[1]{}:t=R\^2\^2a\_1,x\_i=lRx\_i a\_1,\_i=(l-)R\_i a\_1;\
&& G\_[-]{}:x\_i=i\_i\_[-]{},\_i=(\_i +x\_i)\_[-]{};\
&& G\_: x\_i=iR\_i\_,\_i=(R\_i -2lx\_i)\_, while the transformations corresponding to the generators $K_{-1}$ and $M_{ij}$ are the same as in (\[transform\]). In order to obtain the transformation corresponding to $K_{-1}$ and the corresponding integral of motion, one has to make the linear change of the basis (\[change1\]). Thus, the actions (\[PU1\]) and (\[PU2\]) are invariant under $\mathcal{N}=1$ $l$-conformal Galilei superalgebra realized in Newton-Hooke superspace.
The case of a positive cosmological constant can be treated by implementing the formal change of the characteristic time $R\rightarrow iR$. In particular, for a half-integer $l$ the analogues of (\[nied\]) and (\[niedN1\]), \[nied1\] t’=R(t/R),x\_i’(t’)=x\_i(t)/\^[2l]{}(t/R),\_i’(t’)=\_i(t)/\^[2l-1]{}(t/R), yield the action functional \[PU11\] S=dt(x\_i\_[k=1]{}\^[l+]{}(-)x\_i- i\_i\_[k=1]{}\^[l-]{}(-)\_i), while for integer $l$ one gets \[PU21\] S=dt \_[ij]{}(x\_i\_[k=1]{}\^[l]{}(-)\_j- i\_i\_[k=1]{}\^[l]{}(-)\_j). These actions can be viewed as describing $\mathcal{N}=1$ supersymmetric extensions of the Pais-Uhlenbeck oscillator for the particular choice of imaginary frequencies.
It should be stressed that in the flat space limit $R\rightarrow \infty$ all the formulas corresponding to the realizations (\[PU1\]), (\[PU2\]), (\[PU11\]), (\[PU21\]) in the Newton-Hooke superspace correctly reproduce the respective expressions in a flat superspace.
0.5cm [**5. Conclusion**]{} 0.5cm To summarize, in this work we have constructed dynamical realizations of $\mathcal{N}=1$ $l$-conformal Galilei superalgebra in a flat superspace and in the Newton-Hooke superspace. Coordinate transformations which link the realizations were found. The models describe a free higher derivative superparticle and an $\mathcal{N}=1$ supersymmetric extension of the Pais-Uhlenbeck oscillator for the particular choice of its frequencies, respectively.
Turning to possible further developments, it is interesting to investigate whether an $\mathcal{N}=1$ supersymmetric extension of the $l$-conformal Galilei algebra can be realized in systems without higher derivatives in the equations of motion. In this context, it would be interesting to construct a supersymmetric extensions of the models constructed in [@Galajinsky_2], [@Galajinsky_1]. Then it is worth generalizing the results in [@Henkel_2], [@Henkel_3] to incorporate $\mathcal{N}=1$ supersymmetry. A generalization of the analysis in this paper to the case of $\mathcal{N}=2$ $l$-conformal Galilei superalgebras [@Masterov; @Aizawa_1; @Aizawa_2] is worth studying as well.
0.5cm [**Acknowledgements**]{} 0.5cm We thank A. Galajinsky for the comments on the paper. This work was supported by the Dynasty Foundation, Russian Foundation for Basic Research (RFBR) grant 14-02-31139-Mol, the MSE program “Nauka” under project 3.825.2014/K, and TPU grant LRU.FTI.123.2014.
0.5cm [**Appendix A. Infinite-dimensional extension.**]{} 0.5cm
It should be noted that the generators $K_n$ and $G_r$ in (\[algebraGN1\]) form centerless $\mathcal{N}=1$ Neveu-–Schwarz algebra ($\mathcal{NS}$) [@NS], if one takes $n$ and $r$ to be arbitrary integer and half-integer numbers, respectively. It is natural to ask whether an infinite-dimensional extension of the $\mathcal{N}=1$ $l$-conformal Galilei superalgebra is feasible. Such an extension for $\mathcal{N}=1$ Schrödinger superalgebra in $d=1$ has been systematically constructed in [@Henkel_1] as the superalgebra of some functions forming a closed set under the graded Poisson bracket
$$[f,g\}=\frac{\partial f}{\partial q}\frac{\partial g}{\partial p}-\frac{\partial f}{\partial p}\frac{\partial g}{\partial q}+i \frac{\overleftarrow{\partial}f}{\partial\theta}\frac{\overrightarrow{\partial}g}{\partial\theta}.\eqno{(A1)}$$
According to the analysis in [@Henkel_1], one can choose $K_{n}=q\,p^{n+1}$ and $G_r=\sqrt{2}\,q^{\frac{1}{2}}\,p^{r+\frac{1}{2}}\,\theta$ which under (A1) obey the structure relations of $\mathcal{NS}$. Other generators $$C^{(n)}=q^{l}\,p^{n},\qquad L^{(n)}=\frac{1}{\sqrt{2}}\,q^{l-1/2}\,p^{n}\,\theta.\eqno{(A2)}
\nonumber$$ can be unambiguously fixed by imposing the structure relations (\[algebraGN1\]). In contrast to (\[MCE\]) and (\[MCEN1\]) the nonvanishing relations between the generators (A2) read $$[C^{(n)},C^{(m)}\}=l(m-n)M^{m+n}_{1,1},\qquad[L^{(n)},L^{(m)}\}=\frac{i}{2}M^{n+m+1}_{1,1},\eqno{(A3)}$$ where we denoted $$M^{n}_{k,s}=p^{n-1}q^{2lk-s}.\eqno{(A4)}$$ Moreover, the relations involving (A4) and $G_{r}$, $L^{(n)}$, $$[G_r,M^{n}_{k,s}\}=(n-1-(2lk-s)(2r+1))\tilde{M}^{n+r-1}_{k,2s+1},$$ $$[L^{(n)},M^{m}_{k,s}\}=\frac{1}{2}\left((2l-1)(m-1)-2n(2lk-s)\right)\tilde{M}^{m+n-3/2}_{k,2s-2l+3},\eqno{(A5)}$$ yield extra generators $$\tilde{M}^{r}_{k,s}=\frac{1}{\sqrt{2}}p^{r-\frac{1}{2}}q^{2lk-\frac{1}{2}s}\theta.\eqno{(A6)}$$ Along with (\[algebraG\]), (\[algebraGN1\]), (A3), (A5) the structure relations of the desired infinite-dimensional extension involve
& \[M\^[n\_1]{}\_[k\_1,s\_1]{},M\^[n\_2]{}\_[k\_2,s\_2]{}}=((2lk\_1-s\_1)(n\_2-1)-(2lk\_2-s\_2)(n\_1-1))M\^[n\_1+n\_2-2]{}\_[k\_1+k\_2,s\_1+s\_2+1]{},\
& \[K\_[n]{},M\^[m]{}\_[k,s]{}}=(m-1-(2lk-s)(n+1))M\^[n+m]{}\_[k,s]{}, \[\_[k\_1,s\_1]{}\^[r\_1]{},\_[k\_2,s\_2]{}\^[r\_2]{}}=M\_[k\_1+k\_2,(s\_1+s\_2)]{}\^[r\_1+r\_2]{},\
& \[K\_n,\^[r]{}\_[k,s]{}}=(r-1/2-(2lk-s/2)(n+1))\^[n+r]{}\_[k,s]{},\[G\_[r\_1]{},\^[r\_2]{}\_[k,s]{}}=iM\^[r\_1+r\_2]{}\_[k,(s-1)]{},\
& \[C\^[(n)]{},M\^[m]{}\_[k,s]{}}=(l(m-1)-n(2lk-s))M\^[n+m-1]{}\_[k,s-l+1]{},\[L\^[(n)]{},\^[r]{}\_[k,s]{}}=M\^[n+r+1/2]{}\_[k,(s+1)-l]{},\
& \[C\^[(n)]{},\^[r]{}\_[k,s]{}}=(l(r-1/2)-n(2lk-s/2))\^[n+r-1]{}\_[k,s-2l+2]{},\
& \[L\^[(n)]{},M\^[m]{}\_[k,s]{}}=((2l-1)(m-1)-2n(2lk-s))\^[m+n-3/2]{}\_[k,2s-2l+3]{}.
By analogy with [@Henkel_1], one can define the grading of coordinates as follows $$gra(p)=0,\qquad gra(q)=1,\qquad gra(\theta)=\frac{1}{2},$$ with respect to which $$gra(K_n)=gra(G_r)=1,\qquad gra(C^{(n)})=gra(L^{(n)})=l,$$ $$gra(M^{n}_{k,s})=2lk-s,\quad gra(\tilde{M}^{r}_{k,s})=2lk-s/2+1/2.\eqno{(A7)}$$ In Ref. [@Henkel_1], the grading relations (A7) play the crucial role as they allow one to simplify the infinite-dimensional superalgebra considerably. In particular, it has been shown that for $l=1/2$ one can construct the quotient algebra which involves only additional generators $M^{n}_{1,1}$ and $\tilde{M}^{n}_{1,3}$ (for more details see [@Henkel_1]). Unfortunately, the difference in grading (A7) makes an analogous simplification for the superalgebra with $l>1$ problematic.
[nn]{} R. Cherniha, M. Henkel, On nonlinear partial differential equations with an infinite-dimensional conditional symmetry, J. Math. Anal. Appl. [**298**]{} (2004) 487, math-ph/0402059. J. Lukierski, P.C. Stichel, W.J. Zakrzejewski, Exotic Galilean conformal symmetry and its dynamical realizations, Phys. Lett. A [**357**]{} (2006) 1, hep-th/0511259. M. Henkel, J. Unterberger, Supersymmetric extensions of Schrodinger-invariance, Nucl. Phys. B [**746**]{} (2006) 155, math-ph/0512024. J. Lukierski, P.C. Stichel, W.J. Zakrzejewski, Acceleration-extended Galilean symmetries with central charges and their dynamical realizations, Phys. Lett. B [**650**]{} (2007) 203, hep-th/0702179. A. Galajinsky, Remark on quantum mechanics with conformal Galilei symmetry, Phys. Rev. D [**78**]{} (2008) 087701, arXiv:0808.1553. C. Duval, P.A. Horvathy, Non-relativistic conformal symmetries and Newton-Cartan structures, J. Phys. A [**42**]{} (2009) 465206, arXiv:0904.0531. A. Galajinsky, I. Masterov, Remark on quantum mechanics with N=2 Schrodinger supersymmetry, Phys. Lett. B [**675**]{} (2009) 116, arXiv:0902.2910. C. Duval, M. Hassaine, P.A. Horvathy, The geometry of Schrödinger symmetry in gravity background/nonrelativistic CFT, Annals Phys. [**324**]{} (2009) 1158, arXiv:0809.3128. A. Galajinsky, N=2 superconformal Newton-Hooke algebra and many-body mechanics, Phys. Lett. B [**680**]{} (2009) 510, arXiv:0906.5509. J.A. de Azcarraga, J. Lukierski, Galilean superconformal symmetries, Phys. Lett. B [**678**]{} (2009) 411, arXiv:0905.0141. A. Galajinsky, O. Lechtenfeld, Harmonic N=2 Mechanics, Phys. Rev. D [**80**]{} (2009) 065012, arXiv:0907.2242. P.D. Alvarez, J.L. Cortes, P.A. Horvathy, M.S. Plyushchay, Super-extended noncommutative Landau problem and conformal symmetry, JHEP [**0903**]{} (2009) 034, arXiv:0901.1021. A. Galajinsky, Conformal mechanics in Newton-Hooke spacetime, Nucl. Phys. B [**832**]{} (2010) 586, arXiv:1002.2290. R. Cherniha, M. Henkel, The exotic conformal Galilei algebra and nonlinear partial differential equations, J. Math. Anal. Appl. [**369**]{} (2010) 120, arXiv:0910.4822. C. Duval, P. Horvathy, Conformal Galilei groups, Veronese curves, and Newton-Hooke spacetime, J. Phys. A [**44**]{} (2011) 335203, arXiv:1104.1502. S. Fedoruk, E. Ivanov, J. Lukierski, Galilean confromal mechanics from nonlinear realizations, Phys. Rev. D [**83**]{} (2011) 085013, arXiv:1101.1658. S. Fedoruk, J. Lukierski, The algebraic structure of Galilean superconformal symmetries, Phys. Rev. D [**84**]{} (2011) 065002, arXiv:1105.3444. A. Galajinsky, I. Masterov, Remarks on l-conformal extension of the Newton-Hooke algebra, Phys.Lett. B [**702**]{} (2011) 265, arXiv:1104.5115. I. Masterov, N=2 supersymmetric extension of l-conformal Galilei algebra, J. Math. Phys. [**53**]{} (2012) 072904, arXiv:1112.4924. N. Aizawa, P.S. Isaac, Y. Kimura, Highest weight representations and Kac determinant for a class of conformal Galilei algebras with central extension, Int. J. Math. [**23**]{} (2012), 1250118, arXiv:1204.2871. N. Aizawa, N=2 Galilean superconformal algebras with central extension, J. Phys. A [**45**]{} (2012) 475203, arXiv:1206.2708. J. Gomis, K. Kamimura, Schrodinger equation for higher order non-relativistic particles and N-Galilean conformal symmetry, Phys. Rev. D [**85**]{} (2012) 045023, arXiv:1109.3773. K. Andrzejewski, J. Gonera, Nonrelativistic conformal groups and their dynamical realizations, Phys. Rev. D [**86**]{} (2012) 065009, arXiv:1204.5950. A. Galajinsky, I. Masterov, Dynamical realization of l-conformal Galilei algebra and oscillators, Nucl. Phys. B [**866**]{} (2013), arXiv:1208.1403. K. Andrzejewski, J. Gonera, Unitary representations of N-conformal Galilei group, Phys. Rev. D [**88**]{} (2013) 065011, arXiv:1305.4777. K. Andrzejewski, J. Gonera, A. Kijanka-Dec, Nonrelativistic conformal transformations in Lagrangian formalism, Phys. Rev. D [**87**]{} (2013) 065012, arXiv:1301.1531. K. Andrzejewski, J. Gonera, Dynamical interpretation of nonrelativistic conformal groups, Phys. Lett. B [**721**]{} (2013) 319. K. Andrzejewski, J. Gonera, P. Kosiński, P. Maślanka, On dynamical realizations of l-conformal Galilei groups, Nucl. Phys. B [**876**]{} (2013) 309, arXiv:1305.6805. A. Galajinsky, I. Masterov, Dynamical realizations of l-conformal Newton-Hooke group, Phys. Lett. B [**723**]{} (2013) 190, arXiv:1303.3419. N. Aizawa, Z. Kuznetsova, F. Toppan, Chiral and Real N=2 supersymmetric l-conformal Galilei algebras, J. Math. Phys. [**54**]{} (2013) 093506, arXiv:1307.5259. N. Aizawa, Interwining operator for l-conformal Galilei algebras and hierarchy of invariant equations, J. Phys. A [**46**]{} (2013) 405204, arXiv:1308.0121. M. Daszkiewicz, Twist deformation of l-conformal Galilei Hopf algebra, Adv. Stud. Theor. Phys. [**7**]{} (2013) 471, arXiv:1307.1140. R. Lu, V. Mazorchuk, K. Zhao, On simple modules over conformal Galilei algebras, J. of Pure and Appl. Algebra [**218**]{} (2013) 10, arXiv:1310.6284. K. Andrzejewski, A. Galajinsky, J. Gonera, I. Masterov, Conformal Newton-Hooke symmetry of Pais-Uhlenbeck oscillator, Nucl. Phys. B [**885**]{} (2014) 150, arXiv:1402.1297. A. Hosseiny, Possible central extension of non-relativistic conformal algebras in 1+1, J. Math. Phys. [**55**]{} (2014) 061704, arXiv:1403.4537. D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrodinger symmetry, Phys. Rev. D [**78**]{} (2008) 046003, arXiv:0804.3972. K. Balasubramanian, J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. [**101**]{} (2008) 061601, arXiv:0804.4053. M. Henkel, Local scale invariance and strongly anisotropic equilibrium critical system, Phys. Rev. Lett. [**78**]{} (1997) 1940, cond-mat/9610174. J. Negro, M.A. del Olmo, A. Rodriguez-Marco, Nonrelativistic conformal groups, J. Math. Phys. [**38**]{} (1997), 3786. J. Negro, M.A. del Olmo, A. Rodriguez-Marco, Nonrelativistic conformal groups. II. Further developments and physical applications, J. Math. Phys. [**38**]{} (1997), 3810. U. Niederer, The maximal kinematical invariance group of the free Schrodinger equation, Helv. Phys. Acta [**45**]{} (1972) 802. P. Havas, J. Plebańsky, Conformal extensions of the Galilei group and their relation to the Schrödinger group, J. Math. Phys. [**19**]{} (1978) 482. E. Inonu, E.P. Wigner, On the contraction of groups and their representations, Proc. Nat. Acad. Sci. [**39**]{} (1953) 510. H. Bacry, J.M. Lévy-Leblond, Possible kinematics, J. Math. Phys. [**9**]{} (1968) 1605. G.W. Gibbons, C.E. Patricot, Newton-Hooke space-time, Hpp waves and the cosmological constant, Class. Quant. Grav. [**20**]{} (2003) 5225, hep-th/0308200. J.P. Gaunlett, J. Gomis, P.K. Townsend, Supersymmetry and the physical phase space formulation of spinning particles, Phys. Lett. B [**248**]{} (1990) 288. M. Leblanc, G. Lozano, H. Min, Extended superconformal Galilean symmetry in Chern-Simons matter systems, Annals Phys.[**219**]{} (1992) 328, hep-th/9206039. C. Duval, P.A. Horvathy, On Schrodinger superalgebras, J. Math. Phys. [**35**]{} (1994) 2516, hep-th/0508079. S.R. Coleman, J. Wess, B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. [**177**]{} (1969) 2239. C.G. Callan, S.R. Coleman, J. Wess, B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. [**177**]{} (1969) 2247. E.A. Ivanov, V.I. Ogievetsky, The inverse Higgs phenomenon in nonlinear realizations, Teor. Mat. Fiz. [**25**]{} (1975) 164. U. Niederer, The maximal kinematical invariance group of the harmonic oscillator, Helv. Phys. Acta [**46**]{} (1973), 191. A. Pais, G.E. Uhlenbeck, On field theories with nonlocalized action, Phys. Rev. [**79**]{} (1950) 145. A.V. Smilga, Comments on the dynamics of the Pais-Uhlenbeck oscillator, SIGMA [**5**]{} (2009) 017, arXiv:0808.0139. K. Andrzejewski, Hamiltonian formalisms and symmetries of the Pais-Uhlenbeck oscillator, arXiv:1410.0479. M. Ademollo, L. Brink et al, Supersymmetric strings and colour confinement, Phys. Lett. B [**62**]{} (1976) 105.
|
---
author:
- Lyuba Malysheva^^ and Alexander Onipko^^
title: 'From acene to graphene spectrum of $\pi$ electrons with the use of the Green’s function'
---
Introduction.
=============
Free standing graphene, a monoatomic layer of graphite is a new material [@Nov1; @Nov2] that exhibits exceptional electronic quality [@Nov1; @Nov2; @Nov3; @Zhang; @Geim] and a unique nature of charge carriers. On one hand, electrons and holes appear as quasi-particles in condensed-matter physics when the Schrödinger equation directs our prediction of their behavior [@Wallace; @Fujita; @Nakada; @Saito; @W1; @W2; @Thomsen; @Peres1; @prb2007; @W3; @Lyuba1]. On the other hand, they behave as relativistic particles which are subject to the Dirac equation [@Sem; @Khv; @Ando; @Brey; @Gus1; @Gus2]. Much theoretical effort has been focused primarily on the electronic states of infinite graphene. In the continuum approximation, these states can be mapped into the Hamiltonian of 2+1 dimensional quantum electrodynamics with Dirac fermions. This description is based on the cone-like spectrum near the neutrality (zero-energy) points of 2D graphite band structure, see [@Gus2] and references therein. Less attention has been paid to graphene as a molecular species. From this point of view, graphene can be considered as a macromolecule obtained by C-C bonding of acene chains as shown in Fig. 1. In certain aspects, the macromolecule model provides a better approximation for the description of real graphene structures [@Lyuba1], particularly, in the nanometer scale, where the continuous approximation is of a limited value [@Lyuba2].
![$\cal N$-long linear acene as a basic unit of $N$$\times$$\cal N$ sheet of graphene honeycomb lattice (up) and $\cal N$-long cyclacene playing the same role for zigzag ($\cal N$,0) carbon nanotubes (CNTs).[]{data-label="Fig1"}](Fig1pss.eps){width="0.8\linewidth" height="0.63\linewidth"}
As is clear from Fig. 1, $\cal N$-long linear acenes are basic units of $N$$\times$$\cal N$ sheet of graphene honeycomb lattice, whereas $\cal N$-long cyclacenes play the same role for zigzag ($\cal N$,0) carbon nanotubes (CNTs). In this sense, the graphene and acene spectra are intimately interrelated. The electronic structure of acenes, the semi-empirical description of which goes back to the Pariser and Parr classic works [@PP; @P], is understood fairly well [@Hoff; @Kiv; @Houk]. This study elucidates the relationship between the graphene and acene spectra. Keeping in mind subsequent applications, our analysis has been performed with the use of the Green’s function method. All derivations are based on the standard operator equation $$\label{1}
G^{\rm M} =\left[EI-H^{M}\right]^{-1},$$ where $H^{\rm M}$ is the tight-binding Hamiltonian operator of acene (M=A) or graphene (M=G); the electron energy $E$ is in unites of $|t|$, $t$ denotes C-C hopping integral. For the Green’s function matrix elements, notations $\langle m\alpha|G^{\rm A}|m'\beta\rangle$ $\equiv$ $G^{\rm A}_{m\alpha,m'\beta}$ and $\langle mn\alpha |G^{\rm G}|m'n'\beta\rangle$ $\equiv$ $G^{\rm G}_{mn\alpha,m'n'\beta}$ will be used for acene and graphene, respectively; $|m\alpha\rangle$ and $|mn\alpha\rangle$ denote $2p_z$ orbitals at the $\alpha$th carbon atom of the $m$th and $mn$th hexagon, respectively, as shown in Fig. 1. The formal description of acenes and cyclacenes, and graphene and CNT is very similar. However, in the interests of simplicity, only acene–graphene pair will be in focus. An extended discussion that also includes cyclacenes and zigzag CNTs can be found in Ref. [@Lyuba4]
Acene Green’s function.
=======================
For linear acenes, the matrix form of solution to Eq. (\[1\]) is given by [@Klym1] $$\label{6}
G^{\rm A}_{m \alpha,m'\beta} = \frac{2}{{\cal N}+1}\sum_{j=1}^{\cal N}
g_{\alpha,\beta}^j \sin(\xi_j m)\sin(\xi_j m'),$$ where $\xi_j= \pi j/({\cal N}{\rm +}1)$, $j=1,2,...,\cal N$, and $$\label{7}
{\cal D}_jg_{\alpha,\beta}^j =
\left\{
\begin{array}{lr}
E\left[E^2{\rm -}4\cos^2(\xi_j/2){\rm -}1\right ],&
\alpha =\beta=l,r,\\
4\cos^2(\xi_j/2),&\hspace{-0.9cm}
\alpha =l(r), \beta=r(l),\\
\end{array}
\right.$$ where ${\cal D}_j=[E^2-4\cos^2(\xi_j/2)]^2-E^2$. For the rest of matrix elements, we obtain $$G^{\rm A}_{m \lambda,m'
\left\{ \begin{smallmatrix} \lambda \\ \rho \end{smallmatrix} \right \}}
=\frac{\delta_{m,m'} \left\{ \begin{smallmatrix} E \\ 1 \end{smallmatrix} \right \}}{E^2-1}+ \frac{8}{{\cal N}+1}$$ $$\label{8}
\times\sum_{j=1}^{\cal N}
g_{\alpha,\beta}^j \cos^2(\xi_j/2)\sin[\xi_j (m{\rm -}1/2)]\sin[\xi_j (m'{\rm -}1/2)],$$ $${\cal D}_j (E^2-1)g_{\alpha,\beta}^j$$ $$\label{9}
=
\left\{
\begin{array}{lr}
E[E^2{\rm -}4\cos^2(\xi_j/2){\rm +}1],&
\alpha =\beta=\lambda,\rho,\\
2E^2{\rm -}4\cos^2(\xi_j/2),&
\alpha =\lambda(\rho), \beta=\rho(\lambda),\\
\end{array}
\right.$$ and $$G^{\rm A}_{m \alpha,m'\beta}$$ $$\label{10}
= \frac{4}{{\cal N}+1}
\sum_{j=1}^{\cal N}g_{\alpha,\beta}^j\cos(\xi_j/2) \sin[\xi_j (m{\rm -}1/2)]\sin(\xi_j m'),$$ $$\label{11}
{\cal D}_j g_{\alpha,\beta}^j =
\left\{
\begin{array}{lr}
E^2-4\cos^2(\xi_j/2),&
\alpha =\lambda(\rho), \beta=l(r),\\
E,&
\alpha =\lambda(\rho), \beta=r(l).\\
\end{array}
\right.$$
As shown in [@Lyuba1], defined in Eq. (\[7\]) quantities $g^j_{l,l}=g^j_{r,r}$ and $g^j_{l,r}=g^j_{r,l}$ determine the dispersion relation for an $N$$\times$${\cal N}$ sheet of graphene $E(\kappa_j,\xi_j)$ via an equation $$\label{12}
2 g_{l,r}^j \cos \kappa_j = 1-(g_{l,l}^j)^2 +(g_{l,r}^j)^2,$$ where $\kappa_j$ is the second, complementing $\xi_j$ quantum number. $2N$ values of $\kappa^\pm_{j,\nu}$, $\nu$ = $0,1,...,N-1$, for each $j$ classify the graphene spectrum into 4${\cal N}N$ $j,\nu$ levels; see below.
![Band structure of polyacene, $E(\kappa{\rm =}k^+)$ (red) and $E(\kappa{\rm =}k^-)$ (blue), calculated from Eq. (\[13\]) for infinite linear acene (left panel) and cyclacene (right panel). Continuation to imaginary values of wave vectors, $k^\pm\rightarrow \pm\pi\pm i\delta$, $0\leq\delta\leq 2 \ln((1+\sqrt{17})/4)\approx 0.5$, is shown by dashed lines. Solid lines of the right panel for $\kappa\le\pi$ repeats similar calculations of Ref. [@Kiv]. Triangles show energy levels for $\cal N$=7 according to Eq. (\[15\]).[]{data-label="Fig2"}](Fig2pss.eps){width="0.9\linewidth" height="0.6\linewidth"}
By performing summation over $j$, it is possible to express the energy dependence of acene Green’s function in terms of (dimensionless) wave vectors $k^+$ and $k^-$, subjected to equation $$\label{13}
4\cos^2(k^\pm/2)=E(E \pm 1).$$ For example, $G^{\rm A}_{1\lambda(\rho),({\cal N}+1)\lambda(\rho)}= G_{k^+} +G_{k^-} $, where $$\label{14}
G_{k^\pm} =
\frac{1}{2(E\pm1)}\frac{\sin k^\pm}{ \sin({\cal N}+1)k^\pm}.$$ Other matrix elements have similar expressions. As seen from Fig. 2, for the given energy, either both $k^+$ and $k^-$ are real, or one is real, whereas the other one is imaginary. In the latter case, matrix elements which refer to the opposite sites of the chain have two typical terms: One is exponential, $\sim \exp\left(-\sqrt{|E|}{\cal N}\right )$, and the second term has a singular character. Hence, the probability of electron transmission through acenes ($\sim G^{\rm A\,2}_{1,({\cal N}{\rm +}1)}$ [@Chapt]) can occur via partly coherent and partly tunneling mechanisms. Tunneling governed by a single exponential factor, as it takes place in conjugated oligomers [@Chapt; @Mag], is not possible. This is a reflection of a metallic nature of polyacenes [@Hoff; @Kiv].
{width="0.96\linewidth" height="0.24\linewidth"}
Acene electron spectrum.
------------------------
The acene electronic structure is determined by equation $(E^2-1){\cal D}_j=0$; zeros of ${\cal D}_j$ give a four-band spectrum $$\label{15}
E_{c(v)}^{\pm} =+(-) \frac{1}{2}\left[\mp1\pm \sqrt{1+16\cos^2(\xi_j/2)}\right],$$ where each of two conduction ($c$) and two valence ($v$) bands has $\cal N$ levels. Note that two extra levels $E=\pm1$ do not appear in Eq. (\[6\]) as poles of the acene Green’s function. The states with these energies correspond to zero wave function amplitudes at $l$ and $r$ sites and thus, break the acene chain into $\lambda$-$\rho$ “isolated” pairs.
For long chains ${\cal N}$$>>$1 and $|E|<<1$, spectrum (\[15\]) is well approximated by $
E=\pm q_\mu^2$ with $q_\mu$$=\pi\mu/{\cal N}$, $\mu$ = 1,2,... . In the limit ${\cal N}$$\rightarrow$$\infty$, the quantum number $\xi_j$ can be considered as a continuous variable, $\xi_j\rightarrow k$, and $q_\mu\rightarrow q =\pi{\rm -} k$ acquires the meaning of $k$ separation from the zero-energy point at $k=\pi$. The spectrum of polyacene (an infinite acene chain) that comes out from the spectra of a linear acene and cyclacene, is shown in Fig. 2.
Graphene Green’s function.
==========================
Similarly to Eq. (\[6\]), matrix elements of graphene Green’s function can be represented in the form of an expansion $$\label{16}
G^{\rm G}_{mn\alpha,m'n'\beta} =
\frac{2}{{\cal N}+1}\sum_{j=1}^{\cal N} G^j_{n\alpha,n'\beta}
\sin(\xi_j m)\sin(\xi_j m'),$$ where $G^j_{n\alpha,n'\beta} = G^{j+}_{n\alpha,n'\beta} +G^{j-}_{n\alpha,n'\beta}$. Here, we restrict ourselves by representative examples of matrix elements refering to the $l$ and $r$ sites, namely, $$\label{17}
G^{j\pm}_{1l,
\left\{ \begin{smallmatrix} 1 \\ N \end{smallmatrix} \right \}
\left\{ \begin{smallmatrix} l \\ r \end{smallmatrix} \right \}}
=
\frac{
\left\{ \begin{smallmatrix} \displaystyle g_{l,l}^j \sin (\kappa^\pm_{j} N)\\
\displaystyle g_{l,r}^j \sin \kappa^\pm_{j} \end{smallmatrix} \right \}}
{\sin(\kappa^\pm_{j} N)-g_{l,r}^j\sin[\kappa^\pm_{j} (N-1)]},$$ where labels + and $-$ correspond to, respectively, “plus” and “minus” branches of dispersion relation (\[12\]). This relation can be rewritten as $$E_{c(v)}^{j\pm} =$$ $$\label{18}
+(-) \sqrt{1+4\cos^2 (\xi_j/2) \pm 4|\cos (\xi_j/2) \cos (\kappa_j/2)|}.$$ Thus, quantum numbers $\kappa^\pm_{j\nu}$ are determined by poles of $G^{j\pm}_{n\alpha,n'\beta}$ as a function of $\kappa_j^\pm$, i.e., by the roots of equation $\sin \kappa_j^\pm N-g_{l,r}^{j}\sin \kappa^\pm_j(N-1)=0$, or [@Lyuba1] $$\label{19}
\sin \kappa^\pm_jN=\mp2\cos(\xi_j/2)\sin \kappa^\pm_j(N+1/2),$$ which together with Eq. (\[18\]) gives energies of 4$N\cal N$ $j,\nu$ levels. Additionally, the graphene spectrum contains two $N$-fold degenerate levels with $E=\pm1$.
In most cases, it is an interval near the Fermi energy, where interesting physics occurs. Whithin this interval and for $N,{\cal N}>>1$, we have [@Lyuba3] $$\label{20}
E_c^{(j^*\pm\mu)-}= \left \{
\begin{array}{l}
\sqrt{\mu^2 \Delta^{2}_{_{\rm G}} + \kappa^2/4 },\\
\sqrt{(\mu\mp \frac{1}{3})^2 \Delta^{2}_{_{\rm G}} +\kappa^2/4},
\end{array} \right.$$ where $\Delta_{_{\rm G}}$ = $\sqrt{3}\pi/[2({\cal N}$+1)\], $\mu<<\cal N$, and the upper and lower lines refer to an integer and rational value of $2({\cal N}{\rm +}1)/3$, respectively. In the case of integer values of $2({\cal N}{\rm +}1)/3$=$j^*$, say, for $\cal N$=${\cal N}^*$, infinitely long armchair graphene ribbons (GRs) have a metallic spectrum. In contrast, for ${\cal N}$=${\cal N}^*$$\pm1$, the lowest energy mode is $j^*$=$2({\cal N}^*$ $\pm1)/3$, and the spectrum has a band gap ($E_g=2\Delta_{_{\rm G}}/3$), as defined by the lower line in Eq. (\[20\]).
The Green’s functions of GRs with the length $N$ and $\cal N$ = ${\cal N}^*$, ${\cal N}^*$+1 and ${\cal N}^*{\rm -1}$ have a pronouncedly different appearance; see Fig. 3. In what follows, we concentrate on energy intervals below (above) the bottom (top) of the second lowest band of metallic GRs, $|E|$$\le$$ \Delta_{_{\rm G}}$, and whithin the band gap of semiconducting GRs, $|E|$$\le$$E_g/2$. For these energies, the Green’s functions shown in the mid and right panel in Fig. 3 are accurately reproduced by a single member of expansion (\[16\]), namely, by the $j^*$th term. To reproduce the energy dependence of the Green’s function illustrated in the left panel, two terms are needed, partial Green’s functions $j^*$ and $j^*$+1.
![Solutions to Eq. (\[19\]), shown by intersections of black, red, and blue curves, corresponding to $N$ =50, 20, and 6, respectively, with three horizontal lines $2\cos \xi_{j^*}/2$ for ${\cal N}$ = ${\cal N}^*$ (red), ${\cal N}^*$-1 (blue), and ${\cal N}^*{\rm +1}$ (green); ${\cal N}^*$=8. Green curve, representing $N$=3 ($<$$\cal N$/2) does not have intersections in the region of imaginary $\kappa=i\delta$; see text. []{data-label="Fig4"}](Fig4pss.eps){width="0.8\linewidth" height="0.62\linewidth"}
As immediately follows from Eq. (\[20\]), the $E$–$\kappa$ relation can be satisfied only by real and only by imaginary ($\kappa$=$i\delta$) values of the wave vector for metallic and semiconducting GRs, respectively. As a result, an exponential factor $$\label{21}
G^{j^*}_{1 l,Nr}
\sim \exp\left ( -2N\sqrt{(E_g/2)^2-E^2}\right ),$$ appears in the Green’s function of semiconducting GRs. This factor governs the probability of single-mode electron/hole transmission through a potential barrier; see Ref. [@Yu] of this issue. For metallic GRs, the number of poles depends on $N$ as, approximately, $[\sqrt{3}N/({\cal N}{\rm+1)+}1/2]$. For semiconducting GRs with $\cal N$=${\cal N}^*{\rm -1}$, there are no poles at all, but in the case of $\cal N$=${\cal N}^*{\rm +1}$, there is a single pole under the condition $N$$>$$0.9\sqrt{3}$$({\cal N}{\rm +1})\pi{\rm -}1/2$ or, approximately $N$$>$${\cal N}/2$. These peculiarities are illustrated in Fig. 3.
To conclude this report, the spectra and Green’s functions of graphene and its building blocks, acenes have been discussed in parallel that provides a deeper insight into the origin and particularities of graphene electronic structure. An exact analytical expression of graphene Green’s function is found in terms of an expansion over partial Green’s functions. The character of the Green function singularities near the Fermi energy has been examined and shown to be qualitatively different for metallic and semiconducting graphene ribbons.
This work was partly supported by Visby program of the Swedish Institute (SI).
[\[1\]]{} K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, and A.A. Firsov, Science **306**, 366 (2004).
K.S. Novoselov, D. Jiang, T. Booth, V.V. Khotkevich, S.V. Morozov, and A.K. Geim, Proc. Nat. Acad. Sc. **102**, 10451 (2005).
K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, and A.A. Firsov, Nature (London) **438**, 197 (2005).
Y. Zhang, Y.-W. Tan, H.L. Stormer, and P. Kim, Nature (London) **438**, 201 (2005).
A.K. Geim and K.S. Novoselov, Nature Materials **6**, 183 (2007).
P.R. Wallace, Phys. Rev. B **9**, 622 (1947).
M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, J. Phys. Soc. Japan **65**, 1920 (1996).
K. Nakada, M. Fujita, G. Dresselhaus, and M.S. Dresselhaus, Phys. Rev. B **54**, 17954 (1996).
R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College, London, 1998).
K. Wakabayashi, M. Fujita, H. Ajiki, and M. Sigrist, Phys. Rev. B **59**, 8271 (1999).
K. Wakabayashi, Phys. Rev. B **64**, 125428 (2001).
S. Reich, C. Thomsen, and J. Maultzsch, Carbon Nanotubes: Basic concepts and Physical Properties (WILEY-VCH, Weinheim, 2004).
N.M.R. Peres, A.H. Castro Neto, and F. Guinea, Phys. Rev. B **73**, 195411 (2006).
M. Kohmoto and Y. Hasegava, Phys. Rev. B **76**, 205402 (2007).
K. Wakabayashi, Y. Takane, and M. Sigrist, Phys. Rev. Lett. **99**, 036601 (2007).
L. Malysheva and A. Onipko, Phys. Rev. Lett., to be published, arXiv:0801.4155v1 \[cond-mat.mes-hall\].
G.W. Semenoff, Phys. Rev. Lett. **53**, 2449 (1984).
D.V. Khveshchenko, Phys. Rev. Lett. **87**, 206401 (2001).
T. Ando, J. Phys. Soc. Japan **74**, 777 (2002).
L. Brey and H.A. Fertig, Phys. Rev. B **73**, 235411 (2006).
Y.P. Gusynin, S.G. Sharapov, and J.P. Carbotte, Phys. Rev. Lett. **98**, 157402 (2007).
Y.P. Gusynin, S.G. Sharapov, and J.P. Carbotte, Int. J. Mod. Phys. B **21**, 4611 (2007).
L. Malysheva and A. Onipko, arXiv:0802.1522v1 \[cond-mat.mes-hall\].
R. Pariser and R.G. Parr, J. Chem. Phys. **21**, 466 (1953).
R. Pariser, J. Chem. Phys. **24**, 250 (1956).
M. Kertesz abd R. Hoffmann, Solid State Commun. **47**, 97 (1983).
S. Kivelson and O.L. Chapman, Phys. Rev. B **28**, 7236 (1983).
K.N. Houk, P.S. Lee, and M. Nendel, J. Org. Chem. **66**, 5517 (2001).
Yu. Klymenko, unpublished.
L. Malysheva and A. Onipko, arXiv:0804.1761v1 \[cond-mat.mes-hall\].
A. Onipko and L. Malysheva, in: Nano and Molecular Electronics Handbook, edited by S.E. Lyshevski, Nano- and Microscience, Engineering, Technology, and Medicine Series (CRC Press, Taylor & Francis Group, Boca Raton Fl., 2007) chap. 23.
M. Magoga and C. Joachim, Phys. Rev. B **56**, 4722 (1997).
L. Malysheva and A. Onipko, arXiv:0803.1761v1 \[cond-mat.mes-hall\].
Yu. Klymenko, O. Shevtsov, and A. Onipko, arXiv:0804.1761v1 \[cond-mat.mes-hall\].
|
---
author:
- |
\
University of Agder, Kristiansand, Norway\
E-mail:
- |
Stefano Moretti\
University of Southampton, Southampton, UK\
E-mail:
- |
Shoaib Munir\
APCTP, Pohang, South Korea\
E-mail:
- |
Leszek Roszkowski\
[^1] NCBJ, Warsaw, Poland\
E-mail:
title: LHC phenomenology of light pseudoscalars in the NMSSM
---
Introduction
============
Despite the ever tighter exclusion limits on sparticle masses coming from the LHC, supersymmetry still remains our best bet for physics beyond the Standard Model (SM). It is worth pointing out that, although the mass of the discovered Higgs boson[@Aad:2012tfa; @Chatrchyan:2012ufa] is a bit high for supersymmetry, it is consistent with the Minimal Supersymmetric Standard Model (MSSM).
The MSSM, though, suffers from a problem in the dimensionful supersymmetric $\mu$ term which for phenomenological reasons has to be of the same scale as the soft supersymmetry breaking terms. As these terms are a priory unrelated, this poses a problem for high scale model building.
One way to enforce the similarity in scale between the $\mu$ term and the soft supersymmetry breaking terms is to forbid the $\mu$ term and instead introduce a gauge singlet scalar superfield which then can generate the $\mu$ term by getting a Vacuum Expectation Value (VEV) at the Electroweak scale. This VEV will be generated from soft supersymmetry breaking terms and hence the similarity of scales comes out naturally.
This is the idea behind the Next to Minimal Supersymmetric Standard Model (NMSSM) [@Ellwanger:2009dp]. This model has enjoyed a renewed interest with the discovery of the, for the MSSM, somewhat heavy Higgs boson, since the NMSSM contains new tree level contributions to the Higgs mass and hence allows for a 125 GeV Higgs while keeping the fine-tuning at acceptable levels.
With the additional scalar superfield, the NMSSM has three neutral scalars, $H_1,H_2,H_3$ where $M_{H_1}<M_{H_2}<M_{H_3}$, and two pseudoscalars, $A_1,A_2$ with $M_{A_1}< M_{A_2}$, as well as a charged scalar $H^\pm$. The discovered SM-like Higgs, $H_{\rm SM}$, is identified as the doublet-like of $H_1$ or $H_2$.
As we are looking for light pseudoscalars, $A_1$ will always be singlet-like and its mass is basically a free parameter so we will take it to be lighter than $H_{\rm SM}$. Such light pseudoscalars can be detected in several ways. If $M_{A_1}< 10$ GeV they might show up in meson decays. One could also consider direct production, but that seems unlikely to work; only $b\bar b A_1$ production is of interest but at least for $M_{A_1}> 10$ GeV, the dominant decay channel is $A_1\to b\bar b$ rendering a detection very challenging [@Bomark:2014gya].
The remaining hope is indirect production through the decay of heavier particles. In the following we will discuss studies [@Bomark:2014gya; @Bomark:2015fga] of the detection prospects from heavier scalars decaying to $A_1A_1$ or $A_1Z$.
Scan
====
In order to assess the discovery prospects of a light pseudoscalar in the NMSSM parameter space, we employ Bayesian scans using MultiNest-v2.18 [@Feroz:2008xx] coupled with NMSSMTools-v4.2.1 [@Ellwanger:2006rn], Higgsbounds-v4.1.3 [@Bechtle:2008jh; @Bechtle:2011sb; @Bechtle:2013gu; @Bechtle:2013wla], SuperISO-v3.3 [@superiso] and micrOMEGAs-v2.4.5 [@micromegas].
To cover all possibilities, we use two scans; one focusing on what we call the “naturalness region” with low $\tan\beta$ (the ratio of the vacuum expectation values of the two Higgs doublets) and large $\lambda$ (the coupling constant for the singlet scalar to the Higgs doublets) to maximise the tree-level NMSSM specific enhancements of the SM-like Higgs mass, and one scan using wider parameter ranges. We also use separate scans for $H_1$ and $H_2$ being SM-like.
To make sure the SM-like Higgs is acceptable, we require it to be between 122 and 128 GeV where the large range is due to large theoretical uncertainties. We also require the points to comply with the Higgs signal rates for $H_{\rm SM}\to \gamma\gamma$ and $H_{\rm SM}\to ZZ$ as given by CMS in [@CMS-PAS-HIG-14-009]. We do not use ATLAS data here since they at the time of the scan had large deviations from SM values that have later disappeared [@ATLAS-CONF-2014-009; @Aad:2014eha].
LHC studies
===========
To estimate the LHC reach in the studied channels we use MadGraph5\_aMC$@$NLO [@Alwall:2014hca] to generate parton level backgrounds and then Pythia 8.180 [@Sjostrand:2007gs] and FastJet-v3.0.6 [@Cacciari:2011ma] for signal generation, hadronisation and jet clustering. To optimise the sensitivity at low pseudoscalar masses we employ the jet substructure methods of [@Butterworth:2008iy].
Our studies include all channels where scalars are produced and decay to $A_1A_1$ or $A_1Z$, however, as explained in [@Bomark:2014gya], $H_{1,2}\to A_1Z$ and $H_3\to A_1A_1$ have too small rates to be of any interest. This means that we focus on the lighter scalars $H_{1,2}$ decaying to $A_1A_1$ and $H_3\to A_1Z$. The latter is an interesting channel for somewhat heavier pseudoscalars that have not received much attention — due to the small couplings between $H_3$ and the weak vector bosons, this channel can only be studied through gluon fusion (GF) production of the scalar. For the lighter scalars, also vector boson fusion (VBF) as well as Higgsstrahlung ($ZH$ and $WH$) production can be of interest, though the higher rates of GF seems to always be more important than the lower backgrounds of the other channels [@Bomark:2015fga].
Since we are focusing on $M_{A_1}>10$ GeV, BR$(A_1\to b\bar b)$ is always around 0.9 making 4b-jets the dominant final state for the $A_1A_1$ channel — however, the reduction in background in the 2b2$\tau$[^2] final state more then compensates for the factor 9 in the signal rate making it the best channel if the initial scalar is produced through GF or VBF. With $WH$ and $ZH$ the backgrounds are sufficiently suppressed by the additional vector bosons (that we require to decay leptonically) so that the higher rates of $4b$ channel is more beneficial.
Results
=======
If we look at the channels with $H_{\rm SM}\to A_1A_1$, there is a problem in getting points where the channel is kinematically open, especially in the naturalness region this is an issue as $\lambda$ here is large and hence BR$(H_{\rm SM}\to A_1A_1)$ usually becomes so large that it suppresses other Higgs decays below experimentally acceptable rates, this is especially true when $H_1$ is SM-like. This can be seen in Figure \[fig:hSM\], where the left panel is very scarcely populated; the right panel has better coverage but also here can we see a clear upper limit on the cross-section.
Perhaps even more interesting is the possibility to see the non-SM one of $H_1$ and $H_2$ ($H_{\rm non-SM}$) decay to $A_1A_1$. Since $H_{\rm non-SM}$ has no lower limits on other signal rates, BR$(H_{\rm non-SM}\to A_1A_1)$ may well be close to one and hence this might be our best (or only) chance of discovering also this scalar. As can be seen in Figure \[fig:hnonSM\] the prospects for such discovery are rather good: especially when $H_2$ is SM-like, this channel can be very promising.
While GF production of the initial scalar gives the highest signal rates, it also have large backgrounds, therefore it could be interesting to compare to the reach in other channels. Of the other possibilities $WH$ seems like the most promising; $ZH$ has an almost negligible background due to the requirement of a leptonically decaying $Z$ but the signal is also very small, so $WH$ amounts to a better compromise between the two. However, as can be seen in Figure \[fig:WH\], the sensitivity is clearly worse than for GF. One should remember though, that this could be an important complement, especially to measure vector boson couplings of the $H_{\rm non-SM}$.
For somewhat heavier $A_1$s we have to rely on the $H_3\to A_1Z$ channel, which, as can be seen in Figure \[fig:h3A1Z\], does show some promise. It should be remembered that, since this work focuses on light pseudoscalars, the cuts used are generic and rather soft, so a more detailed study of the effects of harder cuts may significantly improve the scope of this channel.
The prospects in all channels are summarised in Table \[tab:Channels\].
Production mode Channels Accessibility Range(GeV)
------------------------- ----------------- --------------- ---------------------
$b\bar b A_1$ —
$H_1\to A_1A_1$ ($H_1$) $gg$, VBF, $VH$ 300/fb $m_{A_1}<63$
$H_1\to A_1A_1$ ($H_2$) $gg$, VBF, $VH$ 30/fb $m_{A_1}<60$
$H_1\to A_1Z$ —
$H_2\to A_1A_1$ ($H_1$) $gg$, VBF 300/fb $60< m_{A_1} <80$
$H_2\to A_1A_1$ ($H_2$) $gg$, VBF, $VH$ 30/fb $m_{A_1}<63$
$H_2\to A_1Z$ —
$H_3\to A_1A_1$ —
$H_3\to A_1Z$ $gg$ 300/fb $60 < m_{A_1} <120$
: List of the $A_1$ production channels included in this study. The second column shows the production mechanisms of interest for the initial scalar, while the third column shows the integrated luminosity at which the $A_1$ can be accessible at the LHC in at least one of these combinations. In the fourth column we provide the mass range within which a signature of $A_1$ can be established in the given channel.[]{data-label="tab:Channels"}
Conclusions
===========
With its neat solution to the $\mu$ problem and improved ability to give a heavy enough Higgs boson, as compared to the MSSM, the NMSSM is a compelling model for new physics. The possible existence of light singlet states also holds great promise for new phenomenology at the LHC.
In this proceedings we have discussed work demonstrating that a light pseudoscalar can be discovered through $H_{1,2}\to A_1A_1$ in large parts of parameter space. This is especially interesting for the non-SM like of $H_1$ and $H_2$ as this might well be our only handle on such a scalar, i.e. seeing $H_{\rm non-SM}\to A_1A_1$ might not only be a way of discovering the pseudoscalar but also the scalar.
For pseudoscalars above the kinematic threshold of the above channels, our best chance might be $H_3\to A_1Z$, which is an interesting but poorly studied channel.
[99]{}
ATLAS Collaboration, G. Aad [*et al.*]{}, Phys.Lett. [**B716**]{}, 1 (2012), 1207.7214.
CMS Collaboration, S. Chatrchyan [*et al.*]{}, Phys.Lett. [**B716**]{}, 30 (2012), 1207.7235.
U. Ellwanger, C. Hugonie, and A. M. Teixeira, Phys.Rept. [**496**]{}, 1 (2010), 0910.1785.
N.-E. Bomark, S. Moretti, S. Munir, and L. Roszkowski, JHEP [**1502**]{}, 044 (2015), 1409.8393.
N. E. Bomark, S. Moretti and L. Roszkowski, arXiv:1503.04228 \[hep-ph\]. F. Feroz, M. Hobson, and M. Bridges, Mon.Not.Roy.Astron.Soc. [**398**]{}, 1601 (2009), 0809.3437.
U. Ellwanger and C. Hugonie, Comput. Phys. Commun. [**177**]{} (2007) 399 \[hep-ph/0612134\]. P. Bechtle, O. Brein, S. Heinemeyer, G. Weiglein, and K. E. Williams, Comput.Phys.Commun. [**181**]{}, 138 (2010), 0811.4169.
P. Bechtle, O. Brein, S. Heinemeyer, G. Weiglein, and K. E. Williams, Comput.Phys.Commun. [**182**]{}, 2605 (2011), 1102.1898.
P. Bechtle [*et al.*]{}, PoS [**CHARGED2012**]{}, 024 (2012), 1301.2345.
P. Bechtle [*et al.*]{}, Eur.Phys.J. [**C74**]{}, 2693 (2014), 1311.0055.
A. Arbey and F. Mahmoudi, Comput.Phys.Commun. [**176**]{}, 367 (2007), 0906.0369.
G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov, Comput.Phys.Commun. [**181**]{}, 1277 (2010), hep-ph/0607059.
CERN Report No. CMS-PAS-HIG-14-009, 2014 (unpublished).
CERN Report No. ATLAS-CONF-2014-009, 2014 (unpublished).
ATLAS Collaboration, G. Aad [*et al.*]{}, Phys.Rev. [**D90**]{}, 112015 (2014), 1408.7084.
J. Alwall [*et al.*]{}, JHEP [**1407**]{}, 079 (2014), 1405.0301.
T. Sj[ö]{}strand, S. Mrenna, and P. Z. Skands, Comput.Phys.Commun. [**178**]{}, 852 (2008), 0710.3820.
M. Cacciari, G. P. Salam, and G. Soyez, Eur.Phys.J. [**C72**]{}, 1896 (2012), 1111.6097.
J. M. Butterworth, A. R. Davison, M. Rubin, and G. P. Salam, Phys.Rev.Lett. [**100**]{}, 242001 (2008), 0802.2470.
[^1]: On leave of absence from the University of Sheffield, UK.
[^2]: Also 2b2$\mu$ could be interesting, that will be the topic of future studies.
|
---
abstract: 'The paper studies lower bounds for the dimensions of projective indecomposable modules for Chevalley groups $G$ in defining characteristic $p$. The main result extending earlier one by Malle and Weigel (2008) determines the modules in question of dimension equal to the order of a Sylow $p$-subgroup of $G$. We also substantially generalize a result by Ballard (1978) on lower bounds for the dimensions of projective indecomposable modules and find lower bounds in some cases where Ballard’s bounds are vacuous.'
address: 'Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via R. Cozzi 53, 20126, Milano, Italy'
author:
- 'A. E. Zalesski'
title: Low dimensional projective indecomposable modules for Chevalley groups in defining characteristic
---
Introduction
============
[characteristic ]{} Let $G$ be a finite group and $p$ a prime number. Let $F$ be a field of $p$ dividing the order $|G|$ of $G$. We always assume that $F$ contains a primitive $m$-root of unity, where $m=|G|/|G|_p$ and $|G|_p$ is the order of a $U$ of $G$. The group algebra $FG$ of $G$ over $F$ viewed as a left $FG$-module is called the regular module and indecomposable direct summands are called [*principal indecomposable $FG$-modules*]{}, customarily abbreviated as PIM’s. These are classical objects of study in the modular theory of finite groups [@BN]. One of the main open problems is to determine their dimensions or at least provide satisfactory information on the dimensions.
This paper studies this problem for finite Chevalley groups in defining $p$. For certain groups of small rank the PIM dimensions have been explicitly computed, but not much is known in general. A rather detailed survey of the current state of the problem and the methods used to attack it is done by J. Humphreys [@Hub].
The results of this paper concentrate mainly on obtaining lower bounds for the PIM dimensions. The absolute lower bound for a PIM dimension for any finite group $G$ is $|G|_p$, and this bound is attained for every Chevalley group. Indeed, every such group has an $FG$-module of dimension $|G|_p$, hence of $p$-defect 0, known as the Steinberg module. This is unique if $G$ is quasi-simple, and this is a PIM. Our use of the term ‘lower bound’ assumes that we exclude modules of $p$-defect 0.
The earliest result on lower bounds for PIM dimensions for Chevalley groups is due to Ballard [@Ba Corollary 5.4], who considered only non-twisted groups. His result is stated in the same shape in [@Hub §9.7]. We show in Section 4 how to extend Ballard’s result for arbitrary twisted groups, and we also obtain a version for it for a parabolic subgroup in place of a Borel subgroup in the original Ballard’s statement.
Ballard’s lower bound is not available for every PIM, for instance, it is useless for any PIM for any non-twisted Chevalley group over the field of order 2. Therefore, it is essential to make it clear when Ballard’s type bound is applicable, and this is desirable to be made in terms of the standard parameterization of PIM’s, specifically, in terms of their socles. Recall that every PIM (for any finite group $G$) has an socle, which determines the PIM. This establishes a bijection between PIM’s and the of $G$, which we refer to here as the standard parametrization. The PIM corresponding in this way to the trivial $FG$-module $1_G$ is called 1-PIM in [@MW].
No further result was known over almost 30 years until Malle and Weigel [@MW] determined the 1-PIM’s of dimension $|G|_p$. They did so for all simple groups $G$ and for all primes dividing $|G|$. For Chevalley groups in characteristic $p$ they suggested a method called the parabolic descent in [@MW]. This allows to bound from below the 1-PIM dimension in terms of Levi subgroups of parabolic subgroups of $G$. The method in its original shape does not work for other PIM’s. In this paper we develop the method further to a level when it can be used for arbitrary PIM’s (in characteristic $p$). This allows to extend the above mentioned result by Malle and Weigel [@MW] as follows:
\[wt1\]Let $G$ be a quasi-simple Chevalley group in defining characteristic $p$, and let $\Phi$ be a $p$-modular PIM of non-zero defect. Then $\dim \Phi>|G|_p$, unless $\Phi$ is a $1$-PIM and $G/Z(G)\cong PSL(2,p)$ or ${}^2G_2(3)\cong {\rm
Aut}\,SL(2,8)$.
The parabolic descent reduces the proof to groups of BN-pair rank at most 2, and for most of them Theorem \[wt1\] is already known to be true. However, for the groups $G=SU(4,p), {}^3D_4(p)$ and ${}^2G_2(3^k)$ the PIM dimensions are not known, although the character tables are known. These are not sufficient to rule out the above three groups, and the parabolic descent method is only partially helpful. In order to prove Theorem \[wt1\] for these groups we also make use of the following simple observation (Lemma \[b1\]): $\dim\Phi= |G|_p\cdot (\chi,1_U^G)=|G|_p\cdot
(\chi,\Gamma)$, where $\chi$ is the character of $\Phi$, $\Gamma$ is a Gelfand-Graev character and $U$ is the of $G$. (Here $1_U$ is the trivial of $U$ and $1_U^G$ is the induced representation.)
I conjecture that for classical groups $G$ of rank $n$ the dimension of a PIM other than the Steinberg one is at least $(n-1)\cdot |G|_p$. Some progress is achieved in this paper by using a new idea based on the analysis of common constituents of the ordinary character of a PIM and the induced module $1_U^G$, where $U$ is a of $G$. Let $W$ be the Weyl group of $G$ viewed as a group with $BN$-pair. In favorable circumstances, in particular, for groups $SL(n,q)$, $n>4$, $E_6(q)$, $E_7(q)$, $E_8(q)$ the PIM dimension is shown to be at least $d\cdot |G|_p$, where $d$ is the minimum dimension of a non-linear of $W$ (Theorem \[th1\]). If $G=SL(n+1,q)$, $n>3$ then the rank of $G$ is $n$, $W\cong S_{n+1}$, the symmetric group, and $d=n$. So in this case the conjecture is confirmed. Note that if $q=2$ then there is a PIM of dimension $n\cdot |G|_p$; if $q>2$ and $n>2$ then there is a PIM of dimension $(n+1)\cdot |G|_p$ [@Tsu].
[**Notation**]{}. ${\mathbb Q}$, ${\mathbb C}$ are the rational and complex number fields, respectively, and ${\mathbb Z}$ is the ring of integers. $F_q$ is the finite field of $q$ elements, and $F$ an of characteristic $p>0$.
If $G$ is a finite group, then $Z(G)$ is the center and $|G|$ is the order of $|G|$. If $p$ is a prime then $|G|_p$ is the $p$-part of $|G|$ and also the order of every of $G$. A $p'$-group means a finite group with no element of order $p$. If $g\in G$ then $|g|$ is the order of $g$. Notation for classical groups are standard.
All and modules are over $F$ or over ${\mathop{\mathbb C}\nolimits}$ (unless otherwise stated). Sometimes we deal with $RG$-modules, where $1\in R\subset
{\mathop{\mathbb C}\nolimits}$ is a subring. In this case all $RG$-modules are assumed to be free as $R$-modules. If the ground field is clear from the context, we take liberty to use the term ‘$G$-module’. All modules are assumed to be finitely generated. Representations of $G$ in characteristic $p$ are called $p$-[*modular*]{}, and those over the complex numbers are called [*ordinary*]{}. The regular of $G$ is denoted by $\rho_G^{reg}$, and the trivial one-dimensional is denoted by $1_G$. We also use $1_G$ to denote the trivial one-dimensional module and its (Brauer) character. If $M$ is an $FG$-module, then ${\mathop{\rm Soc}\nolimits}M$ means the socle of $M$, the sum of all submodules.
The set of characters of $G$ is denoted by ${\mathop{\rm Irr}\nolimits}G$, and ${\mathop{\mathbb Z}\nolimits}{\mathop{\rm Irr}\nolimits}G$ is the ${\mathop{\mathbb Z}\nolimits}$-span of ${\mathop{\rm Irr}\nolimits}G$; elements of ${\mathop{\mathbb Z}\nolimits}{\mathop{\rm Irr}\nolimits}G$ are called [*generalized characters*]{}.
A projective indecomposable $FG$-module is called a PIM and usually denoted by $\Phi$. Every PIM is determined by its socle. The PIM whose socle is $1_G$ is called here 1-PIM, and denoted by $\Phi_1$. More notation concerning projective modules is introduced in Section 3. We set $c_\Phi=\dim \Phi /|G|_p$.
If $\chi$ is a character vanishing at all non-identity $p$-elements then we write $c_\chi =\chi(1)/|G_p|$; this is an integer.
Let $M$ be an $FG$- or $RG$-module. We set $C_M(G)=\{m\in M:gm=m$ for all $g\in G \}$. Thus, $C_M(N)$ is the set of $G$-invariants (or fixed points) on $M$. Let $H$ be a subgroup of $G$ and $N$ a normal subgroup of $H$. Then $C_M(N)$ is an $H$-module, and when it is viewed as an $H/N$-module, it is denoted by $\overline{M}_{H/N}$ (or $\overline{M}$), and called the [*Harish-Chandra restriction of $M$ to*]{} $H/N$. Conversely, given an $H/N$-module $D$, one can view it as an $H$-module with trivial action of $N$. Then the induced $G$-module $D^G$ (when $D$ is viewed as an $H$-module) is denoted by $D^{\# G}$ and called [*Harish-Chandra induced from*]{} $D$. For details see [@CR2 p. 667-668, §70A], where these operations are called generalized restriction and induction. This corresponds to similar operations on characters and Brauer characters. The Harish-Chandra restriction and induction extends by linearity to the class functions on $G$. So if $\chi$ is a class function on $G$ then $\overline{\chi}_{H/N}$ is the corresponding Harish-Chandra restriction of $\chi$ to $H/N$, and if ${\lambda }$ is a class function on $H/N$ then ${\lambda }^{\# G}$ denotes the Harish-Chandra induced class function on $G$. (For the ordinary induction we use notation ${\lambda }^G$.) Let $\eta:G{ \rightarrow }{\mathbb C}$ be a class function on $G$. The formula $({\lambda }^{\#
G},\eta)=({\lambda }, \overline{\eta}_{H/N})$ is an easy consequence of the Frobenius reciprocity and called the [*Harish-Chandra reciprocity*]{}. (This is formula 70.1(iii) in [@CR2 p.668].) Note that $\overline{\eta}_{H/N}$ can be obtained as the truncation of $\eta$. This is defined by $\frac{1}{|N|}\sum_{n\in
N}\eta(hn)$ for $h\in H$ viewed as a function on $H/N$.
Let ${{\mathbf G}}$ be a connected reductive algebraic group. An algebraic group ${{\mathbf G}}{ \rightarrow }{{\mathbf G}}$ is called [*Frobenius*]{} if its fixed point subgroup is finite. We denote Frobenius endomorphisms by $Fr$. So ${{\mathbf G}}^{Fr}=\{g\in {{\mathbf G}}: Fr(g)=g \}$ is a finite group, called here a finite reductive group. If ${{\mathbf G}}$ is simple and simply connected, we refer to $G={{\mathbf G}}^{Fr}$ as a [*Chevalley group*]{}. More notation concerning algebraic groups will be introduced in Section 5. If $G$ is a finite reductive group or an algebraic group, $p$ is reserved for the defining characteristic of $G$.
The methods and main results
============================
In order to guide a reader through the paper, we comment here the machinery used in the proofs.
Parabolic descent
-----------------
A well known standard fact on PIM’s for any finite group $G$ is that the mapping $\Phi{ \rightarrow }{\mathop{\rm Soc}\nolimits}\Phi$ yields a bijection PIM$\,{}_G { \rightarrow }{\mathop{\rm Irr}\nolimits}G$ between the set of PIM’s and the set of equivalence classes of $FG$-modules. In addition, every projective module is a sum of PIM’s. Therefore, every projective module is determined by its socle, and a projective module is a PIM its socle is irreducible.
Let $G$ be a Chevalley group, so $G={{\mathbf G}}^{Fr}$, where ${{\mathbf G}}$ is a simply connected simple algebraic group. The result is the well known Smith-Dipper theorem (see [@Sm], [@Di], [@Cab], [@GLS 2.8.11]):
\[sdt\]Let $G$ be a group with a split BN-pair in characteristic $p$, $P$ a parabolic subgroup with Levi subgroup $L$. Let $V$ be an $FG$-module. Then $\overline{V}_L=C_V(O_p(P))|_L$ is an $L$-module.
In other words, for every parabolic subgroup $P$ of $G$ there is a mapping $\sigma_{G,P}:{\mathop{\rm Irr}\nolimits}G{ \rightarrow }{\mathop{\rm Irr}\nolimits}L$ defined via the Harish-Chandra restriction $V{ \rightarrow }\overline{V}_L
$. Note that $C_V(O_p(P))$ coincides with ${\mathop{\rm Soc}\nolimits}V|_P$ as $p={\rm char}\, F$. The mapping $\sigma_{G,P}$ is surjective (see Lemma \[sm9\]). This allows one to define a surjective mapping $\pi_{G,P}:\,$PIM${}_G{ \rightarrow }\,$PIM$\,_L $ as the composition of the mappings
$$\Phi{ \rightarrow }{\mathop{\rm Soc}\nolimits}\Phi{ \rightarrow }\overline{({\mathop{\rm Soc}\nolimits}\Phi)}_L{ \rightarrow }\Psi,$$ where $\Psi$ is the PIM for $L$ with socle $\overline{({\mathop{\rm Soc}\nolimits}\Phi)}_L$. This is well defined in view of Lemma \[sdt\]. We call this mapping the [*parabolic descent from $G$ to $L$*]{}. (We borrow the term from [@MW] but the meaning of the term is not the same as in [@MW].) One observes that $\pi_{G,L}(\Phi)$ is a direct summand of $\overline{ \Phi}_L=C_{\Phi}(O_p(P))|_L$ (but the equality rarely holds). This implies that $c_\Phi\geq
c_{\Psi}$, where $\Psi=\pi_{G,P}(\Phi)$, see Lemma \[mp1\]. In the special case where $\Phi=\Phi_1$ is the 1-PIM, this fact has been exploited in [@MW]. An attempt to extend it to other PIMs meets an obstacle, specifically, in order the method could work one needs at least to guarantee that $\Psi$ is not a defect zero $FL$-module. We show how to manage with this difficulty in Section 5.
A weak point of the parabolic descent is that it only allows to bound (from below) $\dim \Phi$ in terms of $\dim\Psi$, and can not be used for showing that $c_\Phi$ grows as the rank of $G$ tends to infinity. Nonetheless this is useful for classifying PIM’s of dimension $|G|_p$, which is one of our tasks below.
A formal analog of Lemma \[sdt\] for a projective module would be a claim that if $\Phi$ is a PIM for $G$ then $\overline{\Phi}_L$ is a PIM for $L$. However, this is not true. This is evident from Propositions \[r2\] and \[r55\].
Ballard’s bound revised {#sub1}
-----------------------
Let $B,N$ be the subgroups of $G$ defining the $BN$-pair structure on $G$ (see [@CR2 §69.1]). The group $T:=N\cap B$ is normal in $N$. Set $W=N/T$. Then $B$ is a Borel subgroup of $G$, $U=O_p(B)$ is a of $G$ and $T=N\cap B$ is a maximal torus of $B$. Let $\Phi$ be a PIM for $G$ with socle $M$. Ballard [@Ba Corollary 5.4] proves that $\dim \Phi\geq |W\beta|\cdot |G|_p$, equivalently, $c_\Phi\geq |W\beta|$ in our notation, where $ |W\beta|$ is the size of the $W$-orbit of the (Brauer) character $\beta$ of $T$ afforded by the restriction of ${\mathop{\rm Soc}\nolimits}(M|_B)$ to $T$. He assumes $G$ to be non-twisted. In Section 4 Ballard’s result is generalized to all twisted groups as follows. The conjugacy action of $N$ on $T$ induces an action on ${\mathop{\rm Irr}\nolimits}T $, and for $\beta\in{\mathop{\rm Irr}\nolimits}T$ let $|N\beta|$ denote the size of the $N$-orbit of $\beta$. Let $\Phi$ be a PIM for $G$ with socle $V$ and let $\beta $ be the Brauer character of the $FG$-module $C_V(U)|_T$; it is well known that $\beta\in{\mathop{\rm Irr}\nolimits}T$. Then our version of Ballard’s result states that $c_\Phi\geq |N\beta|$ (Proposition \[r2\]).
The bound $|N\beta|$ is vacuous if $ |N\beta|=1$. For instance, there are PIM’s for which $\beta=1_{T}$, so the bound is vacuous for such PIM’s. In addition, in many cases $ |N\beta|$ is small, and the bound is not sharp. (This happens for instance if $|N/T|=2$ but $G$ is not $SL(2,q)$.)
We find out that the nature of Ballard’s result is not specific for PIM’s. We prove a similar result for arbitrary characters vanishing at all non-trivial $p$-elements, see Proposition \[r2b\]. This implies the result for PIM’s as their characters have this property.
The parabolic descent can be combined with our interpretation of the Ballard bound as follows. Let $P$ be a standard parabolic subgroup of $G$ [@CR2 65.15], and let $L$ be the standard Levi subgroup of $P$ [@CR2 69.14]. Let $\Psi$ be the parabolic descent of $\Phi$. Let $N_L$ be the normalizer of $L$ in $N$, so $N_L$ acts on $L$ via conjugation. For $n\in N_L$ let $\Psi^n$ denote the twist of $\Psi$ by $n$. (For any $FL$-module $X$ one defines $X^n$ to be $X$ with the twisted action of $L$, that is, $l^n(x):=nln{^{-1}}\cdot
x$, where $l\in L$, $x\in X$, $n\in N_L $.) Denote by $|N\Psi|$ the size of the $N$-orbit of $\Psi$. Then our generalization of Ballard’s theorem asserts that $c_\phi\geq
|N\Psi|\cdot c_\Psi$ contains every PIM $ \Psi^n$ $(n\in N_L)$ (Proposition \[r55\]).
Harish-Chandra induction
------------------------
Let $G\in \{SL(n,q),~ n>4,~
E_6(q),~E_7(q),~E_8(q)\}$, and let $r$ be the rank of $G$. Let $\chi$ be the character of a PIM $\Phi\neq St$. We show (Section 6) that the Harish-Chandra theory together with the main result of Malle-Weigel [@MW] yields a lower bound $c_{\Phi}\geq r$. Here is the idea of the proof. In notation of Section \[sub1\], by analysis of the action of $N$ on $T$ (Proposition \[r4\]) and using Ballard’s bound, we deduce that $c_{\Phi}\geq
|N\beta|>r$, whenever $ \beta\neq 1_{T}$. If $1_{T}= \beta$ then Ballard’s bound is vacuous. In this case we first show that $c_\Phi=(\chi,1_B^G)$, see Proposition \[r01a\]. Let ${\lambda }\in{\mathop{\rm Irr}\nolimits}G $ be a common constituent of $\chi$ and $1_B^G$. The Harish-Chandra theory tells us that $({\lambda }, 1_B^G)\geq r$ for the above groups, unless ${\lambda }\in \{St, 1_G\}$. As $St$ is not a constituent of $\chi$, $c_\Phi< r$ implies ${\lambda }=1_G$. So $c_\Phi=(\chi,1_B^G)=(\chi,1_G)$. By general modular representation theory, $(\chi,1_G)\neq 0$ implies $\Phi=\Phi_1$ and $ (\chi,1_G)=1$, and hence $c_\Phi=1$. The groups $G$ for which $c_{\Phi}=1$ have been determined in [@MW].
We expect that this reasoning can be improved to obtain a lower bound for all classical groups, however, this requires much deeper analysis.
Preliminaries
=============
Let $G$ be a finite group of order $|G|$ and $p$ a prime number. Let ${\varepsilon}$ be a primitive $|G|$-root of unity. Any ordinary representation is equivalent to a representation $\phi$ over ${\mathbb Q}({\varepsilon})$, and moreover, over a maximal subring $R $ of ${\mathbb Q}({\varepsilon})$ not containing $1/p$. In addition, $R$ has a unique maximal ideal $I$ such that $F=R /I$ is a finite field of characteristic $p$. Note that $F$ contains a primitive $m$-root of unity, where $m=|G|/|G|_p$. For uniformity one can similarly define $R$ in the algebraic closure of ${\mathbb
Q}$, and then fix this $R $ to deal with all finite groups. The mapping $R { \rightarrow }F$ yields also a surjective of the group of roots of unity in $R $ to the group of roots of unity in $F$, used to define Brauer characters.
Every ordinary representation is equivalent to a representation over $R$. So if $\phi (G)\subset GL(n,R)$ for some $n$, then the natural projection $GL(n,R){ \rightarrow }GL(n,F)$ yields a $p$-modular representation $\overline{\phi}:G{ \rightarrow }GL(n,F)$ called the [*reduction of $\phi$ modulo*]{} $p$. It is well known that the composition factors of $\overline{\phi}$ remain under any field extension of $F$. This can be translated to the language of $RG$- and $FG$-modules, however, it requires to consider only $R
G$-modules that are free as $R$-modules. So [*all $RG$-modules below are assumed to be free as $R$-modules*]{}.
If $G$ is a $p'$-group then, by Dickson’s theorem, the reduction yields a bijection between the isomorphism classes of $ RG$- and $FG$-modules which makes identical the $p$-modular theory with the ordinary theory of $G$. If $p$ divides $|G|$, this is not true anymore, however, there is a rather sophisticated replacement: the reduction modulo $p$ yields a bijection between the isomorphism classes of projective $RG$- and projective $FG$-modules (Swan’ theorem, see [@CR Theorem 77.2]).
Let $M\neq 0$ be a projective $FG$-module. Then the corresponding projective $R G$-module is called [*the lifting of*]{} $M$, which we often denote by $\tilde M$. Obviously, $\dim M$ is equal to the $R $-rank of $\tilde M$. The latter is equal to the dimension of the $KG$-module obtained from $ M_{R }$ by the extension of the coefficient ring to the quotient field $K$ of $R $, so we also write $\dim M$ for the rank of an $R $-module $M$.
For sake of convenience we record the easy observation:
\[ax2\]$(1)$ Let $M=M_1\oplus M_2$ be a direct sum of $FG$- or $R G$-modules. Then $ C_M(G)=
C_{M_1}(G)\oplus C_{M_2}(G) $.
$(2)$ If $H$ is a subgroup of $G$, $M$ a projective $FG$-module with lifting $L$ then $\dim C_M(H)=\dim
C_L(H)$.
Proof. (1) is trivial. As the restriction of a projective $G$-module to $H$ remains projective, it suffices to prove (2) for $H=G$. Then (2) is obvious if $M$ is the regular $FG$-module. By (1), this is implies (2) when $M$ is free, and hence when $M$ is projective. $\Box$
The following is well known (see for instance [@Fein p. 52]):
\[dp8\]Let $G=H\times N$, the direct product of finite groups $H$ and $N$, and let $\Phi,\Psi $ be PIM’s for $H,N$ respectively. Then $\Phi\otimes \Psi$ is a PIM for $G$, and hence $c_{\Phi\otimes
\Psi}=c_\Phi\cdot c_\Psi$.
The lemma asserts that, for a projective $G$-module $M$ and a normal subgroup $N$ of $G$, the $G/N$-module $C_M(N)$ is projective. This is a rather general fact, but it does not seem to be recorded in any standard textbook. The proof below is a variation of that given in [@MW the proof of Proposition 2.2].
\[a2\]Let $H$ be a subgroup of $G$ and $N$ a normal subgroup of $H$. Let $M$ be a projective $FG$-module.
$(1)$ $\overline{M}_{H/N}$ is a projective $F(H/N)$-module.
$(2)$ Let $L$ be the lifting of $ M$. Then $\overline{L}_{H/N}$ is the lifting of $ \overline{M}_{H/N}$.
$(3)$ Let $\chi$ be the character of $L$. Then the truncation $\overline{\chi}_{H/N}$ is the character of $\overline{L}_{H/N}$.
Proof. (1) As $M|_H$ is a projective module, it suffices to prove the statement for $H=G$.
Assuming $H=G$, suppose first that $M$ is the regular $FH$-module. Then $M|_N$ is a free $FN$-module of rank $|H:N|$, and hence $\dim\overline{FH}=|H:N|$. Let $a:=\sum_{n\in N}n\in FN$. Then the mapping $h:x{ \rightarrow }xa$ $(x\in FG )$ is an $FH$-module whose kernel $A$ is spanned by the elements $g(n-1)$ $(n\in N,g\in H)$. Therefore, $h(FH)=FH/A\cong F(H/N)$. As $xa\in \overline{FH}$ and $\dim F(H/N)=|H:N|= \dim\overline{FH}$, it follows that $\overline{FH}$ is isomorphic to $F(H/N)$, the regular $F(H/N)$-module.
Therefore, the lemma is true if $M$ is free (and in this case $\overline{ M}$ is free). Otherwise, let $M'$ be a projective $FG$-module such that $M\oplus
M'=J$, where $J$ is free. Then $\overline{ J}$ is a free $F(G/N)$-module, and $\overline{ M}\oplus \overline{
M'}=\overline{ J}$. As $\overline{ J}$ is a free $F(G/N)$-module, $\overline{ M}$ is projective.
\(2) Let $\pi: L{ \rightarrow }M$ be the reduction map. Then $\pi(\overline{
L})\subseteq \overline{ M}$. By Lemma \[ax2\], $\dim\overline{
L}=\dim\overline{ M}$, as desired.
\(3) follows from the definition of $\overline{\chi}_{H/N}$. $\Box$
For a $p$-group $P$, every projective $FP$-module $M$ is free, that is, a direct sum of copies of the regular $FP$-module, which is the only PIM for $P$ (see [@CR §54, Exercise 1] or [@Fe Ch.III, Corollaries 2.9, 2.10]). This is therefore true for its lifting as well. both $M$ and its lifting have the same rank as $H$-modules, equal exactly to $\dim M/|P|$. Obviously, $\dim C_M(P)=1$ for the regular $FP$-module $M$. This implies the following assertion:
\[y2\]Let $P$ be a $p$-subgroup of $G$, $M$ a projective $FG$-module and $L=M_{K }$. Then $ \dim C_M(P)=(\dim M)/|P|=(\dim L)/|P|=\dim C_L(P)$.
Let $U\in {\rm Syl}_p(G)$ and $M$ a projective $FG$-module. Then $c_M:=(\dim M)/|U|$ is an integer. Some formulas become simpler if one uses $c_M$ instead of $\dim M$. The first equality of Lemma \[y2\] implies:
\[x2\]Let $H$ be a subgroup of $G$ with $(|G:H|,p)=1$. Then $c_M=c_{M|_H}$ for any projective $FG$-module $M$.
Let $M$ be a projective $FG$-module with lifting $L$. For brevity, the character $\chi$ of $L$ is also called the character of $M$. $\chi$ vanishes at the $p$-singular elements (indeed, if $g=su$, where $u\neq 1$ is a $p$-element, $s$ is a $p'$-element and $[s,u]=1$, then $L$ is a direct sum of the eigenspaces of $s$; by the Krull-Schmidt theorem, every $s$-eigenspace is a free $
R{ \langle }u{ \rangle }$-module (as so is $L$), and hence the trace of $su$ is $0$). See [@Fe Ch.IV, Corollary 2.5].
\[t1\]Let $G$ be a finite group and $U\in {Syl}_p(G)$. Let $M$ be a projective $FG$-module with lifting $L$, and $\chi$ the character of $L$. Then $c_M=(\chi|_U,
1_U)$. Moreover, $C_M(U)$ and $C_L(U)$ (the fixed point subspaces of U on $L,M$, resp.) are $N_G(U)$-modules with the same Brauer character.
Proof. As $N_G(U)/U$ is a $p'$-group, $N_G(U)$ splits as $UH$, where $H\cong N_G(U)/U$. Therefore, $M|_H$ and $L|_H$ have the same Brauer characters. Let $\rho:L{ \rightarrow }M$ be the reduction Obviously, $\rho(C_L(U))\subset C_M(U)$. As $\dim C_L(U)=c_M=\dim
C_M(U)$, we have $\rho(C_L(U))=C_M(U)$. $C_M(U)|_H$ and $C_L(U)|_H$ have the same Brauer characters, and the lemma follows. $\Box$
\[om1\]Let $N$ be a normal subgroup of $G$ with $(|G:N|,p)=1$. Let $M$ be an $FG$-module.
$(1)$ ${\mathop{\rm Soc}\nolimits}(M|_N)=({\mathop{\rm Soc}\nolimits}M)|_N $.
$(2)$ Let $M$ be a PIM for $G$ and let $S={\mathop{\rm Soc}\nolimits}M$. Suppose that $S|_N$ is irreducible. Then $M|_N$ is a PIM for $N$.
Proof. (1) Let $X$ be an submodule of $M|_N$. Then so is $gX$ for every $g\in G$, and hence $Y=\sum_{g\in G}gX$ is an $FG$-module, obviously, completely reducible. Therefore, $Y\subseteq {\mathop{\rm Soc}\nolimits}M$. As ${\mathop{\rm Soc}\nolimits}(M|_N)$ is the sum of submodules of $M|_N$, by the above we have ${\mathop{\rm Soc}\nolimits}(M|_N)\subseteq ({\mathop{\rm Soc}\nolimits}M)|_N
$. The converse inclusion follows from Clifford’s theorem.
\(2) By (1), ${\mathop{\rm Soc}\nolimits}(M|_H)$ is irreducible, so $M|_H$ is a PIM (as it is projective). $\Box$
The following lemma is a special case of [@Fe Ch.IV, Lemma 4.26]. Our proof below is different, and somehow, more natural.
\[a4\] Let $N$ be a normal $p$-subgroup of $G$ and let $\Phi$ be a PIM for $G$. Then $\overline{\Phi}:=C_{\Phi}(N)={\mathop{\rm Soc}\nolimits}(\Phi|_N) $ is a PIM for $G/N$ and $c_\Phi =c_{\overline{\Phi}}$.
Proof. By Lemma \[a2\], $\overline{ \Phi}$ is a projective $F(G/N)$-module. As $N$ acts trivially on every $FG$-module, ${\mathop{\rm Soc}\nolimits}\Phi\subseteq \overline{ \Phi}$. In fact, ${\mathop{\rm Soc}\nolimits}\Phi={\mathop{\rm Soc}\nolimits}\overline{ \Phi}$ since $G$ acts in $\overline{\Phi}$ via $G/N$. Recall that a projective module is a PIM its socle is irreducible. Therefore, ${\mathop{\rm Soc}\nolimits}\Phi$ is irreducible as an $FG$-module, and hence as an $F(G/N)$-module. So $\overline{ \Phi}$ is a PIM for $G/N$, as claimed.
Let $U\in {\rm Syl}_p(G)$ and $\overline{ U}=U/N$. Obviously, $C_\Phi(U)=C_{\overline{\Phi} }(\overline{ U})$, so $c_M=\dim C_\Phi(U)=\dim C_{\overline{\Phi}
}(\overline{U})=c_{\overline{\Phi}}$, where $\overline{ \Phi}$ is viewed as $F(G/U)$-module. $\Box$
\[a1\] Let $H$ be a subgroup of $G$ and $M$ a projective $FG$-module. Then $M|_H$ is a projective module, and $c_{M}= \frac{|H|_p}{|G|_p}\cdot c_{M|_H}$.
Proof. The first claim is well known. The second one follows by dividing by $|G|_p$ the left and the right hand sides of the equality $|G|_p\cdot c_{M} =\dim M|_H= c_{M|_H}\cdot |H|_p$. $\Box$
\[mp1\]Let $H\subset G$ be finite groups such that $H$ contains a of $G$, and $N=O_p(H)$. Let $M$ be a projective $FG$-module with socle $S$. Let $D={\mathop{\rm Soc}\nolimits}\overline{S}_{H/N}$ and let $M'$ be a projective $F(H/N)$-module with socle $D$. Then $\overline{M}_{H/N}$ contains a submodule isomorphic to $M'$. In addition, $c_M\geq c_{M'}$.
Proof. As $S\subset M$ and $N$ is a $p$-group, we have ${\mathop{\rm Soc}\nolimits}(S|_H)\subseteq {\mathop{\rm Soc}\nolimits}(M|_H)\subseteq C_M(N)$. Viewing each module as an $F(H/N)$-module, we have $D={\mathop{\rm Soc}\nolimits}(\overline{S}_{H/N})\subseteq {\mathop{\rm Soc}\nolimits}(\overline{M}_{H/N})\subseteq
\overline{M}_{H/N} $, and $\overline{M}_{H/N}$ is projective by Lemma \[a2\]. Therefore, $M'\subseteq \overline{M}_{H/N}$. The additional claim follows from Lemma \[a1\], as $|H|_p=|G|_p$. $\Box$
\[z1\]Let $H\subset G$ be finite groups. Suppose that $(|G:H|,p)=1$ and $\Phi $ is a PIM of dimension $|G|_p$, that is, $c_\Phi=1$. Then $\Psi:=\Phi|_H$ is a PIM for $H$, and $c_\Psi=1$. In addition, ${\mathop{\rm Soc}\nolimits}(\Phi|_H)$ is irreducible.
Proof. This follows straightforwardly from Lemma \[a1\]. $\Box$
\[b1\]Let $G$ be a finite group and $U$ a of $G$. Let $\eta:G{ \rightarrow }{\mathop{\mathbb C}\nolimits}$ be a conjugacy class function such that $\eta(g)=0$ for every $1\neq g\in U$. Let $\tau$ be an character of $U$. Then $(\eta, \tau^G)=\eta(1)\tau(1)/|U|.$ In particular, if $\eta$ is a character of a PIM $\Phi$ then $(\eta, \tau^G)=\tau(1)\cdot
c_{\Phi}$.
Proof. By Frobenius reciprocity we have: $(\eta, \tau^G)=(\eta|_U,
\tau)=\eta(1)\tau(1)/ |U|$.
\[gg1\]Let $G$ be a Chevalley group in defining characteristic $p$, $U$ a of $G$ and let $\eta$ be the character of a $p$-modular PIM $\Phi$. Then $(\eta,
1_U^G)=c_{\Phi} =(\eta, \Gamma)$, where $\Gamma$ denotes a Gelfand-Graev character of $G$.
Note that every Gelfand-Graev character of $G$ is induced from a certain one-dimensional character of $U$, see [@DML] or [@DM].
\[de4\]Let ${\mathbf H } $ be a reductive algebraic group and $Fr$ a Frobenius endomorphism of ${\mathbf H } $. Let ${\mathbf G } $ be the semisimple part of ${\mathbf H } $ and $H={\mathbf H }^{Fr} $, $G={\mathbf G }^{Fr} $. Then:
$(1)$ Every $FH$-module $M$ remains under restriction to $G$. Consequently, if $\Phi$ is a PIM for $H$ then $\Phi|_G$ is a PIM for $G$.
$(2)$ Let $\Psi$ be a PIM for $G$ with character $\eta$ and let ${\lambda }\in{\mathop{\rm Irr}\nolimits}G $. For $h\in H$ denote by ${\lambda }^h$ the $h$-twist of ${\lambda }$. Then $({\lambda }, \eta)=({\lambda }^h, \eta)$. In other words, the rows of the decomposition matrix of $G$ corresponding to $H$-twisted ordinary characters coincide.
Proof. (1) It is known that $H$ is a group with BN-pair [@MT 24.10]. Let $B$ be a Borel subgroup of $H$ and $U=O_p(H)$. By [@C1 Theorem 4.3(c)], $\dim
C_{M}(U)=1$. As $H/G$ is a $p'$-group, $U\subset G$. By Clifford’s theorem, $M|_G$ is completely reducible, and if $M|_G$ is reducible then $\dim C_{M}(U)>1$, which is false.
The additional statement in (1) follows from Lemma \[om1\].
\(2) It follows from (1) that $\Psi=\Psi^h$. So $({\lambda },
\eta)=({\lambda }^h, \eta^h)= ({\lambda }^h, \eta)$. $\Box$
\[m75\]Let $H\subset G$ be finite groups such that $(|G:H|,p)=1$ and $N=O_p(H)$. Let $M$ be a projective $FG$-module with socle $S$ and character $\chi$. Let $D \subseteq{\mathop{\rm Soc}\nolimits}(S|_H)$ be an $F(H/N)$-module and let $\eta$ be the Brauer character of $D$. Let ${\lambda }\in{\mathop{\rm Irr}\nolimits}H/N$. Suppose that $\eta$ is a constituent of ${\lambda }\,({\rm mod}\,p)$ with $d$. Then $(\chi, {\lambda }^{\#
G})\geq d$.
Proof. Let $R$ be the projective $F(H/N)$-module with socle $D$ and character $\rho$. In fact, $R$ is a PIM as $D$ is Then $(\rho,\eta)=1$ by the Brauer reciprocity [@Fe Lemma 3.3]. Furthermore, $(\overline{\chi},\eta)\geq 1$ as $R\subseteq
\overline{M}$. As ${\lambda }\,({\rm mod}\, p)$ contains $\eta$ with $d$, it follows that $(\overline{\chi},{\lambda })\geq d$. By the Harish-Chandra reciprocity [@CR2 70.1(iii)], we have $(\chi,
{\lambda }^{\# G}) = (\overline{\chi}_{H/N}, {\lambda })\geq d$. $\Box$
\[xx1\]Let $G$ be a Chevalley group, and $P$ be a parabolic subgroup. Let $\Phi$ be a PIM with character $ \chi$ and socle $S$. Suppose that ${\mathop{\rm Soc}\nolimits}(S|_P)$ lifts, and let ${\lambda }$ be the character of this lift. Then $(\chi,{\lambda }^G)>0$.
Proof. This is a special case of Proposition \[m75\] with $M=\Phi$ and $H=P$. $\Box$
Lower bounds for PIM dimensions
===============================
Let $G$ be a quasi-simple Chevalley group, and let $B,N $ be subgroups defining a $BN$-pair structure of $G$. Here $B$ is a Borel subgroup of $G$, $U=O_p(B)$ and let $T_0$ be a maximal torus of $B$. Then $W_0=N/T_0$ is the Weyl group of $G$ as a group with $BN$-pair. If $G$ is non-twisted then $W_0$ coincides with $W$, the Weyl group of ${{\mathbf G}}$.
Every character $\beta$ of $T_0$ inflated to $B$ yields a character of $B$, trivial on $U$, which we denote by $\beta_B$. Obviously, $\beta{ \rightarrow }\beta_B$ is a bijection between ${\mathop{\rm Irr}\nolimits}T_0$ and the 1-dimensional characters of $B$ trivial on $U$. Therefore, the induced $\beta_B^{G}$ coincides with $\beta^{\# G}$.
Recall that a PIM $\Phi$ has an socle $S$; so the socle of $\Phi|_B$ contains the socle of $S|_B$.
\[m12\]Let $G$ be a Chevalley group viewed as a group with BN-pair (so $W_0:=W(T_0)$ is the Weyl group of the BN-pair). Let $\chi$ be a character of $G$ vanishing at all unipotent elements $g\neq 1$. Then $c_\chi=(\chi, 1_U^G)= \sum _{\beta }|W_0\beta|\cdot
(\chi,\beta_B^G)=\sum _{\beta }|W_0\beta|\cdot
(\overline{\chi}_{T_0},\beta)$, where $\beta$ runs over representatives of the $W_0$-orbits in ${\mathop{\rm Irr}\nolimits}T_0$. In particular, if $\chi$ is the character of a PIM $\Phi$ then $c_\Phi= \sum
_{\beta }|W_0\beta|\cdot (\overline{\chi}_{T_0},\beta).$
Proof. By Lemma \[b1\], \[gg1\], $|U|\cdot (\chi, 1_U^G)=\chi(1)$. Note that $1_U^G=\oplus _{\beta_0\in {\mathop{\rm Irr}\nolimits}T_0 }\beta_B^G$. Let $\beta'\in {\mathop{\rm Irr}\nolimits}T$. By [@St Theorem 47], $\beta_B^G$ and $\beta_B^{\prime G}$ are equivalent $\beta$ and $ \beta'$ are in the same $W_0$-orbit. $1_U^G=\oplus _{\beta}|W_0:C_{W_0}(\beta)|\beta_B^G$, where $\beta$ runs over representatives of the $W_0$-orbits in ${\mathop{\rm Irr}\nolimits}T_0$. This implies the first equality, while the second one follows from the Harish-Chandra reciprocity formula $(\chi,\beta_B^G)=(\overline{\chi}_{T_0},\beta)$. If $\chi$ is the character of $\Phi$ then $c_\Phi=\chi(1)/|U|$. $\Box$
\[2ya\]Let $S$ be a finite $p$-group with normal subgroup $K$. Let $\chi$ be a character vanishing at all non-identity $p$-elements of $S$. Then $\chi=c_{\chi}\cdot \rho^{reg}_S$ and $c_\chi=c_{\overline{\chi}_{S/K}}$ (where $\rho^{reg}_S$ denotes the regular character of $S$).
Proof. Let $\tau$ be an character of $S$. Then $(\chi,\tau)=\tau(1)\chi(1)/|S|=c_\chi\cdot \tau(1)$, whereas $(\rho^{reg}_S,\tau)=\tau(1)$. So the former claim follows. Let $M$ be the ${\mathop{\mathbb C}\nolimits}S$-module with character $\chi$. $M$ is a free ${\mathop{\mathbb C}\nolimits}S$-module of rank $c_\chi$. The latter claim is obvious for the regular ${\mathop{\mathbb C}\nolimits}S$-module in place of $M$, which implies the lemma. $\Box$
\[x2a\]Let $H$ be a finite group and $U=O_p(H)$. Let $\chi$ be a character of $G$ vanishing at all non-identity $p$-elements of $G$. Then $\overline{\chi}_{H/U}$ vanishes at all non-identity $p$-elements of $H/U$ and $c_\chi=c_{\overline{\chi}_{H/U}}$.
Proof. Let $M$ be a ${\mathop{\mathbb C}\nolimits}H$-module with character $\chi$, and let $M'$ be the fixed point subspace of $U$ on M. Note that $x\in M'$ $x=\frac{1}{|U|}\sum_{u\in U} um$ for some $m\in M$. Let $g\in
H$, $u\in U$. Suppose that the projection of $g$, and hence of $gu$, into $H/U$ is not a $p'$-element (the projections of $gu$ and $g$ in $H/U$ coincide). $\overline{\chi}_{H/U}(g)=\frac{1}{|U|}\sum_{u\in U}\chi(gu)=0$ by assumption, whence the first claim. The equality $c_\chi=c_{\overline{\chi}_{H/U}}$ follows from Lemma \[2ya\].$\Box$
\[r2b\]Let $G,B=UT_0,W_0$ be as in Proposition $\ref{m12}$, and let $\chi$ be a character of $G$ vanishing at all $p$-elements $1\neq
u\in G$. Let $\beta\in{\mathop{\rm Irr}\nolimits}T_0$ and $\beta_B$ the inflation of $\beta$ to $B$. Suppose that $\beta$ is an constituent of $ \overline{\chi}_{B/U}$. Then $c_{ \chi} \geq |W_0\beta|$. In addition, if $\chi(1)=|G|_p$ then $ \overline{\chi}_{B/U}$ is and $W_0$-invariant.
Proof. By [@St Theorem 47], $\beta_B^G$ is equivalent to $w(\beta)_B^G$ for every $w\in W_0$. By the Frobenius reciprocity, $(\chi|_B,\beta_B)=(\chi,\beta_B^G)=(\chi,w(\beta)_B^G)=
(\chi|_B,w(\beta)_B)$. Therefore, both $\beta_B$ and $
w(\beta)_B$ occur in $\chi|_B$ with equal multiplicity. As $w(\beta)_B$ is trivial on $U=O_p(B)$, it follows that $w(\beta)$ is a constituent of $ \overline{\chi}:=\overline{\chi}_{B/U}$. So $\overline{\chi}(1)\geq |W_0\beta|$. As $B/U$ is a $p'$-group, $c_{ \overline{\chi}}= \overline{\chi}(1)$. We know that $c_\chi\geq c_{ \overline{\chi}}$ (Lemma \[2ya\]). So the result follows. This also implies the additional statement, as $1=c_\chi\geq c_{ \overline{\chi}}\geq |W_0\beta|$ means that $\beta$ is $W_0$-stable. $\Box$
\[r2\]Let $G,B=UT_0,W_0$ be as in Proposition $\ref{m12}$. Let $\Phi$ be a PIM with socle $S$ and character $\chi$, and let $\beta_B$ be the Brauer character of ${\mathop{\rm Soc}\nolimits}S|_B$. Then $c_{ \Phi} \geq
|W_0\beta|$.
Proof. Let $\chi$ be the character of $\Phi$, that is, the character of the lifting $M$ of $\Phi$. By Lemma \[t1\], $c_\Phi=(\chi|_U, 1_U)=c_\chi$, and the character of $C_M(U)|_B$ coincides with the Brauer character of $C_\Phi(U)|_B$. Therefore, $\beta_B$ occurs as a constituent of $C_M(U)|_B$ so $(\chi|_B,\beta_B)\geq 1$, and hence $\beta$ is a constituent of $\overline{\chi}_{B/U}$. So the result follows from Proposition \[r2b\]. $\Box$
Remarks. (1) Let ${{\mathbf G}}$ be the algebraic group defining $G$ as $G={{\mathbf G}}^{Fr}$. Then in Proposition \[r2\] $S=V_\mu|_G$ for some ${{\mathbf G}}$-module $V_\mu$, where $\mu$ is the highest weight of $V_\mu$. Moreover, $\beta=\mu|_{T_0}$. If $G$ is a non-twisted Chevalley group then $W(T_0)=W(G)$. Therefore, for non-twisted groups the result coincides with that of Ballard [@Ba Corollary 5.4], see also [@Hub 9.7]. (2) Recall that $\beta_B$ is irreducible, whereas $\overline{\chi}_{B/U}$ may be reducible.
This can be generalized to a parabolic subgroup $P$ in place of a Borel subgroup $B$, and a Levi subgroup $L$ of $P$ in place of $T_0$. However, the statement has to be modified. For this we need to replace $W_0$ by a certain group $\overline{W}_L$, which is contained in $N_G(L)/L$. Specifically, we may assume that $ B\subseteq P$, and that $T_0\subseteq L$. (The equality holds only if $P=B$.) Using the data $B,N,W_0,$ defining the the $BN$-pair structure of $G$, we set $N_L=\{n\in N: nLn{^{-1}}=L\}$. Then $\overline{W}_L=N_L/(N_L\cap L)$. (Note that $\overline{W}_L$ is not the Weyl group of $L$ viewed as a group with $BN$-pair; the latter is $(N_L\cap L)/T^0$.) For a character ${\lambda }\in {\mathop{\rm Irr}\nolimits}L$ and $n\in N_L$ one considers the $n$-conjugate ${\lambda }^n$ of ${\lambda }$. Of course, ${\lambda }^n={\lambda }$ if $n\in L$. Therefore, ${\lambda }^n$ depends only on the coset $w:=n\cdot (N\cap L)$, which is an element of $\overline{W}_L$. So one usually writes ${\lambda }^w$ for $w\in
\overline{W}_L$, with the meaning that ${\lambda }^w={\lambda }^n$ for $n$ from the pullback of $w$ in $N_L$. If $L=B$ then $\overline{W}_L$ is exactly $W_0$.
Recall (see Notation) that $\overline{\chi}_L$ denotes the Harish-Chandra restriction (or the truncation) of $\chi$, and $\overline{\chi}_L$ coincides with $\chi'|_L$, where $\chi'$ is a character of $P$ trivial on $O_p(P)$ and such that $\chi|_P=\chi'+\mu$ for some character $\mu$ whose all constituents are non-trivial on $O_p(P)$.
\[bm1\]Let $P$ be a parabolic subgroup of $G$ and $L$ a Levi subgroup of $P$. Let $\chi$ be a character of $G$. Then $\overline{\chi}_L$ is $\overline{W}_L$-invariant. In particular, if $P=B$ then $\overline{\chi}_{T_0}$ is $W_0$-invariant.
Proof. Let ${\lambda }'$ be an of $ \chi'$, and ${\lambda }={\lambda }'|_L$. By the Frobenius reciprocity, $(\chi, {\lambda }^
{\prime ^G})=(\chi|_P, {\lambda }')=(\chi',{\lambda }')$ as ${\lambda }'$ is trivial on $O_p(P)$. Then $(\chi',{\lambda }')$ equals $(\chi'|_L,
{\lambda })=(\overline{\chi}_L, {\lambda })$. Furthermore, ${\lambda }^{\prime ^G}=w({\lambda }')^{\# G}$ for every $w\in \overline{W}_L$, see [@CR2 70.11]. Hence $(\chi',{\lambda })=(\chi',w({\lambda }))$, and the result follows. $\Box$
The group $N_L$ acts on $L$ by conjugation, and hence $\overline{W}_L$ acts on ${\mathop{\rm Irr}\nolimits}L$. Note that for any finite group $G$ the correspondence $\Phi{ \rightarrow }{\mathop{\rm Soc}\nolimits}\Phi$ is compatible with the group action. In other words, if $h$ is an of $G$ and $\Phi^h$ is the $h$-twist of $\Phi$, then ${\mathop{\rm Soc}\nolimits}\Phi^h=({\mathop{\rm Soc}\nolimits}\Phi)^h$.
\[r55\]Let $G$ be a Chevalley group, and let $P$ be a parabolic subgroup of $G$ with Levi subgroup $L$. Let $\Phi$ be a PIM with socle $S$. Let $S_1={\mathop{\rm Soc}\nolimits}(S|_P)$ and $S_L=S_1|_L$. Let $\Psi$ be the projective $FL$-module with socle $S_L$. Then $c_\Phi\geq
|\overline{W}_L:C_{\overline{W}_L}(S_L)|\cdot c_\Psi$.
Proof. Let $M$ be the lifting of $\Phi. $ Then $\overline{M}_L$ and $\overline{\Phi}_L$ are projective $L$-modules with the same character $\overline{\chi}_L$, see Lemma \[mp1\]. (By convention we call $\overline{\chi}_L$ the character of $\overline{\Phi}_L$.) Note that $O_p(L)=1$, so $\overline{\chi}_L$ coincides with the character in Lemma \[bm1\], which tells us that $\overline{\chi}_L$ is $\overline{W}_L$-invariant. As a projective $FL$- and $ RL$-module is determined by its character, it follows that $\overline{M}_L^w=\overline{M}_L$ and $\overline{\Phi}_L^w=\overline{\Phi}_L$ for every $w\in \overline{W}_L$.
By Lemma \[sdt\], $S_L$ is As $O_p(P)$ acts trivially on $S_L$, it follows that $S_L\subseteq {\mathop{\rm Soc}\nolimits}\overline{\Phi}_L $. Then $\Psi\subseteq \overline{\Phi}_L$. By the comment prior the proposition, the $\overline{W}_L$-orbits of $S_L$ and $\Psi$ are of the same size $l:=|\overline{W}_L:C_{\overline{W}_L}(S_L)|$. As $\overline{\Phi}_L$ is $\overline{W}_L$-invariant, every $\Psi^w$ ($w\in \overline{W}_L$) is in $\overline{\Phi}_L$. Therefore, $\overline{\Phi}_L$ contains at least $l$ distinct PIM’s $\overline{\Psi}^w$. Obviously, $\dim \Psi^w=\dim\Psi$ for $w\in
\overline{W}_L$, so $\dim \overline{\Phi}_L\geq l \cdot \dim\Psi$, and hence $c_{\overline{\Phi}_L}\geq l\cdot c_{\Psi}$. By Lemma \[mp1\], $c_\Phi\geq c_{\overline{\Phi}_L}\geq l\cdot c_{\Psi}$, as required. $\Box$
In the remaining part of this section we discuss the question when the lower bounds provided in Proposition \[r2\] and Proposition \[r55\] are efficient. It is known from Ballard’s paper [@Ba] that the bound is sharp for some PIM’s, and more examples are provided in [@Hub §10.7]. However, in general the bound is not sharp, and, especially for twisted groups, there are some characters $1_{T_0}\neq \beta\in {\mathop{\rm Irr}\nolimits}T_0$ for which the bound is too small for efficient use. The situation is better for some non-twisted Chevalley groups; this will be explained in the rest of this section.
If $C_{W_0}(\beta)=W_0,$ then Proposition \[r2\] gives $c_\Phi\geq 1$, which is trivial. This always happens if $G$ is a non-twisted Chevalley group $G(q)$ with $q=2$, or, in general, if $\beta=1_{T_0}$. In fact, there are more cases where $C_{W_0}(\beta)=W_0.$ In addition, one needs to decide what is the minimum size of $|W_0\beta|$ if it is greater than 1. Thus, we are faced with two problems:
\(1) Determine $\beta\in {\mathop{\rm Irr}\nolimits}T_0$ such that $C_{W_0}(\beta)=W_0,$ and
\(2) Assuming $|W_0\beta|>1$, find a lower bound for $|W_0\beta|$.
We could obtain a full solution to these problems. However, it seems that for the purpose of this paper we need only to describe favourable situations, where the orbit $W_0\beta$ is not too small for every $\beta\neq 1_{T_0}$. Our results in this line are exposed in Propositions \[dn5\], \[cm2\] and \[r4\], where the groups considered are non-twisted. To explain our approach, we therefore assume that $G$ is non-twisted. In this case $W_0$ coincides with the Weyl group $W$ of ${{\mathbf G}}$.
Let ${{\mathbf G}}$ be a simple simply connected algebraic group in defining characteristic $p$, and $G=G(q)$. Let $r$ be the rank of ${{\mathbf G}}$ and let ${\bf T_0}$ be a maximal torus of ${{\mathbf G}}$. Then the action of $W=N_{{\mathbf G}}({\mathbf T_0})$ on ${\bf T_0}$ yields the action of $W$ on ${\mathbb Z}^r$, the group of rational characters of ${\bf
T_0}$. In turn, this yields a $\zeta_0:W{ \rightarrow }GL(r,{\mathbb
Z})$, which we call the natural of $W$. It is well known that $\zeta_0(W)$ is an group (of $GL(n,{\mathop{\mathbb C}\nolimits})$) generated by reflections. (See [@DM 0.31].) The $\zeta_0$ is well understood, see [@St1; @St]. It turns out that, if $\beta\neq 1_{T_0}$, then $|W\beta|$ is not too small provided $\zeta_0(W)$ remains modulo every prime dividing $q-1$. This requires $q$ to be even for $G$ of type $B,C,D$, see Table 3. For a prime $\ell$ dividing $q-1$ denote by $\zeta_\ell$ the obtained from $\zeta_0$ by reduction modulo $\ell$. More precise analysis shows that it is enough that the dual of $\zeta_\ell$, if it is reducible, were fixed point free. This happens for $SL(r+1,q)$, $r>1$, when $\ell$ divides $r+1$, and for $E_6(q),E_7(q),E_8(q)$, see Proposition \[cm2\].
\[du3\]Let $G=G(q)$ be a non-twisted Chevalley group of rank $n$, $T_0$ a split torus, and $W$ the Weyl group of $G$. Let $\zeta_0:W{ \rightarrow }GL(n,{\mathbb Z})$ be the natural of $W$ and let $\zeta_\ell$ denote the reduction of $\zeta_0$ modulo a prime $\ell$. Then $T_0$ has a non-trivial $W$-invariant character there is a prime $\ell$ dividing $q-1$ such that $\zeta_\ell$ fixes a non-zero vector on $F_\ell^n$. In addition, $\zeta_\ell$ is dual to the natural action of $W$ on $T_\ell$, the subgroup of elements of order $\ell$ in $T_0$.
Proof. If $W $ fixes a character $1_{T_0}\neq \beta\in{\mathop{\rm Irr}\nolimits}T_0$ then it fixes any power $\beta^k$ too. So it suffices to deal with the case where the order of $\beta$ is a prime. So let $\ell=|\beta|$ be a prime.
It is well known that $T_0$ is a direct product of cyclic groups of order $q-1$, and hence $T_{\ell}=\{t^{(q-1)/\ell}: t\in T_0\}$. The characters of $T_0$ therefore correspond to elements of ${\mathbb Z}^n/(q-1){\mathbb Z}^n$, and those of $T_\ell$ correspond to elements of ${\mathbb Z}^n/\ell{\mathbb Z}^n$. This yields the reduction mapping $\zeta_0{ \rightarrow }\zeta_\ell$, and the first assertion of the lemma follows.
The additional claim describes $\zeta_\ell$ in terms of the action of $W$ on $T_\ell$. The group $T_0^*:={\mathop{\rm Irr}\nolimits}T_0$ is isomorphic to $T_0$, and the actions of $W$ on $T_0$ and on $T_0^*$ are dual to each other. Let $T_\ell^*=\{t\in
T_0^*:t^\ell=1\}$. Then the action of $W$ on $T_\ell^*$ is dual to the action of $W$ on $
T/T^\ell$. As $T$ is homocyclic, the action of $W$ on $ T/T^\ell$ is equivalent to that on $T_\ell$. $\Box$
\[dn5\]Let $q$ be even, $G=C_n(q)$, $n>1$, or $D^+_n(q) $, $n>3$, and let $1_{T_0}\neq \beta\in {\mathop{\rm Irr}\nolimits}T_0$. Then $|W_0\beta|\geq 2n$.
Proof. The group $W\cong W(B_n)=W(C_n)$, resp., $ W(D_n)$, is a semidirect product of a normal 2-group $A$ of order $2^n$, resp., $2^{n-1}$, and the symmetric group $S_n$. Note that $|A|\geq 2n$. In the reflection $\zeta_0$ the group $\zeta_0(W)$ can be realized by monomial $(n\times n)$-matrices over ${\mathbb Z}$ with diagonal subgroup $A$ and the group $S_n$ as the group of all basis permuting matrices. This group remains under reduction modulo any prime $\ell>2$. In addition, $A$ fixes no non-zero vector on $F_\ell^n$, and if $G=D^+_n(q)$ then $\det a=1$ for $a\in A$.
It suffices to prove the lemma when $|\beta|=\ell$ for every prime divisor $\ell$ of $|T_0|$. As $|T_0|$ is odd, $\ell$ is odd too. So $\zeta_\ell$, the reduction of $\zeta_0$ modulo $\ell$, is an matrix group. Let $0\neq v\in F_\ell^n$. Set $X=C_{W}(v)$ and $Y=A\cap X$. Since $A$ acts fixed point freely on ${\mathbb
Z}^n$, and hence on $F_\ell^n$, it follows that $Y\neq A$. We show that $W:X\geq 2n$. If $Y=1$ then $|W:X|\geq
|A| \geq 2n$. Suppose $Y\neq 1$. If $X=Y$ then $W:X\geq 2|S_n|\geq 2n$. Suppose $X\neq
Y$, and let $S=X/Y$. Then $S\subset S_n$. As $|A:Y|\geq 2$, we have $|W:X|= |A:Y|\cdot
|S_n:S|\geq 2\cdot |S_n:S|$. If $|A:Y|= 2$ then $v$ is a basis vector, and $S\cong S_{n-1}$. Therefore, $|W:X|=2n$. Suppose that $|A:Y|>2$. It is easy to check that this implies that $|Y|=2^i$ with $i>1$ and $X\cong S_i\times S_{n-i}$. This again implies $|W:X|\geq 2n$, as required. $\Box$
In Tables 1, 2, 3 $R$ is an indecomposable root system, and $Z_2$ in Table 2 denotes the cyclic group of order 2. Note that the data in Tables 2,3 are well known for root systems of types $A,B,C,D$, and for types $E_i$, $i=6,7,8$, the data follow from [@Atl; @MAtl].
TABLE 1: The structure of $W(R)$
$R$ $ A_{n-1} $ $B_n,C_n$ $D_n$ $E_6$ $E_7$ $E_8$ $F_4 $ $ G_2$
----- ------------- ----------------- --------------------- ---------------------- ------------------------------ -------------------------- -------------------------- -----------------
$W$ $S_n$ $ 2^n\cdot S_n$ $2^{n-1}\cdot S_n $ $PSp(4,3)\cdot Z_2 $ $Z_2\cdot Sp(6,2)\cdot Z_2 $ $\Omega^+(8,2)\cdot Z_2$ $\Omega^+(4,3)\cdot Z_2$ $Z_2\times S_3$
TABLE 2: The minimum degree of a non-linear of $W(R)$
$R$ $ A_{n-1}$, $B_n,C_n$, $D_n$, $n\neq 4$ $A_3$, $B_4=C_4, D_4$ $E_6$ $E_7$ $E_8$ $F_4 $, $ G_2$
----- ----------------------------------------- ----------------------- ------- ------- ------- ----------------
$d$ $n-1$ $2$ $ 6$ $7 $ $8 $ $2 $
TABLE 3: mod$\,\ell$ irreducibility of the natural of $W(R)$
[|c|c|c|c|c|c|c|c|c|c| ]{}
$R$& $A_{n-1} $ &$B_n,C_n$ &$D_n$&$E_6$&$E_7$&$E_8$&$F_4 $&$ G_2$\
$\ell$ & $(\ell,n)\neq 1$ & $ \ell\neq 2$ & $\ell\neq 2 $ &$\ell\neq 3 $&$\ell\neq 2 $& any $\ell $&$\ell\neq 2$& $\ell\neq 3$ dim &$n-1$&$n$&$n$&$6$&$7$&$8$&$4$&$2$\
\[n1\]Let $d$ be the minimum degree of a non-linear character of $W(R)$. Then $d$ is as in Table $2$.
Proof. If $R=A_{n-1}$ then $W(R)\cong S_n$. The degree formula for of $S_n$ easily implies that $d\geq n-1$ unless $n=
4$, where $d=2$. Let $R=B_n,C_n$ or $D_n$. It is well known that the minimum degree of a faithful of $W(R)$ equals $2n$. As $W(R)/A\cong S_n $ for an abelian normal subgroup $A$, one arrives at the same conclusion as for $S_n$. If $R=F_4$ then $W(R)\cong O^+(4,3)$. This group has a normal series $N_1\subset
N_2\subset W(R)$, where $N_1$ is extraspecial 2-subgroup of order 32, $N_2/N_1$ is elementary abelian of order 9, and $W(R)/N_2$ is elementary abelian of order 4. One observes that $W(R)/N_1$ is isomorphic to $S_3\times S_3$, and this group has an character of degree 2. Groups $Sp(4,3)$, $Sp(6,2)$ and $\Omega^+(8,2)$ are available in [@Atl], so the result follows by inspection. $\Box$
Below $G=E_6(q),E_7(q)$ are groups arising from the simply connected algebraic group.
\[cm2\]Let $G\in \{SL(n,q), n>2, E_6(q),E_7(q),E_8(q), F_4(q), G_2(q)\}$. Let $W$ be the Weyl group of $G$, and $T_0$ a split torus. Then $1_{T_0}$ is the only $W$-invariant character of $T_0$.
Proof. We use Lemma \[du3\] without explicit references to it.
Case 1. $G=SL(n,q)$, $n>2$. Here $W\cong S_{n}$. We can assume that $T_0$ is the group of diagonal matrices. Let $ \zeta_0$ be the usual of $S_{n}{ \rightarrow }GL(n-1, {\mathbb Z})$. Let $\ell $ be a prime dividing $|T_0|.$ If $(\ell, n)=1$ then $ \zeta_0$ remains modulo $\ell$. So assume $(\ell, n)\neq 1$. Then $ \zeta_\ell$ is not completely reducible. It has two composition factors, one is of dimension $n-2$, and the other factor is trivial (see [@KL 5.3.4]). Let $\eta$ be a non-trivial $\ell$-root of unity in $F_q$ and $t=\eta\cdot{\mathop{\rm Id}\nolimits}$; then $t$ is a scalar matrix. Obviously, $t\in T_0$. It follows that $\zeta^*_\ell(W)$ fixes a non-zero vector of $F^{n-1}_\ell$. As $\zeta_\ell^*$ is dual to $\zeta_\ell$, it follows by dimension reason that $\zeta_\ell$ has no fixed vector, unless, possibly, $n-1= 2$. If $n=3$ then $\ell=3$. As $\zeta^*_3$ is faithful and reducible, $\zeta^*_3(W)$ is conjugate with the matrix group $\big\{ \begin{pmatrix}1&*\cr 0
&\pm 1\end{pmatrix}\big\}$. Then $\zeta_3(W)$ fixes no non-zero vector on $F_3^2$.
Case 2. $G= E_6(q)$. Here $\zeta_0(W)\subset GL(6, {\mathbb Z})$ and $W\cong PSp(4,3)\cdot Z_2$. Then $\zeta_0$ remains modulo any prime $\ell\neq 3$, see [@MAtl]. Let $\ell=3$, so $3$ divides $q-1$ and $|Z(G)|=(3,q-1)=3$. Therefore, $W$ fixes a non-identity element of $T_3$. So $T_3\cong F_3^6$ viewed as a $W$-module has a one-dimensional subspace $S$, say, fixed by $W$. As $ PSp(4,3)$ has no non-trivial of degree less than $5$ [@MAtl], it follows that the second composition factor is of degree 5. As $T_3$ is indecomposable, and the quotient $T_3/S$ is irreducible, the dual module $T^*_3$ has no trivial submodule.
Case 3. $G= E_7(q)$. Here $\zeta_0(W)\subset GL(7, {\mathbb Z})$ and $W\cong Z_2\cdot Sp(6,2)\cdot Z_2$. Then $\zeta_0$ remains modulo any prime $\ell>2$. Let $\ell=2$. Then $q$ is odd and $|Z(E_7(q))|=(2,q-1)=2$. Therefore, the module $T_2$ has a non-trivial fixed point submodule $S$. As $T_2$ is indecomposable, and the quotient $T_2/S$ is irreducible, the dual module $T^*_2$ has no trivial submodule.
Case 4. $G= E_8(q)$. Here $\zeta_0(W)\subset GL(8, {\mathbb Z})$ and $W\cong Z_2\cdot \Omega^+(6,2)\cdot Z_2$. Then $\zeta_0$ remains modulo any prime $\ell$, see [@MAtl].
Case 5. $G=F_4(q)$. Then $W\cong W(F_4)\cong O^+(4,3)$ and $\zeta_\ell(W)$ is for $\ell>2$, see Table 2. Note that $O^+(4,3)'\cong SL(2,3)\circ SL(2,3)$. So it suffices to observe that $O^+(4,3)\,$mod$\,2$ fixes no non-zero vector on $F_2^4$. As this is the case for the Sylow 3-subgroup of $W$, the result follows.
Case 6. $G=G_2(q)$. Then $W$ is the dihedral group of order 12. Then $\zeta_\ell(W)$ is modulo any prime $\ell\neq 3$. Let $\ell=3$. Then a Sylow $2$-subgroup of $W$ fixes no non-non-zero vector in $F_3^2$, and hence this is true for the dual action. So again $W$ fixes no element of order 2 of ${\mathop{\rm Irr}\nolimits}T_0$. $\Box$
\[r4\]Let $G\in \{SL(n,q), n>4, E_6(q),E_7(q),E_8(q)\}$. Let $1_{T_0}\neq \beta\in{\mathop{\rm Irr}\nolimits}T_0$. Then $|W \beta|\geq m$, where $m=n, 27,28,120$, respectively.
Proof. By Proposition \[cm2\], $|W\beta|>1$. Let $r$ be the rank of $G$. If $W$ is realized via $\zeta_0$ as a subgroup of $GL(r,{\mathbb
Z}) $ generated by reflections then the stabilizer $C_{W}(v)$ of every vector $v\in {\mathbb Z}^r$ is generated by reflections. Due to a result of J.-P. Serre, see [@KM], this is also true if $W$ acts in $F_\ell^r$ and $\ell$ is coprime to $|W|$. This makes easy the computation of $|W\beta|=|W:C_{W}(v)|$. If $\ell$ divides $|W|$, and $v\in F_\ell^{r_0}$ then $C_{W}(v)$ is not always generated by reflections; see [@KM] where the authors classify all finite groups $H$ such that $C_{H}(V)$ is generated by reflections for every subspace $V$ of $F_\ell^{r}$. Partially we could use the results of [@KM], but it looks simpler to argue in a more straightforward way. For our purpose, in most cases it suffices to know the index of a maximal non-normal subgroup of $W$, which can be read off from [@Atl].
Let $\ell $ be a prime dividing $|\beta|$. As $|W:C_{W}(\beta)|\geq |W:C_{W}(\beta^k)|$, it is sufficient to deal with the case where $|\beta|=\ell$ is a prime. Let $D$ be the derived subgroup of $W$. If $\zeta_\ell (D)$ is irreducible then $|W:C_{W}(\beta)|\geq |D:C_D(\beta)|$, which is not less than the index $m_D$ of a maximal subgroup in $D$.
If $G=SL(n,q)$ then $W_0\cong S_{n} $ and $D\cong {\mathcal A}_{n}$, the alternating group. It is well known that every proper subgroup of ${\mathcal A}_{n}$, $n>4$, is of index at least $n$. So the lemma follows in this case. Let $G=E_6(q)$; then $D\cong SU_4(2)$ and $m_D=27$, see [@Atl]. Let $G=E_7(q)$; then $D\cong Sp_6(2)$ and $m_D=28$ [@Atl]. If $G=E_8(q)$ then $D/Z(D)\cong O^+_8(2)'$ and $m_D=120$ [@Atl]. $\Box$
Remark. If $G=SL(n,q)$, $q$ odd, $n=4$ then there is an element $\beta\neq 1_{T_0}$ with $|W\beta|=3$. Indeed, the Sylow 2-subgroup $X$ of $S_4$ has index 3, and fixes a non-zero vector of $F_2^3$. By Proposition \[cm2\], this vector is not fixed by $S_4$, whence the claim. Let $G=SL(3,q)$. Then $W\cong S_3$, and $\zeta_{\ell}$ is for every $\ell\neq 3$. In this case $m=3$. Let $\ell=3$. It is observed in the proof of Proposition \[cm2\] that $\zeta_3^*(W)$ fixes no non-zero vector. However, the Sylow $3$-subgroup of $W$ fixes a vector $v\neq 0$. Then $|\zeta_3^*(W)v|=2$, so $m=2$.
Parabolic descent
=================
Recall that for a PIM $\Phi$ of a group $G$ we set $c_\Phi=\dim\Phi/|G|_p$. If $H$ is a normal subgroup of $G$ and $M$ is an $FG$-module, then $C_M(H)$, the fixed point submodule for $H$, is viewed as $F(G/H)$-module. The socle of a module $M$ is denoted by Soc$\,M$. Every PIM is determined by its socle. The PIM whose socle is $1_G$ is called here 1-PIM, and denoted by $\Phi_1$.
Let $G$ be a Chevalley group so (see Notation) $G={{\mathbf G}}^{Fr}$, where ${{\mathbf G}}$ is simple and simply connected. Let $P$ be a parabolic subgroup of $G$ with Levi subgroup $L$. The parabolic descent is the mapping $\pi_{G,P}:\Phi{ \rightarrow }\Psi$, where $\Phi$ runs over the set of PIM’s for $G$ and $\Psi$ is a PIM for $L$. (One can extend this to the ${\mathbb Z}$-lattices spanned by PIM’s for $G$ and $L$.) The parabolic descent $\pi_{G,P}:\Phi{ \rightarrow }\Psi$ is determined by the Smith-Dipper mapping $\sigma_{G,L}:{\mathop{\rm Irr}\nolimits}G{ \rightarrow }{\mathop{\rm Irr}\nolimits}L$ defined by $S{ \rightarrow }{\mathop{\rm Soc}\nolimits}\,(S|_{P})$, where $S\in {\mathop{\rm Irr}\nolimits}G$ and the right hand side is viewed as an $FL$-module.
\[sm9\] Let $P$ be a parabolic subgroup of $G$ and $L$ a Levi of $P$. The mappings $\sigma_{G,P}:{\mathop{\rm Irr}\nolimits}G{ \rightarrow }{\mathop{\rm Irr}\nolimits}L$ and $\pi_{G,P}:{\rm
PIM }_G { \rightarrow }{\rm PIM }_L$ are surjective.
Proof. Let $M$ be an $FP$-module trivial on $O_p(P)$. There exists an $FG$-module $R$ such that ${\rm Hom}\,(M^G, R)\neq 0
$. By Frobenius reciprocity [@CR1 10.8], $\dim {\rm
Hom}\,(M^G, R)=\dim {\rm Hom}\,(M, R|_P)$. So $M$ is isomorphic to a submodule $M',$ say, of $R|_P$. So $M'\subseteq {\mathop{\rm Soc}\nolimits}R|_P$. By Lemma \[sdt\], ${\mathop{\rm Soc}\nolimits}R|_P$ is so $M'={\mathop{\rm Soc}\nolimits}R|_P$. This implies the statement for $\sigma_{G,P}$. In turn, this implies the statement for $\pi_{G,P}$ as, both for $G$ and $L$, modules are in natural bijection with PIM’s. $\Box$
Let $L'$ be the subgroup of $L$ generated by all unipotent elements of $L$. If $\Psi$ is a PIM for $L$ then $\Psi|_{L'}$ is a PIM for $L'$ (Proposition \[de4\]). Then one may also consider the mapping $\pi_{G,L'}$ which sends $\Phi$ to $\Psi|_{L'}$. For our purpose this version of the parabolic descent has some advantage. Indeed, there are a parabolic subgroup ${\mathbf P}$ of ${\mathbf G}$ and a Levi subgroup ${\mathbf L}$ of $ {\mathbf P}$ such that $P={\mathbf P}^{Fr}$ and $L={\mathbf L}^{Fr}$. Let ${\mathbf L}'$ denote the semisimple part of ${\mathbf L}$. Then $L'=({\mathbf L}')^{Fr}$. Thus, $L'$ corresponds to a semisimple subgroup ${\mathbf L}'$ of ${{\mathbf G}}$, and hence the of $L'$ can be parameterized in terms of highest weights. This allows us to make more precise control of $\pi_{G,L'}$ in terms of $\sigma_{G,L'}$. By Lemma \[mp1\], $c_\Phi\geq c_{\Psi'}$, where $\Psi'=\pi_{G,L'}(\Phi)$. (This is useful only if $c_{\Psi'}>1$.) The main case where $c_{\Psi'}=1$ (and hence $c_\Psi=1$) is when $\Psi$ is of defect 0. Corollary \[fg8\] below tells us that if $\Phi\neq St$ then $\Psi$ is not of defect 0 for some maximal parabolic subgroup of $G$. In order to prove this, we first turn Theorem \[sdt\] to a shape which allows to control ${\mathop{\rm Soc}\nolimits}(S|_P)$ in terms of $S$, where $S\in {\mathop{\rm Irr}\nolimits}G$. This is necessary mainly for twisted Chevalley groups.
Let ${{\mathbf G}}$ be an algebraic group over $F$ of rank $n$, ${\alpha}_1{,\ldots ,}{\alpha}_n$ be simple roots and ${\omega }_1{,\ldots ,}{\omega }_n$ be the fundamental weights of ${{\mathbf G}}$. Let $D$ denote the Dynkin diagram of ${{\mathbf G}}$ with nodes labeled by $1{,\ldots ,}n$ according to Bourbaki [@Bo]. We denote by $X_{\alpha}$ the root subgroup of ${{\mathbf G}}$ corresponding to a root ${\alpha}$. The dominant weights of ${{\mathbf G}}$ are of shape $\sum a_i{\omega }_i$ for some integers $ a_1{,\ldots ,}a_n\geq
0$; for an integer $q$, those with $0\leq a_i\leq q-1$ for $i=1{,\ldots ,}n$ are called $q$-restricted. The of ${{\mathbf G}}$ are parametrized by the dominant weights. Given a dominant weight $\mu$, we denote by $V_\mu$ the of ${{\mathbf G}}$ correspondimg to $\mu$; this weight $\mu$ is called the highest weight of $V_\mu$. For our purpose we may assume that ${{\mathbf G}}$ is simply connected.
Let $G={{\mathbf G}}^{Fr}$. Usually one takes for $q$ the common absolute value of $Fr$ acting on the weight lattice of ${{\mathbf G}}$, and set $G(q)={{\mathbf G}}^{Fr}$. If $q$ is an integer then $G=G(q)\in
\{SL(n+1,q), SU(n+1,q), Sp(2n,q)$, Spin$\,{}(2n+1,q)$, Spin$\,{}^\pm(2n,q)$, $E_i(q)$, $i=6,7,8$, ${}^2E_6(q)$, $F_4(q),
G_2(q), {}^3D_4(q) \}$. Otherwise, $q^2$ is an integer, and then $G\in \{ {}^2B_2(q),{}^2F_4(q), {}^2G_2(q)\}$.
The of $G$ are parameterized by the dominant weights satisfying certain conditions. More precisely, every of $G$ is the restriction to $G$ of an of ${{\mathbf G}}$ whose highest weight belongs to the set $\Delta
(G)$, defined as follows:
$$\Delta (G)=\begin{cases}a_1{\omega }_1+\cdots +a_n{\omega }_n:~0\leq a_1{,\ldots ,}a_n<q&~ if ~q ~is ~an ~integer,\\ a_i<q\sqrt{1/p}~~ if~~ {\alpha}_i~
long, ~and ~a_i<q\sqrt{p}~~ if ~{\alpha}_i~ is~ short & ~if ~~q~ is~
not~ an~ integer.\end{cases}$$
We refer to the elements of $\Delta (G)$ as dominant weights for $G$. (These are called the basic weights for $G$ in [@GLS 2.8.1].) By [@St Theorem 43] $\Delta (G)$ parameterizes the of $G$ up to equivalence. Therefore, there is a bijection $\Delta (G){ \rightarrow }{\mathop{\rm Irr}\nolimits}G$, so the of $G$ can be written as $\phi_{\lambda }$ for ${\lambda }\in\Delta (G)$. Thus, $\phi_{\lambda }$ extends to a unique of ${{\mathbf G}}$ with highest weight ${\lambda }\in\Delta (G)$, see [@St]. For brevity we refer to ${\lambda }$ as the highest weight of $\phi_{\lambda }$.
Furthermore, there a unique weight ${\lambda }\in\Delta (G) $ with maximal sum $a_1+\cdots +a_n$ (if $q\in{\mathop{\mathbb Z}\nolimits}$ then $a_1=\cdots =
a_n=q-1$). For this ${\lambda }$ $\dim\phi_{{\lambda }}$ is greater than for all other weights in $\Delta (G)$, and equals $|G|_p$, see [@St Corollary of Theorem 46] or [@St1 p. 88]. The corresponding $FG$-module is called here [*the Steinberg module*]{}, and is denoted by $St$. We record this as follows:
If $q$ is not an integer, then this is refined as follows. Set $q_1:=q/\sqrt{p}$; then $q_1$ is an integer.
\[st9\] Define a weight ${\sigma }$ as follows: $\sigma=(q-1)({\omega }_1+\cdots
+{\omega }_n)$ if $q$ is integer, otherwise and $\sigma=(q_1-1){\omega }_1+(2q_1-1){\omega }_2$, $(3q_1-1){\omega }_1+(q_1-1){\omega }_2$, $(q_1-1)({\omega }_1+{\omega }_2)+(2q_1-1)({\omega }_3+{\omega }_4)$, where $q_1:=q/\sqrt{p}$, respectively, for the group $G={}^2B_2(q)$, ${}^2G_2(q)$, ${}^2F_4(q)$.
Then $\dim V_{\sigma }=|G|_p$ and the restriction of $V_{{\sigma }}$ to $G$ is a unique $FG$-module of defect $0$.
A standard result of the theory of finite groups implies that $St$ is a unique $FG$-module of defect 0, and lifts to characteristic 0. It follows that there is a unique character of $G$ of degree divisible by $|G|_p$. This is called [*the Steinberg character*]{}; usually we keep the notation $St$ for this character as well.
For a reductive algebraic group ${{\mathbf G}}$ Smith’s theorem [@Sm] states that if ${\mathbf P}$ is a parabolic subgroup of ${{\mathbf G}}$ with Levi subgroup ${\mathbf L}$ and $V$ is a rational ${{\mathbf G}}$-module then $C_V(O_p({\mathbf P}))$ is an ${\mathbf L}$-module. Furthermore, suppose that the Frobenius endomorphism stabilizes ${\mathbf P}$ and ${\mathbf L}$, and set $P={\mathbf P}^{Fr}$, $L={\mathbf L}^{Fr}$. Then $C_V(O_p({\mathbf P}))=C_V(O_p(P))$ and this is an $FL$-module, see Cabanes [@Cab 4.2].
To every subset $J\subset D$ one corresponds a parabolic subgroup ${\mathbf P}_J$ by the condition $X_{{\alpha}_i}\in {\mathbf P}_J$ for $i\in D$ and $X_{-{\alpha}_i}\in {\mathbf P}_J$ $i\in J$. (These $
{\mathbf P}_J$ are called standard parabolic subgroups. If $J$ is empty, $ {\mathbf P}_J$ is a Borel subgroup.) Note that for a subset $J'\in D$ the inclusion $ {\mathbf P}_J\subset {\mathbf
P}_{J'}$ holds $J\subset J'$; in particular, every ${\mathbf
P}_J$ contains the standard Borel subgroup. Set ${{\mathbf G}}_J={ \langle }X_{\pm
{\alpha}_i}:i\in J{ \rangle }$. Then ${{\mathbf G}}_J$ is the semisimple component of a Levi subgroup ${\mathbf L}_J$ of ${\mathbf P}_J$. If ${\mathbf
P}_J$ and ${\mathbf L}_J$ are $Fr$-stable, then so is ${\mathbf
G}_J$. We set $P_J={\mathbf P}_J^{Fr}$, $L_J={\mathbf L}_J$ and $G_J={{\mathbf G}}_J^{Fr}$; these $P_J$ are called standard parabolic subgroups of $G$. The is known but we have no explicit reference:
\[le1\]$C_V(O_p(P_J))$ is an $FG_J$-module.
Proof. By Lemma \[sdt\], $C_V(O_p(P_J))$ is an $FL_J$-module, so the claim follows from Lemma \[de4\]. $\Box$
There is some advantage of dealing with ${\mathbf G}_J$ in place of ${\mathbf
L}_J$. The result is well known [@Sm]:
\[sd1\] Let ${{\mathbf G}}$ be a simple algebraic group over $F$, $J$ a non-empty set of nodes at the Dynkin diagram of ${{\mathbf G}}$, and ${{\mathbf G}}_J = \langle X_{\pm {\alpha}_j} : j\in J\rangle $. Let $V$ be an ${{\mathbf G}}$-module of highest weight $ {\omega }$, and let $v \in V$ be a vector of weight ${\omega }$. Then $V_J := { \langle }{{\mathbf G}}_J \, v{ \rangle }_{F}$ is an irreducible direct summand of $V|_{{{\mathbf G}}_J}$, with highest weight ${\omega }_{J}=\sum_{j\in J} a_j{\omega }_j$.
If $J$ is connected then ${{\mathbf G}}_J$ is a simple algebraic group of rank $|J|$, and one may think of the fundamental weights of ${{\mathbf G}}_J$ as $\{{\omega }_j:j\in J\}$. Then ${\omega }_{J}$ means $\sum_{j\in J}
a_j{\omega }_j$. If $J$ is not connected, let $J=J_1\cup\cdots\cup J_k$, where $J_1{,\ldots ,}J_k$ are the connected components of $J$. Then ${{\mathbf G}}_J$ is the central product of simple algebraic groups ${{\mathbf G}}_{J_1}{,\ldots ,}{{\mathbf G}}_{J_k} $, where ${{\mathbf G}}_{J_i}$ corresponds to $J_i$ ($i=1{,\ldots ,}k$). Then it is convenient to us to view ${\omega }_J$ as the string $({\omega }_{J_1}{,\ldots ,}{\omega }_{J_k})$. Furthermore, $V_J|_{{{\mathbf G}}_J}$ is the tensor product of the of ${{\mathbf G}}_{J_i}$ with highest weight ${\omega }_{J_i}$ for $i=1{,\ldots ,}k$.
There is a version of Lemma \[sd1\] for finite Chevalley groups. Lemma \[sdt\] is insufficient as it does not tell us how $(\overline{V}_L)|_{G_J}$ depends on $V$ (in notation of Lemma \[sd1\]). If $G=G(q)$ is non-twisted then this is easy to describe. Indeed, every $FG$-module extends to a ${{\mathbf G}}$-module with $q$-restricted highest weight; call it $V$. Then $G_J:={{\mathbf G}}_J^{Fr}$ is a non-twisted Chevalley group corresponding to ${{\mathbf G}}_J$, and the weight ${\omega }_{J}$ is $q$-restricted. Therefore, an ${{\mathbf G}}_J$-module $V_J$ remains as an $FG_J$-module, and can be labeled by ${\omega }_J$. In addition, $G_J$ is the central product of $G_{J_i}:={{\mathbf G}}_{J_i}^{Fr}$.
This argument can be adjusted to obtain a version for twisted Chevalley group but the twisted group case is less straightforward. The matter is that $Fr$ induces a permutation $f$, say, of the nodes of the Dynkin diagram of $G$, which is trivial $G$ is non-twisted. In the twisted case a set $J$ is required to be $f$-stable. If every connected component of $J$ is $f$-stable, then ${\omega }\in \Delta(G)$ implies ${\omega }_J\in
\Delta(G_J)$. So again we can use ${\omega }_J$ to identify $(V_J)|_{G_J}$.
An additional refinement is required if there is a connected component $J_1$, say, of $J$ such that $J_2:=f(J_1)\neq J_1$. If there are roots of different length then, by reordering $J_1,J_2$ we assume that the roots ${\alpha}_i$ with $i\in J_1$ are long. Note that the non-trivial $f$-orbits on $\{1{,\ldots ,}n\}$ are of size $a=2$, except for the case $G={}^3D_4(q)$ where $a=3$ [@St]. Then $G_J:=({{\mathbf G}}_{J_1}\circ {{\mathbf G}}_{J_2})^{Fr}\cong G_{J_1}(q^2)\cong
{{\mathbf G}}_{J_1}^{Fr^2} $ if $a=2$, or $G_J:=({{\mathbf G}}_{J_1}\circ
{{\mathbf G}}_{J_2}\circ {{\mathbf G}}_{J_3})^{Fr}\cong G_{J_1}(q^3)\cong
{{\mathbf G}}_{J_1}^{Fr^3}$ if $a=3$. Thus, in this case the Chevalley group obtained from the $f$-orbit on $J$ is non-twisted and quasi-simple. So one would wish to identify the $V_J|_{G_J}$ in terms of algebraic group weights of ${{\mathbf G}}_{J_1}$ rather than of ${{\mathbf G}}_{J_1}\circ {{\mathbf G}}_{J_2}$ when $a=2$, or ${{\mathbf G}}_{J_1}\circ
{{\mathbf G}}_{J_2}\circ {{\mathbf G}}_{J_3}$ when $a=3$. We do this in the following proposition. For this purpose it suffices to assume that $f$ is transitive on the connected components of $J$. To simplify the language, we call the highest weight of ${{\mathbf G}}_{J_1}$ obtained in this way the highest weight of $V_J$.
\[sd3\]Let $V$ be an ${{\mathbf G}}$-module of highest weight ${\omega }=\sum
a_i{\omega }_i$ such that $V|_{G}$ is (so ${\omega }\in\Delta(G)$). Let $J$ be an $f$-stable set of nodes at $D$, the Dynkin diagram of ${{\mathbf G}}$. Suppose that $J$ is not connected and $f$ is transitive on the connected components of $J$. Set $J_i=f^{i-1}(J_1)$ for $1<i\leq a$ and ${\omega }_{J_i}=\sum_{j\in J_i}a_j{\omega }_j$. Then $G_J\cong G_{J_1}(q^a)$.
Let $\tilde{{\omega }}_J$ be the highest weight of $V_J$ viewed as a $G_{J_1}(q^a)$-module. Then $\tilde{{\omega }}_J\in
\Delta(G_{J_1}^{Fr^(q^a)})$. More precisely, set ${\omega }'_{J_1}=\sum_{j\in J_1}a_{f(j)}{\omega }_{j}$ and, if $a=3$ set ${\omega }''_{J_1}=\sum_{j\in J_1}a_{f^2(j)}{\omega }_{j}$. Then:
If $q$ is an integer then $\tilde{{\omega }}_J={\omega }_{J_1}+q {\omega }'_{J_1}$ for $a=2$, and $\tilde{{\omega }}_J={\omega }_{J_1}+q{\omega }'_{f(J_1)}+q^2{\omega }''_{f^{2} (J_1)}$ for $a=3$;
If $q$ is not an integer then $q^2=p^{2e+1}$ for some integer $e\geq 0$, and $\tilde{{\omega }}_J={\omega }_{J_1}+ p^{e}{\omega }_{f (J_1)}$.
Proof. We consider only $a=2$, as the case $a=3$ differs only on notation. Thus, we show that $G_J:=({{\mathbf G}}_{J_1}\circ
{{\mathbf G}}_{J_2})^{Fr}\cong G_{J_1}(q^2)$. Note that $Fr$ permutes ${{\mathbf G}}_{J_1}$ and $ {{\mathbf G}}_{J_2}$, and acts as follows. Let $x_i\in {{\mathbf G}}_{J_i}$ $(i=1,2)$. If $q=p^e$ is an integer then $Fr(x_1,x_2)=(Fr^e_0 x_2, Fr^e_0x_1)$, where $Fr_0$ is the standard Frobenius endomorphism arising from the mapping $y{ \rightarrow }y^p$ $(y\in F)$. If $q$ is not an integer, then $Fr(x_1,x_2)=(Fr^{e+1}_0 x_2, Fr^{e}_0x_1)$.
So $Fr^{2}$ stabilizes each ${{\mathbf G}}_{J_i}$, and its fixed point subgroup on ${{\mathbf G}}_{J_i}$ is $G_{J_i}(q^2)$. Then $(x_1,x_2)$ is fixed by $Fr$ $x_1\in G_{J_1}(q^2)$ and $x_2=Fr_0^e(x_1)$. So $({{\mathbf G}}_{J_1}\circ
{{\mathbf G}}_{J_2})^{Fr}\cong G_{J_1}(q^2)$, as claimed. Furthermore, $(V_J)|_{{{\mathbf G}}_{J_1}\circ
{{\mathbf G}}_{J_2}}$ is the tensor product of the ${{\mathbf G}}_{J_1}$- and ${{\mathbf G}}_{J_2}$-modules of highest weights ${\omega }_{J_1}$ and ${\omega }_{J_2}$, respectively (as the groups ${{\mathbf G}}_{J_1}$ and $ {{\mathbf G}}_{J_2}$ commute elementwise). One can consider the ${{\mathbf G}}_{J_1}$-module obtained from $W|_{{{\mathbf G}}_{J_1}\circ
{{\mathbf G}}_{J_2}}$ via the ${{\mathbf G}}_{J_1}{ \rightarrow }{{\mathbf G}}_{J_1}\circ
{{\mathbf G}}_{J_2}$ defined by $x_1{ \rightarrow }(x_1,Fr_0^e(x_1))$. Clearly, this is the tensor product of the ${{\mathbf G}}_{J_1}$-modules of highest weights ${\omega }_{J_1}$ and $p^e{\omega }'_{f(J_1)}$.
If $q\in {\mathbb Z}$ then ${\omega }_{J_1}$ and ${\omega }'_{ f(J_1)}$ are $q$-restricted, so $\tilde{{\omega }}_J={\omega }_{J_1}+q{\omega }'_{f (J_1)}$ is $q^2$-restricted, and hence belongs to $\Delta(G_{J_1}(q^2))$.
Suppose that $q\notin {\mathbb Z}$. As $V|_G$ is irreducible, ${\omega }\in\Delta(G)$. This implies that $a_i<p^e$ for $i\in J_1$ and $a_{f(i)}<p^{e+1}$. Then $a_i+p^{e}a_{f(i)}<q^2$, as required. (So the highest weight of the ${{\mathbf G}}_{J_1}$-module in question is ${\omega }_{J_1}+p^{e}{\omega }'_{f ( J_1)}\in \Delta(G_{J_1}(q^2))$.) $\Box$
Remark. In Proposition \[sd3\] $J$ is disconnected, which implies that the BN-pair rank of $G$ is at least 2.
Example. Let ${{\mathbf G}}$ be of type $A_{2n-1+k}$, $k=0,1$, $G=SU_{2n+k}(q)$ and $J=\{1{,\ldots ,}n-1,n+k+1{,\ldots ,}2n-1+k\}$. Then $J_1=\{1{,\ldots ,}n-1\}$ and $G_{J}\cong SL(n,q^2)$. Let $\mu=a_1{\omega }_1+\cdots +a_{2n-1+k}{\omega }_{2n-1+k}$. Then $\tilde{{\omega }}_J=(a_1+qa_{2n-1+k}){\omega }_1'+\cdots
+(a_{n-1}+qa_{n+1+k}){\omega }'_{n-1}$, where ${\omega }'_{1}{,\ldots ,}{\omega }'_{n-1}$ are the fundamental weights of ${{\mathbf G}}_{J_1}=A_{n-1}$.
As above, $D$ denotes the Dynkin diagram of ${{\mathbf G}}$ and $G={{\mathbf G}}^{Fr}$. Recall that the standard parabolic subgroups $P_J$ of $G$ are in bijection with $f$-stable subsets $J$ of the nodes of $D$ (and $P_J=G$ $J=D$).
\[st0\]Let $D=J\cup J'$ be the disjoint union of two $f$-stable subsets. Then ${\mathop{\rm Irr}\nolimits}G{ \rightarrow }{\mathop{\rm Irr}\nolimits}(G_J\times G_{J'} )$ is a bijection. In addition, if $T_J,T_{J'}$ are maximal split tori in $G_J,G_{J'}$, respectively, then $T_JT_{J'}$ is a maximal split torus in $G$.
Proof. The first statement follows from the reasoning in [@GLS p. 79]. The second one follows from [@GLS Theorem 2.4.7(a)], which tells us that $\Pi_i T_{I(i)}$ is a maximal split torus in $G$, where $I(i)$ is the $f$-orbit containing $i$ and $T_{I(i)}$ is a a maximal split torus in $ G_{I(i)})$.
By induction, it follows that a similar statement is true for any disjoint union $D=\cup_iJ_i$ of $f$-stable subsets $J_i$ of $D$, in particular, when $J_i$ is an $f$-orbit for every $i$. (Only this case is explicitly mentioned in [@GLS].) Note that the bijection in Lemma \[st0\] yields a bijection $PIM_G{ \rightarrow }PIM_{G_J\times G_{J'}}$.
\[fg8\]$G={{\mathbf G}}^{Fr}$ be a Chevalley group of $BN$-pair rank at least $2$, and $G\neq {}^2F_4(q)$. Let $V$ be an ${{\mathbf G}}$-module with highest weight $\mu=\sum a_i{\omega }_i\in \Delta (G)$. Suppose that $\emptyset\neq J\subseteq D$ is $f$-stable. Let $P:=P_J$ be the standard parabolic subgroup of $G$ corresponding to $J$, and $L:=L_J$ a Levi subgroup of $P$. Let $V_J=C_V(O_p(P))$.
$(1)$ $\dim V_J=1$ $a_i=0$ for all $i\in J$.
$(2)$ $V_J|_L$ is of defect $0$ $a_i=q-1$ for all $i\in J$. (The latter is equivalent to saying that $M|_{G_J}$ is Steinberg).
$(3)$ Let $J'=D\setminus J$ and $L'$ be a Levi subgroup of $P_{J'}$. If $V_J|_{L}$ and $V_{J'}|_{L'}$ are of defect $0$ then so is $V|_G$. Equivalently, If $V_J|_{G_J}$ and $V_J|_{G_{J'}}$ are Steinberg modules then $V|_G$ is Steinberg.
PS. [parabolic subgroup ]{}
Proof. Let ${\mathbf P}_J$ and ${{\mathbf G}}_J$ be the respective algebraic groups. Then $V_J|_{G_J}=C_V(O_p({\mathbf P}_J))$, and $V_J$ is an ${{\mathbf G}}_J$-module.
\(1) follows from Lemma \[sd1\].
\(2) Suppose that $a_i=q-1$ for all $i\in J$. Then $V_J|_{G_J}$ is Steinberg by Lemma \[st9\]. To prove the converse, observe that if $a_i<q-1$ for some $i\in J$ then $\dim V_J<\dim St$ by Lemma \[st9\] applied to ${\mathbf G}_J$. As $|L|_p=|G_J|_p$, it follows that $V_J$ is of defect 0. This implies (2).
\(3) By (2), $a_i=q-1$ for all $i\in D$, so the claim follows from Lemma \[st9\]. $\Box$
\[st6\]Let $M$ be the Steinberg $FG$-module for a Chevalley group $G={{\mathbf G}}^{Fr}$, and let $B$ be a Borel subgroup of $G$. Then ${\mathop{\rm Soc}\nolimits}(M|_B)=1_B$.
Proof. Let $U$ be the of $B$. As $M$ is projective and $\dim
M=|U|$, it follows that $M|_U$ is the regular $FU$-module. Therefore, $\dim{\mathop{\rm Soc}\nolimits}(M|_B)=1$. Let ${\lambda }$ be the Brauer character of ${\mathop{\rm Soc}\nolimits}(M|_B)$. As $U$ is in the kernel of ${\lambda }$, it can be viewed as an ordinary character of $B$. Let $St$ denote the character of $M$, so this is exactly the Steinberg character. By Proposition \[m75\], $(St,{\lambda }^G)>0$. If ${\lambda }\neq 1_B$ then $({\lambda }^G,1_B^G)=0$ [@CR2 Theorem 70.15A]. In addition, $(St,1_B^G)=1$ [@CR2 Theorem 67.10]. Therefore, ${\lambda }=1_B$, as required. $\Box$
\[sx6\]Let $q\in{\mathop{\mathbb Z}\nolimits}$ and let $V_\mu$ be an ${{\mathbf G}}$-module of highest weight $\mu=\sum a_i{\omega }_i\in\Delta(G)$. Let $B$ be a Borel subgroup of $G={{\mathbf G}}^{Fr}$. Then the are equivalent:
$(1)$ ${\mathop{\rm Soc}\nolimits}(V_\mu|_B)=1_B$.
$(2)$ $a_1{,\ldots ,}a_n\in \{0,q-1\}\ $ and $a_{f(i)}=a_i$ for every $i$.
Proof. Note that the result is well known for $G=SL(2,q)$.
Suppose first that $G\cong SU(3,q)$, so $ {{\mathbf G}}\cong SL(3,F)$. Let $M$ be the module for ${{\mathbf G}}$ of highest weight ${\omega }_1$. Then $M|_{G}$ is isomorphic to the natural module $M'$, say, for $G\cong SU(3,q)$. Let ${\mathbf B} $ be a Borel subgroup of ${{\mathbf G}}$ such that $B={\mathbf B}\cap G$ is a Borel subgroup of $G$. As explained in the discussion after Lemma \[st9\], the socle of ${\mathbf B}$ on $M$ coincides with the socle of $B$. Let $v\in M$ be a vector of highest weight ${\omega }_1$ on $M$. Then ${ \langle }v{ \rangle }$ is ${\mathbf B}$-stable and hence $B$-stable. We can view $M'$ as an $F_{q^2}G$-module endowed by a unitary form, and choose $v\in
M'$. Then $v$ is an isotropic vector. It is easy to see that $T_0$ is isomorphic to the multiplicative group $F^{\times}_{q^2}$ of $F_{q^2}$. Let $\nu\in {\mathop{\rm Irr}\nolimits}T_0$ be the of $B$ on ${ \langle }v{ \rangle }$. Then it is faithful. As $M^*$, the dual of $M$, has highest weight ${\omega }_2$, one can check that the of $T_0$ in $C_{M^*}(B)$ is $\nu^q$. Therefore, the of $T_0$ on $C_{V_\mu}(B)$ is $\nu^{a_1}\nu^{a_2q}$. ${\mathop{\rm Soc}\nolimits}(V_\mu|_B)=1_B$ $\nu^{a_1}\nu^{a_2q}=1_{T_0}$. Let $t$ be a generator of $T_0$. Then $\nu^{a_1}\nu^{a_2q}(t)=t^{a_1+a_2q}$. It is clear that $t^{a_1+a_2q}=1$ either $a_1=a_2=0$ or $a_1=a_2=q-1$. This completes the proof for $G=SU(3,q)$.
Other groups $G$ are of BN-pair rank 2, and hence satisfy the assumptions of Corollary \[fg8\].
$(1){ \rightarrow }(2)$. Let $i\in D$ be such that $a_i\neq 0,q-1$, and let $J$ be the $f$-orbit of $i$. Let $P_J$ and $G_J$ be as in Corollary \[fg8\], and let $ B_J=G_J\cap B$, so $B_J$ is a Borel subgroup of $G_J$.
Suppose first that $|J|=1$. Then $J=\{i\}$ for some $i$, and $G_J=SL_2(q)$ (both in the twisted and non-twisted cases). Then $V_J:=({\mathop{\rm Soc}\nolimits}(V_\mu |_{P_J})|_{G_J}$ is with highest weight $a_i{\omega }_i$. Then ${\mathop{\rm Soc}\nolimits}(V_J|_{B_J})=1_{B_J}$ $a_i\in\{0,q-1\}$. As $1_B=({\mathop{\rm Soc}\nolimits}(V_\mu|_B))$ equals ${\mathop{\rm Soc}\nolimits}(V_J
|_{B_J})$ inflated to $B$, it follows that $a_i\in\{0,q-1\}$.
Let $f(i)=j$, and $i,j$ are not adjacent. If $|J|=2$ then $G_J=SL_2(q^2)$ and the highest weight of $V_J$ as an $FG_J$-module is $a_i+qa_j$. So $a_i+qa_j=q^2-1$ or $0$. Therefore, $a_i=a_j\in\{0,q-1\}$. Similarly, if $|J|=3$ then $G_J=SL_2(q^3)$, so $a_i+qa_{f(i)}+q^2a_{f^2(i)}=q^3-1$ or $0$. This again implies $a_i=a_{f(i)}=a_{f^2(i)}\in\{0,q-1 \}$.
Let $f(i)=j$, $f(j)=i$ and $i,j$ are adjacent. Then $G_J\cong
SU_3(q) $ as $q\in {\mathop{\mathbb Z}\nolimits}$. As ${\mathop{\rm Soc}\nolimits}(V_J|_{B_J})\subseteq {\mathop{\rm Soc}\nolimits}(V_\mu|_B)=1_B$, it follows by the above that $a_i=a_{f(i)}\in
\{0,q-1\}$.
$(2){ \rightarrow }(1)$. Let $J=\{i: a_i=q-1\}$ and $J'=\{i: a_i=0\}$. Let $T_0$ be a maximal torus of $B$. Then ${\mathop{\rm Soc}\nolimits}(V_\mu|_B)={\mathop{\rm Soc}\nolimits}({\mathop{\rm Soc}\nolimits}(V_\mu|_P))|_B)$ as $\dim {\mathop{\rm Soc}\nolimits}(V_\mu|_B)=1$ and ${\mathop{\rm Soc}\nolimits}(V_\mu|_P)$ is Let $B_J$ be a Borel subgroup of $G_J$ and $T_J=T_0\cap B_J$. By Corollary \[fg8\](2), $({\mathop{\rm Soc}\nolimits}(V_\mu|_P)|_{G_J}$ is the Steinberg $FG_J$-module, so, by Lemma \[st6\], $(({\mathop{\rm Soc}\nolimits}(V_\mu|_P)|_{G_J})|_{B_J}=1_{B_j}$, and hence $(({\mathop{\rm Soc}\nolimits}(V_\mu|_P)|_{B_J})|_{T_J}=1_{T_j}$.
Set $T_{J'}=T_0\cap G_{J'}$. Then $T_0=T_JT_{J'}$, see Lemma \[st0\]. Let $B_{J'}$ be a Borel subgroup of $G_{J'}\subset P_{J'}$ containing $T_{J'}$. Then $S:=({\mathop{\rm Soc}\nolimits}(V_\mu|_{P_{J'}})|_{G_{J '}}$ is the trivial $FG_{J'}$-module, so $S|_{B_{J'}}=1_{B_{J'}}$, and hence $S|_{T_{J'}}=1_{T_{J'}}$. Therefore, $({\mathop{\rm Soc}\nolimits}(V_\mu|_{B})|_{T_{J'}}=1_{T_{J'}}$. $({\mathop{\rm Soc}\nolimits}(V_\mu|_{B})|_{T_0}=1_{T_0}$, and then $({\mathop{\rm Soc}\nolimits}(V_\mu|_{B})=1_B$, as required. $\Box$
\[yg8\]Let $V_\mu$ be an ${{\mathbf G}}$-module of highest weight $\mu=\sum
a_i{\omega }_i\in\Delta(G)$, and let $J=\{i: a_i=q-1\}$. Suppose that $q$ is an integer, $J$ is $f$-stable and $a_i\in\{0,q-1\}$ for $i=1{,\ldots ,}n$. Let $B$ be a Borel subgroup of $G$, and let $P=P_J$ be a parabolic subgroup of $G$. Let $\Phi$ be the PIM with socle $V:=V_\mu|_G$ and character $\chi$.
$(1)$ ${\mathop{\rm Soc}\nolimits}(V_\mu|_B)=1_B$.
$(2)$ $(\chi,1_B^G)>0$.
$(3)$ Let $L$ be a Levi subgroup of $P$ and $S:=\pi_{G,P}(V)$ (which is an $FL$-module). Then $S$ is of defect $0$. Furthermore, let $\rho$ be the character of the lift of $S$. Then $\rho$ is a constituent of $1_{B\cap L}^{L}$ and $(\chi, \rho
^{\#G})>0$.
Proof. (1) is contained in Lemma \[sx6\]. (2) This follows from Corollary \[xx1\].
\(3) By Corollary \[fg8\], the $S|_{G_J}$ is the Steinberg module. So $S$ is an $FL$-module of defect 0, and hence lifts to characteristic 0. So $(\chi, \rho ^{\#G})>0$ by Corollary \[xx1\].
Furthermore, by the reasoning in Lemma \[sx6\], ${\mathop{\rm Soc}\nolimits}({\mathop{\rm Soc}\nolimits}(V_\mu|_P)|_B)$ coincides with ${\mathop{\rm Soc}\nolimits}(V_\mu|_B)=1_B$. As $S=({\mathop{\rm Soc}\nolimits}(V_\mu|_P))|_L$, it follows that $S|_{L\cap B}=1_{L\cap B}$. By Lemma \[a2\], where one takes $M=S$, $H=B\cap L$ and $N=O_p(H)$, the truncation $\overline{\rho}$ is the trivial character. By the Harish-Chandra reciprocity, $\rho$ is a constituent of $1_{B\cap L}^{L}$. $\Box$
The lemma will be used in Section 7 in order to determine the PIM’s for $G$ of dimension $|G|_p$.
\[sd7\]Let $G={{\mathbf G}}^{Fr}$ be a Chevalley group of $BN$-pair rank at least $2$, and if $G= {}^2F_4(q)$ assume $q^2>2$. Let $n$ be the rank of ${{\mathbf G}}$, and let $V$ be an $F{{\mathbf G}}$-module of highest weight $0\neq \mu=a_1{\omega }_1+\cdots
+a_n{\omega }_n\in \Delta(G) $. Suppose that $V|_G\neq St$ and for every parabolic subgroup $P$ of $G$ the restriction of ${\mathop{\rm Soc}\nolimits}(V|_P)$ to a Levi subgroup $L$ is either projective, or the semisimple part $L'$ of $L$ is of type $A_1(p)$, $A_2(2)$ or ${}^2A_2(2)$ and ${\mathop{\rm Soc}\nolimits}(V|_P)$ is trivial on $L'$. Then one of the holds:
$(1)$ $G=SL(3,p)$ and ${\omega }=(p-1){\omega }_1$ or $(p-1){\omega }_2;$
$(2)$ $G=Sp(4,p)\cong {\rm Spin}\,(5,p)$ and ${\omega }=(p-1){\omega }_1$ or $(p-1){\omega }_2;$
$(3)$ $G=G_2(p)$ and ${\omega }=(p-1){\omega }_1$ or $(p-1){\omega }_2;$
$(4)$ $G=SU(4,p)$ and ${\omega }=(p-1)({\omega }_1+{\omega }_3);$
$(5)$ $G={}^3D_4(p)$ and ${\omega }=(p-1)({\omega }_1+{\omega }_3+{\omega }_4);$
$(6)$ $G=SU(5,2)$ and ${\omega }={\omega }_1+{\omega }_4;$
In addition, in all these cases ${\mathop{\rm Soc}\nolimits}(V|_B)=1_B$.
Proof. As above, for an $f$-stable subset of nodes of the Dynkin diagram of ${{\mathbf G}}$ we denote by $P_J$ the corresponding parabolic subgroup of $G$, by $G_J$ the standard semisimple subgroup of $P_J$ and set $V_J=C_V(O_p(P_J))$ viewed as an $FG_J$-module.
Suppose first that $G$ is of type ${}^2F_4(q)$, $q^2>2$. Then there are two $f$-orbits on $D$: $J=\{1,4\}$ and $J'=\{2,3\}$. Accordingly, $G$ has two parabolic subgroups $P_J$, $P_{J'}$. By Proposition \[sd3\], the highest weight of $V_J$ is $(a_1+2^ea_4){\omega }'_1$, where ${\omega }'_1$ is the fundamental weight for ${{\mathbf G}}_J$ of type $A_1$ and $2^e=q/\sqrt{2}$. The highest weight of $V_{J'}$ is $a_2{\omega }'_1+a_3{\omega }'_2$, where ${\omega }'_1$, ${\omega }'_2$ are the fundamental weights for ${{\mathbf G}}_{J'}$ of type $B_2$.
If $V_{(a_1+2^ea_4){\omega }'_1}$ is the Steinberg module for $G_{J'}$ then $a_1+2a_4=q^2-1$, whence we have $a_1=q/\sqrt{2}-1$, $a_4=q\sqrt{2}-1$ (as $0\leq
a_1<q/\sqrt{2}$, $0\leq a_4<q\sqrt{2}$). If the restriction of $V_{a_2{\omega }'_1+a_3{\omega }'_2}$ to $ G_{J'}$ is of defect 0 then $V_{a_2{\omega }'_1+a_3{\omega }'_2}|_{ G_{J'}}$ is the Steinberg module, and hence $a_2=q/\sqrt{2}-1$ and $a_3=(2q/\sqrt{2})-1$ by Lemma \[st9\]. Thus, $a_1=a_2=q/\sqrt{2}-1$, and $a_3=a_4=(2q/\sqrt{2})-1$. By Lemma \[st9\], $V |_G$ is the Steinberg module.
So assume that $G\neq {}^2F_4(q)$. By Lemma \[st9\], at least one of $a_1{,\ldots ,}a_n$ differs from $q-1$.
Suppose first that $G=G(q)$ is non-twisted of rank 2. In notation of Lemma \[sd1\] and comments following it, $G_J\cong SL(2,q)$ for $J=\{1\},\{2\}$, so $q=p$, and all these groups are listed in items $(1),(2), (3)$. In addition, $V_J$ is of $a_1{\omega }_1$ or $a_2{\omega }_1$. So the claim about the weights in $(1),(2), (3)$ follows.
Suppose that $G=G(q)$ is non-twisted of rank greater than 2, and let $D$ be the Dynkin diagram of ${{\mathbf G}}$. Then one can remove a suitable edge node from $D$ such that $a_i\neq q-1$ for some $i$ in the remaining set $J$ of nodes. Then the $FG$-module $V_J$ is not projective (Corollary \[fg8\]), and hence we have a contradiction, unless $G_J\in
\{A_1(p),A_2(2)\}$ and $a_i=0$ for $i\in J$. In fact, $G_J\neq
A_1(p)$ as otherwise $|D|=2$. If $G_J\cong A_2(2)\cong SL(3,2)$ then $|D|=3$ and $G=SL(4,2)$. However, in this case, taking $J=\{1,2\}$ and $J'=\{2,3\}$, one obtains $a_i=0$ for $i\in J\cap
J'=D$. Then $\mu=0$, which is false. Suppose that $G$ is twisted. We argue case-by-case.
\(i) $G={}^3D_4(q)$. There are two $f$-orbits $J,J'$ on $D$, where $J=\{1,3,4\}, J'=\{2\}$, and hence $G_{J'}\cong SL(2,q)$ and $G_J\cong SL(2,q^3)$. So the assumption is not satisfied unless $q=p$. This case occurs in item $(5)$. The claim on the weights follows from Corollary \[fg8\].
\(ii) $G={}^2A_{n}(q)$, where $n=2m-1>1$ is odd. Take $J=\{1{,\ldots ,}m-1, m+1{,\ldots ,}2m-1\}$. Then $G_J\cong SL(m,q^2)$. By assumption, $V_J$ is Steinberg. So $a_1=\cdots =a_{m-1}=a_{m+1}=\cdots
=a_{2m-1}=q-1$. Therefore, $a_m<q-1$. Take $J=\{m\}$. Then $G_J\cong A_1(q)$, whence $q=p$ and $a_m=0$. The case $n=3$ is recorded in (4). Let $n>3$. Take $J=\{m-1,m,m+1\}$. Then $G_J\cong
{}^2A_3(q)$, which is a contradiction.
\(iii) $G={}^2A_{n}(q)$, $n=2m$ even. Take $J=\{1{,\ldots ,}m-1, m+2{,\ldots ,}2m\}$. Then $G_J\cong A_{m-1}(q^2)$. Then $a_1=\cdots
=a_{m-1}=a_{m+2}=\cdots =a_{2m}=q-1$. Therefore, $a_{m}a_{m+1}<(q-1)^2$. Then take $J=\{m,m+1\}$. Then $G_J\cong
{}^2A_2(q)$, and hence $q=2$, $a_{m}=a_{m+1}=0$. The case $m=2$ is recorded in item (6). Let $m>2$. Then for $J=\{m-1,m,m+1,m+2\}$ we have $G_J={}^2A_{4}(2)$, which is a contradiction.
\(iv) $G={}^2D_{n}(q)$, $n>3$. Take $J=\{n-1,n\}$. Then $G_J\cong
A_1(q^2)$, and the assumption implies $a_{n-1}=a_n=q-1$, and hence $a_i<q-1$ for some $i<n-1$. Next take $J=\{1{,\ldots ,}n-2\}$, so $G_J=SL(n-1,q)$. This implies $n=4$, $q=2$ and $a_1=a_2=0$. Finally, take $J=\{2,3,4\}$. Then $G_J$ is of type ${}^2A_3(2)$, so this option is ruled out.
\(v) $G={}^2E_6(q)$. Then (using Bourbaki’s ordering of the nodes) the orbits of $f$ are $(1,6)$, $(3,5)$, $(2)$, $(4)$. Take $J=\{1,3,4,5,6\}$. Then $G_J\cong {}^2A_5(q)$, which implies $a_1=a_3=a_4=a_5=a_6=q-1$. So $a_2<q-1$. Next take $J=\{2,4\}$. Then $G_J\cong A_2(q)$, and $V_J$ is not trivial. This is a contradiction. $\Box$
We close this section by two results which illustrate the use of Propositions \[r55\] and \[sd3\].
\[g77\]Let $G\in \{B_n(q),n>2,C_n(q),n>1,D_n(q), n>3,{}^{2}D_{n+1}(q),n>2
\}$. Let $J=\{1{,\ldots ,}n-1\}$. Let $\Phi$ be a PIM with socle $V$ and $S={\sigma }_{G,G_J}(V)$. Then either $S$ is self-dual or $c_\Phi\geq
2c_\Psi$.
Proof. Let $P_J$ be the standard parabolic subgroup of $G$ corresponding to $J$ and let $L:=L_J$ be the standard Levi subgroup of $P_J$. Then $G_J\cong SL(n,q)$. Let $\overline{W}_L$ be as in Proposition \[r55\]. We show first that $|\overline{W}_L|=2$, and a non-trivial element of $\overline{W}_L$ induces the duality $h$ on $L$ (that is, if $\phi\in {\mathop{\rm Irr}\nolimits}L$ then $\phi^h$ is the dual of $\phi$). Indeed, we can assume that $T_0\subset L$, and then $N_G(T_0)$ contains an element $g$ acting on $T_0$ by sending every $t\in T_0 $ to $t{^{-1}}$. Let $h$ be the inner of $G$ induced by the $g$-conjugation. Then $S^h$ is the dual $L$-module of $S$. So the result follows from Proposition \[r55\]. (One can make this clear by using an appropriate basis of the underlying space $V$ of $G$, and by considering, for the group $G_1$ of isometries of $V$, a matrix embedding $GL(n,q){ \rightarrow }G_1 $ sending every $x\in GL(n,q)$ to ${\mathop{\rm diag}\nolimits}(x,{}^{Tr}x{^{-1}})$, where ${}^{Tr}x$ denotes the transpose of $x$.)
A similar result holds for the groups ${}^{2}A_{2n+2}(q),{}^{2}A_{2n+1}(q)$, where $P$ has to be chosen so that $L$ contains $SL(n,q^2)$. In this case $h$ is the duality automorphism following the Galois automorphism.
\[bn1\]Let $k=0,1$ and $G={}^2A_{2n-1+k}
\cong SU(2n+k,q)$. Let $J=\{1{,\ldots ,}n-1,n+k+1{,\ldots ,}2n-1+k\}$ so $G_J\cong SL(n,q^2)$. Let $\mu=a_1{\omega }_1+\cdots
+a_{2n-1+k}{\omega }_{2n-1+k}\in \Delta(G)$, let $V_\mu$ be an ${{\mathbf G}}$-module of highest weight $\mu$, $V=V_\mu|_G$ and $V_J=\pi_{G,G_J}$.
$(1)$ The highest weight of $V_J$ is $\mu'=(a_1+qa_{2n-1+k}){\omega }_1'+\cdots+(a_{n-1}+qa_{n+k+1}){\omega }_{n-1}'
$, where ${\omega }'_1{,\ldots ,}{\omega }_{n-1}'$ are the fundamental weight of $SL(n,F)$.
$(2)$ Let $\Phi$ be the PIM with socle $V$, and $\Psi$ be the PIM for $G_J$ with socle $V_J$. Then either $c_{\Phi}\geq 2c_{\Psi}$, or $a_i=a_{n+i+k}$ for $i=1{,\ldots ,}n-1$.
Proof. (1) follows from Proposition \[sd3\]. (2) Let $1\neq w\in
\overline{W}_L$. The induced by $w$ on $G_J$ sends $g\in
SL(n,q^2)$ to ${}^{Tr}g^{-\gamma}$, where $\gamma$ is the Galois of $F_{q^2}/F_q$ and ${}^{Tr}g$ is the transpose of $g$. Then the mapping $v{ \rightarrow }w(g)v$ $(v\in V_J)$ yields an $FG_J$-module $V'_{J} $ of highest weight $\mu':=(qa_{n-1}+a_{n+k+1}){\omega }_1'+\cdots+(qa_1+a_{2n-1+k}){\omega }_{n-1}'
$. If $V_{J} $ is not isomorphic to $V'_{J}$ then $c_\Phi\geq
2c_\Psi$ by Proposition \[r55\]. If $V_J\cong V'_{J} $ then $a_1+qa_{2n+k-1}\equiv (qa_{n-1}+a_{n+k+1})({\rm mod }\,q^2)\, {,\ldots ,}\,$ $a_{n-1}+qa_{n+k+1}\equiv (qa_1+a_{2n+k-1})({\rm mod }\,q^2)$. As $a_i<q$ for $i=1{,\ldots ,}2n+k$, it follows that $a_i=a_{n+k+i}$ for $i=1{,\ldots ,}n-1$. $\Box$
Harish-Chanra induction and a lower bound for the PIM dimensions
================================================================
The classical result by Brauer and Nesbitt [@BN Theorem 8] (see also [@Fe Ch.IV, Lemma 4.15]) is probably not strong enough to be used for studying PIM dimension bounds for Chevalley groups.
The lemma is one of the standard results on of groups with $BN$-pair, see [@CR2 pp.683 - 684] or [@St §14, Theorem 48]:
\[bb5\]Let $G$ be a Chevalley group viewed as a group with BN-pair, $B$ a Borel subgroup and $W_0$ the Weyl group. Then the induced character $1_B^G$ is a sum of characters $\chi_{\lambda }$ of $G$ labeled by ${\lambda }\in {\mathop{\rm Irr}\nolimits}W_0$, and the multiplicity of $\chi_{\lambda }$ in $1_B^G$ is equal to ${\lambda }(1)$. In other words, $1_B^G=\sum_{{\lambda }\in{\mathop{\rm Irr}\nolimits}W_0}{\lambda }(1)\chi_{\lambda }$.
Recall that $G$ is called a Chevalley group if $ G ={{\mathbf G}}^F$, where ${{\mathbf G}}$ is a simple, simply connected algebraic group.
\[r01a\]Let $G$ be a Chevalley group, $B$ a Borel subgroup of $G$, $U$ the of $B$ and $T_0$ a maximal torus of $B$. Let $W_0$ be the Weyl group of $G$ as a group with $BN$-pair and let $d$ be the minimum degree of a non-linear character of $W_0$. Let $\chi$ be a character vanished at all $1\neq u\in U$ such that $(\chi,1_G)=(\chi,St)=0$. Suppose that the derived subgroup of $W_0$ has index $2$ and, for every non-trivial character $\beta\in {\mathop{\rm Irr}\nolimits}T_0$, either $|W_0\beta|\geq
d$ or $(\chi,\beta_B^G)=0 $, where $\beta_B$ is the inflation of $\beta$ to $B$. Then $\chi(1)\geq d\cdot |G|_p$.
Proof. By Corollary \[gg1\], $\chi(1)=(\chi,1_U^G)\cdot |G|_p$. By Lemma \[m12\], $(\chi,1_U^G)= \sum_{\beta_B}|W_0\beta|(\chi,
\beta_B^G)$. Suppose $\chi(1)<d\cdot |G|_p$. Then, by assumption, $(\chi, 1_U^G)=(\chi,1_B^G)$.
Let $\chi_{\lambda }$ be the constituent of $1_B^G$ corresponding to ${\lambda }\in{\mathop{\rm Irr}\nolimits}W_0$, see Lemma \[bb5\]. Let $m_{\lambda }$ be the multiplicity of $\chi_{\lambda }$ in $\chi$. By Lemma \[bb5\], $$(\chi,1_B^G)=\sum_{{\lambda }\in { {\mathop{\rm Irr}\nolimits}W}}m_{\lambda }\cdot {\lambda }(1)<d .$$ $m_{\lambda }=0$ if ${\lambda }(1)>1$. So $(\chi,1_B^G)=\sum_{{\lambda }\in
{\mathop{\rm Irr}\nolimits}W:{\lambda }(1)=1}m_{\lambda }$. It is well known [@CR2 Theorem 67.10] that $1_G$ and $St$ occur in $1_B^G$ with 1. As $W_0$ has exactly two one-dimensional representations, $\chi_{\lambda }$ is either $St $ or $1_G$. This is a contradiction, as neither $St$ nor $1_G$ is a constituent of $\chi$. $\Box$
Remarks. (1) The values of $d$ are given by Table 2. (2) The derived subgroup of $W_0$ has index 2 $G\in \{A_n(q),D_n(q),
E_6(q)$, $E_7(q)$, $E_8(q)$, ${}^2B_2(q)$, ${}^2G_2(q)\}$. (The last two groups are of BN-pair rank 1, and hence $|W_0|=2$.)
\[r01\]Let $G$ be a Chevalley group, $B$ a Borel subgroup of $G$, and $T_0$ a maximal torus of $B$. Let $W_0$ be the Weyl group of $G$ as a group with $BN$-pair and let $d$ be the minimum degree of a non-linear character of $W_0$. Let $\Phi\neq St$ be a PIM for $G$ with character $\chi$. Suppose that the derived subgroup of $W_0$ has index $2$ and $|W_0\beta|\geq d$ for every non-trivial character $\beta\in {\mathop{\rm Irr}\nolimits}T_0$, where $\beta_B$ is the inflation of $\beta$ to $B$. Then $c_\Phi\geq d$, unless $\Phi=\Phi_1$ and $G=SL(2,p)$, $SL(3,2)$ or ${}^2G_2(3)$.
Proof. We show that the hypothesis of Proposition \[r01a\] holds. Indeed, $\chi (u)=0$ for every $1\neq u\in U$, as $\chi$ is the character of a projective module. As $St$ is a character of a PIM, $St$ is not a constituent of $\chi$. Suppose that $(\chi,1_G)>0$. Then, by orthogonality relations [@Fe Ch. IV, Lemma 3.3], we have $\Phi= \Phi_1$ and $(\chi,1_G)=1$. So $\dim\Phi_1=|G|_p$. This contradicts a result of Malle and Weigel [@MW], unless $G= SL(2,p) $, $SL(3,2)$ or ${}^2G_2(3)$. With these exceptions, the result now follows from Proposition \[r01a\]. $\Box$
\[th1\] Let $G\in \{SL(n,q)$, $n>4$, $ Spin^{+}(2n,q)$, $q$ even, $n>3$, $E_6(q),E_7(q),E_8(q)\}$. Let $\Phi\neq St$ be a PIM with socle $S$.
$(1)$ If ${\mathop{\rm Soc}\nolimits}S|_B\neq 1_B$ then $ c_\Phi\geq m$, where $m=n,2n,27,28,120$, respectively.
$(2)$ Suppose that ${\mathop{\rm Soc}\nolimits}S|_B= 1_B$. Then $c_\Phi\geq d$, where $d=n-1,n-1,6,7,8$, respectively.
Proof. Let $W$ be the Weyl group of $G$.
\(1) By Proposition \[r2\], $c_\Phi\geq |W\beta|$. The lower bound $m$ for $|W\beta|$ is provided in Propositions \[r4\] and \[dn5\]. In particular, $|W\beta|\geq m$ unless $C_W(\beta)=W$. This implies $\beta=1_{T_0}$ by Lemma \[cm2\].
\(2) follows from Proposition \[r01\]. Indeed, let $d$ be the minimum dimension of a non-linear of $W$; by Table $2$, $d$ is as in the statement (2). As $m> d$, it follows from (1) that $|W\beta|> d$ for every $1_B\neq \beta\in {\mathop{\rm Irr}\nolimits}T_0$. For the groups $G$ in the statement the derived subgroup of $W$ is well known to be of index 2. So the hypothesis of Proposition \[r01\] holds, and so does the conclusion. $\Box$
\[or4\]Let $G=
{\rm Spin}^+\,(2n,q),$ $ n>4$, and let $\Phi\neq St$ be a PIM with socle $V=V_\mu|_G$, where $V_\mu$ is a ${{\mathbf G}}$-module with highest weight $\mu =a_1{\omega }_1+\cdots
+a_n{\omega }_{n}$. Then $c_\Phi\geq n-1$.
Proof. Let $J=\{1{,\ldots ,}n-1 \}$ or $ \{1{,\ldots ,}n-2,n \}$. Then $G_J\cong SL(n,q)$ and the highest weight of $V_j$ is $\nu:=a_1{\omega }_1+\cdots +a_{n-1}{\omega }_{n-1}$ or $a_1{\omega }_1+\cdots
+a_{n-2}{\omega }_{n-2}+a_{n}{\omega }_{n}$, respectively. So $V_J$ is not the Steinberg $FG_J$-module at least for one of these two cases. Furthermore, by Theorem \[th1\], if $V_J\neq St$ then $c_{\Psi}\geq n-1$, where $\Psi$ is a PIM for $G_J$ with socle $V_J$. By Lemma \[mp1\], $c_{\Phi} \geq c_{\Psi}$, and the result follows. $\Box$
Remark. If $G= SL(n,q)$ or Spin${}^+(2n,q)$ then it follows from Theorem \[th1\] and Corollary \[or4\] that $c_\Phi{ \rightarrow }\infty$ as $n{ \rightarrow }\infty$, provided $\Phi\neq St$. If $G\in \{Sp(2n,q), Spin(2n+1,q) $, Spin${}^-(2n+2,q)\}$ then a similar argument gives that $c_\Phi\geq n-1$ unless $a_1=\cdots =a_{n-1}=q-1$ (respectively, $c_\Phi\geq n-2$ unless $a_1=\cdots =a_{n-2}=q-1$). However, this does not lead to the same conclusion as above. A similar difficulty arises for the groups $SU(2n,q)$ and $SU(2n+1,q)$.
\[tp1\]Let $G=G(q)$ be a non-twisted Chevalley group, and let $V_\mu$ be a ${{\mathbf G}}$-module with $\mu=\sum a_i{\omega }_i \in \Delta(G)$. Let $\Phi$ be a PIM of $G$ with socle $V_\mu|_G$. Let $J$ be a subset of nodes on the Dynkin diagram of ${{\mathbf G}}$, not adjacent to each other. $a_i\neq 0,q-1$ for $i\in J$. Then $c_\Phi\geq
2^{|J|} $.
Proof. As the nodes of $J$ are not adjacent, it follows that ${{\mathbf G}}_J$ is the direct product of $|J|$ copies of $G_i\cong
SL(2,F)$ for $i\in G$. Furthermore, the Smith correspondent $\sigma_{G,G_J}$ of $V_\mu$ is the tensor product of $FG_i$-modules with highest weight $a_i{\omega }_1$. Let $\Psi=\pi_{G,G_J}(\Phi)$ be the parabolic descent of $\Phi$ to $G_J$. Then $c_{\Psi}\geq 2^{|J|} $ by Lemmas \[sr1\], \[dp8\], and $c_{\Phi} \geq c_{\Psi}$ by parabolic descent. $\Box$
The proof of Proposition \[tp1\] illustrates the fact that the parabolic descent from non-twisted groups $G(q)$ for $q=p$ to minimal parabolic subgroups (distinct from the Borel subgroup) does not work. Indeed, in this case $G_J\cong A_1(p)$; if all coefficients $a_i$ are equal to 0 or $p-1$ then $c_\Psi=1$. One observes that it is possible to run the parabolic descent to subgroups $G_J\cong SL(3,p)$, obtaining tensor product of with highest weight $(p-1){\omega }_1$ or $(p-1){\omega }_2$, and then use known results for PIM’s with such socles.
The Ree groups
==============
In this section we prove Theorem \[wt1\] for Ree groups $G={}^2G_2(q)$, $q^2=3^{2f+1}>3$. Note that $|G|_p=q^{12}=3^{6(2f+1)}$.
Note that if $f=0$ then $G\cong {\rm Aut}\,\SL(2,8)$. For this group the decomposition numbers are known, and $c_\phi=1$ $\Phi=\Phi_1$.
\[re1\] Let $\chi$ be a character of degree $q^6$ vanishing at all unipotent elements of $G$. Suppose that $(\chi,1_G)=0$. Then $\chi=St$.
Proof. Suppose the contrary. Then $(\chi, 1_U^G)=1$ (Corollary \[gg1\]), so there is exactly one $\tau$ of $\chi$ common with $1_U^G$. Recall that $1_U^G=\sum_{\beta\in {\mathop{\rm Irr}\nolimits}T_0 }\beta_B^G$, where $\beta_B$ is the inflation of $\beta$ to $B$. By Proposition \[r2b\], $\tau\in
\beta_B^G $ implies that $\beta$ is $W_0$-invariant. (Or straightforwardly, the degree of every $\beta_B^G$ is $|G:B|=|G|_p+1$, so we can ignore the $\beta$’s with $\beta_B^G$ irreducible. Observe that $\beta_B^G$ is reducible $C_{W_0}(\beta)=W_0$ [@St Theorem 47].) As $T_0$ is cyclic of order $q^2-1$ and $|W_0|=2$, the non-identity element of $W_0$ acts on $T_0$ as $t{ \rightarrow }t{^{-1}}$ ($t\in T_0$). So $C_W(\beta)\neq 1$ implies $\beta^2=1$. In this case $(\beta_B^G,\beta_B^G)=|C_W(\beta)|=2$ ([@St Ex.(a) after Theorem 47]), so there are two constituents. If $\beta=1_{T_0}$ then $\beta_B^G=1_G+St$. If $\beta^2=1_{T_0}\neq \beta$ then the character table in [@Wa] leaves us with exactly two possibilities for $\tau(1)$, which are $d_1=q^4-q^2+1$ and $d_2=q^6+1-d_1=q^6-q^4+q^2$.
The constituents of $\chi$ that are not in $1_U^G$ are cuspidal. (Indeed, every proper parabolic subgroup of $G$ is a Borel subgroup, so every non-cuspidal character has a 1-dimensional constituent under restriction to $U$. So they are in $1_U^G$.) Malle and Weigel [@MW p. 327] recorded the cuspidal character degrees that are less than $|G|_p$, and each degree is greater than $q^6-d_2$. Therefore, $\tau(1)=d_1$. By Lemma \[gg1\], there is exactly one regular character $\gamma$ occurring as a constituent of $\chi$, and $\gamma$ is not a unipotent character (as $St$ is the only unipotent regular character of $G$). The two characters listed in [@MW p. 327] are unipotent (see [@Ca p. 463] for the degrees of unipotent characters of $G$). The remaining regular character degrees are $d_3=(q^2-1)(q^4-q^2+1)=q^6-2q^4+2q^2-1$, and
$d_4=(q^4-1)(q^2-q\sqrt{3}+1)=q^6-q^5\sqrt{3}+q^4-q^2+q\sqrt{3}-1$.
Thus, $\gamma(1)=d_3$ or $d_4$. As $(\chi, 1_U^G)=1$, we have $(\tau,
\chi)=1$. Note that $q^6-d_1-d_3=q^4-q^2$. Other characters which may occur in the decomposition of $\chi$ are of degree $d_5=q(q^4-1)/\sqrt{3}$, $d_6=q(q^2-1)(q^2-
q\sqrt{3}+1)/2\sqrt{3}$ or $d_7=q(q^2-1)(q^2+
q\sqrt{3}+1)/2\sqrt{3}$, see [@Wa]. Each of these degrees is greater than $q^4-q^2$, so $\gamma(1)=d_4$. So
$$q^6=(q^4-q^2+1)+ (q^6-q^5\sqrt{3}+q^4-q^2+q\sqrt{3}-1)+$$ $$\label{e1}
+\frac{aq(q^4-1)}{\sqrt{3}}+ \frac{b q(q^2-1)(q^2+ q\sqrt{3}+1)}{2\sqrt{3}}
+\frac{cq(q^2-1)(q^2- q\sqrt{3}+1)}{2\sqrt{3}},$$
where $a,b,c\geq 0$ are integers, and $a+b+c>0$.
Cancelling the equal terms and dropping the common multiple $q^2-1$, we get
$$\label{e3}
a(q^2+1) + (q^2+1)\frac{b+c}{2}-3(q^2+1)=q\sqrt{3}\cdot (\frac{c-b}{2}-2).$$
Suppose first that $a>2$. Then $(q^2+1)\frac{b+c}{2}\leq
q\sqrt{3}\cdot (\frac{c-b}{2}-2) $, and hence $e:=c-b>0$. Then we have $(q^2+1)b+e\cdot (\frac{q^2+1}{2}-\frac{q\sqrt{3}}{2})\leq
-2q\sqrt{3}$. One easily check that $q^2+1-q\sqrt{3}>0$, which is a contradiction. So $a\leq 2$.
Let $X,Y,m$ be as in the character table of $G$ in [@Wa], so $m=q/\sqrt{3}$, $|X|= 3$ and $|Y|=9$. Then one observes from [@Wa] that the characters of the same degree have the same value at $X$, and also at $Y$. Therefore, $\chi(g)=\tau(g)+\gamma(g)+a\xi_5(g)+b\xi_6(g)+c\xi_7(g)=0$, where $g\in\{X,Y\}$ and $\xi_j(g)$ is the value of any character of degree $d_j$ ($j=5,6,7$). By [@Wa], $\tau (Y)=-\gamma (Y)=1$, $\xi_5(Y)=-m$, $\xi_6(Y)=\xi_7(Y)=m$. So $a\xi_5(g)+b\xi_6(g)+c\xi_7(g)=-am+(b+c)m=0$, whence $a=b+c$. So (\[e3\]) simplifies to
$$\label{e2}
3(q^2+1)((b+c-2)=q\sqrt{3}\cdot (c-b-4).$$
Recall that $a=b+c\leq 2, $ and hence $b+c\neq 0$. If $b+c=2$ then $c=b+4$, which is a contradiction. If $b+c=1$ then the left hand side in $(\ref{e2})$ is not divisible by 9, and hence $q\sqrt{3}=3$. This is not the case. $\Box$
\[re2\] Let $G={}^2G_2(3^{2f+1})$, $q^2=3^{2f+1}$. Let $\Phi\neq St$ be a PIM, and let $\chi$ be the character of $\Phi$. Then $\chi( 1)>q^6$.
Proof. If $(\chi,1_G)=0$, the result follows from Lemma \[re1\]. Otherwise $\Phi=\Phi_1$, and the result for this case follows from [@MW]. $\Box$
Proposition \[re2\] together with certain known results implies Theorem \[wt1\] for groups of BN-pair rank 1. These are $A_1(q),
{}^2A_2(q)$, ${}^2B_2(q)$, ${}^2G_2(q).$ The groups ${}^2G_2(q)$ have been treated above. Below we quote known results on minimal PIM dimensions for the remaining groups of BN-pair rank 1.
\[sr1\][[@Sr]]{} Let $G=SL(2,q)$, $q=p^k$, and let $\Phi\neq St$ be a PIM of $G$ with socle $V_{m{\omega }_1}|_G$, where $V_{m{\omega }_1}$ is an $SL(2,F)$-module with highest weight $m{\omega }_1$ for $0<m<q-1$. Let $m=m_0+m_1p+ \cdots+ m_{k-1}p^{k-1}$ be the $p$-adic expansion of $m$, and $r$ the number of digits $m_i$ distinct from $p-1$. Then $c_\Phi=2^r$. In addition, $c_{\Phi_1}=2^k-1$. In particular, if $q=p$ then $c_\Phi=2$ unless $\Phi=\Phi_1$ with $c_{\Phi_1}=1$.
\[sr1a\][[@CF1]]{} Let $G={}^2B_2(q)$, $q^2>2$, and let $\Phi\neq St$ be a PIM of $G$. Then $c_\Phi\geq 4$.
\[su0\] Let $G=SU(3,p)$, $p>2$, and let $\Phi\neq St$ be a PIM of $G$. Then $ c_\Phi \geq 3 $.
Proof. If $p\equiv 1 \,({\rm mod}\, 3)$ then the PIM dimensions are listed in [@Hu90 Table 1]. The same holds if $p=3$, see the GAP library. Let $p\equiv 2 \,({\rm mod}\, 3)$. Then the formulas in [@Hu81 p. 10], lead to the same conclusion.
\[ra1\] Theorem $\ref{wt1}$ is true for groups of BN-pair rank $1$.
Proof. The result follows from Proposition \[re2\] and Lemmas \[sr1\], \[sr1a\] and \[su0\]. $\Box$
Groups $SU(4,p)$ and ${}^3D_4(p)$
=================================
In this section we prove Theorem \[wt1\] for the above groups.
Some general observations {#74}
-------------------------
Prior dealing with the cases where $G=SU(4,p)$ and ${}^3D_4(p)$, we make some general comments which facilitate computations. These are valid for $q$ in place of $p$, so we do not assume $q$ to be prime until Section 8.2.
Let ${\mathbf H}$ be a connected reductive algebraic group and $H={\mathbf H}^{Fr}$. The Deligne-Lusztig theory partitions characters of $H$ to series usually denoted by ${\mathcal E}_s$, where $s$ runs over representatives of the semisimple conjugacy classes of the dual group $H^*$. (See [@DM p. 136], where our ${\mathcal E}_s$ are denoted by ${\mathcal E}({{\mathbf G}}^F,
(s)_{{{\mathbf G}}^{*F^*}}).)$ The duality also yields a bijection $T{ \rightarrow }T^*$ between maximal tori in $H$ and $H^*$ such that $T^*$ can be viewed as ${\mathop{\rm Irr}\nolimits}T$, the group of linear characters of $T$. If $H=U(n,q)$ or ${}^3D_4(q)$ then $H^*\cong H$.
\[dd4\]Let $H$ be a finite reductive group, $B$ a Borel subgroup of $H$ and $T_0\subset B$ a maximal torus. Let $B=T_0U$, where $U$ is the unipotent radical of $B$. Let $\beta\in{\mathop{\rm Irr}\nolimits}T_0$, and $\beta_B$ its inflation to $B$. Let $s\in T_0^*\subset H^*$ be the element corresponding to $\beta$ under the duality mapping ${\mathop{\rm Irr}\nolimits}T_0{ \rightarrow }T_0^*$.
$(1)$ ${\mathcal E}_s$ contains the set of all constituents of $\beta_B^H$. In addition, if an character $\chi$ of $H$ is a constituent of $1_U^H$ then $\chi\in{\mathcal E}_s$ for some $s\in
T_0^*$.
$(2)$ $\beta_B^H$ contains a regular character and a semisimple character of ${\mathcal E}_s$.
$(3)$ Suppose that ${\mathcal E}_1$ coincides with the set of all constituents of $1_B^G$ and $s\in Z(H^*)$. Then ${\mathcal
E}_s$ coincides with the set of all constituents of $\beta_B^G$.
Proof. (1) By [@Ca Proposition 7.2.4], $\beta_B^H=R_{T_0,\beta} $, where $R_{T_0,\beta} $ is a generalized character of $H$ defined by Deligne and Lusztig (called usually a Deligne-Lusztig character). By definition [@DM p. 136], ${\mathcal E}_s$ contains the set $\{\chi\in
{\mathop{\rm Irr}\nolimits}H:(\chi,R_{T_0,\beta} )\neq 0\}$, which coincides with $\{\chi\in {\mathop{\rm Irr}\nolimits}H:(\chi,\beta_B^H )\neq 0\}$. This implies the first claim. (Note that this statement is a special case of [@Ko Proposition 4.1].) The additional claim follows as $1_U^H=\sum_{\beta\in{\mathop{\rm Irr}\nolimits}T} \beta_B^H$.
\(2) This is a special case of [@Ko Proposition 5.1 and Corollary 5.5].
\(3) By [@DM Proposition 13.30], there exists a one-dimensional character $\hat s$ of $H$ such that the characters of ${\mathcal E}_s$ are obtained from those of ${\mathcal E}_1$ by tensoring with $\hat s$, and $\hat s|_{T_0}=\beta$. It follows that ${\mathcal E}_s$ is the set of all constituents of $1_B^H\otimes \hat s$. As $\hat s|_B=\beta_B$, we have $1_B^H\otimes \hat s=\beta_B^H$, as required. $\Box$
Remark. (3) can be stated by saying that if ${\mathcal E}_1$ coincides with the Harish-Chandra series of $1_B^H$ and $s\in
Z(H^*)$ then ${\mathcal E}_s$ coincides with the Harish-Chandra series of $\beta_B^H$.
\[gg4\] $(1)$ Let $s\in G^*$ be a semisimple element and ${\mathcal E}_s$ the corresponding Lusztig class. Let $\Gamma$ be a Gelfand-Graev character of $G$. Then ${\mathcal E}_s$ contains exactly one character common with $\Gamma$.
$(2)$ Suppose that $s\in T^*_0={\mathop{\rm Irr}\nolimits}T_0\cong T_0$. Let $\beta\in{\mathop{\rm Irr}\nolimits}T_0$ correspond to $s$. Then every regular character of ${\mathcal E}_s$ is a constituent of $\beta_B^G$.
Proof. (1) is well known if ${{\mathbf G}}$ has connected center. In general this follows from results in [@DM Section 14]. Indeed, the character $\chi_{(s)}$ introduced in [@DM 14.40] is a sum of regular characters [@DM 14.46]. By definition of $\chi_{(s)}$, its constituents belong to ${\mathcal E}_s$. In addition, every regular character of $G$ is a constituent of $\chi_{(s)}$ [@DM 14.46]. Therefore, every regular character of ${\mathcal E}_s$ is a constituent of $\chi_{(s)}$. As $(
\chi_{(s)}, \Gamma)=1$ [@DM 14.44], the claim follows.
\(2) By [@Ca 7.4.4], $\beta_B^G=R_{{{\mathbf T}}_0,\beta}$, where $R_{{{\mathbf T}}_0,\beta}$ is the Deligne-Lusztig character. Note that $(R_{{{\mathbf T}}_0,\beta}, \Gamma)=1$ (indeed, it follows from the reasoning in the proof of Lemma 14.15 in [@DM 14.44] that $(R_{{{\mathbf T}}_0,\beta}, \Gamma)=\pm 1$; as $\beta_B^G$ is a character, $(\beta_B^G,\Gamma)\geq 0$). (This also follows from [@Ko2 Proposition 2.1].) So the claim follows from (1).$\Box$
\[nf1\] Let $G=SU(4,q)$, resp., ${}^3D_4(q)$, and let $J=\{1,3\}$, resp., $\{1,3,4\}$ be a set of nodes on the Dynkin diagram of the respective algebraic group ${{\mathbf G}}$. Let $\mu=(q-1)\sum_{j\in J}{\omega }_j$, $V_\mu$ an ${{\mathbf G}}$-module of highest weight $\mu$, and $V=V_\mu |_G$. Then $S:={\sigma }_{G,L}(V)$ is of defect $0$. Let $\Psi$ be a PIM with socle $S$ and let ${\lambda }$ be the character of the lift of $S$. Then $({\lambda }^{\# G},\chi)>0$. In addition, if $c_\Phi=1$ then $({\lambda }^{\# G},\tau)=1$.
Proof. The statement about the defect of $S$ and the inequality $({\lambda }^{\# G},\chi)>0$ is proved in Proposition \[yg8\](3). Moreover, it is shown there that ${\lambda }$ is a constituent of $1_{B\cap L}^L$. Therefore, by transitivity of Harish-Chandra induction [@CR2 Proposition 70.6(iii)], $1_B^G={\lambda }^{\#
G}+{\lambda }'$, where ${\lambda }'$ is some character of $G$. So $\tau$ is a common constituent of ${\lambda }^{\# G}$ and $1_U^G$; as $(\tau,1_U^G)=1$, it follows that $(\tau, {\lambda }^{\# G})=1$. $\Box$
\[u44\]Let $H=U(4,q)$, $B$ a Borel subgroup of $H$ and $T_0\subset B$ a maximal torus. Let $B=T_0U$, where $U$ is the unipotent radical of $B$. Let $\beta\in{\mathop{\rm Irr}\nolimits}T_0$, and $\beta_B$ its inflation to $B$. Let $s\in T_0^*\subset H^*$ be the element corresponding to $\beta$ under the duality mapping ${\mathop{\rm Irr}\nolimits}T_0{ \rightarrow }T_0^*$.
$(1)$ ${\mathcal E}_1$ coincides with the set of all constituents of $1_B^H$.
$(2)$ ${\mathcal E}_s$ coincides with the set of all constituents of $\beta_B^H$.
Proof. (1) Note that $H$ has no cuspidal unipotent character, see [@Ca p. 457]. every unipotent character of $H$ is a constituent of $1_B^G$. This is equivalent to the statement.
\(2) Observe that $H\cong H^*$. In notation of [@No], $A_1,A_9,B_1,C_1,C_3$ are the only conjugacy classes that meet $T^*_0$. If $s\in A_1$ then $s\in Z(H^*)$, and the claim follows Lemma \[dd4\](2). If $s$ belongs to $B_1,C_1,$ or $C_3$ then ${\mathcal E}_s$ contain only regular and semisimple characters, so the claim follows from (2). Let $s\in A_9$. By [@DM 13.23], $|{\mathcal E}_s|$ is equal to the number of unipotent characters of $C_{H^*}(s)$. This group is isomorphic to $U (2,q)\times U(2,q)$, so the number of unipotent characters equals 4. By Lemma \[dd4\](1), the constituents of $\beta_B^H$ are contained in ${\mathcal E}_s$. We show that $\beta_B^H$ has 4 distinct constituents.
Indeed, $(\beta_B^H,\beta_B^H)=|W_s|$, where $W_s$ is the stabilizer of $s$ in $W_0$ (see Section 4 for the definition of $W_0$). It is easy to observe that $W_s$ is an elementary abelian group of order 4. By (2), the regular and semisimple characters occur as constituents of $\beta_B^H$. As $(\beta_B^H,\beta_B^H)=4$ is the sum of squares of the multiplicities of the constituents of $\beta_B^H$, it follows that there are exactly four distinct constituents. This equals $|{\mathcal E}_s| $, the result follows.$\Box$
Let $\Phi$ be a PIM for $G$ with character $\chi$. Suppose $c_\Phi =1$. Then, by Corollary \[gg1\], $(\chi, 1_U^G)=(\chi, \Gamma)=1$, which means that $\chi$ has exactly one constituent common with $1_U^G$, and exactly one constituent common with every Gelfand-Graev character $\Gamma$. Denote them as $\tau$ and $\gamma$, respectively. By Proposition \[yg8\], $(\chi, 1_B^G)>0$; as $(\chi, 1_U^G)=1 $, it follows that $\tau$ is constituent of $1_B^G$. (This also follows from Proposition \[r2b\].) Recall that the constituents of $\Gamma$ are called regular characters in the Deligne-Lusztig theory. In general there are several Gelfand-Graev characters, however, if $G=U(n,q)$ or ${}^3D_4(q)$ then $G$ has a single Gelfand-Graev character (see [@DM 14.29]). Note that by Lemma \[de4\], it suffices to prove that $c_\Phi>1$ for the group $U(4,q)$ in place of $SU(4,q)$.
In order to prove Theorem \[wt1\] for the groups $G=SU(4,p)$ and ${}^3D_4(p)$ we first determine $\tau(1)$ and $\gamma(1)$, and observe that $\tau\neq \gamma$ by Lemma \[dd4\]. Next we express $\chi=\tau+\gamma+\sum \nu_i$, where $\nu_i$ runs over the characters that are neither regular nor in $1_U^G$. In particular, $\nu_i(1)\leq \chi(1)-\tau(1)-\gamma(1)$. As the character table of $G$ is available in [@No] for $U(4,q)$ and in [@DM] for ${}^3D_4(q)$, we obtain a contradiction by inspecting all the possibilities.
The reasoning below does not use much from modular theory. In fact, we prove the following. Let $G$ be either $SU(n,q)$ or ${}^3D_4(p)$, and let $\chi$ be an ordinary character vanishing at all non-semisimple elements of $G$. Let $L$ be a Levi subgroup of $G$ whose semisimple part is isomorphic to $SL(2,q^a)$, where $a=2$ for the former group and $a=3$ for the latter one. Suppose that $\overline{\chi}_L$ is of defect zero. Then $\chi(1)>|G|_p$. (Of course one has to replace $p$ by $q$, and use Lemma \[b1\] in place of Corollary \[gg1\].)
The unitary groups $G=SU(4,p)$
------------------------------
Note that $|G|=p^6(p^4-1)(p^3+1)(p^2-1)$. Let $\Phi$ be a PIM with socle $V$ and character $\chi$. Arguing by contradiction, we suppose that $\dim\Phi=|G|_p$, that is, $c_\Phi=1$.
Let $J=\{1,3\}$ be a set of nodes at the Dynkin diagram of ${{\mathbf G}}=SL(4,F)$. Then $G_J\cong SL(2,p^2)$, see Example prior Lemma \[st0\]. Let $P=P_J$ be a parabolic subgroup of $G$ corresponding to $J$ and $L=L_J$ its Levi subgroup. Let $S={\sigma }_{G,P_J}(V)$ be the Smith-Dipper correspondent of $V$. By Corollary \[fg8\], $S|_{G_J}$ is the Steinberg module. So $S$ is an $FL$-module of defect 0, and hence lifts to characteristic 0. Let ${\lambda }$ be the character of the lift so ${\lambda }(1)=p^2$. By Lemma \[nf1\], $\tau$ is a constituent of ${\lambda }^{\# G}$.
In order to determine $\tau(1)$ we first decompose ${\lambda }^{\# G}$ as a sum of constituents.
\[u4st\]$(1)$ ${\lambda }^{\# G}=St+{\sigma }+{\sigma }'$, where ${\sigma },{\sigma }'$ are characters of $G$ such that ${\sigma }(1)=p^2(p^2+1)$ and ${\sigma }'(1)=p^3(p^2-p+1)$.
$(2)$ Let $\tau$ be a common constituent for $\chi$ and $
1_U^G$. Then $\tau={\sigma }'$, where ${\sigma }'$ is as in $(1)$. In particular, $\tau(1)=p^3(p^2-p+1)$.
Proof. (1) The degrees of constituents of $1_B^G$ are given in [@CKS Proposition 7.22]. As ${\lambda }^{\#
G}(1)=p^2|G:P_J|=p^2(p+1)(p^3+1)$, this implies (1).
\(2) By Corollary \[gg1\], $(\chi,1_U^G)=1$; as mentioned in [@CKS Proposition 7.22], ${\sigma }$ occurs in $1_B^G$ with 2, so $(\chi, {\sigma })=0$, and hence $\tau={\sigma }'$, as stated. $\Box$
We need to write down the degrees of the regular characters of $H:=U(4,p)$. By the Deligne-Lusztig theory, the regular characters of $H$ are in bijection with semisimple conjugacy classes in the dual group $H^*\cong H=U(4,p)$. So we write $\rho_s$ for the regular characters of $H$ corresponding to $s$, a representative of a semisimple conjugacy classes in $H$. Furthermore, $\rho_s(1)=|C_H(s)|_p\cdot |G:C_H(s)|_{p'}$.
The characters of $H$ can be partitioned in classes consisting of characters of equal degree. This has been done in Nozawa [@No], who computed the characters of $H$. Similarly, the elements $g\in H$ can be partitioned in classes consisting of all elements $g'$ such that $C_H(g')$ is conjugate to $C_H(g)$. Below we use Nozawa’s notation $A_1{,\ldots ,}A_{14}, B_1,\ldots $ for such classes. (So each class in question is a union of conjugacy classes.) In Table 4 below the first column lists the semisimple conjugacy classes of $H$ with conjugate centralizers $C_H(s)$, and the second column lists $|C_H(s)|$. In order to extract from [@No] the regular characters we use the above formulas for their degrees. The third column lists $\rho_s(1)$. For reader’s convenience we also identify $\rho_s$ with notation in [@No] in the fourth column. For instance, $\chi_{11}(s)$ for $s\in A_1$ are characters of the same degree $p^6$ depending on a parameter $s$. In computations below we do not use $s$ to parameterize the characters; instead we write $\chi_{11}(1)=p^6$ to tell that [*every*]{} character from the set $\chi_{11}$ takes value $p^6$ at $1\in H$. (Note that the degrees of regular characters of $U(4,q)$ are as in Table 4 with $q$ in place of $p$.)
TABLE 4: Degrees of the regular characters of $H=U(4,p)$
[|l|l|l|c| ]{}
$~~~~~~~~~~~~~~~~~s$& $~~~~~~|C_H(s)| $ &$~~~~~~~~~\rho_s(1)$ &$\rho_s$ in [@No]\
$A_1 $ & $~~~~~~~~~|H|$ & $~~ p^6$ & $\chi_{11}(s) $ $A_6 $ &$~p^3(p+1)^2(p^2-1)(p^3+1)$&$p^3(p-1)(p^2+1)$&$\chi_{13}(s)$\
$A_9 $& $~p^2(p+1)^2(p^2-1)^2$& $~p^2(p^2+1)(p^2-p+1)
$&$\chi_{20}(s)$\
$A_{12} $ & $~p(p+1)^3(p^2-1)$& $~p(p-1)(p^2-p+1)(p^2+1)$&$\chi_{15}(s)$\
$A_{14} $& $(p+1)^4$& $(p-1)^2(p^2-p+1)(p^2+1)$&$\chi_{10}(s)$\
$B_1 $ &$~p(p+1)(p^2-1)^2$& $p(p^2+1)(p^3
+1)$&$\chi_{8}(s)$\
$B_3 $& $(p+1)^2(p^2-1)$& $
(p-1)(p^2+1)(p^3+1)$&$\chi_{6}(s)$\
$C_1$&$~p^2(p^2-1)(p^4-1)$&$p^2(p+1)(p^3+1)$&$\chi_{4}(s)$\
$C_3 $&$(p^2-1)^2$&$(p+1)(p^2+1)(p^3+1)$&$\chi_{2}(s)$\
$D_1$ &$(p+1)(p^3+1)$& $(p^2-1)(p^4-1) $&$\chi_{9}(s)$\
$E_1$& $~p^4-1 $& $(p+1)(p^3+1)(p^2-1)$&$\chi_{5}(s)$\
\[u4q\]Let $G=SU(4,p)$ and $\Phi\neq St$ be a PIM. Then $\dim\Phi>|G|_p$.
Proof. (1) Let $V$ be the socle and $\chi$ the character of $\Phi$. Then $V=V_\mu|_G$, where $V_\mu$ is a ${{\mathbf G}}$-module with $p$-restricted $\mu$. Suppose that $c_\Phi=1$; then Lemma \[sd7\] implies that $\mu=(p-1)({\omega }_1+ {\omega }_3)$.
We have explained in Section \[74\] that $\chi=\tau+\gamma +\sum \nu_i$, where $\tau$ is constituent of $1_U^G$, $\gamma$ is a regular character and $\nu_i$ are some characters that are neither regular nor in $1_U^G$. In particular, $\nu_i(1)\leq \chi(1)-\tau(1)-\gamma(1)$. We keep notation of Section \[74\].
\(2) By Lemma \[de4\], $\Phi=\Psi|_G$, where $\Psi$ is some PIM for $H$. In particular, $\dim \Psi =p^6=|H|_p$. Let $\chi$ be the character of $\Psi$, so $\chi(1)=p^6$. By Corollary \[gg1\], $\chi$ must contain exactly one regular character $\rho$.
\(3) If $(\chi,\gamma)>0$ for a regular character $\gamma$ of $H$ then $\gamma(1)=p(p-1)(p^2+1)(p^2-p+1)$ or $(p-1)^2(p^2+1)(p^2-p+1)$.
Indeed, set $f=p^6-\tau(1)=p^3(p-1)(p^2+1)$. Then $\rho(1)\leq f$. One easily checks that for characters $\rho_s$ in Table 4 $\rho_s(1)>f$ unless $s\in\{A_{6},A_{12},A_{14}\}$.
Observe that $\rho_s(1)=f$ for $s\in A_{6}$. However, $\gamma$ cannot coincide with $\rho_s$ for this $s$, as otherwise $\tau=\tau'|_G$ for some character $\tau'\in{\mathop{\rm Irr}\nolimits}H$ and $\chi=\tau'+\rho_s$; by inspection in [@No], there are non-semisimple elements $g\in G\subset H$ such that $\tau(g)+\rho_s(g)\neq 0$ for $s\in A_{6}$, while $\chi(g)=0$ as $\chi$ is the character of a projective module. Thus, $s\in\{A_{12},A_{14} \} $, so $\rho_s\in \{\chi_{15},\chi_{10}\}$, and (3) follows.
\(4) Thus, $\gamma\in\{\chi_{10},\chi_{15} \} $. Note that $\chi_{10}(1)<\chi_{15}(1)$. Set
$e_1=f-\chi_{15}(1)=p(p^2+1)(p-1)^2$ and
$e_2=f-\chi_{10}(1)=(p^2+1) (p-1)(2p^2-2p+1)$.
Then $\nu_i(1)\leq e_1$ or $e_2$. It follows from Lemma \[u44\](3) that $\cup_{s\in T_0^*
}{\mathcal E}_s= {\mathop{\rm Irr}\nolimits}1_U^G$. Therefore, $\nu_i\in {\mathcal E}_s$ for some semisimple elements $s\in H^*$ for $s\notin T_0^*$.
We recall some facts of character theory of Chevalley groups.
Observe that $A_1,A_9,B_1,C_1,C_3$ are the only conjugacy classes that meet $T_0$. So $\nu\in {\mathcal E}_s$ and $s\notin
\{A_1,A_9,B_1,C_1,C_3\}$. It is well known that $ |{\mathcal
E}_s|=1 $ then $(|C_H(s)|,p)=1$ (this follows for instance from the formulas for regular and semisimple characters in ${\mathcal E}_s$, see [@Ca Ch. 8]). So, if $(|C_H(s)|,p)=1$ then the regular character is the only character of ${\mathcal
E}_s$. This happens $s\in \{A_{14}, B_3,C_3,D_1,E_1\}$, see Table 4. As $\rho$ is the only regular character that is a constituent of $\chi$, in our case $s\notin \{A_{14},
B_3,C_3,D_1,E_1\}$. We are left with the cases $s\in\{A_6,A_{12}\}$. Let $S$ denote the subgroup of $C_H(s)$ generated by unipotent elements. By the Deligne-Lusztig theory, the characters in ${\mathcal E}_s$ are of degree $d\cdot
|H:C_H(s)|_{p'}$, where $d$ is the degree of a unipotent character of $S$. If $s\in A_{12}$ then $S\cong SL(2,p)$, and if $s\in A_6$ then $S\cong SU(3,p)$, see [@No]. The degrees of unipotent characters of these groups are well known to be $1,p$ and $1,p(p-1),p^3$, respectively. Therefore, non-regular characters in ${\mathcal E}_s$ are of degrees $\chi_{19}(1)=(p-1)(p^2+1)$ and $\chi_{17}(1)=p(p-1)^2(p^2+1)$ for $s\in A_6$, and of degree $\chi_{16}(1)=(p-1)(p^2-p+1)(p^2+1)$ for $s\in A_{12}$. Note that $\chi_{16}(1)> \chi_{17}(1)=e_1>\chi_{19}(1)$.
For further use, we write down some character values extracted from [@No]. Let $g\in A_{10}$, $h\in A_{11}$ with the same semisimple parts, and $\chi_i$ are as in the last column of Table 4.
$~~~~~~~~~~~~~(*) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \chi_i(g)+(p-1)\chi_i(h)=
\begin{cases}0& for ~~i=10~~and ~~i=17\\
\pm p&for ~~i=13\\
\pm 1&for ~~i=16~~and ~~i=19\\
\end{cases}
$
Note that the absolute value of $\chi_i(g)+(p-1)\chi_i(h)$ is independent from the choice of an individual character in the set $\chi_i$. This is the reason to consider $\chi_i(g)+(p-1)\chi_i(h)$ instead of computing $\chi$ at $g$ or $h$ in some formulas below.
Case 1. $\gamma (1)=\chi_{15}(1)$.
The $\tau$ above is denoted by $\chi_{13}$ in [@No]. Note that $\chi_{13}(1)+\chi_{15}(1)+\chi_{17}(1)=p^6$. However, $\chi$ is not the sum of characters from these sets as $\chi_{13}+\chi_{15}+\chi_{17}$ does not vanish at elements of some class in $A_{11}$.
It follows that the only possibility is $\chi=\chi_{13}+\chi_{15}+p(p-1) \chi_{19}$. (This is not an actual formula, it only tells that $\chi$ is the sum of $p(p-1)$ characters from the set $\chi_{19}$ and one character from each set $\chi_{13}$ and $\chi_{15}$.) Inspection of [@No] shows that this is false. (One can compute the values of these characters at a regular unipotent element of $H$; it takes zero values for every character from $\chi_{13}$ and $\chi_{15}$ and value $-1$ for every character from $\chi_{19}$. So we have a contradiction.)
Case 2. $\gamma(1)=\chi_{10}(1)$.
Then $e_2=(p^2+1) (p-1)(2p^2-2p+1)=x\cdot \chi_{16}(1)+y\cdot \chi_{17}(1)+z\cdot \chi_{19}(1)$
$=x(p-1)(p^2-p+1)(p^2+1)+yp(p-1)^2(p^2+1)+z(p-1)(p^2+1)$,
where $x,y,z\geq 0$ are integers. Dividing by $(p^2+1)(p-1)$, we get:
$2p^2-2p+1=x(p^2-p+1)+yp(p-1)+z$, or $p(p-1)(2-x-y)=z+x-1$.
An obvious possibility is $x=y=0$, $z=2p^2-2p+1$. Suppose $x+y\geq 1$. If $x+z=1$ then either $z=0, x=y=1$ or $z=1,x=0,y=2$.
Suppose $x+z\neq 1$. Then $0\neq x+z-1\equiv 0\mod p(p-1)$, so $x+z-1=kp(p-1)$ for some integer $k>0$. This implies $2=x+y+k$, so $x+y\leq 1$, and hence $x+y=1$, $k=1$. So either $x=1,y=0$, $z=p(p-1)$ or $x=0,y=1,z=p(p-1)+1$.
Altogether we have five solutions.
\(1) $e_2=(2p^2-2p+1)\chi_{19}(1)$, and $\chi=\chi_{13}+\chi_{10}+(2p^2-2p+1)\chi_{19}$;
\(2) $e_2=\chi_{16}(1)+\chi_{17}(1)$ and $\chi=\chi_{13}+\chi_{10}+\chi_{16}+\chi_{17}$;
\(3) $e_2=2\chi_{17}(1)+\chi_{19}(1)$ and $\chi=\chi_{13}+\chi_{10}+2\chi_{17}+\chi_{19}$;
\(4) $e_2=\chi_{16}(1)+(p^2-p)\chi_{19}(1)$ and $\chi=\chi_{13}+\chi_{10}+\chi_{16}+(p^2-p)\chi_{19}$;
\(5) $e_2=\chi_{17}(1)+(p^2-p+1)\chi_{19}(1)$ and $\chi=\chi_{13}+\chi_{10}+\chi_{17}+(p^2-p+1)\chi_{19}$.
As above, here $(2p^2-2p+1)\chi_{19}$ means the sum of $(2p^2-2p+1)$ characters from the set $\chi_{19}$, and similarly for other cases.
Let $g\in A_{10}$, $h\in A_{11}$ with the same semisimple parts. Next, we compute $\chi(g)-(p-1)\chi(h)$ for $\chi$ in the cases (2) and (3) above. As both $g,h$ are not semisimple, this equals 0. However, computing this for the right hand side in these formulas gives us a non-zero value. This will rule out these two cases.
Case 2: Using (\*), we have $0=\chi(g)+(p-1)(\chi(h)=\chi_{10}(g)+\chi_{13}(g)+\chi_{16}(g)+\chi_{17}(g)
+(p-1)(\chi_{10}(h)+\chi_{13}(h)+\chi_{16}(h)+\chi_{17}(h)) =
\pm p\pm 1\neq 0$. This is a contradiction. Case 3: Similarly, by (\*), we have $0=\chi(g)+(p-1)\chi(h)=\chi_{10}(g)+\chi_{13}(g)+2\chi_{17}(g)+\chi_{19}(g)+
(p-1)(\chi_{10}(h)+\chi_{13}(h)+2\chi_{17}(h)+\chi_{19}(h))=\pm p\pm 1\neq 0$.
In order to deal with cases (1), (4), (5), we compute the character values at the regular unipotent element $u$ (from class $A_5$ of [@No]). We have $\chi_{10}(u)=1$, $\chi_{13}(u)=0$, $\chi_{16}(u)=-1$, $\chi_{17}(u)=0$, $\chi_{19}(u)=1$. The values do not depend on the choice of an individual character in every class $\chi_i$ ($i\in \{10,13,16,17,19\}$.)
Case (1). $\chi(u)=\chi_{10}(u)+\chi_{13}(u)+(2p^2-2p+1)\chi_{19}(u)=
1-(2p^2-2p+1)\neq 0$, a contradiction.
Case (4). $\chi(u_5)=\chi_{10}(u_5)+\chi_{13}(u_5)+\chi_{16}(u_5)+(p^2-p)\chi_{19}(u_5)=
1-1-(p^2-p)\neq 0$, a contradiction.
Case (5). $\chi(u_5)=\chi_{10}(u_5)+\chi_{13}(u_5)+\chi_{17}(u_5)+(2p^2-2p+1)\chi_{19}(u_5)=
1-(2p^2-2p+1)\neq 0$, a contradiction. $\Box$
Remark. Some characters $\chi$ in cases (2) and (3) vanish at non-identity $p$-elements. (In (3) $2\chi_{17}$ may be the sum of two distinct characters of degree $p(p-1)^2(p^2+1)$.)
The groups $G={}^3D_4(p)$
-------------------------
We follows the strategy described in Section \[74\]. We can assume $p>2$ as the decomposition matrix for ${}^3D_4(2)$ is available in the GAP library, and one can read off from there that the minimum valu e for $c_\Phi$ equals 15.
\[d4x\]Let $G={}^3D_4(p)$, and $\Phi\neq St$ be a PIM. Then $c_\Phi\geq 2$.
Proof. Let $V$ be the socle and $\chi$ the character of $\Phi$. Then $V=V_\mu$, where $\mu=(p-1)({\omega }_1+{\omega }_3+{\omega }_4)$. We keep notation of Section \[74\].
Arguing by contradiction, we assume that $c_\Phi=1$, and then we denote by $\tau$ and $\gamma$ characters of $G$ occurring as common constituents of $\chi$ with $1_U^G$ and $\Gamma$, respectively.
We first show that $\tau$ is a unipotent character of degree $p^7(p^4-p^2+1)$. Let $P$ be a parabolic subgroup of $G$ corresponding to the nodes $J=\{1,3,4\}$ at the Dynkin diagram of $G$. Let $L$ be a Levi subgroup of $P$ and $L'$ the subgroup of $L$ generated by unipotent elements. Then $L'\cong SL(2,p^3)$. Let $S={\mathop{\rm Soc}\nolimits}(V_\mu|_P)|_L$ and $\rho$ the character of $S$. By Lemma \[nf1\], $(\chi, \rho^{\# G})>1$. As $\rho^G$ is a part of $1_U^G$, it follows that $(\chi, \rho^G)=1$. In addition, $\tau$ is a constituent of $1_B^G$. The degrees of constituents of $1_B^G$ are given in [@CKS Proposition 7.22]. The order of $G$ is $p^{12}(p^6-1) (p^2-1)(p^8+p^4+1)$, so $\rho^{\# G}(1)=p^3|G:P_J|=p^3(p+1)(p^8+p^4+1)$. From this one easily obtains the lemma (which true for $q$ in place of $p$):
\[d41\]In the above notation, $\rho^{\# G}=St+\rho'_1+\rho'_2+\rho_2$, where $ \rho_2'(1)=p^3(p^3+1)^2/2$, $ \rho_2(1)=p^3(p+1)^2(p^4-p^2+1)/2$ and $ \rho_1'(1)=p^7(p^4-p^2+1)$.
\[d42\] Let $\Phi$ be a PIM with $c_{\Phi}=1$. Then $\tau(1)=p^7(p^4-p^2+1)$.
Proof. By [@CKS Proposition 7.22] or [@CKS p.115], the characters $\rho_2,\rho'_2$ occur in $1_B^G$ with multiplicity greater than 1; therefore, none of them is a constituent of $\chi$, and the claim follows.$\Box$
As $G$ coincides with its dual group $G^*$, we identify maximal tori in $G$ and $G^*$. Following [@DeM] we denote by $s_i$, $i=1{,\ldots ,}15$, the union of the semisimple classes of $G$ with the same centralizer (that is, $C_G(x)$ is conjugate with $ C_G(y)$ for $x,y\in s_i$). In the character table of $G$ in [@DM] a class with representative $s\in s_i$ meets $ T_0$ $i\leq 8$. (The set $s_8$ consists of regular elements.)
\[uc4\] The set $\cup _{i\leq 8}{\mathcal E}_{s_i}\setminus {\mathop{\rm Irr}\nolimits}1_U^G$ consists of two unipotent cuspidal characters.
Proof. The unipotent characters of $G$ have been determined by Spaltenstein [@Spa]. All but two of them are constituents of $1_B^G$.
The characters $\chi_{3,*}$, $\chi_{5,*}$, $\chi_{6,*}$, $\chi_{7,*}$ and $\chi_8$ from $\cup _{i\leq 8}{\mathcal E}_{s_i} $ are either regular or semisimple. So they are in $ 1^G_U $ by Lemma \[dd4\](2). This leaves with $s_i$ for $i=2,4$ (as $s_1=1$). In these cases $C_G(s_2)\cong (SL(2,p^3)\circ SL (2,p))\cdot T_0$ and $C_G(s_4)\cong SL (3,p)\cdot T_0$, where $T_0$ is a split maximal torus in $G$, see [@DeM Proposition 2.2]. For these groups the number of unipotent characters are known to be 4 and 3, respectively. So $|{\mathcal E}_{s_i} |=4$, resp., 3, for $i=2,4$, see [@DM Theorem 13.23]. Let $\beta_i\in{\mathop{\rm Irr}\nolimits}T_0$ be the character corresponding to $s_i$. Then the Harish-Chandra series $\beta_{i,B}^G$ is contained in $ {\mathcal E}_{s_i} $ by Lemma \[dd4\](1). Set $W_i=C_{W_0}(\beta_i)$. Then $W_2\cong
{\mathop{\mathbb Z}\nolimits}_2\times {\mathop{\mathbb Z}\nolimits}_2$ and $W_4\cong S_3$ [@DeM Lemma 3.4]. Moreover, the centralizing algebra of $\beta_{i,B}^G$ is isomorphic to the group algebra of $W_i$ ([@St Exercise in §14 prior Lemma 86]). Therefore, $\beta_{1,B}^G$ consists of 4 distinct constituents, whereas $\beta_{2,B}^G$ has three distinct constituents (two of them occurs with 1 and one constituent occurs with 2). In both the cases $|{\mathcal
E}_{s_i} |$ coincides with the number of distinct constituents in $\beta_{i,B}^G$, and the claim follows. $\Box$
\[d49\]$\gamma(1)\in\{d_1,d_2,d_4\}$, where $d_2=(p-1)^2(p^3+1)^2(p^4-p^2+1)$, $d_4=(p-1)(p^3-1)(p^8+p^4+1)$.
Proof. We first write down the degrees of the regular characters of $H$ in Table 5. Note that one can easily detect the regular characters in the character table of $G$ in [@DeM]. Indeed, the characters in [@DeM] are partitioned in Lusztig series ${\mathcal E}_{s}$ with $s\in
s_i$, $i=1{,\ldots ,}15$. As mentioned in the proof of Lemma \[gg4\], every ${\mathcal E}_s$ has a single regular character. Then its degree equals $|G:C_G(s)|_{p'}\cdot |C_G(s)|_p$, while other characters in ${\mathcal E}_s$ are of degree $|G:C_G(s)|_{p'}\cdot
e$, where $e$ is the degree of a unipotent character in $C_G(s)$. (So the $p$-power part of the character degrees in ${\mathcal
E}_s$ is maximal for the regular character.)
As is explained in Section \[74\], $\gamma (1)\leq p^{12}-
\tau(1)=p^{12}-p^{11}+p^{9}-p^{7}$. One observes that for $s\in
\cup_{i>8}s_i$ only the characters $\chi_{12}(s)$ and $\chi_{15}(s)$ satisfy this inequality. So exactly one of these characters is a constituent of $\chi$, as claimed.
Let $d_1,d_2$ be the degrees of these $\chi_{12}(s)$, $\chi_{15}(s)$, respectively, and set $f_i=p^{12}-p^{11}+p^{9}-p^{7}-d_i$ ($i=1,2$). Then
$f_1=p^{11}+p^9+4p^8+p^7 +2p^6+ 2p^5+4p^4+2p-1$,
$f_2=2p^{9}-2p^8+p^5-2p^4+p^3+p-1=(p-1)(2p^8+p^4-p^3+1).$
Thus, $f_i$ $(i=1,2)$ is a sum of the degrees of non-regular characters that do not belong to $1_U^G $. Let ${\lambda }$ be one of these characters. Note that $G$ has two cuspidal unipotent characters of degrees $e_1=p^3(p^3-1)^2/2$ and $e_2=p^3(p-1)^2(p^4-p^2+1)/4$. They do not belong to $1_U^G $ whereas the other 6 unipotent characters belong to $1_U^G $.
Inspection of the character table of $G$ in [@DeM] shows that non-regular characters that are in ${\mathcal E}_{s_i}$ with $i>8$ are the characters $\chi_{9,1}$, $\chi_{9,qs'}$ and $\chi_{10,1}$ in [@DeM], of degrees $e_3=(p^3-1)(p^2+p+1)(p^4-p^2+1)$, $e_4=p(p^3-1)^2(p^4-p^2+1)$ and $e_5=(p^3-1)(p^8+p^4+1)$, respectively. One observes that $(p^2-p+1)(p^2+p+1)=p^4-p^2+1$ and $(p^2+p+1)(p-1)=p^3-1$. As $p^2+p+1$ is odd, it follows that $p^2+p+1$ divides $e_i$ for every $i=1,2,3,4,5$, and hence $p^2+p+1$ divides $f_j$ for $j=1,2$. However, $f_1({\rm mod }\,
p^2+p+1)=-5$ and $f_2({\rm mod }\, p^2+p+1)=2p-11$. This is a contradiction, which completes the proof. $\Box$
TABLE 5: Degrees of the regular characters of $G={}^3D_4(p)$, $p>2$
[|l|l|c| ]{}
$s$ &$ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,
\, \, \, \, \, \, \, \, \, \, \, \, \, \, \,\rho_s(1)$ &$\rho_s$ in [@DM]\
$s_1 $ & $~~ p^{12}$ & $St $ $s_2 $ &$~p^4(p^8+p^4+1)
$&$\chi_{2,St,St'}(s)$\
$s_3 $ & $~p^3(p+1)(p^8+p^4+1)
$&$\chi_{3,St}(s)$\
$s_{4} $ & $p^3(p^3+1)(p^2-p+1)(p^4-p^2+1)
$&$\chi_{4,St}(s)$\
$s_{5} $& $p(p^3+1)(p^8+p^4+1)
$&$\chi_{5,St}(s)$\
$s_6 $ & $(p+1)(p^3+1)(p^8+p^4+1)
$&$\chi_{6}(s)$\
$s_7 $& $ p^3(p-1)(p^8+p^4+1)$&$\chi_{7,St}(s)$\
$s_8$& $(p-1)(p^3+1)(p^8+p^4+1)$&$\chi_{8}(s)$\
$s_9 $& $p^3(p^3-1)(p^2+p+1)(p^4-p^2+1)$&$\chi_{9,
St}(s)$\
$s_{10}$& $p(p^3-1)(p^8+p^4+1) $&$\chi_{10,St}(s)$\
$s_{11}$& $(p+1)(p^3-1)(p^8+p^4+1)$&$\chi_{11}(s)$\
$s_{12}$& $(p-1)^2(p^3+1)^2(p^4-p^2+1)$&$\chi_{12}(s)$\
$s_{13}$& $(p+1)^2(p^3-1)^2(p^4-p^2+1)$&$\chi_{13}(s)$\
$s_{14}$& $(p^6-1)^2$&$\chi_{14}(s)$\
$s_{15}$& $(p-1)(p^3-1)(p^8+p^4+1)$&$\chi_{15}(s)$\
The minimal PIM’s
=================
In this section we complete the proof of Theorem \[wt1\]. To settle the base of induction, we refer to certain known results for groups of small rank. We first write down some data available in the GAP library:
\[dn1\]Let $G$ be one of the groups below and let $\Phi$ be a PIM for $G$ other than the Steinberg PIM.
$(1)$ If $G=Sp(4,2)$ or $Sp(4,3)$ then $c_{\Phi}\geq 3$.
$(2)$ If $G=Sp(4,3)$ then $c_{\Phi_1}=2$, and $c_{\Phi} \geq 3$ for $\Phi\neq \Phi_1$.
$(3)$ If $G=SU(4,2)$ then ${\rm min}\, c_\Phi\geq 4$. $(4)$ If $G=SU(5,2)$ then ${\rm min}\, c_\Phi\geq 5$.
$(5)$ If $G={}^3D_4(2)$ then $ c_\Phi\geq 15$.
$(6)$ If $G=G_2(2)$ then ${\rm min}\, c_\Phi=5$.
$(7)$ If $G={}^2F_4(2)$ then $ c_\Phi \geq 14 $.
\[sm6\] Let $G$ be one of the groups below and let $\Phi\neq St$ be a PIM for $G$.
$(1)$ Let $G=SL(3,p)$, $p>2$. Then $ c_\Phi \geq 2 $. $(2)$ Let $G=Sp(4,p)$, $p>3$. Then $ c_\Phi \geq 3 $.
$(3)$ Let $G=G_2(p)$, $p>2$. Then $ c_\Phi \geq 6 $.
Proof. The statements (1), (2), (3) follows from the results in [@Hu81], [@Hu80], [@Hub Ch. 18], respectively. $\Box$
[**Proof of Theorem**]{} \[wt1\]. Let $r$ be the BN-pair rank of $G$. If $r=1$ then the result is contained in Proposition \[ra1\]. In general, suppose the contrary, and let $G$ be a counter example, so $r>1$. Let $\Phi\neq St$ be a PIM with $c_\Phi=1$, and let $V$ be the socle of $\Phi$. Then $V$ is and hence $V=V_\mu|_G$ for some module $V_\mu$ for the respective algebraic group ${{\mathbf G}}$. (Here $\mu$ is the highest weight of $V_\mu$.) Let $P$ be a proper parabolic subgroup of $G$, which is not a Borel subgroup of $G$. Then $P=P_J$ for some non-empty set $J$, see Section 5. Let $L_J$ be a Levi subgroup of $P$, let $G_J$ be as in Section 5 and $V_J=C_V(O_p(P))$. Then $G_J$ is a Chevalley group of rank $r_J<r$, and $V_J$ is both as an $FL_J$- and an $FG_J$-module (Lemma \[le1\]). Let $\Psi$ be the PIM for $L_J$ with socle $V_J|_{L_J}$. By Lemma \[mp1\], $c_\Psi=1$, so $\dim \Psi=|L_J|_p$. As $|L_J|_p=|G_J|_p$, the restriction $\Psi|_{G_J}$ is a PIM for $G_J$ of dimension $|G_J|_p$ (Lemma \[om1\](2)). This is a contradiction unless $V_J|_{G_J}$ is isomorphic to the Steinberg module for $G_J$, or else $G_J\in \{A_1(p), A_2(2),{}^2A_2(2)\}$ and $V_J|_{G_J}$ is the trivial $FG_J$-module. As this is true for every parabolic subgroup of $G$, which is not a Borel subgroup, $V$ satisfies the assumption of Lemma \[sd7\]. Therefore, $G$ belongs to the list (1) - (6) of Lemma \[sd7\], or else $G={}^2F_4(\sqrt{2})$. However, for these groups Theorem \[wt1\] is true by Lemmas \[u4q\], \[d4x\], \[dn1\] and \[sm6\], which is a contradiction.
Acknowledgement. The author is indebted Ch. Curtis, J. Humphreys, G. Malle and M. Pelegrini for discussion and remarks.
[hhhhhh]{}
J. Ballard, Projective modules for finite Chevalley groups, [*Trans. Amer. Math. Soc.*]{} 245(1978), 221 - 249.
N. Bourbaki, [*Groupes et algèbres de Lie*]{}, Chaps. IV–VI, Hermann, Paris, 1968.
R. Brauer and Nesbitt, On the modular of groups of finite order, [*Ann. Math*]{}. 42(1941), 556 - 590.
M. Cabanes, Irreducible modules and Levi complements, [*J. Algebra*]{} 90(1984), 84 - 97.
R. Carter, [*Finite groups of Lie type: Conjugacy classes and complex characters*]{}, Wiley, Chichester, 1985.
L. Chastkofsky and W. Feit, On the projective characters in characteristic $2$ of the groups $Suz\, (2^m)$ and $Sp_4(2^n)$, [*Publ. Math. IHES*]{} 51(1980), 9 - 35.
J. Conway, R. Curtis, S. Norton, R. Parker and R. Wilson, [*Atlas of finite groups*]{}, Clarendon Press, Oxford, 1985.
Ch. Curtis, Modular of finite groups with split BN-pair, In: “[*Seminar on algebraic groups and related finite groups*]{}, Lect. notes Math. vol.131, Springer-Verlag, 1970”, ...
Ch. Curtis, N. Iwahori and R. Kilmoyer, Hecke algebras and characters of parabolic type of finite groups with (B,N)-pairs, [*Publ. Math. IHES*]{} 40(1972), 81 - 116.
Ch. Curtis, W. Kantor and G.Seitz, The $2$-transitive permutation of the finite Chevalley groups, [*Trans. Amer. Math. Soc*]{}. 218(1976), 1 - 59.
Ch. Curtis and I. Reiner, [*Representation theory of finite groups and associative algebras*]{}, Interscience, New York, 1962.
C. W. Curtis, I. Reiner, [*Methods of representation theory with applications to finite groups and orders*]{}, Vol. 1, Wiley, New York e.a., 1981.
C. W. Curtis, I. Reiner, [*Methods of representation theory with applications to finite groups and orders*]{}, Vol. 2, Wiley, New York e.a., 1987.
D. Deriziotis and G. Michler, Character table and blocks of finite simple triality group ${}^3D_4(q)$, [*Trans. Amer. Mat. Soc.*]{} 303(1987), 39 - 70.
F. Digne and J. Michel, [*Representations of finite groups of Lie type*]{}, London Math. Soc. Student Texts no.21, Cambridge University Press, 1991.
F. Digne, G. Lehrer and J. Michel, The characters of the group of rational points of a reductive group with non-connected center, [*J. reine angew. Math.*]{} 425(1992), 155 - 192.
R. Dipper, On modules over finite Chevalley groups and parabolic subgroups, [*Comm. Algebra*]{} 10(1982), 1073 - 1088.
R. Dipper, On modules of twisted groups of Lie type, [*J. Algebra*]{} 81(1983), 370 - 389.
R. Fein, Representations of direct products of finite groups, [*Pacific J. Math.*]{} 20(1967), 45 - 58.
W. Feit, [*The representation theory of finite groups*]{}, North-Holland, Amsterdam, 1982.
D. Gorenstein, R. Lyons and R. Solomon, [*The Classification of Finite Simple Groups*]{}, No. 3, Part I, Chapter A, Amer. Math. Soc. Providence, R.I. 1998.
G. Hiss and A.E. Zalesski, Tensoring generalized characters with the Steinberg character, [*Proc. Amer. Math. Soc*]{}. 138(2010), 1907 - 1921.
J. Humphreys, Ordinary and modular characters of $SL(3,p)$, [*J. Algebra*]{} 72(1981), 8 - 16.
J. Humphreys, Ordinary and modular characters of $SU(3,p)$, [*J. Algebra*]{} 131(1990), 425 - 431.
J. Humphreys, Projective modules for $Sp(4,p)$, [*J. Algebra*]{} 104(1986), 80 - 88.
J. Humphreys, [*Modular of finite groups of Lie type*]{}, In: “[*Finite Simple Groups*]{}, II”, Academic press, New York/London, 1980, 279 - 290.
J. Humphreys, [*Modular of finite groups of Lie type*]{}, Cambridge Univ. Press, Cambridge, 2006.
C. Jansen, K.Lux, R. Parker and R. Wilson, [*An Atlas of Brauer characters*]{}, Oxford Science publications, Clarendon Press, Oxford, 1995.
G. Kemper and G. Malle, The finite linear groups with ring of invariants, [*Transformation groups*]{} 2(1997), 57 - 89.
P. Kleidman and M.W. Liebeck, [*The subgroup structure of the classical groups*]{}, Cambridge Univ. Press, Cambridge 1990.
D. Kotlar, On the components of degenerate Gelfand-Graev characters, [*J. Algebra*]{} 173(1995), 348 - 360.
D. Kotlar, The regular components of induced cuspidal characters, [*J. Algebra*]{} 174(1995), 729 - 748.
G. Malle and D. Testerman, [*Linear algebraic groups and finite groups of Lie type*]{}, Cambridge Univ. Press, Cambridge, 2011.
G. Malle and Th. Weigel, Finite groups with minimal $1$-PIM, [*Manuscripta Math.*]{} 126(2008), 315 - 332.
S. Nozawa, On the characters of the finite unitary group $U(4,q^2)$, [*J. Fac. Sci. Univ. Tokyo, Sect. $1$A Math.*]{} 19(1972), 257 - 293.
S. Smith, Irreducible modules and parabolic subgroups, [*J. Algebra*]{} 75(1982), 286 - 289.
N. Spaltenstein, Caractères unipotent de ${}^3D_4(q)$, [*Comment. Math. Helvetici*]{} 57(1982), 676 - 691.
B. Srinivasan, On the modular characters of the special linear group $SL(2,p^n)$, [*Proc. London Math. Soc.*]{} 14(1964), 101 - 114.
T. Springer and R. Steinberg, Conjugacy classes, In: “[*Seminar on algebraic groups and related finite groups*]{}, Lect. notes Math. vol.131, Springer-Verlag, 1970”,
R. Steinberg, [*Lectures on Chevalley groups*]{}, Yale University, New Haven, 1968.
R. Steinberg, [*Endomorphisms of linear algebraic groups*]{}, Memoirs Amer. Math. Soc. 80(1968), 1 - 108.
Pham Huu Tiep and A.E. Zalesski, Mod $p$ reducibility of unramified of finite groups of Lie type. [*Proc. London Math. Soc.*]{} (3) 84(2002), 439 – 472.
Y. Tsushima, On certain projective modules for finite groups of Lie type, [*Osaka J. Math.*]{} 27(1990), 947 - 962.
H.N. Ward, On Ree’s Series of Simple Groups, [*Trans. Amer. Math. Soc*]{}. 121(1966), 62 - 89.
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abstract: |
We investigate the nonuniform motion of a straight screw dislocation in infinite media in the framework of the translational gauge theory of dislocations. The equations of motion are derived for an arbitrarily moving screw dislocation. The fields of the elastic velocity, elastic distortion, dislocation density and dislocation current surrounding the arbitrarily moving screw dislocation are derived explicitely in the form of integral representations. We calculate the radiation fields and the fields depending on the dislocation velocities.\
[**Keywords:**]{} dislocation dynamics; gauge theory of dislocations; radiation.\
author:
- |
Markus Lazar $^\text{a,b,}$[^1]\
\
${}^\text{a}$ Emmy Noether Research Group,\
Department of Physics,\
Darmstadt University of Technology,\
Hochschulstr. 6,\
D-64289 Darmstadt, Germany\
${}^\text{b}$ Department of Physics,\
Michigan Technological University,\
Houghton, MI 49931, USA
title: '[**The gauge theory of dislocations: a nonuniformly moving screw dislocation**]{}'
---
Introduction
============
For many years, the investigation of the nonuniform motion of dislocations has attracted the interest of researchers in different fields such as physics, material science, continuum mechanics, seismology and earthquake engineering. Usually, the motion of dislocations is investigated in the framework of incompatible elastodynamics where the dislocation density tensor and the dislocation current tensor are given as source terms of the elastic fields (see, e.g., @Mura63 [@Mura; @Lardner; @Gunther73]). The behaviour of the motion of a dislocation is somehow particular, because at any time the fields are determined not only by the instantaneous values of the velocity (or higher derivatives of the position with respect to time), but also by the values of these quantities in the past [@Eshelby51; @Eshelby53]. As @Eshelby51 succinctly put it: ‘The dislocation is haunted by its past’. This fact is based on the property that Huygens’ principle is not valid in two dimensions (see, e.g., [@Wl]). Solutions for nonuniformly moving screw dislocations have been given by @Nabarro51 [@Eshelby51; @Eshelby53; @KM64], and @XM80. Results for the nonuniform motion of a gliding edge dislocation have been given by @Mura64 [@XM81], and @Brock82.
Immediately, the question comes up, what is the suitable theory for an improved approach of a continuum theory of dislocation dynamics. Generalized theories of elasticity are, for instance, nonlocal elasticity [@Eringen83; @Eringen02], gradient elasticity [@Mindlin64; @Mindlin72] and dislocation gauge theory [@Edelen83; @Edelen88; @Lazar00; @LA08]. However, in the dynamical version of nonlocal elasticity it is not obvious which dynamical kernel should be used. In the dynamical extension of gradient elasticity [@Aifantis06; @Aifantis07] the choice of ‘dynamical’ gradients and length scales is based on ad-hoc assumptions. On the other hand, dislocation gauge theory is not based on such assumptions. The choice of the dynamical state variables is given by a canonical field theoretical framework [@LA08]. For this reason, dislocation gauge theory seems to be the appropriate theory and it will be used in this paper.
Thus, a very promising and straightforward candidate for an improved dynamical approach of dislocations including scale effects is the translational gauge theory of dislocations [@Edelen83; @Edelen88; @LA08; @LA09]. In the gauge theory of dislocations, dislocations arise naturally as a consequence of broken translational symmetry and their existence is not required to be postulated a priori. Moreover, such a theory uses the field theoretical framework which is well accepted in theoretical physics. From the point of view of theoretical physics, the gauge field approach of dislocations [@Edelen83; @Edelen88; @LA08; @LA09] removes the singularity at the core of a moving dislocation at subsonic as well as at supersonic speed [@Lazar09], the gauge theoretical approach in essence being reminiscent of nonlocality. Up to now, no solution for a nonuniformly moving dislocation has been given in the dislocation gauge theory or another generalized theory of elasticity. The aim of this paper is the examination of a nonuniformly moving screw dislocation, for the first time, in the framework of dislocation gauge theory. We will calculate the retarded expressions of the elastic fields as well as of the dislocation density and dislocation current tensors. This paper provides new insights on the change of the dislocation core structure while the dislocation is moving at sound wave speed.
Gauge theory of dislocations
============================
In this section, we briefly review the gauge theory of dislocations in the form given by @LA08, and @Lazar09.
In dislocation dynamics, the following state quantities of dislocations are of importance[^2] $$\begin{aligned}
\label{disl-den}
T_{ijk}=\beta_{ik,j} - \beta_{ij,k},\qquad
I_{ij}=-v_{i,j} + {\dot{\beta}}_{ij},\end{aligned}$$ which are called the dislocation density tensor and the dislocation current tensor, respectively. They are kinematical quantities of dislocations and they are given in terms of the incompatible elastic distortion tensor $\beta_{ij}$ and incompatible physical velocity of the material continuum $v_i$. Their dimensions are: $[\beta_{ij}]=1$, $[v_{i}]=\text{length}/\text{time}$, $[T_{ijk}]=\text{1}/\text{length}$ and $[I_{ij}]=\text{1}/\text{time}$. The dislocation density and the dislocation current tensors fulfill the translational Bianchi identities $$\begin{aligned}
\label{BI}
\epsilon_{jkl}T_{ijk,l}=0,\qquad
\dot{T}_{ijk} + I_{ij,k}-I_{ik,j}= 0. \end{aligned}$$ In the dynamical translation gauge theory of dislocations, the Lagrangian density is of the bilinear form $$\begin{aligned}
\label{tot-Lag}
\LL =T-W=
\frac{1}{2}\,p_{i}v_{i}
+ \frac{1}{2}\,D_{ij}I_{ij}-\frac{1}{2}\,\sigma_{ij}\beta_{ij}
- \frac{1}{4}\,H_{ijk}T_{ijk}.\end{aligned}$$ The canonical conjugate quantities (response quantities) are defined by $$\begin{aligned}
\label{can-qua}
p_i:=\frac{\pd \LL}{\pd v_i},\qquad
\sigma_{ij}:=-\frac{\pd \LL}{\pd \beta_{ij}} ,\qquad
D_{ij} :=\frac{\pd \LL}{\pd I_{ij}},\qquad
H_{ijk}:=-2\frac{\pd \LL}{\pd T_{ijk}},\end{aligned}$$ where $p_i$, $\sigma_{ij}$, $D_{ij}$, and $H_{ijk}$ are the linear momentum vector, the force stress tensor, the dislocation momentum flux tensor, and the pseudomoment stress tensor[^3], respectively. They have the dimensions: $[p_i]=\text{momentum}/(\text{length})^3
\stackrel{\rm SI}{=}\text{N\,s}/\text{m}^3$, $[D_{ij}]=\text{momentum}/(\text{length})^2
\stackrel{\rm SI}{=}\text{N\,s}/\text{m}^2$, $[\sigma_{ij}]=\text{force}/(\text{length})^2
\stackrel{\rm SI}{=}\text{Pa}$ and $[H_{ijk}]=\text{force}/\text{length}\stackrel{\rm SI}{=}\text{N}/\text{m}$.
The Euler-Lagrange equations derived from the total Lagrangian density $\LL=\LL(v_i,\beta_{ij},I_{ij},T_{ijk})$ are given by $$\begin{aligned}
\label{euler-lag-2}
&E^{\, \Bv}_i(\LL)= \pd_t \frac{\pd \LL}{\pd \dot{v}_i}
+ \pd_j\frac{\pd \LL}{\pd v_{i,j}} - \frac{\pd \LL}{\pd v_{i}} = 0,\\
\label{euler-lag-3}
&E^{\, \Bbeta}_{ij}(\LL)=\pd_t \frac{\pd \LL}{\pd \dot{\beta}_{ij}}
+ \pd_k \frac{\pd \LL}{\pd \beta_{ij,k}} - \frac{\pd \LL}{\pd \beta_{ij}} = 0.\end{aligned}$$ We add to $\LL$ a so-called null Lagrangian, $\LL_N=\sigma^0_{ij}\beta_{ij}-p^0_i v_i$, with the ‘background’ stress $\sigma^0_{ij}$ and the ‘background’ momentum $p^0_i$ as external source fields, which satisfy: $\dot{p}^0_i - \sigma^0_{ij,j}= 0$. In terms of the canonical conjugate quantities (\[can-qua\]), Eqs. (\[euler-lag-2\]) and (\[euler-lag-3\]) take the form $$\begin{aligned}
{2}
\label{inhom-di-1}
D_{ij,j}+ p_i&=p^0_i,
\qquad &&(\text{momentum balance of dislocations}),\\
\label{inhom-di-2}
\dot{D}_{ij} + H_{ijk,k}+ \sigma_{ij}&=\sigma^0_{ij},
\qquad &&(\text{stress balance of dislocations}).\end{aligned}$$ Eqs. (\[inhom-di-1\]) and (\[inhom-di-2\]) represent the dynamical equations for the balance of dislocations. Eq. (\[inhom-di-1\]) represents the momentum balance law of dislocations, where the physical momentum is the source of the dislocation momentum flux. Eq. (\[inhom-di-2\]) represents the stress balance of dislocations. The force stress and the time derivative of the dislocation momentum flux are the sources of the pseudomoment stress. It can be seen in Eqs. (\[inhom-di-1\]) and (\[inhom-di-2\]) that the sources of the dislocation momentum flux and the pseudomoment stress tensors are the so-called effective momentum vector $(p_i-p^0_i)$ and the effective force stress tensor $(\sigma_{ij}-\sigma^0_{ij})$ (see, e.g., [@Edelen88]). These effective fields, which are the difference between the physical field and the background field, drive the dislocation fields.
The conservation of linear momentum appears as an integrability condition from (\[inhom-di-1\]) and (\[inhom-di-2\]). It reads $$\begin{aligned}
{2}
\label{inhom-di}
\dot{p}_i - \sigma_{ij,j}&= 0,
\qquad &&(\text{force balance of elasticity}),\end{aligned}$$ where the time-derivative of the physical momentum vector is the source of the force stress tensor.
The linear, isotropic constitutive relations for the momentum, the force stress, the dislocation momentum flux and the pseudomoment stress are $$\begin{aligned}
\label{con-eq}
p_i&=\rho v_i,\\
\label{con-eq-2}
\sigma_{ij}&= \lambda \delta_{ij} \beta_{kk} + \mu (\beta_{ij}+\beta_{ji}) + \gamma (\beta_{ij}-\beta_{ji}),\\
\label{con-eq-3}
D_{ij}&= d_1 \delta_{ij} I_{kk} + d_2 (I_{ij} + I_{ji}) + d_3 (I_{ij} - I_{ji}),\\
\label{con-eq-4}
H_{ijk}&= c_1 T_{ijk} + c_2 (T_{jki} - T_{kji}) + c_3 (\delta_{ij}T_{llk} - \delta_{ik}T_{llj}),\end{aligned}$$ where $\rho$ is the mass density. Here $\mu, \lambda, \gamma$ are the elastic stiffness parameters, $c_1,c_2,c_3$ denote the resistivity parameters associated with dislocations and $d_1,d_2,d_3$ are the inertia terms associated with dislocation currents. The nine material parameters have the dimensions: $[\mu, \lambda, \gamma]=\text{force}/(\text{length})^2
\stackrel{\rm SI}{=}\text{Pa}$, $[c_1,c_2,c_3]=\text{force}\stackrel{\rm SI}{=}\text{N}$ and $[d_1,d_2,d_3]=\text{mass}/\text{length}\stackrel{\rm SI}{=}\text{kg}/\text{m}$.
The requirement of non-negativity of the energy (material stability) $E=T+W\ge~0$ leads to the conditions of semi-positiveness of the constitutive moduli. Particularly, the constitutive moduli have to fulfill the following conditions [@LA08] $$\begin{aligned}
{3}
\rho&\ge 0,\nonumber \\
\mu&\ge0,\qquad &\gamma&\ge 0,\qquad &3\lambda+2\mu&\ge 0,\nonumber \\
d_2&\ge 0,\qquad &d_3 &\ge 0,\qquad &3 d_1+2d_2&\ge 0,\nonumber \\
\label{IE-c}
c_1-c_2&\ge 0,\qquad &c_1+2c_2&\ge 0, \qquad &c_1-c_2+2 c_3&\ge 0.\end{aligned}$$
If we substitute the constitutive equations (\[con-eq-3\]) and (\[con-eq-4\]) in the equations (\[inhom-di-1\]) and (\[inhom-di-2\]) and use the definitions (\[disl-den\]), we find $$\begin{aligned}
\label{dyn-sys2}
& d_1 ({\dot{\beta}}_{jj,i}-v_{j,ji})
+(d_2+d_3)({\dot{\beta}}_{ij,j}-v_{i,jj})
+(d_2-d_3)({\dot{\beta}}_{ji,j}-v_{j,ji})
+p_i=p^0_i,\\
\label{dyn-sys3}
&d_1\delta_{ij}({\ddot{\beta}}_{kk}-{\dot{v}}_{k,k})
+(d_2+d_3)(
{\ddot{\beta}}_{ij}- {\dot{v}}_{i,j})
+(d_2-d_3)({\ddot{\beta}}_{ji}-{\dot{v}}_{j,i})\nonumber\\
&\quad
+ c_1(\beta_{ik,jk}-\beta_{ij,kk})
+c_2(\beta_{ji,kk}-\beta_{jk,ik}+\beta_{kj,ik}-\beta_{ki,jk})
+c_3\big[\delta_{ij}(\beta_{lk,lk}-\beta_{ll,kk})\nonumber \\
&\quad
+ (\beta_{kk,ji}-\beta_{kj,ki})\big]
+\sigma_{ij}
=\sigma^0_{ij},\end{aligned}$$ which are a coupled system of partial differential equations for the fields $\Bv$ and $\Bbeta$.
Equations of motion of a screw dislocation
==========================================
We now derive the equations of motion for an arbitrarily moving screw dislocation. We consider an infinitely long screw dislocation parallel to the $z$-axis and traveling in the $xy$-plane. The symmetry of such a straight screw dislocation leaves only the following non-vanishing components of the physical velocity vector and elastic distortion tensor (see, e.g., [@Lardner; @Gunther73]): $v_z$, $\beta_{zx}$, $\beta_{zy}$, and for the dislocation density and dislocation current tensors: $T_{zxy}$, $I_{zx}$, $I_{zy}$. The equations of motion of a moving screw dislocation read $$\begin{aligned}
\label{v-z1}
(d_2+d_3)(\dot{\beta}_{zx,x}+\dot{\beta}_{zy,y}-\Delta v_z)+\rho v_z&=\rho
v^0_z,\\
\label{B-zx1}
(d_2+d_3)(\ddot{\beta}_{zx}-\dot{v}_{z,x})+c_1(\beta_{zy,xy}-\beta_{zx,yy})+(\mu+\gamma)\beta_{zx}&=(\mu+\gamma)\beta^0_{zx},\\
(d_2-d_3)(\ddot{\beta}_{zx}-\dot{v}_{z,x})+c_2(\beta_{zx,yy}-\beta_{zy,xy})+(\mu-\gamma)\beta_{zx}&=(\mu-\gamma)\beta^0_{zx},\\
(d_2+d_3)(\ddot{\beta}_{zy}-\dot{v}_{z,y})+c_1(\beta_{zx,xy}-\beta_{zy,xx})+(\mu+\gamma)\beta_{zy}&=(\mu+\gamma)\beta^0_{zy},\\
\label{B-zy2}
(d_2-d_3)(\ddot{\beta}_{zy}-\dot{v}_{z,y})+c_2(\beta_{zy,xx}-\beta_{zx,xy})+(\mu-\gamma)\beta_{zy}&=(\mu-\gamma)\beta^0_{zy},\end{aligned}$$ where $\Delta=\pd_{xx}+\pd_{yy}$. In addition, the condition (\[inhom-di\]) reads now $$\begin{aligned}
\label{EC}
(\mu+\gamma)(\beta_{zx,x}+\beta_{zy,y})=\rho \dot{v}_z.\end{aligned}$$ From the equations (\[B-zx1\])–(\[B-zy2\]) we obtain the following relations $$\begin{aligned}
\label{Rel-c}
\frac{c_1}{\mu+\gamma}&=-\frac{c_2}{\mu-\gamma} ,\\
\label{Rel-d}
\frac{d_2+d_3}{\mu+\gamma}&=\frac{d_2-d_3}{\mu-\gamma} \end{aligned}$$ and we may introduce the quantities $$\begin{aligned}
\ell^2_1&=\frac{c_1}{\mu+\gamma},\\
L^2_1&=\frac{d_2+d_3}{\rho},\\
\tau^2_1&=\frac{d_2+d_3}{\mu+\gamma}.\end{aligned}$$ Here $\ell_1$ and $L_1$ are the ‘static’ and ‘dynamic’ characteristic length scales and $\tau_1$ is the characteristic time scale of the anti-plane strain problem. Due to the conditions (\[IE-c\]), they are non-negative: $\ell_1\ge 0$, $L_1\ge 0$, $\tau_1\ge 0$. The velocity of elastic shear waves is defined in terms of the ‘dynamic’ length scale $L_1$ and the time scale $\tau_1$: $$\begin{aligned}
\label{cT}
c^2_{\TT}=\frac{L_1^2}{\tau_1^2}=\frac{\mu+\gamma}{\rho}.\end{aligned}$$ In a similar way, we introduce the following transversal gauge-theoretical velocity defined in terms of $\ell_1$ and $\tau_1$: $$\begin{aligned}
\label{aT}
a^2_{\TT}=\frac{\ell_1^2}{\tau_1^2}=\frac{c_1}{d_2+d_3},\end{aligned}$$ and we find the relation $$\begin{aligned}
\label{Rel-a}
\frac{\ell_1^2}{L_1^2}=\frac{a_{\TT}^2}{c_{\TT}^2}.\end{aligned}$$ Applying Eq. (\[EC\]), the equations of motion (\[v-z1\])–(\[B-zy2\]) can be given in the form $$\begin{aligned}
\label{EOM-v}
\tau_1^2\ddot{v}_{z}-L_1^2\Delta v_z+v_z&=v^0_z,\\
\label{EOM-Bzx}
\tau_1^2\ddot{\beta}_{zx}-\ell_1^2 \Delta \beta_{zx}
-\tau_1^2 \Big(1-\frac{\ell_1^2}{L_1^2}\Big) \dot{v}_{z,x}
+\beta_{zx}&=\beta^0_{zx},\\
\label{EOM-Bzy}
\tau_1^2\ddot{\beta}_{zy}-\ell_1^2 \Delta \beta_{zy}
-\tau_1^2\Big(1-\frac{\ell_1^2}{L_1^2}\Big) \dot{v}_{z,y}
+\beta_{zy}&=\beta^0_{zy}.\end{aligned}$$
If we assume that the following relation is valid (see also [@Lazar09]) $$\begin{aligned}
\label{L-l}
L_1=\ell_1 ,\end{aligned}$$ then the field equations (\[EOM-v\])–(\[EOM-Bzy\]) decouple to Klein-Gordon equations. Equations (\[Rel-a\]) and (\[L-l\]) give the relation $a_{\TT}=c_{\TT}$, that means that we have only one characteristic velocity $c_{\TT}$ under this assumption. Particularly, the elastic fields fulfill the uncoupled Klein-Gordon equations $$\begin{aligned}
\label{KGE-v-2}
&\big [1+\ell_1^2 \square_{\TT} \big] v_z=v^0_z,\\
\label{KGE-Bzx-2}
&\big [1+\ell_1^2 \square_{\TT} \big] \beta_{zx}
=\beta^0_{zx},\\
\label{KGE-Bzy-2}
&\big [1+\ell_1^2 \square_{\TT} \big] \beta_{zy}
=\beta^0_{zy},\end{aligned}$$ with the following $(1+2)$-dimensional d’Alembert operator (wave operator) $$\begin{aligned}
\square_{\TT}&=\frac{1}{c^2_{\TT}}\, \pd_{tt}-\Delta.
$$ In addition, using (\[disl-den\]) we may derive from (\[KGE-v-2\])–(\[KGE-Bzy-2\]) inhomogeneous Klein-Gordon equations for the non-vanishing components $T_{zxy}$, $I_{zx}$, $I_{zy}$ $$\begin{aligned}
\label{KGE-T}
&\big [1+\ell_1^2 \square_{\TT} \big] T_{zxy}=T^0_{zxy},\\
\label{KGE-Ix}
&\big [1+\ell_1^2 \square_{\TT} \big] I_{zx}
=I^0_{zx},\\
\label{KGE-Iy}
&\big [1+\ell_1^2 \square_{\TT} \big] I_{zy}
=I^0_{zy}.\end{aligned}$$ In field theory, Klein-Gordon equations describe massive fields (see, e.g., [@Rubakov]). That means that a dislocation is a massive gauge field.
From the condition (\[L-l\]), we find for the inertia term of a screw dislocation $$\begin{aligned}
d_2+d_3=\frac{c_1}{c^2_{\TT}}=\rho\, \ell^2_1,\end{aligned}$$ that it is given in terms of the characteristic length scale $\ell_1$.
Nonuniformly moving screw dislocation
======================================
Now we consider a nonuniformly moving screw dislocation at the position ($\xi(t),\eta(t)$) at time $t$. The dislocation line and the Burgers vector $b=b_z$ are parallel to the $z$-axis. The dislocation velocity has components: $V_x=\dot{\xi}(t)$, $V_y=\dot{\eta}(t)$.
At first, we want to find the gauge theoretical solutions of the dislocation density and the dislocation currents for a nonuniformly moving screw dislocation. Thus, we have to solve the equations (\[KGE-T\])–(\[KGE-Iy\]), where the right-hand sides are given by the following sources $$\begin{aligned}
\label{DD-0}
T^0_{zxy}&=b\, \delta(x-\xi(t))\delta(y-\eta(t)), \\
\label{Ix-0}
I^0_{zx}&=V_y \, T^0_{zxy},\\
\label{Iy-0}
I^0_{zy}&=-V_x\, T^0_{zxy}\, .\end{aligned}$$ We consider the situation where the source terms have acted after initial quiescence at $t\rightarrow -\infty$. The solutions for the dislocation density and the dislocation currents are the convolution integrals[^4] $$\begin{aligned}
\label{C-T}
T_{zxy}&=G^{\rm KG}*T^0_{zxy},\\
\label{C-Ix}
I_{zx}&=G^{\rm KG}*I^0_{zx},\\
\label{C-Iy}
I_{zy}&=G^{\rm KG}*I^0_{zy},\end{aligned}$$ where the Green function of the $(1+2)$-dimensional Klein-Gordon equation is defined by $$\begin{aligned}
\big [1+\ell_1^2 \square_{\TT} \big] G^{\rm KG}=\delta(t)\delta(x)\delta(y)\, .\end{aligned}$$ The Green function of the Klein-Gordon equation in $(1+2)$ dimensions is given by [@Iwan] $$\begin{aligned}
\label{G-KG}
G^{\rm KG}
=\frac{1}{2\pi\ell_1^2}\,
\frac{H\big(t-r/c_\TT\big)}{\sqrt{t^2-r^2/c^2_\TT}}\,
\cos\bigg( \frac{\sqrt{c_\TT^2 t^2-r^2}}{\ell_1}\bigg),
\qquad r^2=x^2+y^2 \, ,\end{aligned}$$ where $H$ is the Heaviside step function. Substituting (\[DD-0\])–(\[Iy-0\]) and (\[G-KG\]) into (\[C-T\])–(\[C-Iy\]), the spatial integration can be performed to give the results $$\begin{aligned}
\label{T-s}
T_{zxy}&=\frac{b}{2\pi\ell_1^2}\int_{-\infty}^{t_{{\TT}}}\d t'\,
\frac{1}{S_\TT}\,\cos \bigg(\frac{c_\TT S_\TT}{\ell_1}\bigg),\\
\label{Ix-s}
I_{zx}&=\frac{b}{2\pi\ell_1^2}\int_{-\infty}^{t_{{\TT}}}\d t'\,
\frac{V_y(t')}{S_\TT}\,\cos \bigg(\frac{c_\TT S_\TT}{\ell_1}\bigg),\\
\label{Iy-s}
I_{zy}&=-\frac{b}{2\pi\ell_1^2 }\int_{-\infty}^{t_{{\TT}}}\d t'\,
\frac{V_x(t')}{S_\TT}\,\cos \bigg(\frac{c_\TT S_\TT}{\ell_1}\bigg)\end{aligned}$$ with the notations $$\begin{aligned}
\label{Not-s}
&\bar{x}=x-\xi(t')\, ,\qquad \bar{y}=y-\eta(t')\, ,
\qquad \bar{t}=t-t'\, , \qquad
\bar{R}^2=\bar{x}^2+\bar{y}^2, \nonumber\\
&S^2_{\TT}=\bar{t}^2-\frac{\bar{R}^2}{c^2_{\TT}}\, ,
\qquad t_{\TT}={t}-\frac{\bar{R}}{c_{\TT}}\, .\end{aligned}$$ Note that $t_\TT$ is called retarded time, which is the root of the equation $S_\TT^2=0$ and is less than $t$, and ($\bar{x}, \bar{y})$ is the distance between the field point $(x,y)$ and the position of the dislocation $(\xi,\eta)$. Here, $\xi$ and $\eta$ are the positions in the $x$ and $y$ directions of the dislocation at time $t'$, $V_x$ and $V_y$ are the velocity components of the dislocation at the same time $t'$. The retarded time $t_\TT$ is the time before $t$, when the dislocation caused an excitation of the dislocation field, which moves from $(\xi,\eta)$ to $(x,y)$ in the time $\bar{R}/c_\TT$. In addition, $t-t_\TT=\bar{R}/c_\TT$ is the time when the dislocation fields move from $(\xi,\eta)$ to $(x,y)$ with velocity $c_\TT$. Thus, from each point $\Bx'$, the dislocation fields (\[T-s\])–(\[Iy-s\]) draw contributions emitted at all times $t'$ from $-\infty$ up to $t_\TT$. That is the reason why @Eshelby51 said: ‘The dislocation is haunted by its past’.
Solutions for the elastic fields in terms of potential functions
----------------------------------------------------------------
If we multiply Eqs. (\[KGE-v-2\])–(\[KGE-Bzy-2\]) with $\square_{\TT}$ and use the ‘classical’ result [@Eshelby53; @Lardner], we obtain $$\begin{aligned}
\label{KGE-v-3}
&\big [1+\ell_1^2 \square_{\TT} \big]\square_{\TT} v_z=I^0_{zx,x}+I^0_{zy,y},\\
\label{KGE-Bzx-3}
&\big [1+\ell_1^2 \square_{\TT} \big]\square_{\TT} \beta_{zx}
=T^0_{zxy,y}+\frac{1}{c_{\TT}^2} \dot{I}^0_{zx},\\
\label{KGE-Bzy-3}
&\big [1+\ell_1^2 \square_{\TT} \big] \square_{\TT}\beta_{zy}
=T^0_{zyx,x}+\frac{1}{c_{\TT}^2} \dot{I}^0_{zy},\end{aligned}$$ as a set of fourth-order partial differential equations. Eqs. (\[KGE-v-3\])–(\[KGE-Bzy-3\]) have the two-dimensional form of the Bopp-Podolsky equations [@Bopp; @Podolsky] (see also [@Iwan]) in generalized electrodynamics, introduced by Bopp and Podolsky in order to avoid singularities in electrodynamics. As source terms only the classical dislocation density $T^0_{zxy}$ and dislocation currents $I^0_{zx}$, $I^0_{zy}$ are acting.
Following @Eshelby53, we express the elastic velocity and the elastic distortion in terms of ‘potential’ functions ($F$, $A_x$, $A_y$): $$\begin{aligned}
\label{vz-A}
v_z&=A_{y,x}-A_{x,y},\\
\beta_{zx}&=F_{,y}+\frac{1}{c^2_{\TT}}\, \dot{A}_y,\\
\beta_{zy}&=-F_{,x}-\frac{1}{c^2_{\TT}}\, \dot{A}_x.\end{aligned}$$ Then, if $A_x$, $A_y$ and $F$ are chosen to satisfy the subsidiary condition $$\begin{aligned}
A_{x,x}+A_{y,y}+\dot{F}=0,\end{aligned}$$ the field equations (\[KGE-v-3\])–(\[KGE-Bzy-3\]) become equivalent to the following inhomogeneous Bopp-Podolsky equations $$\begin{aligned}
\label{KGE-F}
&\big [1+\ell_1^2 \square_{\TT} \big]\square_{\TT} F=T^0_{zxy},\\
\label{KGE-Ax}
&\big [1+\ell_1^2 \square_{\TT} \big]\square_{\TT} A_x=-I^0_{zy},\\
\label{KGE-Ay}
&\big [1+\ell_1^2 \square_{\TT} \big]\square_{\TT} A_y=I^0_{zx}.\end{aligned}$$ The solution of these equations for an infinite medium is defined by $$\begin{aligned}
F&=G^{\rm BP}*T^0_{zxy},\\
A_x&=-G^{\rm BP}*I^0_{zy},\\
A_y&=G^{\rm BP}*I^0_{zx},\end{aligned}$$ where $G^{\rm BP}$ is the Green function of the Bopp-Podolsky equation (or wave-Klein-Gordon equation) $$\begin{aligned}
\label{BPE}
\big [1+\ell_1^2 \square_{\TT} \big]\square_\TT G^{\rm BP}=\delta(t)\delta(x)\delta(y).\end{aligned}$$ As in generalized electrodynamics, we solve (\[BPE\]) with the help of two fields $$\begin{aligned}
\label{BPE-2}
\big [1+\ell_1^2 \square_{\TT} \big] G^{\rm BP}=G^\square\, ,\qquad
\square_{\TT}G^\square=\delta(t)\delta(x)\delta(y) \end{aligned}$$ and we get $$\begin{aligned}
G^{\rm BP}=G^\square-\ell_1^2 G^{\rm KG},\end{aligned}$$ where the first field is the Green function of the two-dimensional wave equation [@Wl; @Barton] $$\begin{aligned}
\label{G}
G^\square=\frac{1}{2\pi}\,
\frac{H\big(t-r/c_\TT\big)}{\sqrt{t^2-r^2/c^2_\TT}}\end{aligned}$$ and the second one is the Green function of the Klein-Gordon equation (\[G-KG\]). Finally, we obtain for the Green function of the $(1+2)$-dimensional Bopp-Podolsky equation $$\begin{aligned}
\label{G-BP}
G^{\rm BP}
=\frac{1}{2\pi}\,
\frac{H\big(t-r/c_\TT\big)}{\sqrt{t^2-r^2/c^2_\TT}}\,
\bigg[1-
\cos\bigg( \frac{\sqrt{c_\TT^2 t^2-r^2}}{\ell_1}\bigg)\bigg] \, .\end{aligned}$$
Substituting from (\[DD-0\])–(\[Iy-0\]) for the case of a single dislocation, the spatial integrations can be performed to give the results for the potential functions of a screw dislocation $$\begin{aligned}
\label{F}
F&=\frac{b}{2\pi}\int_{-\infty}^{t_{{\TT}}}\d t'\,
\frac{1}{S_\TT}\,
\bigg[1-\cos \bigg(\frac{c_\TT S_\TT}{\ell_1}\bigg)\bigg],\\
\label{Ax}
A_x&=\frac{b}{2\pi}\int_{-\infty}^{t_{{\TT}}}\d t'\,
\frac{V_x(t')}{S_\TT}\,
\bigg[1-\cos \bigg(\frac{c_\TT S_\TT}{\ell_1}\bigg)\bigg],\\
\label{Ay}
A_y&=\frac{b}{2\pi}\int_{-\infty}^{t_{{\TT}}}\d t'\,
\frac{V_y(t')}{S_\TT}\,
\bigg[1-\cos \bigg(\frac{c_\TT S_\TT}{\ell_1}\bigg)\bigg].\end{aligned}$$ In analogy to electromagnetic field theory, we can call (\[F\])–(\[Ay\]) the (two-dimensional) ‘Liénard-Wiechert potentials’ of a screw dislocation in the gauge theory of dislocations.
Radiation part of the elastic fields
------------------------------------
On the other hand, using (\[Ix-0\]) and (\[Iy-0\]), we want to solve directly the equations (\[KGE-v-3\])–(\[KGE-Bzy-3\]). Consequently, we have to solve the equations $$\begin{aligned}
\label{KGE-v-4}
&\big [1+\ell_1^2 \square_{\TT} \big]\square_{\TT} v_z=
V_y\, T^0_{zxy,x}-V_x\, T^0_{zxy,y},\\
\label{KGE-Bzx-4}
&\big [1+\ell_1^2 \square_{\TT} \big]\square_{\TT} \beta_{zx}
=\frac{\dot{V}_y}{c_\TT^2}\, T^0_{zxy}
+\Big(1-\frac{V^2_y}{c^2_\TT}\Big)\, T^0_{zxy,y}
-\frac{V_x V_y }{c_{\TT}^2}\, T^0_{zxy,x},\\
\label{KGE-Bzy-4}
&\big [1+\ell_1^2 \square_{\TT} \big] \square_{\TT}\beta_{zy}
=-\frac{\dot{V}_x}{c_\TT^2}\, T^0_{zxy}
-\Big(1-\frac{V^2_x}{c^2_\TT}\Big)\, T^0_{zxy,x}
+\frac{V_x V_y }{c_{\TT}^2}\, T^0_{zxy,y},\end{aligned}$$ where the right-hand side is given by (\[DD-0\]). It can be seen that only the elastic distortions contain acceleration parts. The elastic velocity does not depend on dislocation acceleration. Using the Green function (\[G-BP\]), we obtain the result for the elastic distortions $$\begin{aligned}
\label{Bzx-s}
\beta_{zx}&=\frac{b}{2\pi c^2_\TT}\Bigg[
\int_{-\infty}^{t_{{\TT}}}\d t'\,\frac{\dot{V}_y(t')}{S_{{\TT}}}\,
\bigg[1-\cos \bigg(\frac{c_\TT S_\TT}{\ell_1}\bigg)\bigg]
+\frac{\pd}{\pd y}
\int_{-\infty}^{t_{{\TT}}}\d t'\,\frac{c^2_\TT-V^2_y(t')}{S_{{\TT}}}\,
\bigg[1-\cos \bigg(\frac{c_\TT S_\TT}{\ell_1}\bigg)\bigg]\nonumber\\
&\qquad
-\frac{\pd}{\pd x}
\int_{-\infty}^{t_{{\TT}}}\d t'\,\frac{V_x(t')V_y(t')}{S_{{\TT}}}\,
\bigg[1-\cos \bigg(\frac{c_\TT S_\TT}{\ell_1}\bigg)\bigg]
\Bigg],\\
\label{Bzy-s}
\beta_{zy}&=-\frac{b}{2\pi c^2_\TT}\Bigg[
\int_{-\infty}^{t_{{\TT}}}\d t'
\,\frac{\dot{V}_x(t')}{S_{{\TT}}}\,
\bigg[1-\cos \bigg(\frac{c_\TT S_\TT}{\ell_1}\bigg)\bigg]
+\frac{\pd}{\pd x}
\int_{-\infty}^{t_{{\TT}}}\d t'\,\frac{c^2_\TT-V^2_x(t')}{S_{{\TT}}}\,
\bigg[1-\cos \bigg(\frac{c_\TT S_\TT}{\ell_1}\bigg)\bigg]\nonumber\\
&\qquad
-\frac{\pd}{\pd y}
\int_{-\infty}^{t_{{\TT}}}\d t'\,\frac{V_x(t')V_y(t')}{S_{{\TT}}}\,
\bigg[1-\cos \bigg(\frac{c_\TT S_\TT}{\ell_1}\bigg)\bigg]
\Bigg].\end{aligned}$$ The solution of Eq. (\[KGE-v-4\]) for the elastic velocity has the same expression as given in Eq. (\[vz-A\]) with (\[Ax\]) and (\[Ay\]). It can be seen that the fields given by (\[Bzx-s\]) and (\[Bzy-s\]) consist of two parts:
\(a) Field depending on the dislocation velocities $V_x$ and $V_y$ alone and proportional to spatial derivatives – velocity field (b) Field depending on the dislocation accelerations $\dot{V}_x$ and $\dot{V}_y$ and proportional to $1/S_\TT$ – acceleration field or radiation field.
All our results depend on $\ell_1$ which is a characteristic length scale of the gauge theory of dislocations.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author has been supported by an Emmy-Noether grant of the Deutsche Forschungsgemeinschaft (Grant No. La1974/1-3).
T. Mura, Phil. Mag. [**8**]{} (1963) 843. T. Mura, Micromechanics of defects in solids, 2nd edition, Martinus Nijhoff, Dordrecht, 1987. R.W. Lardner, Mathematical Theory of Dislocations and Fracture, University of Toronto Press, Toronto, 1974. H. G[ü]{}nther, Zur Dynamik schneller Versetzungen, Akademie-Verlag, Berlin, 1973. J.D. Eshelby, Phil. Trans. Roy. Soc. London A [**244**]{} (1951) 87. J.D. Eshelby, Phys. Rev. [**90**]{} (1953) 248. W.S. Wladimirow, Equations of mathematical physics, Berlin: Deutscher Verlag der Wissenschaften (in German), 1971.
F.R.N. Nabarro, Phil. Mag. [**7**]{} (1951) 1224.
J. Kiusalaas, T. Mura, On the motion of a screw dislocation, in: Recent Advances in Engineering Science (A. C., Eringen, ed.) Gordon & Breach, New York, 1965, p. 543–464.
X. Markenscoff, J. Elasticity [**10**]{} (1980) 193. J. Kiusalaas, T. Mura, Phil. Mag. [**9**]{} (1964) 1. X. Markenscoff, R.J. Clifton, J. Mech. Phys. Solids [**29**]{} (1981) 253. L.M. Brock, Int. J. Engng. Sci. [**20**]{} (1982) 113.
A.C. Eringen, J. Appl. Phys. [**54**]{} (1983) 4703. A.C.Eringen, Nonlocal Continuum Field Theories, New York: Springer, 2002.
R.D. Mindlin, Arch. Rational. Mech. Anal. [**16**]{} (1964) 51. R.D. Mindlin, J. Elast. [**2**]{} (1972) 217.
A. Kadi[ć]{}, D.G.B Edelen, D.G.B. A Gauge Theory of Dislocations and Disclinations, in: Lecture Notes in Physics, Vol. 174, Springer, Berlin, 1983.
D.G.B Edelen, D.C. Lagoudas, Gauge theory and defects in solids, North-Holland, Amsterdam, 1988.
M. Lazar, Ann. Phys. (Leipzig) [**9**]{} (2000) 461.
M. Lazar, C. Anastassiadis, Phil. Mag. [**88**]{} (2008) 1673. H. Askes, E.C. Aifantis, Int. J. Fract. [**139**]{} (2006) 297. H. Askes, T. Bennett, E.C. Aifantis, Int. J. Meth. Engng. [**72**]{} (2007) 111.
M. Lazar, C. Anastassiadis, Phil. Mag. [**89**]{} (2009) 199. M. Lazar, Proc. R. Soc. A [**465**]{} (2009) 2505. M. Lazar, F.W. Hehl, Cartan’s spiral staircase in physics and, in particular, in the gauge theory of dislocations, Foundations of Physics, DOI: 10.1007/s10701-010-9440-4; \[arXiv:0911.2121\].
V. Rubakov, Classical Theory of Gauge Fields, Princeton University Press, Princeton, 2002.
D. Iwanenko, A. Sokolow, Klassische Feldtheorie, Akademie-Verlag: Berlin, 1953.
F. Bopp, Ann. Phys. (Leipzig) [**38**]{} (1940) 345.
B. Podolsky, Phys. Rev. [**62**]{} (1942) 68.
G. Barton, Elements of Green’s Functions and Propagation, Oxford University Press, 1989.
[^1]: [*E-mail address:*]{} lazar@fkp.tu-darmstadt.de (M. Lazar).
[^2]: We use the usual notations: $\beta_{ij,k}:=\pd_k \beta_{ij}$ and $\dot{\beta}_{ij}:=\pd_t \beta_{ij}$.
[^3]: The moment stress tensor $\tau_{ijk}=-\tau_{jik}$ can be obtained from the pseudomoment stress tensor: $\tau_{ijk}=-H_{[ij]k}$ (see [@LA09; @LH09]).
[^4]: The convolution is defined by: $T_{zxy}(\Bx,t)=G^{\rm KG}*T^0_{zxy}=
\int_{-\infty}^t \d t'\int\d \Bx'
G^{\rm KB}(\Bx,t,\Bx',t')\, T^0_{zxy}(\Bx',t')$.
|
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author:
- 'W. J. Henney[^1]'
title: 'Polarization Profiles of Scattered Emission Lines. [i]{}. General Formalism for Optically Thin Rayleigh Scattering'
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mn [Mon. Not. R. ast. Soc.]{}\
apj [Astrophys. J.]{}\
aa [Astr. Astrophys.]{}\
aj [Astr. J.]{}\
pasp [Publs. astr. Soc. Pacif.]{}
[Polarization Profiles of Scattered]{}\
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[Abstract]{}\
[ A general theoretical framework is developed for interpreting spectropolarimetric observations of optically thin emission line scattering from small dust particles. Spatially integrated and spatially resolved line profiles of both scattered intensity and polarization are calculated analytically from a variety of simple kinematic models. These calculations will provide a foundation for further studies of emission line scattering from dust and electrons in such diverse astrophysical environments as Herbig-Haro objects, symbiotic stars, starburst galaxies and active galactic nuclei. ]{}
[*Subject headings*]{}: ISM: dust, extinction—ISM: jets and outflows—ISM: reflection nebulae—polarization
Introduction
============
The Doppler shifts of optical emission lines which have been scattered by surrounding dust and electrons can provide useful information about the kinematics, geometry, and physical conditions of astrophysical flows. In principle, the scatterers can provide views of the line-emitting gas from different directions, allowing the 3-dimensional velocity of the emitting gas to be determined and revealing sources which are obscured from direct view. Unfortunately, the interpretation of these scattered emission lines is not straightforward since, in general, the geometry of the scattering and the velocity of the scatterers is unknown. If spectropolarimetric observations are available, then determination of the relative orientation and velocity of the source and the scatterers becomes more plausible, since the scattered light will be partially polarized to a degree dependent on the angle of scattering and on the details of the scattering process.
In this paper, a variety of very simple models of the scattering geometry and kinematics are investigated. Polarimetric line profiles are calculated (both spatially integrated and spatially resolved) for the cases of Rayleigh scattering by small dust particles, where the scattering phase function has a particularly simple form and it is feasible to derive analytic expressions for the scattered line profiles.
Previous work on the problem of the scattering of emission lines by dust or electrons has tended to involve detailed numerical radiative transfer modelling, often using Monte Carlo codes. Examples of the use of such techniques can be found applied to, $\!$, electron scattering in Wolf-Rayet winds [@Hil91] or scattering in circumstellar dust shells [@Lef92; @Mea92]. While such approaches have obvious merits, they are limited in their applicability, often needing to be specially tailored for each object, and it is difficult to draw general conclusions from them. A simpler approach is taken by , who model electron scattering in the circumstellar disks of Be stars, but, even here, numerical techniques are employed. In the present paper, numerical calculations are eschewed for the most part and scattered line profiles of intensity and polarization are calculated analytically. The price that must be paid in order to make this approach tractable is to restrict one’s attention to optically thin Rayleigh scattering of narrow emission lines which originate from a point source. The chief advantage of this approach is that it makes it possible to investigate a wide range of models for the geometry and kinematics of the scatterers and to develop a broad understanding of the ways in which these affect the line profiles. This understanding can then provide a firm foundation for further investigation into the effects of different phase functions (dust), thermal broadening (electrons), an extended source, absorption, multiple scattering and other complications. This development will be carried out in further papers of this series and detailed applications of the resultant techniques will be made to particular cases such as upstream dust scattering in Herbig-Haro objects [@Nor91], dust scattering in the jet of the symbiotic star R Aquarii [@Sol92], dust scattering in the superwinds of starburst galaxies [@Sca91] and electron scattering in active galactic nuclei [@Mil91]. Some preliminary results in these areas can be found in .
Rayleigh Scattering {#phase}
===================
Arbitrarily polarized light is fully described by means of the four-component Stokes vector [@Sto52] ${\bf I} \meq [I,Q,U,V]^T$, which can be functionally defined in terms of measurements of the light intensity as follows: $I$ is the total intensity of the light; $Q$ is the difference between the intensities after the light has passed through a linear dichroic plate with its transmitting axis aligned respectively perpendicular and parallel to a reference direction; $U$ is the difference between the intensities with the same plate aligned at $\pm45^{\circ}$ to the reference direction, and $V$ is the difference between the intensities after the light has passed through a right and left handed circular dichroic plate respectively. Unpolarized light has $Q\meq U\meq V \meq 0$ and fully polarized light has $I^2\meq Q^2 \mplus U^2 \mplus V^2$. Partially polarized light has a degree of polarization $$p \meq (Q^2+U^2+V^2)^{1/2}{\big /}I
\label{pdef}$$ and an angle of polarization $\chi$ (with respect to the reference direction in the definition of $Q$ and $U$ above) given by $$\tan 2 \chi \meq U/Q\strut \label{chidef2}.$$ Note that, since one must choose a reference direction corresponding to $\chi\meq0$, there is a certain arbitrariness in the specification of $Q$ and $U$. In astronomy this reference direction is conventionally taken to be North (or sometimes, the galactic plane). The effect on the Stokes vector of a rotation of the reference direction by an angle $\phi$ is to multiply it by the rotation matrix $${\bf M}(\phi) \meq
\left[ \begin{array}{cccc}
1& 0 & 0 & 0\\ 0 & \cos 2 \phi& \sin 2 \phi & 0 \\ 0 & -\sin 2
\phi & \cos 2 \phi& 0 \\ 0 & 0 & 0 & 1 \end{array} \right].$$
In the following discussion, it is imagined that light is emitted from a source at the origin and scattered by a single scatterer at $[x,y,z]$ toward an observer who is in the far field and is looking down the negative $z$-axis (see Figure \[scat1\]).
When the light is scattered, the components of the scattered Stokes vector are linearly related to those of the incident Stokes vector. Hence, the scattering cross-section can be written as a matrix. One way of representing the differential cross-section per solid angle $\Omega $ is then $$\sigmat \meq \frac{3\sigma_0}{16\pi} {\bf K},$$ where $\sigma_0$ is the mean scattering cross-section and the matrix ${\bf K}$ (the scattering kernel; ) contains the direction-dependent scattering information. The matrix $\sigmat$ is normalized so that $$\int_{4\pi} \sigmat {\bf I} \dd\Omega \meq \sigma_0 {\bf I}
\label{4pi}$$ for any Stokes vector ${\bf I}$. If ${\bf I}_0$ is the Stokes vector of the unscattered light at the source (in the direction of the scatterer), then, by considering the solid angle at the source subtended by the scatterer, the scattered Stokes vector will be $${\bf I}_{\rm S} \meq \sigmat{\bf I}_0/R^2,$$ where $R^2\meq x^2
\mplus y^2 \mplus z^2$. If the scatterers are much smaller than the wavelength of the light, then, [*so long as both are specified with respect to the scattering plane*]{}, the Stokes vector of the scattered light is related to that of the incident light via the Rayleigh scattering matrix [@b:Cha60 p 37]: $${\bf R}(\Theta) \meq \half
\left[ \begin{array}{cccc}
\csqbt \mplus 1& \csqbt \!-\! 1 & 0 & 0 \\
\csqbt \mmin 1& \csqbt \mplus 1& 0 & 0 \\ 0 & 0 &
2\cos \Theta & 0 \\ 0 & 0 & 0 & 2\cos \Theta \end{array} \right]
\label{rayleigh}$$ where $\Theta$ is the scattering angle, the angle between the incident and the scattered ray. In general, if the scattering plane is at an angle $\psi$ to the reference direction, then ${\bf R}$ must be pre- and post-multiplied by appropriate rotation matrices ${\bf M}$ to give the scattering kernel ${\bf K}(\Theta,\psi)$. In our example, the scattering kernel is $${\bf K}(\Theta,\psi) \meq {\bf M}(\pi-\psi) {\bf R}(\Theta) {\bf
M}(-\psi\prime)$$ where the angles $\psi$ and $\psi\prime$ are shown in Figure \[scat1\] and the reference direction is taken to be the $y$-axis. If the incident light is unpolarized, then $Q$, $U$, and $V$ are zero in the incident ray, so the ${\bf M}(-\psi\prime)$ rotation need not be carried out and the Stokes vector of the scattered light is given by $$\label{iscat1}
{\bf I}_{\rm S} \meq\frac{\sigma_0 I_0 }{4\pi R^2} {\bf X}(\Theta,\psi),$$ where $${\bf X}(\Theta,\psi) \meq \frac{3}{4} \left[ \begin{array}{c}
\cos^2\Theta \mplus 1\\
-\cos 2 \psi \sin^2\Theta\\
\sin 2 \psi \sin^2\Theta\\
0 \end{array}\right].
\label{ray_phase}$$ The vector scattering phase function ${\bf X}(\Theta,\psi)$ is normalized so that, when integrated over all solid angles, the $I$-component becomes $4\pi$ and all other components become zero. Since single scattering of initially unpolarized light cannot produce any $V$-component to the scattered Stokes vector, the $V$-component will be dropped in the rest of the paper and a reduced Stokes vector ${\bf I} \meq [I,Q,U]^T$ will be used. From equations (\[ray\_phase\]) and (\[pdef\]) it can be seen that the degree of polarization of Rayleigh scattered light is $$p\meq \sin\sq\Theta/(\cos\sq\Theta\mplus 1)
\label{pray}$$ while the angle of polarization $\chi$ is, using equation (\[chidef2\]), $$\chi \meq \phi \pm \pi/2.
\label{chiray}$$ Hence the scattered light is always partially polarized [*perpendicular*]{} to the plane of scattering and is fully polarized for scattering angles of 90$^{\circ}$.
Simple Scattering Geometries {#geom}
============================
In this section, three simple classes of problems are considered as examples of Rayleigh scattering: first, that of a stationary dust cloud scattering light from a moving unpolarized source at its center; second, that of an outflowing dusty wind; and third, that of a free-falling dusty inflow, the last two both scattering light from a stationary unpolarized central source. In the first instance, the scattering cloud/wind/infall is assumed to be spherically symmetric and the light source is assumed to be a point, although the first of these assumptions is subsequently relaxed for the first two cases.
Considering the spatially integrated light first, the three models are equivalent and it is evident on symmetry grounds that the scattered light will have no net polarization. This can be verified by taking the scattered intensity from a single scatterer (eq. \[\[iscat1\]\]) and integrating it over all the scatterers in the cloud. If the number density of scatterers is $n(R)$ and the cloud radius is $\rc$ then this yields $${\bf I}_{\rm S}\meq \frac{3 \sigma_0 I_0 }{16 \pi}
\int_{\mbox{cloud}} \frac{n(R)}{R^2}{\bf X}(\Theta,\psi)
\dd V
\meq \tau I_0
\left[ \begin{array}{c} 1\\0\\0 \end{array} \right]
\meq \tau {\bf I}_0
\label{totint}$$ where $\tau\meq \sigma_0 \int_0^{\rc} n\, \dd R$ is the scattering optical depth to the source. Of course, since $\tau\ll 1$, the scattered light makes a negligible contribution to the total intensity unless the direct light from the source is obscured.
Spatially Integrated Scattered Line Profiles {#velscat}
--------------------------------------------
### Moving Source in a Stationary Scattering Cloud {#mover}
If it is now supposed that the source of light is moving with a velocity $\vecu$ with respect to the (stationary) dust cloud and that the light it emits is monochromatic (of frequency $\nu_0$), then the scattered light will be Doppler shifted by $\Delta\nu\meq(\nu_0/c)
\rhat \cdot \vecu$, where $\rhat$ is the unit vector in the direction of the scatterer from the source. One can define a dimensionless velocity shift[^2] (frequency shift) of the scattered light as $v\meq-\rhat
\cdot \vecu / u_0 \equiv -\cos \gamma$. This is illustrated in
Figure \[vel\_diag\], in which the $x$-axis is taken to be along the projection of $\vecu$ onto the plane of the sky and the angle between $\vecu$ and the $x$-axis is denoted by $\alpha$. It is apparent that all scatterers that lie on the surface of a cone of opening half-angle $\gamma$ “see” the same Doppler shift ($v\meq-\cos\gamma$) in the light from the source. This will be called the “isovelocity cone” and $\xi$ will signify the angle around this cone, measured from a direction parallel to the $y$-axis (see Figure \[vel\_diag\]).
The spatially integrated scattered line shape can best be determined by considering the relationship between the two angular co-ordinate frames $[\gamma,\xi]$ and $[\Theta,\psi]$. Equating Cartesian components of a unit vector in the two co-ordinate frames gives
x & &\
y & &\
z & &
so that, using equation (\[ray\_phase\]), the scattering phase function becomes $${\bf X}(\gamma,\xi)\meq \frac{3}{4}
\left[ \begin{array}{c} a_0 \mplus a_1 \sin\xi
\mplus a_2 \sin\sq\xi \\ b_0 \mplus b_1 \sin\xi \mplus b_2 \sin\sq\xi
\mplus b_3 \cos\sq\xi\\ c_1 \sin\xi \mplus c_4 \sin\xi\cos\xi
\end{array} \right]$$ where[$$\begin{array}{lll}
a_0 \meq\sin\sq\alpha\cos\sq\gamma \mplus 1 ;\!\!& a_1\meq -\half \sin
2\alpha \sin 2\gamma;\!\!& a_2\meq \cos\sq\alpha\sin\sq\gamma;\!\!\\ b_0\meq
\cos\sq\alpha\cos\sq\gamma;\!\!& b_1\meq \half\sin2\alpha\sin2\gamma;\!\!& b_2
\meq\sin\sq\alpha\sin\sq\gamma ;\!\!\\ b_3 \meq -\sin\sq\gamma;\!\! & c_1\meq
-\cos\alpha\sin2\gamma;\!\!& c_4 \meq -2 \sin\alpha\sin\sq\gamma.
\end{array}$$ ]{} To find the scattered intensity in a velocity range $v_1$ to $v_2$ it is necessary to integrate equation (\[iscat1\]) over all the dust between the two isovelocity cones corresponding to $v_1$ and $v_2$. Hence, $${\bf I}(v_1 \rightarrow v_2) \meq \ifac \int_{\gamma =
\pi\mmin\cos^{-1}\!v_1}^{\pi\mmin\cos^{-1}\!v_2}
\int_{\xi = 0}^{2\pi} {\bf X}(\gamma,\xi) \sin\gamma\, \dd\gamma\, \dd\xi.$$ Since ${\dd} v \meq \sin\gamma \,\dd\gamma$, the specific intensity (per unit velocity range) is then $${\bf I}_v \meq \left| \frac{\dd {\bf I}\,}{\dd{\mit v}} \right| \meq \ifac
\int_0^{2\pi} {\bf X}(v,\xi)\, \dd \xi
\meq \frac{3}{8}\tau I_0 \left[ \begin{array}{c}
a_0\mplus \half a_2 \\ \!\!\!b_0\mplus \half
b_2 \mplus \half b_3\!\!\!\\ 0\end{array}\right]
\label{specintens}$$ so that
I\_v && I\_0 { 2$1\mplus v^2$ $1-3v^2$}
,\
Q\_v && I\_0 $1-3v^2$
,\
U\_v &&0
.
The polarization of the scattered light is then $$\label{integvel_p}
p_v\meq \frac{1-3v^2 \cos\sq\alpha}{2\(1\mplus
v^2\)\mplus\(1-3v^2\)\cos\sq\alpha},$$ while the angle of polarization $\chi$ is zero (perpendicular to the projected source velocity) when $Q_v$ is positive, or 90$^\circ$ (parallel to the projected source velocity) when $Q_v$ is negative.[^3] These quantities are plotted in
Figure \[intspec\] for various values of $\alpha$.
It should be noted that equation (\[integvel\_p\]) is more general than equations (\[integvel\]) and (\[integvel\_q\]) and applies to any distribution of scatterers that has cylindrical symmetry about the direction of the source velocity. This is because\[general\] the number density $n(R,\gamma,\xi)$ cancels out when taking the ratio of $Q$ and $I$. (Cylindrical symmetry is still required, since it is assumed when performing the $\xi$ integral in eq. \[\[specintens\]\].)
To illustrate how the intensity profile is modified if the scattering cloud is not spherical, the case of prolate and oblate ellipsoidal clouds will be considered.
#### Ellipsoidal Scattering Clouds:
If the scattering cloud is non-spherical, then the chance of a source photon being scattered is dependent on its initial direction. For a prolate ellipsoid of eccentricity $e$, with symmetry axis aligned with the source velocity, this can be translated into a direction dependent optical depth of $$\frac{\tau(\gamma)}{\tau_\star}\meq\(\frac{1-e^2\sin^2\alpha}
{1-e^2 \cos^2\gamma}\)^{1/2},$$ where $\tau_\star$ is the optical depth along the line of sight from the observer (corresponding to $\gamma\meq\pi/2
\mmin\alpha$). Hence, the intensity of the scattered light is given by $$\begin{aligned}
I_v & \meq & \frac{3}{8}\tau_\star I_0
\(\frac{1-e^2\sin^2\alpha}{1-e^2 v^2}\)^{1/2}\nonumber \\
& & \times
\left\{ 2\(1\mplus v^2\)
\mplus \(1-3v^2\)\cos\sq\alpha\right\}.\end{aligned}$$ A similar treatment for an oblate ellipsoid yields $$\begin{aligned}
I_v &\meq &\frac{3}{8}\tau_\star I_0
\(\frac{1-e^2\sin^2\alpha}{\(1-e^2\)\mplus e^2 v^2}\)^{1/2}\nonumber \\
& & \times
\left\{ 2\(1\mplus v^2\)
\mplus \(1-3v^2\)\cos\sq\alpha\right\}.\end{aligned}$$ The resultant line shapes are plotted for various values of $e$ and
$\alpha$ in Figures \[prol\_vel\] and \[ob\_vel\]. The polarization is not shown since it is the same as for a spherical cloud. It is apparent that prolate ellipsoids produce line shapes in which most of the intensity is concentrated toward $v\meq\pm1$ whereas oblate ellipsoids produce line shapes strongly peaked at $v\meq0$. The reason for this can be seen if one considers the distributions of scatterers in the two cases: in prolate clouds, most of the scatterers are concentrated along the direction of the source velocity, either “upstream” or “downstream”, while in oblate clouds they are concentrated toward the plane perpendicular to the source velocity and hence see little Doppler shift. It will be noticed that the integrated intensity (with respect to $v$) is not the same in each case. This is because the intensities are normalized with respect to $\tau_\star I_0$, where $\tau_\star$ is the scattering optical depth to the source along the line of sight. Generally, $\tau_\star$ is not equal to the angle-averaged optical depth (weighted by the phase function) $\bar{\tau}$, although it is this latter that determines the integrated scattered intensity. For example, a prolate ellipsoid at $\alpha\meq0$ or an oblate ellipsoid at $\alpha\meq\pi/2$ would each have $\tau_\star < \bar{\tau}$.
### Constant Velocity Scattering Wind {#convel1}
If the scattering wind is supposed to flow radially outward at constant speed $u_{\rm w}$, then the Doppler frequency shift in the scattered light will be $\Delta\nu\meq(\nu_0/c)u_{\rm w}(1\mplus \rhat
\cdot \zhat)$, where $\zhat$ is the unit vector along the $z$-axis (from the source toward the observer). In the same manner as for the moving source model, a dimensionless velocity shift $v\meq 1\mplus \rhat
\cdot \zhat $ is introduced which again defines isovelocity cones. In this model, however, the opening half-angle of the cone is the scattering angle $\Theta$ and this is related to the velocity shift by $\cos\Theta\meq v-1$. Determination of the scattered line profile is hence far simpler than in the preceding section, giving $${\bf I}_v \meq \ifac
\int_0^{2\pi} {\bf X}(v,\Psi)\, \dd \Psi
\meq \frac{3}{8}\tau I_0 \left[ \begin{array}{c}
v^2-2v\mplus 2\\ 0\\ 0 \end{array}\right].
\label{specintens2}$$ The scattered line is unpolarized because the isovelocity cones project onto the plane of the sky as circles. This line shape is illustrated in
Figure \[sp\_wind\_line\] and it can be seen that it is identical to the moving source model with $\alpha\meq \pi/2$, except for a shift in the velocity origin. If the wind is not spherically symmetric but is conical or disk-like in form, then the symmetry of the isovelocity cones is broken and polarized line profiles can result.
#### Outflowing Bicone:
The outflow axis of the wind is taken to make an angle $\alpha$ with the plane of the sky and the wind density is assumed to be independent of direction within the two cones of opening half-angle $\delta_{\rm c}$ about this axis, and zero outside the cones. These scattering cones will intersect a given isovelocity cone along 0, 2 or 4 radii with position angles ($\Psi_1 \dots \Psi_4$) given by $$\label{psicone}\begin{array}{ccccc}
\strut\sin\Psi_1&\meq&
\sin\Psi_2&\meq&\frac{\displaystyle\rule[-4pt]{0pt}{14pt}
\cos\delta_{\rm c}-\cos\theta\sin\alpha}{\displaystyle\rule[-4pt]{0pt}{14pt}
\sin\theta \cos\alpha},\\
\strut\sin\Psi_3&\meq&
\sin\Psi_4&\meq&\frac{\displaystyle\rule[-4pt]{0pt}{14pt}
-\cos\delta_{\rm c}-\cos\theta\sin\alpha}{\displaystyle\rule[-4pt]{0pt}{14pt}
\sin\theta\cos\alpha},\\
\end{array}$$ where $\theta\meq\cos^{-1}|v-1|$. The scattered Stokes vector is therefore given by $${\bf I}_v \meq \ifac
\left\{ \int_{\Psi_1}^{\Psi_2} {\bf X}(v,\Psi)\, \dd \Psi \mplus
\int_{\Psi_3}^{\Psi_4} {\bf X}(v,\Psi)\, \dd \Psi \right\},$$ which can be integrated to give the following:
I\_v && I\_0(v\^22v1) F\_1
------------------------------------------------------------------------
\
Q\_v && I\_0 (2vv\^2) F\_2
------------------------------------------------------------------------
\
U\_v&&0
------------------------------------------------------------------------
,
where
F\_1 {
[ll]{} 2&\
22(\_1\_3)&\
0 &
$\left\{\begin{array}{l}\left[\mbox{\bf if} \
\ \ \alpha\mplus\theta > \pi/2 \mplus
\delta \right. \\ \left. \ \mbox{\bf or} \ \ \ \alpha\mplus\theta <
\pi/2 \mmin \delta \right]\end{array}\right. $
\
2\_1 &
.\
\
F\_2 {
[ll]{} 0 &\
2\_12\_3&\
0 &
$\left\{\begin{array}{l}\left[\mbox{\bf if} \ \
\ \alpha\mplus\theta > \pi/2 \mplus \delta \right. \\ \left. \
\mbox{\bf or} \ \ \ \alpha\mplus\theta < \pi/2 \mmin \delta
\right]\end{array}\right. $
\
2\_1 &
.
In Figures \[bicone\_spec1\] and \[bicone\_spec2\], examples of these line shapes are given for narrow ($\delta_{\rm c}\meq0.2$ radians) and wide ($\delta_{\rm
c}\meq1.0$ radians) cones.
#### Outflowing Disk:
If the wind is confined to a thin equatorial disk, of angular half-thickness $\delta_{\rm d}$ then the scattered line profiles can be calculated by subtracting the result for a biconical wind from that of a spherical one so that $\delta_{\rm
d}\meq\pi/2\mmin\delta_{\rm c}$. Neglecting terms higher than quadratic in $\delta_{\rm d}$ (thin disk) then gives
I\_v && \_[d]{} I\_0
\
Q\_v && -\_[d]{} I\_0
,
where $\alpha$ is now the angle between the normal to the disk and the plane of the sky. Examples of these line shapes are given in Figure \[disk\_spec\].
### Free-falling Inflow {#freefall}
If a dusty flow is assumed to fall radially inwards from rest at infinity towards a gravitating body of mass $M_{\star}$ then, neglecting any deceleration, the infall velocity will be given by $u_{\rm in}\meq(R_0/R)^{1/2}u_0$ where $u_0
\meq (2 G M_{\star}/R_0)^{1/2}$ is the velocity at the inner cut off radius $R_0$ (which may be identified with the grain destruction radius). The inflow is supposed to have an outer radius $R_{\rm c}$ and mass conservation dictates that the dust number density have the form $n\meq n_0 (R_0/R)^{3/2}$. The Doppler frequency shift of the scattered light is $\Delta\nu\meq-(\nu_0/c)u_{\rm in}(1\mplus \rhat
\cdot \zhat)$, in a similar manner to the outflowing wind but with the opposite sign. In this case however the dust velocity is not constant, so the dimensionless velocity shift is dependent on the dust radius: $v\meq -(\epsilon_0/\br)^{1/2}(1\mplus\cos\Theta)$, where $\br\meq R/R_{\rm
c}$ and $\epsilon_0 \meq R_0/R_{\rm c}$. This means that the isovelocity surfaces are no longer cones but have the more complex 0[\_0]{}
shapes shown in Figure \[infall\_surf\]. By integrating the scattered intensity over these isovelocity surfaces, it is possible to determine the spatially integrated scattered line profile of the infall as $$\label{in_eq}
I_v\meq \frac{3}{8(1\mmin\ep0^{1/2})}
\times \left\{
\begin{array}{lc}
G_1 & \left[\mbox{\bf if} \ \ v <
-2\ep0^{1/2}\right] \\
& \\
G_2 & \left[ \mbox{\bf
if}\ \ v \geq -2\ep0^{1/2}\right]
\end{array}\right. ,$$ where
G\_1 & & 0 2v$\ep0^{-1/2}\mmin1$ v\^2$\ep0^{-1}\mmin1$\
G\_2 & & 2$-\half v$ 2 2v v\^2 .
Note that the scattered line profiles are unpolarized because the isovelocity surfaces are circularly symmetric from the point of view of the observer. Example line profiles are shown in
Figure \[infall\_spec\] for various values of $\ep0$. It can be seen that for high $\ep0$ (corresponding to a small spread in dust velocities with radius) the profile is similar to that from the constant velocity wind but with the sign of the Doppler shift reversed. As $\ep0$ is decreased there is a greater proportion of slow-moving dust and so the line profile becomes increasingly skewed toward $v\meq 0$.
Spatially Resolved Scattered Line Profiles: Position-Velocity Diagrams
----------------------------------------------------------------------
In this section, an analytic approach is used to calculate the Stokes intensities of the Rayleigh scattered light resolved both spatially and in velocity for two of the classes of models presented above.
### Moving Source in a Stationary Scattering Cloud {#isovel}
For the purposes of this section, the scattering cloud will be taken to be spherical and homogeneous. The observer is looking down the negative $z$-axis (as in Figure \[vel\_diag\]) and the position of a scatterer in the cloud is characterized by dimensionless coordinates $[\bx,\by,\bz]$ where $\bx\meq x/R_{\rm c}$ . Straightforward geometry then shows that the dimensionless Doppler shift of light scattered at $[\bx,\by,\bz]$ is $$v \meq \frac{\bx \cos\alpha \mplus \bz \sin \alpha} {\(\bx^2\mplus
\by^2 \mplus \bz^2\)^{1/2}}.
\label{vgeneral}$$ Hence, solving this equation for $\bz$, a line of sight $[\bx,\by]$ intersects a given isovelocity cone $v$ in zero, one or two places given by $$\begin{array}{ll}
\left\{\bz_{\rm a},\bz_{\rm b} \right\} \meq
\left\{\rule{0cm}{3ex}\right.\!\!\!\! & \bx\sin\alpha\cos\alpha \pm
v\left[\bx^2\(1-v^2\)\right.\\
& \left.\left. \mplus\by^2\(\sin\sq\alpha-v^2\)\right]^{1/2}
\rule{0cm}{3ex}\right\} \mtimes\(v^2-\sin\sq\alpha\)^{-1}
\end{array}
\label{zdef}$$ so long as $v$ satisfies $$\bx \cos\alpha \pm \left[1\mmin\(\bx^2\mplus\by^2\)\right]^{1/2}
< |v| < \(1\mmin \frac{\displaystyle\by^2\cos\sq\alpha}
{\displaystyle\bx^2\mplus\by^2}\)^{1/2},
\label{vlimits}$$ where the positive sign in the equation and inequalities applies to $\bz_{\rm a}$ and the negative sign to $\bz_{\rm b}$. The left-hand side of equation (\[vlimits\]) reflects the fact that most lines of sight will not intersect the source velocity vector, while the right-hand side is the result of the finite size of the scattering cloud. The scattered line profile along a line of sight can then be calculated as $${\bf I}_v \meq \left| \frac{\dd {\bf I}\,}{\dd v}\right| \meq
\frac{\tau F_0}{4\pi\thc^2}
\left( \sum_{i = a,b}
\frac{{\bf X}(\bx,\by,\bz_i)}
{\bx^2\mplus\by^2\mplus\bz_i^2}
\left|\frac{\dd \bz_i}{\dd v}\right| \right)
\label{basic_int}$$ where $F_0$ is the [*flux*]{} of the source and $\thc$ is the apparent angular radius of the scattering cloud. The phase function in Cartesian coordinates can be written as $${\bf X}(\bx,\by,\bz)\meq\frac{3}{4(\bx^2+\by^2+\bz^2)}
\left[ \begin{array}{c}
\bx^2\mplus \by^2 \mplus 2 \bz^2 \\
\bx^2 - \by^2 \\
- 2 \by \bz \end{array} \right]
\label{cartphase}$$ and from equation (\[vgeneral\]) it follows that $$\left|\frac{\dd \bz}{\dd v}\right| \meq
\left| \frac{\(\bx^2\mplus \by^2 \mplus \bz^2\)^{3/2}}
{\(\bx^2\mplus\by^2\)\sin\alpha-\bz\bx\cos\alpha} \right|.
\label{dzdv}$$ It is then necessary to eliminate $\bz$ from equation (\[basic\_int\]) using equation (\[zdef\]). For the general case this leads to very complicated expressions which are best calculated by computer but results for two special cases are presented here.
#### Source Velocity in Plane of the Sky ($\alpha\meq 0$): {#an_isovel}
If the dimensionless velocity shift $v$ lies between $\bx$ and $\bx/\(\bx^2\mplus\by^2\)^{1/2}$, then
I\_v&
& ,\
Q\_v&
& ,\
U\_v&
& ,
otherwise $I_v\meq Q_v \meq U_v \meq 0$. The degree and angle of polarization are given by
p\_v&
& ,\
&& \^[-1]{}$\frac{\by}{\bx}$
.
Note that $\chi$ is independent of $v$, which is true for all values of $\alpha$.
#### Source Velocity Along Line of Sight ($\alpha\meq\pi/2$):
For $\alpha\meq\pi/2$ the scattering geometry has circular symmetry from the point of view of the observer so it is convenient to introduce a dimensionless impact parameter $r\meq(\bx^2\mplus\by^2)^{1/2}$. Then, if $|v|\leq(1\mmin r^2)^{1/2}$,
I\_v&
& ,\
Q\_v&
& ,\
U\_v&
& ,
otherwise $I_v\meq Q_v \meq U_v \meq 0$. The degree of polarization is $$p_v\meq \frac{1\mmin v^2}{1\mplus v^2}$$ and the angle of polarization is still given by equation (\[chi\_v\]). Note that the polarization is independent of position on the cloud for a given velocity, as is the intensity (apart from a scale factor). However, the range of allowed velocities [*is*]{} strongly dependent on the position on the cloud.
#### Example Spectra and Position-Velocity Diagrams:
In Figure \[slits\] the positions on the scattering cloud of six apertures used in constructing example spectra, three slits and three point apertures, are shown. In Figures \[w1:ap:a\]
to \[w1:ap:f\], the spectra and position-velocity diagrams from these apertures are shown for different values of $\alpha$. These spectra include the broadening effects of the frequency profile of the source emission line and of atmospheric seeing. Both are assumed to be Gaussian in form; the source profile is taken to have a FWHM of $0.16
u_0$ and the seeing profile to have a FWHM of $0.11\thc$.
### Constant Velocity Scattering Wind {#convel2}
#### Spherically symmetric wind:
For a constant velocity wind (assuming a constant dust-gas ratio), mass conservation requires the dust number density to be of the form $$n(R) \meq n_0 \(\frac{R_0}{R}\)^2 \meq \frac{n_0 \epsilon_0^2}
{r^2\mplus \bz^2}$$ where $n_0$ is the number density at an inner cut-off radius $R_0\meq\epsilon_0 R_{\rm c}$. The scattering optical depth to the source is then given by $$\tau\meq \sigma_0 n_0 R_{\rm c}\epsilon_0(1\mmin \epsilon_0)$$ so that the the line profiles are given by $${\bf I}_v \meq
\frac{\tau F_0}{4\pi\thc^2}\frac{\epsilon_0}{1\mmin\epsilon_0}
\frac{{\bf X}(\bx,\by,\bz)}
{(r^2\mplus\bz^2)^2}
\left|\frac{\dd \bz}{\dd v}\right|
\label{basic_int2}.$$ Note that in this instance a line of sight can only intersect an isovelocity cone in, at most, one place and that the situation is very similar to the $\alpha\meq\pi/2$ case of § \[isovel\]. Hence, if $(1\mmin(r/\epsilon_0)^2)^{1/2}\leq |1\mmin v| \leq (1\mmin
r^2)^{1/2}$ then
\[wind\] I\_v&&\
p\_v&&,
otherwise $I_v\meq 0$ ($\chi$ is still given by eq. \[\[chi\_v\]\]). As in the $\alpha\meq\pi/2$ case of § \[isovel\], variation in the line profiles with position is chiefly caused by the conditions for intersection between the line of sight and isovelocity cone.
#### Outflowing Bicone:
With a conical wind, the intensity and polarization profiles will still be given by equation (\[wind\]) but the range of allowed velocities for the scattered light from a given line of sight is subject to the additional condition that the line of sight must intersect the cone of scatterers. To simplify matters, only lines of sight that intersect the cone symmetry axis ($\by\meq0$) are considered. In this case, the conditions that there be scattered flux at velocity $v$ from a line of sight $\bx$ are given in Table 1,
[TABLE 1]{}\
[Conditions for there to be Scattered Flux from an Outflowing Bicone at a Point $\bar{x}$, $v$ in Position-Velocity Space]{}\
[c]{}\
\
\
[—See text for explanation of the symbols.]{}
where all three conditions must be satisfied using either entirely the left terms in curly brackets or entirely the right. The position velocity diagrams so obtained (corresponding to a slit placed along the projected axis of the conical outflow) are illustrated in
Figures \[conepv1\] and \[conepv2\] for both a narrow cone and a wide cone at various inclinations to the plane of the sky.
#### Outflowing Disk:
Since the disk is assumed thin, the spatially resolved scattered line profile at each position on the disk will just be a single spike. Hence, the position-velocity diagram for a slit placed along the projected minor axis of the disk ($\by\meq 0$) will be a ridge at $v\meq 1
\mmin \cos\alpha$ for negative $\bx$ and a ridge at $v \meq 1 \mplus
\cos \alpha$ for positive $\bx$, both with a degree of polarization $\sin\sq\alpha/(1\mplus\cos\sq\alpha)$, whereas for a slit along the projected major axis of the disk ($\bx\meq 0$) the position-velocity diagram is merely a ridge at $v\meq 0$ with 100% polarization.
Discussion
==========
The calculations presented in this paper have been selected because they are amenable to analytic treatment and hence represent extreme idealizations of situations likely to be encountered in astrophysical objects. Nevertheless, they demonstrate in a simple fashion the type of scattered line shapes that will be produced in different situations.
Moving Source
-------------
The moving source model (§§ \[mover\] and \[isovel\]) is applicable to any case in which a line-emitting plasma is moving with respect to a dusty environment, such as Herbig-Haro objects. This is discussed in much greater detail in and , Papers [II]{} and [III]{} of this series. It may also furnish an alternative explanation to that proffered in for the enormous line widths observed in knots of the collimated outflow from R Aquarii.
It is found that the spatially integrated line shapes from these models show scattered wings extending a distance equal to the source velocity $u_0$ to both sides of the rest frequency of the line (Figure \[intspec\]). These wings are polarized to a degree dependent on the inclination of the source velocity to the line of sight, being highest when the source is moving in the plane of the sky and zero if the source is moving directly toward or away from the observer. Of course, the Doppler shift of the [*direct*]{} light from the source means that a source moving toward the observer would be seen to have an unpolarized red wing of width $2u_0$ and a source moving away from the observer would have a similar unpolarized blue wing, while a source moving in the plane of the sky would have polarized red and blue wings. In this latter case, the polarization is highest at the extremes of the wings, drops to zero and then rises again toward line center (although in the center dilution by the intrinsic light will reduce the observed polarization substantially). Departure from spherical symmetry of the scattering cloud (Figures \[prol\_vel\] and \[ob\_vel\]) causes the blue- and red-shifted scattered light to become more (less) intense than the unshifted light if the cloud is prolate (oblate).
Of the results for spatially resolved line shapes that are presented in § \[isovel\], it is perhaps those for Aperture A of Figure \[slits\] (corresponding to a narrow slit placed along a diameter of the cloud, parallel to the direction of motion of the source) that are the most interesting (Figure \[w1:ap:a\]). For the source velocity in the plane of the sky, these show a characteristic “double triangle” morphology to the position-velocity diagram of the scattered light. The scattered light is blue-shifted to the right (upstream) of the source and red-shifted to the left (downstream). The polarization is highest for the scattered light that undergoes the largest Doppler shift (red or blue) since it is the dust directly in front of or behind the source that scatters light through $90^{\circ}$, leading to maximum polarization (Eq. \[\[pray\]\]). As the angle $\alpha$ between the source direction and the plane of the sky increases, some upstream dust appears to the left of the source and some downstream dust to the right. Eventually, when the source is moving directly toward or away from the observer, this leads to the situation illustrated in the bottom-right panel of Figure \[w1:ap:a\] in which the position-velocity diagram is symmetrical about $x\meq0$. In this latter case, it is the scattered light which undergoes no Doppler shift which has the highest polarization since the dust in the plane of the sky is now directly to the sides of the motion of the source. Note that, for both $\alpha\meq0$ and $\alpha\meq\pi/2$, the area of the position-velocity diagram in which the direct light from the source will lie ($x\meq0$, $v\meq0$ for $\alpha\meq0$, $v\meq\pm 1$ for $\alpha\meq\pm\pi/2$) is well away from the area of highest polarization so there will be little dilution of the scattered light there. Note also that, unlike in the spatially integrated case, the scattered light is polarized for all inclinations of the source velocity.
Scattering Wind
---------------
These models (§§ \[convel1\] and \[convel2\]) are applicable to any case in which chromospheric emission lines are scattered from a stellar wind. This may be dust scattering, as is possible in evolved late-type stars such as Mira variables [@Ana93; @Lef92], or electron scattering, in Be stars [@Woo93], B\[e\] supergiants [@Boy91], or Wolf-Rayet stars [@Sch92]. They are also relevant to the scattering by dust and electrons in outflows from active galactic nuclei [@Mil90a]. Of course, for electron scattering the thermal broadening of the electron velocity distribution will significantly modify the results presented here for all but the highest Mach number winds (the thermal speed of the electrons will exceed the bulk wind velocity so long as the Mach number $M \leq 2
(m_{\rm H}/m_{\rm e})^{1/2} \msim 80$), a fact that is ignored in . The extension of these results to Thomson scattering from thermal electrons will be presented in a forthcoming paper.
One obvious result is that some asymmetry of the wind is necessary for the integrated scattered line profile to be polarized. For a spherical wind (Figure \[sp\_wind\_line\]), the scattering merely produces an unpolarized red wing, extending up to twice the wind velocity $u_{\rm
w}$. If the wind is concentrated toward the polar directions (Figures \[bicone\_spec1\] and \[bicone\_spec2\]), then, for outflows in the plane of the sky, one finds a polarized red bump at a redshift of $u_{\rm w}$ whose width depends on the degree of collimation of the wind. As the outflow axis is rotated toward the observer (increasing $\alpha$), the bump splits into two, which move apart to redshifts of zero and $2u_{\rm w}$ as $\alpha\rightarrow\pi/2$ and whose polarizations diminish to zero. For equatorially enhanced winds (approximated as a disk; Figure \[disk\_spec\]), the intensity profiles of the scattered lines for a given inclination of the symmetry axis $\alpha$ can be seen to be qualitatively similar to the profiles for a narrow cone (Figure \[bicone\_spec1\]) with inclination $\pi/2\mmin\alpha$, as would be expected. The polarization profiles do not follow this pattern, however, and show a maximum at a red shift of $u_{\rm w}$ and a general decrease in polarization as the disk changes from an edge-on to a face-on orientation.
The spatially resolved position-velocity diagrams are not illustrated for the spherically symmetric wind but they are very similar to those for the moving source model with $\alpha\meq\pi/2$ (bottom-right panels of Figures \[w1:ap:a\]–\[w1:ap:c\]), except for three differences:
1. The Doppler shifts will range from 0 to 2$u_{\rm w}$ instead of from $-u_{\rm w}$ to $u_{\rm w}$.
2. In the spectrogram from Aperture A, there will be an elliptical “hole” in the center because of the inner cut-off to the density distribution.
3. The scattered brightness will fall off more rapidly with distance from the center of the cloud because of the inverse-square density distribution in the wind.
With the conical wind (Figures \[conepv1\] and \[conepv2\]), the position-velocity diagrams are the same as for the spherical wind but with certain portions masked out. The sizes and positions of the non-empty regions of position-velocity space depend on the opening angle and orientation of the cones.
Scattering Inflow
-----------------
The free-falling inflow model (§ \[freefall\]) is applicable to the scattering haloes of young stellar objects, which often show density profiles indicative of an infalling envelope [@Wei92].
The spatially integrated line shapes (Figure \[infall\_spec\]) can be seen to depend sensitively on the value assumed for the ratio of inner to outer dust radius $\ep0$ (see discussion after eq. \[\[in\_eq\]\]). It should be remembered that the dimensionless Doppler shift $v$ is scaled to the infall velocity at the inner cut-off radius, which is proportional to $\ep0^{-1/2}$, so that the velocity scales of Figure \[infall\_spec\] are different if all the inflows are assumed to have the same outer radius and central mass.
Calculations of the spatially resolved line profiles from this model are not presented here because they are too involved for the simple analytic approach used in this paper. In addition, it is likely that many young stellar objects have optically thick haloes [@Whi93] and in such cases the single scattering model presented here is not strictly applicable. These issues will be addressed in Paper [III]{}, in which Monte Carlo simulations of multiple scattering will be made and where the effects of a non-zero angular momentum for the infall will be treated.
Summary
=======
This paper has presented a small collection of line profiles (both intensity and polarization) that result from the optically thin Rayleigh scattering of an emission line by dust in its environment. The calculations have all been performed analytically and this has necessarily restricted the complexity of the models employed. Nonetheless, they form a basis for understanding the ways in which the geometric and kinematic relations between the line source and the scatterers determine the scattered line profiles.
Although, the calculations have been performed for the optically thin case, for one of the kinematic models considered (moving source in a stationary cloud) large optical depths will make very little difference to the scattered line profiles. This is because there is no relative motion between the scatterers themselves and hence the only Doppler shift is that induced by the first scattering. Of course, for the wind and infall models, the optically thick case will be quite different, with much more extended wings to the profiles due to multiple scattering. However, the polarization will also be reduced compared with the single scattering case so that the optically thin models are perhaps more likely to be relevant to sources with observable polarization changes across their line profiles.
One effect that has been ignored in this paper and that can have a significant effect on the line profiles is that of a non-Rayleigh scattering phase function for the dust. This is probably important at optical wavelengths, where the scattering phase function can be quite forward-peaked unless the dust grains are smaller than usual. This issue will considered in depth in Paper III.
I gratefully acknowledge financial support from SERC, UK and CONACyT, México and I would like to thank D. J. Axon and A. C. Raga for many helpful discussions during the course of this work.
[Noriega-Crespo, Calvet & Böhm<1991>]{}
Anandarao, B. G., Pottasch, S. R., & Vaidya, D. B. 1993, A&A, [273]{}, 570
Boyd, C. J., & Marlborough, J. M. 1991, ApJ, [369]{}, 191
Chandrasekhar, S. 1960, (New York: Dover)
Henney, W. J., Axon, D. J., & Raga, A. C. 1994, in preparation (Paper III)
Henney, W. J., Raga, A. C., & Axon, D. J. 1994, ApJ, in press (Paper II)
Henney, W. J. 1992, University of Manchester
Hillier, D. J. 1991, A&A, [247]{}, 455
Lefèvre, J. 1992, A&A, [254]{}, 274
Meaburn, J., Walsh, J. R., & Wolstencroft, R. D. 1992, A&A, [268]{}, 283
Miller, J. S., & Goodrich, R. W. 1990, ApJ, [355]{}, 456
Miller, J. S., Goodrich, R. W., & Matthews, W. G. 1991, ApJ, [378]{}, 47
Noriega-Crespo, A., Calvet, N., & Böhm, K.-H. 1991, ApJ, [379]{}, 676 (NCB)
Pomraning, G. C. 1973, (Oxford, UK: Pergamon)
Scarrott, S. M., Eaton, N., & Axon, D. J. 1991, MNRAS, [252]{}, 12[p]{}
Schulte-Ladbeck, R. E., Nordsiek, K. H., Taylor, M., Bjorkman, K. S., Magalh[ã]{}es, A. M., & Wolff, M. J. 1992, ApJ, [387]{}, 347
Solf, J. 1992, A&A, [257]{}, 228
Stokes, G. G. 1852, Trans. Cambridge Phil. Soc., [9]{}, 399 Reprinted in
Swindell, W., ed. 1975, (Stroudsburg, USA: Dowden, Hutchinson & Ross)
Weintraub, D. A., Kastner, L. H., Zuckerman, B., & Gatley, I. 1992, ApJ, [391]{}, 784
Whitney, B. A., & Hartmann, L. 1993, ApJ, [402]{}, 605
Wood, K., Brown, J. C., & Fox, G. K. 1993, A&A, [271]{}, 492
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abstract: 'We present a weak-lensing analysis of the galaxy cluster CL J1226+3332 at $z\simeq0.89$ using [*Hubble Space Telescope*]{} Advanced Camera for Surveys images. The cluster is the hottest ($>10~$keV), most X-ray luminous system at $z>0.6$ known to date. The relaxed X-ray morphology, as well as its high temperature, is unusual at such a high redshift. Our mass reconstruction shows that on a large scale the dark matter distribution is consistent with a relaxed system with no significant substructures. However, on a small scale the cluster core is resolved into two mass clumps highly correlated with the cluster galaxy distribution. The dominant mass clump lies close to the brightest cluster galaxy whereas the other less massive clump is located $\sim40\arcsec$ ($\sim310$ kpc) to the southwest. Although this secondary mass clump does not show an excess in the X-ray surface brightness, the gas temperature of the region is much higher ($12\sim18$ keV) than those of the rest. We propose a scenario in which the less massive system has already passed through the main cluster and the X-ray gas has been stripped during this passage. The elongation of the X-ray peak toward the southwestern mass clump is also supportive of this possibility. We measure significant tangential shears out to the field boundary ($\sim1.5$ Mpc), which are well described by an Navarro-Frenk-White profile with a concentration parameter of $c_{200}=2.7\pm0.3$ and a scale length of $r_s=78\arcsec\pm19\arcsec$ ($\sim600$ kpc) with $\chi^2/$d.o.f=1.11. Within the spherical volume $r_{200}=1.6$ Mpc, the total mass of the cluster becomes $M(r<r_{200})=(1.4\pm0.2)\times10^{15}~M_{\sun}$. Our weak-lensing analysis confirms that CL1226+3332 is indeed the most massive cluster known to date at $z>0.6$.'
author:
- 'M. JAMES JEE and J. ANTHONY TYSON'
title: 'DARK MATTER IN THE GALAXY CLUSTER CL J1226+3332 AT $Z=0.89$'
---
INTRODUCTION \[section\_introduction\]
======================================
In the hierarchical structure formation paradigm with cold dark matter (CDM), galaxy clusters grow through multiple mergers between groups and smaller clusters of galaxies. Accordingly, on average galaxy clusters found at higher redshifts should be less massive and more irregular. Despite many unresolved issues in detail on how these structures grow over time, because cluster assembly on a large scale is governed by CDM only subject to gravity, it is possible to quantitatively predict a cluster mass function for a given cosmology at a specific epoch through either analytic approaches or numerical simulations. Therefore, comparison of cluster mass functions today and at high redshift has been a powerful tool in constraining cosmological parameters.
Although large samples of high-redshift clusters have been compiled in the past decade through extensive surveys, the principal ambiguity yet to be resolved is how to relate the measured cluster properties to the mass of the system in a quantitatively robust way. This issue becomes more important for the most massive clusters at high redshifts ($z\sim1$). The statistics of these rare systems are extremely sensitive to the matter content ($\Omega_M$) of the universe and its fluctuation ($\sigma_8$). A historic example is MS1054-0321 (hereafter MS1054), the hottest cluster in the Einstein Medium Sensitivity Survey (EMSS) at $z=0.83$. The mere existence of such a massive cluster is thought to be problematic in the $\Omega_M=1$ universe and has been frequently used as an argument for low $\Omega_M<1$ values (Bahcall & Fan 1998; Donahue et al. 1998; Jeltema et al. 2001). It is interesting to note, however, that the intracluster medium (ICM) temperature of MS1054 has been debated (Donahue et al. 1998; Jeltema et al. 2001; Joy et al. 2001; Vikhlinin et al. 2002; Tozzi et al. 2003; Gioia et al. 2004; Jee et al. 2005b; Branchesi et al. 2007) and this challenges the mass estimate based on X-ray observations alone. Moreover, the complicated substructures of MS1054 ubiquitously present in optical, X-ray, and weak-lensing observations make the validity of the hydrostatic equilibrium assumption (commonly questioned even for a relaxed system) more open to doubts. However, weak-lensing analyses, which do not depend on the dynamical state of the cluster, from [*Hubble Space Telescope (HST)*]{} Wide Field Planetary Camera 2 (WFPC2) (Hoekstra et al. 2000) and Advanced Camera for Surveys (ACS) (Jee et al. 2005b) show that MS1054 is indeed a massive system at $z=0.83$.
Presumably, the galaxy cluster CL J1226.9+3332 (hereafter CL1226) at $z=0.89$ is much more massive than MS1054. The recent measurements of the X-ray temperature $10.4\pm0.6$ keV and the total X-ray luminosity ($L_X = 5.12\pm 0.12~\mbox{erg}~\mbox{s}^{-1}$) of the cluster by Maughan et al. (2007) show that the cluster is the hottest, most luminous system at $z>0.6$ known to date. These X-ray properties suggest that CL1226 might be perhaps the most massive $z>0.6$ structure as well; the detailed 3-dimensional mass structure analysis of Maughan et al. (2007) derives an enclosed mass $M(r<0.88 \mbox{Mpc})=5.2_{-0.8}^{+1.0}\times 10^{14}~M_{\sun}$. The relaxed X-ray morphology of the cluster is also remarkable; the cluster’s X-ray surface brightness distribution is symmetric with a single X-ray peak in spatial agreement with the brightest cluster galaxies. This is in stark contrast with the X-ray images of CL0152-0152 (hereafter CL0152; Maughan et al. 2003) and MS1054 (Jeltema et al. 2001). These two high-redshift clusters at a similar redshift of $z\sim0.8$ show multiple peaks in their X-ray images indicative of their active stage of formation. Within the hierarchical structure formation paradigm, a virialized structure like CL1226 with such a high mass is extremely rare at $z=0.89$, when the age of the universe is less than half its current value.
In this paper, we present a weak-lensing study of CL1226 with HST/ACS images. By analyzing weak distortions of background galaxies behind the cluster, we aim to address the following issues. First, we will examine how the lensing mass compares the X-ray value. Although the analysis of Maughan et al. (2007) using both $Chandra$ and XMM-$Newton$ data was careful and certainly more sophisticated than the common isothermal $\beta$ model approach, the X-ray method is by nature still dependent on the validity of the hydrostatic equilibrium assumption. Because the relaxed X-ray morphology of the system supports this hypothesis, it provides an interesting opportunity to compare weak-lensing and X-ray estimates on a fair basis. In addition, Maughan et al. (2007) found that their X-ray mass is $\sim30$% lower than the predicted $M-T$ (Vikhlinin et al. 2006) and $Y_X-T$ (Kravtsov et al. 2006) relation. If the cause of this departure is an on-going merger activity as suggested by Maughan et al. (2007), our weak-lensing mass estimate should improve the cluster’s scaling relation. Second, we will investigate if our two-dimensional mass reconstruction uncovers any substructure not observed in X-rays. In Jee et al. (2005a; 2005b), we witnessed that the weak-lensing mass in CL0152 and MS1054 traces the cluster galaxy distribution very closely whereas the X-ray maps often do not always show the same details. Therefore, it is probable that the weak-lensing analysis of CL1226 might reveal some significant substructures that have not been detected by X-rays. In particular, Maughan et al. (2007) noted that the ICM temperature of the region $\sim40\arcsec$ to the southwest of the X-ray peak is much higher ($12-18~$keV) than the average value ($\sim10.4$ keV). Although this hot region is correlated with the cluster galaxy distribution, the structure does not stand out in X-ray surface brightness. Perhaps either the mass associated with this galaxy group is not significant enough to produce X-ray overdensity, or the X-ray gas has been stripped in a previous pass-through. Our weak-lensing measurement of the mass of this substructure will allow us to address this question.
Throughout the paper, we use a $(h,\Omega_M,\Omega_{\Lambda})=(0.7,0.3,0.7)$ cosmology. All the quoted uncertainties are at the 1-$\sigma$ (68%) level.
OBSERVATIONS \[section\_obs\]
=============================
Data Reduction and Object Detection
-----------------------------------
CL1226 was observed with the Wide Field Camera (WFC) of ACS during April 2004 (PROP ID:9033, PI:Harald Ebeling). The cluster was imaged in F606W and F814W (hereafter $v_{606}$ and $i_{814}$, respectively) in a 2$\times$2 mosaic pattern covering $\sim6\arcmin\times6\arcmin$ with integrated exposure per pointing of $4000$ s. We used CALACS (Hack et al. 2003) to perform low level CCD processing and APSIS (Blakeslee et al. 2003) to create final mosaic images. A Lanczos3 kernel (windowed sinc function) with a 0.05$\arcsec$ output scale was chosen for drizzling (Fruchter and Hook 2002). This combination of drizzling parameters has been extensively tested in our previous weak-lensing analyses (e.g., Jee et al. 2005a), and provides relatively sharp point spread functions (PSF) with small noise correlations between pixels. The pseudo-color composite of the cluster image is shown in Figure \[fig\_cl1226\]. We use the $v_{606}$ and $i_{814}$ images to represent the blue and red intensities, respectively, whereas the mean of the two images is used for the green intensity. Figure \[fig\_cl1226\]a displays the entire $\sim6\arcmin\times6\arcmin$ field of the cluster. Approximately, north is up and east is left. The “feather-like" feature near the northwestern corner is attributed to the internal reflection of bright stellar light. The central $30\arcsec\times30\arcsec$ region marked with a yellow square is magnified in Figure \[fig\_cl1226\]b. The cluster red-sequence galaxies are easily identified by their distinct colors. We also observe many blue strongly lensed arc candidates.
We created a separate detection image by weight-averaging the two passband images with their inverse variance maps and SExtractor (Bertin & Arnouts 1996) was run in dual-image mode; objects were detected by searching for at least 5 contiguous pixels above 1.5 times sky rms values in the detection image while photometry is performed on each passband image. We set [CLEAN=Y]{} with [CLEAN\_PARAM=1.2]{} to let spurious detections around the brightest objects be automatically removed. By visual inspection we identified additional 736 objects that should not be used for weak-lensing. They include stars, diffraction spikes, merged/fragmented objects, missed cosmic rays, stray light near very bright objects, etc. The final catalog contains a total of 11910 objects.
Source Galaxy Selection \[section\_source\_selection\]
------------------------------------------------------
We base our source galaxy selection on the objects’ $v_{606}-i_{814}$ colors and $i_{814}$ magnitudes, assuming that a significant fraction of background galaxies are faint ($i_{814}>24$) and bluer ($v_{606}-i_{814}<1.2$) than the red-sequence of the cluster. This is a common approach in weak-lensing analyses when limited HST colors are available. Although complementary ground-based observations help us to obtain good photometric redshifts for bright objects \[e.g., COSMOS (Scoville et al. 2006) or AEGIS (Davis et al. 2007)\], most lensing signals (especially for high-redshift clusters such as CL1226) come from a very faint population, for which it is difficult to obtain reliable, HST PSF-matched colors from the ground.
We show the $i_{814}$ versus $v_{606}-i_{814}$ color magnitude diagram of CL1226 in Figure \[fig\_cmr\]; we use SExtractor’s MAG\_AUTO and MAG\_ISO values for $i_{814}$ magnitudes and $v_{606}-i_{814}$ colors, respectively. The redshifted 4000 Å break of galaxies at $z=0.89$ is nicely straddled by the $v_{606}$ and $i_{814}$ filters. This makes the cluster galaxies clearly visible at $v_{606}-i_{814}\sim1.8$. down to $i_{814}\sim26$. We define our source population as the $24<i_{814}\lesssim28$ and $v_{606}-i_{814}<1.2$ galaxies with ellipticity errors less than 0.2 in one of the two filters (see \[section\_ellipticity\] for details on the ellipticity measurement). The resulting number density is $\sim124$ $\mbox{arcmin}^{-2}$ (a total of 4745 objects). Considering the high-redshift of the cluster, we suspect that a non-negligible fraction of the cluster members might be bluer than the cluster red-sequence (Butcher & Oemler 1984). Therefore, it is important to estimate how much our source catalog might contain blue cluster members despite the clear presence of the red-sequence. For control fields, we use the Great Observatories Origins Deep Survey (GOODS; Giavalisco et al. 2004) ACS images and the Ultra Deep Field (UDF; Beckwith 2003) ACS images. Because of the large field, the GOODS data serves as a fair comparison sample whereas the UDF data provides good statistics of faint galaxies beyond the detection limit of our cluster observation. The F775W filter (hereafter $i_{775}$) is used in both GOODS and UDF observations instead of the $i_{814}$ filter. Hence, we transformed $v_{606}-i_{775}$ colors to $v_{606}-i_{814}$ colors to maintain the consistency. After selecting galaxies with the same selection criteria ($24<i_{814}\lesssim28$ and $v_{606}-i_{814}<1.2$), we compared the magnitude distribution of these surveys with that of the cluster observation (Figure \[fig\_hist\]). The comparison shows no indication of excess in the cluster field due to the potential blue cluster member contamination. This result is consistent with our results in Jee et al. (2005a; 2005b), where we also examined the possible impact of the blue cluster member contamination to the source catalogs for weak-lensing analyses of the two $z\sim0.83$ clusters.
Ellis et al. (2006) studied the color magnitude relation of CL1226 and reported that a low fraction ($\sim33$%) of the 27 spectroscopically confirmed members possess E or S0 early-type morphologies while some galaxies with late-type morphology lie on the color-magnitude relation defined by the early-type galaxies (four galaxies within 0.1 mag in the $V-K$ color magnitude relation). Because the study is based on a small number of bright cluster members ($K>19.6$), it is difficult to apply the result to faint magnitudes. Our analysis above shows that the fraction of the blue cluster members is negligibly small at least for the population selected by the ($24<i_{814}\lesssim28$ and $v_{606}-i_{814}<1.2$) criteria.
Now with the source catalog at hand, we need to estimate the redshift distribution in order to put our subsequent lensing analyses on the proper scale. We utilize the publicly available photometric redshift catalog of the UDF (Coe et al. 2006). The UDF covers a small area (though twice as large as the Hubble Deep Field), but thanks to the unprecedented depth of HST-based observations in the $B_{435}$, $V_{555}$, $i_{775}$, $z_{850}$, $j_{110}$, and $h_{160}$ filters, it provides high-fidelity photometric redshifts for the galaxies beyond the limiting magnitudes of the CL1226 data. To account for different depth between the cluster field and UDF, we estimate the redshift distribution per each magnitude bin and correct for the difference in the normalized number density. The resulting mean redshift of the source population is determined to be $\bar{z}=1.71$. Of course, this value should not be confused with the [*effective*]{} redshift of the source population because objects at redshifts smaller than the cluster redshift dilute the signal. In fact, in weak-lensing studies this lensing efficiency is expressed in terms of $\beta$: $$\beta = \mbox{max} \left ( 0, \frac{D_{ls}} {D_s} \right ). \label{eqn_beta}$$ where $D_s$ and $D_{ls}$ are the angular diameter distance from the observer to the source and from the lens to the source, respectively. We obtain $<\beta>=0.265$ for the given cosmology, which corresponds to $z_{eff}=1.373$. Another important quantity that affects our subsequent lensing analysis is the width of the redshift distribution, which is often expressed in terms of $<\beta^2>$. Seitz & Schneider (1997) found that under a single redshift source plane assumption the measured shear $g\prime$ is overestimated by $$g\prime = \left[1 + (\frac{<\beta^2>}{<\beta>^2}-1) \kappa \right]g$$ For the current source population, we obtain $<\beta^2>=0.12$ and therefore the measured shear is overestimated by $(1+0.71\kappa)$. This correction becomes increasingly important with lens redshift and should be included in high-redshift cluster lensing analyses.
PSF Modeling and Ellipticity Measurements \[section\_ellipticity\]
------------------------------------------------------------------
Weak-lensing measures a subtle distortion of background galaxy images and therefore it is important to remove any instrumental effect, which can mimic gravitational lensing signals. When geometric distortion and image registration are done carefully, the most important remaining task is PSF modeling. Because anisotropic PSF can induce a false lensing signal and the impact becomes greater for fainter galaxies, which contain more signal, a great amount of efforts and time is invested on studying the PSF of any instrument before a signal is extracted.
Although the ACS PSF is far smaller than what one can achieve from the ground, it still measurably affects the shapes of objects whose sizes at the surface brightness limit are comparable to the PSF. This places great importance on deep imaging (larger size at low surface brightness) and good understanding of the PSF variations. It has been known that ACS PSFs vary in a complex way with time and position (Krist 2003; Jee et al. 2005a; Sirianni et al. 2005). In Jee et al. (2007b), we presented a principal component analysis of the ACS PSF and made the ACS PSF library publicly available based on archival ACS images of stellar fields. In this work, we use the PSF library of Jee et al. (2007b) to model the PSF variation in the CL1226 field. Those who are interested in the method in detail are referred to the paper. Below we briefly describe the procedure and the result specific for our cluster analysis.
We first derive a PSF model for an individual exposure and then shift/rotate the model PSFs with respect to the final mosaic image in a similar way to our image registration procedure. In each exposure, there are typically 8-15 high S/N stars available, which can be used to find the best-matching template from the library. The final PSFs are the results of stacking all the contributing PSFs. In Figure \[fig\_psf\] we show the comparison of the observed $i_{814}$ PSFs with the modeled ones (similar results are obtained for the $v_{606}$ PSFs). The PSF model ($middle$) obtained in this way closely resembles the observed pattern ($left$). The $e_{+}$ versus $e_{\times}$ plot (right) shows that both the centroid $<{\textbf{e}}>\simeq(-7\times10^{-3},8\times10^{-3})$ and the dispersion $<|{\textbf{e}}|^2>^{0.5}\simeq0.02$ of the observed points (diamond) are significantly improved in the residuals (‘+’ symbol). The centroid and dispersion of the residuals are $<\delta {\textbf{e}}>\simeq(-5\times10^{-3},1\times10^{-3})$ and $<|\delta {\textbf{e}}|^2>^{0.5}\simeq0.01$, respectively.
We determine object ellipticities by fitting a PSF-convolved elliptical Gaussian to the images. In theory, this is equivalent to the method proposed by Bernstein & Jarvis (2002) although the implementation is different. Instead of fitting an elliptical Gaussian to an object, they shear the object progressively until it fits a $circular$ Gaussian. This scheme is conveniently implemented by first decomposing galaxy shapes with shapelets (Bernstein & Jarvis 2002; Refregier 2003) and then by applying shear operators to the shapelet coefficients until the object’s quadrupole moments disappear. We adopt the method in our previous analysis (Jee et al. 2005a; 2005b; 2006). We noted in Jee et al. (2007a) however that directly fitting an elliptical Gaussian to the pixelized object reduces aliasing compared to the shapelet formalism, particularly when the object has extended features. The issue was important in Jee et al. (2007a) because the lens was a low-redshift strong-lensing cluster. The ellipticities of the strongly lensed arc(let)s were substantially underestimated if the shapelet formalism is employed. Although the CL1226 cluster images do not show such a large number of arc(let)s, we use the same pipeline of Jee et al. (2007a) in the current analysis. Fitting a PSF-convolved elliptical Gaussian to pixelated images is more numerically stable for faint objects, and also provides straight-forward error estimates in the results. Because we have two passband images available, the finalcombined ellipticities of objects are given as weighted averages.
Figure \[fig\_e\_distribution\] displays the ellipticity distribution of non-stellar objects in the cluster field. We only include the objects with a S/N$\sim5$ or greater at least in one passband. The ellipticity distribution of the $r_h$[^1] $> 0.15\arcsec$ objects is slightly affected after PSF correction (Figure \[fig\_e\_distribution\]a) whereas the change is significant for smaller ($0.1\arcsec<r_h < 0.15\arcsec$) objects (Figure \[fig\_e\_distribution\]b). Note that much of the useful lensing signal comes from this “small” galaxy population (only slightly larger than the instrument PSFs $r_h\sim0.06\arcsec$) for high-redshift clusters. This illustrates that that, even if ACS PSFs are small, one must carefully account for their effects to maximize the full resolving power of ACS.
A potentially important factor in determining object shapes in addition to the PSF correction discussed above is the charge transfer efficiency (CTE) degradation of ACS. Riess and Mack (2004) reported strong evidence for photometric losses in the parallel direction for ACS. Rhodes et al. (2007) studied the CTE-induced charge elongation in the context of weak-lensing studies and established an empirical prescription, where the strength of the elongation is proportional to 1) the distance from the read-out register, 2) the observation time, and 3) the inverse of the S/N of objects. When we assume that equation 10 of Rhodes et al. (2007) is also applicable to the current data, we obtain $\delta e_{+}\sim0.01$ for the faintest object that is also farthest from the read-out register; on average however for all sources throughout the entire field the required correction is $\delta e_{+}\sim0.003$. Hence, the CTE-induced elongation is much smaller than the weak-lensing signal of the cluster and the statistical noise (object ellipticity dispersion) to the extent that the effect can be safely ignored in our subsequent analysis. Heymans et al. (2008) obtained a similar result in their weak-lensing study of the Abell 901/902 supercluster whose ACS data were taken about 1 year after CL1226 was observed. We are also investigating the CTE-induced elongation issue independently by analyzing cosmic-rays. Because cosmic-rays are not affected by the instrument PSF, their study enables us to nicely separate the PSF effect from the CTE degradation effect. Our preliminary result is consistent with that of Rhodes et al. (2007) in the sense that the required correction is negligibly small for the current observation. We will present our result of the CTE-induced elongation study elsewhere along with our future publication of the weak-lensing analysis of the $z\sim1.4$ cluster XMMU J2235.3-2557 (M. Jee et al. in preparation), whose ACS data were taken in the 2005-2006 years and thus are potentially subject to greater bias due to the CTE degradation.
WEAK-LENSING ANALYSIS
=====================
Two-Dimensional Mass Reconstruction \[section\_mass\_map\]
----------------------------------------------------------
One of the easiest ways to visually identify the presence of a lensing signal is to plot a smoothed two-dimensional distribution of the source galaxies’ ellipticity. Massive clusters shear shapes of background galaxies in such a way that they appear on average tangentially aligned toward the center of the clusters. In Figure \[fig\_whisker\] we present this so-called “whisker” plot obtained by smoothing the source galaxy ellipticity map with a FWHM=$20\arcsec$ Gaussian kernel. As in the case of Figure \[fig\_psf\], the length and the orientation of the sticks represent the magnitude and the direction of the weighted mean ellipticity, respectively. An ellipticity with a magnitude of $g=0.1$ is shown at the top with a circle for comparison. The tangential alignment around the center of the cluster (the location of the BCG) is clear.
Many algorithms exist for the conversion of this ellipticity map to the mass density map of the cluster. Due to its simplicity, the classic method of Kaiser & Squires (1993; hereafter KS93) or the real-space version (Fischer & Tyson 1997) still is widely used. The KS93 method is based on the notion that the measured shear $\gamma$ is related to the dimensionless mass density $\kappa$ by the following convolution: $$\kappa ({\textbf{x}}) = \frac{1}{\pi} \int D^*({\textbf{x}}-{\textbf{x}}^\prime) \gamma ({\textbf{x}}^\prime) d^2 {\textbf{x}} \label{k_of_gamma}.$$ where $D^*({\textbf{x}} )$ is the complex conjugate of the convolution kernel $D({\textbf{x}} ) = - 1/ (x_1 - i x_2 )^2$. The method assumes that the shear $\gamma$ is directly measurable whereas in fact it is the reduced shear $g=\gamma/(1-\kappa)$ that we can measure directly. Obviously, in the region where $\kappa$ is small, the assumption is valid. However, near cluster centers, $g$ is often much greater than $\gamma$, and this leads to overestimation of $\kappa$. In addition, it is not straightforward to incorporate measurement errors (i.e., ellipticity errors and shear uncertainties) or priors in the KS93 scheme. Because measured shears in cluster outskirts have much lower significance, the algorithm frequently produces various noise peaks when the smoothing scheme is optimized to reveal significant structures in the cluster center.
These pitfalls are overcome in the new methods such as Marshall et al. (2002) and Seitz et al. (1998), where individual galaxy shapes (not averaged shears) are used and the resulting mass map is regularized. In the current paper, we used the mass reconstruction code of Jee et al. (2007a), who modified the method of Seitz et al. (1998) so that strong-lensing constraints are incorporated. For the current mass reconstruction of CL1226, however, we turned off the strong-lensing capability of the software and utilized only the weak-lensing data.
We present our maximum-entropy mass reconstruction of CL1226 in the left panel of Figure \[fig\_mass\_reconstruction\]. For comparison, we also display the result obtained by the conventional KS93 algorithm in the right panel. For the KS93 method we choose a smoothing scale of FWHM$\sim24\arcsec$. The regularization parameter of Jee et al. (2007a) was adjusted in such a way that the result matches the resolution of the KS93 version at $r\lesssim50\arcsec$. Both mass reconstructions clearly reveal the strong dark matter concentration in the cluster center. However, in the relatively low $\kappa$ region the KS93 algorithm produces many spurious substructures, most of which do not stand out in the maximum-entropy reconstruction. It is certain that both the inadequate (too small kernel) smoothing and the $g\sim\gamma$ approximation of the KS93 algorithm are the causes of the artifacts. Therefore, our interpretation hereafter is based on the result from our maximum-entropy reconstruction.
On a large scale the $\kappa$ field shows that the cluster does not possess any significant substructure. This relaxed appearance is also indicated by the X-ray emission from the cluster (Maughan et al. 2007). However, this symmetric mass distribution is somewhat unusual for a cluster at such a high redshift. In our current hierarchical structure formation paradigm, relaxed clusters are thought to be rare at $z=0.89$, when the universe is at less than half its current age. Our previous weak-lensing analysis of CL0152 and MS1054 (both at $z\sim0.83$) revealed significant substructures composed of several mass clumps suggestive of the active formation of the systems.
On a small scale, however, our mass reconstruction resolves the core into two mass clumps, which are separated by $\sim40\arcsec$ (Figure \[fig\_massxraynum\]a). Comparing the projected masses within $r<20\arcsec$, we estimate that the mass ratio of the two substructures is approximately 3:2. The more massive mass clump \[$M(r<20\arcsec)=(1.3\pm0.1)\times10^{14}~M_{\sun}$\] is located near the brightest cluster galaxy, which is also close to the center of the X-ray peak (Figure \[fig\_massxraynum\]b). The less massive structure \[$M(r<20\arcsec)=(8.5\pm0.6)\times10^{13}~M_{\sun}$\] $\sim40\arcsec$ to the southwest is however not detected in the X-ray surface brightness although we note that the contours near the X-ray peak are slightly elongated toward this secondary mass clump. Because the mass of this structure is significant and comparable to that of the western mass peak (the most massive among the three) of MS1054, the apparent absence of the gas overdensity associated with the structure is counter-intuitive. The weak-lensing mass structure is nevertheless highly consistent with the cluster red-sequence distribution. We display the number density contours of the cluster red-sequence in Figure \[fig\_massxraynum\]c. The secondary mass clump is in good spatial agreement with the cluster red-sequence. Interestingly, Maughan et al. (2007) found that the gas temperature of the region is much higher ($12-18$ keV) than those of the rest \[see Figure \[fig\_massxraynum\]d where we overplot the mass contours on top of the temperature map of Maughan et al. (2007)\]. They suggested that this temperature structure might relate to the possible on-going merger indicated by the cluster galaxy distribution and we agree. Our detailed discussion on the comparison between the mass, X-ray intensity, gas temperature, and galaxy distributions is deferred to \[section\_merger\].
Tangential Shear, Cosmic Shear Effect and Mass Estimation \[section\_mass\_estimation\]
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In addition to the two-dimensional mass reconstruction discussed previously, tangential shear is also a useful measure of total lensing mass. By taking azimuthal averages, one can lower the effect of shot noise and more easily determine the presence of the lensing than in the two-dimensional analysis particularly when the signal is weak. Therefore, many authors prefer to use tangential shear profiles in the estimation of cluster masses assuming an azimuthal symmetry. In the current paper, we present the tangential shear profile of CL1226 first, and then estimate the mass based on the profile. Finally, we also compute the mass using the two-dimensional mass map and compare the results.
Tangential shear is defined as $$g_T = < - g_1 \cos 2\phi - g_2 \sin 2\phi > \label{tan_shear},$$ where $\phi$ is the position angle of the object with respect to the lens center, and $g_{1(2)}$ is a reduced shear $g_{1(2)}=\gamma_{1(2)} /( 1- \kappa )$; true shears $\gamma_1$ and $\gamma_2$ are related to the lensing potential $\psi$ by $\gamma_1=0.5 (\psi_{11} -\psi_{22})$ and $\gamma_2=\psi_{12} =\psi_{21}$. If no lensing is present, the reduced tangential shear $g_T$ must be consistent with zero. The filled circles in Figure \[fig\_tan\_shear\] show the reduced tangential shears of CL1226 centered on the BCG. The lensing signal from the cluster is clear out to the field limit; we note that at $r>180\arcsec$ we cannot complete a circle. The observed reduced shears at $r>50\arcsec$ decrease monotonically with radius. This is in accordance with the mass map of the cluster discussed in \[section\_mass\_map\], which shows no major asymmetric substructures outside the main clump. The diamond symbols represent our measurement of tangential shear when the background galaxies are rotated by 45$\degr$. This test shows that the B-mode signal is consistent with zero as expected.
The error bars in Figure \[fig\_tan\_shear\] include only the statistical uncertainties determined by the finite number of source galaxies in each radial bin. Hoekstra (2003) demonstrated that background structures (cosmic shear effects) are important sources of uncertainties and need to be considered in cluster mass estimation. Following the formalism of Hoekstra (2003), we evaluated the background structure effect on the uncertainties in the tangential shear measurements. The solid line in Figure \[fig\_cs\_effect\] displays the predicted errors $\sigma_{\gamma}$ for the redshift distribution of the source population in the current cosmology. For comparison, we approximately reproduce here the prediction of Hoekstra (2003) for their $20<R<26$ sample ($\bar{z}=1.08$, dashed). Note that the cosmic shear effect is substantially ($\sim50$%) higher in our sample because the mean redshift of our source population is also significantly higher ($\bar{z}=1.71$). However, in the $r\lesssim200\arcsec$ region, where we measure our tangential shears for CL1226, the errors induced by the cosmic shear are still lower than the statistical errors (‘+’ symbol) and thus the effect is minor in our cluster mass estimation Nevertheless, in the following analysis, we include this cosmic shear effect in the quoted errors.
We characterize the reduced tangential shears with three parametric models: singular isothermal sphere (SIS), Navarro-Frenk-White (NFW; Navarro et al. 1997), and non-singular isothermal sphere (NIS). In these parametric fits, we use all the data points in Figure \[fig\_tan\_shear\] unlike the cases of Jee et al. (2005a; 2005b), where we excluded a few tangential shear values near the cluster core to avoid the possible effects of the apparent substructure and blue cluster galaxy contamination. In the current cluster, however, both effects appear to be minor and smaller than the statistical uncertainties. The SIS fit results in the Einstein radius of the cluster $\theta_E=11.7\arcsec\pm0.4\arcsec$ ($\chi^2/$d.o.f=3.61) With $\beta=0.265$ (\[section\_source\_selection\]), the implied velocity dispersion is $\sigma_v=(1237\pm22)~\mbox{km}~\mbox{s}^{-1}$. Assuming energy equi-partition between gas and dark matter, we can also convert this velocity dispersion to the [*lower*]{} limit of the gas temperature $T_X=(9.4\pm0.3)$ keV. Although the SIS model is in general a good description of cluster mass profiles at small radii, numerical simulations show that the mass density of relaxed clusters at large radii should drop faster than $\propto r^{-2}$. Also, near the cluster core it is believed that the density profile is less steep than $\propto r^{-2}$. Therefore, modified density profiles such as an NFW model are a preferred choice over the traditional SIS in the description of cluster mass profiles. For the NFW fit, we obtain $r_s=78\arcsec\pm19\arcsec$ ($\sim604$ kpc) and $c=2.7\pm0.3$ ($\chi^2/$d.o.f=1.11). The comparison of the reduced $\chi^2$ values and the best-fit results in Figure \[fig\_tan\_shear\] show that the cluster’s reduced tangential shear is better described by this NFW model. The large reduced $\chi^2$ value of the SIS fitting is mainly due to the fact that the observed tangential shear does not rise as steeply as the SIS prediction at $r\lesssim50\arcsec$. If we assume a non-singular core instead \[NIS; i.e., $\kappa=\kappa_0/(r^2+r_c^2)^{1/2}$\], the discrepancy is substantially reduced, and we obtain $r_c=9.5\arcsec\pm1.2\arcsec$ ($\sim74$ kpc) and $\kappa_0=7.4\pm0.7$ with $\chi^2$/d.o.f=0.77. We compare the projected mass profiles estimated from these results in Figure \[fig\_mass\_comparison\]. Also plotted is the result based on the two-dimensional mass map of the cluster, for which we lifted the mass-sheet degeneracy $\kappa \rightarrow \lambda \kappa + 1-\lambda$ by constraining $\bar{\kappa}(150\arcsec<r<200\arcsec)$ to be the same as the value given by the NFW result. The discrepancy from difference approaches is small over the entire range of the radii shown here except for the NIS model, which, although similar to the other results at small radii ($r\lesssim70\arcsec$), gives substantially higher masses at large radii (e.g., $\sim15\%$ higher at $r\sim150\arcsec$). Because the $\propto r^{-2}$ behavior of the NIS model at large radii is unrealistic (despite its smallest goodness-of-fit value), we do not consider the result as representative of the overall cluster mass profile. Error bars are omitted to avoid clutter in Figure \[fig\_mass\_comparison\]. For the SIS result, the mass uncertainties are $\sim6.5$% (after we rescale with the reduced $\chi^2$ value) over the entire range. The uncertainty in the NFW mass non-uniformly increases with radii: approximately 5%, 10%, and 15% of the total mass at $r=50\arcsec, 100\arcsec$, and $200\arcsec$.
DISCUSSION
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Comparison with Other Studies
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The X-ray temperature of CL1226 was first measured by Cagnoni et al. (2001) based on short exposure ($\sim10$ ks) $Chandra$ data. They obtained an X-ray temperature of $10_{-3}^{+4}$ keV, a $\beta$ index of $0.770\pm0.025$, and a core radius of $r_c=18\arcsec.1\pm0\arcsec.9$. With an isothermal $\beta$ model assumption, their measurement gives a projected (within a cylindrical volume) mass of $M(r<1~\mbox{Mpc})=(1.4_{-0.4}^{+0.6})\times10^{15} M_{\sun}$, which is consistent with our result. Joy et al. (2001) used Sunyaev-Zeldovich effect (SZE) observations and determined the mass of the cluster to be $M(r<65\arcsec)=(3.9\pm0.5)\times10^{14}~M_{\sun}$ with a SZE temperature of $9.8_{-1.9}^{+4.7}$ keV. This SZE result is also in accordance with our lensing estimate. With XMM-Newton observations, Maughan et al. (2004) derived a virial mass of $(1.4\pm0.5)\times10^{15}~M_{\sun}$ within $r_{200}=1.66\pm0.34~$Mpc. Our NFW fitting results gives $r_{200}=1.64\pm0.10~$Mpc and $M(r<r_{200})=(1.38\pm0.20)\times10^{15}~M_{\sun}$, which are in excellent agreement with the result of Maughan et al. (2004). Maughan et al. (2007) refined their early study of CL1226 by using both deep XMM-Newton and Chandra observations. From the comprehensive analysis of the cluster’s three-dimensional gas and temperature structure, they obtained $r_{500}=(0.88\pm0.05)~$Mpc and $M_{500}=5.2_{-0.8}^{+1.0}\times 10^{14}~M_{\sun}$. The new mass is, however, $\sim30$% lower than our lensing estimate $M(r<0.88~\mbox{Mpc})=(7.34\pm0.71)\times10^{14}~M_{\sun}$. This mass discrepancy is interesting because Maughan et al. (2007) noted that their mass is $\sim30$% below the $M-T$ relation (Vikhlinin et al. 2006) and also the $M-Y_X$ relation (Kravtsov et al. 2006). They suggested the possibility that the on-going merger indicated by their temperature map may lead to the underestimation of the total mass with X-ray methods. Our mass reconstruction resolves the cluster core substructure and supports the merger scenario. If the merger is indeed responsible for the underestimation from the X-ray analysis, the apparent improvement in the $M-T$ relation with the lensing estimate highlights the merits of gravitational lensing for mass estimation, which does not depend on the dynamical state of the system.
Stage of the Merger in the Core of CL1226 \[section\_merger\]
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Galaxy cluster cores frequently possess merging signatures despite their relaxed morphology on a large scale. Even the Coma Cluster, long regarded as the archetype of relaxed clusters, has been found to be composed of many interesting substructures by X-ray and optical observations (e.g., Biviano et al. 1996). Several lines of evidence strongly suggest that the cluster CL1226 is also undergoing an active merger in the cluster core. First, the cluster galaxy distribution is bimodal in the core; the dominant number density peak is close to the X-ray center and the other overdensity is seen to the southwest. Second, the X-ray temperature map shows that the ICM near this southwestern number density peak is significantly higher ($14\sim18$ keV) than in the neighboring region. Although this high-temperature region does not stand out in the X-ray surface brightness map, which reveals only a single peak in spatial agreement with the dominant galaxy number density peak, a scrutiny of both the $Chandra$ and the $XMM-Newton$ images shows that the contours near the X-ray peak is slightly elongated toward the high-temperature region. Third, our weak-lensing analysis confirms that a substantial mass is associated with this southwestern galaxy number density peak. Therefore, it is plausible that the hydrodynamic interaction between the two substructures is responsible for the high temperature ($14\sim18$ keV) feature.
It is puzzling that the southwestern weak-lensing mass clump is not detected in the X-ray surface brightness. As mentioned in \[section\_mass\_map\], the mass of this substructure is comparable to that of the western weak-lensing mass peak of MS1054, which is the most massive of the three dominant peaks of the cluster and has its own distinct X-ray peak. As a possible cause, we suggest the possibility that the substructure has passed through the other more massive structure from the eastern side. If it is a slow encounter, the gas of the system could be severely stripped during this penetration. A shock, on the other hand, can propagate ahead of the gas core and leave trails of hot temperature as observed in the current case. Of course, the collisionless galaxies and dark matter of the system are expected to survive the core passthrough as observed unlike the gas system. A famous example is the “bullet” cluster 1E0657-56 at $z\simeq0.3$ (Clowe et al. 2006; Bradac et al. 2006).
We noticed a very similar case in our weak-lensing and X-ray study of MS1054 (Jee et al. 2005b). The eastern substructure of MS1054, whose presence is clear both in the cluster galaxy and dark matter distribution, is conspicuously absent in the X-ray observations. Moreover, the MS1054 temperature map of Jee et al. (2005b) shows that the gas in the region of the eastern halo is notably higher ($\gtrsim10$ keV) than the average temperature ($\sim8.9$ keV). Hence in Jee et al. (2005b) we proposed that the cluster galaxies now observed in the eastern mass clumps might have passed through the central mass clump from the southwest. Intriguingly, a significant fraction of the star forming galaxies (four out of the five brightest IR galaxies) are found to exist near the eastern mass clump of MS1054 (Bai et al. 2007). The result can be interpreted as indicating a recent star formation triggered by this hypothesized merger. Although we have not performed a parallel study for the star-formation properties of the CL1226 galaxies yet, the existing features in the X-ray surface brightness, gas temperature, mass, and galaxy distribution is suggestive of the similar merger scenario.
Is the above post-merger picture the unique scenario that explains the observed features? If the peculiar temperature structure of CL1226 were absent or we can attribute it to something else, one can also consider the possibility that the secondary mass clump might not be massive enough to produce X-ray emission, but still detected in weak-lensing simply because it forms a line-of-sight superposition with the already massive dark matter halo of the primary cluster. Obviously, in this hypothesized configuration the background density can boost the lensing signal even if the mass of the southwestern clump by itself is not significant. To explore this possibility quantitatively, we estimated the expected X-ray temperature of the secondary clump for this scenario in the following way. First, we created a new radial density profile of the main cluster from the mass map by excluding the azimuthal range that contains the southewestern substructure. Second, we subtracted this new radial density profile from the original mass map. Finally, we measured the mass of the secondary clump from this subtracted mass map. Of course, this procedure overestimates the contribution (i.e., boosting effect) of the primary cluster because, even if we avoided the southwestern region in the creation of the new radial profile, the $\kappa$ value in the other azimuthal range is still the sum of the two subclusters. Hence, here we assume an extreme case, where the primary cluster is dominant in mass. Then, within $r=30\arcsec$ ($\sim232$ kpc), the total projected mass of the secondary cluster would be $\sim5\times10^{13} M_{\sun}$. Assuming isothermality with $\beta_X=0.7$, this mass is translated into $T_X \sim 2.6$ keV. Given the depth ($\sim72$ ks for each detector) of the observation, a subcluster with this cool core would have been easily identified in the $XMM-Newton$ image of CL1226 (Maughan et al. 2007). Even for the relatively shallow $Chandra$ data ($\sim50$ ks when the two datasets ObsID 3180 and 5014 are combined), we predict $\sim200$ counts within a $r=10\arcsec$ aperture, which would give a significance of $\sim8~\sigma$. Therefore, even if we disregard the temperature structure, we are not likely to be observing a very low mass cluster whose lensing efficiency is enhanced due to the primary cluster halo.
SUMMARY AND CONCLUSIONS
=======================
We have presented a weak-lensing study of the galaxy cluster CL1226 at $z=0.89$. The cluster is a very interesting, rare system because, despite its high redshift, it has a relaxed morphology in X-ray surface brightness and an unusually high gas temperature. Our HST/ACS-based weak-lensing analysis of the cluster provides the dark matter distribution in unprecedented detail and allows us to measure the mass profiles out to the virial radius of the cluster.
Our two-dimensional mass reconstruction shows that on large scales the dark matter distribution is consistent with a relaxed system with no significant substructures. However, viewed in detail, the cluster core is resolved into two mass clumps. This bimodality of the core mass structure is also seen in the cluster galaxy distribution. The dominant mass clump lies close to the BCG whereas the other less massive one is located $\sim40\arcsec$ to the southwest. This secondary mass clump does not stand out in the X-ray surface brightness although the temperature of the region is much higher than in the rest of the cluster. When the significant mass associated with the substructure is considered, the absence of the corresponding X-ray excess in the region is puzzling. Therefore, we propose that we may be observing the system after the less massive subcluster passed through the main cluster. It is possible that the X-ray gas of the less massive system might have been stripped due to the ram pressure. The slight elongation of the X-ray peak toward the southwestern mass clump is also supportive of this scenario. These features are similar to the ones that we observed in MS1054, another massive galaxy cluster at $z=0.83$, where we proposed a similar possibility.
We measure significant shear signals out to the field boundary ($\sim200\arcsec$), which indicates that the cluster is indeed massive as already implied by its high X-ray temperature. Fitting an NFW profile to the reduced tangential shears gives $r_{200}=1.64\pm0.10~$Mpc and $M(r<r_{200})=(1.38\pm0.20)\times10^{15}~M_{\sun}$, where the error bars include both statistical and cosmic-shear induced systematic uncertainties. Although the predicted velocity dispersion and X-ray temperature from the lensing result are consistent with previous work, our cluster mass is $\sim30$% higher than the recent XMM-$Newton$ and $CHANDRA$ analysis of Maughan et al. (2007), who interestingly pointed out that their mass estimate is $\sim30$% below the $M-T$ and $Y_X-T$ scaling relations. If the on-going merger is indeed the cause of the underestimation of the total mass with the X-ray method (Maughan et al. 2007), the apparent improvement in the $M-T$ relation with the lensing estimate highlights the advantages of gravitational lensing for mass estimation, which does not depend on the dynamical state of the system.
Our weak-lensing study confirms that CL1226 is indeed the most massive cluster at $z>0.6$ known to date. In the hierarchical structure formation paradigm, a CL1226-like cluster is extremely rare at $z=0.89$. Because the abundance of such a massive system is sensitive to the matter density ($\Omega_M$) and its fluctuation ($\sigma_8$), the current result and future lensing measurements of other high-redshift massive clusters \[e.g., XMMU J2235.3-2557 at $z=1.39$ (Mullis et al. 2005) and XCS J2215.9-1738 at $z=1.45$ (Stanford et al. 2006; Hilton et al. 2007) \] will provide useful constraints on the normalization of the power spectrum.
M. James Jee acknowledges support for the current research from the TABASGO foundation presented in the form of the Large Synoptic Survey Telescope Cosmology Fellowship. We thank Ben Maughan for allowing us to use his X-ray temperature map of CL1226.
Bahcall, N. A. & Fan, X. 1998, , 504, 1 Bai, L., et al. 2007, , 664, 181 Beckwith, S., Somerville, R., Stiavelli M., 2003, STScI Newsletter vol 20 issue 04 Blakeslee, J. P., Anderson, K. R., Meurer, G. R., Ben[í]{}tez, N., & Magee, D. 2003, ASP Conf. Ser. 295: Astronomical Data Analysis Software and Systems XII, 12, 257 Bernstein, G. M. & Jarvis, M. 2002, , 123, 583 Bertin, E., & Arnouts, S. 1996, , 117, 393 Biviano, A., Durret, F., Gerbal, D., Le Fevre, O., Lobo, C., Mazure, A., & Slezak, E. 1996, , 311, 95 Brada[č]{}, M., et al. 2006, , 652, 937 Branchesi, M., Gioia, I. M., Fanti, C., & Fanti, R. 2007, , 472, 727 Butcher, H., & Oemler, A., Jr. 1984, , 285, 426 Cagnoni, I., Elvis, M., Kim, D.-W., Mazzotta, P., Huang, J.-S., & Celotti, A. 2001, , 560, 86 Clowe, D., Brada[č]{}, M., Gonzalez, A. H., Markevitch, M., Randall, S. W., Jones, C., & Zaritsky, D. 2006, , 648, L109 Coe, D., Ben[í]{}tez, N., S[á]{}nchez, S. F., Jee, M., Bouwens, R., & Ford, H. 2006, , 132, 926 Davis, M., et al. 2007, , 660, L1 Donahue, M., Voit, G. M., Gioia, I., Lupino, G., Hughes, J. P., & Stocke, J. T. 1998, , 502, 550 Ellis, S. C., Jones, L. R., Donovan, D., Ebeling, H., & Khosroshahi, H. G. 2006, , 368, 769 Fischer, P., & Tyson, J. A. 1997, , 114, 14 Fruchter, A. S., & Hook, R. N. 2002, , 114, 144 Gao, L., Navarro, J. F., Cole, S., Frenk, C., White, S. D. M., Springel, V., Jenkins, A., & Neto, A. F. 2007, ArXiv e-prints, 711, arXiv:0711.0746 Giavalisco, M., et al. 2004, , 600, L93 Gioia, I. M., Braito, V., Branchesi, M., Della Ceca, R., Maccacaro, T., & Tran, K.-V. 2004, , 419, 517 Hack, W., Busko, I., & Jedrzejewski, R. 2003, ASP Conf. Ser. 295: Astronomical Data Analysis Software and Systems XII, 295, 453 Hoekstra, H., Franx, M., & Kuijken, K. 2000,, 532, 88 Hoekstra, H. 2003, , 339, 1155 Jee, M. J., White, R. L., Ben[í]{}tez, N., Ford, H. C., Blakeslee, J. P., Rosati, P., Demarco, R., & Illingworth, G. D. 2005a, , 618, 46 Jee, M. J., White, R. L., Ford, H. C., Blakeslee, J. P., Illingworth, G. D., Coe, D. A., & Tran, K.-V. H. 2005b, , 634, 813 Jee, M. J., White, R. L., Ford, H. C., Illingworth, G. D., Blakeslee, J. P., Holden, B., & Mei, S.2006, , 642, 720 Jee, M. J., et al. 2007a, , 661, 728 Jee, M. J., Blakeslee, J. P., Sirianni, M., Martel, A. R., White, R. L., & Ford, H. C. 2007b, , 119, 1403 Jeltema, T. E., Canizares, C. R., Bautz, M. W., Malm, M. R., Donahue, M., & Garmire, G. P. 2001, , 562, 124 Joy, M., et al. 2001, , 551, L1 Kaiser, N., & Squires, G. 1993, , 404, 441 Kravtsov, A. V., Vikhlinin, A., & Nagai, D. 2006, , 650, 128 Krist, J. 2003, $Instrument$ $Science$ $Report$ $ACS$ 2003-06 Marshall, P. J., Hobson, M. P., Gull, S. F., & Bridle, S. L. 2002, , 335, 1037 Maughan, B. J., Jones, L. R., Ebeling, H., Perlman, E., Rosati, P., Frye, C., & Mullis, C. R. 2003, , 587, 589 Maughan, B. J., Jones, L. R., Ebeling, H., & Scharf, C. 2004, , 351, 1193 Maughan, B. J., Jones, C., Jones, L. R., & Van Speybroeck, L. 2007, , 659, 1125 Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, , 490, 493 Refregier, A. 2003, , 338, 35 Rhodes, J. D., et al. 2007, , 172, 203 Riss, A. & Mack, J. 2004, $Instrument$ $Science$ $Report$ $ACS$ 2004-006 Schneider, P., van Waerbeke, L., Jain, B., & Kruse, G. 1998, , 296, 873 Scoville, N. 2006, Bulletin of the American Astronomical Society, 38, 1211 Seitz, C., & Schneider, P. 1997, , 318, 687 Seitz, S., Schneider, P., & Bartelmann, M. 1998, , 337, 325 Sirianni, M., et al. 2005, , 117, 1049 Tozzi, P., Rosati, P., Ettori, S., Borgani, S., Mainieri, V., & Norman, C. 2003, , 593, 705 Vikhlinin, A., VanSpeybroeck, L., Markevitch, M., Forman, W. R., & Grego, L. 2002, , 578, L107 Vikhlinin, A., Kravtsov, A., Forman, W., Jones, C., Markevitch, M., Murray, S. S., & Van Speybroeck, L. 2006, , 640, 691
![Mass, X-ray, and galaxy number density contours in the central $150\arcsec\times150\arcsec$ region of CL1226. (a) Our weak-lensing mass reconstruction resolves the core substructure, which consists of the dominant clump near the BCG and the less massive, but distinct, clump to the southwest. (b) The presence of this secondary mass peak is hard to identify in the adaptively smoothed (Ebeling 2005) $CHANDRA$ X-ray contours although the slight elongation of the X-ray peak toward the secondary mass peak is marginally suggestive of this feature. (c) The number density contours of the red-sequence candidates (smoothed with a FWHM$=20\arcsec$ Gaussian) show that the galaxy distribution is similar to the mass distribution. (d) Mass contours overlaid on the temperature map of Maughan et al. (2007). The alignment is approximate. The gas temperature in the region where the southwestern mass clump is detected is unusually high ($12\sim18$ keV). \[fig\_massxraynum\]](f9.eps){width="12cm"}
![Projected mass profile of CL1226. We compare the results from the various methods discussed in \[section\_mass\_estimation\]. The discrepancy is small over the entire range of the radii shown here except for the NIS model, which, although similar to the other results at small radii ($r\lesssim70\arcsec$), gives substantially higher masses at large radii (e.g., $\sim15\%$ higher at $r\sim150\arcsec$). Note that for the parameter-free method the azimuthal average is estimated from a complete circle only at $r\lesssim180\arcsec$ (blue dashed). Error bars are omitted to avoid clutter. For the SIS result, the mass uncertainties are $\sim6.5$% (after we rescale with the reduced $\chi^2$ value) over the entire range. The uncertainty in the NFW mass non-uniformly increases with radii: approximately 5%, 10%, and 15% of the total mass at $r=50\arcsec, 100\arcsec$, and $200\arcsec$. \[fig\_mass\_comparison\]](f12.eps){width="12cm"}
[^1]: $r_h$ represents a half light radius.
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abstract: 'Using archival RXTE data we derive the 2-16 keV Fourier-resolved spectra of the Atoll source u in a sequence of its timing states as its low QPO frequency spans the range between 6 and 94 Hz. The increase in the QPO frequency accompanies a spectral transition of the source from its island to its banana states. The banana-states’ Fourier-resolved spectra are well fitted by a single blackbody component with $kT \sim 2-3$ keV depending on the source position in the color – color diagram and the Fourier frequency, thus indicating that this spectral component is responsible for the source variability on these timescales. This result is in approximate agreement with similar behavior exhibited by the Z sources, suggesting that, as in that case, the bounday layer – the likely source of the thermal component – is supported by radiation pressure. Furthermore, it is found that the iron line at $\sim$6.6 keV, clearly present in the averaged spectra, not apparent within the limitations of our measurements in the frequency-resolved spectra irrespective of the frequency range. This would indicate that this spectral component exhibits little variability on time scales comprising the interval $10^{-2}-10^2$ seconds. In the island state the single blackbody model proved inadequate, particularly notable in our lowest frequency band ($0.008-0.8$ Hz). An absorbed powerlaw or an additive blackbody plus hard powerlaw model was required to obtain a satisfactory fit. Statisics do not allow unambiguous discrimination between these possible scenarios.'
author:
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- 'C.R. Shrader$^{1,4}$, P. Reig$^{2,3}$ and D. Kazanas$^1$'
title: 'Fourier resolved spectroscopy of u: New Insights into Spectral and Temporal Properties of Low-Mass X-ray Binaries'
---
Introduction
============
Rapid X-ray variability and spectral distributions are powerful probes of the physics of accretion flows onto compact objects, i.e. onto neutron stars and black holes. More than two decades of persistent efforts to probe and comprehend the physics involved in these flows led to the accumulation of a wealth of data of both their spectra and time variations. Significant progress in both these directions has been made over the last decade, in particular, with the wealth of data provided by the Rossi X-Ray Timing Explorer (RXTE). Its large collecting area and high time resolution have facilitated detailed studies on the rapid variability and the spectra of a large sample of sources. However, the accompanying improvement in the understanding of the dynamics and radiation emission physics associated with these accretion powered sources has progressed considerably slower than the accumulation of the data and [ their ensuing classification within semi-coherent phenomenological models.]{}
Concerning the differentiation between the spectral and timing characteristics of accreting neutron stars and black holes, the former appeared to present higher complexity in both their spectral and timing properties than the latter. This additional complexity is generally attributed to the presence of the neutron star surface which can complicate the dynamics of accretion and as a result the ensuing spectral and timing properties. In the temporal domain, the power density spectra (PDS) of neutron star low-mass X-ray binaries (LMXBs) exhibit a variety of features ranging from narrow quasi-periodic oscillations (QPOs) to broad noise components (for a review, see Wijnands (2001); also van der Klis (2006) and references therein), only some of which appear to have corresponding features in the black hole case. In the spectral domain, while Comptonization is presumably the predominant process for producing the observed X-ray emission above $\sim 1$ keV, the neutron-star LMXB spectra are quite complex, more so than those of accreting black hole candidates. This is likely due to the fact that a number of components contribute to the emission in this energy range and that their individual spectra are hard to disentangle.
A simplified way to classify the (apparently complex) spectra of these sources has been that of the color-color (or color - intensity) diagrams (Hasinger & van der Klis 1989); these exhibit, instead of a detailed spectral decomposition, the ratios of fluxes in adjacent bands (colors) within the detector’s range (color - color diagrams) or one color as a function of the total flux of the source (color - intensity diagrams). The large body of accumulated data then made it clear that in response to changes in luminosity, these sources cover a certain trajectory in this color - color (or color - intensity) space whose shape has since been used to classify them: The most luminous LMXBs, $L_x \sim 10^{38}$ erg s$^{-1}$, cover a Z-shape track in color - color space, hence the name Z-sources. The less luminous sources, $L_x \sim 10^{37}$ erg s$^{-1}$, tend to cover a circular-like path in same space, leading to their designation as atoll sources. At the same time, it was found that there exists a correspondence between their spectral and timing properties, with the frequencies of timing features, such as QPOs or breaks, generally increasing in response to increases in the X-ray flux and presumably the accretion rate.
The physics governing the spectral and timing properties of Z or atoll sources and their differences remains obscure, although as early as 1984, Mitsuda et al. (1984) noticed the possible decomposition of their spectra into hard and soft components with the hard one being the more variable of the two. A great advance in the understanding of these spectra was made recently with a refinement of the arguments put forward by Mistuda et al. (1984). This involves the implementation of a novel technique that combines the spectral and timing properties of these sources in an effort to produce a coherent picture of the dynamics of these sources.
This technique is known as Fourier Resolved Spectroscopy (FRS) (e.g. Revnivtsev et al. 1999; Zycki 2003; Papadakis et al. 2005, 2006). In short, instead of producing the power spectra as a function of photon energy, as it is customary, this technique accumulates the variability amplitudes over several well defined frequency bands for each energy bin to produce the energy spectra of a source for the specific frequency bands considered. This particular method of presenting the data facilitates the identification of the variable spectral components as well as the time scales over which the variability occurs; as such, it allows certain immediate insights into the dynamics responsible for the emission of the specific spectral components. It was pointed out that this method is particularly suited for the study of spectral features that result from X-ray reprocessing where the light crossing time provides a natural frequency filter; in this case FRS can provide straightforward insights about the geometry of the reprocessing medium. So far, this technique has been applied successfully to galactic black hole candidates (Revnivtsev et al. 1999; Gilfanov et al. 2001; Sobolewska & Zycki 2006; Reig et al. 2006), to neutron-star LMXBs (Gilfanov & Revnivtsev et al 2005) and also to AGN (Papadakis et al. 2005, 2006) yielding insights into variability of different spectral components and infer their spatial arrangements.
Recently, the same technique has been applied to the spectro - temporal data of accreting neutron stars and more specifically to those of Z-sources (Gilfanov et al. 2003; Revnivtsev & Gilfanov 2006). This analysis did confirm the earlier tentative decomposition of the Z-source spectra into a variable hard component and a soft less variable one. The normalization of these components varies with the accretion rate as do their spectra to produce the well known color-color diagram of the Z-sources. In particular, the softer of the two components was identified with the accretion disk and the harder one with the boundary layer of these sources. The temperature of the former was found to change slowly in response changes in luminosity, as expected. The temperature of the boundary layer component remained constant, independent of luminosity and of Fourier frequency. This supports the conjecture that the boundary layer is dominated by radiation pressure. It was also found (Revnivtsev & Gilfanov 2006) that the disk component contributes to the spectra only at the lowest Fourier frequencies. This is consistent with the findings of Reig et al. (2006) pertaining to the lack of variability of the disk component around a transient black hole LMXB 4U 1543-47. This separation and the identification of these two components in the spectra was possible only because of the combined spectral - timing analysis.
In the present paper we extend the application of the method of Fourier Resolved Spectroscopy to the spectra of a source of the atoll class (i.e. neutron stars accreting at rates $\sim 10$ times lower than those of Z-sources). The trajectories of these sources in the color-color diagram have a different shape from those of the Z-sources as discussed above. They are known to produce a sequence of states of increasing luminosity known as island, lower banana and upper banana states. Variations in the spectral properties between these states are accompanied by corresponding changes in their timing properties (van der Klis 2006). The source under study is 4U 1728–34, a neutron star LMXB system that is considered to be one of the proto-typical examples of an atoll source. It exhibits variability on a variety timescales, from months down to milliseconds (e.g. Di Salvo et al 2001) as well as exhibiting Type-I bursting behavior.
In §2 we outline the details of our data selection and reduction procedures, while in §3 we describe the FRS methodology. In §4 we present our results for u offering our interpretation of the variability of individual spectral components. In §5 we present our general conclusions, including insights gained into accreion geometry and overall physical system configuratons associated with the separate states.
Observations and Data Reduction
===============================
Data obtained with RXTE can be collected and telemetered to the ground in many different ways depending on the intensity of a source and the spectral and timing resolution desired. The specific observational modes are selected by the observer and may change during the overall observation. This project requires a temporal sampling of at least twice the highest frequency band to be probed, and enough spectral resolution to separate out the different continuum components as well as to crudely study possible iron line signatures. Since this is an archival study, we had to examine the various datasets for the epochs of interest in order to identify suitable data acquistion modes. Typically, we found that the event-by-event data sampled at 125 microseconds over 64 onboard spectral channels covering the full bandpass of the PCA, suited our purposes. Practically, in order to ensure homogeneity in the reduction process and for signal-to-noise considerations the energy resolution was restricted to be between 16–24 channels covering the energy band 2-16 keV.
Light curves were extracted for each onboard channel range using the current RXTE software[^1], binned at a resolution of 0.0078125 s. We then divided the data into 256-s segments and, following the prescription of Reig et al (2006), we obtained the Fourier resolved spectra of the source in the following broad frequency bands: 0.008-0.8 Hz, 0.8-8 Hz and 8-64 Hz. In total, we examined a total of approximately 54.3 ks of data from the RXTE archives at 7 epochs (see Table \[log\]).
Spectral Index – QPO Frequency Correlation
------------------------------------------
It is well established that in accreting neutron star systems, the X-ray luminosity and their spectral and/or timing quantities (X-ray colors, QPO frequencies) tend to correlate with the spectral state of a an individual source, but that these correlations are not identical for the ensemble of sources (see e.g. van der Klis 2006). Furthermore, for a given source, the correlations are much more pronounced over short (hours to days) than on longer time scales (van der Klis 2000 and references therein). Therefore, the use of hardness ratios and source intensity as tracers of the source spectral state can become blurred. In addition, the problem of gain changes associated with the aging of the PCU detectors aboard $RXTE$ makes even the determination and comparison of X-ray colors at different epochs difficult.
To avoid these difficulties we have opted to use one of the QPO frequencies as a proxy for the state of the source. As has been previously pointed out, e.g. van der Klis (2006), the frequencies of all PDS features, most notably QPOs, correlate with the source intensity and also with its spectral state. Furthermore, there is a noted correspondence in the sequence of the spectral and timing states with source intensity between accreting neutron stars and black hole systems.
Motivated by the correlation between the (low-frequency) QPO centroid frequency and photon index in several black-hole XRBs (Vignarca et al. 2003), we have opted to use the corresponding QPO frequency of u as a tracer of the configuration changes in the accretion flow. Although as noted above, the neutron star spectra are more complicated than those of the black holes, we find that the correlation between the spectral and timing properties of u and similar systems (van der Klis 2006) is sufficiently robust to allow the use of the QPO frequency as an indicator of the spectral state. In fact, Titarchuk & Shaposhnikov (2005) found a correlation between the QPO frequency and their spectral index parameter $\Gamma$ in their analysis of u. However, given that its spectrum is more complicated than those of galactic black holes one should not consider this correlation to be universal.
Conventional Spectral and Timing Analysis
=========================================
Fig 1 shows typical energy spectra of the source corresponding to the observational periods selected. The spectra were extracted from PCA [*Standard 2*]{} mode data, and in this case no Fourier frequency decomposition has been applied. The response matrix and background models were created using the standard HEADAS software, version 5.3. The number of detectors (PCU) that were switched on for each observation varies, and, in order to be able to compare the spectra, we divided them by the respective number of PCUs.
The spectral analysis was performed using the XSPEC analysis package version 11.3.1. We have added systematic errors of 1% to all channels and have restricted our analysis to the 2-16 keV band only (to match the energy band used in the case of the Fourier-resolved spectra). The results are listed in Table \[avsptab0\]. The errors quoted for the best fit values correspond to the 90% confidence limit for one interesting parameter.
We analyzed data at seven different epochs as indicated in Table 1. The 21 and 26 September 1997, 25 January 1999 and 9 February 2001, observations were selected because they span a wide range in both the QPO and break frequencies, $\nu_1$ and $\nu_{\rm b}$, respectively, as well as the spectral parameter $\Gamma$ of [@tita05]. As mentioned above we inferred the spectral state of ufor a given observation based on the pair ($\Gamma$, $\nu_{\rm b}$). In three cases, 7 June 2001, 18 February 1996, and 19 August, 1999 the source was known to be in a particlular region of the color-color diagram based on results in the published literature.
As shown in Table 2 the average spectra comprise a number of continuum and atomic transition components: (a) A low temperature disk blackbody of innermost temperature $T_{\rm in} \simeq 1$ keV. (b) A blackbody component of $T_{\rm bb} \sim 2$ keV attributed to the boundary layer. (c) A power law with index $\Gamma \simeq 2$, presumably the result of the Comptonization of the disk or boundary layer photons by the accretion disk corona. (d) An Fe line feature at $E_{\rm c} \sim 6.6$ keV and, (e) An Fe line edge at $E_{\rm edge} \sim 9.5$ keV. In figs \[is\]a,b. we present the overall spectrum as the decomposition of these three components for two epochs when the source was in its island (09 Feb. 2001) and upper banana (21 Sep. 1997) states. The disk blackbody component is not required in the spectrum of the island state (IS) of Feb 09, 2001 but it is necessary to fit the lower banana (LB) and upper banana (UB) branch spectra. The blackbody temperature $T_{\rm bb}$ appears to increase as the source moves from the IS to the UB while its normalization traces the overall increase in the source counts along the above sequence. The progression from the IS to the LB and UB states is also accompanied by an increase of the power law normalization and its index $\Gamma$. The latter is usually interpreted as a steepening of the Comptonization spectrum due to the cooling of the Comptonizing electrons by the increased soft (blackbody) photon flux.
We have also found that the source variability properties change markedly with the spectral state. First, as shown in Table 1 the break and QPO frequencies increase monotonically as the source flux and power law index $\Gamma$ increase. In Fig 3 we present the PSD of the source in three different spectral states. The island state exhibits the highest RMS variability at all frequency bands, its $\nu P_{\nu}$ spectrum exhibiting a maximum at the limit of the searched frequencies ($\sim 500$ Hz). In the lower banana state the $\nu P_{\nu}$ spectrum rises from the low frequencies to a plateau (i.e. $\nu P_{\nu}\propto \nu^0$ at $\nu \sim 20$ Hz) and the RMS variability is lower at all frequencies than that of the island state. Finally, in the upper banana state the shape remains qualitatively similar to that of the LB but with greatly reduced amplitude and the additional appearance of excess power at frequencies lower than 0.3 Hz. We believe that this sequence in the timing properties is suggestive of the progressive influence of the effects of radiation pressure in the accretion process (negligible in the IS but progressively important towards the UB state). It is reasonable to consider the possibility that the radiation feedback on the flow could damp the short scale variability by the diffusion of photons through the accreting plasma; inundating the system with soft photons that could conceivably also reduce the overall variability amplitude [@kazhua].
Fourier Resolved Spectral Analysis
==================================
In this section we present the results of our Fourier resovled spectral analysis for each state and compare the resulting best-fit parameters to those of the previous section (i.e, those obtained from the average energy spectra). As before, the [ XSPEC]{} package was used for the model fitting. In all cases we added an absorption component which we fixed at $3\times10^{22}$ cm$^{-2}$ (e.g. Narita, Grindlay & Barret, 1999). A uniform systematic error of 1% was added quadratically to the statistical error of all Fourier spectra in each energy channel. Errors quoted for the best-fit values correspond to the 90% confidence limit for one interesting parameter.
The virtue of the frequency-resolved spectra is that they receive significant contribution only from the spectral components that are variable on the time scales sampled; the FRS do not represent photon rates but rather variability amplitudes at a given energy. Therefore, through Fourier-resolved spectroscopy we can investigate whether different spectral components in the overall spectrum of the source (e.g. blackbody, Comptonization, iron line) are variable and at which frequency. In general, the interpretation of the FRS is not unique (see Papadakis et al. 2005, Reig et al. 2006 for some insights). For spectral features that result from the reprocessing of higher energy ($E\ge 7$ keV) continuum, such as the iron Fe K$\alpha$ line and the Compton reflection “hump”, one plausible interpretation of the Fourier-resolved analysis is that the light crossing delay $R/c$ (where $R$ is the size of the reprocessing region) filters out variability of these features on frequencies higher than $\sim c/R$. We would note that while this interpretation may be the most straightforward one, other interpretations, such as scenarios involving screening geometries are also possible e.g. ([@revn99]).
In the following subsections we describe the results of our analysis for each of the spectral states of the source. In addition to the four sets of data mentioned above, we analyzed data for three additional epochs for which the spectral state of the source was previously documented. Our objective is to compare the results obtained above with the traditional designations of island and banana states in order to characterize the spectral and timing properties of each state. We searched in the literature for observations for which u was clearly identified as being in one of the characteristic atoll states. u was in the island state on June 7, 2001 [@migl03], upper banana state on August 19, 1999 [@pira00] and lower banana state on February 18, 1996 [@salv01].
The Island State
----------------
The source was in the island state on Feb. 09, 2001, and, as noted by Migliari et al. (2003) u also appeared to be in the island state during the June 7, 2001 observations. We obtained the 64-channel, 1-ms data sets from those observations and formed the FRS as described in §3. We found that a single-temperature blackbody model led to fits which were unacceptable, particularly at our low-frequency band. In particular, residuals at the low- and high-energy channels were evident. Improved fits were provided with an additive combination of the blackbody plus a powerlaw. The powerlaw required is rather flat $\Gamma \simeq 0.5$, but is is poorly constrained, particulary in the Feb. 09, 2001 observation. The reason is that in those model fits the blackbody is the predominant component except in the (several) highest energy energy bins.
The blackbody component in these cases is characterized at each frequency band by a temperature of about $kT=1.0-1.7$ keV without an obvious trend. The spectral decomposition parameters for the island-state FRS are given in Tables \[ISplfits\] and \[ISbbplfits\]. As summarized there and shown in fig. \[island\], the Feb. 9, 2001 FRS, for example, can be fitted well by a blackbody plus powerlaw spectrum with temperature $k T \sim 1.5-1.7$ keV, similar in magnitude to the temperature $T_{\rm bb}$ of the conventional spectra attributed to emission by the boundary layer. [ The Power Law (energy) spectral indices]{} are quite hard ($\simeq 0.6$) although they are marginally constrained. The same spectra can actually be fit by a simple power law absorbed at low energies, without the presence of a black body component. A summary of such fits is given in Table \[ISplfits\], for an absorbing column of $N_H = 3 \times 10^{22}$ cm$^{-2}$. Also see \[island\]. The fits exhibit no particular trend with Fourier frequency.
We also tested several other simple models. A powerlaw model with low-energy cutoff was unable to constrain the inflection energy value and broken powerlaw fits led to unacceptable results.
There seems to be a general trend in the island state FRS towards hard or even positively sloped spectral-energy distributions, i.e. a trend variablity to increase with energy. This is particularly notable for example in the low-frequency spectra for both island state epochs (Tables \[ISplfits\] and \[ISbbplfits\]), and it is consistent with the RMS variability versus energy analysis of [@revn00]. This is reminiscent of Blazar variability.
While none of the FRS results require an iron line - upper limits on its equivalent width range from a few hundred to a thousand eV - it is clearly seen in the conventional spectra for 3 of the 4 epochs denoted in Table 2; specifically the EW determination indicates a $\sim 10\sigma$ significance in several cases. The FRS are harder than the averaged spectra, indicating again that the hard emission may be the predominant component of the source variability.
Finally, we note that although the island-state count rate was lower by about a factor of 10 than the banana-state cases, the rms variability in the island state is significantly higher. Thus, the statistics of the island-state FRS are comparable or superior to those of some banana state epochs.
The Lower Banana state
----------------------
On Jan. 25, 1999 as well as on Feb. 18, 1996 (also see di Salvo et al. 2001) the source intensity along with the PDS break frequency at about 16 Hz, indicated that u was in the the lower banana state. A single blackbody model provides acceptable fits for all three of our frequency ranges (our fit results are summarized in Table \[frstab1\]). Its temperature is consistent with that of the time averaged spectra, indicating that it is the boundary layer component which is responsible for the source variability. The absence of the power law and Fe line components that are present in the time averaged spectral fits, suggests that these components are not variable on the time scales examined. The normalization of this component is comparable in the low and high frequencies and slightly lower at the mid-frequency range, indicating that all frequencies contribute roughly equally to the variations of this component.
The single blackbody fit to the FRS for various banana spectral states, and for the three frequency bands have characteristic temperatures that span the nominal kT$\simeq 2-3$ keV range. However, no unambiguous trends with temperature versus frequency are seen.
The Upper Banana State
----------------------
The source was found to be in the upper banana state on Sept. 26 and Sept. 21 1996, as determined by the value of the PSD break frequency $\nu_b$. We also obtained data for the upper banana state noted during August 19, 1999 by (Piraino et al. 2000). In this case, we found that the RMS variations above $\sim$8 Hz (high-frequency) were apparently small as we were unable to obtain useful signal-to-noise for any of our FRS bands for these data. The overall count rate on the other hand was high, $\sim 10$ times that for the island state. As discussed above a single blackbody provides a good fit to the FRS without the need for additional components at any of the frequency bands. The amplitude of this component appears to increase with decreasing frequency in the 1996 observations while remained the same during the 1999 one, thus indicating that this attribute of the spectra is not characteristic of the particular state.
Discussion
==========
We have investigated the spectro-temporal characteristics of the atoll X-ray binary u. To this end we have produced the energy and power spectra of the source in its different spectral states, i.e. island (IS), lower banana (LB) and upper banana (UB) and also the Fourier-resolved spectra (FRS) of each state for three different frequency bands.
Fits of the time average spectra of the source require a combination of a blackbody (to model the boundary layer emission), a multicolor blackbody (to model emission by the accretion disk), a power law (to model coronal emission), an Fe line and a low-energy galactic-absorption column of $N_H = 3\times 10^{22}$ cm$^{-2}$. The results for the conventional spectra are given in Table \[avsptab0\]. As the source crosses from the island to the lower and upper banana states the source spectra change significantly (see Fig. 1). Our spectral decomposition indicates for this sequence the component exhibiting the largest parameter range is the multi-color disk. While its presence is not required in the island state, it is necessary for acceptable fits in the other states with approximately constant temperature and highly variable normalization. Unfortunately, the RXTE spectra do not cover the energy regime $E < 2$ keV, necessary for the more accurate determination of this component. Also variable is the index $\Gamma$ of the power law component; the increase of $\Gamma$ with the total flux of the source is generally attributed to the decrease in the temperature of the Comptonizing medium by the increase of soft thermal photon flux and the resulting Compton cooling effects. The temperature of the blackbody component exhibits an increase between the island and the banana states (referring to the blackbody plus powelaw fits to the island state), but it appears to remain constant once the banana states are reached. This suggests a behavior different from that of the Z-sources. Gilfanov et al. (2003) find from their FRS analysis that the temperature of this component remains constant, suggesting that the sources’ boundary layer is radiation pressure dominated. Following the same argument, the increase of the thermal component temperature with increasing source flux leads to the conclusion that in the atoll sources the boundary layer is not radiation dominated in the island state, while it appears to become so in the banana states. Finally, the EW of the Fe line appeared to remain constant to within statistics throughout the source’s spectral sequence.
The variability properties of the source, as manifest by the source power-density spectra, change significantly with its spectral state: The overall RMS fluctuations across all frequencies decrease with increasing source flux; in addition the PDS shape changes as the source moves from the island to its banana states (see Fig. 3). Finally, in the general case, the frequency of features in the PDS such as breaks and QPOs increase along the island to upper banana sequence. We suggest that the decrease in variability is due to the effects of radiation pressure on the accretion flow, which through diffusion damps the high frequency components and the overall amplitude.
We have elaborated further on the study of the spectro-temporal properties of the source by producing its Fourier-resolved spectra at three frequency bands for each of the source’s spectral states, namely $0.008 - 0.8$ Hz, $0.8 - 8$ Hz and $8 - 64$ Hz. While interesting features in the Fourier domain are known to occur at higher frequencies (i.e. kHz QPO) our data did not allow the application of this technique to these frequencies with any significance. The source state was determined either from prior analysis gleaned from the published literature or was defined by the spectral hardness–break frequency correlation. In the latter cases, we mostly relied on the measurements and analysis of [@tita05], but in several instances we measured these quantities ourselves. Several notable trends are gleaned from our analysis and are identified below.
The results of our FRS analysis can be summarized as follows:
\(i) The iron line at $\sim$6.6 keV, distinctly present in the conventional spectra ($\simeq 10 \sigma$ for the banana state cases), is not apparent in any of our frequency-resolved spectra. For example, when we fix the line energy at 6.6 keV and the width to 0.6 keV, we derive only upper limits on the equivalent widths (values ranged from about 50 to 600 ev). This suggests that, at leaset within the limitations of our measurements, there may be no significant Fe line variability on time scales [ between $10^{-2} - 10^2$ s]{}. This is in agreement with the lack of correlation between the Fe K$\alpha$ equivalent width and QPO frequencies reported by [@tita05]. These authors concluded that the size of the Fe K$\alpha$ emitting region is not of the same order as that of the X-ray continuum emission, the former possibly being orders of magnitude larger than the latter. On the other hand, a different view has been promoted by Miniutti et al. (2003) with respect to the Fe line variability in AGN. In as much as the geometry and physics of Fe line production in these very diverse objects is similar, such an interpretation should also be considered.
\(ii) No disk component is present in the FRS, indicating as in the case of the iron line, that the accretion disk emission is significantly less variable at the frequencies sampled by our study, i.e. 0.08-64 Hz. This result agrees with those found in other neutron-star [@revn06] and black-hole [@reig06] binaries.
\(iii) We have examined five data sets corresponding to the banana state spectral/temporal configuration. Our FRS reveal that the temperature of the thermal spectral component is consistent with being constant across frequencies and states. Unfortunately, this statement cannot be made with high degree of confidence for the UB states as the decrease in the variability amplitude makes the determination of the Fourier resolved parameters of the spectra difficult.
\(iv) Two observations corresponding to the island state reveal evidence for a spectral hardening at the lowest frequencies examined. The island state are generally harder than the corresponding frequency averaged spectra. Single blackbody model fits to the island state FRS, were unsatisfactory, with a significant high-energy residual appearing, particularly for our lowest frequency interval. The addition of a hard powerlaw component leads to improved fits to the data, although the slope is poorly constrained. [ We have also tried an alternative models, the most satisfactory of which was a single power law with a low energy neutral absorpion by a column of $N_H = 3 \times 10^{22}$ cm$^{-2}$. It was found that this presents an good fit to the data with the corresponding parameters as given in Table \[ISplfits\]. In general,]{} the FRS are notably harder than the conventional spectra for the island state, indicating a substantial contribution of the high-energy emission to the source variablity, at least for this spectral state, a situation not unlike that found in other studies as previously referenced.
\(v) The results of our FRS analysis are in general agreement with those of Gilfanov et al. (2003) of Z-sources, in that the FRS spectra are dominated by a single blackbody component of constant temperature when $\dot{M}\sim 0.1-1 \; \dot M_{\rm Edd}$, that is, for Z sources and atoll sources in the banana state. This is attributed to emission by the boundary layer which in addition is radiation pressure dominated hence the independence of the temperature on the source flux (the local flux is always that of Eddington). The temperature of this component is consistent with that of the blackbody component in the time average spectra.
\(vi) Finally, we have extended the study of the Fourier-resolved spectra of neutron-star low-mass X-ray binaries to the lower luminosity end. Previous studies had concentrated on [ higher luminosity states of these objects]{}, i.e., Z-sources or atoll sources in their soft/high spectral state [@gilf03; @gilf05; @revn06]. The X-ray luminosity in the 2–16 keV range of u in the island state, assuming a distance of 4.3 kpc [@fost86] is $\sim 3\times 10^{36}$ erg s$^{-1}$, i.e. $\sim 10^{-2} L_{\rm Edd}$, about one order of magnitude lower than previous studies. We found that the temperature of the blackbody component of the conventional spectrum of the island state is significantly lower than that of the corresponding FRS when the powerlaw plus blackbody model is applied. The most straightforward interpretation of this fact is that the conventional island-state spectrum includes contribution from a component of lower temperature and little variability.
Future applicaton of Fourier-Resolved spectroscopy to low-mass X-ray binaries, incorporating both larger data sets and additional objects of the representative subclasses offers the posiblity for furher insight into each of these issues.
[[**Acknowledgements**]{}]{}
The authors wish to acknowledge Nikolai Shaposhnikov for providing the [@tita05] spectral-QPO frequency correlation results in tabular form. We also thank the referee, Mikhail Revnivtsev for a thorough reading of the manuscript and a number of useful suggestions. This work has made use of data obtained through the High Energy Astrophysics Science Research Center of the NASA Goddard Space Flight Center. Part of this work was supported by the General Sectreteriat of Research and Technology of Greece.
[999]{}
di Salvo, T., Méndez, M., van der Klis, M., Ford, E., Robba, N.R. 2001, ApJ, 546, 1107
Ford, E., & van der Klis, M., 1998, ApJ, 506, L39
Foster, A. J., Fabian, A. C., Ross, R. R. 1986, MNRAS, 221, 409
Gilfanov, M., Churazov, E. & Revnivtsev, M. 2000, MNRAS, 316, 923
Gilfanov, M., Revnivtsev, M., & Molkov, S., A&A, 410, 217
Gilfanov, M., & Revnivtsev, M. 2005, Astr, Nachr., 326, 812
Kazanas, D. & Hua, X.-M., 1999, ApJ, , 519, 750
Mendez, M., & van der Klis, M., 1999, ApJ, 517, L51.
Migliari et al., 2003, MNRAS, 342, L67
Narita, T., Grindlay, J.E., and Barret, D., bull AAS, 31, 904
Papadakis, I. E., Kazanas, D., & Akylas, A. 2005, ApJ, 631, 727
Papadakis, I.E., Ioannou, Z., & Kazanas, D., 2006, ASNA, 327, 1047
Piraino, S., Santangelo, A., & Kaaret, P., 2000, A&A, 360, L35
Reig, P., Papadakis, I.E., Shrader, C.R., & Kazanas, D., 2006, ApJ, 644, 424
Revnivtsev, M., Gilfanov, M., & Churazov, E. 1999, A&A, 347, L23
Revnivtsev, Mikhail G., Borozdin, Konstantin N., Priedhorsky, William C.,& Vikhlinin, Alexey 2000, Ap.J., 230, 955
Revnivtsev, M., Gilfanov, M., & Churazov, E. 2001, A&A, 380, 502
Revnivtsev, M. G., Gilfanov, M. R. 2006, A&A, 453, 253
Sobolewska, M.A., & Zycki, P.T., 2006, MNRAS, 370, 405
Titarchuk, L. & Shaposhnikov, N. 2005, ApJ, 626, 298
van der Klis, M., 2000, ARA&A, 38, 717
van der Klis, M., 2006, in “In: Compact stellar X-ray sources”, Ed. by W. Lewin & M. van der Klis. Cambridge Astrophysics Series, No. 39. Cambridge University Press.
Vignarca, F., Migliari, S., Belloni, T., Psaltis, D., van der Klis, M. 2003, A&A, 397, 729
Wijnands, R., 2001, AdSpR, 28, 469
Zycki, P., 2003, ASPC, 290, 135
[lccccc]{} Epoch &$\Gamma$ &$\nu_{\rm b}$ &Source &2-16 keV &Exposure\
& &(Hz) &state &flux (c/s)$^*$ &time(ks)\
Feb 9, 2001 &1.40 &0.75 & IS+ &640 &4.40\
Jan 25, 1999 &2.07 &12.54 & LB+ &1824 &11.3\
Sep 26, 1997 &2.88 &24.2 & UB+ &1659 &12.8\
Sep 21, 1997 &5.54 &41.55 & UB+ &1953 &13.5\
Jun 7, 2001 & 1.46 & 0.9 &IS &445 &6.91\
Feb 18, 1996 &2.41 &12.21 &LB &1380 &3.33\
Aug 19, 1999 &3.35 & 24.1 &UB &2600 &2.05\
[lcccc]{} Observation &Feb 9, 2001 &Jan 25, 1999 &Sep 26, 1997 &Sep 21, 1997\
Flux$^1$ (erg s$^{-1}$ cm$^{-2}$) &1.4 &4.1 &3.8 &4.6\
\
$T_{\rm in}$ (keV) &– &1.7$^{+1.1}_{-0.5}$ &1.06$\pm$0.03 &1.4$\pm$0.1\
norm &– &5$^{+10}_{-4}$ &110$\pm$15 &45$^{+20}_{-10}$\
\
$kT_{\rm bb}$ (keV) &1.27$\pm$0.05 &2.3$\pm$0.4 &2.1$\pm$0.1 &2.2$\pm$0.1\
norm ($\times 10^{-2}$)&0.34$\pm$0.08 &1.3$^{+0.5}_{-0.1}$ &2.8$\pm$0.1 &3.3$\pm$0.3\
\
$\Gamma$ &1.75$\pm$0.04 &2.03$\pm$0.08 &2.1$\pm$0.2 &2.3$^{+0.2}_{-1.4}$\
norm &0.28$\pm$0.04 &1.0$\pm$0.2 &0.4$\pm$0.2 &0.42$\pm$0.05\
\
E$_{\rm c}$ (keV) &6.6$\pm$0.2 &6.6$\pm$0.2 &6.5$\pm$0.3 &6.6$\pm$0.3\
$\sigma$ (keV) &0.5$\pm$0.4 &0.6$\pm$0.2 &0.8$\pm$0.3 &0.7$\pm$0.3\
EW (eV) &100$\pm$60 &150$\pm$15 &180$\pm$20 &140$\pm$15\
\
E$_{\rm edge}$ (keV)&– &9.5$\pm$0.5 &9.6$\pm$0.6 &9.9$\pm$0.6\
$\tau$ &– &0.04$\pm$0.02 &0.04$\pm$0.03 &0.04$\pm$0.03\
\
\
[lcc]{} Observation &09/02/01 &07/06/2001\
\
$\Gamma$ &1.7$\pm$0.1 &1.8$\pm$0.2\
norm ($\times 10^{-2}$) &2.0$\pm$0.3 &0.9$\pm$0.4\
\
$\Gamma$ &1.7$\pm$0.1 &1.7$\pm$0.2\
norm ($\times 10^{-2}$) &3.4$\pm$0.6 &1.6$\pm$0.5\
\
$\Gamma$ &1.6$\pm$0.3 &1.4$\pm$0.6\
norm ($\times 10^{-2}$) &3.0$\pm$1.5 &1.0$\pm$0.8\
$\chi^2$/dof &1.5/52 &0.91/46\
[lcc]{} Observation &09/02/01 &07/06/2001\
\
$kT_{\rm bb}$ (keV) &1.7$\pm$0.1 &1.0$\pm$0.3\
norm ($\times 10^{-3}$) &0.7$\pm$0.1 &0.19$\pm$0.04\
$\Gamma$ &0.6$\pm$0.3 &0.4$\pm$0.7\
norm ($\times 10^{-3}$) &0.55$\pm$0.06 &0.35$\pm$0.07\
\
$kT_{\rm bb}$ (keV) &1.5$\pm$0.1 &1.7$\pm$0.3\
norm ($\times 10^{-3}$) &1.0$\pm$0.1 &0.5$\pm$0.1\
$\Gamma$ &0.6$^f$ &0.4$^f$\
norm ($\times 10^{-3}$) &1.4$\pm$0.1 &0.32$\pm$0.06\
\
$kT_{\rm bb}$ (keV) &1.7$\pm$0.6 &1.0$^{+2.0}_{-0.5}$\
norm ($\times 10^{-3}$) &0.9$^{+0.9}_{-0.4}$ &0.9$\pm$0.3\
$\Gamma$ &0.6$^f$ &0.4$^f$\
norm ($\times 10^{-3}$) &1.6$\pm$0.4 &0.8$\pm$0.2\
$\chi^2$/dof &0.90/51 &0.73/43\
F-test prob.$^*$($\times 10^{-2}$) &$7\times 10^{-8}$ &$7\times 10^{-6}$\
\
\
[lccccc]{} Observation &Feb 18, 1996 &Jan 25, 1999 &Sep 26, 1997 &Sep 21, 1997 &August 19, 1999\
\
$kT_{\rm bb}$ (keV) &2.7$\pm$0.4 &2.27$\pm$0.04 &2.27$\pm$0.04 &2.14$\pm$0.05 &2.4$\pm$0.2\
norm ($\times 10^{-3}$) &1.0$\pm$0.1 &4.8$\pm$0.1 &5.0$\pm$0.1 &3.7$\pm$0.1 &1.8$\pm$0.1\
\
$kT_{\rm bb}$ (keV) &2.10$\pm$0.08 &2.02$\pm$0.05 &2.3$\pm$0.2 &2.1$\pm$0.4 &2.5$\pm$0.5\
norm ($\times 10^{-3}$) &2.6$\pm$0.1 &3.07$\pm$0.07 &1.5$\pm$0.2 &1.4$\pm$0.3 &2.2$\pm$0.4\
\
$kT_{\rm bb}$ (keV) &2.18$\pm$0.09 &2.15$\pm$0.07 &2.2$\pm$0.4 &2.2$\pm$0.3 &2.8$\pm$0.6\
norm ($\times 10^{-3}$) &4.2$\pm$0.2 &4.6$\pm$0.1 &2.2$\pm$0.3 &2.5$\pm$0.3 &4.0$\pm$0.8\
$\chi^2$/dof &0.95/37 &1.80/64 &1.05/53 &0.94/55 &0.95/39\
[^1]: http://heasarc.gsfc.nasa.gov/docs/software/lheasoft/
|
---
abstract: |
[The renormalized expectation value of the energy-momentum tensor for a scalar field with any mass $m$ and curvature coupling $\xi$ is studied for an arbitrary homogeneous and isotropic physical initial state in de Sitter spacetime. We prove quite generally that $\langle T_{ab}\rangle$ has a fixed point attractor behavior at late times, which depends only on $m$ and $\xi$, for any fourth order adiabatic state that is infrared finite. Specifically, when $m^2+\xi R>0$, $\langle
T_{ab} \rangle$ approaches the Bunch-Davies de Sitter invariant value at late times, independently of the initial state. When $m=\xi=0$, it approaches instead the de Sitter invariant Allen-Folacci value. When $m=0$ and $\xi\ge 0$ we show that this state independent asymptotic value of the energy-momentum tensor is proportional to the conserved geometrical tensor $^{(3)}H_{ab}$, which is related to the behavior of the quantum effective action of the scalar field under global Weyl rescaling. This relationship serves to generalize the definition of the trace anomaly in the infrared for massless, non-conformal fields. In the case $m^2+\xi R=0$, but $m$ and $\xi$ separately different from zero, $\langle T_{ab} \rangle$ grows linearly with cosmic time at late times. For most values of $m^2$ and $\xi$ in the tachyonic cases, $m^2+\xi R < 0$, $\langle T_{ab} \rangle$ grows exponentially at late cosmic times for all physically admissable initial states. ]{}
address: |
$^{1}$ Department of Physics Wake Forest University, Winston-Salem, North Carolina, 27109\
$^{2}$ T-8, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545\
$^{3}$ Centro de Astrobiología, CSIC-INTA, Carretera de Ajalvir Km. 4, 28850 Torrejó n, Madrid, Spain
author:
- 'Paul R. Anderson$^{1,2}$ [^1], Wayne Eaker$^{1}$ [^2], Salman Habib$^{2}$ [^3], Carmen Molina-París$^{2,3}$ [^4], and Emil Mottola$^{2}$ [^5]'
date: 'LA-UR-00-52; May 23, 2000'
title: Attractor states and infrared scaling in de Sitter space
---
\#1\#2
Introduction
============
Quantum field theory in curved spacetimes does not contain in itself a unique specification of the quantum state of the system [@bi-da]. Even in Minkowski spacetime, where the existence of the Poincaré group singles out a special state, the Minkowski vacuum, it is certainly of interest to consider states that are non-invariant under Poincaré transformations, since they contain all the information about the physical excitations and dynamics of the theory. Such non-vacuum states are also necessary in a general initial value formulation of the back-reaction problem in both curved and flat spacetimes. In flat space the initial value problem for arbitrary physically allowable states has been formulated and studied for both QED and scalar $\Phi^4$ theory in the large $N$ limit, principally for time varying but spatially homogeneous mean fields [@chkm; @HKMP].
The simplest situation in which the back-reaction problem can be studied in curved spacetime is that of a free scalar field in a spatially homogeneous and isotropic Robertson-Walker (RW) cosmology, where the geometry is characterized by just one non-trivial function of time. The wave equation for a free scalar field in such a geometry can be separated and expressed in terms of a complete set of time dependent mode functions. The general initial value problem is specified by giving initial data for this complete set at a given initial time. The back-reaction of the quantum scalar field(s) on the RW geometry can be studied then by constructing the renormalized expectation value of the energy-momentum tensor $\langle T_{ab}
\rangle$ of the field(s) and solving (numerically) the semi-classical Einstein equations, augmented by higher derivative terms required by renormalization[@wald; @star; @fhh; @a1; @wada; @sa]. As in the flat space examples, this semi-classical back-reaction problem becomes exact in the large $N$ limit, with $N$ the number of identical scalar fields [@tom].
As a prelude to the dynamical back-reaction problem in cosmological spacetimes it is necessary to study non-vacuum states first in fixed RW backgrounds. The maximally symmetric de Sitter spacetime is of particular interest. Most previous work has focused on maximally symmetric $O(4,1)$ de Sitter invariant states or the special $O(4)$ invariant state found by Allen [@a]. Since the universe is not globally $O(4,1)$ invariant, a more generic set of initial conditions, consistent only with RW symmetry and general principles of renormalization of $\langle T_{ab} \rangle$ is required for cosmology. The investigation of these much weaker requirements and specification of the general initial value problem for back-reaction calculations was initiated in Ref. [@hmp].
In this paper we study the behavior of the renormalized $\langle
T_{ab} \rangle$ for arbitrary physically admissable spatially homogeneous and isotropic states in a fixed de Sitter background. We argue in Section III that such states must be fourth order adiabatic states [@bi-da] that also possess an infrared finite two-point function. In de Sitter space the wave equation for free scalar fields can be solved exactly for arbitrary values of the mass and the curvature coupling. Its solutions depend only on the wave number $k$ of the mode and the parameter $\nu^2 ={9\over 4}-m^2 \alpha^2 -12\xi$, with $R=12\alpha^{-2}$ the constant scalar curvature of de Sitter spacetime. For $\Re (\nu)<{3\over 2}$, corresponding to $m^2+\xi R>0$, we prove that for [*all*]{} UV and IR physically allowed initial states the renormalized value of $\langle T_{ab}
\rangle$ at late times asymptotically approaches that of the Euclidean or Bunch-Davies de Sitter invariant state [@c-t; @t; @d-c; @bu-da]. The conformally invariant scalar field ($m=0$, $\xi={1\over 6}$) falls into this class.
The case $\nu = {3\over 2}$ corresponding to $m^2 \alpha^2 +12\xi =0$ is more delicate. If $m$ and $\xi$ are separately zero (the massless, minimally coupled case), then we prove that the renormalized $\langle
T_{ab} \rangle$ for all physically admissable states approaches the Allen-Folacci de Sitter invariant value [@a-f; @f; @k-g]. Numerical evidence for this result was found previously in Ref. [@hmp]. In this paper we provide an analytic proof that late time attractor behavior occurs for [*all*]{} physically admissable RW states, when $m^2 \ge 0$ and $\xi \ge 0$. If $m^2 \alpha^2 +12\xi =0$ but $m^2$ and $\xi$ are not separately zero (so that one of them is negative) we prove that $\langle T_{ab} \rangle$ grows linearly in RW comoving (cosmic) time without bound, and this asymptotic behavior is independent of the state of the field. Finally, and in contrast, in the case $\nu>{3\over
2}$, corresponding to $m^2+\xi R<0$, $\langle T_{ab} \rangle$ depends sensitively on the state and, for most values of $m$ and $\xi$, grows exponentially at late times for all states. This case is of considerably less physical relevance, since it corresponds to a tachyonic field theory with no stable vacuum state.
The asymptotic approach of $\langle T_{ab} \rangle$ to a de Sitter invariant form, independently of the lower symmetry of the initial data when $\Re(\nu)\le {3\over 2}$ is a striking result. Certainly no such attractor behavior of $\langle T_{ab} \rangle$, independent of initial conditions occurs in Minkowski space for any mass. One may regard this result as a kind of cosmic “no hair” theorem for scalar quantum fields in de Sitter space. For $\Re(\nu) < {3\over 2}$ it is in accord with one’s classical intuition that any initial energy density satisfying the weak energy condition ($\varepsilon +p>0$) is redshifted away by the exponential de Sitter expansion, although as we will see, the redshifting of the quantum $\langle T_{ab} \rangle$ is not that of classical matter or radiation. At asymptotically late times what is left behind is a kind of frozen quantum vacuum energy “condensate,” satisfying the de Sitter invariant equation of state $p=-\varepsilon$. This result justifies the choice of the Bunch-Davies vacuum in calculations of quantum fluctuations of [*free*]{} fields, [*i.e.*]{} without back-reaction, in a long-lived de Sitter expansion phase of inflationary cosmological models. For $\nu = {3\over 2}$, $m
= \xi = 0$, the approach of $\langle T_{ab} \rangle$ to the de Sitter invariant Allen-Folacci value is perhaps more surprising. As shown in Section IV one expects the leading order contribution of the modes to $\langle T_{ab} \rangle$ in this case to be constant in comoving time at late times. In fact this occurs if $m$ and $\xi$ are not separately zero for all the modes. However when $m$ and $\xi$ are both zero, the leading order contributions to $\langle T_{ab} \rangle$ of all the modes [*except the spatially homogeneous one*]{}, for an arbitrary physically admissable state have exactly zero coefficient, the subleading contributions redshift away, and we are left only with the with the de Sitter invariant Allen-Folacci constant value at late times. The finite difference from the Bunch-Davies value may be attributed entirely to the constant behavior of the spatially homogeneous mode contributing to the vacuum energy condensate in de Sitter space.
In all those cases for which $\Re(\nu)\le {3\over 2}$ when $\langle
T_{ab} \rangle$ approaches a de Sitter invariant value at late times, the quantum expectation value loses all its initial state dependence and hence its asymptotic value must be determined purely by the background geometry. When the mass of the field vanishes, the existence of only one covariantly conserved local geometrical tensor of adiabatic order four in de Sitter space, namely $^{(3)} H_{ab}$ given by Eq. (\[eq:Hthr\]) below, permits us to identify the asymptotic value of $\langle T_{ab} \rangle$ in the vacuum energy condensate with this tensor. Since $^{(3)}H_{ab}$ cannot be derived by variation of a covariant local action, but corresponds instead to a certain well defined non-local term in the quantum effective action [@amm], the asymptotic vacuum energy condensate of the quantum field is determined by the global or extreme infrared properties of de Sitter space. The form of the non-local effective action is determined by the trace anomaly in conformally flat spacetimes. Since the approach of $\langle T_{ab} \rangle$ to a de Sitter invariant value occurs for all $\xi \ge 0$, the existence of this term in the effective action for massless fields is much more general than the strict definition of the trace anomaly in the conformally invariant case. Hence the asymptotic late time behavior of $\langle T_{ab} \rangle$ in de Sitter space can be used to define a generalized trace “anomaly” coefficient in the massless but non-conformally invariant cases, $\xi \neq {1\over 6}$. As we show by consideration of the covariant $\zeta$ function method [@bi-da], this coefficient is exactly the same as that which determines the infrared response of the vacuum condensate to global Weyl rescalings. Hence the significance of the state-independence of the vacuum energy condensate in de Sitter space is that it determines certain conformal properties of non-conformal field theories in the extreme infrared.
The paper is organized as follows. In Section \[sec:scalar\] we give the expectation value of the energy-momentum tensor as a mode sum for an arbitrary homogeneous and isotropic, physically admissable state with a non-zero initial particle number of the scalar field. In Section \[sec:analytic\] we analyze the late time behavior of expectation value of the energy-momentum tensor in de Sitter space in flat spatial sections and show that it approaches the de Sitter invariant Bunch-Davies value for all $\Re (\nu) < {3\over 2}$, independently of the initial state. In Section \[sec:nu32\] we analyze the limit $\Re (\nu) \rightarrow
{3\over 2}$ in closed spatial sections in order to keep careful track of the spatially homogeneous mode in a discrete basis, and show how the Allen-Folacci fixed point at late times is obtained for the massless minimally coupled field. We also investigate the asymptotic behavior of the energy-momentum tensor for arbitrary mass and curvature coupling when $\nu \ge {3\over 2}$. In Section \[sec:numerical\] we illustrate the analytic results with numerical studies, investigating in particular the interesting case when $\nu$ is slightly smaller than ${3\over 2}$. We find that for many states when $\nu$ is only slightly smaller than ${3\over 2}$ the energy-momentum tensor first approaches the Allen-Folacci value and only much later approaches the Bunch-Davies value. In Section \[sec:scaling\] we consider the geometric significance of the state independent asymptotic behavior of $\langle T_{ab} \rangle$, relating it to the quantum effective action which determines the behavior of $S_{\mathrm
eff}$ under global Weyl scaling and providing the generalization of the trace “anomaly” in the infrared, for $\xi \neq {1\over
6}$. Section \[sec:discussion\] contains some discussion and final conclusions. There are two Appendices. Appendix A completes the proof of the Bunch-Davies attractor behavior in the cases of integer and pure imaginary $\nu$, while Appendix B contains a discussion of the simple harmonic oscillator in the limit of vanishing frequency, which shares many features with the spatially constant mode in de Sitter spacetime.
Scalar field in a RW background {#sec:scalar}
===============================
The metric for a general RW spacetime can be written in conformal time $\eta$ in the form $${\mathrm{d}}s^2 = a^2(\eta) \left(-{\mathrm{d}}\eta^2
+ \frac{{\mathrm{d}}r^2}{1 - \kappa r^2}
+ r^2 {\mathrm{d}} \Omega^2 \right)
\; . \label{eq:metricRW}$$ Here $a(\eta)$ is the scale factor and $\kappa =0,+1,-1$ corresponds to the cases of flat, spherical, and hyperbolic spatial sections, respectively. Throughout we use units such that $\hbar = c = 1$ and the Misner, Thorne, and Wheeler [@MTW] conventions for the curvature tensors, $R^a_{bcd} = \Gamma^a_{bd,c} - ...$ and $ R_{ab} = R^c_{acb}$.
We consider in this paper a free quantum scalar field $\Phi$ with the quadratic action $$\begin{aligned}
S = -{1\over 2}\int\,
{\mathrm{d}}^4 x \sqrt{-g}
\, \left[(\nabla_a \Phi)g^{ab}
(\nabla_b \Phi)
+ m^2\Phi^2 + \xi R \Phi^2\right]
\; ,
\label{action}\end{aligned}$$ where $\nabla_a$ denotes the covariant derivative, $R$ is the scalar curvature, and $g\equiv \det(g_{ab})$. The mass $m$ and curvature coupling $\xi$ are allowed to have any real value. The wave equation for $\Phi$ obtained by varying this action is $$\begin{aligned}
\left[-{\,\raise.5pt\hbox{$\mbox{.09}{.09}$}\,}
+ m^2 + \xi R\right] \Phi(\eta, {\bf x}) = \left[ {1\over a^4} {\partial\over
\partial \eta}\left(a^2 {\partial\over \partial \eta}\right) - \frac{1}{a^2}
\Delta^{(3)} + m^2 + \xi R\right]\Phi = 0\; ,
\label{waveq}\end{aligned}$$ with $\Delta^{(3)}$ the covariant spatial Laplacian. For spacetimes with the metric (\[eq:metricRW\]) the field $\Phi$ can be expanded as a mode sum in the form, [@bi-da] $$\Phi(\eta,{\bf x}) = \frac{1}{a(\eta)} \int {\mathrm{d}}
\tilde\mu({\bf k}) \left[ a_{\bf k} Y_{\bf k}({\bf x}) \psi_k (\eta)
+ a_{\bf k}^\dagger Y_{\bf k}^*({\bf x})
\psi^*_k (\eta) \right]
\; ,$$ where the integration measure is given by $$\begin{aligned}
\int {\mathrm{d}} \tilde\mu({\bf k})
\equiv
\left\{
\begin{array}{lll}
\int {\mathrm{d}}^3 {\bf k} \;\; & {\rm if}& \kappa = 0 \; ,
\vspace{0.2cm}
\nonumber \\
\int_0^{\infty} {\mathrm{d}} k \sum_{l,m} \;\; & {\rm if}&
\kappa = -1 \; ,
\vspace{0.2cm}
\nonumber \\
\sum_{k,l,m} \;\; & {\rm if} & \kappa = +1 \; ,
\end{array}
\right.\end{aligned}$$ and the spatial part of the mode functions $Y_{\bf k}({\bf x})$ obeys the equation $$-\Delta^{(3)} Y_{\bf k}({\bf x}) = (k^2 - \kappa) Y_{\bf k}({\bf x})
\; ,$$ with $k = 1, 2, \dots$ in the case of closed spatial sections, $\kappa = +1$. The time dependent part of the mode functions $\psi_k$ obeys the equation $${\psi_k}'' + \left[k^2 + m^2 a^2 + \left(\xi
- {1\over 6}\right) a^2 R\right] \psi_k = 0
\;,
\label{eq:mode}$$ where primes denote derivatives with respect to the conformal time variable $\eta$, and the scalar curvature in a general RW spacetime is given by $$R = 6 \left(\frac{a''}{a^3} + \frac{\kappa}{a^2} \right)
\;.$$ For the quantum field to satisfy the canonical commutation relations, the creation and annihilation operators are required to obey the commutation relations $[a_{\bf k},a^{\dagger}_{\bf k'}]=
\delta_{{\bf k}{\bf k'}}$, whereupon the $\psi_k$ must obey the Wronskian condition $$\psi_k {\psi^{*}_k}' - \psi^*_k \psi'_k = i\,. \label{eq:wronskian}$$
The components of the unrenormalized energy-momentum tensor (energy density and trace) are given by [@Bunch]
$$\begin{aligned}
\varepsilon_{u} &=& -\langle {T^0}_0\rangle_u =
\frac{1}{4 \pi^2 a^4}\int {\rm{d}}\mu(k)
(2 n_k + 1) \left\{|\psi'_k|^2 + (k^2+m^2 a^2) |\psi_k|^2\right.
\nonumber \\
& &\left.\;\;\;\; + \; \left(6\xi - 1\right) \left[\frac{a'}{a}
(\psi_k {\psi^{*}_k}' + \psi_k^* \psi'_k) - \left(
\frac{a^{\prime 2}}{a^2}- \kappa\right)
|\psi_k|^2\right]\right\} \; ,
\label{eq:T00u}\end{aligned}$$
and $$\begin{aligned}
-\varepsilon_{u} + 3p_{u} &=& \langle T\rangle_{u} =
\frac{1}{2 \pi^2 a^4}\int {\rm{d}}
\mu(k)
(2 n_{k} + 1)
\left\{ -m^2 a^2 |\psi_k|^2
+ \left(6\xi - 1\right)\left[-|\psi'_k|^2
+ \frac{a'}{a} (\psi_k {\psi^{*}_k}' + \psi_k^* {\psi'_k})
\right]\right.
\nonumber \\
& &
\left.+ \left(6\xi-1\right)\left[k^2+m^2 a^2 + \left(\frac{a''}{a}-
\frac{a'^2}{a^2}\right) + \left( \xi-\frac{1}{6} \right) a^2
R\right] |\psi_k|^2 ]\right\} \label{eq:Tru}
\; .\end{aligned}$$
where we have allowed an arbitrary number of particles in the initial state, $n_k = \langle a^{\dagger}_{\bf k} a_{\bf k}\rangle$, and the scalar measure ${\mathrm{d}}\mu (k)$ is given by $$\begin{aligned}
\int {\mathrm{d}} \mu(k) \equiv
\left\{
\begin{array}{lll}
\int_0^{\infty}
{\mathrm{d}} k \; k^2 \;\;\;
& {\rm if}& \kappa = 0, -1 \; ,
\vspace{0.2cm}
\nonumber \\
\sum_{1}^{\infty} k^2 \;\;\; &{\rm if}&
\kappa = +1 \; .
\end{array}
\right.\end{aligned}$$ As we are considering spatially homogeneous and isotropic initial states (consistent with the RW symmetry), $n_k$ depends only on the magnitude $k$ of the spatial wave vector $\bf k$. Expectation values of the bilinears $\langle a_{\bf k} a_{\bf k}\rangle$ and $\langle
a^{\dagger}_{\bf k} a^{\dagger}_{\bf k}\rangle$ in a general state need not be considered since they can be removed by a time-independent Bogoliubov transformation at the initial time [@chkm]. Hence these initial state correlations may be parameterized instead by the initial data on the mode functions $\psi_k$, together with the non-negative set of $n_k$, with no loss of generality.
Since the expectation value of the unrenormalized energy-momentum tensor $\langle T_{ab}\rangle_{u}$ is quartically divergent, a procedure for defining a finite, renormalized expectation value must be given. We will follow the adiabatic regularization method [@P; @ParkerFulling; @FP; @FPH]. In this method the renormalization counterterms are constructed using a fourth order WKB expansion for the mode functions. We denote these counterterms by $\langle T_{ab}\rangle_{ad}$. They are given in Refs. [@Bunch] and [@AP]. The renormalized energy-momentum tensor is then $$\langle T_{ab}\rangle_{ren} \,=\, \langle T_{ab}\rangle_{u} -
\langle T_{ab}\rangle_{ad}
\;.
\label{eq:trenorm}$$ This subtraction scheme is not manifestly covariant in form, since space and time are treated quite differently. However, adiabatic regularization is equivalent to a covariant point splitting procedure in which the points are split only in the spacelike hypersurface of constant $\eta$ [@AP; @Birr], and the [*value*]{} of the renormalized $\langle T_{ab} \rangle$ obtained by this procedure is the same as in a strictly covariant one. Hence this subtraction procedure does correspond to adjustment of counterterms to the quantum effective action, and $\langle T_{ab}\rangle_{ren}$ is covariantly conserved. As discussed in detail in Ref. [@AP] the adiabatic terms in all cases consist of an integral rather than a sum over $k$. The reason is that subtraction corresponds to purely local counterterms in the effective action, and thus must be independent of the global compactness or non-compactness of the spatial sections.
For an arbitrary homogeneous and isotropic state to be physically admissible the renormalized energy-momentum tensor defined by the adiabatic order four subtractions in (\[eq:trenorm\]) must be both ultraviolet and infrared finite. In a general RW spacetime, ultraviolet finiteness for a field with non-conformal coupling to the scalar curvature requires that the particular solution of the mode equation (\[eq:mode\]) in a general physical state must match the fourth order adiabatic form at large $k$, with the deviations from the fourth order WKB form falling faster than $k^{-4}$. Likewise the initial number $n_k$ must fall faster than $k^{-4}$ at large $k$, for the mode sums (integrals) to be ultraviolet convergent. This is equivalent to the requirement that the two-point function of the scalar field have the vacuum Hadamard form [@r-l; @n-o; @junker; @lindig] to sufficiently high order in the short distance expansion as the points approach one another. As long as there are two linearly independent complex oscillatory solutions to the equation (\[eq:mode\]), the Wronskian normalization condition (\[eq:wronskian\]) can be imposed and the state will be free of any infrared divergences. However, when $m^2 + \xi R \rightarrow 0$ for some low $k$ in de Sitter space no such complex oscillatory solutions to (\[eq:mode\]) exist and the infrared finiteness requirement on the both the energy-momentum tensor and the two-point function of the physical state becomes non-trivial, as we discuss in detail in Section IV.
A useful variation of the method of adiabatic regularization has been developed by two of us [@AE]. In this method one first computes a quantity $\langle {T_{ab}}\rangle_{d}$, obtained by expanding the adiabatic counterterms $\langle
{T_{ab}}\rangle_{ad}$ in inverse powers of $k$ and truncating at order $k^{-3}$. The same renormalized energy-momentum tensor defined in Eq. (\[eq:trenorm\]) is separated into the sum of two [*finite*]{} terms by adding and subtracting the simplified form of the divergent counterterms $\langle {T_{ab}}\rangle_{d}$ $$\begin{aligned}
\langle {T_{a b}}\rangle_{ren} &=& \langle {T_{ab}}\rangle_{n}
+ \langle {T_{ab}}\rangle_{an}
\; , \nonumber \\
\langle {T_{ab}}\rangle_{n} &=& \langle {T_{ab}}\rangle_{u}
- \langle {T_{ab}}\rangle_{d}
\; , \nonumber \\
\langle {T_{ab}}\rangle_{an} &=& \langle {T_{ab}}\rangle_{d}
- \langle {T_{ab}}\rangle_{ad}
\; . \label{eq:deftan}\end{aligned}$$ The full expressions for $\langle T_{ab}\rangle_d$ and $\langle
T_{ab}\rangle_{an}$ are given in Ref. [@AE] for a general RW spacetime. The advantage of this splitting is that $\langle T_{ab}\rangle_{n}$ and $\langle T_{ab}\rangle_{an}$ are separately conserved, and moreover, $\langle T_{ab}\rangle_{an}$ may be computed analytically in terms of the scale factor $a(\eta)$ and its derivatives [@AE]. Thus the state dependence of the renormalized $\langle T_{ab}\rangle_{ren}$ resides completely in $\langle T_{ab}\rangle_{n}$, which can be computed numerically.
The Bunch-Davies Attractor for $\Re (\nu) <{3\over 2}$ {#sec:analytic}
======================================================
We restrict our consideration henceforth to the particular maximally symmetric RW background of de Sitter space. The geometry of de Sitter spacetime can be described in a number of different coordinate systems. If $\kappa =0$ the spatial sections are flat and the scale factor is $$a(\eta) =- \frac{\alpha}{\eta} \;,\;\;\; -\infty < \eta < 0\; ,
\;\;\; \kappa =0 \; ,
\label{eq:ak0}$$ with $\alpha$ a real, positive constant, and $R = 12\alpha^{-2}$. If $\kappa =+1$ then the scale factor is $$a(\eta) = \alpha\,\sec\eta \; ,\;\;\;\;
-\frac{\pi}{2} < \eta < \frac{\pi}{2} \;, \;\;\; \kappa =+ 1 \; ,
\label{eq:ak1}$$ which is equivalent to $a(\eta)=\alpha \csc \eta$ with $0 < \eta < \pi$. To simplify the notation we will generally use dimensionless units where $\alpha = 1$ and $R= 12$, restoring the dimensionful quantities when it is instructive to do so.
The asymptotic behavior of the energy-momentum tensor does not depend on $\kappa$ so the bulk of the analysis will be carried out in the flat ($\kappa = 0$) coordinates. However, when we turn to the massless, minimally coupled limit in the next section, it will become useful to have a discrete $k$ basis in order to separate out the $k=1$ spatially homogeneous mode explicitly, since it is the most infrared sensitive. No confusion should be caused by our use of the same symbol $\eta$ for conformal time in both cases of flat and closed spatial sections, since we make use of only the $\kappa = 0$ coordinates in this Section and only the $\kappa = +1$ coordinates in the next Section. We will not make use of the spatially open ($\kappa = -1$) coordinates in this paper.
For the case of Eq. (\[eq:ak0\]) the general solution to the mode equation can be written as [@bu-da] [^6] $$\psi_k(\eta) =
\frac{1}{2} (-\pi\eta)^{1\over 2} e^{\frac{i\nu\pi}{2}}
\left[c_1(k) H_\nu^{(1)} (-k\eta)
+ c_2(k) H_\nu^{(2)}(-k\eta) \right]
\; ,
\label{eq:psiK0}$$ where the $H_\nu^{(1),(2)}$ are Hankel functions and $$\nu^2 \equiv \frac{9}{4} - m^2 \alpha^2 - 12 \xi \equiv -\gamma^2
\; .
\label{eq:nu}$$ The latter notation is useful in the case $\nu^2 <0$ so that $\nu =
i\gamma$ is purely imaginary. When $\nu^2 > 0$ we will choose $\nu$ to be the positive root of (\[eq:nu\]). From Eq. (\[eq:psiK0\]) we see that solutions to the mode equation in de Sitter space depend on $m$ and $\xi$ only through their dependence on the parameter $\nu$. Note that because of the minus sign in the arguments of the Hankel functions, it is the function $H_\nu^{(1)}$ that corresponds to a positive frequency mode in the large $k$ limit. The normalization of the mode function in (\[eq:psiK0\]) has been chosen so that the Wronskian condition (\[eq:wronskian\]) becomes simply $$|c_1(k)|^2 - |c_2(k)|^2 = 1
\; .
\label{eq:c1c2K0}$$ The Bunch-Davies (BD) state is defined by the choice, $c_1 = 1$ and $c_2 = 0$ (with $n_k =0$) for all $k$. The renormalized energy-momentum tensor in the BD state is given by [@d-c; @bu-da] $$\begin{aligned}
\langle T_{ab}\rangle_{BD} &=&
-\frac{g_{ab}}{64 \pi^2} \left\{m^2\left[m^2 +
\left(\xi - \frac{1}{6}\right) R\right]
\left[\psi \left(\frac{3}{2} + \nu \right)
+
\psi \left(\frac{3}{2} - \nu \right)
- \log \left(\frac{12 m^2}{R}\right) \right]
\right.
\nonumber \\
&-& \left.
m^2 \left(\xi-\frac{1}{6}\right) R
- \frac{1}{18} m^2 R
- \frac{1}{2} \left(\xi - \frac{1}{6}\right)^2 R^2
+ \frac{R^2}{2160} \right\}
\; ,
\label{eq:tmunu-bd}\end{aligned}$$ where $\psi(z) = {d \log \Gamma (z)\over dz}$ is the digamma function.
That this finite value of $\langle T_{ab}\rangle_{BD}$ coincides with the renormalized $\langle T_{ab}\rangle_{ren}$ defined by the adiabatic subtraction in (\[eq:trenorm\]) follows from the fact that the BD state is an allowed fourth order adiabatic state. This may be checked by comparing the asymptotic expansion of the exact BD mode function, $\frac{(-\pi\eta)^{1\over 2}}{2}H_\nu^{(1)}(-k\eta)$, for large $k$ with the fourth order adiabatic mode function $$\psi_k^{(4)}(\eta) = \frac {1} {\sqrt {2 W_k^{(4)}}}
\exp\left( - i\int^{\eta}\,
W_k^{(4)}(\eta')\,{\rm d}\eta'\right)\,,
\label{eq:psiadb}$$ with $W_k^{(4)}(\eta)$ the fourth order adiabatic frequency. It is given explicitly in de Sitter space in flat conformal time coordinates by $$W_k^{(4)}(\eta) = k + {1 \over 2 k\eta^2}\left(\frac{1}{4} - \nu^2\right)
- {1 \over 8 k^3 \eta^4}\left(\frac{1}{4} - \nu^2\right)
\left(\frac{25}{4} - \nu^2\right) + {\cal O}\left( {1\over k^5}\right)\,,$$ up to the required order at large $k$.
For the general state with $c_2 \neq 0$ to remain fourth order adiabatic, we must have for large values of $k$ $$c_2(k) = \frac{C(k)}{k^4} \; , \label{eq:ckbound}$$ for some complex function $C(k)$ which vanishes in the limit $k\rightarrow
\infty$. This condition is necessary for an arbitrary (spatially homogeneous) state to posses a finite energy-momentum tensor after the fourth order adiabatic subtraction defined by (\[eq:trenorm\]). Likewise the same condition of finite $\langle
T_{ab}\rangle$ requires us to restrict the average number of particles $\langle a^{\dagger}_{\bf k} a_{\bf k}\rangle = n_k$ by $$n_k = \frac{N(k)}{k^{4}} \; , \label{eq:nkbound}$$ for some real function $N(k)$ which vanishes in the limit $k\rightarrow \infty$. The two ultraviolet conditions $$\lim_{k\rightarrow \infty} \vert C(k)\vert =
\lim_{k\rightarrow\infty} N(k) =0\,, \label{eq:CNlim}$$ on the physically allowed states guarantee that the Green’s function for the scalar field is locally of the Hadamard form [@r-l; @n-o; @junker; @lindig], and that the divergences of $\langle T_{ab} \rangle$ match those of the fourth order adiabatic vacuum, and are removed by the adiabatic subtraction procedure. We will call any state which satisfies the conditions (\[eq:ckbound\]), (\[eq:nkbound\]), and (\[eq:CNlim\]), together with the Wronskian condition (\[eq:c1c2K0\]), a UV admissible physical state [^7].
We shall require also that the arbitrary physical state possess a two-point function and energy-momentum tensor which are free of any infrared divergences. Because the canonical dimension of $\langle
T_{ab}\rangle$ is four whereas that of $\langle
\Phi(x)\Phi(x')\rangle$ is two, the conditions (\[eq:ckbound\]) and (\[eq:nkbound\]) which require finiteness of $\langle T_{ab}
\rangle_{ren}$ are more restrictive in the UV, whereas the condition of finiteness of $\langle \Phi(x)\Phi(x')\rangle$ is more restrictive in the IR. These two sets of conditions will be sufficient to demonstrate that the energy-momentum tensor for [*any*]{} UV and IR admissible physical state approaches the BD value at late times for $\Re (\nu)< {3\over 2}$.
To understand why such a result is to be expected and outline the more detailed proof which we give below, let us observe that at late times $\eta \rightarrow 0^-$, the general state mode function (\[eq:psiK0\]) behaves like $$\psi_k \sim (-\eta)^{{1\over 2}\, -\nu} \sim a^{\nu -{1\over 2}} \; .
\label{eq:psilate}$$ Substituting this into (\[eq:T00u\]) and (\[eq:Tru\]) shows that to leading order at late times the contributions to the mode sums of the unrenormalized energy-momentum tensor behave like $(-\eta)^{3-2\nu} \sim a^{2\nu -3}$ for $\nu$ real. Since the renormalization counterterms are state independent [@Bunch], the state dependent terms are the same in the unrenormalized and renormalized energy-momentum tensor. One can perform all the UV renormalization in the BD state at a fixed time and collect the remaining finite state dependent terms which are unaffected by the subtraction procedure, and they all fall off at least as fast as $(-\eta)^{3-2\nu}$ as $\eta \rightarrow 0^-$ for $\Re (\nu) < {3\over
2}$.
These remaining finite state dependent terms in the energy density and trace are expressible as integrals over the wave number $k$ with the general form $$I (\eta) = \int_0^{\infty}\, {{\rm d}k \over k}\ R(k)\ S(-k\eta)\,,
\label{eq:genint}$$ where $R(k)$ is one of the four state dependent, but time independent functions $$\begin{aligned}
\vert c_2(k)\vert^2 &&(1 + 2n_k),\nonumber\\
\Re [c_1(k) c_2(k)]&&(1 + 2n_k),\nonumber\\
\Im [c_1(k) c_2(k)]&&(1 + 2n_k), \nonumber\\ && n_k, \end{aligned}$$ and $S(z=-k\eta)$ is a product of the state independent Bessel functions and their derivatives. An explicit basis for the twelve products of Bessel functions $S_i (z)$ for $i = 1,\dots, 12$ which appear in the integrals is given in Table 1. The essential point is that all the state dependent mode integrals of the form (\[eq:genint\]) are [*uniformly*]{} convergent for all $\eta$ (including $\eta =0$) at both their lower limit, $k=0$, and their upper limit $k=\infty$, due to the IR and UV finiteness of the state. Hence the limit of $\eta \rightarrow 0^-$ can be taken [*inside*]{} the integral over $k$. Since, as Table 1 shows, all the $S_i(z)$ behave like $$S_i(z) \rightarrow s_{i, 0}\ z^{\beta_i}\,,
\qquad {\rm with} \qquad \beta_i > 0\,,
\label{eq:Slim}$$ as $z\rightarrow 0$, for $\Re (\nu) < {3\over 2}$, it follows that $$\lim_{\eta\rightarrow 0^{-}} I(\eta) =
\int_0^{\infty}\, {{\rm d}k \over k}\ R(k)\ \lim_{\eta\rightarrow
0^{-}} S(-k\eta) = 0\, ,$$ and all the state dependent contributions to $\langle T_{ab} \rangle_{ren}$ vanish at late times.
The validity of bringing the limit inside the integral depends on the uniform convergence of the integral at both its upper and lower limits. In the form (\[eq:genint\]) the behavior of the $S(-k \eta)$ factor at small arguments (and the absence of any IR divergence from the $R(k)$ factor) clearly guarantees the uniform convergence at the lower limit. However, the change of variables $z= -k\eta$ brings the integral (\[eq:genint\]) into the form $$I(\eta) =
\int_0^{\infty}\, {{\rm d}z \over z}\ R\left({z\over -\eta}\right)\ S(z)\,.
\label{eq:genintz}$$ In this form it is clear that the uniform convergence of the integral at its upper limit is guaranteed by the falloff of the state dependent mode functions at large $k$, namely $$\lim_{k \rightarrow \infty} R(k) =
\lim_{\eta \rightarrow 0^-}R \left({z\over -\eta}\right) = 0
\; .
\label{eq:Rlim}$$ Equations (\[eq:Slim\]) and (\[eq:Rlim\]) guarantee that both the IR and UV contributions go to zero as $\eta\rightarrow 0^{-}$ for $\Re (\nu) < {3\over 2}$.
In fact, we can go one step further by using the information from the fourth order adiabatic nature of the state $$\lim_{k
\rightarrow \infty} k^4 R(k) = 0\,,
\label{eq:adbR}$$ to conclude that the UV contribution to the integral from $z \sim 1$, and very large $k \sim (-\eta)^{-1}$ in (\[eq:genintz\]) falls faster than $(-\eta)^4$ as $\eta \rightarrow 0^{-}$. This is faster than the IR contribution which falls only as $(-\eta)^{\beta_i}$ if $\beta_i
\le 4$. If $\beta_i > 4$ then both the IR and UV contributions fall faster than $(-\eta)^4$. Hence, the $S_i(z)$ for which $\beta_i > 4$ are subdominant at late times. Thus we conclude that the leading order state dependent terms at late times are those with the smallest $\beta_i$. These give an IR dominant, ([*i.e.*]{} finite $k$, $z\ll
1$) contribution of the form, $$I(\eta) \rightarrow
(-\eta)^{\beta_i} \int_0^{\infty}\, {\rm d}k \ k^{\beta_i -1}\ R(k)
\rightarrow (-\eta)^{3- 2\nu} \int_0^{\infty}\, {\rm d}k \ k^{2-
2\nu}\ R(k)\,,
\label{eq:Ilim}$$ as $\eta\rightarrow
0^{-}$. This last integral is guaranteed to converge at its upper limit for all $\Re (\nu) \ge -{1\over 2}$, and in particular for $0
\le \Re (\nu) \le {3\over 2}$ by (\[eq:adbR\]). Hence for all $\Re
(\nu) < {3\over 2}$ the state dependent contributions go to zero as $a^{2\nu -3}$ at late times. The limiting case when $\nu \rightarrow
{3\over 2}$ is special because then the state dependent IR contributions apparently does not go to zero at late times. This case will be considered separately in the next Section.
We will now make the proof more explicit by giving the form of all of the state dependent terms of the scalar field in de Sitter space, and analyzing the IR and UV contributions in detail. If we make use of Eq. (\[eq:c1c2K0\]), we find that in an arbitrary state $$\begin{aligned}
\langle T_{ab} \rangle_{ren} &=& \langle T_{ab}
\rangle_{BD} + \langle T_{ab} \rangle_{SD} \;,\end{aligned}$$ where $\langle T_{ab}\rangle_{SD}$ is the finite state dependent term, depending on the coefficients $c_1(k), c_2(k)$, and $n_k$, which may be expressed as an integral over the wave number $k$ in the form $$\begin{aligned}
\langle T_{ab} \rangle_{SD} &=& \frac{1}{4 \pi^2} \int_0^{\infty }
{\mathrm{d}} k
\; I_{a b}(k,\eta) \; . \label{eq:Iab}\end{aligned}$$ The explicit expressions for the integrand $I_{ab}$ depend on whether $\nu$ is real and not an integer, real and an integer, or pure imaginary, although the result will be the same for all $\Re (\nu)<
{3\over 2}$. In the rest of this Section we restrict our discussion to the case where $\nu$ is real and not equal to an integer. The cases of integer and imaginary values of $\nu$ are covered in Appendix A. For real $\nu$ after some regrouping of terms we have
$$\begin{aligned}
{I^0}_0 &=& A_1(k) \left[S_1 + \frac{1}{4} (9 + 4 m^2 - 48 \xi)
S_4 - \frac {3}{2} (4 \xi -1) S_7 + S_{10} \right] \nonumber \\
& & + A_2(k) \left[S_2 + \frac{1}{4} (9 + 4 m^2 - 48 \xi) S_5 -
\frac{3}{2} (4 \xi -1) S_8 + S_{11} \right] \nonumber \\
& & + A_3(k) \left[S_3 + \frac{1}{4} (9 + 4 m^2 - 48 \xi) S_6 -
\frac{3}{2} (4 \xi -1) S_9 + S_{12} \right]
\; ,\qquad {\rm and} \label{eq:I00} \\
I &=& A_1(k) \left[-2(6 \xi - 1) S_1 - \frac{1}{2}\left(9 + 8 (3 \xi - 1) m^2
- 102 \xi + 288 \xi^2\right)S_4 + 3(6 \xi - 1) S_7 + 2(6 \xi -1) S_{10}
\right] \nonumber \\
& & + A_2(k) \left[-2(6 \xi - 1) S_2
- \frac{1}{2}\left(9 + 8 (3 \xi - 1) m^2
- 102 \xi + 288 \xi^2\right)S_5 + 3(6 \xi - 1) S_8 + 2(6 \xi -1) S_{11}
\right] \nonumber \\
& & + A_3(k) \left[-2(6 \xi - 1) S_3 - \frac{1}{2}\left(9
+ 8 (3 \xi - 1) m^2
- 102 \xi + 288 \xi^2\right)S_6 + 3(6 \xi - 1) S_9 + 2(6 \xi -1) S_{12}
\right]
\; ,
\label{eq:I}\end{aligned}$$
with
$$\begin{aligned}
A_1(k) &=& -\frac{\pi}{2 k} \left[\csc^2(\nu \pi) \left((1+ 2 n_k) |c_2|^2 +
n_k\right) + \frac{1}{2} \left(1-\cot^2(\nu\pi)\right)(1+2 n_k)
(c_1 c_2^* + c_1^* c_2)\right. \nonumber \\
& & \left. + \frac{i}{2} \cot(\nu\pi) (1+2 n_k)(c_1 c_2^* - c_1^* c_2)
\right]
\; ,\\
A_2(k) &=& \frac{\pi}{k} \left[\cot(\nu \pi) \csc(\nu\pi) \left((1+ 2 n_k)
|c_2|^2 + n_k\right) - \frac{1}{2} \cot(\nu\pi)\csc(\nu\pi)(1+2 n_k)
(c_1 c_2^* + c_1^* c_2) \right.\nonumber \\
& & \left. + \frac{i}{2} \csc(\nu\pi) (1+2 n_k) (c_1 c_2^* - c_1^* c_2)
\right]
\; , \\
A_3(k) &=& -\frac{\pi}{2k} \left[\csc^2(\nu\pi) \left((1+ 2 n_k) |c_2|^2 + n_k
\right) - \frac{1}{2} \csc^2(\nu\pi)(1+2 n_k)
(c_1 c_2^* + c_1^* c_2) \right]\, ,
\label{eq:Adef}\end{aligned}$$
and the $S_i(-k\eta)$ composed of various products of $J_\nu(-k\eta)$, $J_{-\nu}(-k\eta)$ and their derivatives. Making use of the general formula for the product of two Bessel functions [@gradrhyz], $$J_\mu(z) J_\nu(z) = \sum_{p=0}^{\infty}\frac{(-1)^p
\left(\frac{z}{2}\right)^{\nu+\mu+2p}\Gamma(\nu+\mu+2p+1) }
{p! \Gamma(\nu+\mu+p+1) \Gamma(\nu+p+1) \Gamma(\mu+p+1)} \,,
\label{eq:JJ}$$ we can expand the $S_i$ in power series of the form, $$S_i(z) = \sum_{p=0}^{\infty} s_{i,p}\ z^{2p + \beta_i} \;, \label{eq:Si}$$ with $z = -k\eta$. The explicit expressions for $\beta_i$ and $S_i$ for the case of real, non-integer $\nu$ are given in Table 1.
[|c||c|c|]{}
------------------------------------------------------------------------
$\;\;i\; \; $ &$\; \; \; \; \beta_i \; \; \; \; $&$\; \; S_i(z)\; \; \;$\
------------------------------------------------------------------------
1 & $5 + 2 \nu$ & $z^5 J_\nu^2(z)$\
------------------------------------------------------------------------
2 & $5 $ & $z^5 J_\nu(z) J_{-\nu}(z)$\
------------------------------------------------------------------------
3 & $5 - 2 \nu$ & $z^5 J_{-\nu}^2(z)$\
------------------------------------------------------------------------
4 & $3 + 2 \nu$ & $z^3 J_\nu^2(z)$\
------------------------------------------------------------------------
5 & $3 $ & $z^3 J_\nu(z) J_{-\nu}(z)$\
------------------------------------------------------------------------
6 & $3 - 2 \nu$ & $z^3 J_{-\nu}^2(z)$\
------------------------------------------------------------------------
7 & $3 + 2 \nu$ & $z^4 \frac{\rm d}{{\rm d}z} J_\nu^2(z)$\
------------------------------------------------------------------------
8 & $3 $ & $z^4 \frac{\rm d}{{\rm d}z}
\left(J_\nu(z) J_{-\nu}(z)\right)$\
------------------------------------------------------------------------
9 & $3 - 2 \nu$ & $z^4 \frac{\rm d}{{\rm d}z} J_{-\nu}^2(z)$\
------------------------------------------------------------------------
10 & $3 + 2 \nu$ & $z^5 \left(\frac{\rm d}{{\rm d}z} J_\nu (z)\right)^2$\
------------------------------------------------------------------------
11 & $3 $ & $z^5 \left(\frac{\rm d}{{\rm d}z} J_\nu(z)\right)
\left(\frac{\rm d}{{\rm d}z} J_{-\nu}(z)\right)$\
------------------------------------------------------------------------
12 & $3 - 2 \nu$ & $z^5 \left(\frac{\rm d}{{\rm d}z} J_{-\nu}(z)\right)^2$\
[Table 1]{}
We are interested in the behavior of the finite state dependent terms, $\langle T_{ab} \rangle_{SD}$, in the limit $\eta \rightarrow 0^- $. To investigate this limit in detail it is useful to break up the integral in Eq. (\[eq:Iab\]) into the three parts $$\langle T_{ab} \rangle_{SD}
= \frac{1}{4 \pi^2} \int_0^\lambda
{\mathrm{d}} k \; I_{ab}(k,\eta)
+ \frac{1}{4 \pi^2}\int_\lambda^{-Z/\eta} {\mathrm{d}} k\; I_{ab}(k,\eta)
+ \frac{1}{4 \pi^2}\int_{-Z/\eta}^{\infty} {\mathrm{d}} k\;
I_{ab}(k,\eta)
\; .
\label{eq:thrI}$$ Here $\lambda$ is a finite positive constant. For $k>\lambda$ we can make use of the UV conditions (\[eq:ckbound\]), (\[eq:nkbound\]), and (\[eq:CNlim\]) in the second and third integrals. Thus, the most infrared sensitive integral is the first one. The positive constant $Z$ is arbitrary, provided only that $Z > -\lambda\eta$, which is always satisfied for fixed $\lambda$ and small enough $-\eta$. Hence the second integral provides the bulk of the contribution of the state dependent wave numbers that have redshifted outside the de Sitter horizon at late times. The third integral is the contribution of the state dependent terms still within the horizon which go to zero very rapidly at late times due to the UV conditions. If the $\eta \rightarrow 0^-$ limit is uniform then all three integrals should vanish unconditionally in this limit, [*i.e.*]{} without any restrictions on the arbitrary parameters $\lambda$ and $Z$.
We begin by analyzing the first integral in (\[eq:thrI\]). We are considering only fourth order adiabatic states that are IR admissible. Hence the $k$ integration converges at its lower limit and we can expand the integrand for this first integral in a series of the form (\[eq:Si\]) and interchange the order of summation and integration. Each term in the resulting sums contains an integral over $k$ that is finite by assumption and a factor of $(-\eta)^{2 p +
\beta_i}$ where $p =0, 1, 2\dots$ is a non-negative integer. In fact, since at late times, $-\lambda \eta \ll 1$ for any finite $\lambda$, it is sufficient to consider only the leading $p=0$ term. From Table 1 it is clear that for $\nu < {3\over 2}$, $\beta_i > 0$ for all $i$. Hence in the limit $\eta \rightarrow 0^-$, the integrand vanishes and therefore, the first integral on the right hand side of Eq. (\[eq:thrI\]) vanishes in this late time limit.
For the second integral in (\[eq:thrI\]) we may utilize the expansion (\[eq:JJ\]) again. Since the integration is between finite limits for finite $\eta$ we can exchange the order of summation over $p$ and integration over $k$. Having done so we can then bound each term in the sums by taking the absolute value of its factors. Inspection of the expressions (\[eq:I\]) and (\[eq:Adef\]) indicates that the result is a linear combination of terms involving integrals of the three possible forms $$\begin{aligned}
{\cal I}_1&=& \int_\lambda^{-Z/\eta} \frac{{\rm d}k}{k} \ |c_2(k)|^2\
(1 + 2 n_k)
\,(-k\eta)^{2 p + \beta_i}\,, \nonumber \\
{\cal I}_2&=& \int_\lambda^{-Z/\eta} \frac{{\rm d}k}{k}\
|c_1(k) c_2(k)|\ (1 + 2n_k)
\,(-k\eta)^{2 p + \beta_i} \,, \nonumber \\
{\cal I}_3&=& \int_\lambda^{-Z/\eta} \frac{{\rm d}k}{k}\ n_k
\,(-k\eta)^{2 p + \beta_i}
\; ,
\label{eq:J12}\end{aligned}$$ multiplied by constant coefficients. Because of (\[eq:CNlim\]) for any $\lambda$ $$\begin{aligned}
\vert C(k)\vert &<& C\,,\nonumber\\
N(k)&<& N\,,\end{aligned}$$ for some real positive numbers $C$ and $N$, depending on $\lambda$. With these bounds we can bound the values of $|c_2(k)|$ and $n_k$, and use Eq. (\[eq:c1c2K0\]) to bound $|c_1(k)|$ for all $k \ge \lambda$ as follows $$\begin{aligned}
|c_2(k)| &<& \frac{C}{k^4}\; , \nonumber \\
1 \le |c_1(k)| &=& \left[1 + \vert c_2(k)\vert^2\right]^{1\over 2}
\le 1 + \vert c_2(k)\vert^2 < 1 + {C^2\over k^8} \le 1 + \frac{C^2}{\lambda^8}\; ,\nonumber \\
n_k &<& \frac{N}{k^4} \nonumber \\
1 + 2 n_k &<& 1 + \frac{2N}{k^4} \le 1 + \frac{2 N}{\lambda^4} \; .
\label{eq:c1c2nkbounds} \end{aligned}$$ Using these bounds the integrals in Eq. (\[eq:J12\]) can be bounded as follows $$\begin{aligned}
{\cal I}_1&< & C^2 (1 + 2N/\lambda^4) \int_\lambda^{-Z/\eta} \,{\rm d}
k\,k^{2p +\beta_i - 9}
(-\eta)^{2p + \beta_i}\nonumber\\
&=& \frac{C^2(1 + 2N/\lambda^4)}{\beta_i + 2p - 8} \left[(-\eta)^8 Z^{2p + \beta_i - 8}
- (-\eta)^{2p + \beta_i} \lambda^{2p + \beta_i - 8} \right] \nonumber \\
{\cal I}_2&< & C(1+C^2/\lambda^8)(1+ 2N/\lambda^4) \int_\lambda^{-Z/\eta} \, {\rm d}k\,
k^{2p +\beta_i - 5} (-\eta)^{2p + \beta_i}\nonumber\\
&=& \frac{ C(1+C^2/\lambda^8)(1+ 2N/\lambda^4)}{\beta_i + 2p - 4}
\left[(-\eta)^4 Z^{2p + \beta_i - 4} - (-\eta)^{2p + \beta_i}
\lambda^{2p + \beta_i - 4}\right]\nonumber \\
{\cal I}_3&< & N \int_\lambda^{-Z/\eta} \,{\rm d}k\,k^{2p +\beta_i - 5}
(-\eta)^{2p + \beta_i}\nonumber\\
&=& \frac{N}{\beta_i + 2p - 4} \left[
(-\eta)^4 Z^{2p + \beta_i - 4}-
\lambda^{2p + (-\eta)^{2p + \beta_i} \beta_i - 4}\right]\,.
\label{eq:J12value}\end{aligned}$$ Each of these bounds vanishes in the limit $\eta \rightarrow 0^-$. Therefore each of the terms appearing in the second integral in (\[eq:thrI\]) vanishes in the late time limit for $\Re (\nu) <{3\over 2}$.
If $\nu = {1\over 2}$ some of the terms will have vanishing denominators and should be interpreted according to the limiting relation $$\lim_{q \rightarrow 0} { \left(-{Z\over \eta}\right)^q - \lambda^q \over q} =
\log \left({Z\over -\eta\lambda}\right) \,,$$ but the appearance of these logarithms does not change the result since they are always multiplied by at least $(-\eta)^{2p +\beta_i}$ which vanishes for $\Re (\nu) <{3\over 2}$ for $p = 0, 1, 2\dots$. The cases $\nu = 0, 1$ also involve logarithms in the Bessel function expansions but the result that the second integral in (\[eq:thrI\]) vanishes in the late time limit is unchanged. One may consider also the case when $\nu = i \gamma$ is pure imaginary, where the forms of the coefficients (\[eq:Adef\]) and bilinears $S_i(z)$ in the table change somewhat, with again the same result. For completeness these cases are treated in detail in Appendix A.
Finally, for the third integral on the right hand side of Eq.(\[eq:thrI\]) we do not expand the Bessel functions in powers of $(-k\eta)$. Instead we change the integration variable to $z = -k
\eta$ and find integrals of the forms $$\begin{aligned}
{\cal J}_1 &=& (-\eta)^8 \int_Z^{\infty} \frac{{\rm d}z}
{z^9}\ \Big|C\left(\frac{z}{-\eta}\right)\Big|^2\
\left[ 1 + 2\left({-\eta\over z}\right)^4 N
\left(\frac{z}{-\eta}\right)\right] \
S_i(z)\,, \nonumber \\
{\cal J}_2 &=& (-\eta)^4 \int_Z^{\infty} \frac{{\rm d}z}
{z^5}\ (\Re\ {\rm or}\ \Im)
\left\{c_1\left({z\over -\eta}\right) \left[C^*\left(\frac{z}{-\eta}\right)
\right]
\right\}\
\left[1 + 2\left({-\eta\over z}\right)^4
N\left(\frac{z}{-\eta}\right) \right]\ S_i(z)\,, \nonumber \\
{\cal J}_3 &=& (-\eta)^4 \int_Z^{\infty} \frac{{\rm d}z}
{z^5}\ N\left(\frac{z}{-\eta}\right) S_i(z)\,.\end{aligned}$$ Since we are considering fourth order adiabatic states these integrals are finite for all $-\infty <\eta \le 0$. Furthermore, the integrands of these integrals are finite throughout the entire integration range for $-\infty < \eta \le 0$. With the lower limit $Z > 0$ fixed, it is clear that we can evaluate the integrals at $\eta = 0$ by first taking the limit $\eta \rightarrow 0^-$ of the integrands and then computing the integrals. Since the integrands all vanish in the limit $\eta
\rightarrow 0^-$, the integrals do as well. Then the third integral in (\[eq:thrI\]) vanishes in the late time limit for $\Re (\nu)
<{3\over 2}$. From the adiabatic four conditions (\[eq:CNlim\]) these UV contributions go to zero faster than $(-\eta)^4$ for any fourth order adiabatic state.
Therefore we have proven that for $\Re (\nu) < {3\over 2}$ all the integrals in Eq. (\[eq:thrI\]) vanish in the limit $\eta
\rightarrow 0^-$ for an arbitrary state that is both infrared and ultraviolet finite. The fact that all three integrals vanish at late times, [*independently*]{} of the parameters $\lambda$ and $Z$ which we introduced to control the IR and UV contributions of the mode integral verifies that the total mode integral does converge uniformly in $\eta$, as expected. Hence all the state dependent contributions vanish at late times, $\eta \rightarrow 0^-$, and we have proven that in this limit $\langle T_{a b} \rangle_{ren} \rightarrow \langle T_{a
b} \rangle_{BD}$. Thus, we conclude that for $\Re(\nu)<{3\over 2}$ the value of the energy-momentum tensor in any physically admissable homogeneous and isotropic RW quantum state asymptotically approaches the Bunch-Davies value in de Sitter space at late times.
Moreover, from inspection of (\[eq:J12value\]) we observe that the contribution from the upper limit at $Z$ in the second integral always falls faster than $(-\eta)^4$, while the contribution from the lower limit at $\lambda$ falls off only as $(-\eta)^{\beta_i}$ as $\eta
\rightarrow 0^-$. This must be the same order as the first IR integral so that the arbitrary parameter $\lambda$ drops out of the final result. Hence our detailed evaluation has verified that the leading order state dependent corrections to the BD expectation value at late times come from the terms in $\langle T_{a b} \rangle_{ren}$ with the smallest $\beta_i$. From Table 1 these are the $i=6, 9$, and $12$ terms. Collecting these terms from (\[eq:I\]) and (\[eq:Adef\]), and the numerical coefficients from the Bessel function product formula (\[eq:JJ\]), we conclude that the leading order behavior of the energy density and trace at late times are given by
$$\begin{aligned}
\varepsilon &\rightarrow & a^{2\nu -3} \left[\left(\nu -{1\over 2}\right)^2
+ m^2 + 2(6\xi -1)(\nu -1)\right] \int_0^{\infty}\ {\rm d}k\ k^{2-2\nu} R(k)
\,, \qquad {\rm and}\label{eq:lateps}\\
T &\rightarrow& 2 a^{2\nu -3} \left\{-m^2 + (6\xi-1) \left[-\left(\nu
-{1\over 2}\right)^2 + 2\nu + m^2 + 2(6\xi -1)\right]\right\}
\int_0^{\infty}\ {\rm d}k\ k^{2-2\nu} R(k)\,,
\label{eq:lateT}\end{aligned}$$
respectively, with $$R(k) = {2^{2\nu - 3}\csc^2 (\pi\nu)\over \pi \left[\Gamma (-\nu + 1)\right]^2}
\left\{ (1+ 2 n_k) \left[|c_2(k)|^2 - \Re (c_1 c_2^*)\right] + n_k \right\} \,.$$ The leading order state dependent contribution at late times is $a^{2
\nu -3}$ from the IR part of the mode integral, which falls off very slowly if $\Re (\nu) \rightarrow {3\over 2}$. We examine this latter limit in detail in the next Section.
The Allen-Folacci Attractor for $\Re (\nu) \rightarrow {3\over 2}$ {#sec:nu32}
==================================================================
In the analysis of the previous Section we saw that the terms with the slowest falloff at late times were those with the smallest $\beta_i =
3 - 2 \nu$ which behave like $a^{2 \nu - 3}$ for $\nu$ real and positive, and the coefficient of this falloff is controlled by the finite $k$, IR part of the mode integral. To examine the limit $\nu
\rightarrow {3\over 2}$ carefully, it is easiest to work with closed spatial sections and a discrete set of mode functions in order to treat the most infrared sensitive, spatially homogeneous $k=1$ mode separately from the rest, instead of dealing with an infrared sensitive continuous mode integral.
The scale factor for $\kappa = +1$ is given by Eq. (\[eq:ak1\]) and under the variable substitution $\zeta= i \tan \eta$ the mode equation (\[eq:mode\]) becomes Legendre’s differential equation. Hence the general solutions may be expressed in terms of associated Legendre functions $P^{\pm k}_{-{1\over 2} + \nu}(\zeta)$. Since as conventionally defined these functions have a cut discontinuity on the real axis from $-1$ to $1$ if $k$ is an odd integer, we write the fundamental complex valued solution for real $\nu$ in the form $$\begin{aligned}
f_k(\eta) &=&
\left[{\Gamma\left(k + \frac{1}{2} + \nu\right) \Gamma\left(k +
\frac{1}{2} - \nu\right)\over 2}\right]^{1\over 2} \exp
\left(-{ik\pi\over 2} \epsilon (\eta)\right) P^{-k}_{-{1\over 2} +
\nu} (i \tan \eta) \nonumber\\ &=& \left[{\Gamma\left(k + \frac{1}{2}
+ \nu\right) \Gamma\left(k + \frac{1}{2} - \nu\right)\over
2}\right]^{1\over 2} {e^{-ik\eta}\over k!}\,F\left(\frac{1}{2} + \nu ,
\frac{1}{2} - \nu ;k+1; {1 -i\tan \eta\over 2}\right)\,,
\label{eq:hyper} \end{aligned}$$ where $\epsilon (\eta) = \theta(\eta)
- \theta(-\eta)= \pm 1$ is the sign function and $F$ is the hypergeometric function. The phase factor depending on the sign of $\eta$ for odd $k$ removes the discontinuity in the $P^{-k}_{-{1\over
2} + \nu}$ function as $\eta$ approaches zero from positive or negative values, as the second form of (\[eq:hyper\]) makes clear, since the $F$ function is an analytic function of $1-\zeta\over 2$, with no branch cuts for $\zeta$ on the imaginary axis. With the normalization factors chosen as in (\[eq:hyper\]) the Wronskian condition $$f_k {f_k^*}' - f_k^* f_k' =i\,,$$ is satisfied. Hence the general solution of the mode equation (\[eq:mode\]) is $$\psi_k(\eta) = \alpha_k
f_k(\eta) + \beta_k f_k^*(\eta)\,, \label{eq:psik}$$ with $$|\alpha_k|^2 - |\beta_k|^2 = 1\,. \label{eq:AkBk}$$ The Bunch-Davies state is given by $\alpha_k = 1$ and $\beta_k = 0$.
Now as $\nu \rightarrow {3\over 2}$ inspection of (\[eq:hyper\]) shows that all the $f_k$ for $k>1$ are regular. In fact, in that limit the hypergeometric series for $F$ terminates and the $f_k
(\eta)$ become the elementary functions $$\lim_{\nu \rightarrow {3\over 2}}f_k(\eta) =
\frac{e^{-i k \eta}}{[2 k(k^2 - 1)]^{1\over 2}}\, (k + i\tan \eta)\,,\qquad
k = 2, 3, \dots
\; ,
\label{eq:psiknon}$$ so they can be treated in the Bunch-Davies state $\alpha_k = 1,
\beta_k = 0$ with no difficulty. However in this limit the $k=1$ mode function is singular and must be treated separately. The behavior of the $k=1$ mode as $\nu \rightarrow {3\over 2}$ is similar to that of a simple harmonic oscillator mode as its frequency goes to zero, [*i.e.*]{} in the limit where the harmonic oscillator becomes a free particle. The zero frequency limit in this simple flat space analogy is reviewed in Appendix B. Just as in that case, one can construct regular solutions to the mode equation in the limit $\nu \rightarrow
{3\over 2}$ by taking suitable linear combinations of $f_1$ and $f_1^*$. In fact, the limiting form of $f_1 (\eta)$ can be found from Sec. 2.3.1 of Ref. [@Bate], which gives $$f_1(\eta) \rightarrow {\sec \eta \over 2 \sqrt{{3\over 2} - \nu}}
-{i\over 2} \sqrt{{3\over 2} - \nu}\ \sec \eta\ (\eta + \sin\eta\,\cos\eta) +
\dots
\; .$$ We have neglected terms in the real part of $f_1$ that are of order $\sqrt{{3\over 2} - \nu}$. We have also neglected all terms that go to zero faster than this as $\nu \rightarrow {3\over 2}$. Extracting the scale factor $a(\eta) = \sec\eta$ we now define the real functions $u$ and $v$ by $$\begin{aligned}
u(\eta)&\equiv &{1\over a(\eta)}\lim_{\nu \rightarrow
{3\over 2}} \left\{\sqrt{{3\over 2} - \nu}
\ (f + f^*)\right\} = 1\,,\nonumber\\
v(\eta) &\equiv & {i\over a(\eta)}\ \lim_{\nu \rightarrow
{3\over 2}} \left\{{1\over 2\sqrt{{3\over 2} - \nu}}\
(f - f^*)\right\} =
{\eta + \sin\eta\,\cos\eta\over 2} \,.
\label{uvdeS}\end{aligned}$$
We next define new coefficients $$\begin{aligned}
A &\equiv& -i\ \lim_{\nu \rightarrow {3\over 2}}
\left\{ \sqrt{{3\over 2} - \nu}\ (\alpha_1 - \beta_1)
\right\}
\; ,
\nonumber\\
B &\equiv& \lim_{\nu \rightarrow {3\over 2}}
\left\{ {1\over 2\sqrt{{3\over 2} - \nu}}\ (\alpha_1 + \beta_1)
\right\}
\; ,
\label{ABdeS}\end{aligned}$$ which are finite in this limit. With these definitions the normalization is such that $$A^*B - B^*A = i
\; ,$$ and the limit of the general $k=1$ mode function may be written $$\lim_{\nu \rightarrow {3\over 2}}\psi_1(\eta) = \sec\eta\ (Av + Bu)
= a(\eta) \left[ {A\over 2} \ (\eta + \sin\eta\,\cos\eta)+ B\right]\,.
\label{eq:psionelim}$$ One can also define the time-independent Hermitian operators, $$\begin{aligned}
Q &\equiv & {1\over 2\sqrt{{3\over 2} - \nu }}
\left[(\alpha_1 + \beta_1)\ a_1 +
(\alpha_1 + \beta_1)^*\ a_1^{\dagger}\right] \rightarrow
Ba_1 + B^* a_1^{\dagger}
\,,\nonumber\\
P &\equiv & -i \sqrt{{3\over 2} - \nu}\ \left[(\alpha_1 - \beta_1)\ a_1
- (\alpha_1 - \beta_1)^*\ a_1^{\dagger}\right]
\rightarrow Aa_1 + A^* a_1^{\dagger}\,,
\label{eq:QPdef}\end{aligned}$$ obeying the canonical commutation relations, $[Q, P] = i$. They also remain finite in the limit $\nu \rightarrow {3\over 2}$.
We will consider the limit $\nu \rightarrow {3\over 2}$ of the energy density in the particular order of fixing the mass of the scalar field to be $m=0$ and letting the dimensionless parameter $\xi$ approach zero, since this case is relevant for the infrared scaling analysis of Section VI. From (\[eq:nu\]) with $m=0$ and $\xi$ small we have $$\xi \simeq {(3 - 2 \nu)\over 8} \rightarrow 0
\;.
\label{eq:xizero}$$ The complementary case, $\xi =0$ and $m^2 \rightarrow 0$ is similar and has been discussed in Refs. [@a-f] and [@k-g]. From (\[eq:T00u\]) we read the energy density in the $k=1$ mode for $m=0$ $$\varepsilon_1 = -\langle {T^0}_0\rangle_{k=1} =
{(1 + 2n_1)\over 4\pi^2 a^4}\left\{ |\psi_1'|^2 + |\psi_1|^2
+ (6 \xi -1) [\tan\eta\ (\psi_1\psi_1^{*\prime} + \psi_1^*\psi_1')
- (\tan^2\eta - 1)|\psi_1|^2]\right\}\,.$$ We are interested first in the asymptotic form of this energy density at late times, $\eta \rightarrow {\pi \over 2}$, and then in the limiting form of the resulting expression as $\xi \rightarrow 0$ according to (\[eq:xizero\]). The asymptotic late time limit of the mode function $f_1 (\eta)$ for any $\nu > 0$ can be found from the inversion transformation of the hypergeometric function, given by formula 2.1.4 (17) of [@Bate]. We find that $$\begin{aligned}
f_1 \left(\eta \rightarrow {\pi\over 2}\right) &\rightarrow &
\left[ {\Gamma \left( {3\over 2} - \nu\right) \over 2 \Gamma \left( {3\over 2}
+ \nu\right)}\right]^{1\over 2} {\Gamma (2 \nu) \over \Gamma
({1\over 2} + \nu)}
(-i)\left({i\sec \eta\over 2}\right)^{\nu - {1\over 2}}
\sim a^{\nu - {1\over 2}}\, ,\\
f_1' &\rightarrow & \left(\nu - {1\over 2}\right)\ \sec\eta\ f_1 \sim
a^{\nu + {1\over 2}}\, , \end{aligned}$$ as $ \eta \rightarrow {\pi\over2}$, which is the same late time behavior in terms of the scale factor that we found in the spatially flat coordinates. Notice that at $\nu = {3\over 2}$ the phase factor cancels and these limiting forms of the oscillatory mode functions become real, while the $\Gamma$ function has a pole singularity there. Thus, keeping only the leading behavior as $\nu$ approaches $3\over 2$ we find that the late time limits of the mode function and its derivative are $$\begin{aligned}
\psi_1 \left(\eta \rightarrow {\pi\over 2}\right) &\rightarrow &
{\alpha_1 + \beta_1 \over 2\sqrt{{3\over 2} - \nu}}\ a^{\nu - {1\over 2}}\,,
\nonumber\\
\psi_1' \left(\eta \rightarrow {\pi\over 2}\right) &\rightarrow &
{\alpha_1 + \beta_1 \over 2\sqrt{{3\over 2} - \nu}}\ a^{\nu + {1\over 2}}\,.
\label{eq:psikone}\end{aligned}$$ Thus, the dominant terms in the $k=1$ energy density $\varepsilon_1$ in this limit are $|\psi_1'|^2 $ and the terms involving ${a'\over a} = \tan \eta$, and we obtain $$\begin{aligned}
\lim_{\eta \rightarrow {\pi \over 2}} \varepsilon_1 =
\left[{3(1 + 2 n_1)\over 32\pi^2 }|\alpha_1 + \beta_1|^2 +
{\cal O}(3 - 2\nu)\right] a^{2\nu -3}\,.
\label{eq:epsone}\end{aligned}$$ The singularity in the $\Gamma$ function has canceled against the $\xi$ in the numerator. The remaining coefficient is clearly state dependent. From this asymptotic form of the energy density in the $k=1$ mode at late times it is clear that for $\nu$ close to but still less than $3\over 2$, the state-dependent energy density $\varepsilon_1$ goes to zero at late times, albeit very slowly, which is consistent with our previous flat section analysis.
The corresponding expression for the trace in the $k=1$ mode is $$-\varepsilon_1 + 3p_1 =
{(1 + 2n_1)\over 2\pi^2 a^4} (6 \xi -1)\left\{-|\psi_1'|^2
+ [\tan\eta\ (\psi_1\psi_1^{*\prime} + \psi_1^*\psi_1')
+ [\sec^2\eta + 2(6 \xi -1)\sec^2\eta ]|\psi_1|^2]\right\}\,.$$ Substituting the late time asymptotic forms (\[eq:psikone\]) to leading order in $\xi$ as before yields $$-\varepsilon_1 + 3p_1 \rightarrow \left[-{3(1 + 2n_1)\over 8\pi^2 }
|\alpha_1 + \beta_1|^2 + {\cal O}(3 - 2\nu)\right] a^{2\nu -3}
\rightarrow -4\varepsilon_1\, .
\label{eq:trone}$$ Hence, $p_1 \rightarrow -\varepsilon_1$ and the contribution from this mode is de Sitter invariant at late times for any initial physical state.
Since the Bunch-Davies state is $\alpha_k =1$ and $\beta_k = n_k = 0$ for all $k$ and this state has a renormalized $\langle T_{ab} \rangle$ which is strictly time independent for all $\nu$, the time dependent contribution in (\[eq:epsone\]) and (\[eq:trone\]) for $\nu <
{3\over 2}$ in the $k=1$ mode must be canceled by a time dependent contribution from all the other modes in the renormalized energy-momentum tensor. In other words, the late time behavior of all the $k>1$ modes in the BD state must be $$\varepsilon_{BD}\Big\vert_{k >1} \rightarrow \varepsilon_{BD} -
\left[{3\over 32\pi^2 } + {\cal O}(3 - 2\nu)\right] a^{2\nu -3} \; ,
\label{eq:kgone}$$ for $\nu$ close to $3\over 2$. The pressure for the $k>1$ modes is obtained from this by the de Sitter invariant relation $p=-\varepsilon$.
The subtraction of the second term in (\[eq:kgone\]) can be understood in a different way. Consider the short distance expansion of the BD two-point function for $\nu < {3\over 2}$, namely [@k-g] $$G_{BD}(x,x') \rightarrow {1\over 8\pi^2}\left[ {1\over 1 - Z} - \log (1-Z)
+ {1\over {3\over 2} - \nu}\right]\,,
\label{BDG}$$ for the de Sitter invariant bi-scalar, $Z(x,x') \rightarrow 1$ and $\nu$ close to $3\over 2$. The constant term is singular at $\nu =
{3\over 2}$, but it gives a finite contribution to the energy density from the $\xi G_{ab} \phi^2$ term in the energy-momentum tensor. In the dimensionless units we are using this is equal to $${3\over 8\pi^2}\ \xi\ \left({R\over 12}\right)^2{1\over {3\over 2} - \nu} =
{3\over 32\pi^2}\,,
\label{eq:BDUV}$$ for $\xi \rightarrow 0$ according to (\[eq:xizero\]). The last constant term in (\[BDG\]) is absent in the short distance expansion of the Allen-Folacci two-point function [@a-f], and in the Bunch-Davies two-point function it comes entirely from the $k=1$ mode in the BD state. Hence the contribution to the energy density of the $k>1$ modes in the BD state does not contain (\[eq:BDUV\]), and must equal the full $\varepsilon_{BD}$ minus (\[eq:BDUV\]) from the $k=1$ mode, which is equivalent to (\[eq:kgone\]) at $\nu = {3\over 2}$.
With the $k=1$ and $k>1$ mode contributions separated we are now in a position to take the limit of $\nu \rightarrow {3\over 2}$. Apparently we would obtain a state dependent contribution from (\[eq:epsone\]). However, inspection of (\[eq:psionelim\]) shows that requiring the $k=1$ mode function to be regular as $\nu \rightarrow
{3\over 2}$ is equivalent to requiring that $A$ and $B$ remain finite in this limit. From (\[ABdeS\]) we then have $$\alpha_1 + \beta_1 \rightarrow 2B\,\sqrt{{3\over 2} - \nu} \rightarrow 0\,.$$ Therefore the apparently state dependent contribution (\[eq:epsone\]) from the $k=1$ mode [*vanishes*]{} in any infrared finite state parameterized by finite $A$ and $B$, while the $k>1$ contribution is still given by (\[eq:kgone\]), and we have the result $$\varepsilon_R =\lim_{\xi \rightarrow 0}\varepsilon_{BD}(\xi)\Big\vert_{m =0}
- {3\over 32\pi^2}
\left({R\over 12}\right)^2= \varepsilon_{AF}
\; ,$$ in any $(A,B)$ “vacuum" state with $\alpha_k =1, \beta_k = n_k = 0$ for $k>1$, upon restoring physical units. The difference here is just that required to give the Allen-Folacci (AF) renormalized expectation value, provided the limit of the BD value is taken in the same order of $m=0$, $\xi \rightarrow 0$ that we have evaluated (\[eq:epsone\]). Since from (\[eq:tmunu-bd\]) $$\varepsilon_{BD}(\xi)\Big\vert_{m =0} = {3\over 16\pi^2}
\left({R\over 12}\right)^2
\left[{1\over 180} - {1\over 6}(6\xi -1)^2\right]\,,$$ and both the BD and AF values are de Sitter invariant, $$\begin{aligned}
\lim_{\xi \rightarrow 0}\langle T_{ab}\rangle_{ren}(\xi)\Big\vert_{m =0} &=&
- {3 g_{ab}\over 16\pi^2}\left({R\over 12}\right)^2
\left( {1\over 180} - {1\over 6} - {1\over 2}\right)\nonumber\\
&=& \frac {119\ }{960\pi^2} \left({R\over 12}\right)^2 g_{ab}
= \frac{119\ R^2}{138\, 240 \pi^2}g_{ab} = \langle T_{ab}\rangle_{AF}
\; ,\end{aligned}$$ and we obtain the de Sitter invariant AF result for all $(A,B)$ “vacuum" states in the massless, minimally coupled case.
With the above careful analysis of the spatially homogeneous mode and the independence of the asymptotic value of its energy-momentum tensor in an arbitrary $k=1$ infrared finite state characterized by $A$ and $B$, it is now straightforward to carry out the proof of the attractor behavior of the AF state for an arbitrary UV finite physical state with $m=\xi=0$, by allowing the $k>1$ modes to have $\beta_k$ and $n_k$ different from zero. Substituting (\[eq:psik\]), (\[eq:psiknon\]), (\[eq:psionelim\]), and (\[eq:psikone\]) into (\[eq:T00u\]), (\[eq:Tru\]) with $m=\xi=0$, and using (\[eq:AkBk\]) gives
$$\begin{aligned}
\varepsilon = -\langle {T^0}_0 \rangle_{ren} &=&
-\langle {T^0}_0\rangle_{AF}
+(1+ 2n_1) \frac{|A|^2 \cos^6 \eta}{\pi^2}
\nonumber \\
& &+
\; \frac{1}{4 \pi^2}
\sum_{k=2}^{\infty} \left\{ \left[
2 n_k + 2 (1+2n_k) |\beta_k|^2
\right]
[k^3 \cos^4 \eta
+ k ( -\cos^4 \eta + \frac{1}{2} \cos^2 \eta)] \right.
\nonumber \\
& & +
\;
(1+ 2n_k) \left[
(\beta_k \alpha_k^* e^{2 i k \eta} + \beta_k^* \alpha_k e^{-2ik\eta})
k ( -\cos^4 \eta + \frac{1}{2} \cos^2 \eta)
\right.
\nonumber \\
& & \left. \left.
+ \; i
\; (\beta_k \alpha_k^* e^{2ik\eta} - \beta_k^* \alpha_k e^{-2ik\eta})
k^2 \cos^3 \eta
\sin \eta \right]\right\}
\; ,
\label{eq:t0032} \\
\langle T\rangle_{ren} &=& \langle T\rangle_{AF}
+ (1+2n_1) \frac{2 |A|^2 \cos^6 \eta}{\pi^2}
\nonumber
\\
&& - \; \frac{1}{4 \pi^2} \sum_{k=2}^{\infty}
\left\{
\left[ 2 n_k + 2 (1 + 2n_k) |\beta_k|^2
\right]
k
\cos^2 \eta
\right. \nonumber \\
& &
+
\; (1+2n_k)
\left[(\beta_k \alpha_k^* e^{2ik\eta} + \beta_k^* \alpha_k e^{-2ik\eta})
(-2 k^3 \cos^4 \eta + k \cos^2 \eta)
\right.
\nonumber
\\
&& \left. \left.
+ \; 2 i \; (\beta_k \alpha_k^* e^{2ik\eta}
- \beta_k^* \alpha_k e^{-2ik\eta}) k^2 \cos^3 \eta
\sin \eta \right]
\right\}
\label{eq:tr32}
\; .\end{aligned}$$
Provided the $k$ sums converge, it is clear that all the state dependent terms contain at least one factor of $a^{-2} = \cos^2\eta$, and so vanish in the limit of $\eta \rightarrow {\pi \over
2}$. However, the requirement that the state be fourth order adiabatic just guarantees this convergence, for the same reason as in the previous analysis in spatially flat coordinates. Indeed we have $$\begin{aligned}
\vert\beta_k\vert &= & \frac {C(k)} {k^4}\,,\nonumber\\
n_k &=& \frac {N(k)} {k^4}\,,\end{aligned}$$ for some $C(k)$ and $N(k)$ that vanish as $k\rightarrow \infty$. This is sufficient to guarantee the absolute convergence of all terms in the sums. Since all state dependent terms are multiplied by at least two powers of $\cos \eta = a^{-1}$, which vanishes in the late time limit $\eta \rightarrow {\pi \over 2}$, we conclude that any UV and IR admissible state of the massless, minimally coupled scalar field has an energy-momentum tensor which approaches the AF value, $\langle T_{ab} \rangle_{AF}$ in the late time limit $\eta \rightarrow {\pi\over 2}$.
If one considers the contributions of the state dependent terms to the energy density and trace from the $k>1$ modes, for $\nu$ not exactly $3\over 2$, the kinematics is essentially the same as our previous analysis of the $k=1$ mode. Their contribution also falls off proportional to $\xi |\alpha_k + \beta_k|^2 a^{2\nu - 3}$ at late times for $\nu$ close to $3\over 2$. However, there is no compensating large factor coming from the pole in the $\Gamma$ function normalization constant as there is for the $k=1$ mode. Hence, the coefficient of this slow fall off goes to zero as the massless, minimally coupled limit is approached. That is, exactly at $m=\xi=0$ when the contribution of the $k>1$ modes no longer falls off and they could in principle contribute to the asymptotic value of $\langle
T_{ab}\rangle_{ren}$ at late times, at that very point their coefficient vanishes identically and they make no contribution at all. Thus, at precisely $m=0$ and $\xi =0$ the difference between the BD and AF values can be attributed entirely to the additional condensate in the spatially homogeneous $k=1$ mode alone, and there are no slow transient modes in our explicit analysis of the massless, minimally coupled energy and trace.
These considerations are relevant to the case when $\nu = {3\over 2}$ but $m$ and $\xi$ are separately non-zero. The calculation is almost identical but the conclusion is different, since now the finite $k>1$ mode sum is multiplied by a coefficient $\xi$ which does [*not*]{} vanish. The entire mode sum of state independent terms from $k = 2$ up to $a(\eta)$ do not fall off at late times and in fact all add up, to give a contribution proportional to $\langle \Phi^2\rangle
\sim\sum_{k=2}^a k^{-1} \sim \log a$ in $\langle T_{ab}\rangle_{ren}$, which grows linearly in comoving time. The explicit expression is most conveniently calculated by using (\[eq:deftan\]) to divide the energy-momentum tensor into a state dependent numerical part, $\langle
T_{ab} \rangle_n$ and a state independent analytic part, $\langle
T_{ab} \rangle_{an}$. They are separately conserved [@AE]. The quantities $\langle T_{ab} \rangle_d $ and $\langle T_{ab}
\rangle_{an} $ are given in Ref. [@AE]. By substituting Eqs. (\[eq:psikone\]) and (\[eq:psiknon\]) into Eqs. (\[eq:T00u\]) and (\[eq:Tru\]), and subtracting the expression for $\langle T_{ab} \rangle_d $ given in Ref. [@AE], we find that $\langle T_{ab} \rangle_n$ approaches a state dependent, finite constant in the limit $\eta \rightarrow {\pi\over 2}$. However, the quantity $\langle T_{ab} \rangle_{an} $ has a term that is proportional to $\langle \Phi^2\rangle$. At late times this term behaves as $\log a$ and dominates. In fact
$$\begin{aligned}
\varepsilon_{ren}
&\rightarrow & -\frac{3 \xi}{4\pi^2} \log (\cos \eta)
+ q_1
\; , \label{eq:q1} \\
\langle T \rangle_{ren} &\rightarrow & {3 \xi \over {\pi^2}}
\log (\cos \eta)
+ q_2
\;, \label{eq:q2}\end{aligned}$$
with the constants $q_1$ and $q_2$ dependent on the state of the field and constrained by the conservation equation. Since $\log (\cos \eta)
\rightarrow t$ in comoving time, the energy-momentum tensor grows linearly with $t$ at late times. For $\xi >0$ (and $m^2<0$) the linear growth in comoving time decreases the effective cosmological “constant", whereas for $\xi <0$ (and $m^2>0$) it increases it. In either case the back-reaction of the energy-momentum of the quantum scalar field for $m^2 + \xi R = 0$ (but $m^2 = -\xi R \neq 0$) on the geometry certainly cannot be neglected at late times since $\langle T_{ab}\rangle$ grows without bound for any physical state. The fact that a massive non-minimal field with $\xi <0$ can induce an effective cosmological “constant" due to inflationary particle production was noted in Ref. [@sh] using a different approach. It is an interesting open question whether this linearly growing behavior (in proper time) carries over to the physically more relevant case of one-loop gravitons, since the mode functions for gravitons in a particular gauge obey the same equation in a RW spacetime as do the mode functions for a massless minimally coupled scalar field [@Grischuk].
Finally, we note that for all real values of $\nu$, the analysis of the $k=1$ mode in this Section may be extended to all of the higher $k$ modes, since the late time behavior of the higher $k$ modes is determined by that of the hypergeometric function in (\[eq:hyper\]), which gives $$f_k \left(\eta \rightarrow {\pi\over 2}\right) \rightarrow
\left[ {\Gamma \left( k+ {1\over 2} - \nu\right) \over 2 \Gamma
\left( k + {1\over 2} + \nu\right)}\right]^{1\over 2}
{\Gamma (2 \nu) \over \Gamma ({1\over 2} + \nu)}
{(-i)^k\over k!}\left({i\sec \eta\over 2}\right)^{\nu - {1\over 2}} \,.$$ Thus all of the higher $k$ modes behave as $a^{\nu - {1\over 2}}$ and give the (unrenormalized) energy density the leading order late time behavior $$a^{2\nu - 3}\left[\left(\nu -{3\over 2}\right)^2 + 12\xi (\nu -1) + m^2\right]
\left[{\Gamma(2\nu)\over 2^{\nu}\Gamma\left(\nu + {1\over 2}\right)}\right]^2
\sum_{k = 1}^{\infty} (1 + 2n_k){\Gamma \left( k+ {1\over 2} - \nu\right)
\over \Gamma \left( k + {1\over 2} + \nu\right)(k!)^2}\vert\alpha_k +
e^{i \pi (k + \nu -{1\over 2})}\beta_k\vert^2\,,
\label{eq:div}$$ which grows (or shrinks) like $a^{2\nu - 3}$ for an arbitrary physical state unless either the coefficient vanishes or $\alpha_k + e^{i\pi \left(k
+ \nu - {1\over 2}\right)}\beta_k = 0$ for all $k$. However, the latter is impossible since it is inconsistent with the requirement that all physically allowable states be fourth order adiabatic states, which requires that $\alpha_k \rightarrow 1$ and $\beta_k \rightarrow
0$ sufficiently fast as $k \rightarrow \infty$, in order for the renormalized energy density to be UV finite. Since the fourth order adiabatic subtraction behaves at most like a constant at late times, the $a^{2\nu - 3}$ behavior is not affected by the UV subtraction, and indeed the sum in (\[eq:div\]) converges at large $k$ for any UV admissable state. Hence after the adiabatic four subtraction the leading order late time behavior indicated by (\[eq:div\]) survives unless the first bracket in front of the entire expression vanishes. The quantity in this bracket is identical to the factor in (\[eq:lateps\]) found in the flat spatial section analysis of the last Section. The corresponding quantity for the trace is given by the factor in curly brackets in (\[eq:lateT\]), which is a similar combination of $\nu, \xi$, and $m^2$. If we require that both of these factors vanish identically, to eliminate the leading order behavior in all components of $\langle T_{ab}\rangle_{ren}$, then these two conditions plus the defining relation (\[eq:nu\]) give $$\begin{aligned}
m^2 &=& - \frac{\nu (2 \nu - 3) (2 \nu - 1)}{4 (\nu - 2)}\; , \nonumber \\
\xi &=& \frac{(2 \nu - 3)}{8 (\nu - 2)}
\; .\end{aligned}$$ Thus, except for $\nu = 2$, for any given $\nu$ there is always one value of $m^2$ and one value of $\xi$ for which the coefficients of these leading order terms in $\langle T_{ab}\rangle_{ren}$ vanish. The next to leading order terms go like $a^{2\nu -5}$, and grow without bound at late times in any case when $\nu > {5\over 2}$.
The above analysis implies that, for most values of $m^2$ and $\xi$, when $\nu > {3\over 2}$ the leading order terms in the components of the energy-momentum tensor grow without bound like $a^{2\nu - 3}$ in de Sitter space for any physically admissable initial state of the scalar field. These values of $\nu$ correspond to the purely tachyonic cases $m^2 + \xi R <0$.
Numerical Studies {#sec:numerical}
=================
In this Section we display numerical results for various values of $\nu$. All results are given in dimensionless units where $\alpha =
1$ and $R = 12$. The quantum state in each case is a fourth order adiabatic state matched to the vacuum at some initial time $\eta_0$. That is, we choose initial conditions for the mode function and its first derivative to be $$\begin{aligned}
\psi_k (\eta_0) &=& \psi_k^{(4)}(\eta_0)\,,\nonumber\\
{{\rm d} \psi_k\over {\rm d}\eta}(\eta_0) &=& {{\rm d}
\psi_k^{(4)}\over {\rm d}\eta}(\eta_0)\,,\end{aligned}$$ with $\psi_k^{(4)}$ the fourth order adiabatic mode function defined by (\[eq:psiadb\]) with phase measured from the initial time $\eta_0$. The initial time was chosen to be $t_0 =1$ in comoving coordinates, [*i.e.*]{} $\sec \eta_0 = \cosh (1) = 1.54308\dots$.
For $\; \Re (\nu)<{3\over 2}$ the proof in Section \[sec:analytic\] states that the energy-momentum tensor approaches the Bunch-Davies value (\[eq:tmunu-bd\]) at late times for an arbitrary physically admissable state. This occurs due to a redshifting of the state dependent part of the energy-momentum tensor. The initial state dependent transient contributions fall off like $$a^{2 \nu -3} = (\cosh t)^{2 \nu - 3} \;.$$ Thus the characteristic time to approach the BD value is $$\tau = \frac{1}{3 - 2 \nu}\,.$$ In Fig. 1 we plot the renormalized energy density for these adiabatic initial conditions with $m=0$ and $\xi = {1\over 7}$. For this relatively large value of $\xi$, the characteristic time $\tau$ is of order one and the energy density approaches the BD value within one expansion time.
For smaller values of $\xi$ the initial value transients persist for longer times. Our analysis in the previous Section shows that the $k=1$ contribution is essential for the shift from the BD to AF value as $\xi \rightarrow 0$. Since the $k=1$ mode in an arbitrary physical state contributes to the energy density the value $\varepsilon_1$ given by (\[eq:epsone\]), up to terms which fall off like $a^{-2}$, for arbitrary $\alpha_1, \beta_1$, and $n_1$, while the BD state has $\alpha_1 =1, \beta_1 = n_1 =0$, it is clear that the difference of the renormalized energy density from the BD value coming from this mode is $$\varepsilon_R - \varepsilon_{BD} = \varepsilon_1 - {3\over 32\pi^2}
\left({R\over 12}\right)^2 a^{2 \nu - 3}\,.$$ For the adiabatic initial conditions here we find that $$\psi_1(\eta_0) = {1 \over \sqrt{2 W_1^{(4)}(\eta_0)}}
\approx {(\alpha_1 + \beta_1) \sec\eta_0\over
2 \sqrt{{3\over 2} - \nu}}\,.$$ Since $W_1^{(4)}(\eta_0)$ remains finite as $\xi \rightarrow 0$, provided that $\eta_0 \neq 0$, it follows that in this limit $|\alpha_1 + \beta_1|^2 \sim 3 - 2 \nu \sim \xi$ also goes to zero. Hence for small $\xi$, $\varepsilon_R$ goes to a value close to ( but slightly larger than) the AF value after a time of order one. This is observed in both Figs. 2 and 3 for $\xi = {1\over 100}$ and $\xi = {1\over 1000}$, respectively.
Further, the energy-momentum tensor contains a term proportional to $\xi \langle \Phi^2\rangle$ which would grow linearly in comoving time for $\nu$ very close to $3\over 2$, except for the factor of $a^{2 \nu
- 3}$ that damps it to zero at very late times $t \gg \tau$. Hence on times $1 < t \ll \tau$, where the $a^{2 \nu - 3}$ factor is essentially constant, we should expect this linear growth of $\varepsilon$ in comoving time with a slope proportional to $\xi$, given by (\[eq:q1\]). This behavior is demonstrated in Figs. 2 and 3. In the latter case $\xi$ is so small that $\alpha_1 + \beta_1$ and the slope are nearly vanishing and the energy density stays close to the AF value until times of order $\tau = 125$, which is much larger than the times shown in Fig. 3. When $\xi =0$ (still keeping $m=0$) both $\alpha_1 + \beta_1$ and the slope vanishes identically, so the energy-momentum tensor goes to the AF value and remains there. This demonstrates that the limit $\Re (\nu)\rightarrow {3\over 2}$ is quite continuous when viewed at finite times with well-defined physical initial conditions, although the late time limit is discontinuous.
In the case $\nu = {3\over 2}$ there are two distinct behaviors depending on whether $m$ and $\xi$ are separately vanishing or not. In the massless, minimally coupled case we proved in Section \[sec:nu32\] that the Allen-Folacci value is a fixed point at late times. In Figure 4 we show the approach of the energy density, pressure, and trace to their Allen-Folacci values for the massless minimally coupled field. The field is in an “n-particle” state with $n_1 = n_2 = 2$ and $n_k = 0$ for all $k > 2$. In Figure 5 we show the behavior of the energy density and trace for the case $m = 0.3$ and $\xi = - 0.0075$ when the field is in a fourth order adiabatic vacuum state.
When $\Re (\nu) > {3\over 2}$ the analysis at the end of Section \[sec:nu32\] shows that the leading order state dependent terms in the energy-momentum tensor will generally grow exponentially with time. This is illustrated in Fig. 6. The exponential growth that occurs here is similar to the well known one for the classical scalar field when $m$ and $\xi$ have values such that $\nu>{3\over
2}$ [@dolgov; @ford]. Such fields are tachyonic and presumbably of little physical interest unless interactions are added to stabilize them.
Infrared Scaling and the Generalized Anomaly {#sec:scaling}
============================================
We have found that for all $\Re (\nu)<{3\over 2}$ the renormalized expectation value of $\langle T_{ab} \rangle$ approaches the de Sitter invariant Bunch-Davies value for any physically admissable initial state, whereas it approaches the de Sitter invariant Allen-Folacci value for any physically admissable initial state in the massless, minimally coupled case. Since all the initial state dependence vanishes asymptotically, these state independent de Sitter invariant fixed point values for $\langle T_{ab} \rangle_{ren}$ must be purely geometrical in origin. Indeed, both the Bunch-Davies point-splitting calculation and the Hadamard calculation of Allen-Folacci rely only on the properties of the two-point function of the scalar field $G(x,x')$ for $x' \rightarrow x$. Hence, the BD and AF asymptotic values of $\langle T_{ab} \rangle_{ren}$ are certainly “pseudo-local” in the terminology of Ref. [@dfcb], [*i.e.*]{} they are expressible in terms of purely [*local*]{} functions of the RW scale factor $a(\eta)$ and its derivatives.
If we specialize now to zero mass, $m=0$, then on simple dimensional grounds the asymptotic $\langle T_{ab} \rangle_{ren}$ can be expressed purely in terms of local conserved tensors of fourth adiabatic order. Although we have used adiabatic subtraction methods to renormalize $\langle T_{ab} \rangle$ it is known that the value of $\langle T_{ab}
\rangle_{ren}$ so obtained is equal to that in a fully covariant procedure such as dimensional regularization or covariant point-splitting [@AP; @Birr]. In a fully covariant procedure, which yields a local conserved tensor of fourth adiabatic order, only the Riemann tensor together with its covariant derivatives and contractions can appear. Hence $\langle T_{ab} \rangle_{ren}$ for $m=0$ must be expressible entirely in terms of such local geometrical tensors.
In four dimensions the only such local tensors are linear combinations of $^{(1)}H_{ab}$, $^{(2)}H_{ab}$, and $^{(3)}H_{ab}$, where $$\begin{aligned}
^{(1)}H_{ab} &\equiv& {1\over \sqrt{-g}} {\delta\over\delta g^{ab}}
\int\, \sqrt{-g}\,R^2\,{\rm d}^4x
\nonumber \\
&=& 2 g_{ab} \sq R - 2\nabla_a\nabla_b R
+ 2 R R_{ab} - {1\over 2} g_{ab} R^2\,,
\label{eq:Hone}\end{aligned}$$ vanishes in de Sitter spacetime and $$^{(3)}H_{ab} = R_a^cR_{cb} - {2\over 3}RR_{ab} -{1\over 2} R_{cd}R^{cd}g_{ab}
+ {1\over 4} R^2 g_{ab}\,.
\label{eq:Hthr}$$ In RW spacetimes, which are all conformally flat, the tensor $^{(2)}H_{ab}$ is proportional to $^{(1)}H_{ab}$ and hence vanishes as well. Therefore the only non-trivial fourth order conserved geometrical tensor in de Sitter spacetime is $^{(3)} H_{ab}$ and we conclude that the fixed point BD and AF values found in our previous analysis are proportional to $$^{(3)}H_{ab} = {R^2\over 48}\ g_{ab} = 3 \left({R\over 12}\right)^2 g_{ab}\,.
\label{eq:Hthrds}$$ Furthermore, since $$^{(3)} H_{ab}\,g^{ab} = - R_{ab}R^{ab} + {R^2\over 3}
={1\over 2} \left( R_{abcd}R^{abcd} - 4 R_{ab}R^{ab} + R^2\right)
\equiv {G\over 2} \; ,$$ in RW spacetimes (where the Weyl tensor vanishes), the coefficient of $^{(3)} H_{ab}$ is proportional to the coefficient of the Gauss-Bonnet integrand in the trace of $\langle T_{ab} \rangle$. Such a term in the trace is known to correspond to a non-local but nevertheless fully covariant action and this action is precisely the same as that generated by the trace anomaly of free conformal fields [@amm]. Since we have obtained fixed point results for the asymptotic values of $\langle T_{ab} \rangle_{ren}$ for massless fields in de Sitter space which are purely geometrical and proportional to $^{(3)}H_{ab}$, even for [*non*]{}-conformal massless fields, we can give a definite meaning to the value of the proportionality coefficient and the non-local anomaly-like term in the effective action even when $\xi
\neq {1\over 6}$.
Let us define the generalized anomaly coefficient by fixing the normalization $$\lim_{t \rightarrow \infty} \langle T_{ab} \rangle_{ren}
= -{Q^2\over 16\pi^2}\,
^{(3)}H_{ab} = -{3 Q^2\over 16\pi^2}\left({R\over 12}\right)^2 g_{ab}\,.
\label{conom}$$ With this normalization we find from the asymptotic value of $\langle
T_{ab} \rangle_{ren}$ for a scalar field in de Sitter space that $$Q^2 = \left\{ \begin{array}{ll}
Q^2_{BD} &= {1\over 180} - {1\over 6} (6\xi -1)^2, \qquad m=0, \qquad \xi > 0\,,
\\
Q^2_{AF} &= {1\over 180} - {1\over 6} - {1\over 2} = -{119\over 180},
\qquad m=0, \qquad \xi = 0\,.
\end{array} \right.
\label{Qval}$$ The value of $Q^2$ for a conformally invariant field ($m=0,\xi={1\over6}$) is $1\over 180$ and corresponds to the pure trace anomaly coefficient. The first member of (\[Qval\]) provides the generalization of this coefficient away from the conformal case. The discontinuous behavior at $\xi=0$ has been discussed in Ref. [@k-g]. As we have seen it arises from the singular behavior of the spatially constant zero mode of the massless, minimally coupled field, which is non-oscillatory and hence cannot be quantized as a Fock mode in the same fashion as the oscillatory modes. We discussed this discontinuity in detail in Section \[sec:nu32\].
The connection of the tensor $^{(3)}H_{ab}$ with the trace anomaly may be seen from the general form of the effective action for the anomaly in a conformally flat space with metric [@amm] $$g_{ab} = e^{2\sigma} \bar g_{ab}\,,
\label{conf}$$ namely $$S_{\mathrm eff} = -{Q^2\over 16\pi^2} \int\,
{\mathrm d}^4x \sqrt{-\bar g}
\left[\sigma \bar\Delta_4 \sigma + {1\over 2} \left(\overline G- {2\over 3}
{\,\raise.5pt\hbox{$\overline{\mbox{.09}{.09}}$}\,}
{\overline R}\right)\sigma\right]\,,
\label{eq:effact}$$ where $\Delta_4 = \sq ^2 + 2R^{ab} \nabla_a \nabla_b - {\textstyle
\frac{2}{3}} R {\sq}^2 + {\textstyle \frac{1}{3}} (\nabla^a R)
\nabla_a$ is the unique fourth order differential operator acting on scalars which is conformally covariant. A fully covariant but non-local form of the effective action (\[eq:effact\]) can be obtained by solving $\sqrt{-g}\left(G - {2\over 3} \sq R\right) =
\sqrt{-\bar g} \left(\bar G - {2\over 3} \sqb \bar R\right) +4
\sqrt{-\bar g} \bar \Delta_4 \sigma$ for $\sigma$. In that non-local form all reference to the separation of the metric into background and conformal factor as in (\[conf\]) disappears.
The energy-momentum tensor following from the variation of the local form of the effective action (\[eq:effact\]) with respect to the background metric $\bar g_{ab}$, is given by Eq. (2.9) of Ref. [@amm]. The form of this energy-momentum tensor simplifies considerably on the Einstein space $R\times S^3$, where $\overline G =
{\,\raise.5pt\hbox{$\overline{\mbox{.09}{.09}}$}\,} {\overline R} =
0$. If we set $\sigma=\log a(\eta)$ then this is equivalent to evaluating the components of this energy-momentum tensor in a general RW spacetime with closed spatial sections ($\kappa=+1$). Using the expressions for $^{(1)}H_{ab}$ and $^{(3)}H_{ab}$ in a general RW space in terms of $a(\eta)$ and its derivatives, we quickly find that $$T_{ab}[\sigma] = -{2\over \sqrt{-g}} {\delta S_{\mathrm eff}\over
\delta \bar g^{ab}}= {Q^2\over 16\pi^2}\left[ {1\over 18}\, ^{(1)}H_{ab} -
^{(3)}H_{ab} + ^{(3)}H_{ab}\Big\vert_{R\times S^3}\right]\,.
\label{varT}$$ Hence the tensor $^{(3)}H_{ab}$ which is called “accidentally conserved" in Ref. [@bi-da] is associated in fact with the existence of a non-local covariant effective action related to the trace anomaly, which has the local form (\[eq:effact\]) when the metric is conformally decomposed as in (\[conf\]).
The last term in (\[varT\]) would not have been present had we varied the fully covariant but non-local form of the anomalous effective action, in which all reference to the background $\bar
g_{ab}$ drops out. Equivalently, it is just canceled if we add to (\[varT\]) the Casimir energy on $R \times S^3$, which is determined by the anomaly by a further conformal transformation from flat space [@brown]. In either case, we should drop this last term in (\[varT\]) which depends on the arbitrary background $\bar g_{ab}$. Evaluating the first two terms on de Sitter space we find that it is exactly the value of the asymptotic form of the renormalized energy-momentum tensor $\langle T_{ab} \rangle$ in de Sitter space (\[conom\]), found previously, with the value of $Q^2$ for the massless scalar given by (\[Qval\]). Thus, the effective action (\[eq:effact\]), associated with the conformal trace anomaly, appears in the effective action for a massless scalar field, even for [*non*]{}-conformal couplings, $\xi\neq {1\over 6}$.
Since it is associated with the anomaly, the physical significance of $S_{\mathrm eff}$ in the quantum effective action for the scalar field is that it determines the scaling behavior of the field theory under global Weyl transformations of the background space. Using the fact that the Euclidean effective action is given by $I_{\mathrm
eff}=-S_{\mathrm eff}$ and that the Euclidean continuation of de Sitter space is $S^4$ with Euler number $\chi=2$, we can vary $I_{\mathrm eff}$ with respect to a constant $\sigma =\sigma_0$ and obtain $${\partial I_{\mathrm eff}\over \partial \sigma_0} = \alpha
{{\rm d} I_{\rm eff}\over {\rm d}\alpha}
= {Q^2\over 32 \pi^2}\int_{S^4}\, {\mathrm d}^4x\,
\sqrt{g} \; G = Q^2 \chi = 2Q^2
\; ,
\label{eq:scalI}$$ since ${{\rm d} \over {\rm d}\sigma_0} = {{\rm d}\over {\rm d}
\log\alpha}$ is a global rescaling of the $S^4$. As a check, this relation can be verified explicitly by the $\zeta$ function evaluation of the Euclidean effective action, $$I_{\mathrm eff}= {1\over 2}\; {\rm Tr \, \log}
\left(-{\,\raise.5pt\hbox{$\mbox{.09}{.09}$}\,} + m^2 + \xi R\right) =
-{1\over 2}{{\mathrm d}\zeta\over {\mathrm d}s}
\Big\vert_{_{s=0}}\,,
\label{eq:Euceff}$$ where the generalized zeta function for the Euclidean continuation of the wave operator appearing in the Tr $\log$ is $$\zeta (s) = \sum_{n=0}^{\infty} d_n \lambda_n^{-s}\; ,
\label{zdef}$$ in terms of its eigenvalues $\lambda_n$ with degeneracy $d_n$ on $S^4$. This sum is convergent for $\Re (s)>2$ and defines a meromorphic function of $s$ which is analytic near $s=0$, where its derivative is required. Introducing a mass scale $\mu$ to keep $\zeta (s)$ dimensionless for all $s$ leads in a standard calculation to [@mazmot] $$-{1\over 2}{{\mathrm d}\zeta\over {\mathrm d}s}
\Big\vert_{_{s=0}} = -\zeta (0)
\log (\mu\alpha) + I_1(\nu)\,,
\label{Ieff}$$ where $I_1(\nu)$ is a certain finite function of $\nu$, which from the definition of $\nu$ (\[eq:nu\]) becomes independent of $\alpha$ when $m=0$, and the value $\zeta (0)$ is given by [@mazmot] $$-\zeta (0) = {1\over 12}\left( -\nu^4 + {1\over 2}\nu^2
+ {17\over 240}\right) = {1\over 90} - {1\over 3}(6\xi -1)^2
\; ,$$ when $m=0$. By making use of (\[Qval\]), (\[eq:Euceff\]), and (\[Ieff\]), we find that $$\alpha{{\mathrm d}\over {\mathrm d}\alpha}
I_{\mathrm eff}(m=0) = -\zeta (0) = 2Q^2
\; ,$$ is the behavior of the effective action for a massless scalar field with $\xi>0$ under global Weyl rescaling of the metric, exactly as predicted by(\[eq:scalI\]) and the previous discussion based on the anomalous action (\[eq:effact\]).
When $m=0$ and $\xi \rightarrow 0$, the integral representation of the function $I_1(\nu)$ develops a logarithmic singularity, which can be traced to the vanishing of the $n=0$ eigenvalue in the expression (\[zdef\]) for $\zeta (s)$. In this case the $n=0$ mode must be excluded from the ultraviolet regulated sum over modes, which has the effect of [*adding*]{} one unit to $\zeta (0)$ in the infrared scaling behavior of the effective action [@ammnp], and accounts for the addition of $-{1\over 2}$ in $Q^2$ in the minimally coupled case. Hence the discontinuous behavior of $Q^2$ found in (\[Qval\]) by our analysis of the asymptotic attractor behavior of the energy-momentum tensor in de Sitter space is precisely the same as that occuring in the effective action under global Weyl rescalings when $m=0$ and $\xi \rightarrow 0$.
Note also that the global Weyl variation is given in terms of the trace of the energy-momentum tensor by $$\alpha{{\mathrm d}\over {\mathrm d}\alpha}
I_{\mathrm eff}(m=0) = -\int_{S^4}\, {\mathrm d}^4x\,\sqrt{g} \;
\langle T \rangle_{ren} = -{8\pi^2\over 3}\alpha^4
\langle T \rangle_{ren}\,.$$ Since the $\zeta$ function method is fully covariant, the renormalized trace $\langle T \rangle_{ren}$ computed in this way cannot contain non-covariant contributions, and must be expressible entirely in terms of local curvature invariants.
Finally we may consider relaxing the condition $m=0$. If $m^2>0$, the asymptotic form of the energy-momentum tensor for an arbitrary initial state is given by the BD value. However, if we expand the BD result (\[eq:tmunu-bd\]) in powers of $R/m^2$, we find that it contains [*no*]{} adiabatic order four $R^2$ terms, beginning instead with $R^3/m^2$. Mathematically, this is because all terms up to fourth adiabatic order have been removed by the ultraviolet regulating procedure of point splitting or adiabatic subtraction. Hence the coefficient of $^{(3)}H_{ab}$ at asymptotically late times in an arbitrary physical state is given by $$Q^2 = 0 \; \; \; {\rm for} \; \; \; m^2 > 0
\; ,
\label{eq:Qmzero}$$ and no anomalous $S_{\mathrm eff}$ term appears in the quantum effective action for a massive field. This is consistent with the interpretation of the anomalous term (\[eq:effact\]) in the effective action of the scalar field as an [*infrared*]{} effect, since the fluctuations of a massive field decouple at large distances or late times, and should induce only strictly irrelevant operators in the effective action in the far infrared, which are suppressed by positive powers of $R/m^2$.
Only in the strictly conformal case, $m=0$ and $\xi = {1\over 6}$ is the infrared effective action equal to that obtained by ultraviolet methods, such as the $a_2$ coefficient in the Schwinger-DeWitt proper time expansion. However, our analysis of the fixed point behavior of $\langle T_{ab}\rangle_{ren}$ in de Sitter space shows that its coefficient is connected with the global or extreme infrared scaling of the effective action, and this asymptotic behavior does not depend on the field being conformally invariant. The asymptotic attractor behavior of the energy-momentum tensor in de Sitter space defines an infrared scaling coefficient that reduces to the trace anomaly coefficient in the conformal case, but is a much more general concept than the trace anomaly coefficient, since it is well-defined for all massless fields, conformal or not. It is well-defined even for massive fields, although as (\[eq:Qmzero\]) shows, it vanishes in this case.
Discussion and Conclusions {#sec:discussion}
==========================
In this paper we have considered a quantum scalar field in a fixed de Sitter background. We have studied the late time behavior of the renormalized energy-momentum tensor and have found two important cases in which, for arbitrary physically admissable states, the energy-momentum tensor approaches a particular value at late times. The values approached are those of the energy-momentum tensor in the Bunch-Davies and Allen-Folacci states. Thus, in this sense, these special quantum states behave as fixed point attractors.
In the case $\Re (\nu) < {3\over 2}$ we have shown that for all fourth order adiabatic states that are infrared finite the energy-momentum tensor approaches the BD value at late times. The longest time scale for the state dependent terms to redshift away is $\tau=(3 -2
\nu)^{-1}$. This has been numerically verified for various values of $m$ and $\xi$ when the fields are in a fourth order adiabatic state. For the case in which $0 < {3\over 2} - \nu << 1$ and $m$ and $\xi$ are small, we numerically observe a more complicated behavior. The energy-momentum tensor quickly approaches the AF value and grows then linearly with comoving time. Our analytic proof implies that it must then slowly decay to the BD value. It is worth noting that for $\Re
(\nu) > 0$ the redshift of the state dependent terms in the quantum expectation value of $T_{ab}$ is slower than one might guess from the redshifting of classical matter or radiation, [*i.e.*]{} $a^{-3}$ or $a^{-4}$, respectively.
For the case $\nu={3\over 2}$ we have to distinguish two different possibilities. If the field is massless and minimally coupled then we have proven that, for all fourth order adiabatic states that are infrared finite, the energy-momentum tensor approaches the AF value at late times. This is true for both vacuum and initially populated states. For any other values of $m$ and $\xi$ the renormalized energy-momentum tensor grows linearly with comoving time, indicating that back-reaction effects need to be taken into account. The sign of the linear growth depends on the sign of $\xi$.
For the tachyonic cases $\nu>{3\over 2}$ there is no attractor state. Instead, for most values of $m^2$ and $\xi$ the renormalized energy-momentum tensor grows like $a^{2\nu-3}$ at late times for all physically admissable states. Thus back-reaction effects again need to be taken into account.
For the cases in which either the Bunch-Davies or Allen-Folacci states serve as attractors in the above sense and the mass, $m$ of the field is zero, we have shown how these results are connected to the appearance of a certain non-local term in the quantum effective action for the scalar field. This term gives rise to the local geometrical tensor $^{(3)}H_{ab}$ in the asymptotic form of $\langle T_{ab}
\rangle$ at late times, and also determines the global scaling behavior of the effective action for massless fields. Determining this scaling behavior and relating it to the asymptotic $\langle T_{ab}
\rangle$ in de Sitter space has allowed us to generalize the notion of the trace anomaly to massless, non-conformally coupled scalar fields, in the sense that the coefficient of this non-local term in the effective action is well defined even for $\xi \neq {1\over 6}$.
The interplay between the UV and IR properties of the state and the mode sums contributing to the energy-momentum tensor is a theme running through all of these results. We had to be careful to remove the UV divergences from the unrenormalized $\langle T_{ab} \rangle$ in order to analyze the late time limit. Since we have found state independent de Sitter invariant results in both the $\nu < {3\over 2}$ and massless, minimally coupled cases, and since all the state dependence resides in the finite $k$ modes, their form at high $k$ being restricted by the requirement of matching the adiabatic order four vacuum, it is clear that state independent results for $\langle
T_{ab}\rangle_{ren}$ are possible only because the contribution to the BD or AF expectation value comes from arbitrarily large $k$ at very late times. In fact, inspection of the renormalized expectation value of $\langle T_{ab}\rangle$ expressed as a mode sum, after the fourth order adiabatic subtraction has been made, shows that the finite contribution comes from $k \sim a$, [*i.e.*]{} when the physical wavelength of the mode is of order of the de Sitter horizon. At very late times, this corresponds to arbitrarily large values of the coordinate wave number $k$.
The finite difference between the BD value and the AF value in the $m=\xi =0$ case comes entirely from the $k=1$ mode in closed spatial sections, which is a purely IR effect. This leads to a finite discontinuity in the infrared scaling properties of a massless field since the value of the energy-momentum tensor at $m = \xi = 0$ is different from its value in the limit $m=0$ and $\xi \rightarrow 0$. The appearance of the $^{(3)}H_{ab}$ tensor and the corresponding non-local action is quite generic for massless fields; its coefficient $Q^2$ vanishes only if the mass is non-zero. Hence we should expect that, although they are certainly not conformal, gravitons will also contribute to this same infrared effective action with a finite value of $Q^2$, which can be determined in the same way by a background de Sitter calculation of their quantum $\langle T_{ab} \rangle$ at late times. We plan to present the results of this calculation in a future publication.
It is also interesting to note that the coefficient of the generalized trace anomaly, $Q^2$, is not generically positive, in contrast to all the previously known examples of massless conformal fields [@Duff]. It appears that the reason for this is that a positive $Q^2$ comes from the ultraviolet behavior of $\langle T_{ab}
\rangle$ for conformal fields, while the infrared behavior of non-conformal fields can contribute a negative value. For any $Q^2
\neq 0$ the new term in the effective action leads to dynamics for the RW scale factor which is quite different from the Einstein theory, and remains to be investigated in a full dynamical back-reaction calculation.
Several of us would like to thank V. Sahni for helpful discussions, and the Institute for Nuclear Theory, University of Washington, where some of this work was completed. P. R. A. would like to thank T-8, Los Alamos National Laboratory for its hospitality. This work was supported in part by grant numbers DMR-9403009 and PHY-9800971 from the National Science Foundation. It was also supported in part by contract number W-7405-ENG-36 from the Department of Energy.
See for example, N. D. Birrell and P. C. W. Davies, [*Quantum Fields in Curved Space*]{}, Cambridge University Press, Cambridge, England (1982), and references contained therein. F. Cooper and E. Mottola, Phys. Rev. D[**36**]{}, 3114 (1987); F. Cooper, S. Habib, Y. Kluger, E. Mottola, J. P. Paz, and P. R. Anderson, [*ibid*]{} [**50**]{}, 2848 (1994); F. Cooper, S. Habib, Y. Kluger, and E. Mottola, [*ibid*]{} [**55**]{}, 6471 (1997). S. Habib, Y. Kluger, E. Mottola, and J. P. Paz, [*Phys. Rev. Lett.*]{} [**76**]{}, 4660 (1996). R. M. Wald, Ann. Phys. (N.Y.) [**110**]{}, 472 (1978). A. A. Starobinsky, Phys. Lett. [**91B**]{}, 99 (1980). M. V. Fischetti, J. B. Hartle, and B. L. Hu, Phys. Rev. D[**20**]{}, 1772 (1979). P. R. Anderson, Phys. Rev. D[**28**]{}, 271 (1983); D[**29**]{}, 615 (1984); D[**32**]{}, 1302 (1985); D[**33**]{}, 1567 (1986). S. Wada, Phys. Rev. D[**31**]{}, 2470 (1985). W.-M. Suen and P. R. Anderson, Phys. Rev. D[**35**]{}, 2940 (1987). E. T. Tomboulis, Phys. Lett. [**70B**]{}, 361 (1977). B. Allen, Phys. Rev. D[**32**]{}, 3136 (1985). S. Habib, C. Molina-París, and E. Mottola, Phys. Rev. D[**61**]{}, 24010 (2000). N. A. Chernikov and E. A. Tagirov, Ann. Inst. Henri Poincare [**A9**]{}, 109 (1968). E. A. Tagirov, Ann. Phys. (N.Y.)[**76**]{}, 561 (1973). J. S. Dowker and R. Critchley, Phys. Rev. D[**13**]{}, 3224 (1976). T. S. Bunch and P. C. W. Davies, Proc. R. Soc. A[**360**]{}, 117 (1978). B. Allen and A. Folacci, Phys. Rev. D[**35**]{}, 3771 (1987). A. Folacci, J. Math. Phys. [**32**]{}, 2813 (1991); [*ibid*]{} [**32**]{}, 2828 (1991). K. Kirsten and J. Garriga, Phys. Rev. D[**48**]{}, 567 (1993). I. Antoniadis, P. O. Mazur, and E. Mottola, Phys. Rev. D[**55**]{}, 4770 (1997). C. W. Misner, K. S. Thorne, and J. A. Wheeler, [*Gravitation*]{} (Freeman, San Francisco, 1973). T. S. Bunch, J. Phys. A: Math. Gen. [**13**]{}, 1297 (1980). L. Parker, Ph. D. thesis, Harvard University, 1966 (Xerox University Microfilms, Ann Arbor, Michigan, No. 73-31244). L. Parker and S. A. Fulling, Phys. Rev. D [**9**]{}, 341 (1974). S. A. Fulling and L. Parker, Ann. Phys. (N. Y.) [**87**]{}, 176 (1974). S. A. Fulling, L. Parker, and B. L. Hu, Phys. Rev. D [**10**]{}, 3905 (1974). P. R. Anderson and L. Parker, Phys. Rev. D[**36**]{}, 2963 (1987). N. D. Birrell, Proc. R. Soc. London. B[**361**]{}, 513 (1978). C. Lüders and J. E. Roberts, Comm. Math. Phys. [**134**]{}, 29 (1990). A.-H. Najmi and A. C. Ottewill, Phys. Rev. D[**32**]{}, 1942 (1985). W. Junker, DESY-95-144, [*Doctoral Thesis*]{}, hep-th/9507097. J. Lindig, Phys. Rev. D[**59**]{}, 064011 (1999). P. R. Anderson and W. Eaker, Phys. Rev. D [**61**]{} 024003 (2000). I. S. Gradshteyn and I. M. Ryzhik, [*Table of Integrals, Series, and Products*]{} (Academic Press, New York, 1980). , Vol. I, Bateman Manuscript Project, A. Erdelyi, ed., McGraw-Hill, New York (1953). V. Sahni and S. Habib, Phys. Rev. Lett. [**81**]{}, 1766 (1998). L. P. Grishchuk, Zh. Eksp. Teor. Fiz [**67**]{}, 825 (1974) (Sov. Phys. [*JETP*]{}, [**40**]{}, 409 (1975)). A. D. Dolgov, in [*The Very Early Universe*]{}, edited by G. W. Gibbons, S. W. Hawking, and S. T. C. Siklos (Cambridge University Press, Cambridge, 1983). L. H. Ford, Phys. Rev. D[**35**]{}, 2339 (1987). P. C. W. Davies, S. A. Fulling, S. M. Christensen, and T. S. Bunch, Ann. Phys. [**109**]{}, 108 (1977). L. S. Brown and J. P. Cassidy, Phys. Rev. D[**16**]{}, 1712 (1977). B. Allen, Nucl. Phys. B[**226**]{}, 228 (1983); P. O. Mazur and E. Mottola, Nucl. Phys. B[**278**]{}, 694 (1986). I. Antoniadis, P.O. Mazur, and E. Mottola, Nucl. Phys. B[**388**]{}, 627 (1992). M. J. Duff, Nucl. Phys. B[**125**]{}, 334 (1977).
The cases of imaginary and integer values of $\nu$
==================================================
In this appendix we show that the proof in Section III works for imaginary values of $\nu$ as well as for $\nu = 0,1$. For imaginary values of $\nu$ it is useful to write $\nu = i \gamma$ with $\gamma$ a positive real number. Then the formulas in Eq. (\[eq:I00\]) and (\[eq:I\]) are the same as before. However the values of the $A_i$ are different. They are given by $$\begin{aligned}
A_1 &=& \frac{\pi}{2 k} e^{-\gamma \pi} \left[{\rm coth}(\gamma\pi)
{\rm csch}(\gamma\pi) ((1+2 n_k)|c_2|^2 + n_k) - \frac{1}{2}
{\rm coth}(\gamma\pi)
{\rm csch}(\gamma\pi) (1+ 2n_k) (c_1 c_2^* + c_1^* c_2) \right.
\nonumber \\
& & \left. - \frac{1}{2} {\rm csch}(\gamma\pi) (1+ 2n_k)
(c_1 c_2^* - c_1^* c_2) \right]
\; ,
\nonumber\\
A_2 &=& -\frac{\pi}{k} e^{-\gamma \pi} \left[(1+ {\rm csch}^2
(\gamma\pi)((1+2 n_k)
|c_2|^2 + n_k) - \frac{1}{2} {\rm csch}^2(\gamma\pi) (1+ 2n_k)
(c_1 c_2^* + c_1^* c_2) \right]
\; ,
\nonumber\\
A_3 &=& \frac{\pi}{2 k} e^{-\gamma \pi} \left[{\rm coth}(\gamma\pi)
{\rm csch}
(\gamma\pi)
((1+2 n_k)|c_2|^2 + n_k) - \frac{1}{2} {\rm coth}(\gamma\pi)
{\rm csch}(\gamma\pi)
(1+ 2n_k) (c_1 c_2^* + c_1^* c_2) \right. \nonumber \\
& & \left. + \frac{1}{2} {\rm csch}(\gamma\pi) (1+ 2n_k)
(c_1 c_2^* - c_1^* c_2) \right] \; .
\label{eq:A123im}\end{aligned}$$ The last terms in the expressions for $A_1$ and $A_3$ are imaginary. Substitution into Eqs. (\[eq:I00\]) and (\[eq:I\]) shows that the contributions of these terms to the energy-momentum tensor cancel.
Noting that $\nu$ is imaginary, one sees that the argument for the vanishing of the first integral in Eq. (\[eq:thrI\]) is unchanged. Since the terms in the integrand are being bounded for the second integral one must take the real part of $\beta_i$ and substitute that for $\beta_i$ in Eqs. (\[eq:J12\]) and (\[eq:J12value\]). After making this substitution it is still the case that each term in (\[eq:J12value\]) vanishes in the limit $\eta \rightarrow 0^-$. The argument for the vanishing of the third integral in Eq.(\[eq:thrI\]) is unchanged when $\nu$ is imaginary. Thus, the proof is valid for imaginary values of $\nu$.
For $\nu = n= 0,1$ we use the identity $$\begin{aligned}
H_n^{(1)}(z) = J_n(z) + i N_n(z)
\; ,
\nonumber \\
H_n^{(2)}(z) = J_n(z) - i N_n(z)
\; ,
\label{eq:h12jn}\end{aligned}$$ and then use the well known power series solutions for $N_0$ and $N_1$ to write $$N_n(z) = \frac{2}{\pi} J_n(z) \log(z) + P_n(z)
\; ,$$ where $P_n(z)$ is a series of the form $$P_n(z) = \sum_{j=0}^{\infty} b_{nj} z^{2j-n} \; .$$ Because of the $\log(z)$ term it is useful to write $$\frac{{\rm d} N_n(z)}{{\rm d} z} = \frac{2}{\pi}
\frac{{\rm d} J_n(z)}{{\rm d}z} \log(z) + Q_n(z)
\; ,$$ with $Q_n(z)$ a series of the form $$Q_n(z) = \sum_{j=0}^{\infty} \epsilon_{nj} z^{2j-n-1} \;.$$ We can conclude that Eqs. (\[eq:I00\]) and (\[eq:I\]) remain the same but with different expressions for $A_i$, $S_i$, and $\beta_i$. The new expressions for the $A_i$ are given by $$\begin{aligned}
A_1 &=& -\frac{\pi}{2k}
\left[\left(1 + \frac{4}{\pi^2} (\log z)^2 \right) ((1+2 n_k)|c_2|^2 + n_k)
+ \frac{1}{2} \left(1 - \frac{4}{\pi^2} (\log z)^2 \right)(1+2 n_k) (c_1 c_2^* + c_1^* c_2)
\right. \nonumber \\
& & \left. + \frac{2 i}{\pi} \log z \,(1+ 2n_k) (c_1 c_2^* - c_1^* c_2)\right]
\; ,
\nonumber \\
A_2 &=& -\frac{\pi}{k} \left[\frac{2}{\pi} \log z \, ((1+2 n_k)|c_2|^2 + n_k)
- \frac{1}{\pi} \log z \,(1+2 n_k) (c_1 c_2^* + c_1^* c_2)
+ \frac{i}{2} (1+ 2n_k) (c_1 c_2^* - c_1^* c_2)\right]
\; ,
\nonumber \\
A_3 &=& -\frac{\pi}{2k} \left[((1+2 n_k)|c_2|^2 + n_k)
- \frac{1}{2} (1+2 n_k) (c_1 c_2^* + c_1^* c_2)\right]
\;.
\label{eq:A123int}\end{aligned}$$ Note that the $A_i$ are now functions of both $k$ and $\log(z)$. The new expressions for the $\beta_i$ and $S_i$ are given in Table 2.
[|c||c|c|]{}
------------------------------------------------------------------------
$\;\;i\; \; $ &$\; \; \; \; \beta_i \; \; \; \; $&$\; \; \; S_i\; \; \;$\
------------------------------------------------------------------------
1 & $5 + 2n$ & $z^5 J_n^2(z)$\
------------------------------------------------------------------------
2 & $5 $ & $z^5 J_n(z) P_n(z)$\
------------------------------------------------------------------------
3 & $5 - 2n$ & $z^5 P_n^2(z)$\
------------------------------------------------------------------------
4 & $3 + 2n$ & $z^3 J_n^2(z)$\
------------------------------------------------------------------------
5 & $3 $ & $z^3
J_n(z) P_n(z)$\
------------------------------------------------------------------------
6 & $3 - 2n$ & $z^3 P_n^2(z)$\
------------------------------------------------------------------------
7 & $3 + 2n$ & $z^4 \frac{\rm d}{{\rm d}z} J_n^2(z)$\
------------------------------------------------------------------------
8 & $3 $ & $z^4
\left(\frac{\rm d}{{\rm d}z} J_n(z)\right) P_n(z) + J_n(z) Q_n(z))$\
------------------------------------------------------------------------
9 & $3 - 2n$ & $2 z^4 P_n(z) Q_n(z)$\
------------------------------------------------------------------------
10 & $3 + 2n$ & $z^5
\left(\frac{\rm d}{{\rm d}z} J_n(z)\right)^2 $\
------------------------------------------------------------------------
11 & $3 $ & $z^5
\left(\frac{\rm d}{{\rm d}z} J_n(z)\right) Q_n(z)$\
------------------------------------------------------------------------
12 & $3 - 2n$ & $z^5 Q_n^2(z)$\
[Table 2]{}
Substitution these expressions into Eqs. (\[eq:I00\]) and (\[eq:I\]) shows that for the first integral in (\[eq:thrI\]) the expressions are of the same form, except that some terms have factors of $\log(z)$. However these do not prevent the terms in the first integral on the right hand side of (\[eq:thrI\]) from vanishing asymptotically. For the second integral there are still terms of the form given in Eq. (\[eq:J12\]). However, some of them also have factors of $\log(z)$. Inserting factors of $\log(z)$ into Eq. (\[eq:J12value\]) and computing the integrals, one finds that the terms all vanish in the limit $\eta \rightarrow 0^-$. The factors of $\log(z)$ also do not affect the asymptotic vanishing of the third integral on the right in Eq. (\[eq:thrI\]).
Therefore, in all cases where $\Re (\nu) < {3\over 2}$ the quantity $\langle
T_{ab} \rangle_{SD}$ vanishes in the limit $\eta \rightarrow 0^-$.
The Harmonic Oscillator with $\omega \rightarrow 0$
===================================================
In this Appendix we give a detailed discussion of the harmonic oscillator with vanishing frequency, pointing out the analogy with the $\nu \rightarrow {3\over 2}$ limit in de Sitter space. Consider a simple harmonic oscillator with Hamiltonian $$H = {1\over 2}\dot \phi^2 + {1\over 2} \omega^2\phi^2\,.$$ The single degree of freedom $\phi$ can be quantized by introducing the operator representation $$\phi (t) = \langle\phi (t)\rangle + a\psi(t) + a^{\dagger}\psi^*(t)\,,$$ where $a^{\dagger}$ and $a$ are creation and destruction operators obeying $$[a, a^{\dagger}] = 1\,.$$ The canonical commutation relations $[\phi, \dot \phi] = i$ are satisfied provided the mode function $\psi$ obeys the Wronskian condition (\[eq:wronskian\]). The equation of motion for the Heisenberg operator $\phi (t)$ implies that the mode functions also satisfy $$\ddot \psi + \omega^2 \psi = 0\,.
\label{eom}$$ In order to satisfy the Wronskian condition we choose the fundamental normalized complex solution to this equation to be $$f (t)\equiv {1 \over \sqrt{2\omega}}\ e^{-i\omega t}\,.
\label{fmode}$$ The general solution satisfying both the Wronskian condition and (\[eom\]) can be written as the linear combination $$\psi (t) = \alpha f (t)+ \beta f^*(t)\,,$$ where $$|\alpha|^2 - |\beta|^2 = 1\,.$$ Thus, up to an irrelevant overall phase, the Bogoliubov coefficients can be parameterized in terms of two real parameters in the form $$\begin{aligned}
\alpha &=& \cosh\theta\,,\nonumber\\
\beta &=& \sinh\theta\ e^{-i\delta}\,.
\label{ABog}\end{aligned}$$
We could keep the c-number expectation value $\langle\phi(t)\rangle$ non-zero in general, which would lead to the general Gaussian state with displaced origin that we considered in some earlier papers [@HKMP]. Since we do not require it in our free field de Sitter calculation, we specialize to the case where $\langle\phi(t)\rangle =
\langle\dot\phi(t)\rangle = 0$ in the following. With this restriction the field operator can be written simply as $$\phi (t)= a\psi(t) + a^{\dagger}\psi^*(t)\,.$$ We make use of the freedom in the $\alpha$ and $\beta$ coefficients to require $$\begin{aligned}
\langle aa\rangle &=& \langle a^{\dagger} a^{\dagger}\rangle = 0
\; ,
\nonumber\\
\langle a a^{\dagger}\rangle &=& \langle a^{\dagger}a\rangle
+\ 1 = {\sigma + 1\over 2}\,,\end{aligned}$$ with no loss of generality. The positive real parameter $\sigma \ge 1$ has the interpretation $\sigma = 1 + 2 n$, with $n$ the average number of “particles" in the $a$ basis. Thus we need the three parameters $\theta$, $\delta$, and $\sigma$ to specify the general semi-classical (coherent) state of the field.
Using the definitions above it now follows that the average energy in this state is $$E = \langle H\rangle = {\omega\over 2}\ \sigma\ (1 + 2|\beta|^2)\,.$$ Note that we have [*not*]{} insisted that the state be an eigenstate of the number operator $a^{\dagger} a$ so $n$ is only the [*average*]{} number of particles in the state, which can take on any non-negative value and need not be an integer. If $n > 0$ and the state is a Gaussian, consistent with all connected correlations vanishing except the two-point function, then it turns out that the state is necessarily a mixed state [@HKMP].
We are interested in the nature of the “vacuum" state in the limit $\omega
\rightarrow 0$, [*i.e.*]{} when our single degree of freedom becomes that of a [*free*]{} particle. In this limit we can no longer retain the complex oscillating mode function basis since their normalization diverges due to the $(2\omega)^{-{1\over 2}}$ factor in (\[fmode\]). However, the mode equation $\ddot \psi =0$ clearly possesses the regular [*real*]{} solutions $u =1$ and $v=t$, which we can obtain from the complex oscillatory solutions by taking appropriate linear combinations in the limit of vanishing frequency, [*i.e.*]{} $$\begin{aligned}
u(t) &\equiv & \lim_{\omega \rightarrow 0} \left\{ \sqrt{\omega\over 2}
(f + f^*)\right\} = 1\,,\nonumber\\
v(t) &\equiv & i\ \lim_{\omega \rightarrow 0} \left\{{1\over \sqrt{2\omega}}
(f - f^*)\right\} = t\,.
\label{Auvdef}\end{aligned}$$ These definitions are analogous to Eq. (\[uvdeS\]) of the text, with $\omega$ replacing $3-2\nu$.
The general linear combination of modes can then be rewritten in the form $$\psi (t) = {(\alpha + \beta) \over \sqrt{2\omega}} u
- i {\sqrt \omega \over 2}(\alpha - \beta) v = Bu + Av \rightarrow At + B\,,$$ where the quantities $$\begin{aligned}
A &\equiv& -i\ \lim_{\omega \rightarrow 0} \left\{ \sqrt{\omega\over 2} (\alpha - \beta)
\right\}
\; ,
\nonumber\\
B &\equiv& \lim_{\omega \rightarrow 0} \left\{ {1\over\sqrt{ 2\omega}} (\alpha + \beta)
\right\}
\; ,
\label{ABcoef}\end{aligned}$$ are analogous to those defined by (\[ABdeS\]) of the text. They also satisfy $$A^*B - B^*A = i (|\alpha|^2 - |\beta|^2) = i\,.
\label{ABcond}$$ We can also define the time-independent Hermitian operators $$\begin{aligned}
Q &\equiv & {1\over\sqrt{ 2\omega}}\left[(\alpha + \beta)\ a +
(\alpha + \beta)^*\ a^{\dagger}\right] \rightarrow Ba + B^* a^{\dagger}\,,\nonumber\\
P &\equiv & -i \sqrt{\omega\over 2}\ \left[(\alpha - \beta)\ a
- (\alpha - \beta)^*\ a^{\dagger}\right] \rightarrow Aa + A^* a^{\dagger}\,,\end{aligned}$$ analogous to (\[eq:QPdef\]) in the de Sitter case. They obey the canonical commutation relations $$[Q, P] = i\,,
\label{com}$$ so that in the limit $\omega \rightarrow 0$ $$\phi (t) \rightarrow u(t) Q + v(t)P = Q + tP\,,$$ and in the same limit $$E = {1\over 2} \langle P^2\rangle = {\sigma \over 4}\lim_{\omega \rightarrow 0}
\left\{\omega \ |\alpha - \beta|^2\right\} = {\sigma \over 2} |A|^2\,.
\label{ener}$$ Since $\sigma$ is just an overall factor and we are interested in pure vacuum-like states, we can set $\sigma = 1$.
We see that depending on exactly how we take the limit $\omega
\rightarrow 0$, we can get many different values for the energy. If we tried to keep the pure positive frequency solution $\psi = f$, [*i.e.*]{} $\alpha =1$ and $\beta = 0$, then the energy would vanish in the limit $\omega \rightarrow 0$. However, by computing the correlation function, $\langle \phi (t)\phi(t')\rangle$, we would find that the correlator [*diverges*]{} in this limit. In our simple example it is clear why. The ordinary ground state vacuum wave function of the simple harmonic oscillator is a Gaussian with a width proportional to $\omega^{-{1\over 2}}$. Hence $\langle Q^2\rangle \sim
\omega^{-1}$ in this state, which becomes non-normalizable in the limit, unless we were to put the particle in a box of finite volume. But this option is unavailable to us if the “particle” is a mode of a field and the range of the field variable $\phi$ is $(-\infty, \infty)$. Thus, we must reject the positive frequency harmonic oscillator vacuum in the limit $\omega \rightarrow 0$ since it is a non-normalizable state.
In order for the state to remain normalizable and the mode functions bounded, the quantities $A$ and $B$ must remain finite in the limit of vanishing $\omega$. Inspection of the definitions (\[ABcoef\]) shows that this requires that the original Bogoliubov coefficients $|\alpha|$ and $|\beta|$ go to infinity. In fact, using (\[ABog\]) in the definitions of $A$ and $B$ a simple calculation shows that the Bogoliubov parameters $\theta$ and $\delta$ must behave like $$\begin{aligned}
\theta &\rightarrow &-{1\over 2}\log \left({\omega\over 2}\right) +
\log \vert A\vert\,,\nonumber\\
\delta &\rightarrow &\pi + \omega T
\; ,\end{aligned}$$ as $\omega \rightarrow 0$, so that $$\begin{aligned}
A &=& -i\vert A \vert\,,\qquad {\rm and}\nonumber\\
B &=& {1\over 2 \vert A \vert} + {i\over 2} \vert A\vert\ T
\; ,\end{aligned}$$ remain finite in that limit. Thus, the two complex numbers $A$ and $B$ obeying (\[ABcond\]) and the state they characterize can be specified by the two real numbers $|A| \ge 0$ and $T$, up to an unobservable overall phase. Comparison of these expressions with the corresponding equations (4.7) and (4.13) in the paper by Allen and Folacci [@a-f] shows that our $A$ here is equivalent to Allen and Folacci’s $2A$. This is because of a factor of 2 difference in the normalization of their time dependent mode $v(t)$.
Since $\theta \rightarrow \infty$, in a sense the normalizable “vacuum” state with finite $\langle P^2\rangle = |A|^2$ and finite $\langle Q^2\rangle = |B|^2$ is infinitely far from the usual ground state vacuum of the harmonic oscillator, and is [*not*]{} an eigenstate of the free particle Hamiltonian at $\omega = 0$. Instead, it is just a Gaussian state centered at $\langle Q\rangle = 0$ and $\langle P \rangle = 0$ with normalized wave function [@HKMP] $$\Psi (q) = {1 \over (2\pi |B|^2)^{1\over 4}}\exp
\left( - {|A|\over 2 B^*}\ q^2 \right)
\; ,
\label{gaus}$$ in the position representation where $Q\Psi = q\Psi $ and $P\Psi = -i
{{\rm d}\over {\rm d}q}\Psi$. This state is time reversal invariant if and only if $T = 0$, in which case $B = {1 \over 2 |A|}$ becomes real. On the other hand, an eigenstate of the free Hamiltonian $P^2/2$ is necessarily non-normalizable since diagonalizing $P$ requires infinite spread in $Q$. These remarks carry over equally well to the $k=1$ mode in the de Sitter case.
There is no reason to diagonalize $H$ in the field theory case as Allen and Folacci do in their Eqs. (4.26) to (4.28) [@a-f], since the energy-momentum tensor is time dependent in general, and back-reaction has been neglected in our simple calculations. Diagonalizing just the matter Hamiltonian without taking into account the metric is no better than a leading order mean field approximation to the much more difficult full problem with metric fluctuations taken into account. Given that what we are doing is only a semi-classical approximation to this full problem in any case, what makes the most sense from our perspective is to take well-defined normalizable initial adiabatic states for the scalar field and allow them to evolve as they will. Our initial state trial wave functional (\[gaus\]) then remains a bounded Gaussian functional, consistent with semi-classical mean field methods (although it may well spread over time depending on the dynamics).
As should be clear, the non-normalizable positive frequency vacuum with $\alpha =1$ and $\beta =0$ is analogous to the Bunch-Davies vacuum, while the normalizable state related to it by an infinite Bogoliubov transformation (but only a finite shift in zero point energy) is analogous to the Allen-Folacci states parameterized by $A$ and $B$ or $|A|$ and $T$. However, whereas the energy of the normalizable state is finite and arbitrary, depending on the arbitrary $|A|^2$ coefficient in (\[ener\]), and the non-normalizable positive frequency state has zero energy in this simple harmonic oscillator example, the converse is true at late times in de Sitter space. This is due to the different kinematics and form of the energy-momentum tensor of the scalar field in the de Sitter case. Comparison of the energy (\[ener\]) of the harmonic oscillator example and the corresponding $k=1$ late time energy density in the de Sitter case with $m=0$ and $\xi$ small, [*viz.*]{} $$\varepsilon_1 \rightarrow {1\over 2a^2} \left({{\rm d}\phi_1\over {\rm d}\eta}
\right)^2
+ 3 \xi \phi_1^2 = {1\over 4\pi^2 a^2}\left({{\rm d}v\over {\rm d}\eta}
\right)^2
P^2 + {3 \xi\over 2 \pi^2} (Q + vP)^2
\; ,
\label{eq:epsosc}$$ shows that the difference is the redshift factor of $a^{-2}$ multiplying the infrared finite $\langle P^2\rangle$ in the de Sitter case. Thus, whereas this can take on the finite state dependent value $|A|^2$ in the harmonic oscillator example, its asymptotic late time value vanishes in de Sitter space when multiplied by $a^{-2}$, and becomes state independent in any infrared finite state. Further, in the simple harmonic oscillator the usual ground state becomes an eigenstate of $P$ and has zero energy as $\omega \rightarrow 0$, since $\omega^2 \langle (Q+vP)^2 \rangle \sim
\omega \rightarrow 0$ drops out of the energy in this limit, whereas in the de Sitter case, since $\xi$ goes to zero only [*linearly*]{} with $3 - 2 \nu$, the Bunch-Davies expectation value receives a finite contribution from the $k=1$ mode from the infinite spread in $\xi
\langle Q^2 \rangle$ which has no $a^{-2}$ redshift factor in (\[eq:epsosc\]).
[^1]: electronic address: anderson@wfu.edu
[^2]: electronic address: eaker@alumni.wfu.edu
[^3]: electronic address: habib@lanl.gov
[^4]: electronic address: carmen@t6-serv.lanl.gov
[^5]: electronic address: emil@lanl.gov
[^6]: In Ref. [@bu-da] the arguments of the Hankel functions are given as $k\eta$ rather than $-k\eta$. We have chosen to use non-negative arguments to avoid complications that result from the fact that these functions have branch cuts along the negative real axis.
[^7]: In fact, the requirement of fourth order adiabatic states is slightly more restrictive than UV finiteness of the energy-momentum tensor in de Sitter space, since $C(k)$ can go to a non-zero constant at large $k$ and the mode sums still converge. This is associated with the vanishing of the tensor, $^{(1)}H_{ab}$, defined in Eq. (\[eq:Hone\]) in de Sitter space. If the field is non-conformally coupled, a state with $C(k) \rightarrow $ constant would lead to a divergent energy-momentum tensor if the spacetime is not exactly de Sitter. If the field is conformally coupled and the spacetime is not exactly homogeneous and isotropic, then the energy-momentum tensor would again be divergent. Thus the most general physically acceptable UV states are fourth order adiabatic states.
|
---
abstract: 'Using first-principles calculations based on density functional theory, we have studied the mechanical, electronic, and magnetic properties of Heusler alloys, namely, Ni$_{2}BC$ and Co$_{2}BC$ ($B$ = Sc, Ti, V, Cr and Mn as well as Y, Zr, Nb, Mo and Tc; $C$ = Ga and Sn). On the basis of electronic structure (density of states) and mechanical properties (tetragonal shear constant), as well as magnetic interactions (Heisenberg exchange coupling parameters), we probe the properties of these materials in detail. We calculate the formation energy of these alloys in the (face-centered) cubic austenite structure to probe the stability of all these materials. From the energetic point of view, we have studied the possibility of the electronically stable alloys having a tetragonal phase lower in energy compared to the respective cubic phase. A large number of the magnetic alloys is found to have the cubic phase as their ground state. On the other hand, for another class of alloys, the tetragonal phase has been found to have lower energy compared to the cubic phase. Further, we find that the values of tetragonal shear constant show a consistent trend : a high positive value for materials not prone to tetragonal transition and low or negative for others. In the literature, materials, which have been seen to undergo the martensite transition, are found to be metallic in nature. We probe here if there is any Heusler alloy which has a tendency to undergo a tetragonal transition and at the same time possesses a high spin polarization at the Fermi level. From our study, it is found that out of the four materials, which exhibit a martensite phase as their ground state, three of these, namely, Ni$_{2}$MnGa, Ni$_{2}$MoGa and Co$_{2}$NbSn have a metallic nature; on the contrary, Co$_{2}$MoGa exhibits a high spin polarization.'
author:
- 'Tufan Roy$^{1}$, Dhanshree Pandey$^{1}$ and Aparna Chakrabarti$^{1,2}$'
title: 'Probing the possibility of coexistence of martensite transition and half-metallicity in Ni and Co-based full Heusler Alloys : An ab initio Calculation'
---
Introduction
============
Heusler alloys are intermetallic compounds with interesting fundamental properties and possible practical applications. Heusler alloys are typically known to be of two types: full-Heusler alloys (FHA) and half-Heusler alloys (HHA). The full-Heusler alloys, which are having a formula $A_{2}BC$ (with a stoichiometry 2:1:1, e.g. Ni$_{2}$MnGa), commonly exhibit a $L2_{1}$ structure. This has four interpenetrating face-centered (fcc) sub-lattices, for each of the atoms $A$(Ni), $A$(Ni), $B$(Mn) and $C$(Ga). On the other hand, the� half-Heusler alloys with a formula $ABC$ (with a stoichiometry 1:1:1, e.g. NiMnSb) typically assume a $C1_{b}$ structure where one of the four fcc sub-lattices remains unoccupied. Since the discovery of the prototype FHA, Ni$_{2}$MnGa[@PMB49PJW], various studies have been carried out which show that this alloy, having long-range ferromagnetic interaction, possesses various interesting physical properties. For example, Ni$_{2}$MnGa exhibits magnetic field induced strain (MFIS) and magnetic shape memory alloy (MSMA) property[@Ullako96; @Murray00; @Sozinov02], magnetoresistance effect (MRE)[@APL86CB] as well as magnetocaloric effect (MCE)[@APL76FXH].
The Heusler alloys are interesting from both the points of view of possible technological application as well as fundamental science. Hence, these have been enjoying the attention of the researchers - theoreticians and experimentalists alike. Further, the basic drawbacks of the prototype Heusler alloy, Ni$_{2}$MnGa, in terms of technological application are its brittleness and the low martensite transition temperature. Therefore, following the discovery of and studies on Ni$_{2}$MnGa, various FHAs have been synthesized, characterized and studied in the last two decades, as is observed in the literature. Many new FHA materials, till date, have been predicted from ab initio calculations as well. It has been observed that the face-centered-cubic phase is the high-temperature or the so-called austenite phase of these materials. Some of these FHAs has been seen to undergo a tetragonal distortion at a lower temperature. This first-order displacive transition, generally known as martensite transition, where the volumes of the unit cell of both the austenite and martensite phases are close to each other, is typically connected to the SMA property exhibited by these alloys. Among the FHAs, in terms of the electronic structure, there are various categories. While some of these alloys prefer to be metallic, some are found to be semiconducting, and some are having a large spin-polarization at the Fermi level.[@PMB49PJW; @electronic-str; @PRL50RAdG; @PT54WEP]
If the FHAs are magnetic in nature, their properties can change when a magnetic field is applied which can be of interest in terms of potential technological application. Hence, specially, magnetic shape memory alloys are gaining increasing interest. Therefore, detailed studies of magnetic configurations, properties and interactions are of particular importance. In literature, various magnetic ground state configurations are observed in case of full-Heusler alloys. While some alloys are even non-magnetic, many of these exhibit a long-range ferromagnetic ordering and are expected to show MSMA property. In many of these MSMAs, there is presence of a delocalized-like common d-band formed by the d-electrons of the $A$ and $B$ atoms, which are both typically first-row transition metal atoms.[@PRB28JK] Additionally, there is also an indirect RKKY-type exchange mechanism[@RKKY], primarily mediated by the electrons of the $C$ atoms, which plays an important role in defining the magnetic properties of these materials.[@PRB28JK; @PRB77ES; @PMB49PJW; @PRB72ATZ; @JALCOM632TR] Further, it has also been observed that some of these alloys including Mn$_{2}$NiGa even show long-range ferrimagnetism and also anti-ferromagnetism.[@APL87liu; @EPL80bar; @PRB87KRP; @APL98IG; @JMMM401TR]
Hence, it is clear from the literature, that in terms of different physical, including, structural (mechanical), electronic and magnetic properties, the full-Heusler alloys show a rich variety. Further, as has been mentioned above, it has been of particular interest that out of all the full-Heusler alloys, only a few undergo the martensite transition. These alloys are prone to a cubic to tetragonal distortion when temperature is lowered and generally exhibit the technologically important SMA property. These FHAs in general are found to be metallic in nature. On the other hand, it has been observed that there is another group of full-Heusler alloys which are half-metallic-like in nature, with a much reduced density of states (DOS) at the Fermi level in case of one of the spin channels. These materials generally do not show the tendency of undergoing a tetragonal distortion and also showing the SMA property. However, an application in the field of spintronics is a possibility for these materials. From both the points of view of fundamental understanding as well as technological application, it can be interesting to probe the similarities and differences in magnetic, bulk mechanical, and electronic properties of these two categories of materials. It will also be interesting to see if there is any FHA which has a tendency to undergo a tetragonal transition and at the same time possesses a high spin polarization at the Fermi level.
Keeping this in mind, in the present paper, we probe the magnetic, bulk mechanical, and electronic properties of a series of Ni and Co-based full Heusler alloys using density functional theory (DFT) based ab initio calculations. The choice of these two systems (Ni and Co-based FHAs) is due to the following facts. First and foremost, it has been seen that typically, a large amount of work on the FHAs are on Ni and Co-based compounds. It is also seen in the literature that while most of the Ni-based FHAs show MSMA property, many of the Co-based FHAs exhibit large spin-polarization at the Fermi level. It has also been pointed out in the literature, that while the magnetic interactions are somewhat different in the Ni and Co-based FHAs, the total energy variation for an austenite to martensite phase transition is similar.[@AEM14MS] Hence, a comparative study may be interesting and also important for detailed understanding of the properties of these alloys. The primary interest is to study the possibility of the tetragonal transition versus a high spin polarization at the Fermi level. Further, we look for ferromagnetic materials so that realization of MSMA property is possible. In what follows, first, we give a brief account of the methods we used and then we present the results and discussion. In the end, the results of this work are summarized and conclusions are drawn.
Method
======
The full-Heusler alloys, as for example, Ni$_{2}$MnGa, commonly assume an ordered $A_{2}BC$ structure, where typically $A$, $B$ are elements with d-electrons and $C$ are elements with s,p electrons. In the cubic high-temperature austenite phase, Ni$_2$MnGa has a $L2_{1}$ structure that consists of four interpenetrating face-centered-cubic (fcc) sub-lattices with origin at fractional positions, (0.25,0.25,0.25), (0.75,0.75,0.75) (0.5,0.5,0.5), and (0.0,0.0,0.0). In $L2_1$ structure of Ni$_2$MnGa, the Ni atoms occupy the first and second sub-lattices. On the contrary, Mn and Ga occupy the third and fourth sub-lattices, respectively. In this paper, we have carried out calculations on Ni and Co-based systems. So Ni and Co are taken as $A$ atom and $C$ = Ga as well as Sn. As for the $B$ atom, we have taken into consideration and consequently tested the electronic stability of the first five atoms of the first as well as second rows of the transition metal atoms. Therefore, Sc, Ti, V, Cr and Mn as well as Y, Zr, Nb, Mo and Tc are considered as the $B$ atom.
The equilibrium lattice constants of all these alloys are obtained by full geometry optimization using Vienna Ab Initio Simulation Package (VASP)[@VASP] which has been used in combination with the projector augmented wave (PAW) method[@PAW] and the generalized gradient approximation (GGA) over the local density approximation (LDA) for the exchange-correlation functional.[@PBE] GGA is used because it accounts for the density gradients, and hence, for most of the Heusler alloy systems, it has been found that it yields results which are in better agreement with experimental data compared to the results of LDA. We have used an energy cutoff of 500eV for the planewaves. The final energies have been calculated with a $k$ mesh of 15$\times$15$\times$15 for the cubic case and a similar number for the tetragonal case. The energy and the force tolerance for our calculations were 10 $\mu$eV and 10 meV/Å, respectively. The formation energies ($E_{form}$), as calculated[@VASP] by the equation below, has been critically analysed to establish the electronic stability of the alloys. $$E_{form} = E_{tot} - \Sigma_{i} c_{i}E_{i}$$ where $i$ denotes different types of atoms present in the unit cell of the material system and $E_{i}$ are the standard state energies of the corresponding atoms $i$.[@VASP] The optimized geometries of the systems studied are compared with the results obtained in the literature, wherever results are available, and these match well with earlier data as discussed in the section on Results and Discussion.
The response of a material to an applied stress is associated with the elastic constants of the material. Both stress ($\sigma$) and strain ($\epsilon$) in a material have three tensile as well as three shear components. Therefore, the linear elastic constants form a 6$\times$6 symmetric matrix. We have $\sigma_{i}$ = $C_{ij}$ $\epsilon_{j}$ for small stresses, $\sigma$, and strains, $\epsilon$. Calculations of the mechanical properties of the materials involve the variation of total energy of the system induced by the strain.[@VASP] Elastic constants of all the materials are evaluated from the second derivative of the energy with respect to the strain tensor. The number of $k$-points and the energy cut-off have been increased from the values used in SCF calculations till the convergence of the mechanical properties of each individual material has been achieved. Mesh of $k$-points has been taken as 15$\times$15$\times$15 and energy cut-off of 500 eV as per the convergence requirement.
It is well known that, all-electron calculations are more reliable for the prediction of magnetic properties particularly for the systems containing first row transition elements. Hence, to calculate and understand in detail the magnetic as well as electronic properties, we have carried out relativistic spin-polarized all-electron calculations of all the systems, geometries being optimized by VASP.[@VASP] These calculations have been performed using full potential linearized augmented planewave (FPLAPW) program[@Wien2k] with the generalized gradient approximation (GGA) for the exchange-correlation functional.[@PBE] For obtaining the electronic properties, the Brillouin zone (BZ) integration has been carried out using the tetrahedron method with Blöchl corrections.[@Wien2k] An energy cut-off for the plane wave expansion of about 17 Ry is used ($R_{MT}$$K_{max}$ = 9.5). The cut-off for charge density is $G_{max}$= 14. The number of $k$ points for the self-consistent field (SCF) cycles in the reducible (irreducible) BZ is about 8000 (256) for the cubic phase and about 8000 (635) for the tetragonal phase. The convergence criterion for the total energy $E_{tot}$ is about 0.1 mRy per atom. The charge convergence is set to 0.0001.
To gain further insight into the magnetic interactions of some of the magnetic materials, we calculate and discuss their Heisenberg exchange coupling parameters. The geometries optimized by VASP have been used for these calculations. We use the Spin-polarized-relativistic Korringa-Kohn-Rostoker method (SPR-KKR) to calculate the Heisenberg exchange coupling parameters, J$_{ij}$, within a real-space approach, which is proposed by Liechtenstein et al[@JMMM67AIL] and implemented in the SPR-KKR programme package.[@sprkkr] The mesh of k points for the SCF cycles has been taken as 21$\times$21$\times$21 in the BZ. The angular momentum expansion for each atom is taken such that lmax=3. In addition, in terms of the Heisenberg exchange coupling parameters we derive the Curie temperature (T$_{C}$) following the literature[@JPCM23MM].
Results and Discussion
======================
Geometry Optimization and Electronic Stability
----------------------------------------------
[**Lattice parameter and Atomic number $Z$ of $B$ atom**]{} - For the cubic phase, the $L2_{1}$ structure has been assumed for all the structures studied here, namely, Ni$_{2}BC$ and Co$_{2}BC$ ($B$ = Sc, Ti, V, Cr and Mn as well as Y, Zr, Nb, Mo and Tc; $C$ = Ga and Sn). The geometry has been optimized to obtain the converged lattice parameter. Figure 1 and Figure 2 show the variation of this lattice parameter as a function of $Z$ of $B$ elements of $A_{2}BC$ alloy ($A$ = Ni, Co; $C$ = Ga, Sn). The $B$ atoms correspond to the period IV of the periodic table (first row transition metal atoms; Sc etc) and the period V (second row transition metal atoms; Y etc). Therefore, in Figures 1 and 2, we have mentioned the period numbers IV and V in the legends. For the Ni$_{2}BC$ materials, it is observed, as the atomic number of $B$ elements increases, with a saturating trend towards higher $Z$, lattice parameters of the materials decrease for a fixed row of the periodic table (Figure 1). This may happen due to the increasing electro-negativity and decreasing atomic radius of atoms across the column. Further, for these materials we observe (in the left panel of Figure 1) a sudden increase in the lattice parameter value for $Z$ of $B$ = 24 (i.e. Cr atom). It is to be noted that out of all the materials studied here, a deviation from the ferromagnetic nature is expected for Ni$_{2}$CrSn as well as Ni$_{2}$CrGa which was reported earlier.[@JMMM401TR] These two materials have lower energy for a long-range [*intra-sublattice*]{} anti-ferromagnetic (AFM) ordering compared to the ferromagnetic (FM) one. This difference in the long-range magnetic interaction between the FHAs with $B$ = Cr and the other $B$ atoms indicates that the Cr-based materials are possibly of a different class compared to the rest of the FHAs, considered here. This may be the reason behind the deviation from the observed trend. It is to be further noted that for the AFM ordering the lattice parameter (shown by a black square) is even larger compared to the corresponding FM phase. However, the long-range magnetic interaction in all the materials with $A$ = Co is ferromagnetic. For these Co-based materials the lattice parameters show a smooth decrease as we increase the $Z$ of the $B$ atom (Figure 2). As discussed above for Ni-based systems, this may, again, be due to the increase of electronegativity and decrease of atomic radius across the column which may have led to the shrinkage of the electron cloud around the $B$ atom, and consequently, of the whole unit cell. It has been observed that there is an increase in the lattice parameter values when we go from period IV to period V, for both Ni and Co-based materials. Increase of atomic radius is observed across the row, going from period IV to V, and the above-mentioned trend may be because of that.
[**Formation energy and Atomic number $Z$ of $B$ atom**]{} - Figure 3 as well as Figure 4 suggest that the formation energy is negative for all of the materials with $C$ atom = Ga, except Ni$_{2}$TcGa which is having marginally positive formation energy. It is to be noted that Co$_{2}$TcGa has marginally negative formation energy and hence in reality may not be stable electronically. It is well-known that negative formation energy signifies stability; more negativity indicates more stability of the material. A few of the materials, $A_{2}BC$, where $C$ = Sn, like Ni$_{2}$MoSn, Ni$_{2}$TcSn, Co$_{2}$CrSn, Co$_{2}$MoSn and Co$_{2}$TcSn, are having positive formation energy. These calculated values and the prediction that these particular materials are electronically unstable, matches with the results wherever available in the literature.[@MGThesis] The AFM phases for Ni$_{2}$CrGa and Ni$_{2}$CrSn have very close but slightly smaller $E_{form}$ compared to the respective FM case. It is observed from both the figures that, overall, there is a trend of electronic stability decreasing as $Z$ of $B$ atom increases. However, for $B$ = Mn, the stability has increased compared to the previous $B$ element. This suggests an interesting preferance for the Mn atom in the $B$ position for the Heusler alloys with $L2_{1}$ structure in both the cases when $A$ atom is Ni or Co. Similar is the case with $B$ = Zr. It is seen that, for Y, i.e. the first atom of the second row of the transition metal atoms (period V) at the $B$ position, the formation energy is somewhat unfavorable compared to the next case of $B$ = Zr. The origin of this has been found to be electronic in nature - analysis of the density of states for $B$ = Zr indicates a lowering of binding energy in this material compared to the $B$ = Y case. It is observed that the contribution from the majority spin density of states of the Co atom plays an important role. For our further studies, we concentrate only on the materials which, from our calculations, are found to be electronically stable.
[**Electronic Stability of the Tetragonal phase**]{} - We calculate the difference between the energy of the cubic ($E_{C}$) and tetragonal ($E_{T}$) phases of all the electronically stable materials. Figure 5 shows this energy difference, $\Delta E$ = $E_{T}$ - $E_{C}$ (in units of meV per atom), of some typical materials as a function of the ratio of lattice constants $c$ and $a$. From our calculations, we find that only a few materials, among all the electronically stable and magnetic FHAs we study here, exhibits the tetragonal phase as a lower energy state. Among all the Ni-based alloys, we find that Ni$_{2}$MoGa and Ni$_{2}$CrSn possess a lower energy for the tetragonal phase over the cubic phase similar to Ni$_{2}$MnGa and Ni$_{2}$CrGa, both of which have a tetragonal ground state, as has already been shown in the literature. From Figure 5, we also observe that Ni$_{2}$VGa and Ni$_{2}$VSn have very flat energy curves with no clear minimum in the $\Delta E$ versus $c$/$a$ curve. Further, we observe that though in case of Ni$_{2}$MnSn, there is a cubic ground state, there is also a very subtle signature of a tetragonal phase which is evident from the clear asymmetric nature of the $\Delta E$ versus $c$/$a$ curve for this material (Figure 5). It is observed that Ni$_{2}$MoGa exhibits a non-magnetic state as ground state. Further, Ni$_{2}$CrGa and Ni$_{2}$CrSn are likely to possess an [*intra-sublattice*]{} AFM phase as a ground state. It is to be noted that, in this paper, our concentration will be only on the alloys which will have FM phase in its ground state. So, for Ni-based alloys, further on, we will discuss only about Ni$_{2}$MnGa and Ni$_{2}$MnSn.
Out of all the Co-based alloys studied here, we observe that only two alloys are likely to show energetically lower martensite phase over the cubic austenite phase. Out of these two, while Co$_{2}$NbSn is known in the literature[@JPSJ58SF], Co$_{2}$MoGa is not reported till date. From Figure 5, it is clearly seen that for Co$_{2}$MoGa a significant energy difference exists between the cubic and tetragonal phases. This indicates that a martensite phase transition is possible in this material. It is expected that the martensite transition temperature for Co$_{2}$MoGa will be higher than Ni$_{2}$MnGa since the energy difference $\Delta E$ between the austenite and martensite phase of the former is evidently much larger than that of the latter. This expectation is because it is argued in the literature, that, relative values of $\Delta E$ can be indicative of the relative values of martensite transition temperatures of two alloys.[@AEM14MS; @PRB78SRB] A cubic ground state is observed for many Co-based alloys, including the two well-known Co-based half-metallic-like materials Co$_{2}$MnSn[@JALCOM645MB] and Co$_{2}$MnGa[@PRB71YK]. It is interesting to note that there are three more alloys which show a state of cubic symmetry having a lowest energy while the tetragonal phases of these materials are energetically very close (within 25 meV) to the respective austenite phases. The $\Delta E$ versus $c$/$a$ plots of these materials, namely, Co$_{2}$VGa, Co$_{2}$CrGa and Co$_{2}$TcGa are included in Figure 5 which clearly depict this energetic aspect.
Table 1. Martensite transition temperature of the four ferromagnetic materials[^1]
Material $(c/a)_{eq}$ $\Delta E$(meV/atom) $T_{M}$(K) $|\Delta{V}|$/V( %)
-------------- ------------------------ ---------------------- ------------------------ ---------------------
Ni$_{2}$MnGa 1.22 6.05 70.18 0.30
$1.22^{b}$, $1.22^{c}$ $6.18^{b}$ $70.51^{b}$, $210^{d}$
Ni$_{2}$MoGa 1.27 11.08 128.58 0.36
Co$_{2}$MoGa 1.37 19.58 227.13 1.96
Co$_{2}$NbSn 1.11 10.63 123.33 0.08
$233^{e}$
In Table 1 we report the tetragonal transition temperature ($T_{M}$) of those materials which are expected to exhibit tetragonal transition. These are calculated based on $\Delta E$ using the conversion factor 1 meV = 11.6 K. As discussed above, these $T_{M}$ values are not to be considered as the absolute values of the transition temperature. These values are listed here to only give a trend of the relative transition temperatures for different materials as has been done in the literature earlier.[@AEM14MS; @PRB78SRB] Out of the four ferromagnetic materials (magnetic aspect of the materials is discussed in detail in the next subsection) showing the possibility of a tetragonal transition, we find that Ni$_{2}$MnGa has the lowest $T_{M}$ value; on the other hand, Co$_{2}$MoGa is expected to have the highest transition temperature. The optimized $c$/$a$ values for all the four materials are given in Table 1. Co$_{2}$MoGa is found to have the highest value of about 1.4. Furthermore, volume conservation between the cubic and the tetragonal phases (Table 1) as well as an energetically lower tetragonal phase (Figure 5) which are observed here are generally indicative of the martensite transition. Therefore, from the present study, among the materials studied here, two Ni-based and two Co-based FHAs are likely to exhibit MSMA property. Out of which one Ni and one Co-based alloys are well-known MSMA materials, namely Ni$_{2}$MnGa and Co$_{2}$NbSn. On the other hand, Co$_{2}$MoGa and Ni$_{2}$MoGa are two new materials which are also likely to exhibit martensite transition.
Magnetic Properties
-------------------
Table 2. Magnetic properties of austenite phase of Co-based materials; $Z_{t}$ is the total number of valence electrons[^2]
Material $\mu_t$ $Z_{t}$-24 $\mu_A$ $\mu_B$ $\mu_C$ $T_C$(K) P(%)
-------------- --------------------------------- ------------ --------------------------------- ------------ --------------- --------------------------- ------
Co$_{2}$ScGa 0.00 0 0.00 0.00 0.00 - -
0.25$^{b}$
Co$_{2}$TiGa 1.00 1 0.62 $-$0.14 $-$0.01 161 97
0.75$\pm 0.03$$^{c}$,0.82$^{d}$ 0.40$^{b}, $0.40$\pm 0.1$$^{c}$ 130$\pm3$$^{c}$,128$^{d}$
Co$_{2}$VGa 2.00 2 0.95 0.18 $-$0.01 356 100
1.92$^{e}$, 2.00$^{f}$ 0.91$^{f}$ 352$^{e}$, 368$^{f}$
Co$_{2}$CrGa 3.03 3 0.77 1.59 $-$0.05 418 92
3.011$^{g}$, 3.01$^{g}$ 0.90$^{g}$ 1.28$^{g}$ $-$0.07$^{g}$ 419$^{g}$, 495$^{g}$
Co$_{2}$MnGa 4.10 4 0.77 2.73 $-$0.07 586 68
4.04$^{h}$ 700$^{h}$
Co$_{2}$YGa 0.00 0 0.00 0.00 0.00 - -
Co$_{2}$ZrGa 1.00 1 0.61 $-$0.11 $-$0.01 166 94
Co$_{2}$NbGa 2.00 2 1.04 $-$0.01 0.00 397 100
1.39$^{i}$, 2.00$^{j}$
Co$_{2}$MoGa 2.93 3 1.22 0.51 $-$0.01 180 86
Co$_{2}$TcGa 3.95 4 1.37 1.26 $-0.04$ 711 71
Co$_{2}$ScSn 1.05 1 0.67 $-$0.14 $-$0.02 207 80
Co$_{2}$TiSn 2.00 2 1.07 $-$0.06 0.00 409 100
1.96$^{e}$ 371$^{e}$
Co$_{2}$VSn 3.00 3 1.08 0.89 $-$0.02 291 100
1.21$^{e}$, 1.80$^{f}$ 95$^{e}$, 103$^{f}$
Co$_{2}$MnSn 5.03 5 0.98 3.23 $-$0.06 897 76
5.02$^{f}$ 0.885$^{f}$ 3.25$^{f}$ 899$^{f}$
Co$_{2}$YSn 1.05 1 0.67 $-$0.10 $-$0.02 162 79
Co$_{2}$ZrSn 2.00 2 1.10 $-$0.09 0.01 449 100
1.46$^{e}$ 448$^{e}$
Co$_{2}$NbSn 1.98 3 0.97 0.07 0.01 37 -66
0.69$^{e}$ 105$^{e}$
[**Total and Partial Magnetic Moments**]{} - After analysing in detail the electronic stability of the austenite phase as well as the relative energetics of the martensite phase of the materials, we now discuss in detail the magnetic properties of the cubic phase of the Ni and Co-based materials, only which are ferromagnetic in nature. The materials which are electronically unstable are not discussed further as well. As observed earlier[@JMMM401TR] for Ni$_{2}$CrGa, it is seen that Ni$_{2}$CrSn too is expected to show [*intra-sublattice*]{} anti-ferromagnetism; though in both cases, partial moment on Cr atom is seen to be significant. Further, a first-principles calculation by Sasioglu et al on a series of materials, like Pd$_{2}$MnZ, Cu$_{2}$MnZ, show that the magnetic moment is mainly confined on the Mn sublattice for these alloys which contain Mn as the $B$ atom, and a very small moment is induced on the Pd or Cu atom.[@PRB77ES] Similarly, in the Ni-based materials studied here, the magnetism in this class of materials is expected to arise primarily due to the $B$ element. This is because, by itself, Ni carries a very small magnetic moment and, Ga as well as Sn are having almost zero moment. Often the moment of the $A$ atom is seen to be strongly influenced by the $B$ atom as is observed in the literature[@PRB77ES; @PRB88AC] as well as for the Ni-based alloys studied here, except the $B$ = Mn materials, namely, Ni$_{2}$MnGa and Ni$_{2}$MnSn. Ni$_{2}$VSn, which has a very flat minimum in the $\Delta E$ versus $c$/$a$ curve, has a very small moment as well. We find that, out of all the Ni-based materials, only two Ni$_{2}BC$ alloys, namely, Ni$_{2}$MnGa and Ni$_{2}$MnSn exhibit the FM nature and they have very similar total moments. While the total moment is 4.10 $\mu_{B}$ in the former, it is 4.09 $\mu_{B}$ in the latter. Each of Ga and Sn has negligible moments in both cases. As for Ni atom, the moment on this atom is larger in the former alloy (0.36) compared to the latter material (0.24). Further, it is noted that Mn atom has a much larger moment in case of Ni$_{2}$MnSn (3.64) compared to the case of Ni$_{2}$MnGa (3.41). Though the common $B$ atom has a difference of moment of 0.23 $\mu_{B}$, due to the reasonably lower moment of the Ni atom in case of Ni$_{2}$MnSn, the total moments of these two materials are found to be very close to each other.[@PRB70ES; @JPCM11AA] These findings can be supported by the observed larger lattice parameter of Ni$_{2}$MnSn compared to that of Ni$_{2}$MnGa. Due to the larger lattice parameter in the former, the delocalization of the $3d$ electrons of Mn atom is expected to decrease, leading to a larger and more atomic-like partial moment on the same. The larger lattice constant in Ni$_{2}$MnSn leads to the decrease in the overlap between the Mn and Ni atoms, as is evident from the relative DOS too, as discussed later. This may be the likely reason as to why the moment of the Ni atom decreases in case of Ni$_{2}$MnSn in comparison to Ni$_{2}$MnGa, as discussed in the literature.[@JPCM11AA]
Table 2 gives the total and partial moments for the electronically stable Co-based materials. The values available from the literature are also listed in the table for a few materials, wherever available and we note that the matching is very good with the existing calculated data. With the experimental results the matching is reasonably good. It is seen that each of Ga and Sn has negligible moments in all cases. We observe that, as opposed to the materials with $A$ = Ni atom, those with $A$ = Co atom have significant contribution from the $A$ atom to the total moment. However, when the $B$ atom is non-magnetic, in a few cases, the moment on the Co atom is zero or much less compared to its bulk moment. As the moment on the $B$ atom increases, the moment of the Co atom gets larger but always largely underestimated compared to the value of its bulk moment (about 1.7$\mu_{B}$). This is expected due to the delocalized-like common d-band between the $A$ and $B$ atoms.[@PRB28JK] For period IV, when $B$ = Cr and Mn, there is a slight decrease in the partial moment of the Co atom, but not in the total moment value of the respective systems. For all the Co-based alloys, the moments are very close to an integer value and this is generally the signature of a half-metallic material. Further, from Table 2, we note that these FHAs follow the Slater-Pauling rule as is seen earlier in case of many Co-based FHAs.[@JPD40HCK; @PRB66IG] As a consequence of this rule, we get an almost linear variation of the magnetic moment with the atomic number of $B$ elements for all the Co-based materials. It is seen that there is a deviation from the Slater-Pauling rule only for Co$_{2}$NbSn which has been observed and explained in the literature.[@JPD40HCK] Due to the electronic structure, all the Co-based compounds are seen to exhibit (Table 2) a high spin polarization at the Fermi level ($E_{F}$) in comparison to the Ni-based compounds. From our calculations, Ni$_{2}$MnGa and Ni$_{2}$MnSn have spin-polarizations $\sim$28 and $\sim$21%, respectively.
[**Heisenberg Exchange Coupling Parameters**]{} - To gain insight into the magnetic interactions in detail we calculate and present the results of our calculation of the Heisenberg exchange coupling parameters, $J_{ij}$, ($i$ and $j$ being pairs ($A$, $A$) and ($A$, $B$)) for the alloys which are likely to exhibit tetragonal distortion and consequently martensite transition. We also show the same parameters for some other related alloys for the purpose of comparison. Most of the materials chosen for presentation have relatively large moment on the $B$ atom so that the ($A$, $B$) exchange interactions are always significant. In the left panels of Figures 6, 7 and 8, the $J_{ij}$ parameters for different compounds are plotted. The right panels give the interaction parameters, $J_{ij}^{bare}$, which are $J_{ij}$ parameters, divided by the product of the moments of the $i$ and $j$ atoms.
Figure 6 gives these parameters for Ni$_{2}$MnGa and Ni$_{2}$MnSn which exhibit the effect of the change of the $C$ atom, and consequently the lattice parameter. Figure 7 contains the values for Co$_{2}$MnGa and Co$_{2}$MoGa to understand what is the role of the $B$ = Mn over the Mo atom. Similarly, we plot the exchange parameters for alloys Co$_{2}$MnSn and Co$_{2}$NbSn in Figure 8. The results match well with the literature wherever the data are available.[@PRB70ES; @PRB88AC] It is seen that there is a RKKY[@RKKY] type of interaction for the ($A$, $A$) and ($B$, $B$) pairs.[@PRB28JK; @PRB77ES] The oscillatory behavior of the $J_{ij}$ parameters as a function of distance between the atoms $i$ and $j$ (normalized with respect to the lattice constant), is a well-known signature of the same. Further, it is seen that between the $A$ and the $B$ atoms there is a signature of a significant direct interaction whenever $B$ has a strong moment. From Figure 6, we observe that the direct interaction between Ni-$B$ atom is stronger in case of Ni$_{2}$MnGa compared to the case of Ni$_{2}$MnSn. This is due to the increased lattice constant and hence weak coupling in case of the latter. It is also found that the direct interaction is somehwat stronger than the RKKY interaction in case of both the materials, though it is quite clear that, as expected and observed in the literature[@PRB28JK; @PRB77ES], both these types of magnetic interactions are important in these materials.
The magnetic interactions in the Co-based materials shown here exhibit a somewhat similar trend. It is found that for the materials favoring tetragonal transition, Co$_{2}$MoGa (Figure 7) and Co$_{2}$NbSn (Figure 8), the direct $A$-$B$ magnetic interaction is small and to some extent comparable to the indirect RKKY-type interaction between $A$ atoms. In the former material, as is evident from Table 2, the moment on Mo atom is somewhat larger compared to that on the Nb atom in case of the latter material. Consequently, the strength of the direct interaction in the former alloy, between $B$ atom (Mo) and $A$ atom (Co) is found to be slightly more. From Figures 7 and 8, we observe that for the materials Co$_{2}$MnGa and Co$_{2}$MnSn, having high magnetic moments (4.10 and 5.03 $\mu_{B}$, respectively), which are of the order of the moments possessed by the two Ni-based materials discussed above, the direct $A$-$B$ interaction is much stronger compared to the RKKY-type indirect interactions ($A$-$A$ or $B$-$B$). A decrease of partial moment of the Mn atom is observed in case of Co$_{2}$MnGa over Ni$_{2}$MnGa (2.73 in Co$_{2}$MnGa versus 3.41 $\mu_{B}$ in Ni$_{2}$MnGa). But due to larger moment on the $A$ = Co over Ni atom, the direct interaction strength between Mn and the respective $A$ atoms, is much larger in the former material, as seen from top left panels of Figures 6 and 7. Hence, the delocalized common $3d$ band between Mn and Co atom in case of Co$_{2}$MnGa is expected to be more effective compared to the case of Ni$_{2}$MnGa. We observe from Figures 6,7 and 8 that, relatively, the RKKY-type indirect interactions are slightly stronger for the $C$ = Sn over Ga atom. It is to be noted here that Sn atom has one valence electron more than Ga.
It is known that the magnetic interactions in the Heusler alloy materials having a large moment on $B$ atom, comprise of a large contribution from the $A$-$B$ direct interaction. At the same time, contribution of the the $A$-$A$ and $B$-$B$ indirect RKKY type of interaction is important as well. The materials, which have high moment on the $B$ atom, typically exhibit a large $A$-$B$ direct interaction when compared to the strength of RKKY-type interaction. However, when the $J_{ij}^{bare}$ parameters are analyzed, for the majority of the materials, it is seen that, both the direct $A$-$B$ interaction and RKKY-type interaction between $i$ and $j$ ($i$ and $j$ being $A$ and $B$) atoms, are somewhat similar in strength. This observation reiterates the fact that the magnetic exchange interactions are not only the function of $i$-$j$ distances (as is evident from Figures 6 to 8), but also of the individual magnetic moments of the $i$ and $j$ atoms which becomes clear when we present the $J_{ij}^{bare}$ plots.
Based on the $J_{ij}$ parameters of the six materials discussed above, the Curie temperatures of the materials within a mean-field approximation[@JPCM23MM] have been calculated to probe further into the possible MSMA property. For Ni$_{2}$MnGa and Ni$_{2}$MnSn, the Curie Temperature ($T_{C}$) values are 410 and 365 K, respectively. These values as well as the $J_{ij}$ parameters match quite well with both experimental and calculated values reported in the literature.[@PRB70ES; @PRB88AC] We note that the experimental values are generally underestimated compared to the theoretical values. This is because of the mean-field-approximation. For some of the Co-based materials $T_{C}$ values are presented in Table 2. We observe that the calculated values match very well with the previously reported data, wherever these results are available. It is interesting to note that due to the small values of $J_{ij}$ for Co$_{2}$NbSn the value of the Curie temperature for this material is very low and this is consistent with the experimentally observed room-temperature paramagnetism in this material. Similarly, due to weak RKKY-type interaction of pair ($A$, $A$) with a somewhat comparable $A$-$B$ direct interaction, the $T_{C}$ value for another Co-based material, which has been predicted from our present work, namely, Co$_{2}$MoGa, is expected to be below room temperature as well.
Bulk Mechanical Properties
--------------------------
Table 3. Bulk mechanical properties of austenite phase of Ni-based materials[^3]
Material C$_{11}$(GPa ) C$_{12}$(GPa ) C$_{44}$(GPa) C$\prime$(GPa) B(GPa) G$_{V}(GPa)$ G$_{R}(GPa)$ G$_{V}$/B
-------------- ---------------- ---------------- ---------------------- --------------------- ------------------------ -------------- -------------- -----------
Ni$_{2}$VGa 193.20 183.32 109.36 4.94 186.62 67.59 11.56 0.36
Ni$_{2}$MnGa 165.41 159.45 113.67 2.98 161.44 69.39 7.16 0.43
152.0$^{b}$ 143$^{b}$ 103$^{b}$ 4.5$^{b}$ 146$^{b}$ 63.6$^{b}$
Ni$_{2}$MoGa 197.36 206.36 103.40 -4.5 203.36 60.24 -12.04 0.30
Ni$_{2}$MnSn 161.02 137.46 92.56 11.78 145.31 60.25 24.73 0.41
158.1$^{c}$ 128.5$^{c}$ 81.3$^{c}$, 87$^{d}$ 14.8$^{c}$, 8$^{d}$ 138.4$^{c}$, 140$^{d}$
Table 4. Bulk mechanical properties of austenite phase of Co-based materials[^4]
Material C$_{11}$(GPa ) C$_{12}$(GPa ) C$_{44}$(GPa) C$\prime$(GPa) B(GPa) G$_{V}(GPa)$ G$_{R}(GPa)$ G$_{V}$/B
-------------- ---------------- ---------------- --------------- ---------------- ----------- -------------- -------------- -----------
Co$_{2}$VGa 266.52 162.12 126.83 52.20 196.92 96.98 80.68 0.49
198$^{b}$
Co$_{2}$CrGa 233.02 182.82 136.77 25.1 199.56 92.10 49.20 0.46
Co$_{2}$MnGa 254.87 165.27 142.69 44.80 195.14 103.53 76.14 0.53
199$^{c}$
Co$_{2}$MoGa 180.92 163.60 114.10 8.66 169.38 71.92 19.44 0.42
Co$_{2}$TcGa 249.54 186.10 123.87 31.72 207.25 87.01 57.29 0.42
Co$_{2}$MnSn 234.63 136.75 119.05 48.94 169.38 91.01 75.68 0.54
Co$_{2}$NbSn 164.95 184.99 80.80 -10.02 178.31 44.47 -30.77 0.25
After presenting the results of magnetic properties of the austenite phase of the materials, now, we discuss about the bulk mechanical properties of the same phase since from technological application point of view, these properties are important. For demonstrating the differences, we concentrate on eight typical FM materials and present the detailed results of the same. Three of these FM materials are likely to undergo a tetragonal distortion at low temperature and these materials are Ni$_{2}$MnGa, Co$_{2}$MoGa, Co$_{2}$NbSn. The other group of alloys consists of materials having a cubic ground state. Among these alloys, the following compounds have been considered - Co$_{2}$VGa, Co$_{2}$CrGa, Co$_{2}$MnGa, Co$_{2}$MnSn and Ni$_{2}$MnSn, which are expected to have a cubic symmetry at the lowest temperature. Tables 3 and 4 contain the bulk mechanical properties of these above-mentioned Ni and Co-based alloys, respectively. Values of few other materials are also listed in Tables 3 and 4, for comparison. It is observed that the overall agreement with the values from the literature is reasonably good.
There are three independent elastic constants for a cubic structure. These are C$_{11}$, C$_{12}$ and C$_{44}$. These three elastic constants can be found by calculating energies for three different types of strain on the unit cell of the system under equilibrium. From these three linearly independent energy versus strain data, we can find out C$_{11}$, C$_{12}$ and C$_{44}$. The applied strains have the form as ($\delta$, $\delta$, $\delta$, 0, 0, 0), (0, 0, 0, $\delta$, $\delta$, $\delta$) and ($\delta$, $\delta$, (1+$\delta$)$^{-2}$-1, 0, 0, 0). $\delta$ has been taken in the range of -0.02 to +0.02 in steps of 0.005. To start with, we calculate the equilibrium lattice parameter $($a$_{0}$$)$ as well as equilibrium volume $($V$_{0}$$)$, and the corresponding energy is considered as the equilibrium energy $($E$_{0}$$)$. Then strain is applied to the system. Under this strained condition, the energy ($E$) is calculated and subsequent to that, the elastic constants are obtained from our calculations as discussed below. The energies $\frac{E-E_0}{V_0}$ are plotted as a function of applied strain and fitted with a fourth order polynomial. The second order coefficient of the fit gives the elastic constants. The mechanical stability criteria for the cubic crystal are as follows: C$_{11}$ $>$ $0$, C$_{44}$ $>$ $0$, C$_{11}$$-$C$_{12}$ $>$ $0$ and C$_{11}$$+2$C$_{12}$ $>$ $0$. From the tables containing elastic constants we can see that first, second and fourth conditions are satisfied for all the materials listed here, but the third condition is not satisfied by some of the materials.
[**Tetragonal shear constant (C$^\prime$)**]{} - This is defined as 0.5$\times$(C$_{11}$$-$C$_{12}$). A value of C$^\prime$ which is close to zero or negative indicates that the material is mechanically unstable and prone to tetragonal distortion. It is clear from Table 3, that for Ni$_{2}$MnGa, as expected, the tetragonal shear constant is quite close to zero. For Ni$_{2}$MoGa the value of C$^\prime$ is found to be negative, which indicates that it has a mechanically unstable cubic austenite phase, which corroborates the result presented in Figure 5. It is interesting to note that Ni$_{2}$VGa has a value of C$^\prime$ close to that of Ni$_{2}$MnGa. From Figure 5, we observe that it has a flat region near $c$/$a$ = 1, in the energy versus $c$/$a$ curve. Hence, in this case the ground state symmetry can not be properly ascertained as is evident from our results. It is important to mention here that the stoichiometric Ni$_{2}$MnSn is a material which is not known to undergo martensite transition and we find that the tetragonal shear constant has a slightly larger positive value of about 12 compared to Ni$_{2}$MnGa.
From Table 4, it is observed that two Co-based materials show a negative or close to zero value for C$^\prime$. Out of these, Co$_{2}$NbSn is known to exhibit non-cubic distortion.[@JPSJ58SF] Among the materials, namely, Co$_{2}$MoGa, Co$_{2}$VGa, Co$_{2}$CrGa, Co$_{2}$CrGa, Co$_{2}$MnSn, Co$_{2}$TcGa, except the first alloy all others have a large positive value for the $C^{\prime}$ constant. It is also observed from Figure 5 that, for Co$_{2}$MoGa, there is a clear indication of the tetragonal phase being the lowest energy state. This is not the case for the other materials. So the combined study of energetics and bulk mechanical properties of all the materials[@TRunpubl] indicates that the only two Co$_{2}BC$ materials which are likely to undergo tetragonal transition and to exhibit SMA property are Co$_{2}$NbSn and Co$_{2}$MoGa.
[**Inherent Crystalling Brittleness (ICB)**]{} - The calculated values of bulk modulus, $B$, have been listed in Tables 3 and 4 for Ni and Co-based materials, respectively. The isotropic shear modulus, $G$, is related to the resistance of the material to the plastic deformation. In literature, it has been shown[@hill] that the value of $G$ lies in between the values of shear modulii given by formalisms of Voigt ($G_{V}$)[@voigt] and Reuss ($G_{R}$)[@reuss], which means $G$ = ($G_{V}$ + $G_{R}$)/2. As has been discused for austenite phase of Ni$_{2}$MnGa in our previous work (Ref.) the experimental $G$ value is close to the calculated $G_{V}$ value while $G_{R}$ value remains largely underestimated. This occurs due to the small positive value of $C^{\prime}$. This particular aspect of similar FHAs, showing martensite transition, has already been discussed in detail in the literature.[@JALCOM632TR]. Following this observation, we consider $G_{V}$ value as the value of shear modulus ($G$) though it is generally considered to be the higher limit of the same. Further, a simple and empirical relationship, given by Pugh[@PM45SFP], proposes that the plastic property of a material is related to the ratio of the shear and bulk modulus of that particular material. A high value (greater than $\sim$ 0.57) of ratio of shear and bulk modulus, namely, $G$/$B$, is connected with the inherent crystalline brittleness of a bulk material. A value below this critical number phenomenologically signifies that the material’s ICB is low. From Table 3, we find that the values for Ni-based materials are below this critical value and hence the ICB of these materials is low though the well-known FHA, Ni$_{2}$MnGa, has a somewhat higher value compared to the materials containing platinum in place of Ni.[@JALCOM632TR] We note that Ni$_{2}$MoGa show a negative value of C$^\prime$, as well as the lowest value of $G$/$B$ among all and hence it is expected to have a low ICB. So from energetics (Figure 5) and bulk mechanical points of view, it is a promising material, though not from magnetic point of view. Table 4 lists the bulk mechanical properties for the Co-based FHAs studied here. The two materials which are likely to show a tetragonal ground state (namely, Co$_{2}$NbSn and Co$_{2}$MoGa) are expected to exhibit ICB smaller or comparable to Ni$_{2}$MnGa ($G$/$B$ = 0.25 and 0.42, respectively). All the rest of the Co$_{2}BC$ alloys have $G$/$B$ values comparable to or larger than that of Co$_{2}$NbSn.
[**Cauchy Pressure**]{} - We now focus on the value of Cauchy Pressure, $C^{p}$, which is defined as $C^{p}$ = $C_{12}$ - $C_{44}$. In Figure 9, we plot the available data for $C^{p}$ versus $G_{V}$/$B$ calculated in case of some of the Ni and Co-based compounds. We find that overall, there is a clear trend of inverse (linear) relationship between $G_{V}$/$B$ and $C^{p}$. In the literature also, it is observed that, the higher the $C^{p}$, the lower the ratio $G$/$B$. Interestingly, this type of nearly-linear inverse relationship between the Cauchy pressure and the $G$/$B$ ratio seems to be a rather general observation as observed in the literature for various types of materials.[@JALCOM632TR; @cpvsgbyb]
Finally, after analyzing the bulk mechanical as well as the magnetic properties and the energetics, out of all the materials studied, only two new materials, namely, Ni$_{2}$MoGa and Co$_{2}$MoGa, emerge to be promising in terms of application as an SMA material. However, due to the absence of any magnetic moment in Ni$_{2}$MoGa, this material is not expected to be suitable as an MSMA material. A low $T_{C}$ indicates an absence of ferromagnetism in Co$_{2}$MoGa at room temperature as is observed in case of Co$_{2}$NbSn.
Electronic Properties: Density of States
----------------------------------------
### Analysis of Total and Atom-Projected Partial DOS
After discussing the energetics, magnetic and bulk mechanical property of the cubic austenite phase, we now analyse the electronic property in terms of the total and partial density of states of different atoms of various materials. We have carried out calculations on all the eletronically stable materials[@TRunpubl] but here we concentrate on and present the results of the austenite phases of eight typical FM materials as discussed above. These materials are Ni$_{2}$MnGa, Co$_{2}$MoGa, Co$_{2}$NbSn, which are to exhibit a tetragonal symmetry as well as Co$_{2}$VGa, Co$_{2}$CrGa, Co$_{2}$MnGa, Co$_{2}$MnSn and Ni$_{2}$MnSn, which are to possess a cubic symmetry, at the lowest possible temperature. We will discuss the total and atom-projected DOS of these systems in this section.
[**Total DOS**]{}:
It is seen that the valence band width for all the materials is about the same, which is roughly about 6 eV (Figures 10 to 13). The two-peak structure in the DOS for both Ni and $C$ atoms indicating about substantial hybridization among these atoms is evident from Figure 10. A two-peak structure is observed at the Fermi level, this indicates a strong hybridization between the Ga and Ni atoms. This covalent interaction between the Ga $4p$ and Ni $3d$ minority electrons plays a crucial role in the stability. We note here that the overlapping and the two-peak structure of the DOS is prominent in case of $A$ and $C$ atoms of Co$_{2}$ZrGa, which has a highly negative formation energy (Figure 4). On the other hand, in Co$_{2}$CrGa, the two-peak structure and the overlapping of DOS for both the $A$ and $C$ atoms are not quite substantial and the formation energy is low as well (Figure 4).[@TRunpubl] Further, Figure 13 depicts the DOS for $B$ = Y and Zr. These plots indicate a lowering of binding energy in the $B$ = Zr material compared to the $B$ = Y case. This corroborates the trend of the formation energy values of these materials. It is to be noted that the contribution from the $A$ atom plays a crucial role in this. Zayak et al[@PRB72ATZ] have earlier shown that the stability of the Ni$_{2}$MnGa type Heusler alloys is closely related to the minority DOS at the Fermi level as has been argued in other cases as well.[@PRB84CL]
There is a large exchange splitting observed for systems which have Mn as the $B$ atom. From Figures 10 to 12, we observe that for $A_{2}BC$ systems ($A$ = Ni, Co; $B$ = Mn; $C$ = Ga; Sn), the occupied DOS of the $B$ atom is dominated by the majority spin whereas the unoccupied DOS is dominated by the minority spin. For Ni$_{2}$MnGa and Ni$_{2}$MnSn, the majority DOS of the Mn atom is centred around -1.2 eV and -1.5 eV, whereas the minority DOS of the same atom is centred around 1.5 eV and 1.3 eV for the respective systems. For Co$_{2}$MnGa and Co$_{2}$MnSn, the position of the occupied majority spin DOS for Mn atom is at about -0.7 eV and -1.1 eV, respectively, while the position of the unoccupied minority DOS peak is at about 1.8 eV and 1.6 eV. For Co$_{2}$CrGa also, we observe a large separation between the occupied majority DOS peak (at about -0.1 eV) of Cr atom and unoccupied minority DOS peak (at about 1.7 eV). Next, we analyze the partial DOS of few of the important $A$, $B$ atoms, to understand the nature of DOS close to the Fermi level.
[**Partial DOS**]{} :
[*Ni Atom*]{} - The DOS in case of the two Ni-based alloys are similar (Figure 10). However, since Sn atom contains one extra velence electron compared to the Ga atom in the $C$ position, the peak positions of the total DOS of the Mn atom are shifted towards lower energy in case of the materials with $C$ = Sn. At this point, it is worth-mentioning that it has already been discussed in the literature that a rigid band model is a suitable model to understand the trends when the $C$ atom is changed.[@JPCM11AA] We further observe that a similar situation is seen to arise when the $A$ atom is changed from Ni to Co, which is discussed below.
[*Co Atom*]{} - Co has one velence electron less than Ni. Hence, a larger contribution of Co-derived levels compared to Ni-derived levels in the unoccupied part of the respective DOS is expected. Figures 11 to 13 depict this. When DOS of Ni$_{2}$Mn$C$a is compared with Co$_{2}$Mn$C$, it is clearly evident (Figures 10, 11 and 12). Among the materials with $C$ = Ga and $A$ = Co, only Co$_{2}$MoGa is a material which is likely to show a martensite transition (Figure 5). It is seen that it has the first unoccupied DOS peak very close to the $E_{F}$ (Figure 11). Among the materials with $C$ = Sn and $A$ = Co, only Co$_{2}$NbSn is known to be prone to distortion[@JPSJ58SF] and it has an unoccupied DOS peak close to $E_{F}$ as well. When we analyze the DOS of Co$_{2}$MnGa (Figure 11) and Co$_{2}$MnSn (Figure 12), which do not show the tendency of a tetragonal distortion as well as are known to possess high spin polarization at the Fermi level, we observe that the first unoccupied DOS is further away from $E_{F}$ compared to the materials which are prone to tetragonal distortion, namely Co$_{2}$MoGa (Figure 11) and Co$_{2}$NbSn (Figure 12).
[*Mn Atom*]{} - There are four out of eight materials which contain Mn atom in the $B$ position. When we compare the total DOS of the Mn atom at the $B$ position, in all the four materials considered here, it is clearly seen that majority of the DOS of the down spin occupies the unoccupied region above the Fermi level, while the up spin electrons primarily have negative binding energies. As opposed to the down spin DOS, which has one major peak in all the four cases, the up spin electrons typically occupy two energy ranges, one around 1 and one around 3 eV below $E_{F}$. Due to one extra electron in Sn atom compared to the Ga atom in the $C$ position, the peak positions of the total DOS of the Mn atom are shifted towards lower energy in case of the materials with $C$ = Sn. To elaborate, first we compare the DOS of the up spin of Mn in the four alloys Ni$_{2}$MnGa, Ni$_{2}$MnSn, Co$_{2}$MnGa and Co$_{2}$MnSn in the cubic phase. While DOS of Mn atom in the unoccupied part peaks at about 1.5 eV in case of $C$ = Ga, it peaks around 1.2 eV when $C$ = Sn. The corresponding peak positions for Co$_{2}$MnGa and Co$_{2}$MnSn are at about 1.8 and 1.6 eV, respectively. In case of the up spin DOS, there are two ranges of predominant DOS in all the four materials. For the first such range, which is closer to the Fermi energy, the peak positions are at about -1.3, -1.5, -0.7 and -1.1 for Ni$_{2}$MnGa, Ni$_{2}$MnSn, Co$_{2}$MnGa and Co$_{2}$MnSn, respectively. For the range which is at a much higher binding energy, the peak positions are at about -3.2, -3.2, -2.5 (also one slightly weaker one at -2.8) and -2.5 (also one slightly weaker one at -2.8) eV for Ni$_{2}$MnGa, Ni$_{2}$MnSn, Co$_{2}$MnGa and Co$_{2}$MnSn, respectively. It is to be noted that the peaks of the DOS are not sharp but broad ones, with shoulders on either or one of the sides.
### Electronic Stability of the Tetragonal phase from DOS
After discussing the electronic property of the cubic austenite phase, we now analyse the electronic property in terms of the density of states of different materials as a function of $c$/$a$. We concentrate on the eight typical FM materials as discussed above. A tetragonal distortion has been imposed on all these eight materials. To highlight the difference between the two symmetries, we will concentrate on the detailed results of cases with $c$/$a$ = 1, 1.05 and 1.10. The aim is to understand the electronic stability or instability of the tetragonal phase of these compounds from the DOS results.
[**Ni$_{2}$MnGa versus Ni$_{2}$MnSn**]{} :
Figure 14 contains the density of states of the cubic and tetragonal phases, with $c$/$a$ varying from 1 to 1.10 in steps of 0.05 for materials Ni$_{2}$MnGa and Ni$_{2}$MnSn. First we will analyse Figure 14 for the cubic phase. We observe that there is a peak at around -0.2 eV for Ni$_{2}$MnGa and at around -0.5 eV for Ni$_{2}$MnSn, respectively. As has been established in the literature, this peak, which is close to the $E_{F}$, has negative binding energy. This peak is derived from the electrons of the Ni atoms with down spin having $e_{g}$ symmetry and is known to play a crucial role in the stabilization of the tetragonal phase in case on Ni$_{2}$MnGa.[@JPCM11AA] The density of states of the down spin electrons with $t_{2g}$ symmetry of these $A$ atoms corresponds to the peaks with reasonably higher binding energy. This is the case for both the materials. On the other hand, detailed investigation suggests that the $B$ atom = Mn has negligible contribution near the Fermi level; both for the up and for the down spin. The up spin electrons of $A$ atom also do not significantly contribute to the DOS at around -0.2 and -0.5 eV for Ni$_{2}$MnGa and Ni$_{2}$MnSn, respectively.
As $c$/$a$ increases in case of these materials, there are some systematic changes in the density of states, clearly visible from the lower panels of the Figures 14 to 17. For Ni$_{2}$MnGa, it is seen that the peak near the Fermi level, at about -0.2 eV, derived from the down spin DOS, has been split into two peaks. This has been observed and argued about in detail in the literature.[@JPCM11AA; @JPCM11PJB; @SSC18JCS] As a result of tetragonal distortion, the degeneracy of the sub-bands near the Fermi level is lifted. As a consequence, a redistribution of the density of states of the $3d$ electrons and in turn a reduction of free energy occurs. This is the so-called band Jahn-Teller effect which is known to result in the lowering of energy under tetragonal distortion in many FHAs including Ni$_{2}$MnGa.[@JPCM11AA; @JPCM11PJB] For Ni$_{2}$MnSn as well, it is seen that the most prominent change being the spilitting of the peak at about -0.5 eV[@JPCM11AA] upon the tetragonal distortion as seen from Figure 14. However, it is well-known that stoichiometric Ni$_{2}$MnSn is not expected to have tetragonal ground state. It has been argued in the literature that the band Jahn-Teller effect is sensitive to the DOS at the $E_{F}$ in the cubic phase. The closeness of the degenarate peak for Ni$_{2}$MnGa (at about -0.2 eV with respect to $E_{F}$) over Ni$_{2}$MnSn (at about -0.5 eV below $E_{F}$) is an indication of the possibility of tetragonal distortion in the former.[@JPCM11AA; @JPCM11PJB] We find that the density of states at $E_{F}$ in case of cubic phase of Ni$_{2}$MnGa is relatively more in comparison to Ni$_{2}$MnSn, which is evident from the relative position of the $e_{g}$ peak near the Fermi level in the two materials (Figure 14).
[**Co$_{2}$MoGa versus Co$_{2}$MnGa**]{} :
Figure 15 gives the plot of DOS with different $c$/$a$ values. We note that in the literature the stability of the martensite phase for a Co-based system (Co$_{2}$NiGa) has been explained to be due to the lowering of energy of the system under the tetragonal deformation compared to the cubic structure.[@ActaMat58RA] In case of Co$_{2}$MoGa there is a large peak in the minority DOS just above the Fermi level (at about +0.3 eV with respect to $E_{F}$). Detailed analysis shows that this peak at the minority DOS has contributions from all three atoms i.e. (Co, Mo, Ga), but the major contribution comes from the $e_{g}$ levels of $3d$ electrons of Co. We find that these Co $e_{g}$ levels have a major role in the stabilization of the tetragonal phase, similar to the Ni $e_{g}$ levels in case of Ni$_{2}$MnGa. Hence, in case of Co$_{2}$MoGa also, band Jahn-Teller distortion plays a significant role. The down spin DOS close to the Fermi level is high, which leads to the instability of the cubic phase of this material unlike Co$_{2}$MnGa. In case of the latter material, the minority DOS almost vanishes at $E_{F}$. Further, here, the $e_{g}$ levels of Co atom are located farther away from the Fermi level (at about +1.0 eV with respect to $E_{F}$). For both the materials $B$ atom does not contribute to the minority DOS at the Fermi level. However, primarily the $B$ atom only contributes significantly to the large DOS of the up spin electrons at the Fermi level for Co$_{2}$MoGa. The peak positions of the DOS in the unoccupied part of the energy for Co$_{2}$MoGa and Co$_{2}$MnGa are different due to the hybridization of Co atom with the Mo and Mn atoms, respectively. This led to the difference in the electronic characters of these two materials.
[**Co$_{2}$NbSn versus Co$_{2}$MnSn**]{} :
Figure 16 gives the plot of DOS with different $c$/$a$ values. As in case of Ni$_{2}$MnGa, for Co$_{2}$NbSn also we can find same type of evolution of density of states as a function of $c$/$a$ and Co $3d$ $e_g$ states play the key role in the tetragonal transition. In austenite phase of Co$_{2}$NbSn the Co $3d$ $e_g$ peak is at just above Fermi level (at about +0.05 eV). Under tetragonal distortion this peak is split into two: one part being above the Fermi level and another one being below the Fermi level. This splitting lowers the energy of the system and the tetragonal phase tends to be the ground state structure compared to the cubic structure. In the literature[@JPSJ58SF], it has been observed that at the Fermi energy, the contribution to DOS mainly comes from the $3d$ bands of Co and Nb. We observe from Figure 12 that Sn atoms also contribute. In the cubic phase Co atom has a single large peak just above the $E_{F}$. But under tetragonal distortion, this single peak is split into two and the energy of the tetragonal phase is lower compared to the cubic phase. It has been observed[@JPSJ58SF] that the band Jahn-Teller distortion is the cause of the structural transition. As seen from Figures 12 and 16, for Co$_{2}$MnSn, the peak due to Co $3d$ $e_g$ levels is located at a higher energy (at about +0.8 eV) compared to Co$_{2}$NbSn. After the application of the tetragonal distortion this single peak is split into two (Figure 16). But in this case after splitting both the peaks lie above $E_{F}$ which does not yield to lowering of energy.
[**Co$_{2}$CrGa versus Co$_{2}$VGa**]{} :
From Figure 17, it is observed that the single peak of the Co $3d$ $e_g$ above $E_{F}$ is split for Co$_{2}$VGa. However, after splitting both the peaks lie above $E_{F}$. So there is no lowering of energy of the system possible under tetragonal deformation. For Co$_{2}$CrGa, the Co $3d$ $e_g$ single peak of cubic phase is located at an even higher (positive) energy with respect to the Fermi level. In this case also lowering of energy under tetragonal deformation is not possible, which is consistent with the results presented in Figure 5.
Finally, we find that in all the materials, as a result of tetragonal distortion, the degeneracy is lifted for the $3d$ sub-bands in the minority spin channel of $A$ atoms, present closest to the Fermi level. Subsequently, as a result of this band Jahn-Teller effect, a redistribution of the density of states of these $3d$ electrons occurs. In all the materials, which favor a tetragonal deformation, a substantial density of states very close to the Fermi level has been observed. In other words, the band Jahn-Teller effect is found to be, as expected, quite sensitive to the DOS at or close to the $E_{F}$ in the cubic phase. As a result of the redistribution of the DOS, under tetragonal distortion, due to closeness of peak in DOS to the $E_{F}$, the energy gets lowered in these materials. Consequently, the possibility of martensite transition is found to be high. Further, for these materials, a negative or very close to zero value of tetragonal shear constant, C$^\prime$, has also been observed, as is expected from the literature. For all the other materials, under tetragonal distortion, the splitting of the $3d$ minority spin levels is observed as well but the peak is away from the $E_{F}$ resulting in a reduced density of states at $E_{F}$. Therefore, the lowering of free energy is not possible, which renders the tetragonal transition unlikely. The observation regarding the cubic ground state for these materials is further corroborated by the relatively large and positive values of $C\prime$ for all these materials.
Summary and Conclusion
======================
It has been of particular interest that out of all the Ni and Co-based full-Heusler alloys studied so far in the literature, only some materials undergo the martensite transition and these generally show the technologically important magnetic shape memory alloy property. These full Heusler alloys in general are found to be metallic in nature. On the other hand, it has been observed that there is another group of FHAs which are half-metallic-like in nature, with a much reduced density of states at the Fermi level in case of one of the spin channels and these materials generally do not exhibit MSMA property. It has been observed earlier that while most of the Ni-based FHAs show MSMA property, many of the Co-based FHAs exhibit large spin-polarization at the Fermi level.
Therefore, in this paper, using first-principles calculations based on density functional theory, we study in detail the bulk mechanical, magnetic and electronic properties of a series of Ni and Co-based full Heusler alloys, namely, Ni$_{2}BC$ and Co$_{2}BC$ ($B$ = Sc, Ti, V, Cr and Mn as well as Y, Zr, Nb, Mo and Tc; $C$ = Ga and Sn). After establishing the electronic stability from the formation energy and subsequent full geometry optimization, we carry out the calculation of different properties to probe and understand, the similarities and differences in the properties of these materials. We analyze the data in detail to see if among these materials there is any FHA which has a tendency to undergo a tetragonal transition and at the same time possesses a high spin polarization at the Fermi level.
Out of all the electronically stable compounds of the total forty Ni and Co-based materials, most of the Ni-based materials are expected to show a non-magnetic ground state. On the other hand, Ni$_{2}$MnGa and Ni$_{2}$MnSn as well as all the Co-based materials are ferromagnetic in nature. Further, from the Heisenberg exchange interaction parameters, it is seen that the materials exhibit similar nature in terms of the relative contributions of the direct and RKKY-type nature of the magnetic interactions. The trend of the calculated values of Curie temperature for various materials, obtained from the $J_{ij}$ parameters, matches reasonably well with the literature wherever data are available. From the point of view of bulk mechanical properties, the values of tetragonal shear constant show consistent trend: high positive for materials not prone to tetragonal transition and low or negative for others. A general trend of nearly-linear inverse relationship between the Cauchy pressure and the $G$/$B$ ratio is predicted for both the Ni and Co-based materials.
It is observed that the Ni-based materials are typically metallic in nature. However, all the Co-based alloys exhibit a significant spin polarization at the Fermi level. Most of the Ni-based materials have a $3d$ band of the minority spin of the $A$ atoms close to and below the $E_{F}$. On the other hand, the peak position of the same band is above the $E_{F}$ for the Co-based materials. We observe that, in both the cases of Ni and Co-based materials, these $3d$ levels play an important role in deciding the ground state. Further, the replacements of the $A$, $B$, $C$ sites of the $A_{2}BC$ materials by different atoms, indicate that in general a rigid band model explains the differences in the electronic structure of both the Ni and Co-based materials to a large extent. This model along with the hybridization between atoms, further supports the results of partial and total moments of these systems. The relation between the closeness of the peak corresponding to the $e_{g}$ levels of the $3d$ down spin electrons of the $A$ atom to the $E_{F}$ and the tendency of lowering of energy upon tetragonal distortion is consistent across all the Ni and Co-based materials.
Finally, from our study on the two categories of materials, it is clear that out of all the materials which we study here, only four FHAs show a tendency of undergoing martensite transition. Out of these four materials, which have a conventional Heusler alloy structure and exhibit a clear possibility of finding a tetragonal phase as their ground state, three of them, namely, Ni$_{2}$MnGa, Ni$_{2}$MoGa and Co$_{2}$NbSn have a metallic nature as is observed in case of majority of the MSMA material; on the other hand, from our calculations, Co$_{2}$MoGa is expected to emerge as a [*shape memory alloy with high spin polarization at the Fermi level*]{}. This interesting finding awaits a suitable experimental validation.
Acknowledgments
===============
Authors thank P. D. Gupta, P. A. Naik and G. S. Lodha for facilities and encouragement throughout the work. The scientific computing group, computer centre of RRCAT, Indore and P. Thander are thanked for help in installing and and support in running the codes. S. R. Barman, A. Arya, S. B. Roy and C. Kamal are thanked for discussion. TR thanks HBNI, RRCAT for financial support.
P. J. Webster, K. R. A. Ziebeck, S. L. Town, M. S. Peak, Phil. Mag. B, [**49**]{}, 295 (1984). K. Ullakko, J. K. Huang, C. Kantner, R. C. O’Handley, and V. V. Kokorin, Appl. Phys. Lett. [**69**]{}, 1966 (1996). S. J. Murray, M. Marioni, S. M. Allen, R. C. O’Handley, T. A. Lograsso, Appl. Phys. Lett. [**77**]{}, 886 (2000). A. Sozinov, A. A. Likhachev, N. Lanska, and K. Ullakko, Appl. Phys. Lett. [**80**]{}, 1746 (2002); A. Sozinov, N. Lanska, A. Soroka, and W. Zou, Appl. Phys. Lett. [**102**]{}, 021902 (2013). C. Biswas, R. Rawat, S. R. Barman, Appl. Phys. Lett., [**86**]{}, 202508 (2005). F. X. Hu, B. Shen, J. Sun, Appl. Phys. Lett., [**76**]{}, 3460 (2000); F. X. Hu, J. Sun, G. Wu, B. Shen, J. Appl. Phys., [**90**]{}, 5216 (2001). S. Amari, R. Mebsout, S. Mecabih, B. Abbar, B. Bouhafs, Intermetallics, [**44**]{}, 26 (2014); Y. Wu, B. Wu, Z. Wei, Z. Zhou, C. Zhao, Y. Xiong, S. Tou, S. Yang, B. Zhou, Y. Shao, Intermetallics, [**53**]{}, 26 (2014); B. Kong, X. R. Chen, J. X. Yu, C. L. Cai, J. Alloys and Comp., [**509**]{}, 2611 (2011); V. Kanchana, G. Vaitheeswaran, Y. Ma, Y. Xie, A. Svane, O. Eriksson, Phys. Rev. B, [**80**]{}, 125108 (2009); L. Banisla, A. I. Mallick, M. M. Raja, A. K. Nigam, B. S. D. C. S. Varaprasad, Y. K. Takahashi, A. Alam, K. G. Suresh, K. Hono, Phys. Rev. B, [**91**]{}, 104408 (2015). R. A. de Groot, F. M. Mueller, P. G. van Engen, K. H. J. Buschow, Phys. Rev. Lett., [**50**]{}, 2024 (1983). W. E. Pickett, J. S. Moodera, Physics Today, [**54**]{}, 39 (2001). J. Kübler, A. R. Williams, C. B. Sommers, Phys. Rev. B, [**28**]{}, 1745 (1983). M. A. Ruderman and C. Kittel, Phys. Rev., [**96**]{}, 99 (1954); T. Kasuya, Prog. Theor. Phys., [**16**]{}, 45 (1956); K. Yosida, Phys. Rev., [**106**]{}, 893 (1957). E. Sasioglu, L. M. Sandratskii, P. Bruno, Phys. Rev. B, [**77**]{}, 064417 (2008). A. T. Zayak, P. Entel, K. M. Rabe, W. A. Adeagbo, M. Acet, Phys. Rev. B, [**72**]{}, 054113 (2005). T. Roy, M. E. Gruner, P. Entel, A. Chakrabarti, J. Alloy. and Comp., [**632**]{}, 822 (2015) and references therein. G. D. Liu, J. L. Chen, Z. H. Liu, X. F. Dai, G. H. Wu, B. Zhang, X. X. Zhang, Appl. Phys. Lett., [**87**]{}, 262504 (2005); G. D. Liu, X. F. Dai, S. Y. Yu, Z. Y. Zhu, J. L. Chen, G. H. Wu, H. Zhu, J. Q. Xiao, Phys. Rev. B, [**74**]{}, 054435 (2006). S. R. Barman, S. Banik, A. K. Shukla, C. Kamal, Aparna Chakrabarti, Euro. Phys. Lett., [**80**]{}, 57002 (2007); S. R. Barman, Aparna Chakrabarti, Phys. Rev. B, [**77**]{}, 176401 (2008); S. Singh, M. Maniraj, S. W. D’Souza and S. R. Barman, Appl. Phys. Lett., [**96**]{}, 081904 (2010). K. R. Priolkar, P. A. Bhobe, D. N. Lobo, S. W. D’Souza, S. R. Barman, A. Chakrabarti, S. Emura, Phys. Rev. B, [**87**]{}, 144412 (2013). I. Galanakis, E. Sasioglu, Appl. Phys. Lett., [**98**]{}, 102514 (2011). T. Roy, A. Chakrabarti, J. Mag. and Mag. Mater., [**401**]{}, 929 (2016). M. Siewert, M. E. Gruner, A. Hucht, H. C. Herper, A. Dannenberg, A. Chakrabarti, N. Singh, R. Arroyave, P. Entel, Adv. Eng. Mat., [**14**]{}, 530 (2012). G. Kresse, J. Furthmüller, Phys. Rev. B, [**54**]{}, 11169 (1996); G. Kresse, D. Joubert, Phys. Rev. B, [**59**]{}, 1758 (1999); VASP 5.2 programme package is fully integrated in the MedeA® platform (Materials Design, Inc.) with a graphical user interface enabling the computation of the properties. P. E. Blochl, Phys. Rev. B, [**50**]{}, 17953 (1994). J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett., [**77**]{}, 3865 (1996). P. Blaha, K. Schwartz, G. K. H. Madsen, D. Kvasnicka and J. Luitz, WIEN2K, An Augmented Plane Wave plus Local Orbitals Program for Calculating Crystal Properties, (Karlheinz Schwarz, Tech. Universität, Wien, Austria), 2002, ISBN 3-9501031-1-2. A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, V. A. Gubanov, J. Magn. Magn. Mater., [**67**]{}, 65 (1987). The Munich SPR-KKR package, version 5.4, H. Ebert et al; http://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR. M. Meinert, J. M. Schmalhorst, G. Reiss, J. Phys. : Cond. Matt., [**23**]{}, 036001 (2011); M. Meinert, J. M. Schmalhorst, G. Reiss, J. Phys. : Cond. Matt., [**23**]{}, 115005 (2011) and the references therein. Michael Gillessen, “Ma$\beta$geschneidertes und Analytik-Ersatz Uber die quantenchemischen Untersuchungen einiger ternarer intermetallischer Verbindungen”, PhD thesis, University of Aachen, (2009). S. Fujii, S. Ishida, J. Phys. Soc. Jpn., [**58**]{}, 3657 (1989). S. R. Barman, A. Chakrabarti, S. Singh, S. Banik, S. Bhardwaj, P. L. Paulose, B. A. Chalke, A. K. Panda, A. Mitra, A. M. Awasthi, Phys. Rev. B, [**78**]{}, 134406 (2008) and references therein. M. Baral, S. Banik, A. Chakrabarti, D. M. Phase, T. Ganguli, J. Alloys and Comp., [**645**]{}, 117 (2015) and references therein. Y. Kurtulus, R. Dronskowski, G. D. Samolyuk, V. P. Antropov, Phys. Rev. B, [*71*]{}, 014425 (2005). M. B. Sahariah, S. Ghosh, C. S. Singh, S. Gowtham, R. Pandey, J. Phys. : Cond. Matt., [**25**]{}, 025502 (2013). V. N. Antonov, B. N. Harmon, L. V. Bekenov, A. P. Shpak, A. N. Yaresko, Phys. Rev. B, [**71**]{}, 174428, (2005). A. W. Carbonari, R. N. Saxena, W. Pendl, J. Mestnik Filho, R. N. Attili, M. Olzon-Dionysio, S.D. de Souza, J. Magn. Magn. Mater. [**163**]{}, 313, (1996). K. R. A. Ziebeck, P. J. Webster, J. Phys. Chem. Solids. [**35**]{}, 1, (1974). T. Sasaki , T. Kanomata , T. Narita , H. Nishihara , R. Note , H. Yoshida , T. Kaneko, J. Alloys and Comp.[**317**]{}, 406, (2001). P. G. van Engen, K. H. J. Buschow, and M. Erman, J. Magn. Magn. Mater.[**30**]{}, 374 (1983). J. Kübler, G. H. Fecher, C. Felser, Phys. Rev. B **76**, 024414, (2007). R. Y. Umetsu, K. Kobayashi,A. Fujita, K. Oikawa, R. Kainuma, K. Ishida, N. Endo, K. Fukamichi, A. Sakuma, Phys. Rev. B [**72**]{}, 214412, (2005). B. S. D. Ch. S. Varaprasad, A. Rajanikanth, Y. K. Takahashi, K. Hono, Acta Mater. [**57**]{}, 2702, (2009). K.H.J. Buschow, P.G. van Engen, J. Magn. Magn. Mater. **25**, 90, (1981). H. C. Kandpal, G. H. Fecher, C. Felser, J. Phys. D: Appl. Phys., [**40**]{}, 1507 (2007). I. Galanakis, P. H. Dederichs, N. Papanikolaou, Phys. Rev. B, [**66**]{}, 174429 (2002). E. Sasioglu, L. M. Sandratskii, P. Bruno, Phys. Rev. B, [**70**]{}, 024427 (2004) and references therein. A. Ayuela, J. Enkovaara, K. Ullakko, R. M. Nieminen, J. Phys. : Cond. Mat., [**11**]{}, 2017 (1999) and references therein. A. Chakrabarti, M. Siewert, T. Roy, K. Mondal, A. Banerjee, M. E. Gruner, P. Entel, Phys. Rev. B, [**88**]{}, 174116 (2013). J. Worgull, E. Petti, J. Trivisonno, Phys. Rev. B, **54**, 15695 (1996). S. Agduk, G. Gokoglu, J. Alloys and Comp. **511**, 9, (2012). T. Kanomata, Y. Chieda, K. Endo, H. Okada, M. Nagasako, K. Kobayashi, R. Kainuma, R. Y. Umetsu, H. Takahashi, Y. Furutani, H. Nishihara, K. Abe, Y. Miura, M. Shirai, Phys. Rev. B **82**, 144415 (2010) T. Roy, D. Pandey and A. Chakrabarti (unpublished). R. Hill, Proc. Phys. Soc. A, [**65**]{}, 349 (1952). W. Voigt, Ann. d. Phys. u. Chem. N. F., [**38**]{}, 573 (1889). A. Reuss, Zeit. Angew. Math. Mech. (ZAMM), [**9**]{}, 49 (1929). S. F. Pugh, Phil. Mag., [**45**]{}, 823 (1954). H. Niu, X. Q. Chen, P. Liu, W. Xing, X. Cheng, D. Li, Y. Li, Scientific Reports, [**2**]{}, 718 (2012). C. -M. Li, H. -B. Luo, Q. -M. Hu, R. Yang, B. Johansson, L. Vitos, Phys. Rev. B, [**84**]{}, 024206 (2011). P. J. Brown, A. Y. Bargawi, J. Crangle, K. -U. Neumann, K. R. A. Ziebeck, J. Phys.: Cond. Matt., [**11**]{}, 4715 (1999). J. C. Suits, Solid State Comm., [**18**]{}, 423 (1976). R. Arroyave, A. Junkaew, A. Chivukula, S. Bajaj, C.-Y. Yao, A. Garay ,Acta Mater., [**58**]{}, 5220 (2010).
![ Variation of lattice parameter as a function of $Z$ of $B$ elements for Ni$_{2}BC$ alloy ($C$ = Ga, Sn); X=$B$ atoms being first five transition metal elements of period IV (left panel) and V (right panel). []{data-label="fig:1"}](Figure1.eps)
![ Variation of lattice parameter as a function of $Z$ of $B$ elements for Co$_{2}BC$ alloy ($C$ = Ga, Sn); X=$B$ atoms being first five transition metal elements of period IV (left panel) and V (right panel). []{data-label="fig:2"}](Figure2.eps)
![ Variation of formation energy as a function of $Z$ of $B$ elements for Ni$_{2}BC$ alloy ($C$ = Ga, Sn); X=$B$ atoms being first five transition metal elements of period IV (left panel) and V (right panel). []{data-label="fig:3"}](Figure3.eps)
![ Variation of formation energy as a function of $Z$ of $B$ elements for Co$_{2}BC$ alloy ($C$ = Ga, Sn); X=$B$ atoms being first five transition metal elements of period IV (left panel) and V (right panel). []{data-label="fig:4"}](Figure4.eps)
![ Energy difference between the crystal structures with tetragonal (T) and cubic (C) symmetries, of some typical materials represented as $E_{T} - E_{C}$ (in units of meV per atom), as a function of the ratio of lattice constants $c$ and $a$. The energies have been normalized with respect to the energy of the respective cubic austenite phase of each material. []{data-label="fig:5"}](Figure5.eps)
![ $J_{ij}$ parameters between different atoms of Ni$_{2}$MnGa and Ni$_{2}$MnSn as a function of distance between the atoms $i$ and $j$ (normalized with respect to the respective lattice constant). []{data-label="fig:6"}](Figure6.eps)
![ $J_{ij}$ parameters between different atoms of Co$_{2}$MnGa and Co$_{2}$MoGa as a function of distance between the atoms $i$ and $j$ (normalized with respect to the respective lattice constant). []{data-label="fig:7"}](Figure7.eps)
![ $J_{ij}$ parameters between different atoms of Co$_{2}$MnSn and Co$_{2}$NbSn as a function of distance between the atoms $i$ and $j$ (normalized with respect to the respective lattice constant). []{data-label="fig:8"}](Figure8.eps)
![Cauchy pressure, $C^{p}$, versus $G_{V}$/$B$; a linear fitting of all the data is carried out and shown here. An inverse linear-type relation is seen to exist between the two parameters (see text). []{data-label="fig:9"}](Figure9.eps)
![ The left and right set of panels depict the density of states of Ni$_{2}$MnGa and Ni$_{2}$MnSn materials, respectively. From top to bottom panel, first the total density of states as a function of energy has been plotted. Next panel shows the partial density of states of the Ni atom. Partial density of states of the Mn atom and the $C$ atom are shown in the third and the fourth panels. The DOS of the $C$ atom is multiplied by a factor of 10. The Fermi level is at 0 eV. []{data-label="fig:10"}](Figure10.eps)
![ The left and right set of panels depict the density of states of Co$_{2}$MoGa and Co$_{2}$MnGa materials, respectively. From top to bottom panel: first the total density of states as a function of energy has been plotted. Next panel shows the partial density of states of the Co atom. Partial density of states of the $B$ atom and the Ga atom are shown in the third and the fourth panels. The DOS of the $C$ atom is multiplied by a factor of 10. The Fermi level is at 0 eV. []{data-label="fig:11"}](Figure11.eps)
![ The left and right set of panels depict the density of states of Co$_{2}$NbSn and Co$_{2}$MnSn materials, respectively. From top to bottom panel: first the total density of states as a function of energy has been plotted. Next panel shows the partial density of states of the Co atom. Partial density of states of the $B$ atom and the Sn atom are shown in the third and the fourth panels. The DOS of the $C$ atom is multiplied by a factor of 10. The Fermi level is at 0 eV. []{data-label="fig:12"}](Figure12.eps)
![ The left and right set of panels depict the density of states of Co$_{2}$YGa and Co$_{2}$ZrGa materials, respectively. From top to bottom panel: first the total density of states as a function of energy has been plotted. Next panel shows the partial density of states of the Co atom. Partial density of states of the $B$ atom and the Ga atom are shown in the third and the fourth panels. The DOS of the $C$ atom is multiplied by a factor of 10. The Fermi level is at 0 eV. []{data-label="fig:13"}](Figure13.eps)
![ The density of states as a function of energy has been plotted for the cubic and tetragonal phases, with $c$/$a$ varying from 1 to 1.10 in steps of 0.05 for materials Ni$_{2}$MnGa and Ni$_{2}$MnSn in left and right panels, respectively. Panels below show the down spin density near the Fermi level in an expanded scale for respective materials. The Fermi level is at 0 eV. []{data-label="fig:14"}](Figure14.eps)
![ The density of states as a function of energy has been plotted for the cubic and tetragonal phases, with $c$/$a$ varying from 1 to 1.10 in steps of 0.05 for materials Co$_{2}$MoGa and Co$_{2}$MnGa in left and right panels, respectively. Panels below show the down spin density near the Fermi level in an expanded scale for respective materials. The Fermi level is at 0 eV. []{data-label="fig:15"}](Figure15.eps)
![ The density of states as a function of energy has been plotted for the cubic and tetragonal phases, with $c$/$a$ varying from 1 to 1.10 in steps of 0.05 for materials Co$_{2}$NbSn and Co$_{2}$MnSn in left and right panels, respectively. Panels below show the down spin density near the Fermi level in an expanded scale for respective materials. The Fermi level is at 0 eV. []{data-label="fig:16"}](Figure16.eps)
![ The density of states as a function of energy has been plotted for the cubic and tetragonal phases, with $c$/$a$ varying from 1 to 1.10 in steps of 0.05 for materials Co$_{2}$VGa and Co$_{2}$CrGa in left and right panels, respectively. Panels below show the down spin density near the Fermi level in an expanded scale for respective materials. The Fermi level is at 0 eV. []{data-label="fig:17"}](Figure17.eps)
[^1]: Comparison with the experimental or theoretical data, wherever results are available\
$^{b}$Ref. $^{c}$Ref. $^{d}$Ref. $^{e}$Ref.
[^2]: Comparison with experiments or previous calculations, wherever data are available\
$^{b}$Ref.; $^{c}$Ref.; $^{d}$Ref.; $^{e}$Ref.; $^{f}$Ref.; $^{g}$Ref. $^{h}$Ref. $^{i}$Ref. $^{j}$Ref.
[^3]: Comparison with experiments or previous calculations, wherever data are available\
$^{b}$Ref. $^{c}$Ref. $^{d}$Ref.
[^4]: Comparison with experiments or previous calculations, wherever data are available\
$^{b}$Ref. $^{c}$Ref.
|
---
abstract: |
In the present paper we consider a 3-dimensional differentiable manifold $M$ equipped with a Riemannian metric $g$ and an endomorphism $Q$, whose third power is the identity and $Q$ acts as an isometry on $g$. Both structures $g$ and $Q$ determine an associated metric $f$ on $(M, g, Q)$. The metric $f$ is necessary indefinite and it defines isotropic vectors in the tangent space $T_{p}M$ at an arbitrary point $p$ on $M$.
The physical forces are represented by vector fields. We investigate physical forces whose vectors are in $T_{p}M$ on $(M, g, Q)$. Moreover, these vectors are isotropic and they act along isotropic curves. We study the physical work done by such forces.
address:
- '$^{1}$Department of Mathematics and Informatics, Agricultural University of Plovdiv, 12 Mendeleev Blvd., Bulgaria 4000'
- '$^{2}$Department of Mathematics and Informatics, Agricultural University of Plovdiv, 12 Mendeleev Blvd., Bulgaria 4000'
author:
- 'Dimitar Razpopov$^{1}$ and Georgi Dzhelepov$^{2}$'
title: The value of the work done by an isotropic vector force field along an isotropic curve
---
Introduction
============
The physical work and the physical force on differentiable manifolds have a great application in physics. Vector fields are often used to model a force, such as the magnetic or gravitational force, as it changes from one point to another point. As a particle moves through a force field along a curve $c$, the work done by the force is the product of force and displacement. There are some papers concerning physical results on light-like (degenerate) objects of differentiable manifolds ([@duggal], [@Korpinar] and [@robert]).
The object of the present paper is a 3-dimensional differentiable manifold $M$ equipped with a Riemannian metric $g$ and a tensor $Q$ of type $(1, 1)$, whose third power is the identity and $Q$ acts as an isometry on $g$. Such a manifold $(M, g, Q)$ is defined in [@dzhelepov2] and studied in [@AE-dok], [@fil-dok], [@dok-raz-dzhe] and [@dzhelepov1]. Also, we consider an associated metric $f$, which is introduced in [@dzhelepov1]. The metric $f$ is necessary indefinite and it determines space-like vectors, isotropic vectors and time-like vectors in the tangent space $T_{p}M$ at an arbitrary point $p$ on $M$.
We investigate physical forces whose vectors are in $T_{p}M$ on $(M, g, Q)$. Moreover, these vectors are isotropic with respect to $f$ and they act along isotropic curves. We study the physical work done by such forces.
Preliminaries
=============
Let $M$ be a $3$-dimensional Riemannian manifold equipped with an endomorphism $Q$ in the tangent space $T_{p}M$, $p \in M$. Let the local coordinates of $Q$ with respect to some coordinate system form the circulant matrix: $$ (Q_{i}^{j})=\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0 \\
\end{pmatrix}.$$ Then $Q$ has the property $$\label{q3}
Q^{3}={\textrm{id}}.$$ Let $g$ be a positive definite metric on $M$, which satisfies the equality $$\label{ff1}
\quad g(Qr, Qi)=g(r,i).$$ In and further $r, i, w$ will stand for arbitrary vectors in $T_{p}M$.
Such a manifold $(M, g, Q)$ is introduced in [@dzhelepov2].
It is well-known that the norm of every vector $i$ is given by $\|i\|=\sqrt{g(i,i)}.$ Then, having in mind (\[ff1\]), for the angle $\varphi = \angle(i,Qi)$ we have $$ \cos \varphi=\frac{g(i,Qi)}{g(i,i)}.$$ In [@dzhelepov2], for $(M, g, Q)$, it is verified that the angle $\varphi$ is in $[0,\frac{2\pi}{3}]$. If $\varphi \in (0, \frac{2\pi}{3})$, then the vector $i$ form a basis $\{i,Qi, Q^{2}i\}$, which is called a $Q$-basis of $T_{p}M$.
The associated metric $f$ on $(M,g,Q)$, determined by $$\label{f3}
f(r,i)=g(r,Qi)+g(Qr,i).$$ is necessary indefinite [@dzhelepov1].
A vector $r$ in $T_{p}M$ is isotropic with respect to $f$ if $$\label{f4}
f(r,r)=0.$$
In every $T_{p}M$, for $(M, g, Q)$, there exists an orthonormal $Q$-basis $\{i, Qi, Q^{2}i\}$ ([@dzhelepov2]). From , (\[f3\]) and (\[f4\]) we state the following
Let $\{i, Qi, Q^{2}i\}$ be an orthonormal $Q$-basis of $T_{p}M$. If $r=ui+vQi+qQ^{2}i$ is an isotropic vector, then its coordinates satisfy $$\label{f5}
uv+vq+qu=0.$$
An isotropic (null) curves $c: r=r(t)$ are those whose tangent vectors are everywhere isotropic, i.e., $$\label{is-curve}
f(dr, dr)=0.$$
The physical forces are represented by vector fields. We investigate physical forces whose vectors are in $T_{p}M$ on $(M, g, Q)$. Moreover, these vectors are isotropic and they act along isotropic curves. We study the physical work done by such forces.
The work in $T_{p}M$
====================
We consider an orthonormal $Q$-basis $\{i, Qi, Q^{2}i\}$ in $T_{p}M$ on $(M, g, Q)$.
Let $p_{xyz}$ be a coordinate system such that the vectors $i$, $Qi$ and $Q^{2}i$ are on the axes $p_{x}$, $p_{y}$ and $p_{z}$, respectively. So $p_{xyz}$ is an orthonormal coordinate system.
The curve $c$ is determined by $$\label{ff4}
c:r(t)=x(t)i+y(t)Qi+z(t)Q^{2}i,$$ where $t\in [\alpha, \beta]\subset \mathbb{R}$.
Let $c$ be an isotropic smooth curve. Thus equalities , , and imply $$\label{f7}
dxdy+dydz+dxdz=0.$$
We determine a vector force field $$\label{force}
F(x,y,z)=P(x,y,z)i+R(x,y,z)Qi+S(x,y,z)Q^{2}i,$$ where $P=P(x, y, z)$, $R=R(x, y, z)$, $S=S(x, y, z)$ are smooth functions.
Let the vector field $F$ be isotropic. Hence following we get $$\label{f6}
PR+RS+SP=0.$$
Work $A$ done by a force $F$, with respect to $f$, moving along a curve $c$ is given by $$\label{f8}
A=\int_{c}f(F,dr),$$ where $$\label{f9}
dr=dxi+dyQi+dzQ^{2}i.$$
Case (A) Let $F$ and $c$ are both isotropic and they are on the same direction. Since $c$ is a trajectory of $F$ we have that the vectors $F$ and $dr$ are collinear. Therefore their coordinates satisfy $$\label{f10}
\frac{dx}{P}=\frac{dy}{R}=\frac{dz}{S}=\frac{1}{k},$$ where $k\neq0$ is a function. From and it follows $F=kdr$. Then, having in mind and , we get $dA =f(kdr,dr)= kf(dr,dr)=0$, i.e., $A=0$.
Case (B) Now, we consider the case when $F$ and $c$ are both isotropic but they are on different directions.
From , and it follows $$\label{work}
A=\int_{c}\big[P(dy+dz)+R(dx+dz)+S(dx+dy)],$$ and hence $$\label{work2}
A=\int_{\alpha}^{\beta}\big[P(y'+z')+R(x'+z')+S(x'+y')]dt.$$
- Let $dx+dy=0$. From (\[f7\]) we have $dx=dy=0$ and $dz\neq 0$. Then $dr=dzQ^{2}i$. Therefore, using , we get $$\label{f10*}
A=\int^{\beta}_{\alpha}\big(P(k_{1},k_{2},t)+R(k_{1},k_{2},t)\big)dt,$$ where $k_{1}$ and $k_{2}$ are specific constants.
- Let $P+R=0$. From we have $P=R=0$ and hence $S\neq 0$. In this case equalities and imply $$\label{f11}
A = \int^{\beta}_{\alpha}S(x(t),y(t),z(t))[x'(t)+y'(t)]dt.$$
- Let $dx+dy\neq 0$ and $P+R\neq0$. With the help of and we get $$\label{f12}
dz=-\frac{dxdy}{dx+dy}, \quad S=-\frac{PR}{P+R}.$$ We use , and and obtain that the work $A$ is determined by $$\label{f13}
A = \int^{\beta}_{\alpha}\frac{(Py'-Rx')^{2}}{(P+R)(x'+y')}dt.$$
From Case (A) and Case (B) we state the following
Let $f$ be the associated metric on $(M,g,Q)$. Let $p_{xyz}$ be a coordinate system such that the vectors $i$, $Qi$ and $Q^{2}i$ of the orthonormal $Q$-basis in $T_{p}M$ are on the axes $p_{x}$, $p_{y}$ and $p_{z}$, respectively. Let $F$ be an isotropic vector force field moving along an isotropic curve $c$. Let $A$ be the work done by $F$. Then
- $A$ is zero if $F$ and $c$ are on the same direction;
- $A$ is if $F$ and $c$ are on different directions and $dx+dy=0$;
- $A$ is if $F$ and $c$ are on different directions and $P+Q=0$;
- $A$ is if $F$ and $c$ are on different directions and $dx+dy\neq 0$ and $P+R\neq0$.
Work in a $2$-plane
===================
Now we consider an arbitrary $2$-plane $\alpha=\{i,Qi\}$ in $T_{p}M$. We suppose that the angle $\varphi=\angle(i,Qi)$ belongs to the interval $(0, \frac{2\pi}{3}]$. On $\alpha$ we construct a coordinate system $p_{xy}$ such that $i$ is on the axis $p_{x}$ and $j$ is on the axis $p_{y}$, where $$\label{defw}
j=\frac{1}{\sin \varphi}(-\cos \varphi i+Qi).$$ We assume that $\|i\|=1$ and then $p_{xy}$ is an orthonormal coordinate system.
In [@sf] it is proved the following
Let $f$ be the associated metric on $(M,g,Q)$ and let $\alpha=\{i, Qi\}$ be an arbitrary $2$-plane in $T_{p}M$. Let the vector $j$ be defined by and $p_{xy}$ be a coordinate system such that $i\in p_{x}$, $j\in p_{y}$. Then the equation of the circle $ c:\ f(w, w)=a^{2}$ in $\alpha$ is given by $$\label{18}
(\cos\varphi)x^{2}+\frac{(1-\cos\varphi)(1+2\cos\varphi)}{\sin\varphi}xy-\frac{\cos^{2}\varphi}{1+\cos\varphi}y^{2}=\frac{a^{2}}{2}\ ,$$ where $\varphi\in(0, \frac{2\pi}{3}]$.
Let $w=ui+vj$ be an isotropic vector, i.e., $f(w,w)=0$. Therefore, with the help of , we obtain $$\label{ff5}
\cos^{2}\varphi\big(\frac{y}{x}\big)^{2}-\sin\varphi(1+2\cos\varphi)\frac{y}{x}-(1+\cos\varphi)\cos\varphi=0.$$ The discriminant of (\[ff5\]) is $$D = (1+\cos \varphi)(1+3\cos \varphi).$$ Then we get the following cases:
**Case (A)** If $\varphi \in(\arccos (-\frac{1}{3}), \frac{2\pi}{3})$, then $D<0$. There is no isotropic directions in $T_{p}M$.
**Case (B)** If $\varphi=\arccos(-\frac{1}{3})$, then $D=0$. We have one isotropic straight line $c:\ y=\sqrt{2}x$. Then the force $F$ and the curve $c$ both are on one isotropic direction and the work $A$ of the force $F$ along $c$ is zero.
**Case (C)** If $\varphi\in\big(0, \frac{\pi}{2}\big)\bigcup\big(\frac{\pi}{2}, \arccos(-\frac{1}{3})\big)$, then $D>0$. We have two isotropic directions which generate two straight lines: $$ c_{1}: y = k_{1}x, \quad c_{2}: y=k_{2}x, \quad x\in[\alpha,\beta],$$ where $k_{1}$ and $k_{2}$ are solutions of the equation (\[ff5\]) for $\frac{y}{x}$.
- If $F$ is on $c_{1}$, then the work of $F$ along $c_{1}$ is zero. Similarly, if $F$ is on $c_{2}$, then the work of $F$ along $c_{2}$ is zero.
- We suppose that $F$ is on $c_{2}$ but $F$ acts on $c_{1}$. Then $$\label{ff6}
F(x,y)=P(x,y)(i+k_{2}j), \quad dr=dt(i+k_{1}j).$$ Bearing in mind and we calculate $$\label{ff7}
\begin{array}{ll}
g(i,Qi)=g(Qi,i)=\cos\varphi, & g(i,Qj)=g(Qj,i)=\frac{\cos\varphi-\cos^{2}\varphi}{\sin\varphi},\\ g(j,Qi)=g(Qi,j)=\sin\varphi, & g(j,Qj)=g(Qj,j)=-\frac{\cos^{2}\varphi}{1+\cos\varphi}.
\end{array}$$ On the other hand the solutions $k_{1}$ and $k_{2}$ of satisfy equalities $$\label{ff9}
k_{1}+k_{2}=\frac{\sin\varphi(1+2\cos\varphi)}{\cos^{2}\varphi}, \quad k_{1}k_{2}=-\frac{1+\cos\varphi}{\cos\varphi}.$$ Using , , , (\[ff7\]) and (\[ff9\]) we find $$A=\frac{1+3\cos\varphi}{\cos^{2}\varphi}\int^{\beta}_{\alpha} P(t,k_{1}t)dt.$$
- Similarly, if $F$ is on $c_{1}$ and $F$ acts on $c_{2}$ we get $$ A=\frac{1+3\cos\varphi}{\cos^{2}\varphi}\int^{\beta}_{\alpha} P(t,k_{2}t)dt.$$
**Case (D)** Finally, the condition $\varphi=\frac{\pi}{2}$ applied to yields $j=Qi$. Then $i$ and $j$ are isotropic vectors. Therefore, from and it follows:
- $F=P(t,0)Qi$, $dr=(dt)i$. The work is $A=\int^{\beta}_{\alpha}P(t,0)dt$.
- $F=P(0,t)i$, $dr=(dt)Qi$. The work is $A=\int^{\beta}_{\alpha}P(0,t)dt$.
[llll]{} $\varphi$ & $F$ acts on & trajectory of $F$ & $A$\
$(\arccos (-\frac{1}{3}), \frac{2\pi}{3})$ & - & no is. curves & -\
$\arccos(-\frac{1}{3})$ & $c:\ y=\sqrt{2}x$ & $c:\ y=\sqrt{2}x$ & 0\
$\big(0, \frac{\pi}{2}\big)\bigcup\big(\frac{\pi}{2}, \arccos(-\frac{1}{3})\big)$ & $c_{1}: y = k_{1}x$ & $c_{1}: y=k_{1}x$ & 0\
$\big(0, \frac{\pi}{2}\big)\bigcup\big(\frac{\pi}{2}, \arccos(-\frac{1}{3})\big)$ & $c_{2}: y = k_{2}x$ & $c_{2}: y=k_{2}x$ & 0\
$\big(0, \frac{\pi}{2}\big)\bigcup\big(\frac{\pi}{2}, \arccos(-\frac{1}{3})\big)$ & $c_{1}: y = k_{1}x$ & $c_{2}: y=k_{2}x$ & $A=\frac{1+3\cos\varphi}{\cos^{2}\varphi}\int^{\beta}_{\alpha} P(t,k_{1}t)dt$\
$\big(0, \frac{\pi}{2}\big)\bigcup\big(\frac{\pi}{2}, \arccos(-\frac{1}{3})\big)$ & $c_{2}: y = k_{2}x$ & $c_{1}: y=k_{1}x$ & $A=\frac{1+3\cos\varphi}{\cos^{2}\varphi}\int^{\beta}_{\alpha} P(t,k_{2}t)dt$\
$\frac{\pi}{2}$ & $c_{1}: x=0$ & $c_{2}: y=0$ & $A=\int^{\beta}_{\alpha}P(t,0)dt$\
$\frac{\pi}{2}$ & $c_{1}: y=0$ & $c_{2}: x=0$ & $A=\int^{\beta}_{\alpha}P(0,t)dt$.\
The results in Case (A) – Case (D) are summarized in Table \[tab:1\].
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by project ”17-12 Supporting Intellectual Property” of the Center for Research, Technology Transfer and Intellectual Property Protection, Agricultural University of Plovdiv, Bulgaria.
[9]{}
I. Dokuzova 2017 Almost Einstein manifolds with circulant structures, [*J. Korean Math. Soc.*]{} [**54**]{} (5), 1441-1456.
I. Dokuzova 2018 On a Riemannian manifolds with a circulant structure whose third power is the identity, [*Filomat*]{} [**32**]{} (10), 3529-3539.
I. Dokuzova, D. Razpopov and G. Dzhelepov 2018 Three-dimensional Riemannian manifolds with circulant structures, [*Adv. Math., Sci. J.*]{} [**7**]{} (1), 9–16.
K. L. Duggal and B. Sahin 2010 [*Differential Geometry of Lightlike Submanifolds*]{}, Frontiers in Mathematics, (Basel: Birkhäuser), p. 488.
G. Dzhelepov 2018 Spheres and circles in the tangent space at a point on a Riemannian manifold with respect to an indefinite metric,[*Novi Sad J. Math.*]{} [**48**]{} (1), 143-150.
G. Dzhelepov, I. Dokuzova and D. Razpopov 2011 On a three-dimensional Riemannian manifold with an additional structure, [*Plovdiv. Univ. Paisii Khilendarski Nauchn. Trud. Mat.*]{} [**38**]{} (3), 17–27.
G. Dzhelepov, D. Razpopov and I. Dokuzova 2010 Almost conformal transformation in a class of Riemannian manifolds, [*In: Research and Education in Mathematics, Informatics and their Applications - REMIA 2010, Proc. Anniv. Intern. Conf.*]{} Plovdiv, Bulgaria, 125–128.
T. Korpınar and R. C. Demirkol 2017 Energy on a timelike particle in dynamical and electrodynamical force fields in De-Sitter space, [*Revista Mexicana de Fısica*]{} [**63**]{}, 560-568.
Robert H. Wasserman 2004 [*Tensors and Manifolds: With Applications to Physics*]{}, (New York: Oxford University Press), p. 447.
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---
abstract: 'In this paper we investigate the role of native geometry on the kinetics of protein folding based on simple lattice models and Monte Carlo simulations. Results obtained within the scope of the Miyazawa-Jernigan indicate the existence of two dynamical folding regimes depending on the protein chain length. For chains larger than 80 amino acids the folding performance is sensitive to the native state’s conformation. Smaller chains, with less than 80 amino acids, fold via two-state kinetics and exhibit a significant correlation between the contact order parameter and the logarithmic folding times. In particular, chains with N=48 amino acids were found to belong to two broad classes of folding, characterized by different cooperativity, depending on the contact order parameter. Preliminary results based on the Gō model show that the effect of long range contact interaction strength in the folding kinetics is largely dependent on the native state’s geometry.'
author:
- 'P. F. N. Faisca and M. M. Telo da Gama'
title: Native geometry and the dynamics of protein folding
---
Introduction
============
It is well known that most small (from $\sim$ 50-120 amino acids), single domain proteins fold via a two-state (single exponential) kinetics, without observable folding intermediates and with a single transition state associated with one major free energy barrier separating the native state from the unfolded conformations [@JACKSON; @PLAXCO0; @KAYA0]. For this reason small protein molecules are particularly well suited for investigating the correlations between folding times and the native state equilibrium properties, a major challenge for those working in protein research.
The energy landscape theory predicts that the landscape’s rugedeness plays a fundamental role in the folding kinetics of model proteins: The existence of local energy minima, that act as kinetic traps, is responsible for the overall slow and, under some conditions (as the temperature is lowered towards the glass transition temperature), glassy dynamics [@WOLYNES]. On the other hand, rapid folding is associated with the existence of a smooth, funnel-shaped energy landscape [@ONUCHIC]. In Refs. [@PLAXCO1; @GILLESPIE] Plaxco [*et. al.*]{} and Gillespie and Plaxco have provided experimental evidence that the folding energy landscape of single domain proteins is extremly smooth even at considerably low temperatures. Therefore differences in the landscape’s ‘topography’ cannot account for the vast range of folding rates as observed in real proteins [@LEE; @DOBSON]. However, a strong correlation (r=0.94) was found between the so-called contact order parameter, CO, and the experimentally observed folding rates in a set of 24 non-homologous single domain proteins [@PLAXCO2]. The CO measures the average sequence separation of contacting residue pairs in the native structure relative to the chain length of the protein $$CO=\frac{1}{LN}\sum_{i,j}^N \Delta_{i,j}\vert i-j \vert,
\label{eq:no1}$$ where $\Delta_{i,j}=1$ if residues $i$ and $j$ are in contact and is 0 otherwise; $N$ is the total number of contacts and $L$ is the protein chain length. The empirical observation that the CO correlates well with the folding rates of single domain proteins, exhibiting smooth energy landscapes, strongly suggests a geometry-dependent kinetics for such two-state folders.
The connection between the CO and the dominant range of residue interactions brings back an old, well-debated issue in the protein folding literature, that of the role of local (i.e. close in space and in sequence) and long range (i.e. close in space but distant along the sequence) inter-residue interactions in the folding dynamics. Several results appear to agree on the idea that long range (LR) contacts play an active role in stabilizing the native fold [ [@GO; @GUTIN; @BAKER; @GROMIHA0]]{}. In what regards the folding kinetics, results reported in Refs. [@GO; @BAKER; @WETLAUFER; @MOULT] suggest that local contacts increase the folding speed, relative to LR contacts, while results in Ref. [@GUTIN] suggest an opposite trend. In Ref. [@GROMIHA2] Gromiha and Selvaraj have analysed explicitly the contribution of LR contacts in determining the folding rates of 23 (out of the 24) two-state folders studied by Plaxco [*et al*]{} [@PLAXCO2]. These authors proposed the so-called long range order (LRO) parameter, measuring the total number of long range contacts relative to the protein chain length, as an alternative way of quantifying the native structure geometry. In fact, the LRO parameter correlates as well as the CO with the folding rates of the two-state folders analysed in Ref. [@PLAXCO2].
The majority of protein folding theory is based, not only on results for real proteins such as those outlined above, but also on a vast number of findings obtained within the scope of simple lattice models and Monte Carlo (MC) simulations. Although lattice models do not encompass the full complexity of real proteins they are non trivial and capture fundamental aspects of the protein folding kinetics [@SFERMI].
In the present study we investigate through Monte Carlo folding simulations of simple lattice models, such as the Miyazawa-Jernigan model and the Gō model, the dependence of two-state folding kinetics on the native state geometry.
LATTICE MODELS \[sec:secno2\]
=============================
In a lattice model the protein is reduced to its backbone structure: amino acids are represented by beads of uniform size, occupying the lattice vertices, and the peptide bond, that covalently connects amino acids along the polypeptide chain, is represented by sticks, with uniform length, corresponding to the lattice spacing. We model proteins as three-dimensional, self-avoiding chains of $N$ beads. To mimick amino acid interactions we use either the Miyazawa-Jernigan model or the Gō model.
The Miyazawa-Jernigan model
---------------------------
In the Miyazawa-Jernigan (MJ) model the energy of a conformation defined by the set of bead coordinates $\lbrace \vec{r_{i}} \rbrace$ is given by the contact Hamiltonian $$H(\lbrace \sigma_{i} \rbrace,\lbrace \vec{r_{i}} \rbrace)=\sum_{i>j}^N
\epsilon(\sigma_{i},\sigma_{j})\Delta(\vec{r_{i}}-\vec{r_{j}}),
\label{eq:no2}$$ where $\lbrace \sigma_{i} \rbrace$ represents an amino acid sequence, and $\sigma_{i}$ stands for the chemical identity of bead $i$. The contact function $\Delta$ is $1$ if beads $i$ and $j$ are in contact but not covalently linked and is $0$ otherwise. The interaction parameters $\epsilon$ are taken from the $20 \times 20$ MJ matrix, derived from the distribution of contacts of native proteins [@MJ].
The Gō model
------------
In the Gō [@GO] model only native contacts, i.e. contacts that are present in the native state, contribute to the energy of a conformation defined by $\lbrace \vec{r_{i}} \rbrace$. In this case the contact Hamiltonian is $$H(\lbrace \vec{r_{i}} \rbrace)=\sum_{i>j}^N
B_{ij} \Delta(\vec{r_{i}}-\vec{r_{j}}),
\label{eq:no3}$$ where the contact function $\Delta (\vec{r_{i}}-\vec{r_{j}})$, is unity only if beads $i$ and $j$ form a non-covalent native contact and is zero otherwise. Since the Gō model ignores the protein sequence chemical composition the interaction energy parameter is $B_{ij}=-\epsilon$.
Simulation details
==================
Our folding simulations follow the standard MC Metropolis algorithm [@METROPOLIS] and, in order to mimick protein movement, we use the kink-jump move set, including corner flips, end and null moves as well as crankshafts [@BINDER].
Each MC run starts from a randomly generated unfolded conformation (typically with less than 10 native contacts) and the folding dynamics is traced by following the evolution of the fraction of native contacts, $Q=q/Q_{max}$, where $Q_{max}$ is the total number of native contacts and $q$ is the number of native contacts at each MC step. The folding time $t$, is taken as the first passage time (FPT), that is, the number of MC steps corresponding to $Q=1.0$.
The folding dynamics is studied at the so-called optimal folding temperature, the temperature that minimizes the folding time as measured by the mean FPT.
The sequences studied within the context of the MJ model were prepared by using the design method developed by Shakhnovich and Gutin (SG) [@SG] based on random heteropolymer theory and simulated annealing techniques. All targets studied are maximally compact structures found by homopolymer relaxation.
Numerical results \[sec:secno3\]
================================
Evidence for two folding regimes in protein folding
---------------------------------------------------
Figure \[figure:no1\]1(a) shows the dependence on time $t$, of the folding probability $P_{fold}(t)$, for chain lengths $N=27,36,48,54,64,80,100$. Five target structures were considered per chain length and thirty SG sequences were prepared according to the method described in Ref. [@SG].
$P_{fold}(t)$, the probability of the chain having visited its target after time $t$, was computed as the fraction of (150) simulation runs, which ended at time $t$. Two distinct folding regimes were identified depending on the chain length. We name the regime observed for $N<80$ the [*first regime*]{} while that corresponding to $N \ge 80$ is the [*second regime*]{}. We have investigated the contribution of each target to the folding probability curve and found that for $N \ge 80$ the folding performance is sensitive to target conformation, with some targets being more foldable than others as shown in Figure \[figure:no1\]1(b) for N=100. For $N<80$ targets are equally foldable since all folding probability curves are consistent with asymptotic values $P_{fold}\rightarrow 1$.
In order to investigate if kinetic relaxation in the first regime is well described by a single exponential law we have calculated the dependence of $\ln
(1-P_{fold})$ on the time coordinate $t$. Remember that in a two-state process the reactant concentration (the equivalent in our simulations to the fraction of unfolded chains) is proportional to $\exp^{-t/\tau}$ where $\tau$ is the so-called relaxation time. Therefore, if first regime kinetics is single exponential $\ln (1-P_{fold})$ $\it vs.$ $t$ should be a straight line with slope=$-1/\tau$. Results reported in Figure \[figure:no2\]2 show that single-exponential folding is indeed a very good approximation for the folding kinetics of small lattice-polymer proteins.
Contact order and the lattice-polymer model kinetics
----------------------------------------------------
In a recent study [@PFN2] we analysed the folding kinetics of $\approx 5000$ SG sequences and $100$ target conformations distributed over the chain lengths $N=36,48,54,64$ and $80$. Targets were selected in order to cover the observed range of CO ($\approx 0.12<$ CO $<0.26$). Results reported in Ref. [@PFN2] show a significant correlation ($r=0.70-0.79$) between increasing CO and longer logarithmic folding times. In Ref. [@JEWETT] Jewett [*[et al.]{}*]{} found a similar corelation ($r=0.75$) for a 27-mer lattice polymer modeled by a modified Gō-type potential. In a recent study, Kaya and Chan [@KAYA2] studied a modified Gō type model, with specific many-body interactions, and found folding rates that are very well correlated ($r=0.91$) with the CO and span a range that is two orders of magnitude larger than that of the corresponding Gō models with additive contact energies. These results support the empirical relation found between the contact order and the kinetics of two-state folders.
Contact order and structural changes towards the native fold in the Miyazawa-Jernigan model
-------------------------------------------------------------------------------------------
In order to investigate if native geometry as measured by the CO promotes, or does not promote, different folding processes, eventually leading to different folding rates, we have analysed the dynamics of 900 SG sequences with chain length $N=48$, distributed over nine target structures with low (0.126, 0.127, 0.135), intermediate (0.163, 0.173, 0.189) and high (0.241, 0.254, 0.259) contact order. The averaged trained sequence energy shows very little dispersion ranging from -25.11$\pm$0.03 to -26.16$\pm$0.02. Within this target set the folding time and the contact order correlate well ($r=0.82$) although the dispersion of folding times is small as reported in Figure \[figure:no3\]3.
The contact map is a $N \times N$ matrix with entries $C_{ij}=1$ if beads $i$ and $j$ are in contact (but not covalently linked) and are zero otherwise. Figure \[figure:no4\]4 shows the contact maps of targets T1 (CO=0.126), T2 (CO=0.189) and T3 (CO=0.259) respectively. One could argue that high-CO targets are associated with longer logarithmic folding times because they have predominantly LR contacts which, given the local nature of the move set used to simulate protein movement, eventually take a longer time to form. Let the contact time $t_{0}$ be the mean FPT of a given contact averaged over 100 MC runs. The longest contact time ($\ln t_{0}=12.24$) observed for target T3 is two orders of magnitude shorter than T3’s folding time ($\ln t=17.59$) and the sum of all contact times is $\ln(\sum_{i=1}^{57}t_{0}^{i})=15.51$, much lower than the observed folding time. Thus, the fact that T3 and other high-CO structures have predominantly LR contacts cannot justify, [*per se*]{}, their higher folding times.
The contact map provides a straightforward way to compute the frequency $\omega_{ij}=t_{ij}/t$ with which a native contact occurs in a MC run, $t_{ij}$ being the number of MC steps corresponding to $C_{ij}=1$ and $t$ the folding time. For each target studied we computed the mean frequency of each native contact $<\omega_{ij}>$ averaged over 100 simulation runs, and re-averaged $<\omega_{ij}>$ over the number of native contacts in each interval of backbone distance (we measure backbone distance in units of backbone spacing). We have found that while for the low-CO targets the backbone frequency decreases monotonically with increasing backbone distance, for the intermediate and high-CO targets such dependence is clearly nonmonotonic. Figure \[figure:no5\]5 illustrates this behaviour for model structures T1, T2 and T3 elements of the low, intermediate and high-CO target sets respectively. A possible explanation for this behavior, that we have ruled out, is that of a negative correlation between the frequency and the energy of a contact; Could the most stable contacts be the most frequent ones? We found modest correlation coefficients $r=0.63$ and $r=0.65$ for targets T1 and T3 respectively and therefore we conclude that the observed behaviour is not energy driven.
In Table I we show the dependence of the contact time, averaged over contacts in each interval of backbone distance, on the backbone distance for model targets T1 and T3. Since the average contact times, over a given range, are similar for these extreme model structures, the differences in the frequencies reported in Figure \[figure:no5\]5 must necessarilly distinguish different cooperative behaviors.
Results outlined above suggest that two broad classes of folding mechanisms exist for small MJ lattice polymer protein chains. What distinguishes these two classes is the presence, or absence, of a monotonic decrease of contact frequency with increasing contact range that is related to different types of cooperative behaviour. The monotonic decrease of contact frequency with increasing backbone distance is a specific trait of low-CO structures. In this case folding is also less cooperative and is driven by backbone distance: Local contacts form first while LR contacts form progressively later as contact range increases.
Contact order, long-range contacts and protein folding kinetics in the Gō model
-------------------------------------------------------------------------------
The energy landscapes of Gō-type polymers are considerably smooth because in the Gō model the only favourable interactions are those present in the native state. Therefore such models are adequate for investigating the dependence of protein folding kinetics on target geometry.
In this section we investigate the contribution of LR and local interactions to the folding kinetics of targets T1, T2 and T3 (Figure \[figure:no4\]4) in the following way: The total energy of the native structure is kept constant but the relative contributions of LR and local interactions are varied over a broad range. With the above costraint the energy of a conformation is given by $$H(\lbrace \vec{r_{i}} \rbrace, \sigma)=C_{LR}(\sigma) H_{LR}(\lbrace \vec{r_{i}} \rbrace)+
C_{L}(\sigma)H_{L}(\lbrace \vec{r_{i}} \rbrace),
\label{eq:no4}$$ where $C_{LR}({\sigma})={\sigma}/\lbrack(1-\sigma)Q_{L}+\sigma(1-Q_{L})\rbrack$ and $C_{L}({\sigma})=(1-{\sigma}) / \lbrack (1-\sigma)Q_{L}+\sigma(1-Q_{L}) \rbrack$; $Q_{L}$ is the fraction of local native contacts and $H_{LR}(L)$ is given by equation \[eq:no3\]. The parameter $\sigma$ varies from 0 (only local contacts contribute to the total energy) to 1 (only LR contacts contribute to the total energy).
The constraint of fixed native state energy is enforced to rule out differences in the folding dynamics driven by the stability of the native state.
Preliminary results reported in Figure \[figure:no6\]6 show the dependence of the logarithmic folding time, averaged over 100 simulations runs, on the parameter $\sigma$ for the three native geometries. For $\sigma < 0.15$ we have not observed folding of the target T3 and no folding was observed for the target T2 if $\sigma < 0.10$.
The behaviour exhibited by target T3 is easily explained: since approximately 80 percent of T3’s native contacts are LR there is little competition between LR and local contacts. Moreover, such competition is not significantly enhanced when one varies $\sigma$ towards unity. However, the effect of decreasing $\sigma$ is equivalent to that of ‘switching off’ the LR contacts, that is, to force a structure to fold with only approximatly 20 per cent of its total native contacts resulting in longer folding times and for $\sigma < 0.15$ folding failure. More intriguing are the results obtained for the low and intermediate-CO target structures, T1 and T2 respectively. The curves are qualitatively similar (with a minimum at $\sigma >0.5$) but closer inspection reveals an important difference, namely: for $\sigma < 0.5$ the dependence of the folding time on $\sigma$ is much stronger for the intermediate-CO target, T2. Indeed, in this case one observes a remarkable three-order of magnitude dispersion of logarithmic folding times, ranging from $\log (t)=5.62$ (for $\sigma=0.65$) to $\log(t)=8.50$ (for $\sigma=0.10$). We stress, however, that for both targets the kinetics is more sensitive (in the sense that the folding rate decreases more rapidly) to lowering $\sigma$: LR contacts appear therefore to have a crucial/vital role, by comparison with local contacts, in determining the folding rates of small Gō-type lattice polymers and this effect depends on target geometry.
Conclusions and final remarks
=============================
By using different target structures in MC simulations of protein folding we have identified two distinct folding regimes depending on the chain length. In close agreement with experimental observations we found a first regime that describes well the folding of small protein molecules and whose kinetics is single exponential. Folding of protein chains with more than 80 amino acids, on the other hand, belongs to a dynamical regime that appears to be target sensitive with some targets being more highly foldable than others. In this case we ascribe folding failure to existing kinetic traps but we have not been able to carry out our simulations for long enough times in order to observe escape and succesfull folding.
Because the additive MJ lattice polymer model fails to exhibit the remarkable dispersion of folding rates observed in real proteins one should interpret the results for the dependence of folding times on contact order parameter with caution. However, our results strongly suggest that the geometry driven cooperativity is rather robust and this implies an increase in folding times for increasing cooperativity.
We have analysed the role of LR contacts in the folding kinetics of small Gō-type lattice polymers and found a considerably strong dependence on target geometry. In particular, we have found that targets with a similar fraction of LR contacts (that is, targets with similar LRO parameter) and different contact order exhibit considerably different folding rates when LR contacts are destabilized energetically with respect to local contacts. We are currently investigating this issue and results will be published elsewhere (in preparation). This result may provide a clue to understanding the increadible dispersion of folding rates exhibited by real two-state folders: one can expect to observe longer folding times if the distribution of contact energies in real proteins is such that local contacts are, on average, more stable than LR contacts for specific native folds.
Acknowledgements
================
P.F.N.F. would like to thank Fundação para a Ciência e Tecnologia for financial support through grant No. BPD10083/2002.
[99]{}
[**[FIGURES]{}**]{}
\[figure:no1\]
\[figure:no2\]
\[figure:no3\]
\[figure:no4\]
\[figure:no5\]
\[figure:no6\]
[**[FIGURE CAPTIONS]{}**]{}
Figure 1. Dependence of the folding probability, $P_{fold}$, on $\log(t)$. (a) For each of the chain lengths $N=27,36,48,64,80$ and $100$ five target structures were considered and 30 sequences were designed per target. $P_{fold}$, the probability of the chain having visited its target after time $t$, was computed as the fraction of simulation runs that ended in time $t$. (b) Separate contribution of each of the 100 bead long targets for the dependence of $P_{fold}(t)$ on $\log(t)$ [@FAISCA].
Figure 2. Evidence for single exponential folding kinetics for chain length $N=48$. The correlation coeficient between the logarithmic fraction of unfolded chains and ‘reaction’ time is $r \approx 0.97$ for target T4 and $r \approx 0.99$ for the remaining targets.
Figure 3. Dependence of the logarithmic folding times, $\ln_{e} t$, on the contact order parameter ($r \approx 0.82$).
Figure 4. [Contact maps of targets T1 (a), T2 (b) and T3 (c). Each square represents a native contact. We divide the 57 native contacts into two classes: LR contacts are represented by filled squares and correspond to contacts between beads for which the backbone separation is 10 or more backbone units. Local contacts are represented by white squares. There are 23 LR contacts in structure T1, 21 in structure T2 and 44 in structure T3.]{}
Figure 5. The backbone frequency, $<\omega_{\vert i-j \vert}>$, as a function of the backbone separation for the low-CO, high-CO and intermediate-CO target. The backbone frequency is the mean value of $<\omega>$ averaged over the number of contacts in each interval of backbone separation.
Figure 6. Dependence of the logarithmic folding time $\log_{10} t$ on the parameter sigma. The parameter $\sigma$ varies from 0 (only local contacts contribute to the total energy) to 1 (only LR contacts contribute to the total energy).
[**[TABLES]{}**]{}
-------- ----------------- ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ ------------------
Target
$T1$ 8.14 $\pm$ 0.12 10.67 $\pm$ 0.21 11.47 $\pm$ 0.07 11.69 $\pm$ 0.13 11.39 $\pm$ 0.07 - 11.58 $\pm$ 0.10 -
$T3$ 7.66 $\pm$ 0.10 11.14 $\pm$ 0.18 10.88 $\pm$ 0.10 11.60 $\pm$ 0.06 12.06 $\pm$ 0.05 12.24 $\pm$ 0.08 11.82 $\pm$ 0.05 11.61 $\pm$ 0.05
-------- ----------------- ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ ------------------
: The averaged contact time, $\ln_{e}<t_{0}>$, as a function of the backbone separation [@FAISCAPRE][]{data-label="tab:tabno1"}
|
---
abstract: 'Recently, Shi et al. (Phys. Rev. A, 2015) proposed Quantum Oblivious Set Member Decision Protocol (QOSMDP) where two legitimate parties, namely Alice and Bob, play a game. Alice has a secret $k$ and Bob has a set $\{k_1,k_2,\cdots k_n\}$. The game is designed towards testing if the secret $k$ is a member of the set possessed by Bob without revealing the identity of $k$. The output of the game will be either “Yes” (bit $1$) or “No” (bit $0$) and is generated at Bob’s place. Bob does not know the identity of $k$ and Alice does not know any element of the set. In a subsequent work (Quant. Inf. Process., 2016), the authors proposed a quantum scheme for Private Set Intersection (PSI) where the client (Alice) gets the intersected elements with the help of a server (Bob) and the server knows nothing. In the present draft, we extended the game to compute the intersection of two computationally indistinguishable sets $X$ and $Y$ possessed by Alice and Bob respectively. We consider Alice and Bob as rational players, i.e., they are neither “good” nor “bad”. They participate in the game towards maximizing their utilities. We prove that in this rational setting, the strategy profile $((cooperate, abort), (cooperate, abort)$) is a strict Nash equilibrium. If $((cooperate, abort), (cooperate, abort)$) is strict Nash, then fairness as well as correctness of the protocol are guaranteed.'
author:
- Arpita Maitra
date: 'Received: date / Accepted: date'
title: Quantum secure two party computation for set intersection with rational players
---
[**Keywords:**]{} Set Intersection; Quantum Secure Computation; Rationality; Nash Equilibrium
Introduction {#intro}
============
Secure Multiparty Computation (SMC) [@Gordon; @AL11; @GroceK] is an important primitive in cryptology. It has wide applications in electronic voting, cloud computing, online auction etc. Recent trend of the theoretical research in this direction is to combine game theory with cryptology.
Cryptography deals with ‘worst case’ scenario making the protocols secure against various type of adversarial behaviours. Those are characterized as semi honest, malicious and covert adversarial models. In game theory, a protocol is designed against the rational deviation of a party. Rational parties are neither completely “malicious” nor they are fully “honest”. They participate in the game in the motivation towards maximizing their utilities. So placing cryptographic protocols in rational setting empowers more flexibility to the adversary. It seems more practical as in real world most of the people prefer to be rational rather being completely “good” or “bad”.
Recently, Brunner and Linden [@BL] showed a deep link between quantum physics and game theory. They showed that if the players use quantum resources, such as entangled quantum particles, they can outperform the classical players. In [@maitra] the concept of rationality has been introduced in quantum secret sharing. In the present draft, we incorporate the idea of rationality in Secure Computation for Set Intersection (SCSI).
In classical domain this SCSI problem has been studied extensively [@HZ1; @HZ2; @JN10]. It has various applications in dating services, data-mining, recommendation systems, law enforcement etc.
In SCSI, two parties, Alice and Bob, hold two sets $X$ and $Y$ respectively. The sets are assumed to be computationally indistinguishable from each other. Alice and Bob exchange some informations between themselves so that at the end of the protocol, either Alice or Bob (suggested by the protocol) gets $X \cap Y$.
However, the hardness assumptions that Diffie–Hellman (DDH) problem [@Diffi-Hellman], discret logarithm (DL) problem [@Stingson] are computationally hard, have been proven to be vulnerable in quantum domain [@Shor].
In quantum paradigm, Shi et al. [@Shi] proposed a variant of this problem and named it as [*[Quantum Oblivious Set Member Decision Protocol]{}*]{} (QOSMDP). According to the protocol, Alice has a secret element $k$ and Bob holds a set $\{k_1,k_2.\cdots,k_n\}$ of $n$ elements. Now, Bob wants to know if the secret $k$ of Alice is the member of his set. However, Alice does not allow Bob to identify that element. Simultaneously, Bob resists Alice to know a single element except $k$, if it is in the set, of the set.
The authors of [@Shi] commented that it can be exploited to compute the cardinality of the set intersection or union which is the direct consequence of the protocol. Even in [@Shi1], the authors suggested a quantum scheme for Private Set Intersection (PSI) where the client (Alice) gets $X\cap Y$ with the help of a server (Bob) and the server knows nothing. They establish the security of their protocol in “honest but curious” adversarial model.
Contrary to this, in the present draft, we exploit the idea to compute the set intersection in rational setting where the players are trying to maximize their utilities. We show that the strategy profile $((cooperate, abort), (cooperate, abort))$ achieves strict Nash equilibrium in this initiative. We also show that if ((cooperate, abort), (cooperate, abort)) is a strict Nash, then fairness as well as correctness of the protocol are obvious.
In this regard, we like to point out that the procedure described in [@Shi; @Shi1] to detect [*[measure and resend]{}*]{} attack by Bob, requires a more detailed analysis and certain revision. In [@Shi], to detect the attack, the author inserted $l-1$ decoy states encoded as $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_i\right\rangle}})$, where $j_i\in Z_{N}^*$. Each $j_i$ is represented by $\log_2 N$ bits. In [@Shi1], the same type of encoding is exploited to mask the set elements of the client’s set. In both the papers, it is commented that if Bob tries to measure those states, he will introduce noise in the channel which can be detected by Alice by measuring the states in $\{\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_i\right\rangle}}),\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|j_i\right\rangle}})\}$ basis. However, the oracle $O_t$ maps the state $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_i\right\rangle}})$ into $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+(-1)^{q_t(j_i)}{\ensuremath{\left|j_i\right\rangle}})$. Based on the value of $q_t(j_i)\in \{0,1\}$ (not known to Alice), the state will be either $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_i\right\rangle}})$ or $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|j_i\right\rangle}})$. Thus, it requires further clarification how Alice can distinguish the attack from the application of the oracle by measuring the registers in $\{\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_i\right\rangle}}), \frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|j_i\right\rangle}})\}$ basis. To avoid such a security related issue, we modify the protocol accordingly.
Preliminaries
=============
In this section, we discuss the concepts of computational indistinguishability of two distribution ensembles, functionality, rationality, fairness, correctness and equilibrium used in this work.
Computational Indistinguishability
----------------------------------
In communication complexity, two distribution ensembles $X=\{X(a,\lambda)\}_{a\in\{0,1\}^*}$ and $Y=\{Y(a,\lambda)\}_{a\in\{0,1\}^*}$ (where $\lambda$ is the security parameter which usually refers to the length of the input), are computationally indistinguishable if for any non-uniform probabilistic polynomial time algorithm $D$, the following quantity is a negligible function in $\lambda$:
$$\delta (\lambda)=\left|\Pr _{a\gets X(a,\lambda)}[D(a)=1]-\Pr _{a\gets Y(a,\lambda)}[D(a)=1]\right|$$ for every $\lambda\in N$.
In other words, two ensembles are computationally indistinguishable implies that those can not be distinguished by polynomial-time algorithms looking at multiple samples taken from those ensembles.
Functionality
-------------
In classical domain and in two party setting, a functionality $\mathcal{F}=\{f_\lambda\}_{\lambda\in\mathbb{N}}$ is a sequence of randomized processes, where $\lambda$ is the security parameter and $f_\lambda$ maps pairs of inputs to pairs of outputs (one for each party). Explicitly, we can write $f_\lambda=(f_\lambda^{1}, f_\lambda^{2})$, where $f_\lambda^{1}$ represents the output of the first party, say Alice. Similarly, $f_\lambda^{2}$ represents the output of the second party, say Bob. The domain of $f_\lambda$ is $X_\lambda \times Y_\lambda$, where $X_\lambda$ (resp. $Y_\lambda$) denotes the possible inputs of the first (resp. second) party. If $|X_\lambda|$ and $|Y_\lambda|$ are polynomial in $\lambda$, then we say that $\mathcal{F}$ is defined over polynomial size domains. If each $f_\lambda$ is deterministic, we say that each $f_\lambda$ as well as the collection $\mathcal{F}$ is a function [@GK11].
Rationality {#rat}
-----------
Rationality of a player is defined over its utility function $U\in \{u_1,u_2,\cdots,u_n\}$ and its preferences. Each $u_i$, $i \in \{0,1,\cdots,n\}$ is associated with the possible outcomes of the game. The outcomes and corresponding utilities for $2$ players’ set intersection game are described in Table \[table: OutcomesRSS\].
$f_A$ (resp. $f_B$) represents the functionality generated at the place of Alice (resp. Bob) and $U_A$ (resp. $U_B$) represents the utility function of Alice (resp. Bob). Let $\mathcal{F}=X\cap Y$ and $\perp=\emptyset$
[llll]{} $f_A$ & $f_B$ & $U_A(f_A, f_B)$ & $U_B(f_A, f_B)$\
$f_A=\mathcal{F}$ & $f_B$=$\mathcal{F}$ & $U_A^{TT}$ & $U_B^{TT}$\
\
$f_A=\perp$ & $f_B=\perp$ & $U_A^{NN}$ & $U_B^{NN}$\
\
$f_A=\mathcal{F}$ & $f_B=\perp$ & $U_A^{TN}$ & $U_B^{NT}$\
\
$f_A=\perp$ & $f_B=\mathcal{F}$ & $U_A^{NT}$ & $U_B^{TN}$\
\
Here, $TT$, $TN$, $NT$, $NN$ imply
- both Alice and Bob obtain “True” output, i.e., correct and complete values of $\mathcal{F}$.
- Alice (resp. Bob) obtains “True” output where Bob (resp. Alice) obtains “Null” output,
- Alice (resp. Bob) obtains “Null” output but Bob (resp. Alice) gets “True” output,
- both obtain “Null” output
respectively. In this work, we assume that Alice (resp. Bob) has the following order of preferences. $$\mathcal{R}_1 : U^{TN}> U^{TT}>U^{NN}>U^{NT}.$$ That is they prefer to compute the true value of the functionality by herself or himself alone than to compute the true value by both. However, they find it better to compute a null value at both of their ends than to compute a null value by himself or herself when the opponent gets a true value.
Here, one should emphasize that each rational party is only interested to get the complete value of functionality $\mathcal{F}$.
Fairness
--------
A rational player, being selfish, desires an unfair outcome, i.e., he or she always tries to compute the true value of the functionality by himself or herself alone. Therefore, the basic aim of a game when the players are rational should be to achieve fairness.
A formal definition of fairness in the context of a (2,2) Rational Secret Sharing (RSS) protocol was presented by Asharov and Lindell [@AL]. We modify this definition accordingly for our present setting.
Let $\sigma$ be the strategy suggested by the protocol and $\sigma'$ be any deviated strategy. Suppose, Alice has a strategy profile $(\sigma_A,\sigma'_A)$. Similarly Bob has a strategy profile $(\sigma_B,\sigma'_B)$. A game is said to be completely fair if for every arbitrary alternative strategy $\sigma'_A$ followed by Alice, the following holds: $$\begin{aligned}
\Pr[f_A=\mathcal{F}|A=\sigma'_A, B=\sigma_B] \\
< \Pr[f_A=\mathcal{F}|A=\sigma_A, B=\sigma_B].\end{aligned}$$
Here $A$ (resp. $B$) implies the event that Alice (resp. Bob) follows a strategy.
Similarly, for Bob we can write $$\begin{aligned}
\Pr[f_B=\mathcal{F}|A=\sigma_A, B=\sigma'_B] \\
< \Pr[f_B=\mathcal{F}|A=\sigma_A, B=\sigma_B].\end{aligned}$$
In terms of utility function, a game achieves fairness if and only if for a party, the following holds: $$\begin{aligned}
U^{TT} \geq E[U (\mathcal{O}_i)], \end{aligned}$$ where, E(U) is the expected utility value of the player for the input $i$, $i\in\{1,\cdots,n\}$ and $\mathcal{O}_i$ is the corresponding outcome.
Correctness
-----------
A formal definition of correctness in the context of a (2,2) RSS protocol was presented by Asharov and Lindell [@AL]. We modify this definition for the setting as follows:
(Correctness): Let $\sigma$ be the strategy suggested by the protocol and $\sigma'$ be any deviated strategy. Let Alice has a strategy profile $(\sigma_A,\sigma'_A)$. Similarly Bob has a strategy profile $(\sigma_B,\sigma'_B)$. A game is said to be correct if for every arbitrary alternative strategy $\sigma'_B$ followed by Bob, the following holds: $$\begin{aligned}
\Pr[f_A\not\in \{\mathcal{F},\perp\}|A=\sigma_A, B=\sigma'_B]= 0\end{aligned}$$
Here $A$ (resp. $B$) implies the event that Alice (resp. Bob) follows a strategy.
Similarly, for Bob we can write $$\begin{aligned}
\Pr[f_B\not\in \{\mathcal{F},\perp\}|A=\sigma'_A, B=\sigma_B]=0\end{aligned}$$
Equilibrium
-----------
Let $\Gamma$ be a mechanism designed for $n$ players for a certain purpose. Let $\overrightarrow{\sigma}$ be the set of suggested strategies for that $n$ number of players in the mechanism $\Gamma$. That is $\overrightarrow{\sigma}=\{\sigma_1,\sigma_2,\cdots,\sigma_n\}$, where $\sigma_i$ is the suggested strategy for a player $P_i$, $i\in\{1,2,\cdots,n\}$. Then $\overrightarrow{\sigma}$ in the mechanism $(\Gamma,\overrightarrow\sigma)$ is said to be in Nash equilibrium when there is no incentive for a player $P_i$, $i\in\{1,2,\cdots,n\}$ to deviate from the suggested strategy, given that everyone else is following his or her strategy. Thus we can define Strict Nash Equilibrium as follows.
(Strict Nash Equilibrium) The suggested strategy $\overrightarrow{\sigma}$ in the mechanism $(\Gamma,\overrightarrow{\sigma})$ is a strict Nash equilibrium if for every $P_i$ and for any strategy $\sigma'_i$, we have $u_i (\sigma'_i,\overrightarrow{\sigma}_{-i} )<u_i (\overrightarrow{\sigma})$.
Here, $\overrightarrow{\sigma}_{-i}=\{\sigma_1,\sigma_2,\cdots,\sigma_{i-1},\sigma_{i+1},\sigma_n\}$, i.e., the set of the suggested strategies for the players excluding $i$-th player.
Explicitly, a mechanism is in strict Nash equilibrium when the payoff achieved by a player following the suggested strategy will be more than the payoff achieved by the player following any deviated strategy conditional on the event that all other players follow the suggested strategies.
Revisiting the protocol in [@Shi]
=================================
In [@Shi] the protocol for set member decision problem is described as follows. Alice has a secret $k$ and Bob possesses a set $Y=\{k_1,k_2,\cdots,k_n\}$ such that each $k_i$ belongs to the set $\mathbb{Z}_{N}^*=\{1,2,\cdots,N-1\}$. Now, Bob prepares an $N$ element database in a way so that the $j$-th element $p(j)=1$ if and only if $j=k_i(i\in[1,n])$ and $p(j)=0$ otherwise. He now selects $l$ bits $r_1,r_2,\cdots,r_l\in\{0,1\}$ uniformly at random and generates another variable $q_t(j)=p(j)\oplus r_t$ where $t$ varies from $1$ to $l$ and $j$ varies from $1$ to $N-1$. $l$ is the security parameter. Alice and Bob fix $p(0)=0$ and $q_1(0)=q_2(0)=\cdots=q_l(0)=0$ a priori. Table $2$ shows the $N-1$ elements database created from the set $\{k_1,k_2,\cdots,k_n\}$.
$j$ $p(j)$ $q_1(j)$ $q_2(j)$ $\cdots$ $q_l(j)$
---------- ---------- ---------- ---------- ---------- ----------
1 0 $0+r_1$ $0+r_2$ $\cdots$ $0+r_l$
2 0 $0+r_1$ $0+r_2$ $\cdots$ $0+r_l$
3 0 $0+r_1$ $0+r_2$ $\cdots$ $0+r_l$
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$k_1$ 1 $1+r_1$ $1+r_2$ $\cdots$ $1+r_l$
$k_1+1$ $0$ $0+r_1$ $0+r_2$ $\cdots$ $0+r_l$
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$k_2$ 1 $1+r_1$ $1+r_2$ $\cdots$ $1+r_l$
$k_2+1$ $0$ $0+r_1$ $0+r_2$ $\cdots$ $0+r_l$
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$k_n$ 1 $1+r_1$ $1+r_2$ $\cdots$ $1+r_l$
$k_n+1$ $0$ $0+r_1$ $0+r_2$ $\cdots$ $0+r_l$
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$N-1$ 0 $0+r_1$ $0+r_2$ $\cdots$ $0+r_l$
Alice now generates $l$ $M(=\log_2 N)$[^1] qubit registers. One register contains the qubit $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|k\right\rangle}})$ and the remaining $l-1$ registers contain the decoy states $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_1\right\rangle}}), \frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_2\right\rangle}}),\cdots,\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_{l-1}\right\rangle}})$ where $j_i\in \mathbb{Z}_N^*$. Here, ${\ensuremath{\left|0\right\rangle}}$ represents ${\ensuremath{\left|0\right\rangle}}^{\otimes M}$ and each ${\ensuremath{\left|j\right\rangle}}$ is an $M$ qubit string.[^2] Alice sends all these $l$ registers to Bob after a random permutation. She keeps the record of the permutation. Bob now operates an oracle $O_t$ on each register. The matrix representation of the oracle is as follows. $$O_t=
\begin{bmatrix}
(-1)^{q_t(0)} & & & \\
& (-1)^{q_t(1)} & & \\
&&\ddots &\\
& & &(-1)^{q_t(N-1)}
\end{bmatrix}$$ The oracle transforms the $l$-th register $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j\right\rangle}})$ to $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+(-1)^{q_l(j)}{\ensuremath{\left|j\right\rangle}})$. Bob returns all those registers to Alice. After getting back the registers, Alice measures the decoy registers in $\{\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_i\right\rangle}}),\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|j_i\right\rangle}})\}$ basis as she knows the $j_i$ value associates with each register. If any error, which indicates the cheating of Bob, is found, Alice aborts the protocol. Otherwise she will proceed for the next step.
In the second phase, Alice takes the register which contains $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+(-1)^{q_t(k)}\\
{\ensuremath{\left|k\right\rangle}})$ and operates a SWAP gate $U_{swap}$ on the $1$st and $i$-th $1$ of the bit pattern for $k$, $i\in[2,M]$. She then operates a CNOT gate $U_{cnot}$ on the $1$st and $i$-th $1$. These operations are continued until the bit string for $k$ reduces to ${\ensuremath{\left|1\right\rangle}}\otimes {\ensuremath{\left|0\right\rangle}}^{\otimes M-1}$. Thus after these consecutive operations the final state reduces to ${\ensuremath{\left|\pm\right\rangle}}{\ensuremath{\left|0\right\rangle}}^{\otimes M-1}$.
Alice now measures the first particle in $\{{\ensuremath{\left|+\right\rangle}},{\ensuremath{\left|-\right\rangle}}\}$ basis. If she gets ${\ensuremath{\left|+\right\rangle}}$, she concludes that $q_t(k)=0$. If she obtains ${\ensuremath{\left|-\right\rangle}}$, she concludes that $q_t(k)=1$. Alice sends the value of $t$ and the value of $q_t(k)$ to Bob. Bob checks $p(k)=q_t(k)\oplus r_t$ for that $t$. If $p(k)=1$, Bob concludes that $k$ is a set member of his set.
In the following section we use this idea to compute set intersection of two computationally indistinguishable sets $X$ and $Y$ holding by Alice and Bob respectively in rational setting.
Proposed Protocol
=================
In this section we describe the protocol. We assume that Alice and Bob, two rational players, possess two sets $X=\{x_1,x_2,\cdots,x_n\}$ and $Y=\{y_1,y_2,\cdots,y_m\}$ respectively where $x_i, y_i \in \mathbb{Z}_N^*$. The cardinality of $X$ and $Y$ are $n$ and $m$ respectively and are common knowledge to both of the parties. The sets are computationally indistinguishable.
Now, the players want to compute the intersection of their respective sets. They do not like to reveal any other elements except the intersected ones of their respective sets to the opponent. Each of them has the order of preferences $\mathcal{R}_1 ($\[rat\]).
The functionality $\mathcal{F}$ for this game can be defined as $$\begin{aligned}
\mathcal{F}=(X \cap Y, X \cap Y)\end{aligned}$$
Our protocol is described in Algorithm 2. The protocol $\bf\Pi$ calls a subroutine $QKeyGen$ to generate a random bit-stream of length $l$, where $l$ is the security parameter. Bob knows the entire bit-stream whereas Alice knows some fraction of this. Our $QKeyGen$ is described in Algorithm 1. The idea of $QKeyGen$ comes from [@Yang].
For the protocol $\bf\Pi$ we assume that (cooperate, abort) is the suggested strategy profile for each of the players. That is each player is supposed to follow the protocol and abort if he or she identifies any deviation of his or her respective opponent.
\[qkeygen\]
1. Bob and Alice share $2l$ entangled states of the form $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}_{B}{\ensuremath{\left|\phi_0\right\rangle}}_{A}+{\ensuremath{\left|1\right\rangle}}_{B}{\ensuremath{\left|\phi_1\right\rangle}}_{A})$, where, ${\ensuremath{\left|\phi_{0}\right\rangle}}_{A}=\cos{(\frac{\theta}{2})}{\ensuremath{\left|0\right\rangle}}+\sin{(\frac{\theta}{2})}{\ensuremath{\left|1\right\rangle}}$ and ${\ensuremath{\left|\phi_{1}\right\rangle}}_{A}=\cos{(\frac{\theta}{2}){\ensuremath{\left|0\right\rangle}}}-\sin{(\frac{\theta}{2})}{\ensuremath{\left|1\right\rangle}}$. Here, subscript B stands for Bob and subscript A stands for Alice. $\theta$ may vary from $0$ to $\frac{\pi}{2}$.
2. Bob measures his qubits in $\{{\ensuremath{\left|0\right\rangle}}_{B}, {\ensuremath{\left|1\right\rangle}}_{B}\}$ basis, whereas Alice measures her qubits either in $\{{\ensuremath{\left|\phi_{0}\right\rangle}}_{A},{\ensuremath{\left|\phi_{0}^{\perp}\right\rangle}}_{A}\}$ basis or in $\{{\ensuremath{\left|\phi_{1}\right\rangle}}_{A}, {\ensuremath{\left|\phi_{1}^{\perp}\right\rangle}}_{A}\}$ basis randomly.
3. If Bob measures ${\ensuremath{\left|0\right\rangle}}$, he encodes the bit $r_t$, $t\in [1,2l]$ as $0$. If Bob measures ${\ensuremath{\left|1\right\rangle}}$, he encodes the bit $r_t$, $t\in [1,2l]$ as $1$.
4. If the measurement result of Alice gives ${\ensuremath{\left|\phi_{0}^{\perp}\right\rangle}}$, she concludes that the bit at Bob’s end must be $1$. If it would be ${\ensuremath{\left|\phi_{1}^{\perp}\right\rangle}}$, the bit must be $0$.
5. Bob and Alice execute classical post-processing in the motivation to check the error in the channel from randomly chosen $l$ bits. If the error remains below the pre-defined threshold, Bob and Alice continues the protocol. Otherwise they abort.
6. The remaining $l$ bits stream is retained by Bob. Bob knows the whole stream, whereas Alice generally knows several bits of the stream.
Before going to the main protocol $\bf\Pi$, we like to explain how Alice gets fraction of the stream. In this direction, we have to calculate the success probability of Alice to guess a single bit possessed by Bob.
As Bob measures his qubits only in $\{{\ensuremath{\left|0\right\rangle}}_{B}, {\ensuremath{\left|1\right\rangle}}_{B}\}$ basis, he will get either ${\ensuremath{\left|0\right\rangle}}$ with probability $\frac{1}{2}$ or ${\ensuremath{\left|1\right\rangle}}$ with probability $\frac{1}{2}$. When Bob gets ${\ensuremath{\left|0\right\rangle}}$, Alice should get ${\ensuremath{\left|\phi_0\right\rangle}}$. If she chooses $\{{\ensuremath{\left|\phi_0\right\rangle}}_{A}, {\ensuremath{\left|\phi_0^{\perp}\right\rangle}}_{A}\}$ basis, she will get ${\ensuremath{\left|\phi_0\right\rangle}}$ with probability $1$ and never gets ${\ensuremath{\left|\phi_0^{\perp}\right\rangle}}$. However, if she chooses $\{{\ensuremath{\left|\phi_1\right\rangle}}_{A}, {\ensuremath{\left|\phi_1^{\perp}\right\rangle}}_{A}\}$ basis, she will get either ${\ensuremath{\left|\phi_1\right\rangle}}$ with probability $\cos^2\theta$ or ${\ensuremath{\left|\phi_1^{\perp}\right\rangle}}$ with probability $\sin^2\theta$. We formalize all the conditional probabilities in the following table.
--------- ---------------------------------------------- ------------------------------------------------------ ---------------------------------------------- ------------------------------------------------------
A=${\ensuremath{\left|\phi_0\right\rangle}}$ A=${\ensuremath{\left|\phi_0^{\perp}\right\rangle}}$ A=${\ensuremath{\left|\phi_1\right\rangle}}$ A=${\ensuremath{\left|\phi_1^{\perp}\right\rangle}}$
$ B=0 $ $\frac{1}{2}.1$ $\frac{1}{2}.0$ $\bf{\frac{1}{2}.\cos^2\theta}$ $\bf{\frac{1}{2}.\sin^2\theta}$
$B=1$ $\bf{\frac{1}{2}.\cos^2\theta}$ $\bf{\frac{1}{2}.\sin^2\theta}$ $\frac{1}{2}.1$ $\frac{1}{2}.0$
--------- ---------------------------------------------- ------------------------------------------------------ ---------------------------------------------- ------------------------------------------------------
According to the protocol, when Alice gets ${\ensuremath{\left|\phi_0^{\perp}\right\rangle}}$, she outputs $1$. And when she gets ${\ensuremath{\left|\phi_1^{\perp}\right\rangle}}$, she outputs $0$. Thus, the success probability of Alice to guess a bit in $l$ bits stream can be written as [$\Pr(A=B)$ $$\begin{aligned}
\label{sucprob}
&=&\Pr(A=0,B=0)+\Pr(A=1,B=1) \nonumber\\
&=&\Pr(B=0).\Pr(A=0|B=0)+\Pr(B=1).\Pr(A=1|B=1)\\
&=&\frac{1}{2}.\Pr(A=\phi_1^{\perp}|B=0)+\frac{1}{2}.\Pr(A=\phi_0^{\perp}|B=1).\end{aligned}$$]{} From the above table, we can see that the success probability of Alice becomes $\frac{\sin^2\theta}{2}$. Thus, Alice knows $\frac{\sin^2\theta}{2}$ fraction of the whole stream possessed by Bob.
Now, we describe the protocol $\bf\Pi$ for set intersection in Algorithm 2.
1. Alice and Bob possess two sets $X=\{x_1,x_2,\cdots,x_n\}$ and $Y=\{y_1,y_2,\cdots,y_m\}$ respectively where $x_i, y_i \in \mathbb{Z}_N^*$; $N\gg2\max(n,m)$ [@Shi1]; $u$ is the number of intersected elements. Hence, $u\leq \min(n,m)$
2. Bob now prepares an $N-1$ element database. Any element of the database $p(j)=1$ if and only if $j=y_i, i\in[1,m]$ and $0$ otherwise.
3. Bob calls the sub-routine $QKeyGen$ and prepares a sequence of random bits $r_1, r_2,\cdots, r_l$, where $l$ is the security parameter and $l\geq 2n$. Alice knows $\frac{\sin^2{\theta}}{2}$ fraction of those bits, where $\theta\in [0,\frac{\pi}{4}]$.
4. Bob generates a variable $q_t(j)=p(j)\oplus r_t$, where $t$ varies from $1$ to $l$ and $j$ varies from $1$ to $N-1$.
5. Alice and Bob set $p(0)=q_1(0)=q_2(0)=\cdots=q_l(0)=0$ apriori.
6. Alice inserts $n$ check states prepared in $\{{\ensuremath{\left|0\right\rangle}},{\ensuremath{\left|1\right\rangle}}\}$ or $\{\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|1\right\rangle}}),\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|1\right\rangle}})\}$ basis randomly. She keeps the record of the positions of the check elements.
7. Alice prepares $2n$ $M(=\log N)$bits registers.
8. For check registers, Alice does the followings
- If the check element is ${\ensuremath{\left|0\right\rangle}}$, Alice prepares ${\ensuremath{\left|0\right\rangle}}^{\otimes M}$
- If the check element is ${\ensuremath{\left|1\right\rangle}}$, Alice prepares ${\ensuremath{\left|k_s\right\rangle}}{\ensuremath{\left|0\right\rangle}}^{\otimes {M-1}}$; ${\ensuremath{\left|k_s\right\rangle}}={\ensuremath{\left|1\right\rangle}}$, $s\in[1,M]$.
- If the check element is $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}\pm {\ensuremath{\left|1\right\rangle}})$, Alice prepares $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}\pm {\ensuremath{\left|j\right\rangle}})$, where ${\ensuremath{\left|0\right\rangle}}={\ensuremath{\left|0\right\rangle}}^{\otimes M}$ and ${\ensuremath{\left|j\right\rangle}}={\ensuremath{\left|k_s\right\rangle}}{\ensuremath{\left|0\right\rangle}}^{\otimes{M-1}}$; $k_s=1$ and $s\in\{1,2,\cdots,M\}$.
9. In case of remaining $n$ actual registers, Alice prepares $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_i\right\rangle}})$, $j\in X$ and $i\in[1,n]$. Here, ${\ensuremath{\left|0\right\rangle}}={\ensuremath{\left|0\right\rangle}}^{\otimes M}$ and ${\ensuremath{\left|j_i\right\rangle}}={\ensuremath{\left|k_1k_2\cdots k_M\right\rangle}}$, $k_s\in\{0,1\}$, $s\in\{1,2,\cdots,M\}$.
10. Alice sends those registers to Bob. If the number of registers exceed $2n$, Bob aborts the protocol.
11. Bob operates the oracle $O_t$ [@Shi] on each register and sends those back to Alice. The oracle converts the state $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_i\right\rangle}})$ to $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+(-1)^{q_t(j_i)}{\ensuremath{\left|j_i\right\rangle}})$.
12. Alice then selects the check registers and measures those in their respective bases.
13. If error is found, she aborts the protocol. Otherwise she will continue.
14. Alice selects the actual registers, i.e., those registers which contain the actual set elements.
15. Alice operates $U_{swap}$ followed by $U_{cnot}$ on the first $1$ and $2$nd $1$st $1$ of each register. These operations will be continued till all $M-1$ bits except $k_1$ of each register, reduces to $0$. In this way, the final state of each register becomes ${\ensuremath{\left|\pm\right\rangle}}{{\ensuremath{\left|0\right\rangle}}}^{\otimes {M-1}}$.
16. Alice measures the first particles of each register in $\{{\ensuremath{\left|+\right\rangle}},{\ensuremath{\left|-\right\rangle}}\}$ basis. If she gets ${\ensuremath{\left|+\right\rangle}}$, she concludes that $q_t(j_i)=0$ and if she gets ${\ensuremath{\left|-\right\rangle}}$, she concludes that $q_t(j_i)=1$.
17. Alice repeats steps $14$, $15$ for all $n$ registers.
18. Alice conveys the values of $q_t$ and $t$ to Bob for each n registers.
19. Bob executes $q_t\oplus r_t$ for that $j$. Note that here Alice knows $j$, but Bob does not. As $r_t$ remains same for each $j$, the value of $q_t(j)$ remains same for each $j$ and flips if $j$ is the set element of $Y$. Thus. without knowing the value of $j$ Bob can conclude if the element is in his set. If he finds $p(j)=0$, he declares that the element is not in his set. If $p(j)=1$, he declares that the element is his set.
20. Alice checks if the declaration of Bob is correct by finding the value of $p(j)$ for those $t$ for which she has $r_t$. If she finds any cheating of Bob, she aborts the protocol.
21. Alice declares the value of the elements for which Bob gets $p(j)=1$.
22. Bob checks if the element indeed lies in his set. If not, he aborts the protocol.
Security Analysis {#sec}
=================
The security criterion of the proposed protocol demands that at the end of the protocol, both Alice and Bob will get $X\cap Y$. However, Alice will not get any element from $Y\setminus (X\cap Y)$ and Bob will not get any element from $X\setminus (X\cap Y)$. Thus, it is quite natural that Alice (resp. Bob) will choose such a strategy which provides her or him the elements from $X\setminus (X\cap Y)$ (resp. $Y\setminus (X\cap Y)$). Hence, without loss of generality, we can discard all other strategies which do not provide such information to Alice (resp. Bob). In the present draft, considering optimised guessing probabilities for Alice and Bob, we show that this security criterion is maintained when the players are rational.
For the security analysis of our protocol we first show if the preferences of Alice and Bob follow the order of $\mathcal{R}_1$, then $((cooperate, abort),(cooperate, abort))$ is a strict Nash equilibrium in the protocol $\bf\Pi$. Then we will show in this initiative both the parties know only $\mathcal{F}$ and nothing else. We also prove that if $((cooperate, abort), (cooperate, abort))$ is a strict Nash, then fairness as well as correctness of the protocol are preserved automatically.
\[theo1\] In the key establishment phase, $QKeyGen$, of the protocol, for each key bit $r_t$ ($1\leq t\leq 2l$) a dishonest Bob ($\mathcal{B^*}$) can successfully guess if honest Alice ($\mathcal{A}$) gets a conclusive result with probability at most $\frac{1}{2}$.
Honest Alice ($\mathcal{A}$) follows Algorithm $1$. According to $QKeyGen$, Alice will outputs $r_t=0$, when she chooses $\{{\ensuremath{\left|\phi_1\right\rangle}},{\ensuremath{\left|\phi_1^{\perp}\right\rangle}}\}$ basis and gets ${\ensuremath{\left|\phi_1^\perp\right\rangle}}$. Similarly, Alice outputs $r_t=1$ when she chooses $\{{\ensuremath{\left|\phi_0\right\rangle}},{\ensuremath{\left|\phi_0^{\perp}\right\rangle}}\}$ basis and gets ${\ensuremath{\left|\phi_0^\perp\right\rangle}}$. Each of these two events happens with probability $\frac{1}{2}\sin^2{\theta}$.
Alice and Bob are two distant parties. If we assume no signalling from Alice to Bob, then the basis choice of Alice becomes completely random to Bob. The success probability of honest Bob to guess $r_t$ as a conclusive outcome of Alice is $\frac{1}{2}\sin^2{\theta}$. Dishonest Bob ($\mathcal{B}^*$) always tries to maximize this probability so that he can identify the positions of the bits where Alice gets conclusive results. This information might help him to cheat Alice further. Thus, $$\begin{aligned}
\Pr_{guess}[\mathcal{B^*}=\mathcal{A}] \leq \frac{1}{2}\end{aligned}$$
\[min\_ent1\] In the key establishment phase of the protocol, for honest Bob ($\mathcal{B}$) and dishonest Alice ($\mathcal{A^*}$), Alice can successfully guess each of the key bits $r_t$ (for all $1 \leq t \leq 2l$) with probability at most $0.85$.
At the beginning of key establishment phase dishonest Alice ($\mathcal{A}^*$) and honest Bob ($\mathcal{B}$) share $2l$ copies of entangled pairs. The $t$-th copy of the state is given by ${\ensuremath{\left|\psi\right\rangle}}_{\mathcal{B}_t\mathcal{A}^*_t }= \frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}_{\mathcal{B}_t}{\ensuremath{\left|\phi_0\right\rangle}}_{\mathcal{A}^*_t }+ {\ensuremath{\left|1\right\rangle}}_{\mathcal{B}_t}{\ensuremath{\left|\phi_1\right\rangle}}_{\mathcal{A}_t})$, where $t$-th subsystem of Alice and Bob is denoted by $\mathcal{A}^*_t$ and $\mathcal{B}_t$ respectively.
At Alice’s side the reduced density matrix is of the form $$\rho_{\mathcal{A}^*_t} ={\mathrm{Tr}}_ {\mathcal{B}_t}[{\ensuremath{\left|\psi\right\rangle}}_{\mathcal{B}_t\mathcal{A}^*_t}{\ensuremath{\left\langle\psi\right|}}] = \frac{1}{2}({\ensuremath{\left|\phi_0\right\rangle}}{\ensuremath{\left\langle\phi_0\right|}}+{\ensuremath{\left|\phi_1\right\rangle}}{\ensuremath{\left\langle\phi_1\right|}}).$$
At Step $2$, Bob measures each of his part of the state ${\ensuremath{\left|\psi\right\rangle}}_{\mathcal{B}_t\mathcal{A}_t^*}$ in $\{{\ensuremath{\left|0\right\rangle}},{\ensuremath{\left|1\right\rangle}}\}$ basis. Let $\rho_{\mathcal{A}^*_t|r_t}$ denotes the state at Alice’s side after Bob’s measurement. However, Bob does not communicate his measurement result to Alice. Thus, in this case, for $r_t = 0$, we have $\rho_{\mathcal{A}^*_t|r_t =0} =\frac{1}{2}({\ensuremath{\left|\phi_0\right\rangle}}{\ensuremath{\left\langle\phi_0\right|}}+{\ensuremath{\left|\phi_1\right\rangle}}{\ensuremath{\left\langle\phi_1\right|}})=\rho_{\mathcal{A}_t^*}$. Similarly, for $r_t =1$ we have, $$\begin{aligned}
\rho_{\mathcal{A}^*_t|r_t =1} & = {\mathrm{Tr}}_{\mathcal{B}_t}[{\ensuremath{\left|\psi\right\rangle}}_{\mathcal{B}_t\mathcal{A}^*_t}{\ensuremath{\left\langle\psi\right|}}]\\
& = {\mathrm{Tr}}_{\mathcal{B}_t}[\frac{1}{2}({\ensuremath{\left|0\right\rangle}}{\ensuremath{\left|\phi_0\right\rangle}} + {\ensuremath{\left|1\right\rangle}}{\ensuremath{\left|\phi_1\right\rangle}})_{\mathcal{B}_t\mathcal{A}^*_t}({\ensuremath{\left\langle0\right|}}{\ensuremath{\left\langle\phi_0\right|}}+{\ensuremath{\left\langle1\right|}}{\ensuremath{\left\langle\phi_1\right|}})]\\
& = \frac{1}{2}({\ensuremath{\left|\phi_0\right\rangle}}{\ensuremath{\left\langle\phi_0\right|}}+{\ensuremath{\left|\phi_1\right\rangle}}{\ensuremath{\left\langle\phi_1\right|}}\\
&=\rho_{\mathcal{A}_t^*}.\end{aligned}$$
This implies $\rho_{\mathcal{A}^*_t|r_t} = \rho_{\mathcal{A}^*_t}$. As there is no communication between Alice and Bob, due to non-signalling principle we can claim that Alice can guess Bob’s measurement outcome with probability at most $\frac{1}{2}$. This implies, if Alice’s optimal guessing strategy is described by the POVM $\{E_z\}_{0\leq z\leq 1}$ then, $$\begin{aligned}
\Pr_{guess}[r_t|\rho_{\mathcal{A}^*_t}] &= \sum_{r_t}\frac{1}{2}{\mathrm{Tr}}[E_{r_t}\rho_{\mathcal{A}^*_t|r_t}]\\
& = \frac{1}{2}{\mathrm{Tr}}[\sum_{r_t}E_{r_t}\rho_{\mathcal{A}^*_t}]\\
& = \frac{1}{2}.\end{aligned}$$
However, Alice has the information that if Bob measured ${\ensuremath{\left|0\right\rangle}}$, i.e., $r_t=0$, her state must collapse to ${\ensuremath{\left|\phi_0\right\rangle}}$. Similarly, if Bob measured ${\ensuremath{\left|1\right\rangle}}$, i.e., $r_t=1$, her state collapses to ${\ensuremath{\left|\phi_1\right\rangle}}$. Thus, if she could distinguish ${\ensuremath{\left|\phi_0\right\rangle}}$ and ${\ensuremath{\left|\phi_1\right\rangle}}$ optimally, she can guess $r_t$ optimally. Now, we try to find if this can be done with probability greater than $\frac{1}{2}$.
This distinguishing probability has a nice relationship with the trace distance between the states [@Wilde17]. According to this relation we have, $$\begin{aligned}
\Pr_{guess}[r_t|\rho_{\mathcal{A}^*_i}] & \leq \frac{1}{2}(1+\frac{1}{2}||{\ensuremath{\left|\phi_0\right\rangle}}{\ensuremath{\left\langle\phi_0\right|}} - {\ensuremath{\left|\phi_1\right\rangle}}{\ensuremath{\left\langle\phi_1\right|}}||_1)\\
& = \frac{1}{2}(1+\sqrt{1 - F({\ensuremath{\left|\phi_0\right\rangle}}{\ensuremath{\left\langle\phi_0\right|}}, {\ensuremath{\left|\phi_1\right\rangle}}{\ensuremath{\left\langle\phi_1\right|}})})\\
& = \frac{1}{2}(1+\sin{\theta})\\
& = \frac{1}{2} + \frac{1}{2}\sin{\theta}.\end{aligned}$$
This implies that Alice can successfully guess the value of $r_t$ with probability at most $\frac{1}{2} + \frac{1}{2}\sin{\theta}$.
From the above calculation, it is clear that when $\theta\rightarrow \frac{\pi}{2}$, Alice can get the full information about the stream $r_1,r_2,\cdots, r_l$, and hence can compute $\mathcal{F}$ by herself alone. So, we fix the range of $\theta$ in between $[0,\frac{\pi}{4}]$. For this range of $\theta$, the maximum success probability that Alice can achieve is $0.85$.
(Serfling [@Serfling]) \[serf\] Let $\{x_1,x_2,\cdots,x_n\}$ be a list of values in $[a,b]$ (not necessarily distinct). Let $\overline{x}=\frac{1}{n}\sum_i x_i$ be the average of these random variables. Let $k$ be the number of random variables $X_1,X_2,\cdots,X_k$ chosen from the list without replacement. Then for any value of $\delta>0$, we have $\Pr\left[|X-\overline{x}| \geq \delta \right] \leq \exp\left(\frac{-2\delta^2 kn}{(n-k+1)(b-a)}\right),$ where $X=\frac{1}{k}\sum_i X_i$.
If $k\rightarrow 0$, $\exp\left(\frac{-2\delta^2 kn}{(n-k+1)(b-a)}\right)\rightarrow 1$. For $k=\frac{n}{2}$, we can approximate the probability as $\Pr\left[|X-\overline{x}| \geq \delta \right] \leq \exp\left(-2\delta^2 n\right).$ Based on the above two theorems and one lemma, we now prove Nash equilibrium of our protocol.
If $(cooperate, abort)$ is the suggested strategy profile for each party and if the parties have the order of preferences $\mathcal{R}_1$, then $((cooperate, abort),(cooperate, abort))$ is a strict Nash equilibrium in the protocol $\bf {\Pi}$ conditioning on $m+n<\frac{N-1}{2}$ and $\theta\in[0,\frac{\pi}{4}]$.
Let us consider the deviations of the players from the suggested strategy. It should be noted that when one party deviates, another party follows the protocol. Now, at first, let Alice deviates from the suggested strategy in the motivation to get $Y\setminus (X\cap Y)$.
1. Alice’s activities start from step $6$ of algorithm $\bf{\Pi}$. In step $6$, let Alice inserts $n$ extra elements which is not in $X$ but in $\mathbb{Z}_N^*$ along with $n$ check states in the motivation to get $X'\cap Y$, where $X'$ is the set containing $n$ actual elements of set $X$ and $n$ fake elements chosen from $\mathbb{Z}_N^*$. Thus, according to the protocol, Alice now sends $3n$ registers to Bob.
Let there exists $X'\cap C_2$, where $C_2=Y\setminus (X\cap Y)$. In this case, Alice will successfully extract those intersected elements in step $19$ of the protocol $\bf{\Pi}$. And hence, security of the protocol will be compromized.
However, Bob knows the cardinality of set $X$. So if Alice tries to send more than $2n$ registers, Bob aborts the protocol and both of them will end up with utility $U^{NN}$. As a result, Alice should have no incentive to choose the above deviation.
In this regard, one most important thing is that if the cardinalities of $X$ and $Y$ would not be a common knowledge, then choosing the above strategy, Alice might extract some elements from $Y\setminus (X\cap Y)$ causing security loophole in the protocol.
2. Let us now consider a situation when Alice wants to mount the above attack conditioning that the cardinality of her set is a common knowledge. According to the protocol, Bob aborts if he finds more than $2n$ registers coming from Alice. Hence, in this case, Alice will send $2n$ elements from $\mathbb{Z}_N^*$ and no check elements. However, in such situation, Alice can not detect the cheating of Bob.
If we assume that Alice sends a few check elements, then also there remains a non-negligible probability for Bob to cheat Alice. This is an immediate instantiation of Serfling Lemma \[serf\]. The cardinalities of two sets (one for error checking and another for continuing the protocol) should almost be equal. Hence, Alice has to send at least $n$ check registers to Bob. This resists Alice to send fake elements. If she tries to send fake elements, she has to cut the actual elements. And as a result both will end up with $U^{NN}$.
3. Any type of deviation of Alice in steps $12$, $13$, $14$, $15$, $16$, $17$ and $18$ such as avoiding error checking, operating $U_{swap}$ and $U_{cnot}$ improperly or conveying wrong values of $t$ and $q_{t}$ to Bob will lead her to wrong values of the functionality. So she should have no incentive to deviate in those rounds of the protocol $\bf{\Pi}$.
Maximum what Alice can do in this phase is to send wrong value of $q_t$ for which she has $r_t$. This is because, in such case she can calculate $p(j)$ by herself alone and get the information if the element is in $Y$. By telling wrong value for $q_t$ she can make Bob to calculate wrong value of $p(j)$ and as a result Bob will end up with the functionality $\mathcal{F}$ with less elements.
However, if $p(j)=0$, Alice should have no incentive to say a wrong $q_t$ as in this case, she knows that the element is not in $Y$. Moreover, if she conveys a wrong $q_t$ for that $j$, then $p(j)$ at Bob’s place becomes $1$. And in that case, she has to reveal that element which is in $X$ but not in $Y$. According to the security criterion of the protocol, this situation is not desirable at all. So, she can only communicate wrong $q_t$ for which she found $p(j)=1$. But by fixing the value of $\theta \in [0,\frac{\pi}{4}]$, we allow Alice to know a few bits of the sequence $r_1r_2\cdots r_l$. Thus, she can only mount such attack for a few elements, but not all.
By choosing this deviation Alice can not even violate the correctness criterion of the protocol. That is Alice can not make Bob to believe in a wrong element as the intersected one. She also can not resist Bob to know most of the elements in $\mathcal{F}$. The above strategy slightly deviates from the suggested strategy for which we get $f=(\mathcal{F}, \mathcal{F})$. Thus this strategy is essentially the same as suggested strategy and does not constitute any deviation.
4. The round, in which deviation may become advantageous to Alice, is step $21$ of protocol $\bf{\Pi}$. In step $21$, instead of announcing a correct element of set $X$, she may declare a wrong value for which Bob obtains $p(j)=1$. In this way, she can get the correct values of $\mathcal{F}$ and can deceive Bob to believe in wrong values of $\mathcal{F}$.
To do this, Alice chooses an element from $\mathbb{Z}_N^*$. But as she does not want to reveal any element except the intersected ones from her set $X$ to Bob, she has to choose an element $e$ from $\mathbb{Z}_N^*$ such that $e\neq x_i$; $x_i\in X$.
Now, let $S=\mathbb{Z}_N^*\setminus X$, $C_1=\mathbb{Z}_N^*\setminus (X\cup Y)$ and $C_2=Y\setminus (X \cap Y)$. One can write $$\begin{aligned}
S=C_1\cup C_2\end{aligned}$$
Now, Alice always has to choose an element $e \in S$. If $e$ is a set member of both $S$ and $C_1$, Bob aborts the protocol as he finds that $e\notin Y$. In that case neither Alice nor Bob gets $\mathcal{F}$. The utility functions for Alice and Bob becomes $U_A^{NN}$ and $U_B^{NN}$ respectively.
However, if $e$ belongs to $S$ and $C_2$, Bob can not distinguish if $e\in X\cap Y$ or $e\in Y\setminus (X \cap Y)$. So he does not abort the protocol as $e \in Y$. In this case, Alice knows the correct elements but Bob ends up with wrong elements which is effectively equivalent to obtaining no element as this element $e$ neither belongs to $X \cap Y$ nor is in $X\setminus (X \cap Y)$. Thus, in this case, the utility of Alice becomes $U_A^{TN}$.
Moreover, correctness of the protocol is violated, i.e., $$\begin{aligned}
\Pr[f_B\not\in \{\mathcal{F},\perp\}|A=\sigma'_A, B=\sigma_B]\neq 0.\end{aligned}$$
However, Alice does not know $Y$. Let $u$ be the number of intersected elements. Thus, $|C_1|=N-1-n-m+u$ and $|C_2|=m-u$. Probability that $e$ is in set $C_1$ is $$\begin{aligned}
\Pr(e\in C_1)&=&\frac{N-1-n-m+u}{N-1-n}\\
&=& 1-\frac{m-u}{N-1-n}
\end{aligned}$$ Probability that $e$ is in set $C_2$ is $$\begin{aligned}
\Pr(e\in C_2)&=&\frac{m-u}{N-1-n}\\
\end{aligned}$$ Now, $$\begin{aligned}
n+m&\leq& 2\max(n,m)\\
\Rightarrow n+m-u&\leq& 2\max(n,m); u\leq \min(n,m)\\
\Rightarrow n+m-u&\ll&N-1\\
\Rightarrow m-u&\ll&N-1-n
\end{aligned}$$ Let $\epsilon=\frac{m-u}{N-1-n}$. Then, with probability $1-\epsilon$, in step $21$ of protocol $\bf{\Pi}$, for some of $p(j)$s, Alice will choose $e$ from the set $C_1$. In that case, Bob immediately aborts the protocol. Hence, the protocol will be terminated and Alice will not get any intersected elements further [^3]. So, the expected utility $E(U_A)$ over this deviation can be expressed as $$\begin{aligned}
E(U_A)=\Pr(e\in C_2) U_A^{TN} + \Pr(e\in C_1) U_A^{NN}\end{aligned}$$ Depending on the values of $U_A^{TN}$, $U_A^{TT}$ and $U_A^{NN}$ we can fix the values of $m$, $n$ and $N$ in such a way so that $$\begin{aligned}
\label{eqn}
\Pr(e\in C_2) U_A^{TN} + \Pr(e\in C_1) U_A^{NN}< U_A^{TT}\end{aligned}$$ For example, let $U_A^{TN}=1$, $U_A^{NN}=0$ and $U_A^{TT}=\frac{1}{2}$. Then equation \[eqn\] reduces to $$\begin{aligned}
\Pr(e\in C_2) < \frac{1}{2}\\
\Rightarrow \frac{m-u}{N-1-n}<\frac{1}{2}\\
\Rightarrow 2(m-u)< N-1-n\end{aligned}$$ Putting $u=0$, we get $$\begin{aligned}
2m<N-1-n.\end{aligned}$$ Thus, if $m+n<\frac{N-1}{2}$, then for any value of $u=\min(n,m)$, we can bound Alice to choose cooperation over such deviation. Hence, we can write [$$\begin{aligned}
&U_A&((cooperate,abort), (cooperate,abort))\\
>&U_A&(deviation, (cooperate,abort))\end{aligned}$$]{}
Now, we focus on Bob’s deviations. Bob will deviate in the motivation to extract the elements from $X\setminus (X\cap Y)$. The possible deviation in this case is to tell a wrong value of $p(j)$ at step $19$, i.e., when he gets $p(j)=0$, he declares $p(j)=1$. Bob can declare $p(j)=1$ for all $n$ registers. In this case, he actually comes to know the values of all the set elements of $X$. Hence, the security criterion that Bob should not be allowed to know the set elements of Alice other than the intersected ones, is violated.
However, Alice possesses some bits of the stream $r_1r_2\cdots r_l$. Alice gets the values of $q_t$ for all $n$ registers. For the cases where she knows $r_t$, she can easily calculate the value of $p(j)$ and can check if those values match with the values declared by Bob. If not, Alice aborts the protocol without announcing any element further (step $20$).
On the other hand, Bob has at most a random guess about $t$, i.e., the position of Alice’s conclusive result. So, if he tries to declare wrong values for $p(j)$, with probability $\frac{1}{2}$, he will be caught by Alice and protocol will be terminated.
Again, Alice does not know the whole bit stream. So, she can not calculate $p(j)$ for all $n$ registers by herself alone and will not be able to get $\mathcal{F}$ completely by knowing $q_
t$ only. In this case, she can get a very few elements which is equivalent to obtaining $\perp$. Hence, both Alice and Bob will end up with utility $U^{NN}$. According to $\mathcal{R}_1$, $U_B^{TT}> U_B^{NN}$, Bob has no incentive to follow the deviation. Rather he prefers cooperation. Thus, for Bob also we can write [$$\begin{aligned}
&U_B&((cooperate,abort), (cooperate,abort))\\
>&U_B&((cooperate,abort), deviation)\end{aligned}$$]{} This completes the proof.
If $((cooperate, abort), (cooperate, abort))$ is a strict Nash, then fairness of the protocol is guaranteed.
The strategy vector ((cooperate, abort), (cooperate, abort)) is strict Nash implies that $$\begin{aligned}
\Pr[f_A=\mathcal{F}|A=\sigma'_A, B=\sigma_B] \\
< \Pr[f_A=\mathcal{F}|A=\sigma_A, B=\sigma_B].\end{aligned}$$
where $\sigma'_A$ denotes any deviation by Alice. Similarly, for Bob we can write $$\begin{aligned}
\Pr[f_B=\mathcal{F}|A=\sigma_A, B=\sigma'_B] \\
< \Pr[f_B=\mathcal{F}|A=\sigma_A, B=\sigma_B].\end{aligned}$$ where $\sigma'_B$ denotes any deviation by Bob.
If $((cooperate, abort), (cooperate, abort))$ is a strict Nash, then correctness of the protocol is guaranteed.
The proof is immediate. As ((cooperate, abort), (cooperate, abort)) is a strict Nash, $$\begin{aligned}
\Pr[f_A\not\in \{\mathcal{F},\perp\}]=\Pr[f_B\not\in \{\mathcal{F},\perp\}]= 0.\end{aligned}$$
Our next job is to prove the security when Alice (resp. Bob) can mount an active attack. By the word “active attack”, we want to mean that though Alice and Bob follow the suggested strategies, but exploiting the advantage of quantum theory, they may steal some information which is restricted by the protocol. We now show that Alice as well as Bob know only $\mathcal{F}$ and nothing else in this rational setting. The analysis goes on the same line of [@Shi].
The goal of Alice and Bob is to know the elements, other than the intersected ones, of the sets of their respective opponents. In Algorithm $1$, Bob sends one part of each entangled pairs to Alice. So Alice can mount an active attack in this phase. By considering optimal POVM, she can increase her success probability to guess a bit of the bit-stream $r_1r_2\cdots r_l$. This attack has been considered in theorem \[min\_ent1\].
In protocol $\bf{\Pi}$, Bob does not send any elements to Alice. However, Alice sends all the set elements in an encrypted form to Bob. So, in this phase, it is Bob who can perform an active attack on Alice’s system.
One most simple and common attack is measure and resend attack. In this attack model Bob measures each register in some bases and then prepares that register in the measured state. In this way he tries to extract some information about the set elements of Alice.
To detect such type of attack we set $n$ check registers. Those are prepared either in $\{{\ensuremath{\left|0\right\rangle}},{\ensuremath{\left|1\right\rangle}}\}$ or in $\{\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|1\right\rangle}}),\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|1\right\rangle}})\}$ basis. As Bob does not know the positions of those check registers, he can not bypass those and performs the attack only on the actual registers containing the set elements. So, checking the error rate for the check registers, Alice can identify the attack.
Due to no cloning [@wootters; @dieks] theorem and Heisenberg uncertainty principle, Bob can not distinguish the check registers with probability one. If he tries to distinguish them, he must incorporate some noise in the system.
On the other hand, Alice knows the bases for those states. So, she can measure the returning registers in perfect bases, i.e., either in $\{{\ensuremath{\left|0\right\rangle}},{\ensuremath{\left|1\right\rangle}}\}$ or in $\{\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|1\right\rangle}}), \frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|1\right\rangle}})\}$ basis. Now, the oracle $O_t$, $t\in[1,l]$, converts
- ${\ensuremath{\left|0\right\rangle}}$ to ${\ensuremath{\left|0\right\rangle}}$
- ${\ensuremath{\left|1\right\rangle}}$ which is encoded as ${\ensuremath{\left|k_s\right\rangle}}{\ensuremath{\left|0\right\rangle}}^{\otimes {M-1}}$; $k_s=1$ and $s\in[1,M]$ , to $(-1)^{q_t(k_s)}{\ensuremath{\left|k_s\right\rangle}}\\ {\ensuremath{\left|0\right\rangle}}^{\otimes {M-1}}$
- $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}\pm {\ensuremath{\left|1\right\rangle}})$ which is encoded as $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}\pm {\ensuremath{\left|j\right\rangle}})$, where ${\ensuremath{\left|0\right\rangle}}={\ensuremath{\left|0\right\rangle}}^{\otimes M}$ and ${\ensuremath{\left|j\right\rangle}}={\ensuremath{\left|k_s\right\rangle}}{\ensuremath{\left|0\right\rangle}}^{\otimes{M-1}}$; $k_s=1$ and $s\in\{1,2,\cdots,M\}$, to $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}} \pm (-1)^{q_t(k_s)}{\ensuremath{\left|j\right\rangle}})$
Hence, if Alice sends ${\ensuremath{\left|0\right\rangle}}$, in case of no attack, she should get ${\ensuremath{\left|0\right\rangle}}$. If she gets ${\ensuremath{\left|1\right\rangle}}$, she concludes that the attack has been mounted. Same thing happens for ${\ensuremath{\left|1\right\rangle}}$. However, the attack can not be distinguished for the states $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|1\right\rangle}})$ and $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|1\right\rangle}})$. This is because, without any attack the state $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|1\right\rangle}})$ may convert to $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|1\right\rangle}})$ and vice versa due to the effect of the oracle $O_t$. But Alice can always detect the noise in $\{{\ensuremath{\left|0\right\rangle}},{\ensuremath{\left|1\right\rangle}}\}$ basis. If it is above the threshold, Alice aborts the protocol.
In this regard, we like to emphasize that in [@Shi] it is commented that this attack can be identified by measuring the returned decoy states in $\{\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_i\right\rangle}}),\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|j_i\right\rangle}})\}$ bases, $i\in [1,l]$, $j_i\in Z_N^*$. Bob will measures the states in computational basis. If he gets ${\ensuremath{\left|j_i\right\rangle}}$, he can prepare the state as $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_i\right\rangle}})$ or $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|j_i\right\rangle}})$. As a result Alice can not detect if the attack performed in the system. However, if he gets ${\ensuremath{\left|0\right\rangle}}$, he can not create perfect superposition. And Alice can detect the attack measuring the states in $\{\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_i\right\rangle}}),\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|j_i\right\rangle}})\}$ basis. However, the oracle has been designed in such a way so that it can map $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_i\right\rangle}})$ into $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|j_i\right\rangle}})$. Alice does not know $q_t(j)$. So it is not possible for her to determine when she would get $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|j_i\right\rangle}})$ or $\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|j_i\right\rangle}})$ apriori. In that case, it is not very clear how Alice can distinguish if the attack has been performed or the oracle has been operated on the states. To avoid such ambiguity, we prepare the check elements in $\{{\ensuremath{\left|0\right\rangle}},{\ensuremath{\left|1\right\rangle}}\}$ and $\{\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|1\right\rangle}}),\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|1\right\rangle}})\}$ bases randomly so that Alice can distinguish whether attack has been mounted or oracle has been operated checking the noise in $\{{\ensuremath{\left|0\right\rangle}},{\ensuremath{\left|1\right\rangle}}\}$ basis.
In [@Shi], the authors analyze a more complicated attack known as entanglement measure attack. In this attack model Bob combines an ancillary state with each register. He then performs a unitary operation on the register and the ancillary state. The unitary operation $Q$ is described as follows. $$\begin{aligned}
Q_{AB}{\ensuremath{\left|0\right\rangle}}_A{\ensuremath{\left|0\right\rangle}}_B&=&\sqrt{\eta}{\ensuremath{\left|0\right\rangle}}_A{\ensuremath{\left|\phi_0\right\rangle}}_B+\sqrt{1-\eta}{\ensuremath{\left|V_0\right\rangle}}_{AB}\\
Q_{AB}{\ensuremath{\left|v\right\rangle}}_A{\ensuremath{\left|0\right\rangle}}_B&=&\sqrt{\eta}{\ensuremath{\left|v\right\rangle}}_A{\ensuremath{\left|\phi_v\right\rangle}}_B+\sqrt{1-\eta}{\ensuremath{\left|V_v\right\rangle}}_{AB}\\
\end{aligned}$$ Thus, one can write $$\begin{aligned}
Q_{AB}\frac{1}{\sqrt{2}}({\ensuremath{\left|0\right\rangle}}_A+{\ensuremath{\left|v\right\rangle}}_A){\ensuremath{\left|0\right\rangle}}_B&=&\sqrt{\eta}({\ensuremath{\left|0\right\rangle}}_A{\ensuremath{\left|\phi_0\right\rangle}}_B
+{\ensuremath{\left|v\right\rangle}}_A{\ensuremath{\left|\phi_v\right\rangle}}_B)\\
&+&\sqrt{1-\eta}({\ensuremath{\left|V_0\right\rangle}}_{AB}+{\ensuremath{\left|V_v\right\rangle}}_{AB}).
\end{aligned}$$ Where $A$ stands for Alice’s system and $B$ stands for Bob’s system; $\eta$ is some probability. Here, $$\begin{aligned}
\langle{0}{\phi_0}|{V_0}\rangle_{AB}=\langle{v}{\phi_v}|{V_v}\rangle_{AB}=\langle{0}{\phi_0}|{V_v}\rangle_{AB}= \langle{v}{\phi_v}|{V_0}\rangle_{AB}=0.\\
\end{aligned}$$ After applying the oracle he then sends the registers back to Alice and keeps the ancillary systems with him. He measures the ancillary systems to extract the information about the states of the registers. In this initiative, Shi et al. [@Shi] bound the amount of information extracted by Bob by fixing the threshold value sufficiently small.
In our case, we redefine this attack models as follows. $$\begin{aligned}
Q_{AB}({\ensuremath{\left|\psi\right\rangle}},{\ensuremath{\left|W\right\rangle}}) = \sqrt{\eta}{\ensuremath{\left|\psi\right\rangle}}{\ensuremath{\left|E_{00}\right\rangle}} + \sqrt{1-\eta}{\ensuremath{\left|\psi\right\rangle}}^{\perp}{\ensuremath{\left|E_{01}\right\rangle}}\\
Q_{AB}({\ensuremath{\left|\psi\right\rangle}}^{\perp},{\ensuremath{\left|W\right\rangle}}) = \sqrt{1-\eta}{\ensuremath{\left|\psi\right\rangle}}{\ensuremath{\left|E_{10}\right\rangle}} + \sqrt{\eta}{\ensuremath{\left|\psi\right\rangle}}^{\perp}{\ensuremath{\left|E_{11}\right\rangle}}\end{aligned}$$ where, ${\ensuremath{\left|\psi\right\rangle}}$ is any arbitrary qubit of the form $\cos{\frac{\theta}{2}}{\ensuremath{\left|0\right\rangle}}+\sin{\frac{\theta}{2}}{\ensuremath{\left|1\right\rangle}}$, $\theta\in [0,\frac{\pi}{2}]$ and ${\ensuremath{\left|\psi\right\rangle}}^{\perp}$ is the orthogonal state of ${\ensuremath{\left|\psi\right\rangle}}$. $W$ is the ancillary state inserted by Bob. $E_{u,v}$, $u,v\in\{0,1\}$, are the states possessed by Bob after the application of $Q_{AB}$. Here, we assume that $$\begin{aligned}
\langle E_{00}|E_{01}\rangle=\langle E_{10}|E_{11}\rangle=0.\end{aligned}$$ For the check elements in $\{{\ensuremath{\left|0\right\rangle}},{\ensuremath{\left|1\right\rangle}}\}$ basis, the above equations reduce to $$\begin{aligned}
Q_{AB}({\ensuremath{\left|0\right\rangle}},{\ensuremath{\left|W\right\rangle}}) = \sqrt{\eta}{\ensuremath{\left|0\right\rangle}}{\ensuremath{\left|E_{00}\right\rangle}} + \sqrt{1-\eta}{\ensuremath{\left|1\right\rangle}}{\ensuremath{\left|E_{01}\right\rangle}}\\
Q_{AB}({\ensuremath{\left|1\right\rangle}},{\ensuremath{\left|W\right\rangle}}) = \sqrt{1-\eta}{\ensuremath{\left|0\right\rangle}}{\ensuremath{\left|E_{10}\right\rangle}} + \sqrt{\eta}{\ensuremath{\left|1\right\rangle}}{\ensuremath{\left|E_{11}\right\rangle}}\end{aligned}$$ When Bob returns those registers to Alice, Alice measures those in $\{{\ensuremath{\left|0\right\rangle}},{\ensuremath{\left|1\right\rangle}}\}$ basis. She knows when ${\ensuremath{\left|0\right\rangle}}$ (resp. ${\ensuremath{\left|1\right\rangle}}$) has been sent. If she gets the orthogonal states of the states sent to Bob, she concludes that the attack has been mounted and aborts the protocol.
Communication Complexity of the Protocol
========================================
In this section, we compute the communication complexity of the proposed protocol. Like most of the quantum protocol, in our protocol also we have to communicate qubits as well as classical bits. So, we can divide the communication complexity into two parts; one is quantum communication complexity and another is classical communication complexity.
In $QKeyGen$ part the total communication complexity is $2l$, as $2l$ entangled qubits are sent from Bob to Alice. The total quantum communications required in the protocol $\bf\Pi$ is $4n$; $2n$ quantum registers are sent from Alice to Bob and $2n$ quantum registers are returned back from Bob to Alice. Thus, the overall quantum communication complexity is $(4n+2l)$.
After error estimation phase, Alice finds $q_t$ and $t$ for each $n$ registers. The value of $t$ is expressed in $\log l$ bits and $q_t\in\{0,1\}$. So the total communicated bits in step $17$ is $n (\log l+1)$. In step $18$, Bob declares the value of $p(j)\in\{0,1\}$ for each $n$ registers. Thus, in step $18$, there are $n$ classical communications. In step $19$, Alice announces the value of the set elements for which Bob declared $p(j)=1$. We assume that there are $u$ intersected elements. So, if we express the value of each $x_i\in X$ in $\log N$ bits, then the total number of communicated bits should be $u\log N$. Hence, the overall classical communication complexity becomes $(n(\log l+2)+u\log N)$.
In this regard, we like to compare the communication complexity with other similar or classical schemes, to put the protocol in perspective.
In classical domain Freedman, Nissim and Pinkas [@freedman] studied set intersection problem in semi-honest setting. The sets in their protocol include $0$. The communication complexity of the protocol is $O(m_X+m_Y)$, where $m_X$ and $m_Y$ are the cardinality of the sets considered.
Hazay and Lindell [@Hazay] revisited the set intersection problem in the motivation to propose an efficient protocol against a more realistic adversary than semi-honest adversary. In this direction, they proposed two protocols; one against a malicious adversary and other against a covert adversary. Both the protocols are constant round and incur the communication of $O(m_X.p(n)+m_Y)$ group elements, where $p(n)$ is polynomial in the security parameter $n$.
The protocol proposed by Dachman-Soled, Malkin, Raykova and Yung [@dachman] for set intersection in the presence of malicious adversaries incurs communication of $O(m_Y n^2 log^2 m_X + m_X n)$ group elements.
In quantum domain, the communication complexity of [*Quantum Oblivious Set Member Decision Problem*]{} by Shi et al. [@Shi], is constant, i.e., $O(1)$. However, one should note that none of these schemes considered rational adversaries.
Conclusion
==========
In the present draft we propose a two party protocol for computing set intersection securely in quantum domain. The parties, Alice and Bob, have two sets $X$ and $Y$ which are computationally indistinguishable from each other. In classical domain this problem has been considered in [@HZ1; @HZ2; @JN10]. However, the hardness assumptions exploited in those works are proven to be vulnerable in quantum domain.
In quantum domain Shi et al. [@Shi] proposed a variant of this problem and named it as [*Quantum Oblivious Set Member Decision Protocol*]{} (QOSMDP). We extend this problem to compute set intersection of two computationally indistinguishable sets. We consider rational setting as rational setting is more realistic than being completely honest or completely malicious.
In rational setting, we prove that if $(cooperate, abort)$ is the suggested strategy profile for each of the two players, then $((cooperate, abort), (cooperate, abort))$ is a strict Nash equilibrium in our protocol. Following the lines of security proof of [@Shi], we also show that in this initiative Alice and Bob only know $\mathcal{F}$ and nothing else, i.e., Alice does not know any element of the set $Y\setminus(X \cap Y)$ and Bob does not know any element from the set $X\setminus(X \cap Y)$. We also prove that if $((cooperate, abort), (cooperate, abort))$ is a strict Nash, then fairness and correctness of the protocol are preserved.
S. D. Gordon, C. Hazay, J. Katz, Y. Lindell. Complete fairness in secure two-party computation. [*Proceedings of the 40th Annual ACM symposium on Theory of computing (STOC)*]{}, 413–422, ACM Press, (2008).
G. Asharov, R. Canetti, C. Hazay. Towards a Game Theoretic View of Secure Computation [*Advances in Cryptology - EUROCRYPT 2011*]{}, LNCS [**[6632]{}**]{}, 426–445, (2011).
A. Groce, J. Katz. Fair computation with rational players. [*Advances in Cryptology - EUROCRYPT 2012*]{}, 81–98, Springer Berlin Heidelberg, (2012).
N. Brunner, N. Linden. Bell nonlocality and Bayesian game theory [*Nature Communications*]{} [**4**]{}, 2057, (2013).
A. Maitra, S. J. De, G. Paul, A. Pal, Proposal for quantum rational secret sharing. [*Phys. Rev. A*]{}, [**92**]{}, 022305 (2015). C. Hazay, K. Nissim, Efficient Set Operations in the Presence of Malicious Adversaries. [*Journal of Cryptology*]{}, [**25**]{}, 383–433, (2012). C. Hazay, Oblivious Polynomial Evaluation and Secure Set-Intersection from Algebraic PRFs. TCC 2015, LNCS 9015, 90–120, (2015). S. Jarecki, X. Liu, Efficient Oblivious Pseudorandom Function with Applications to Adaptive OT and Secure Computation of Set Intersection. TCC 2009, LNCS 5444, 577–594, (2009). W. Diffie and M. E. Hellman, New Directions in Cryptography. IEEE Transactions on Information Theory, [**22**]{}, 644–654, (1976). D. Stinson. Cryptography Theory and Practice. Chapman & Hall / CRC, Third Edition, (2005). P. W. Shor, Algorithms for Quantum Computation: Discrete Logarithms and Factoring, [*Foundations of Computer Science (FOCS) 1994*]{}, 124–134, IEEE Computer Society Press, (1994).
R -H Shi, Y. Mu, H. Zhong, S. Zhang, Quantum oblivious set-member decision protocol [*Phys. Rev. A*]{}, [**92**]{}, 022309, (2015).
R -H Shi, Y. Mu, H. Zhong, J. Cui, S. Zhang, An efficient quantum scheme for Private Set Intersection [*Quant. Inf. Process.*]{}, [**15**]{}, 363–371, (2016). S. D. Gordon, C. Hazay, J. Katz and Y. Lindell. Complete Fairness in Secure Two-Party Computation. [*Journal of the ACM (JACM)*]{}, [**58**]{} (6), (2011).
G. Asharov, Y. Lindell. Utility Dependence in Correct and Fair Rational Secret Sharing. [*Journal of Cryptology.*]{} [**24**]{}, 157–202, (2010).
W. K. Wootters, W. H. Zurek, A single quantum cannot be cloned, [*Nature*]{}, [**299**]{}, 802803, (1982).
D. Dieks, Communication by EPR devices, [*Phys. Lett. A*]{}, [**92 (6)**]{}, 271272, (1982).
Y. G. Yang, S. J. Sun, P. Xu, J. Tiang, Flexible protocol for quantum private query based on B92 protocol. [*Quantum Inf. Process*]{}, [**13**]{}, 805–813, (2014). Mark M. Wilde, Quantum Information Theory, Cambridge University Press, 2, (2017). R. J. Serfling, Probability inequalities for the sum in sampling without replacement, [*Ann. Stat.*]{}, [**2**]{}, 39, (1974). M. Freedman, K. Nissim, B. Pinkas, Efficient private matching and set-intersection, EUROCRYPT’04, LNCS 3027, 1–19, Springer, Berlin, (2004). C. Hazay, Y. Lindell, Efficient protocols for set intersection and pattern matching with security against malicious and covert adversaries, In 5th TCC. LNCS 4948, 155–175, Springer, Berlin, (2008). D. Dachman-Soled, T. Malkin, M. Raykova, M. Yung, Efficient robust private set intersection, ANCS, LNCS 5479, 125–142, Springer, Berlin, (2009).
[^1]: For the brevity of notation, in the rest of the paper, we will write $\log(.)$ instead of $\log_2(.)$
[^2]: $M$ qubit string can be written as the tensor product of $M$ individual qubits. As $j\in \mathbb{Z_N^*}$, $j$ can be expressed in $M=\log N$ bits. Each bit corresponds to a qubit. Thus ${\ensuremath{\left|j\right\rangle}}$ can be written as ${\ensuremath{\left|d_M\right\rangle}}^{\otimes M}$, $d_M\in\{0,1\}$ and $M\in[1,\log N]$.
[^3]: We assume that there is no payoff for a player who deviates from the game and gets partial knowledge about the functionality. In this case, partial knowledge is considered as no knowledge or $\perp$.
|
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abstract: 'In this article, we investigate the image and preimage of the important combinatorial sets such as central sets, $C$-sets, and $J_\delta$-sets which play an important role in the study of combinatorics under certain partial semigroup homomorphism. Using that we prove certain results which deal with the existence of $C$-set which are not central in partial semigroup framework.'
address: 'Department of Mathematics, University of Kalyani, Kalyani, Nadia-741235 West Bengal, India'
author:
- Arpita Ghosh
title: A study on some combinatorial sets in Partial Semigroups
---
introduction
============
The notion of the central subset of $\mathbb{N}$ was originally introduced by Furstenberg [@F81] in terms of a topological dynamical system. Before defining central sets let us start with original Central Sets Theorem due to Furstenberg
(Original Central Sets Theorem) Let $l\in\mathbb{N}$ and for each $i\in\{1,2,\ldots,l\}$, and let $\langle y_{i,n}\rangle_{n=1}^{\infty}$ be a sequence in $\mathbb{Z}$. Let $C$ be a central subset of $\mathbb{N}$. Then there exist sequences $\langle a_{n}\rangle_{n=1}^{\infty}$ in $\mathbb{N}$ and $\langle H_{n}\rangle_{n=1}^{\infty}$ in $\mathcal{P}_{f}(\mathbb{N})$ such that
- for all $n$, $\max H_{n} < \min H_{n+1}$ and
- for all $F\in\mathcal{P}_{f}(\mathbb{N})$ and all $i\in\{1,2,\ldots,l\},$ $$\sum_{n\in F}(a_{n}+\sum_{t\in H_{n}}y_{i,t})\in C$$.
[@F81 Proposition 8.21].
This theorem has several combinatorial consequences such as Rado’s theorem which deals with the regularity of the system of integral equations. Central sets in natural numbers are known to have substantial combinatorial structure. For example, any central set contains arbitrary long arithmatic progressions, all finite sums of distinct terms of an infinite sequence.
In [@BH90], Vitaly Bergelson and Neil Hindman came with an algebraic characterization of the central subsets of natural numbers in terms of the algebra of $\beta \mathbb{N}.$ Analogously, the notion of a central set has been defined in an arbitrary discrete semigroup $S.$ To define the central subsets in a semigroup, we recall a brief introduction of the algebraic structure of $\beta S$ for a discrete semigroup $(S, \cdot)$.
We take the points of $\beta S$ to be the ultrafilters on $S$, identifying the principal ultrafilters with the points of $S$ and thus pretending that $S\subseteq\beta S$. Given $A\subseteq S$ let us set, $\overline{A}=\{p\in\beta S\mid A\in p\}$. Then the set $\{\overline{A}\mid A\subseteq S\}$ is a basis for a topology on $\beta S$. The operation $\cdot$ on $S$ can be extended to the Stone-Čech compactification $\beta S$ of $S$ so that $(\beta S,\cdot)$ is a compact right topological semigroup (meaning that for any $p\in\beta S$, the function $\rho_{p}:\beta S\rightarrow\beta S$ defined by $\rho_{p}(q)=q\cdot p$ is continuous) with $S$ contained in its topological center (meaning that for any $x\in S$, the function $\lambda_{x}:\beta S\rightarrow\beta S$ defined by $\lambda_{x}(q)=x \cdot q$ is continuous). Given $p,q\in\beta S$ and $A\subseteq S$, $A\in p \cdot q$ if and only if $\{x\in S \mid x^{-1} \cdot A\in q\}\in p$, where $x^{-1} \cdot A=\{y\in S \mid x\cdot y\in A\}$.
A nonempty subset $I$ of a semigroup $(T,\cdot)$ is called a left ideal of ${T}$ if $T\cdot I\subset I$, a right ideal if $I\cdot T\subset I$, and a two-sided ideal (or simply an ideal) if it is both a left and a right ideal. A minimal left ideal is a left ideal that does not contain any proper left ideal. Similarly, we can define a minimal right ideal.
Any compact Hausdorff right topological semigroup $(T,\cdot)$ has a smallest two sided ideal
$$\begin{array}{rl}
K(T) = & \bigcup\{L \mid L\text{ is a minimal left ideal of }T\}\\
= &\bigcup\{R \mid R\text{ is a minimal right ideal of }T\}
\end{array}$$
Given a minimal left ideal $L$ and a minimal right ideal $R$, $L\cap R$ is a group, and in particular, contains an idempotent. An idempotent in $K(T)$ is called a minimal idempotent. If $p$ and $q$ are idempotents in $T$, we write $p\leq q$ if and only if $p\cdot q=q\cdot p=p$. An idempotent is minimal with respect to this relation if and only if it is a member of the smallest ideal. See [@HS12] for an elementary introduction to the algebra of $\beta S$ and for any unfamiliar details.
Now we recall the central sets of a semigroup as
Consider a discrete semigroup $S$ and a subset $A$ of $S.$ Then $A$ is central if and only if there is an idempotent in $K(\beta S) \cap \overline{A}.$
In [@HS12], the Central Sets Theorem was extended to arbitrary semigroups by allowing the choice of countably many sequences at a time. More extended version of the Central Sets Theorem considering all sequences at one time has been established in [@DHS08]. The sets which satisfy the conclusion of the above Central Sets Theorem are the objects that matter. They have several applications in the study of all non-trivial partition regular systems of homogeneous integral equations. Hindman and Strauss understand their importance and named them as $C$-sets. Also, they found similar kind of algebraic characterizations of $C$-sets in terms of the algebra of $\beta S$ and $J$-sets, which was also introduced by them. We recall the definition of $C$-set which will be useful later.
Let $(S,+)$ be a commutative semigroup and let $A\subseteq S$ and $\mathcal{T}=S^{\mathbb{N}}$, the set of sequences in $S$. The set $A$ is a $C$-set if and only if there exist functions $\alpha:\mathcal{P}_{f}(\mathcal{T})\rightarrow S$ and $H:\mathcal{P}_{f}(\mathcal{T})\rightarrow\mathcal{P}_{f}(\mathbb{N})$ such that
\(1) if $F,G\in\mathcal{P}_{f}(\mathcal{T})$ and $F\subsetneq G$, then $ \max H(F) < \min H(G)$ and
\(2) whenever $m\in\mathbb{N},G_{1},G_{2},\ldots,G_{m}\in\mathcal{P}_{f}(\mathcal{T}),G_{1}\subsetneq G_{2}\subsetneq\ldots\subsetneq G_{m}$ and for each $i\in\{1,2,\ldots,m\}$ , $f_{i}\in G_{i}$, one has $\sum_{i=1}^{m}(\alpha(G_{i})+\sum_{t\in H(G_{i})}f_i(t))\in A$.
In this article, the main object of study is partial semigroups. In [@AG18], the author extended the Central Sets Theorem obtained by taking all possible sequences in commutative adequate partial semigroup. It states as
\[adeprod\] Let $(S,\ast)$ be a commutative adequate partial semigroup and let $C$ be a central subset of $S$. Let $\mathcal{T}_S$ be the set of all adequate sequences in $S.$ There exist functions $\alpha:\mathcal{P}_{f}(\mathcal{T}_S)\rightarrow S$ and $H:\mathcal{P}_{f}(\mathcal{T}_S)\rightarrow\mathcal{P}_{f}(\mathbb{N})$ such that
- $F,G\in\mathcal{P}_{f}(\mathcal{T}_S)$ and $F\subsetneq G$, then $\max H(F) < \min H(G)$ and
- whenever $m\in\mathbb{N}$, $G_{1},G_{2},\ldots,G_{m}\in\mathcal{P}_{f}(\mathcal{T}_S)$, $G_{1}\subsetneq G_{2}\subsetneq\ldots\subsetneq G_{m}$, and for each $i\in\{1,2,\ldots,m\}$, $f_i \in G_{i}$, one has $\prod_{i=1}^{m}(\alpha(G_{i})\ast\prod_{t\in H(G_{i})}f_i(t))\in C$.
[@AG18 Theorem 2.4].
In the same article, the author also characterized those sets in any adequate commutative partial semigroups which satisfy the new version of the Central Sets Theorem [@AG18 Theorem 2.4] and introduced the analogous notion of $C$-set and $J$-set, namely respectively $C$-set and $J_{\delta}$-set in adequate partial semigroup as
\[cset\] Let $(S,\ast)$ be a commutative adequate partial semigroup and $A$ be a subset of $S.$ Let $\mathcal{T}_S$ be the set of all adequate sequences in $S.$\
a) $A$ is said to be a $C$-set if and only if there exist functions $\alpha:\mathcal{P}_{f}(\mathcal{T}_S)\rightarrow S$ and $H:\mathcal{P}_{f}(\mathcal{T}_S)\rightarrow\mathcal{P}_{f}(\mathbb{N})$ such that
- if $F,G\in\mathcal{P}_{f}(\mathcal{T}_S)$ and $F\subsetneq G$, then $\max H(F) < \min H(G)$ and
- whenever $m\in\mathbb{N}$, $G_{1},G_{2},\ldots,G_{m}\in\mathcal{P}_{f}(\mathcal{T}_S)$, $G_{1}\subsetneq G_{2}\subsetneq\ldots\subsetneq G_{m}$ and for each $i\in\{1,2,\ldots,m\}$, $f_i \in G_{i}$, one has$\prod_{i=1}^{m}(\alpha(G_{i})\ast\prod_{t\in H(G_{i})}f_i(t)\in A$.
b\) $A$ is a $J_{\delta}$-set if and only if whenever $F\in\mathcal{P}_{f}(\mathcal{T}_S)$, $W\in\mathcal{P}_{f}(S)$, there exist $a\in\sigma(W)$ and $H\in\mathcal{P}_{f}(\mathbb{N})$ such that for each $f\in F$, $\prod_{t\in H}f(t)\in\sigma(W\ast a)$ and $a\ast\prod_{t\in H}f(t)\in A$.\
c) $J_{\delta}(S)=\{p\in\delta S:$ for all $A\in p\,,\,A$ is a $J_{\delta}$-set$\}$.
For a semigroup $S$, the $C$-sets, defined in purely combinatorial terms, are characterized as members of idempotents in $J(S)$. In [@AG18], the author studied a similar kind of properties in the context of adequate partial semigroups and introduced the set $J_{\delta}(S)$. In this article, we first show that for a commutative adequate partial semigroup sets are ideals in $\delta S$ ([*cf.*]{} Theorem 3.4). Next, we study the behaviour of these combinatorial sets under a surjective partial semigroup homomorphism ([*cf.*]{} Theorem 5.5., Theorem 5.6., Theorem 5.7., and Theorem 5.9.). With the help of these results we able to construct a way to construct enormous amount of $C$-sets which are not central.
$\mathcal{T}_S=$ The set of all adequate sequences in $S.$
Preliminaries
=============
In this section, we recall some definitions and results from partial semigroup context. For more details, the readers are referred to [@HS12]. We start with the following definition
(Partial semigroup) A partial semigroup is defined as a pair $(G, \ast)$ where $\ast$ is an operation defined on a subset $X$ of $G \times G$ and satisfies the statement that for all $x$, $y$, $ z$ in $G$, $(x \ast y) \ast z = x \ast (y \ast z) $ in the sense that if either side is defined, so is the other and they are equal. A partial semigroup is commutative if ${x\ast y=y\ast x}$ for every ${(x,y)\in X}$.
Partial semigroups which arise naturally our minds.
Any semigroup is an obvious example of a partial semigroup.
Let us consider ${G=\mathcal{P}_f({\mathbb{N}})}= \{F \mid \emptyset \neq F \subseteq \mathbb{N}$ and $F$ is finite$\}$ and let ${X=\{(\alpha,\beta)\in G \times G \mid \alpha\cap\beta=\emptyset\}}$ be the family of all pairs of disjoint sets, and let ${\ast:X\rightarrow G}$ be the union. It is easy to check that this is a commutative partial semigroup. We shall denote this partial semigroup as $(\mathcal{P}_f({\mathbb{N}}),\uplus)$.
Next, we define homomorphisms between partial semigroups.
Let $(S, \ast)$ and $(T, \ast^{\prime})$ be partial semigroups and let $h : S \rightarrow T.$ Then $h$ is a partial semigroup homomorphism if and only if whenever $y \in \phi_S(x),$ one has that $h (y) \in \phi_T(h(x))$ and $h(x \ast y)= h(x) \ast^{\prime} h(y).$ Here $\phi_S(s)$ stands for the set $\{t\in S \mid s\ast t$ is defined in $S\}$.
For a semigroup, it is known that there is a notion of compactification, namely the Stone-$\check{C}$ech compactification, which is a compact right topological semigroup. One can do the same thing for any partial semigroup but that won’t be a semigroup. To get a compact topological semigroup out of a partial semigroup we recall the following definitions.
Let $(S,\ast)$ be a partial semigroup.\
(a) For $s\in S$, $\phi_S(s)=\{t\in S \mid s\ast t$ is defined in $S\}$.\
(b) For $H\in\mathcal{P}_{f}(S)$, $\sigma_S(H)=\bigcap_{s \in H}\phi_S(s)$.\
(c) $(S,\ast)$ is adequate if and only if $\sigma_S(H)\neq\emptyset$ for all $H\in\mathcal{P}_{f}(S)$.\
(d) $\delta S=\bigcap_{x\in S}cl_{\beta S}(\phi_S(x))=\bigcap_{H\in\mathcal{P}_{f}(S)}cl_{\beta S}(\sigma_S(H))$.
So, the partial semigroup $(\mathcal{P}_f({\mathbb{N}}),\uplus)$ is adequate. We are specifically interested in adequate partial semigroups as they lead to an interesting subsemigroup $\delta S$ of $\beta S$, the Stone-Čech compactification of $S$ which is itself a compact right topological semigroup. Notice that adequacy of $S$ is exactly what is required to guarantee that $\delta S\neq\emptyset$. If $S$ is, in fact, a semigroup, then $\delta S=\beta S$.
Now we recall some of the basic properties of the operation $\ast$ in $\delta S$.
Let $(S,\ast)$ be a partial semigroup. For $s\in S$ and $A\subseteq S$, $s^{-1}A=\{t\in\phi_S(s) \mid s\ast t\in A\}$.
Let $(S,\ast)$ be a partial semigroup, let $A\subseteq S$ and let $a,b,c\in S$. Then $c\in b^{-1}(a^{-1}A)$ if and only if both $b\in\phi_S(a)$ and $c\in(a\ast b)^{-1}A$. In particular, if $b\in\phi_S(a)$, then $b^{-1}(a^{-1}A)=(a\ast b)^{-1}A$.
[@HM01 Lemma 2.3]
Let $(S,\ast)$ be an adequate partial semigroup.\
(a) For $a\in S$ and $q\in\overline{\phi_S(a)}$, $a\ast q=\{A\subseteq S \mid a^{-1}A\in q\}$.\
(b) For $p\in\beta S$ and $q\in\delta S$, $p\ast q=\{A\subseteq S \mid \{a^{-1}A\in q\}\in p\}$.
Let $(S,\ast)$ be an adequate partial semigroup.\
(a) If $a\in S$ and $q\in\overline{\phi_S(a)}$, then $a\ast q\in\beta S$.\
(b) If $p\in\beta S$ and $q\in\delta S$, then $p\ast q\in\beta S$.\
(c) Let $p\in\beta S,q\in\delta S$, and $a\in S$. Then $\phi_S(a)\in p\ast q$ if and only if $\phi_S(a)\in p$.\
(d) If $p,q\in\delta S$, then $p\ast q\in\delta S$.
[@HM01 Lemma 2.7].
\[1.12\] Let $(S,\ast)$ be an adequate partial semigroup and let $q\in\delta S$. Then the function $\rho_{q}:\beta S\rightarrow\beta S$ defined by $\rho_{q}(p)=p\ast q$ is continuous.
[@HM01 Lemma 2.8].
Let $p\in\beta S$ and let $q,r\in\delta S$. Then $p\ast(q\ast r)=(p\ast q)\ast r$.
[@HM01 Lemma 2.9].
Let $p=p \ast p \in \delta S$ and let $A \in p$. Then $A^{\ast} = \{x \in A \mid x^{-1}A \in p\}$.
Given an idempotent $p \in \delta S$ and $A \in p$, it is immediate that $A^{\ast} \in p$.
\[1.15\] Let $p= p \ast p \in \delta S$, let $A \in p$, and let $x \in A^{\ast}$. Then $x^{-1}A^{\ast} \in p$.
[@HM01 Lemma 2.12].
As a consequence of the above results, we have that if $(S, \ast)$ is an adequate partial semigroup, then $(\delta S, \ast)$ is a compact right topological semigroup. Being a compact right topological semigroup, $\delta S$ contains idempotents, left ideals, a smallest two-sided ideal, and minimal idempotents. Thus $\delta S$ provides a suitable environment for considering the notion of central sets and it defines as
Let $(S,*)$ be an adequate partial semigroup. A set $C\subseteq S$ is [*central*]{} if and only if there is an idempotent $p\in\overline C\cap K(\delta S)$.
In the Central sets Theorem for semigroup, we have studied that a necessary condition for the central sets along the line of sequences. Here, in the partial semigroup setting we can expect the similar thing for an adequate sequences instead of the normal one.
The notion of adequate sequence plays an important role in the study of central sets theorem for the partial semigroups. So, we recall the definition as
Let $(S,\ast)$ be an adequate partial semigroup and let $\left\langle y_{n}\right\rangle _{n=1}^{\infty}$ be a sequence in $S$. Then $\left\langle y_{n}\right\rangle _{n=1}^{\infty}$ is adequate if and only if $\prod_{n\in F}y_{n}$ is defined for each $F\in\mathcal{P}_{f}(\mathbb{N})$ and for every $K\in\mathcal{P}_{f}(S)$, there exists $m\in\mathbb{N}$ such that $FP(\left\langle y_{n}\right\rangle _{n=m}^{\infty})\subseteq\sigma_S(K)$.
Let $W_1, W_2 \in \mathcal{P}_f(S)$, then define $W_1 \ast W_2=
\{w_1 \ast w_2 \mid w_1 \in W_1, w_2 \in W_2 \; \text{and} \; w_1 \ast w_2 \; \text{is defined}\}$.
The sets which satisfy the new version of Central Sets Theorem \[adeprod\] in adequate partial semigroup are said to be $C$-sets.
Properties of $J_{\delta}$-sets
===============================
This section concerns with the close look up on the $J_{\delta}$-sets and comes with some essential properties. We start with two essential lemmas.
\[prod ade\] Let $(S, \ast)$ be an adequate partial semigroup. Let $f$ be an adequate sequence in $S$ and let $\langle H_n \rangle_{n=1}^{\infty}$ be a sequence in $\mathcal{P}_f(\mathbb{N})$ such that $\max H_n < \min H_{n+1}$ for each $n \in \mathbb{N}.$ Define $g :\mathbb{N} \rightarrow S$ such that for each $n \in \mathbb{N}$, $g(n)= \prod_{t \in H_n} f(t).$ Then $g$ is an adequate sequence in $S.$
To see that $g$ is an adequate sequence, let $G \in \mathcal{P}_f(\mathbb{N})$ and let $H= \bigcup_{n \in G}H_n.$ Then $H \in \mathcal{P}_f(\mathbb{N}).$ Since $f$ is adequate then $\prod_{t \in H} f(t)$ is defined. Now $\prod_{n \in G}g(n)= \prod_{n \in G} \prod_{t \in H_n} f(t)= \prod_{t \in H} f(t),$ then $\prod_{n \in G} g(n)$ is defined. Let $K \in \mathcal{P}_f(S)$ be given. Then there exist $m \in \mathbb{N}$ such that $FP(\langle f(t) \rangle_{t=m}^{\infty}) \subseteq \sigma_S (K).$ Now we want to show that for $K \in \mathcal{P}_f(S),$ there exists $m \in \mathbb{N}$ such that $FP(\langle g(t) \rangle_{t=m}^{\infty}) \subseteq \sigma_S (K).$ Let $N \in \mathcal{P}_f(\{n \in \mathbb{N} : n \geqslant m\}).$ Then for each $n \in N$, $\min H_m \leqslant \min H_n.$ Let $H^{\prime}= \bigcup_{n \in N}H_n.$ Then $H^{\prime} \in \mathcal{P}_f(\{n \in \mathbb{N} : n \geqslant m\})$ and hence $\prod_{t \in H^{\prime}}f(t) \subseteq \sigma_S(K).$ Now $\prod_{n \in N}g(n)= \prod_{n \in N}(\prod_{t \in H_n}f(t))= \prod_{t \in H^{\prime}}f(t) \in \sigma_S(K).$ Therefore, $g$ is an adequate sequence.
\[J prod\]
Let $S$ be an adequate commutative partial semigroup, let $A$ be a $J_{\delta}$-set in $S.$ Let $F \in \mathcal{P}_f(\mathcal{T}_S),$ and let $\langle H_n \rangle_{n=1}^{\infty}$ be a sequence in $\mathcal{P}_f(\mathbb{N})$ such that for each $n$, $\max H_n < \min H_{n+1}.$ Then for all $W \in \mathcal{P}_f(S),$ there exist $a \in \sigma_S (W)$ and $G \in \mathcal{P}_f(\mathbb{N})$ such that for all $f \in F$, $\prod_{k \in G} \prod_{t \in H_k} \in \sigma_S(W \ast a)$ and $a \ast \prod_{k \in G} \prod_{t \in H_k} \in A. $
For $f \in F,$ define $g_f : \mathbb{N} \rightarrow S$ by $g_f(k)=\prod_{t\in H_k}f(t).$ By Lemma \[prod ade\], $g_f$ is an adequate sequence in $S.$ Now since $A$ is a $J_{\delta}$-set, then for $W \in \mathcal{P}_f(S)$ there exist $a \in \sigma_S(W)$ and $G \in \mathcal{P}_f(\mathbb{N})$ such that for each, $f \in F$, $\prod_{k \in G}g_f(k) \in \sigma_S(W \ast a)$ and $a \ast \prod_{k \in G}g_f(k) \in A.$ These imply $\prod_{k \in G} \prod_{t \in H_k} \in \sigma_S(W \ast a)$ and $a \ast \prod_{k \in G} \prod_{t \in H_k} \in A. $
\[partition r\] Let $S$ be an adequate commutative partial semigroup. Let $A$ be a $J_{\delta}$-set in $S,$ and assume that $A=A_1 \cup A_2.$ Either $A_1$ is a $J_{\delta}$-set in $S$ or $A_2$ is a $J_{\delta}$-set in $S.$
Suppose the conclusion of the statement of the theorem is false. That is both $A_1$ and $A_2$ are not $J_\delta$-sets. Then, we can pick $F_1, F_2 \in \mathcal{P}_f(\mathcal{T}_S)$ and $W_1, W_2 \in \mathcal{P}_f(S)$ such that for all $i \in \{1,2\}$ and for all $d \in \sigma_S(W_i)$ and $K \in \mathcal{P}_f(\mathbb{N})$ such that there exist $f \in F_i$ such that $d \ast \prod_{t\in K}f(t) \notin A_i \cap \sigma_S(W_i).$ Now, set $W = W_1 \cup W_2$ and $F = F_1 \cup F_2.$ Assume that $F = \{f_1, \cdots, f_p \}.$
Next, consider $B = \{w \mid w \; \text{is a word of length}\; n \; \text{over} \; \{ 1, \cdots,p\} \}.$ For each $w = b_1 \cdots b_n \in B$ define a function $g_w : \mathbb{N} \to S$ given by $$g_w(y) = \prod_{1 \leq i \leq n} f_{b_i}(ny +i)$$ where $y \in \mathbb{N}.$ Then, Lemma \[prod ade\] guarantees that for each $w \in B$, $g_w$ is an adequate sequence. Define $G = \{g_w \mid w \in B \} \subseteq \mathcal{P}_f(\mathcal{T}_S).$ Since, $A$ is a $J_{\delta}$-set then for this $G$ and the set $W$ chosen earlier we get $a \in \sigma_S(W)$ and $H \in \mathcal{P}_f(\mathbb{N})$ such that for each $w \in B$ $$a \ast\prod_{y \in H} g_w(y) \in A \cap \sigma_S(W).$$
By [@HS12 Lemma 14.8.1], we can pick $n \in \mathbb{N}$ such that whenever the set $B$ is 2-colored there is a variable word $w(v)$ beginning and ending with a constant and without successive occurrences of $v$ such that $\{w(l): l \in \{ 1, \cdots, p\} \}$ is monochromatic.
Define a $2$-coloring $\phi : B \to \{ 1,2\}$ on the set $B$ given by $$\phi(w) =\begin{cases} 1, & \text{if} \; a \ast\prod_{y \in H} g_w(y) \in A_1 \cap \sigma_S(W) \\ 2, & \text{otherwise} \end{cases}.$$ So, we can pick a variable word $w(v)$ beginning and ending with a constant and without successive occurrences of $v$ such that $\{w(l): l \in \{ 1, \cdots, p\} \}$ is monochromatic. Without loss of generality, we assume that $\phi(w(l)) =1.$ In other words, we assume that for each $l \in \{1, \cdots, p\}$, $$\label{par}
a \ast \prod_{y \in H}g_{w(l)}(y) \in A_1 \cap \sigma_S(W).$$
Suppose $w(v) = b_1\cdots b_n$ and $r$ is the total number of occurrence of $v$ in the word $w(v).$ Then we can write down the set $\{j \in \{1, \cdots, n\} \mid b_j \in \{ 1, \cdots, p\} \} = \bigcup_{i =1}^{r+1}L(i)$ and $\{j \in \{1, \cdots, n\}\mid b_j = v\} = \{s(1), \cdots, s(r) \}$ for some function $s : \{ 1, \cdots, r\} \to \mathbb{N}$ such that $max L(x) < s(x) < min L(x+1)$. Precisely, the function $s$ refers the position function for $v$ in $w(v).$ The construction of $L(i)$’s can be understand by the following example. Let $w(v) = 2v31v12.$ Then $r =2$, $L(1) =\{1\},$ $L(2) =\{ 3,4\},$ and $L(3) = \{ 6,7\}.$ So, $L(i)$ is the set of $j \in \{ 1, \cdots, n\}$ such $b_j \in \{1, \cdots, p \}$ and $b_j$’s are in between $(i-1)^{th}$ and $i^{th}$ occurrences of $v$ in $w(v).$ Next, we try to understand the term $a \ast \prod_{y \in H}g_{w(l)}(y).$ Using the previous discussion so far we have $$\begin{aligned}
a \ast \prod_{y \in H}g_{w(l)}(y)& = a \ast \prod_{y \in H} \prod_{1 \leq i \leq n}f_{b_i}(ny +i) \\ & =a \ast \prod_{y \in H}\prod_{k=1}^{r+1} \prod_{i \in L(k)} f_{b_i}(ny+i) \ast \prod_{y \in H} \prod_{i=1}^{r} f_l(ny + s(i)) \\ & =d \ast \prod_{t \in K}f_l(t)
\end{aligned}$$
Where $$d = a \ast \prod_{y \in H}\prod_{k=1}^{r+1} \prod_{i \in L(k)} f_{b_i}(ny+i)\; \text{and}\; K = \bigcup_{i=1}^r \{ny + s(i)\mid y \in H \}$$. Then definitely $K$ is a finite subset of $\mathbb{N}.$ Also, tells us $W \ast a \ast \prod_{y \in H}g_{w(l)}(y)$ is defined i.e. $W \ast d \ast \prod_{t \in K}f_l(t)$ is defined. Using associativity, this implies $W \ast d$ is defined. Hence $d \in \sigma_S(W)$ and $d \ast \prod_{t\in K}f_l(t) \in A_1 \cap \sigma_S(W) \subset A_1 \cap \sigma_S(W_1).$ This a contradiction to our assumption.
\[ideal\] Let $(S, \ast)$ be an adequate commutative partial semigroup. Then $J_{\delta }(S) $ is a closed two sided ideal of $\delta S.$
Let $A$ be a $J_{\delta}$-set in $S.$ Then Theorem \[partition r\] yields the $J_{\delta}$-sets are partition regular. Next, we claim that $J_{\delta}(S) \neq \emptyset$. To prove the claim, we consider the sets $\mathfrak{R} = \{ A \subseteq S \mid A \; \text{a is } \; J_{\delta}\text{-set}\}$ and $\mathfrak{A} = \{ \phi_S(s) \mid s \in S\}.$ Observe that for $\sigma_S(F) = \cap_{s \in F}\phi_S(s)$ we always can find a $J_{\delta}$-set $A$ such that $A \subseteq \sigma_S(F).$ This gives that for any finite intersections of the members of $\mathfrak{A}$ lies inside $\mathfrak{R}^{\uparrow}.$ Therefore, using Theorem [@HS12 Theorem 3.11] we get a ultrafilter $p$ of $S$ such that $\mathfrak{A} \subseteq p \subseteq \mathfrak{R}^{\uparrow}.$ The first inclusion implies that for each $s \in S,$ $\phi_S(s) \in p$, therefore, $p \in \delta S.$ This forces $p \in J_{\delta}(S).$ Hence, the claim is established. Next, we want to show that $J_{\delta}(S)$ is closed. Let $p \in \delta S \setminus J_{\delta}(S),$ then there exists $B \in p$ such that $B$ is not a $J_{\delta}$-set, therefore, $\overline{B} \cap J_{\delta}(S) = \emptyset ,$ i.e., $J_{\delta}(S) $ is closed.
We want to prove that $J_{\delta }(S)$ is a two sided ideal of $\delta S,$ i.e., for all $p \in \delta S$, $q \in J_{\delta }(S)$ imply that $ p \ast q \in J_{\delta}(S)$ and $q \ast p \in J_{\delta}(S).$
Let $A \in p \ast q.$ We want to show that $A$ is a $J_{\delta }$-set.
Since $A \in p \ast q,$ then $\{a \in S \mid a^{-1}A \in q\} \in p. $ Pick $a \in S$ such that $a^{-1}A \in q.$ Now $q \in J_{\delta }(S),$ then $a^{-1}A$ is a $J_{\delta}$-set. Therefore, for $F \in \mathcal{P}_f(\mathcal{T}_S)$ and $W \in \mathcal{P}_f(S),$ there exist $b \in \sigma_S(W \ast a)$ and $H \in \mathcal{P}_f(\mathbb{N})$ such that $\prod_{t \in H}f(t) \in \sigma_S (W \ast a \ast b)$ and $b \ast \prod_{t \in H}f(t) \in a^{-1}A= \{d \in \phi_S(a) \mid a \ast d \in A\}.$ This implies $a \ast b \ast \prod_{t \in H}f(t) \in A.$ Now define $c=a \ast b.$ Then for $F \in \mathcal{P}_f(\mathcal{T}_S)$ and $W \in \mathcal{P}_f(S),$ there exist $c \in \sigma_S(W)$ and $H \in \mathcal{P}_f(\mathbb{N})$ such that for each $f \in F$, $\prod_{t \in H}f(t) \in \sigma_S (W \ast c)$ and $c \ast \prod_{t \in H}f(t) \in A.$ Therefore, $A \in p \ast q.$
Now let $A \in q \ast p.$ We want to show that $A$ is a $J_{\delta}$-set.
Let $B=\{a \in S \mid a^{-1}A \in p \}.$ Since $A \in q \ast p,$ then $\{a \in S \mid a^{-1}A \in p \} \in q.$ This implies $B \in q.$ Now $q \in J_{\delta}(S),$ then $B$ is a $J_{\delta}$-set. Let $F \in \mathcal{P}_f(\mathcal{T}_S)$ and $F=\{f_1, f_2, \cdots, f_k\}$. Let $W_{0} \in \mathcal{P}_f(S),$ then there exist $b \in \sigma_S(W_{0})$ and $H \in \mathcal{P}_f(\mathbb{N})$ such that for each $f \in F$, $\prod_{t \in H}f(t) \in \sigma_S(W_{0} \ast b) $ and $b \ast \prod_{t \in H}f(t) \in B.$ Therefore, $(b \ast \prod_{t \in H}f(t))^{-1}A \in p$ for all $f \in F$ and this yields $\bigcap\limits_{f \in F}(b \ast \prod_{t \in H}f(t))^{-1}A \in p.$ Now since $p \in \delta S,$ then $\sigma_S(W) \in p$ for all $W \in \mathcal{P}_f(S).$ Therefore, $\sigma_S(W) \cap \bigcap\limits_{f \in F}(b \ast \prod_{t \in H}f(t))^{-1}A \in p$ for all $W \in \mathcal{P}_f(S).$ So it must be nonempty. Take $W= W_{0} \ast F_0,$ where $F_0=\{b \ast \prod_{t \in H}f_1(t),b \ast \prod_{t \in H}f_2(t), \cdots,b \ast \prod_{t \in H}f_k(t)\} $ and pick $$c \in \sigma_S(W_0 \ast F_0) \cap \bigcap\limits_{f \in F}(b \ast \prod_{t \in H}f(t))^{-1}A.$$ Now choose $a=b \ast c.$ Then for each $f \in F$, $\prod_{t \in H}f(t) \in \sigma_S(W_0 \ast a)$ and $a \ast \prod_{t \in H}f(t) \in A.$ This suffices $A$ is a $J_{\delta}$-set.
Let $(S, \ast )$ be an adequate commutative partial semigroup. Let $A \subseteq S.$ If $A$ is a central set in $S,$ then $A$ is a $C$-set in $S.$
Since $A$ is a central set in $S,$ then there is an idempotent $p$ such that $p \in K(\delta S) \cap \bar{A}.$ By lemma \[ideal\], $J_{\delta}(S)$ is a two sided ideal of $\delta S. $ Then $K(\delta S) \subseteq J_{\delta }(S).$ Therefore, by [@AG18 Theorem 3.4], $A$ is a $C$-set in $S.$
Construction of $J_{\delta}$-sets
=================================
This section deals with some technical construction of $J_{\delta}$-sets. We start with some definitions.
Let $\omega$ denotes the first infinite ordinal and each ordinal is the set of its predecessors. In particular, $[0]=\emptyset$ and for $n \in \mathbb{N}$, $[n]=\{0,1,\cdots, n-1\}.$
If $f$ is a function and domain$(f)=[n] \in \omega,$ then for all $x \in S$, $f^{\frown}x = f \cup \{ (n, x)\}$. Precisely, it means we extend the domain of $f$ to $[n+1]$ by defining $f(n)=x.$
Let $T=\{f \mid f : [n] \rightarrow S \},$ i.e., $T$ is a set of functions whose domains are members of $\omega.$ For each $f \in T,$ define $B_f(T)=\{ x \mid f^{\frown}x \in T\}.$
Using these notions in hand we can stated the following
\[words\] Let $S$ be an adequate partial semigroup and let $p \in \delta S.$ Then $p$ is an idempotent if and only if for each $A \in p$ there is a nonempty set $T$ of functions such that
- For all $f \in T,$ domain$(f)\in \omega$ and range$(f) \subseteq A.$
- For all $f \in T$, $B_f(T) \in p.$
- For all $f \in T$ and all $x \in B_f(T)$, $B_{f^{\frown}x}(T) \subseteq x^{-1}B_f(T).$
We claim that $p$ is an idempotent element in $\delta S,$ i.e., $\{x \in S \mid x^{-1}A \in p\} \in p.$ Now there is given $A \in p$ and $T$ which satisfies the above conditions. Pick $f \in T.$ Then $B_f(T) \in p.$ Therefore, if we will prove that $B_f(T) \subseteq \{x \in S \mid x^{-1}A \in p\},$ then our claim will be proved. Let $x \in B_f(T),$ then $f^{\frown}x \in T. $ Therefore, by (b) $B_{f^{\frown}x}(T) \in p.$ Now by (c), $B_{f^{\frown}x}(T) \subseteq x^{-1}B_f(T) \subseteq x^{-1}A.$ Thus $x^{-1}A \in p,$ and so $p$ is an idempotent element.
Conversely, let $p$ be an idempotent element in $\delta S$ and let $A \in p.$ For any $B \in p, $ let define $B^*=\{x \in B \mid x^{-1}B \in p\}.$ Then $B^* \in p$ and by [@HM01 Lemma 2.12], for $x \in B^*$, $x^{-1}B^* \in p.$ To prove the result we inductively construct a filtration of $T = \bigcup\limits_{[n] \in \omega} T_n$ where $T_n=\{f \in T \mid \text{domain}(f)=[n]\}$ and for each $f\in T_n,$ define $B_f=B_f(T).$ Note that $T_{\emptyset} = \{ 0\}$. Now set $B_{\emptyset} = A^*.$
Now let $[n] \in \omega$ and assume that we have defined $T_k$ for $k \leq n$ and defined $B_f$ for $f \in T_k$ such that
- $T_k $ is a set of functions with domain $[k] $ and range contained in $A.$
- If $f \in T_k$ and $x \in B_f,$ then $B_f \in p$ and $x^{-1}B_f \in p.$
- If $k < n$, $f \in T_k,$ and $x \in B_f,$ then $B_{f^{\frown}x}=(x^{-1}B_f)^*.$
If $n=0$, $T_0=\{\emptyset\}.$ Then (a) is trivially true. For (b), $B_{\emptyset}=A^* \in p$ and if $x \in A^*,$ then $x^{-1}A^* \in p.$ Again (c) is vacuously true.
Now, we will check the hypotheses for $T_{n+1}.$
$T_{n+1}=\{f \in T \mid \text{domain}(f)=[n+1] \}.$ So, for any $f \in T_{n+1}$ can be written as $f = {f|_{[n]}}^{\frown}x$ where $f(n)=x.$ So, we can write $T_{n+1}=\{f^{\frown} x \mid f \in T_n \; \text{and} \; x \in B_f\}.$ Now let $g \in T_{n+1},$ then we can take $g=f^{\frown}x$ for $f \in T_n$ and $x=g(n).$ Now by (b), for $f \in T_n$ and $x \in B_f$, $x^{-1}B_f \in p.$ Let $B_g=(x^{-1}B_f)^*.$ Now let $y \in B_g= (x^{-1}B_f)^*,$ then $y^{-1}B_g \in p.$
Therefore, the hypotheses are true for $T_{n+1}.$
Now we are in a position to construct a huge amount of $J_{\delta}$-sets using the set $B_f(T)$ and some particular idempotent elements in the algebra $\delta S.$ Precisely, we have
Let $(S, \ast)$ be an adequate partial semigroup and let $A \subseteq S.$ If there is an idempotent $p \in \bar{A} \cap J_{\delta }(S).$ Then there is a non-empty set $T$ of functions such that:
- For all $f \in T$, domain$(f)\in \omega$ and range$(f) \subseteq A.$
- For all $f \in T$ and all $x \in B_f(T)$, $B_{f^{\frown}x}(T) \subseteq x^{-1}B_f(T).$
- For all $F \in \mathcal{P}_f(T)$, $\bigcap_{f \in F}B_f(T)$ is a $J_{\delta }$-set.
Moreover, there is a downward directed family $\langle A_F\rangle_{F \in I}$ of subsets of $A$ such that:
- For all $F \in I$ and all $x \in A_F,$ there exists $G \in I$ such that $A_G \subseteq x^{-1}A_F.$
- For each $\mathcal{F} \in \mathcal{P}_f(I)$, $\bigcap_{F \in \mathcal{F}}A_F$ is a $J_{\delta}$-set.
Since $p$ is an idempotent element in $\delta S$ and $A \in p,$ then by Lemma \[words\], (i) and (ii) are true and for each $f \in F$, $B_f(T) \in p.$ Therefore, $\bigcap_{f \in F}B_f(T) \in p.$ Now since $p \in J_{\delta}(S),$ then $\bigcap_{f \in F}B_f(T)$ is a $J_{\delta}$-set. This concludes part (iii).
For the existence of a downward directed family we do the following construction. Let $I \in \mathcal{P}_f(T).$ For $F \in I,$ define $A_F=\bigcap_{f \in F}B_f(T).$ For part a), consider $G=\{f^{\frown}x \mid f \in F\}.$ Now by Lemma \[words\], for each $f \in F$, $B_{f^{\frown}x}(T) \subseteq x^{-1}B_f(T).$ This implies that $A_G \subseteq x^{-1}A_F.$ Next, for part b), using the previous arguments, we obtain $A_F$ is a $J_{\delta}$-set. Given $\mathcal{F} \in \mathcal{P}_f(I),$ if $H= \bigcup \mathcal{F},$ then $\bigcap_{F \in \mathcal{F}}A_F=A_H. $ Therefore, $A_H$ is a $J_{\delta }$-set.
and also, we can readily reduce
Let $(S, \ast)$ be a countable adequate partial semigroup and let $A \subseteq S.$ If there is an idempotent $p \in \bar{A} \cap J_{\delta}(S),$ then there is a decreasing sequence $\langle A_n \rangle_{n=1}^{\infty}$ of subsets of $A$ such that:
- For all $n \in \mathbb{N}$ and all $x \in A_n,$ there exists $m \in \mathbb{N}$ such that $A_m \subseteq x^{-1}A_n.$
- For all $n \in \mathbb{N}$, $A_n$ is a $J_{\delta}$-set.
Let $S$ be countable. Then $T$ is also countable. So identify $T$ as $\{f_n \mid n \in \mathbb{N}\}$. For $n \in \mathbb{N},$ let $A_n=\bigcap_{k=1}^n B_{f_k}(T).$ Then each $A_n$ is a $J_{\delta}$-set. Let $x \in A_n.$ Now pick $m \in \mathbb{N}$ such that $\{{f_k}^{\frown}x \mid k \in \{1,2, \cdots, n\}\} \subseteq \{f_1,f_2, \cdots, f_m\}.$ Then $A_m \subseteq x^{-1}A_n.$
Construction of C-sets which are not central
============================================
It is known that the combinatorial sets such as central sets, $C$-sets, and $J_{\delta}$-sets have an extra importance in the study of combinatorics. Also, it is elementary from the definition that every central sets are $C$-sets. In this section, we try to find out a way to construct a $C$-set which are not central. Before go into that we start with few basic facts.
\[semigroup\] Let $A$ and $B$ be semigroups and let $f: A \rightarrow B$ be a surjective homomorphism. If $A$ has a smallest ideal, show that $B$ does as well and that $K(B)=f(K(A)).$
Let $b \in B$ and $y \in f(K(A)).$ Since $f $ is surjective, there exists $a \in A$ such that $f(a)=b.$ Now since $y \in f(K(A)),$ then there exists $x \in K(A)$ such that $f(x)=y.$ As $a \cdot x \in K(A),$ then $b \cdot y= f(a) \cdot f(x)=f(a \cdot x) \in f(K(A)).$ Therefore, $f(K(A))$ is a left ideal in $B.$ Similarly, we can show that it is a right ideal in $B.$
$K(A)=\bigcup\{L \mid L \; \text{is a minimal left ideal of} \; A\}.$ Then, it follows that $$f(K(A))= \bigcup\{f(L)\mid L \; \text{is a minimal left ideal of} \; A\}.$$ Let $J \subseteq f(L),$ where $J$ is an ideal of $B.$ Then $f^{-1}(J) \subseteq L.$ But $L$ is minimal left ideal of $A,$ then $J=f(L).$ Therefore, $f(L)$ is minimal left ideal of $B.$ Then $f(K(A)) \subset K(B).$ But $K(B)$ is the smallest ideal, then $f(K(A))=K(B).$
\[6.0.2\] Let $f : A \rightarrow B$ be a semigroup homomorphism. If $A$ has a smallest ideal, then $f(K(A))=K(f(A)).$
If $f : A \rightarrow B$ is a semigroup homomorphism, then $f : A \rightarrow f(A)$ is a surjective homomorphism. Then by Lemma \[semigroup\], if $A$ has smallest ideal, then $f(A)$ also has smallest ideal and $f(K(A))= K(f(A)).$
\[semihomo\] Let $S$ and $T$ be adequate partial semigroups. Let $h : S \rightarrow T$ be a surjective partial semigroup homomorphism, i.e., $h(S)=T.$ Let $\tilde{h} : \beta S \rightarrow \beta T$ be the continuous extension of $h.$ Then $\tilde{h}(\delta S) \subseteq \delta T$ and $\tilde{h} \mid_{\delta S}$ is a semigroup homomorphism.
For the proof, see [@HS12 Theorem 4.22.3].
\[rmk\] By Lemma \[semihomo\] and Corollary \[6.0.2\], $\tilde{h}(K(\delta S))= K(\tilde{h}(\delta S)).$
\[Jset\] Let $S$ and $T$ be two adequate commutative partial semigroups, let $h: S\rightarrow T$ be a surjective partial semigroup homomorphism. If $A$ is a $J_{\delta}$-set in $S,$ then $h(A)$ is a $J_{\delta}$-set in $T.$
Given $A$ is a $J_{\delta}$-set in $S.$ Let $F \in \mathcal{P}_f(\mathcal{T}_T)$ and $W \in \mathcal{P}_f(T).$ Pick $k: T \rightarrow S$ such that $h\circ k(x)=x$ locally (pointwise). Construct $G=k(F)=\{k\circ f \mid f \in F\}$ and $W_S= k(W).$ Then $G \in \mathcal{P}_f(\mathcal{T}_S)$ and $W_S \in \mathcal{P}_f(S).$ Since $A$ is a $J_{\delta}$-set in $S,$ then there exist $a \in \sigma_S (W_S)$ and $H \in \mathcal{P}_f(\mathbb{N})$ such that for each $g \in G$,
$$\label{2}
\prod_{t \in H}g(t) \in \sigma_S(W_S \ast a) \; \mbox{and} \; a \ast \prod_{t \in H}g(t) \in A.$$
Choose $b \in T$ such that $b=h(a)$ and $$h(a) \in h (\sigma_S(W_S)) \subseteq \sigma_T(h(W_S)) = \sigma_T(h \circ k(W))= \sigma_T(W)$$ Now from one part of the condition (6.1), we have $$h(\prod_{t \in H}g(t) )\in h(\sigma_S(W_S \ast a))$$ which implies $\prod_{t \in H}h(g(t)) \in \sigma_T(h(W_S) \ast h(a)),$ and, so that $\prod_{t \in H}h\circ k \circ f(t) \in \sigma_T(W \ast b).$ This implies $\prod_{t \in H}f(t) \in \sigma_T(W \ast b)$ and the other part of the condition gives us that $h(a \ast \prod_{t \in H}g(t)) \in h(A).$ Since $h$ is partial semigroup homomorphism, then we get, $h(a) \ast \prod_{t \in H}h(g(t)) \in h(A)$ which clearly implies that $b \ast \prod_{t \in H}f(t) \in h(A).$ Therefore, $h(A)$ is a $J_{\delta}$-set in $T.$
\[central\] Let $S$ and $T$ be two adequate commutative partial semigroups, let $h: S\rightarrow T$ be a surjective partial semigroup homomorphism. Let $\tilde{h} : \beta S \rightarrow \beta T$ be the continuous extension of $h.$ If $\tilde{h}: \delta S \rightarrow \delta T$ induces surjective map then
- $\tilde{h}(K(\delta S))= K(\delta T).$
- If $A$ is a central set in $S,$ then $h(A)$ is a central set in $T.$
- If $A$ is a central set in $T,$ then $h^{-1}(A)$ is a central set in $S.$
\(a) Given that $\tilde{h}: \delta S \rightarrow \delta T$ is surjective semigroup homomorphism. Being compact right topological semigroup $\delta S$ has a smallest two sided ideal, then by Lemma \[semigroup\], $\delta T$ has also smallest two sided ideal and $K(\delta T)= \tilde{h}(K(\delta S)).$
\(b) Let $A$ be a central set in $S,$ then there is an idempotent $p$ such that $p \in K(\delta S) \cap \overline{A}.$ Since $\tilde{h} \mid_{\delta S}$ is a semigroup homomorphism by Lemma \[semihomo\] and hence $\tilde{h}(p)$ is an idempotent element and by part (a), $\tilde{h}(K(\delta S))= K(\delta T).$ Therefore, $\tilde{h}(p)$ is contained in $K(\delta T),$ and, so $\tilde{h}(p)$ is an idempotent element in $\overline{h(A)} \cap K(\delta T).$
\(c) Pick an idempotent element $p$ in $K(\delta T) \cap \overline{A}.$ By part (a), pick $q \in K(\delta S)$ such that $\tilde{h}(q)=p.$ Now pick a minimal left ideal $L$ of $ \delta S$ such that $q \in L.$ Then $L \cap \tilde{h}^{-1}(\{p\})$ is a compact subsemigroup of $\delta S.$ So there is an idempotent $r \in L \cap \tilde{h}^{-1}(\{p\}).$ Since $A \in p$ and $\tilde{h}(r)=p$, then $h^{-1}(A) \in r.$ Thus, $h^{-1}(A)$ is a central set in $S.$
\[cset\] Let $S$ and $T$ be two adequate commutative partial semigroups, let $h: S\rightarrow T$ be a surjective partial semigroup homomorphism. Let $\tilde{h} : \beta S \rightarrow \beta T$ be the continuous extension of $h.$
- $\tilde{h}(J_{\delta}(S)) \subseteq J_{\delta }(T)$
- If there is an idempotent $p \in \overline{A} \cap J_{\delta}(S),$ then $h(A)$ is a $C$-set in $T.$
- If $p \in \overline{A} \cap \tilde{h}(J_{\delta }(S)),$ then $h^{-1}(A)$ is a $C$-set in $S .$
\(a) Let $p \in J_{\delta }(S).$ We need to show that $\tilde{h}(p) \in J_{\delta}(T).$ Now let $A \in \tilde{h}(p).$ Then $h^{-1}(A) \in p.$ Therefore, $h^{-1}(A)$ is a $J_{\delta}$-set in $S.$ Now by Lemma \[Jset\], $A=h(h^{-1}(A))$ is a $J_{\delta}$-set in $T.$ Therefore, $\tilde{h}(p) \in J_{\delta}(T).$
\(b) Let $p \in \overline{A} \cap J_{\delta}(S).$ Since $\tilde{h} \mid_{\delta S}$ is a semigroup homomorphism by Lemma \[semihomo\] and hence $\tilde{h}(p)$ is an idempotent element in $\delta T.$ Now since $p \in \overline{A},$ then $h(A) \in \tilde{h}(p),$ and since $p \in J_{\delta}(S),$ then $p \in \delta S $ such that for all $B \in p$, B is a $J_{\delta}$-set in $S.$ Therefore, $\tilde{h}(p) \in \delta T$ where $h(B) \in \tilde{h}(p)$ and $h(B)$ is a $J_{\delta}$-set in $T$ by Lemma \[Jset\].
Now let $D\in \tilde{h}(p).$ We want to show that $D$ is a $J_{\delta}$-set in $T.$ Now for a surjective map $h$, $D=h(h^{-1}(D)).$ Since $h^{-1}(D) \in p,$ then $h^{-1}(D)$ is a $J_{\delta}$-set in $S$ and again by Lemma \[Jset\] $D$ is a $J_{\delta}$-set in $T.$ Therefore, we get an idempotent element in $\overline{h(A)} \cap J_{\delta}(T).$ Then by Theorem [@AG18 Theorem 3.4], $h(A)$ is a $C$-set in $T.$
\(c) Let $p$ be an idempotent element such that $p \in \overline{A} \cap \tilde{h}(J_{\delta }(S)).$ By Theorem \[ideal\], $J_{\delta}(S)$ is closed two sided ideal of $\delta S.$ Now being a closed subset of compact set $\delta S$, $J_{\delta }(S)$ is compact and $J_{\delta}(S) \cap \tilde{h}^{-1}(\{p\})$ is a compact subsemigroup of $\delta S,$ where $\tilde{h}^{-1}(\{p\})=\{r \in \delta S \mid \tilde{h}(r)=p\}.$ Therefore, there is an idempotent $q \in J_{\delta}(S) \cap \tilde{h}^{-1}(\{p\}).$ Since $A \in p$ and $\tilde{h}(q)=p,$ then $h^{-1}(A) \in q.$ Then by Theorem [@AG18 Theorem 3.4], $h^{-1}(A)$ is a $C$-set in $S.$
The following corollary is a direct consequence of Theorem \[central\] and Theorem \[cset\].
\[cor\] Let $S$ and $T$ be two adequate commutative partial semigroups, let $h: S\rightarrow T$ be a surjective partial semigroup homomorphism. Let $\tilde{h} : \beta S \rightarrow \beta T$ be the continuous extension of $h$ and $\tilde{h}: \delta S \rightarrow \delta T$ induces surjective map. If there is an idempotent $p \in \overline{A} \cap \tilde{h}(J_{\delta }(S)),$ where $A$ is not a central set in $T,$ then $h^{-1}(A)$ is a $C$-set in $S$ but not a central set in $S.$
Define $h:(\mathcal{P}_f(\mathbb{N}), \uplus) \to (\mathbb{N},+) $ by $h(A)=|A|,$ where $|A|$ is cardinality of the set $A.$ Then, $h$ is a surjective partial semigroup homomorphism. Let $\tilde{h} : \beta \mathcal{P}_f(\mathbb{N}) \rightarrow \beta \mathbb{N}$ be the continuous extension of $h.$ Then, $\tilde{h}(\delta \mathcal{P}_f(\mathbb{N}))= \beta \mathbb{N}.$ In [@HS9 Theorem 2.8] Hindman and Strauss gave an example which is a $C$-set in $(\mathbb{N}, +),$ but not central. Since the set $A$ produced in [@HS9 Theorem 2.8] is a $C$-set, then there is an idempotent element $p \in \overline{A} \cap J(\mathbb{N},+)$ by [@HS09 Theorem 2.5]. As $J_{\delta}(\mathbb{N}, +) =J(\mathbb{N}, +),$ therefore, we have that $p \in \overline{A} \cap J_{\delta}(\mathbb{N}, +).$ So, in our case, $\; \tilde{h}(J_{\delta}(\mathcal{P}_f(\mathbb{N}), \uplus))= J_{\delta}(\mathbb{N}, +)$. Thus, Theorem \[cset\] yields $h^{-1}(A)$ is a $C$-set, but by Corollary \[cor\], it is not central set in $(\mathcal{P}_f(\mathbb{N}), \uplus).$
[9]{} V. Bergelson and N. Hindman, *Nonmetrizable topological dynamics and Ramsey Theory*, Trans. Amer. Math. Soc. [**320**]{} (1990), 293-320.
D. De, N. Hindman, and D. Strauss, *A new and stronger central sets theorem*, Fund. Math. [**199**]{} (2008), no. 2, 155-175.
H. Furstenberg, *Recurrence in ergodic theory and combinatorial number theory*, Princeton University Press, Princeton, 1981. A. Ghosh, *A generalised central sets theorem in partial semigroups*, Semigroup Forum (2018), https://doi.org/10.1007/s00233-018-9977-7. A. Hales and R. Jewett, *Regularity and positional games*, Trans. Amer. Math, Soc. 106(1963), 222-229.
N. Hindman and R. McCutcheon, *VIP systems in partial semigroups*, Discrete Math. [**240**]{} (2001), 45-70. N. Hindman and D. Strauss, *A simple characterization of sets satisfying the Central Sets Theorem*, New York J. Math. 15 (2009) 405–413. N. Hindman and D. Strauss, *Sets satisfying the Central sets theorem*, Semigroup Forum [**79**]{} (2009), 480-506.
N. Hindman and D. Strauss, *Algebra in the Stone-Čech compactification: theory and applications, second edition*, W. de Gruyter and Co., Berlin, 2012.
J. McLeod, *Some notions of size in partial semigroups*, Topology Proc. 25 (2000), Summer, 317-332 (2002).
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abstract: |
This is a comment to the paper, Limitations on the superposition principle: superselection rules in non-relativistic quantum mechanics by C Cisneros et al 1998 Eur. J. Phys. 19 237. doi:10.1088/0143-0807/19/3/005.\
The proof that the authors construct for the limitation on the superposition of state vectors corresponding to different sectors of the Hilbert space, partitioned by a superoperator has a flaw as outlined below.
address: '$^1$National Institute of Science Education and Research, Bhubaneswar, India'
author:
- 'Namit Anand$^{1}$'
title: 'Comment on Limitations on the superposition principle: superselection rules in non-relativistic quantum mechanics'
---
Introduction
============
In Ref. [@ref1], section 2.4(Impossibility of superposing states belonging to different coherent sectors), the authors construct a proof using a superoperator G that commutes with all observables of the system. The eigenstates of operator G are of the form $|g_m;\alpha_m \rangle$ where $g_m$ and $\alpha_m$ are labels to distinguish different eigenvectors all of which are non-degenerate, i.e. $\langle g_m;\alpha_m | g_n;\alpha_n \rangle = 0$ if $m \neq m$ .\
In the next section(2.4), they construct a vector $|u\rangle$ = $\sum_m u_m |g_m;\alpha_m\rangle$ superposing the eigenstates of G and claim that since G commutes with every other observable, $|u\rangle$ should be an eigenstate of every operator in a complete set of commuting observables of the system.\
Now we know that if $G|a\rangle = a|a\rangle$ and $G|b\rangle = b|b\rangle$ are eigenvectors of an operator G, then $\lambda_1 |a\rangle+ \lambda_2 |b\rangle$ is not an eigenvector unless they have the same eigenvalue since: $$G\left(\lambda_1 |a\rangle+ \lambda_2 |b\rangle\right)$ = $ a \lambda_1 |a\rangle+ b \lambda_2 |b\rangle \neq \lambda \left( \lambda_1 |a\rangle+ \lambda_2 |b\rangle \right)$$ (for some $\lambda$) unless a = b, which is not true in general and definitely not true for the hermitian operators with non-degenerate eigenvalues in the paper.
It is true that if G commutes with every other observable and has non-degenerate eigenvalues, then all such obervables share the same eigenvectors. But $|u\rangle$ itself is not an eigenvector of G as proved above. And so $|u\rangle$ cannot be an eigenvector of other observables too. To summarize, the proof of superselection rules proposed is thus is not valid for the case discussed by the authors. Whether a similar proof can be constructed is still open.
References {#references .unnumbered}
==========
[10]{} C Cisneros et al 1998 *Eur. J. Phys* 19 237, doi:10.1088/0143-0807/19/3/005, Limitations on the superposition principle: superselection rules in non-relativistic quantum mechanics.
|
---
author:
- Wooyoung Chin
bibliography:
- 'wooyoung.bib'
title: 'An Exposition on Wigner’s Semicircular Law'
---
Introduction
============
If $A$ is an $n \times n$ Hermitian matrix, then $A$ is diagonalizable and all eigenvalues of $A$ are real. We denote the eigenvalues of $A$, counted with multiplicities, as $$\lambda_1(A) \ge \cdots \ge \lambda_n(A).$$ [(For details, see Subsection \[subsec:spectra\_basic\].)]{} We define the *spectral distribution* of $A$ as the Borel probability measure $$\mu_A := \frac{1}{n} \sum_{i=1}^n \delta_{\lambda_i(A)}$$ on ${\mathbf{R}}$. If $X$ is a random $n \times n$ Hermitian matrix, then $\mu_X$ is a random Borel probability measure on ${\mathbf{R}}$. [(For details, see Subsection \[subsec:rand\_hermitian\].)]{}
The Borel probability measure ${\mu_{\mathrm{sc}}}$ on ${\mathbf{R}}$ given by $${\mu_{\mathrm{sc}}}(dx) = \frac{1}{2\pi} \sqrt{(4-x^2)_+}\,dx$$ is called the *semicircle distribution*. Here, $x_+ := x \vee 0 = \max\{x,0\}$.
Since the seminal work [@Wig55] by Wigner, there have been many theorems that assume $W_1,W_2,\ldots$ to be random $1 \times 1$, $2 \times 2, \ldots$ Hermitian matrices satisfying certain conditions, and show that $\mu_{W_n}$ converges in some sense to ${\mu_{\mathrm{sc}}}$. Let us call such theorems *semicircular laws*.
In the main part of this paper, we study semicircular laws assuming joint independence of the upper triangular entries (we include the diagonal in both the upper and the lower triangles). We first prove a semicircular law (Theorem \[thm:sclaw\]) with rather weak assumptions. In particular, we don’t require the entries to be identically distributed, and we allow the entries to deviate from unit variance. It is notable that other than the mostly standard reduction steps, our proof is just a simple application of the moment method.
After proving the main theorem, we apply the theorem to obtain another semicircular law (Theorem \[thm:gauss\_sclaw\]) which more or less assumes that the sum of a row converges in distribution to the standard normal distribution. This theorem allows the entries to have infinite variances.
The appendices make up about two thirds of this paper. There we provide a self-contained and rigorous account of the details (including the measure-theoretic ones) involved in the main part of the paper. In the main part of the paper, we refer to the appendix whenever we need a fact given there. After that, for completeness we provide a proof of a semicircular law (little weaker than the one proved in the main part, but still stronger than the laws appear in many textbooks) which uses the Stieltjes transform method.
We assumed no prior knowledge more advanced than one-semester courses in probability theory and combinatorics. A total newcomer to the field might want to read the Appendices \[sec:prob\_meas\_r\]–\[sec:concen\_meas\] first, then read the main part, and then go to Appendix \[sec:unit\_reduction\]–\[sec:stieltjes\].
Now we state our main theorem.
\[thm:sclaw\] For each $n \in {\mathbf{N}}$, let $W_n = ({w_{ij}^{(n)}})_{i,j=1}^n$ be a random $n \times n$ Hermitian matrix [(see Definition \[def:rand\_hermitian\])]{} whose upper triangular entries are jointly independent, have mean zero, and have finite variances. We assume that $W_1, W_2, \ldots$ are defined on the same probability space. If $$\label{eq:sclaw_var}
{\lim_{n\to\infty}\frac{1}{n}
\sum_{i=1}^n \biggl|
\sum_{j=1}^n \Bigl( {\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}- \frac{1}{n} \Bigr)
\biggr|
=0},$$ $$\label{eq:sclaw_rowbdd}
{\lim_{n\to\infty}\frac{1}{n}
\sum_{i=1}^n \biggl( \sum_{j=1}^n {\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}- C \biggr)_+
=0} \qquad \text{for some finite $C \ge 0$,}$$ and $$\label{eq:sclaw_lind}
{\lim_{n\to\infty}\frac{1}{n} \sum_{i,j=1}^n {\operatorname{\mathbf{E}}\bigl[\,|{w_{ij}^{(n)}}|^2{\mathrel{;}}|{w_{ij}^{(n)}}|>{\epsilon}\,\bigr]}=0}
\qquad \text{for every ${\epsilon}>0$,}$$ then ${\mu_{W_n}}{\Rightarrow}{\mu_{\mathrm{sc}}}$ as $n \to \infty$ a.s.
One sufficient condition for and to hold is $${\lim_{n\to\infty}\frac{1}{n} \sum_{i,j=1}^n \Bigl|{\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}-\frac{1}{n}\Bigr| =0}.$$ Note that we can take $C = 1$ to show . An even more special case is when we have ${\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}= 1/n$ for all $n = 1,2,\ldots$ and $i,j = 1,\ldots,n$. This case is Theorem 2.9 in [@BS10]. If there is some finite $C \ge 0$ such that $$\label{eq:abs_rowbdd}
\sum_{j=1}^n {\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}\le C \qquad
\text{for all $n = 1,2,\ldots$ and $i = 1,\ldots,n$,}$$ then holds for the same $C$. This case is more or less equivalent to Corollary 1 in [@GNT15], which is proved first for matrices with Gaussian entries, and then generalized to arbitrary matrices by proving an analogue of the Lindeberg universality principle for random matrices. In this paper, we will prove Theorem \[thm:sclaw\] directly by the moment method without appealing to the universality principle.
Theorem \[thm:sclaw\] assumes no dependence between $W_1$, $W_2$, …, yet it asserts an a.s. convergence. This is in contrast to some versions of the semicircular law where only convergence in probability is asserted (e.g. [@AGZ10 Theorem 2.1.1]), or $\sqrt{n}W_n$ is assumed to be the top left $n \times n$ minor of a fixed infinite random Hermitian matrix (e.g. [@Tao12 Theorem 2.4.2]). If $\mu_1,\mu_2,\ldots$ are Borel probability distributions on a separable metric space $S$, and $c \in S$, then the following two statements are equivalent:
1. $X_n \to c$ a.s. whenever $X_1,X_2,\ldots$ are random elements of $S$ defined on a common probability space such that each $X_n$ has distribution $\mu_n$;
2. $\sum_{n=1}^\infty \mu_n\bigl(\{\, x \in S \mid d(x,c) > {\epsilon}\,\}\bigr)
< \infty$ for all ${\epsilon}> 0$.
This can be shown using the Borel-Cantelli lemmas [@Bil12 Theorem 4.3 and 4.4]. This type of strong convergence is possible in Theorem \[thm:sclaw\] because of a strong concentration of measure result we will use.
The rest of the paper is organized as follows. In Section \[sec:prelim\_reductions\], we will first reduce Theorem \[thm:sclaw\] to a form with stronger assumptions. Then we will see that the reduced semicircular law follows from some moment computations. In Section \[sec:trees\], we will develop a tool needed for the moment computation, and in Section \[sec:comp\_moments\], we will perform the actual moment computation. In Section \[sec:gaussian\], we will derive the aforementioned semicircular law which assumes Gaussian convergence of the sum of a row.
Preliminary reductions {#sec:prelim_reductions}
======================
Assume that $W_n$ satisfies the conditions of Theorem \[thm:sclaw\].
Convergence in expectation is enough
------------------------------------
If we have $\operatorname{\mathbf{E}}{\mu_{W_n}}{\Rightarrow}{\mu_{\mathrm{sc}}}$ [(for the meaning of $\operatorname{\mathbf{E}}\mu_{W_n}$, see Theorem \[thm:expect\_prob\])]{}, then $$\lim_{n\to\infty} \operatorname{\mathbf{E}}\biggl[\int_{\mathbf{R}}f\,d{\mu_{W_n}}\biggr] = \int_{\mathbf{R}}f\,d{\mu_{\mathrm{sc}}}$$ for all continuous and bounded $f\colon{\mathbf{R}}\to{\mathbf{R}}$. By the concentration inequality Theorem \[thm:concen\_spec\] for spectral measures and the Borel-Cantelli lemma, we have $$\lim_{n\to\infty} \int_{\mathbf{R}}f_{p,q}\,d{\mu_{W_n}}= \int_{\mathbf{R}}f_{p,q}\,d{\mu_{\mathrm{sc}}}\qquad \text{a.s.}$$ for all $p,q \in {\mathbf{Q}}$ with $p < q$, where $f_{p,q}\colon{\mathbf{R}}\to{\mathbf{R}}$ is $1$ on $(-\infty,p\,]$, $0$ on $[\,q,\infty)$, and linear on $[\,p,q\,]$. This implies that ${\mu_{W_n}}{\Rightarrow}{\mu_{\mathrm{sc}}}$ a.s. by Theorem \[thm:ch\_weak\_conv\]. Therefore, it is enough to show $\operatorname{\mathbf{E}}{\mu_{W_n}}{\Rightarrow}{\mu_{\mathrm{sc}}}$.
Truncation
----------
Since holds, we have positive integers $n_1 < n_2 < \ldots$ such that $$\frac{1}{n} \sum_{i,j=1}^n \operatorname{\mathbf{E}}\bigl[\,|{w_{ij}^{(n)}}|^2{\mathrel{;}}|{w_{ij}^{(n)}}|>1/k\,\bigr]
\le 1/k$$ for all $n \ge n_k$ for each $k \in {\mathbf{N}}$. If we let $\eta_n = 1$ for all $n \in \{1,\ldots,n_1-1\}$, and $\eta_n = 1/k$ for all $n \in \{n_k,\ldots,n_{k+1}-1\}$ for each $k \in {\mathbf{N}}$, then $\eta_n \to 0$ and $${\lim_{n\to\infty}\frac{1}{n} \sum_{i,j=1}^n \operatorname{\mathbf{E}}\bigl[\,|{w_{ij}^{(n)}}|^2{\mathrel{;}}|{w_{ij}^{(n)}}|>\eta_n\,\bigr] =0}.$$
Let $W_n' := \bigl({w_{ij}^{(n)}}\operatorname{\mathbf{1}}(|{w_{ij}^{(n)}}| \le \eta_n)\bigr)_{i,j=1}^n$. Since $$\frac{1}{n}\operatorname{\mathbf{E}}\bigl[\operatorname{tr}(W_n-W_n')^2\bigr]
= \frac{1}{n}\sum_{i,j=1}^n \operatorname{\mathbf{E}}\bigl[\,|{w_{ij}^{(n)}}|^2 {\mathrel{;}}|{w_{ij}^{(n)}}|>\eta_n\,\bigr]
\to 0 \qquad \text{as $n \to \infty$},$$ it is enough to show $\operatorname{\mathbf{E}}\mu_{W'_n} {\Rightarrow}{\mu_{\mathrm{sc}}}$ by Corollary \[cor:pertrub\_frob\_norm\_exp\] and Theorem \[thm:ch\_weak\_conv\].
Centralization
--------------
For each $n = 1,2,\ldots$ and $i,j = 1,\ldots,n$, let $$v_{i,j}^{(n)} := {w_{ij}^{(n)}}\operatorname{\mathbf{1}}(|{w_{ij}^{(n)}}| \le \eta_n)
- \operatorname{\mathbf{E}}\bigl[\,{w_{ij}^{(n)}}{\mathrel{;}}|{w_{ij}^{(n)}}| \le \eta_n\,\bigr],$$ and let $W_n' - \operatorname{\mathbf{E}}W_n' := (v_{ij}^{(n)})_{i,j=1}^n$. Since $$\begin{gathered}
\frac{1}{n}\sum_{i,j=1}^n \bigl|\operatorname{\mathbf{E}}\bigl[\,
{w_{ij}^{(n)}}{\mathrel{;}}|{w_{ij}^{(n)}}|\le\eta_n \,\bigr]\bigr|^2
= \frac{1}{n}\sum_{i,j=1}^n \bigl|\operatorname{\mathbf{E}}\bigl[\,
{w_{ij}^{(n)}}{\mathrel{;}}|{w_{ij}^{(n)}}|>\eta_n \,\bigr]\bigr|^2 \\
\le \frac{1}{n}\sum_{i,j=1}^n \operatorname{\mathbf{E}}\bigl[\,
|{w_{ij}^{(n)}}|^2 {\mathrel{;}}|{w_{ij}^{(n)}}|>\eta_n \,\bigr]
\to 0 \qquad \text{as $n \to \infty$,}\end{gathered}$$ it is enough to show $\operatorname{\mathbf{E}}\mu_{W'_n-\operatorname{\mathbf{E}}W'_n} {\Rightarrow}{\mu_{\mathrm{sc}}}$ by Corollary \[cor:pertrub\_frob\_norm\_exp\] and Theorem \[thm:ch\_weak\_conv\].
We claim that $W'_n-\operatorname{\mathbf{E}}W'_n$ satisfies all conditions $W_n$ is supposed to satisfy in Theorem \[thm:sclaw\]. The fact that and still hold even if we replace $W_n$ by $W'_n-\operatorname{\mathbf{E}}W'_n$ follows from the following: $$\begin{split}
\sum_{i,j=1}^n &\bigl|{\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}- \operatorname{\mathbf{Var}}\bigl[v_{ij}^{(n)}\bigr]\bigr| \\
&= \sum_{i,j=1}^n \Bigl( \operatorname{\mathbf{E}}\bigl[\, |{w_{ij}^{(n)}}|^2 {\mathrel{;}}|{w_{ij}^{(n)}}| > \eta_n \,\bigr]
+ \bigl|\operatorname{\mathbf{E}}\bigl[\, {w_{ij}^{(n)}}{\mathrel{;}}|{w_{ij}^{(n)}}| \le \eta_n \,\bigr]\bigr|^2 \Bigr) \\
&\le 2 \sum_{i,j=1}^n \operatorname{\mathbf{E}}\bigl[\, |{w_{ij}^{(n)}}|^2 {\mathrel{;}}|{w_{ij}^{(n)}}| > \eta_n \,\bigr].
\end{split}$$ The condition for $W'_n-\operatorname{\mathbf{E}}W'_n$ easily follows from the bound $|{w_{ij}^{(n)}}| \le \eta_n$. Since $|v_{i,j}^{(n)}| \le 2\eta_n$ for all $n = 1,2,\ldots$ and $i,j = 1,\ldots,n$, by doubling $\eta_1$, $\eta_2$, …we have $\eta_n \to 0$ and $|v_{i,j}^{(n)}| \le \eta_n$. Thus, from now on, we can assume that $|{w_{ij}^{(n)}}| \le \eta_n$ for some $\eta_1, \eta_2, \ldots > 0$ satisfying $\eta_n \to 0$.
Rescaling
---------
Fix $n \in {\mathbf{N}}$. We will choose a number $0 \le c_{ij}^{(n)} \le 1$ for each $i,j = 1,\ldots,n$ so that $c_{ij}^{(n)} = c_{ji}^{(n)}$ always hold. Start by letting $c_{ij}^{(n)} = 1$ for all $i,j = 1,\ldots,n$. We start with the first row and the first column. If $\sum_{j=1}^{n} \operatorname{\mathbf{Var}}\bigl[w_{1j}^{(n)}\bigr] \le C$, then do nothing. Otherwise, lower $c_{11}^{(n)},\ldots,c_{1n}^{(n)}$ (not below $0$) so that $$\sum_{j=1}^n \Bigl[\bigl(c_{1j}^{(n)}\bigr)^2
\operatorname{\mathbf{Var}}\bigl[w_{1j}^{(n)}\bigr]\Bigr] = C,$$ and let $c_{j1}^{(n)} := c_{1j}^{(n)}$ for all $j = 1,\ldots,n$. We note that at this point we have $$\sum_{i,j=1}^{n} \Bigl[\bigl(1 - \bigl(c_{ij}^{(n)}\bigr)^2\bigr)
\operatorname{\mathbf{Var}}\bigl[w_{ij}^{(n)}\bigr]\Bigr]
\le 2 \biggl(\sum_{j=1}^n \operatorname{\mathbf{Var}}\bigl[ w_{1j}^{(n)} \bigr] - C\biggr)_+.$$
Assume that $k \in \{2,\ldots,n\}$, and that we’ve examined up to $(k-1)$-th row. If $$\sum_{j=1}^{k-1} \Bigl[\bigl(c_{kj}\bigr)^2\operatorname{\mathbf{Var}}\bigl[w_{kj}^{(n)}\bigr]\Bigr]
+ \sum_{j=k}^n \operatorname{\mathbf{Var}}\bigl[w_{kj}^{(n)}\bigr] \le C,$$ then do nothing. Otherwise, lower $c_{k1}^{(n)},\ldots,c_{kn}^{(n)}$ (not below $0$) so that $$\sum_{j=1}^n \Bigl[\bigl(c_{kj}^{(n)}\bigr)^2
\operatorname{\mathbf{Var}}\bigl[w_{kj}^{(n)}\bigr]\Bigr] = C,$$ and let $c_{jk}^{(n)} = c_{kj}^{(n)}$ for all $j = 1,\ldots,n$. At this point we have $$\sum_{i,j=1}^{n} \Bigl[\bigl(1 - \bigl(c_{ij}^{(n)}\bigr)^2\bigr)
\operatorname{\mathbf{Var}}\bigl[w_{ij}^{(n)}\bigr]\Bigr]
\le 2 \sum_{i=1}^k \biggl(\sum_{j=1}^n
\operatorname{\mathbf{Var}}\bigl[ w_{ij}^{(n)} \bigr] - C\biggr)_+.$$ This can be shown by an induction on $k$. After completing the whole process, we are left with numbers $0 \le c_{ij}^{(n)} \le 1$ such that $$\label{eq:rescale_rowbdd}
\sum_{j=1}^n \Bigl[\bigl(c_{ij}^{(n)}\bigr)^2
\operatorname{\mathbf{Var}}\bigl[w_{ij}^{(n)}\bigr]\Bigr] \le C$$ for all $i = 1,\ldots,n$, and $$\label{eq:rescale_change}
\sum_{i,j=1}^{n} \Bigl[\bigl(1 - \bigl(c_{ij}^{(n)}\bigr)^2\bigr)
\operatorname{\mathbf{Var}}\bigl[w_{ij}^{(n)}\bigr]\Bigr]
\le 2 \sum_{i=1}^n \biggl(\sum_{j=1}^n
\operatorname{\mathbf{Var}}\bigl[ w_{ij}^{(n)} \bigr] - C\biggr)_+.$$
Let $\widehat{W}_n = \bigl( c_{ij}^{(n)} w_{ij}^{(n)} \bigr)_{i,j=1}^n$. Since $(1-c)^2 \le 1-c^2$ holds for any $0 \le c \le 1$, we have $$\begin{gathered}
\frac{1}{n} \operatorname{\mathbf{E}}\bigl[ \operatorname{tr}(W_n - \widetilde{W}_n)^2 \bigr]
= \frac{1}{n}\sum_{i,j=1}^n \Bigl[
\bigl(1 - c_{ij}^{(n)}\bigr)^2 {\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}\Bigr] \\
\le \frac{2}{n} \sum_{i=1}^n \biggl(\sum_{j=1}^n
\operatorname{\mathbf{Var}}\bigl[ w_{ij}^{(n)} \bigr] - C\biggr)_+
\to 0 \qquad \text{as $n \to \infty$}\end{gathered}$$ by . Thus, by Corollary \[cor:pertrub\_frob\_norm\_exp\] and Theorem \[thm:ch\_weak\_conv\], it is enough to show $\operatorname{\mathbf{E}}\mu_{\widehat{W}_n} {\Rightarrow}{\mu_{\mathrm{sc}}}$.
The altered matrix $\widehat{W}_n$ has an advantage over $W_n$ that holds. Also, the modulus of each entry of $\widehat{W}_n$ is still bounded by $\eta_n$. We claim that $\widehat{W}_n$ also satisfies all conditions $W_n$ is assumed to satisfy in Theorem \[thm:sclaw\]. First, each entry of $\widehat{W}_n$ obviously has mean zero. Also, since each entry of $\widehat{W}_n$ has modulus less than or equal to the corresponding entry of $W_n$, the condition is satisfied by $\widehat{W}_n$. The condition for $\widehat{W}_n$ obviously holds as we have an even stronger property . Finally, for $\widehat{W}_n$ follows from and the fact that is satisfied by $W_n$. This proves our claim, and so from now on, we can also assume that is true.
Reduction to moment convergence
-------------------------------
On top of the assumptions of Theorem \[thm:sclaw\], we now also have the following.
1. There are $\eta_1,\eta_2,\ldots > 0$ with $\lim_{n\to\infty} \eta_n = 0$ such that $|{w_{ij}^{(n)}}| \le \eta_n$ for all $n = 1,2,\ldots$ and $i,j = 1,\ldots,n$.
2. There is some finite $C \ge 0$ such that holds.
Since $|{w_{ij}^{(n)}}| \le \eta_n$, every eigenvalue of $W_n$ has absolute value at most $n\eta_n$. So, $\operatorname{\mathbf{E}}{\mu_{W_n}}$ is supported on $[-n\eta_n,n\eta_n]$, and in particular $\operatorname{\mathbf{E}}{\mu_{W_n}}$ has moments of all orders. As $$\biggl|\sum_{k=1}^\infty \frac{1}{k!} \int_{\mathbf{R}}x^k \,{\mu_{\mathrm{sc}}}(dx) r^k \biggr|
\le \sum_{k=1}^\infty \frac{|2r|^k}{k!} < \infty$$ for any $r \in {\mathbf{R}}$ by the ratio test, ${\mu_{\mathrm{sc}}}$ is determined by its moments by [@Bil12 Theorem 30.1]. Thus, by the moment convergence theorem [@Bil12 Theorem 30.2], it is enough to show $$\lim_{n \to \infty} \int_{\mathbf{R}}x^k \,\operatorname{\mathbf{E}}{\mu_{W_n}}(dx) = \int_{\mathbf{R}}x^k \,{\mu_{\mathrm{sc}}}(dx)
\qquad \text{for all $k = 1,2,\ldots$.}$$
For each $k = 1,2,\ldots$, since there are continuous bounded $g_{k,n}: {\mathbf{R}}\to {\mathbf{R}}$ with $g_{k,n}(x) = x^k$ for all $x \in [-n\eta_n,n\eta_n]$, we have $$\int_{\mathbf{R}}x^k \,\operatorname{\mathbf{E}}{\mu_{W_n}}(dx) = \int_{\mathbf{R}}g_{k,n} \,d\operatorname{\mathbf{E}}{\mu_{W_n}}= \operatorname{\mathbf{E}}\int_{\mathbf{R}}g_{k,n} \,d{\mu_{W_n}}= \frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}W_n^k.$$ On the other hand, we can directly compute the moments of ${\mu_{\mathrm{sc}}}$ as follows.
For any $m=1,2,\ldots$, we have $$\int_{\mathbf{R}}x^{2m} \,{\mu_{\mathrm{sc}}}(dx) = \frac{1}{m+1} \binom{2m}{m}.$$
A trigonometric substitution $x = 2\cos\theta$ gives $$\begin{split}
\int_{\mathbf{R}}x^{2m} \,{\mu_{\mathrm{sc}}}(dx)
&= \frac{1}{2\pi} \int_{-2}^2 x^{2m} \sqrt{4-x^2} \,dx
= \frac{2}{\pi} \int_{-\pi}^0 2^{2m}
\cos^{2m}\theta \sin^2\theta \,d\theta\\
&= \frac{2^{2m+1}}{\pi}
\left[ \int_{-\pi}^0 \cos^{2m}\theta \,d\theta
- \int_{-\pi}^0 \cos^{2m+2}\theta \,d\theta \right].
\end{split}$$ As $$\int_{-\pi}^0 \cos^{2l} \theta \,d\theta
= \frac{1}{2^{2l+1}} \int_{-\pi}^\pi
(e^{i\theta}+e^{-i\theta})^{2l} \,d\theta
= \frac{\pi}{2^{2l}} \binom{2l}{l}$$ for any $l=1,2,\ldots$, we have $$\int_{\mathbf{R}}x^{2m} \,{\mu_{\mathrm{sc}}}(dx)
= 2\binom{2m}{m} - \frac{1}{2} \binom{2m+2}{m+1}
= \frac{1}{m+1} \binom{2m}{m}.$$
Note that $\int_{\mathbf{R}}x^k \,{\mu_{\mathrm{sc}}}(dx) = 0$ whenever $k \in {\mathbf{N}}$ is odd. Thus, it is enough to show that $$\label{eq:odd_moment_conv}
\lim_{n\to\infty} \frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}W_n^k = 0
\qquad \text{for all odd $k \in {\mathbf{N}}$,}$$ and that $$\label{eq:even_moment_conv}
\lim_{n\to\infty} \frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}W_n^k = \frac{1}{k/2+1} \binom{k}{k/2}
\qquad \text{for all even $k \in {\mathbf{N}}$.}$$ These will be proved in Section \[sec:comp\_moments\] by using the content of Section \[sec:trees\].
Trees and products of variances {#sec:trees}
===============================
Our graphs will be undirected. We allow graphs to have loops, but don’t allow them to have multiple edges. Let $G$ be a finite graph. For any $n \in {\mathbf{N}}$, denote by $I(G,n)$ the collection of all injections from $V(G)$ into $\{1,\ldots,n\}$. Given any $F \in I(G,n)$ and $e \in E(G)$ with ends $u,v$, we let $$\rho_{e,F}^{(n)} := \operatorname{\mathbf{Var}}\bigl[ w_{F(u)F(v)}^{(n)} \bigr].$$ It is well-defined since each $W_n$ is Hermitian. Then we let $$P(G,F) := \prod_{e \in E(G)} \rho_{e,F}^{(n)}.$$ Here $P$ stands for “product.” Also, the notation $\rho_{e,F}^{(n)}$ will no longer appear.
\[lem:tree\_bdd\] If $T$ is a finite tree with $m$ edges, $u \in V(T)$, $n \in {\mathbf{N}}$, and $i \in \{1,\ldots,n\}$, then $$\label{eq:tree_bdd}
\sum_{\substack{F \in I(T,n)\\F(u)=i}} P(T,F) \le C^m.$$
If $m = 0$, then obviously holds. (We define the product of zero terms as $1$.) To proceed by induction, assume that holds for $m$, and let $T$ be a tree with $m+1$ edges. Choose any leaf $w$ of $T$ different from $u$, and let $x$ be the only vertex of $T$ adjacent to $w$. Since $$\begin{split}
\sum_{\substack{F \in I(T,n)\\F(u)=i, F(x)=j}} P(T,F)
&\le \sum_{\substack{H \in I(T\setminus w,n)\\H(u)=i, H(x)=j}}
\Bigl(P(T\setminus w, H) \sum_{l=1}^n
\operatorname{\mathbf{Var}}\bigl[w_{jl}^{(n)}\bigr]\Bigr)\\
&\le C\sum_{\substack{H \in I(T\setminus w,n)\\H(u)=i, H(x)=j}}
P(T\setminus w, H)
\end{split}$$ for all $j \in \{1,\ldots,n\}$, we have $$\begin{split}
\sum_{\substack{F \in I(T,n)\\F(u)=i}} P(T,F)
&= \sum_{j=1}^n
\sum_{\substack{F \in I(T,n)\\F(u)=i, F(x)=j}} P(T,F) \\
&\le C \sum_{j=1}^n
\sum_{\substack{H \in I(T\setminus w,n)\\H(u)=i, H(x)=j}}
P(T\setminus w, H) \\
&= C \sum_{\substack{H \in I(T\setminus w,n)\\H(u)=i}}
P(T\setminus w, H) \le C^{m+1}
\end{split}$$ by the induction hypothesis.
\[lem:tree\_cont\] For any finite tree $T$, $$\label{eq:tree_cont}
\lim_{n\to\infty} \frac{1}{n}
\sum_{F \in I(T,n)} P(T,F) = 1.$$
Let $m := E(T)$. If $m = 0$, then obviously holds. To proceed by induction, assume that the result holds for trees with $m$ edges, and let $T$ be a tree with $m+1$ edges. Let $u \in V(T)$ be a leaf of $T$, and $w$ be the only vertex of $T$ adjacent to $u$ in $T$. Note that $$\begin{gathered}
\label{eq:tree_cont_free}
\Biggl| \frac{1}{n} \sum_{F \in I(T,n)} P(T,F)
- \frac{1}{n} \sum_{H \in I(T\setminus u, n)} \biggl( P(T\setminus u, H)
\sum_{i=1}^n \operatorname{\mathbf{Var}}\bigl[w_{H(w)i}^{(n)}\bigr] \biggr) \Biggr| \\
\le \frac{1}{n} \sum_{H \in I(T\setminus u, n)}
\biggl( P(T\setminus u, H) \sum_{v \in V(T\setminus u)}
\operatorname{\mathbf{Var}}\bigl[w_{H(w)H(v)}^{(n)}\bigr] \biggr) \\
\le \frac{(m+1)\eta_n^2}{n} \sum_{H \in I(T\setminus u,n)}
P(T\setminus u, H) \to 0 \qquad \text{as $n \to \infty$,}
\end{gathered}$$ by the induction hypothesis. By Lemma \[lem:tree\_bdd\], we have $$\begin{gathered}
\label{eq:tree_cont_bdd}
\biggl| \frac{1}{n} \sum_{H \in I(T\setminus u, n)}
\biggl( P(T\setminus u, H)
\sum_{i=1}^n \biggl( \operatorname{\mathbf{Var}}\bigl[w_{H(w)i}^{(n)}\bigr] - \frac{1}{n}
\biggr) \biggr) \biggr|\\
= \frac{1}{n} \biggl| \sum_{j=1}^n \biggl[ \sum_{i=1}^n
\biggl(\operatorname{\mathbf{Var}}\bigl[w_{ji}^{(n)} \bigr]
- \frac{1}{n} \biggr) \cdot
\sum_{\substack{H \in I(T\setminus u,n)\\H(w)=j}}
P(T\setminus u, H) \biggr] \biggr|\\
\le \frac{C^m}{n} \sum_{j=1}^n \biggl| \sum_{i=1}^n \biggl(
\operatorname{\mathbf{Var}}\bigl[w_{ij}^{(n)}\bigr] - \frac{1}{n} \biggr) \biggr|
\to 0 \qquad \text{as $n \to \infty$.}
\end{gathered}$$ Combining , , and the fact that $$\lim_{n\to\infty} \frac{1}{n} \sum_{H \in I(T\setminus u,n)}
P(T\setminus u, H) = 1,$$ we can conclude that holds.
Computation of moments {#sec:comp_moments}
======================
Fix a $k \in {\mathbf{N}}$. Let us call any $n$-tuple $(i_0,\ldots,i_k)$ with $i_0 = i_k$ a *closed walk of length $k$*. If ${\mathbf{i}}= (i_0,\ldots,i_k)$ is a closed walk, we let $G({\mathbf{i}})$ be the graph (possibly having loops but having no multiple edges) with the vertex set $V({\mathbf{i}}) := \{i_0,\ldots,i_k\}$ and the edge set $$E({\mathbf{i}}) := \bigl\{\, \{i_{t-1},i_t\} \bigm| t = 1,\ldots,k \,\bigr\}.$$ Two closed walks ${\mathbf{i}}= (i_0,\ldots,i_k)$ and ${\mathbf{j}}= (j_0,\ldots,j_k)$ are said to be *isomorphic* if for any $s,t = 0,\ldots,k$ we have $i_s = i_t$ if and only if $j_s = j_t$. If $t \in {\mathbf{N}}$, then a *canonical closed walk of length $k$ on $t$ vertices* is a closed walk ${\mathbf{c}}= (c_0,\ldots,c_k)$ with $V({\mathbf{c}})=\{1,\ldots,t\}$ such that
1. $c_0 = c_k = 1$ and
2. $c_t \le \max\{c_0,\ldots,c_{t-1}\}+1$ for each $t = 1,\ldots,k$.
Let $\Gamma(k,t)$ denote the set of such walks. It is straightforward to show that any closed walk is isomorphic to exactly one canonical closed walk. For any ${\mathbf{c}}\in \Gamma(k,t)$, let $L(n,{\mathbf{c}})$ denote the set of all closed walks $(i_0,\ldots,i_k)$ with $i_0,\ldots,i_k \in \{1,\ldots,n\}$ which are isomorphic to ${\mathbf{c}}$.
Note that $$\label{eq:trace_sum}
\begin{split}
\frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}W_n^k &= \frac{1}{n} \sum_{i_0,\ldots,i_k = 1}^n
\operatorname{\mathbf{E}}\biggl[ \prod_{s=1}^k w_{i_{s-1}i_s}^{(n)} \biggr] \\
&= \sum_{t=1}^{k+1} \sum_{{\mathbf{c}}\in \Gamma(k,t)}
\sum_{(i_0,\ldots,i_k) \in L(n,{\mathbf{c}})}
\operatorname{\mathbf{E}}\biggl[ \prod_{s=1}^k w_{i_{s-1}i_s}^{(n)} \biggr].
\end{split}$$ Here the upper bound of $t$ is (rather arbitrarily) set to $k+1$ since $\Gamma(k,t)$ is empty for any $t > k+1$. We will compute $$\sum_{(i_0,\ldots,i_k) \in L(n,{\mathbf{c}})}
\operatorname{\mathbf{E}}\biggl[ \prod_{s=1}^k w_{i_{s-1}i_s}^{(n)} \biggr]$$ for each $t \in {\mathbf{N}}$ and ${\mathbf{c}}\in \Gamma(k,t)$.
\[lem:no\_walk\_once\] Let $t \in {\mathbf{N}}$ and ${\mathbf{c}}= (c_0,\ldots,c_k) \in \Gamma(k,t)$. If ${\mathbf{c}}$ walks on some edge $\{i,j\}$ exactly once, i.e. $\{c_{s-1},c_s\} = \{i,j\}$ for exactly one $s \in \{1,\ldots,k\}$, then $$\operatorname{\mathbf{E}}\biggl[ \prod_{s=1}^k w_{i_{s-1}i_s}^{(n)} \biggr] = 0$$ for any $n \in {\mathbf{N}}$ and $(i_0,\ldots,i_k) \in L(n,{\mathbf{c}})$.
Since the upper triangular entries of $W_n$ are jointly independent, $\prod_{s=1}^k w_{i_{s-1}i_s}^{(n)}$ can be broken into ${w_{ij}^{(n)}}$ or $w_{ji}^{(n)}$, and a random variable independent from $w_{ij}$. Since $\operatorname{\mathbf{E}}{w_{ij}^{(n)}}= 0$, the desired conclusion follows.
\[lem:no\_mult\_cycle\] Let $t \in {\mathbf{N}}$ and ${\mathbf{c}}= (c_0,\ldots,c_k) \in \Gamma(k,t)$. Assume that ${\mathbf{c}}$ doesn’t walk on any edge exactly once, i.e. for each $s = 1,\ldots,k$ there is a $r \in \{1,\ldots,k\}$ with $r \ne s$ such that $\{c_{s-1},c_s\} = \{c_{r-1},c_r\}$. Then we have $t \le k/2+1$, and the following hold.
1. If $t < k/2 + 1$, then $$\label{eq:iso_sum_zero}
{\lim_{n\to\infty}\frac{1}{n} \sum_{(i_0,\ldots,i_k) \in L(n,{\mathbf{c}})} \operatorname{\mathbf{E}}\biggl[
\prod_{s=1}^k w_{i_{s-1}i_s}^{(n)} \biggr] =0}.$$
2. If $t = k/2 + 1$, then $$\label{eq:iso_sum_one}
\lim_{n\to\infty} \frac{1}{n}
\sum_{(i_0,\ldots,i_k) \in L(n,{\mathbf{c}})} \operatorname{\mathbf{E}}\biggl[
\prod_{s=1}^k w_{i_{s-1}i_s}^{(n)} \biggr] = 1.$$
As each edge of $G({\mathbf{c}})$ is walked on at least twice by ${\mathbf{c}}$, the graph $G({\mathbf{c}})$ has at most $k/2$ edges. Since $G({\mathbf{c}})$ is a connected graph with $t$ vertices, we have $t \le k/2+1$, and $G({\mathbf{c}})$ has a spanning tree $S$ with $t-1$ edges. If ${\mathbf{i}}= (i_0,\ldots,i_k) \in L(n,{\mathbf{c}})$, then there is an injection $F_{\mathbf{i}}\colon\{1,\ldots,t\} \to \{1,\ldots,n\}$ with $i_s = F_{\mathbf{i}}(c_s)$ for all $s = 0,\ldots,k$.
\(i) Assume $t < k/2+1$. Using the bound $|{w_{ij}^{(n)}}| \le \eta_n$, and the fact that ${\mathbf{c}}$ walks on any edge of $G({\mathbf{c}})$ at least twice, we can derive $$\operatorname{\mathbf{E}}\biggl[\prod_{s=1}^k \bigl|w_{i_{s-1}i_s}^{(n)}\bigr|\biggr]
\le \eta_n^{k-2(t-1)} P(S, F_{\mathbf{i}}).$$ Note that $\lim_{n\to\infty} \eta_n^{k-2(t-1)} = 0$ since $t < k/2+1$. Since the map $L(n,{\mathbf{c}}) \to I(S,n)$ given by ${\mathbf{i}}\mapsto F_{\mathbf{i}}$ is a bijection, we have $$\begin{gathered}
\frac{1}{n} \sum_{(i_0,\ldots,i_k) \in L(n,{\mathbf{c}})} \operatorname{\mathbf{E}}\biggl|
\prod_{s=1}^k w_{i_{s-1}i_s}^{(n)} \biggr|
\le \frac{\eta_n^{k-2(t-1)}}{n} \sum_{{\mathbf{i}}\in L(n,{\mathbf{c}})} P(S, F_{\mathbf{i}}) \\
= \frac{\eta_n^{k-2(t-1)}}{n} \sum_{F \in I(S,n)} P(S,F)
\to 0 \qquad \text{as $n \to \infty$}
\end{gathered}$$ by Lemma \[lem:tree\_cont\].
\(ii) Assume $t = k/2+1$. Since $S$ has $k/2$ edges and each edge of $S$ is walked on twice by ${\mathbf{c}}$, we see that $S = G({\mathbf{c}})$. As each edge of $G({\mathbf{c}})$ is traversed once in each direction, i.e. for each $s = 1,\ldots,k$ there is an $r \in \{1,\ldots,k\}$ with $r \ne s$ such that $c_{s-1} = c_r$ and $c_s = c_{r-1}$, we have $$\begin{gathered}
\frac{1}{n} \sum_{(i_0,\ldots,i_k) \in L(n,{\mathbf{c}})} \operatorname{\mathbf{E}}\biggl[
\prod_{s=1}^k w_{i_{s-1}i_s}^{(n)} \biggr]
= \frac{1}{n} \sum_{{\mathbf{i}}\in L(n,{\mathbf{c}})} P(S,F_{\mathbf{i}}) \\
= \frac{1}{n} \sum_{F \in I(S,n)} P(S,F) \to 1
\qquad \text{as $n \to \infty$}
\end{gathered}$$ by Lemma \[lem:tree\_cont\].
Lemma \[lem:no\_walk\_once\] and \[lem:no\_mult\_cycle\] tell us that we have if and only if ${\mathbf{c}}$ doesn’t walk on any edge exactly once and $t = k/2+1$. Otherwise, we have . If $k$ is odd, then $k/2+1$ is not an integer, and so we cannot have $t = k/2+1$. So, for any odd $k \in {\mathbf{N}}$, we have $${\lim_{n\to\infty}\frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}W_n^k =0}$$ by .
Assume that $k$ is even. Let $U$ be the set of all ${\mathbf{c}}\in \Gamma(k,k/2+1)$ which traverses each edge of $G({\mathbf{c}})$ twice. Then by Lemma \[lem:no\_mult\_cycle\] (ii) and , we have $$\lim_{n\to\infty} \frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}W_n^k = |U|.$$
A *Dyck path* of length $k$ is a finite sequence $(x_0,\ldots,x_k)$ satisfying the following:
1. $x_0 = x_k = 0$;
2. $x_s \ge 0$ for all $s=0,\ldots,k$;
3. $|x_s - x_{s-1}| = 1$ for all $s = 1,\ldots,k$.
Given an ${\mathbf{c}}= (c_0,\ldots,c_k) \in U$, let $D({\mathbf{c}}) := (x_0,\ldots,x_k)$ where $x_s$ is the distance between $1$ ($=c_0$) and $c_s$ in $G({\mathbf{c}})$. Then it is clear that $D({\mathbf{c}})$ is indeed a Dyck path, and it is not difficult to see that $D$ is a bijection from $U$ to the set of all Dyck paths of length $k$. It is well-known that there are exactly $\frac{1}{k/2+1}\binom{k}{k/2}$ Dyck paths of length $k$; see [@vLW01 Example 14.8]. Thus, we indeed have $$\lim_{n\to\infty} \frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}W_n^k
= \frac{1}{k/2+1} \binom{k}{k/2}.$$ This finishes the proof of the semicircular law Theorem \[thm:sclaw\].
Gaussian convergence {#sec:gaussian}
====================
The paper [@Jun18] considers real symmetric random matrices $W_1, W_2, \ldots$ with size $1 \times 1$, $2 \times 2$, …whose upper triangular entries are i.i.d. In that paper, it is shown that if the sum of a row of $W_n$ converges in distribution to the standard normal distribution $N(0,1)$ as $n \to \infty$, then ${\mu_{W_n}}{\Rightarrow}{\mu_{\mathrm{sc}}}$ as $n \to \infty$ a.s. We prove this fact generalized to random matrices with non-i.i.d. entries in this section. By doing so, we will demonstrate how one can apply Theorem \[thm:sclaw\] to obtain a semicircular law for random matrices whose entries might have infinite variances.
The type of convergence described in the following fact will appear many times in this section.
\[prop:unif\_array\] Let $S$ be a topological space, $m_1,m_2,\ldots \in {\mathbf{N}}$, and $(s_{ni})_{i=1}^{m_n}$ be a finite sequence in $S$ for each $n \in {\mathbf{N}}$. For any $s \in S$, the following two conditions are equivalent:
1. $s_{ni_n} \to s$ as $n \to \infty$ for any choice of $i_n \in \{1,\ldots,m_n\}$ for each $n \in {\mathbf{N}}$;
2. for any neighborhood $N$ of $s$, there exists some $n_0 \in {\mathbf{N}}$ such that $s_{ni} \in N$ for all $n \ge n_0$ and $i = 1,\ldots,m_n$.
We omit the straightforward proof.
The following is the main result of this section.
\[thm:gauss\_sclaw\] For each $n \in {\mathbf{N}}$, let $W_n = ({w_{ij}^{(n)}})_{i,j=1}^n$ be a random $n \times n$ real symmetric matrix whose upper triangular entries are jointly independent and have symmetric distributions. We assume that $W_1, W_2, \ldots$ are defined on the same probability space. Assume that $(W_n)_{n \in {\mathbf{N}}}$ is a null array in the sense that $w_{i_nj_n}^{(n)} {\Rightarrow}0$ as $n \to \infty$ for any choice of $i_n,j_n \in \{1,\ldots,n\}$ for each $n \in {\mathbf{N}}$. If also $\sum_{j=1}^n w_{i_nj}^{(n)} {\Rightarrow}N(0,1)$ for any choice of $i_n \in \{1,\ldots,n\}$ for each $n \in {\mathbf{N}}$, then ${\mu_{W_n}}{\Rightarrow}{\mu_{\mathrm{sc}}}$ as $n \to \infty$ a.s.
The following two facts will be used in the proof of Theorem \[thm:gauss\_sclaw\].
\[thm:gauss\_conv\] For each $n \in {\mathbf{N}}$, let $X_{n1},\ldots,X_{nn}$ be jointly independent real-valued random variables. Assume that $X_{ni_n} {\Rightarrow}0$ as $n \to \infty$ regardless of how we choose $i_n \in \{1,\ldots,n\}$ for each $n \in {\mathbf{N}}$. Then $\sum_{i=1}^n X_{ni} {\Rightarrow}N(0,1)$ as $n \to \infty$ if and only if the following conditions hold:
1. $\lim_{n\to\infty} \sum_{i=1}^n \operatorname{\mathbf{P}}(|X_{ni}|>{\epsilon}) = 0$ for all ${\epsilon}>0$;
2. $\lim_{n\to\infty} \sum_{i=1}^n \operatorname{\mathbf{E}}[\,X_{ni}{\mathrel{;}}|X_{ni}|\le1\,]
= 0$;
3. $\lim_{n\to\infty} \sum_{i=1}^n \operatorname{\mathbf{Var}}[X_{ni}\operatorname{\mathbf{1}}(|X_{ni}|\le1)] = 1$.
See [@Kal02 Theorem 5.15].
\[thm:bern\_ineq\] Suppose that $X_1,\ldots,X_n$ are independent real-valued random variables, each with mean $0$, and each bounded by $1$. If $S = X_1 + \cdots + X_n$, then $$\operatorname{\mathbf{P}}(S \ge x)
\le \exp \Bigl[ -\frac{x^2}{2(\operatorname{\mathbf{E}}[S^2] + x)} \Bigr]
\qquad \text{for any $x > 0$}.$$
The proof of [@Bil99 M20] with a slight change works.
By Theorem \[thm:gauss\_conv\], we have $$\lim_{n\to\infty} \sum_{j=1}^n \operatorname{\mathbf{P}}(|w_{i_nj}^{(n)}| > {\epsilon}) = 0$$ for any choice of $i_n \in \{1,\ldots,n\}$ for each $n \in {\mathbf{N}}$, for any ${\epsilon}> 0$. Then Proposition \[prop:unif\_array\] implies $$\label{eq:gauss_sclaw_prob_on}
{\lim_{n\to\infty}\frac{1}{n} \sum_{i,j=1}^n \operatorname{\mathbf{P}}(|{w_{ij}^{(n)}}|>{\epsilon}) =0} \qquad \text{for all ${\epsilon}>0$.}$$
Let $v_{ij}^{(n)} := {w_{ij}^{(n)}}\operatorname{\mathbf{1}}(|{w_{ij}^{(n)}}|\le1)$ and $W'_n = (v_{ij}^{(n)})_{i,j=1}^n$. Since $$\label{eq:gauss_sclaw_rank}
\operatorname{rank}(W_n-W'_n) \le \sum_{i,j=1}^n \operatorname{\mathbf{1}}(|{w_{ij}^{(n)}}|>1)
\le 2\sum_{1\le i\le j\le n} \operatorname{\mathbf{1}}(|{w_{ij}^{(n)}}|>1),$$ by bounding $\sum_{1\le i\le j\le n} \operatorname{\mathbf{1}}(|{w_{ij}^{(n)}}|>1)$ from above we would be able to apply Theorem \[thm:rank\_ineq\]. For any given ${\epsilon}> 0$, we have some $n_0 \in {\mathbf{N}}$ such that $$\sum_{1\le i\le j\le n} \operatorname{\mathbf{P}}(|{w_{ij}^{(n)}}|>1) \le {\epsilon}n/2
\qquad \text{for all $n \ge n_0$}$$ by . Since $\operatorname{\mathbf{1}}(|{w_{ij}^{(n)}}|>1)$, $1\le i\le j\le n$, are jointly independent, Bernstein’s inequality (Theorem \[thm:bern\_ineq\]) implies $$\begin{split}
\operatorname{\mathbf{P}}\biggl( \sum_{1\le i\le j\le n} &\operatorname{\mathbf{1}}(|{w_{ij}^{(n)}}|>1) \ge {\epsilon}n \biggr) \\
&\le \operatorname{\mathbf{P}}\biggl( \sum_{1\le i\le j\le n}
\bigl(\operatorname{\mathbf{1}}(|{w_{ij}^{(n)}}|>1)-\operatorname{\mathbf{P}}(|{w_{ij}^{(n)}}|>1)\bigr) \ge {\epsilon}n/2 \biggr) \\
&\le \exp\biggl( -\frac{{\epsilon}^2n^2/4}
{\sum_{1\le i\le j\le n}\operatorname{\mathbf{P}}(|{w_{ij}^{(n)}}|>1)+{\epsilon}n/2} \biggr) \\
&\le \exp\biggl( -\frac{{\epsilon}^2n^2/4}{{\epsilon}n}\biggr)
= \exp(-{\epsilon}n/4).
\end{split}$$ As $\sum_{n=1}^\infty \exp(-{\epsilon}n/4) < \infty$ for each ${\epsilon}> 0$, we have $${\lim_{n\to\infty}\frac{1}{n} \sum_{1\le i\le j\le n} \operatorname{\mathbf{1}}(|{w_{ij}^{(n)}}|>1) =0} \qquad \text{a.s.,}$$ and therefore $${\lim_{n\to\infty}\frac{1}{n} \operatorname{rank}(W_n-W'_n) =0} \qquad \text{a.s.}$$ by . By Theorem \[thm:rank\_ineq\], it now suffices to show $\mu_{W'_n} {\Rightarrow}{\mu_{\mathrm{sc}}}$ as $n\to\infty$ a.s.
We claim that $W'_n$ satisfies all conditions of Theorem \[thm:sclaw\]. Since each ${w_{ij}^{(n)}}$ is symmetric, each entry of $W'_n$ has mean zero. By Theorem \[thm:gauss\_conv\], we have $$\lim_{n\to\infty} \sum_{j=1}^n
\operatorname{\mathbf{Var}}\bigl[v_{i_nj}^{(n)}\bigr] = 1$$ for any choice of $i_n \in \{1,\ldots,n\}$ for each $n \in {\mathbf{N}}$. So, by using Proposition \[prop:unif\_array\], we can see that the conditions \[eq:sclaw\_var\] and \[eq:sclaw\_rowbdd\] with ${w_{ij}^{(n)}}$ replaced by $v_{ij}^{(n)}$ hold. Finally, the condition with ${w_{ij}^{(n)}}$ replaced by $v_{ij}^{(n)}$ follows from $$\sum_{i,j=1}^n \operatorname{\mathbf{E}}\bigl[\,|v_{ij}^{(n)}|^2{\mathrel{;}}|v_{ij}^{(n)}|>{\epsilon}\,\bigr] \le \sum_{i,j=1}^n
\operatorname{\mathbf{P}}\bigl(|v_{ij}^{(n)}|>{\epsilon}\bigr)$$ and .
Probability measures on ${\mathbf{R}}$ {#sec:prob_meas_r}
======================================
Weak convergence
----------------
Let $\Pr({\mathbf{R}})$ denote the set of all Borel probability measures on ${\mathbf{R}}$. We equip $\Pr({\mathbf{R}})$ with the smallest topology that makes $\mu \mapsto \int_{\mathbf{R}}f\,d\mu$ continuous for all continuous bounded $f:{\mathbf{R}}\to{\mathbf{R}}$. Then we equip $\Pr({\mathbf{R}})$ with the Borel $\sigma$-algebra.
Note that if $\mu,\mu_1,\mu_2,\ldots \in \Pr({\mathbf{R}})$, then we have $\mu_n {\Rightarrow}\mu$ if and only if $\mu_n \to \mu$ under the topology of $\Pr({\mathbf{R}})$.
If $F$ and $G$ are distribution functions, then the Lévy distance between $F$ and $G$ is defined by $$L(F,G) := \inf\{\, {\epsilon}>0 \mid
F(x-{\epsilon})-{\epsilon}\le G(x) \le F(x+{\epsilon})+{\epsilon}\text{ for all $x \in {\mathbf{R}}$}\,\}.$$
It is not difficult to show that $L$ is indeed a metric on $\Pr({\mathbf{R}})$. For any given $\mu \in \Pr({\mathbf{R}})$, let $F_\mu$ denote the distribution function of $\mu$.
\[thm:ch\_weak\_conv\] If $\mu$, $\mu_1$, $\mu_2$, $\ldots \in \Pr({\mathbf{R}})$, then the following are equivalent:
1. $\mu_n {\Rightarrow}\mu$;
2. $\int_{\mathbf{R}}f_{p,q}\,d\mu_n \to \int_{\mathbf{R}}f_{p,q}\,d\mu$ for all $p,q \in {\mathbf{Q}}$ with $p < q$, where $f_{p,q}\colon{\mathbf{R}}\to{\mathbf{R}}$ is the function which has value $1$ on $(-\infty,p\,]$, has value $0$ on $[\,q,\infty)$, and is linear on $[\,p,q\,]$;
3. $L(F_{\mu_n},F_\mu) \to 0$.
\(i) implies (ii): Directly follows from the definition of convergence in distribution.
\(ii) implies (i): Assume that $\int_{\mathbf{R}}f_{p,q}\,d\mu_n \to \int_{\mathbf{R}}f_{p,q}\,d\mu$ for all $p,q \in {\mathbf{Q}}$ with $p < q$, and let $F,F_1,F_2,\ldots$ be the distribution functions of $\mu,\mu_1,\mu_2,\ldots$. Let $x$ be any continuity point of $\mu$, and let ${\epsilon}> 0$ be given. Since $F$ is right continuous, we have $F(x+\delta) \le F(x) + {\epsilon}$ for some $\delta > 0$. If we choose any $p,q \in {\mathbf{Q}}$ with $x < p < q < x+\delta$, then $$\limsup_{n \to \infty} F_n(x) \le \lim_{n \to \infty}
\int_{\mathbf{R}}f_{p,q} \,d\mu_n = \int_{\mathbf{R}}f_{p,q} \,d\mu \le F(x+\delta)
\le F(x) + {\epsilon}.$$ As $F$ is also left continuous at $x$, a similar reasoning yields $F(x)-{\epsilon}\le \liminf_{n \to \infty} F_n(x)$. Since ${\epsilon}> 0$ is arbitrary, we have $F_n(x) \to F(x)$.
\(i) implies (iii): Let ${\epsilon}> 0$ be given. Choose continuity points $x_0,\ldots,x_k \in {\mathbf{R}}$ of $F_\mu$ such that $x_0 < \cdots < x_k$, $F_\mu(x_0) \le {\epsilon}$, $F_\mu(x_k) \ge 1-{\epsilon}$, and $$\max\{|x_1-x_0|,\ldots,|x_k-x_{k-1}|\} \le {\epsilon}.$$ Let $N \in {\mathbf{N}}$ be such that $n \ge N$ implies $$\max\{|F_{\mu_n}(x_0) - F_\mu(x_0)|,\ldots,
|F_{\mu_n}(x_k)-F_\mu(x_k)|\} \le {\epsilon}.$$ Let $x \in {\mathbf{R}}$ be arbitrarily given. If $x > x_k$, then $$F_{\mu}(x+{\epsilon}) + {\epsilon}\ge 1 \ge F_{\mu_n}(x)$$ for any $n \in {\mathbf{N}}$. If $x \in (x_{i-1},x_i]$ where $i \in \{1,\ldots,n\}$, then $$F_\mu(x+{\epsilon}) + {\epsilon}\ge F_\mu(x_i) + {\epsilon}\ge F_{\mu_n}(x_i)
\ge F_{\mu_n}(x)$$ for any $n \ge N$. If $x \le x_0$, then $$F_\mu(x+2{\epsilon})+2{\epsilon}\ge F_\mu(x_0) + {\epsilon}\ge F_{\mu_n}(x_0)
\ge F_{\mu_n}(x)$$ for any $n \ge N$. Similarly we can show that $F_\mu(x-2{\epsilon})-2{\epsilon}\le F_{\mu_n}(x)$ for all $x \in {\mathbf{R}}$ and $n \ge N$. So, $L(F_{\mu_n},F_\mu) \le 2{\epsilon}$ for all $n \ge N$. As ${\epsilon}> 0$ was arbitrary, we have $L(F_{\mu_n},F_\mu) \to 0$.
\(iii) implies (i): Let $x \in {\mathbf{R}}$ be a continuity point of $F_\mu$, and let ${\epsilon}> 0$ be given. Since $F_\mu$ is continuous at $x$, there is a $\delta \in (0,{\epsilon}/2)$ such that $|F_\mu(x) - F_\mu(y)| \le {\epsilon}/2$ for all $|y - x| \le \delta$. Let $N \in {\mathbf{N}}$ be such that $L(F_{\mu_n},F_\mu) < \delta$ for all $n \ge N$. Then, $$F_{\mu}(x-\delta) - \delta \le F_{\mu_n}(x) \le
F_{\mu}(x+\delta) + \delta.$$ for all $n \ge N$. Now observe that $$F_\mu(x)-{\epsilon}\le F_\mu(x-\delta)-{\epsilon}/2 \le F_\mu(x-\delta) -
\delta \le F_{\mu_n}(x)$$ and $$F_{\mu_n}(x) \le F_\mu(x+\delta)+\delta \le F_\mu(x+\delta) +
{\epsilon}/2 \le F_\mu(x) + {\epsilon}$$ for all $n \ge N$. Since ${\epsilon}> 0$ was arbitrary, $F_{\mu_n}(x) \to F_\mu(x)$.
Expected probability measures
-----------------------------
For any Borel $A \subset {\mathbf{R}}$, the map $e_A\colon\Pr({\mathbf{R}}) \to [0,1]$ defined by $e_A(\mu) := \mu(A)$ is measurable.
For any $x \in {\mathbf{R}}$ and ${\epsilon}> 0$, let $f_{x,x+{\epsilon}}\colon{\mathbf{R}}\to{\mathbf{R}}$ be the map which is $1$ on $(-\infty,x\,]$, $0$ on $[\,x+{\epsilon},\infty)$, and linear on $[x,x+{\epsilon}]$. Then the map $\mu \mapsto \int_{\mathbf{R}}f_{x,x+{\epsilon}}\,d\mu$ is continuous, and so measurable. Since $\int_{\mathbf{R}}f_{x,x+1/n} \,d\mu \to \mu((-\infty,x])$ as $n \to \infty$ by bounded convergence, the map $e_{(-\infty,x]}$ is measurable for any $x \in {\mathbf{R}}$. Let $\mathcal{C}$ be the collection of all Borel $A \subset {\mathbf{R}}$ such that $e_A$ is measurable. If $A_1,A_2,\ldots \in \mathcal{C}$ are disjoint, then $$e_{\bigcup_{n=1}^\infty A_n} := \sum_{n=1}^\infty e_{A_n}$$ is measurable, and so $\bigcup_{n=1}^\infty A_n \in \mathcal{C}$. If $A \in \mathcal{C}$, then $e_{{\mathbf{R}}\setminus A} = 1 - e_A$ is measurable, and so ${\mathbf{R}}\setminus A \in \mathcal{C}$. These show that $\mathcal{C}$ is a $\lambda$-system containing $(-\infty,x\,]$ for all $x \in {\mathbf{R}}$. As the rays $(-\infty,x\,]$ form a $\pi$-system that generates the Borel $\sigma$-algebra of ${\mathbf{R}}$, the $\pi$-$\lambda$ theorem concludes the proof.
\[thm:expect\_prob\] Let $\mu$ be a random element of $\Pr({\mathbf{R}})$. Then there exists a unique $\operatorname{\mathbf{E}}\mu \in \Pr({\mathbf{R}})$ satisfying $$F_{\operatorname{\mathbf{E}}\mu}(x) = \operatorname{\mathbf{E}}[F_\mu(x)]$$ for all $x \in {\mathbf{R}}$. The probability measure $\operatorname{\mathbf{E}}\mu$ satisfies $$\label{eq:expect_prob_int}
\int_{\mathbf{R}}f \, d\operatorname{\mathbf{E}}\mu = \operatorname{\mathbf{E}}\left[\int_{\mathbf{R}}f \,d\mu\right]$$ for all continuous and bounded $f: {\mathbf{R}}\to {\mathbf{R}}$.
As uniqueness is easy, we only need to show the existence. Define $F:{\mathbf{R}}\to {\mathbf{R}}$ by $f(x) := \operatorname{\mathbf{E}}[F_\mu(x)]$. Since $F_\mu$ is surely nondecreasing, $f$ is nondecreasing. Since $F_\mu(n) \to 1$ and $F_\mu(-n) \to 0$ as $n \to \infty$ surely, $f(n) \to 1$ and $f(-n) \to 0$ as $n \to \infty$ by bounded convergence. If $x_1 \ge x_2 \ge \cdots$ and $x_n \to x \in {\mathbf{R}}$, then $F_\mu(x_n) \to F_\mu(x)$ surely by the right continuity of distribution functions, and so $f(x_n) \to f(x)$ by bounded convergence. This shows that $f$ is right continuous, and so the proof that $f$ is a distribution function is finished.
Let $\operatorname{\mathbf{E}}\mu$ denote the Borel probability measure on ${\mathbf{R}}$ with distribution $f$. For any $-\infty < a \le b < \infty$, we have $$(\operatorname{\mathbf{E}}\mu) ((a,b]) = f(b) - f(a) = \operatorname{\mathbf{E}}[F_\mu(b) - F_\mu(a)]
= \operatorname{\mathbf{E}}[\mu((a,b])].$$ If $(a_1,b_1], (a_2,b_2],\ldots$ are disjoint, then $$(\operatorname{\mathbf{E}}\mu)\biggl(\bigcup_{n=1}^\infty (a_n,b_n]\biggr)
= \operatorname{\mathbf{E}}\biggl[\mu\biggl(\bigcup_{n=1}^\infty (a_n,b_n]\biggr)\biggr]$$ by monotone convergence. Since any open subset of ${\mathbf{R}}$ is a countable union of disjoint open intervals, and any open interval is a countable union of disjoint bounded intervals of the form $(a,b]$, we see that $$(\operatorname{\mathbf{E}}\mu)(U) = \operatorname{\mathbf{E}}[\mu(U)]$$ holds for any open $U \subset {\mathbf{R}}$.
Now we show . By linearity of integral and expectation, we may assume that $f$ is nonnegative. For each $t \ge 0$, let $U_t := \{\,x \in {\mathbf{R}}\mid f(x) > t\,\}$. We want to apply Tonelli’s theorem to the map $G:\Pr({\mathbf{R}}) \times [0,\infty)
\to [0,1]$ given by $(\nu, t) \mapsto \nu(U_t)$, so we first show that this map is jointly measurable. For each $n \in {\mathbf{N}}$, let $G_n:\Pr({\mathbf{R}}) \times [0,\infty) \to [0,1]$ by $$G_n(\nu,t) := \nu\bigl(U_{\lceil t2^{n}\rceil2^{-n}}\bigr).$$ Since $\nu \mapsto \nu(U)$ is measurable for any open $U$, each $G_n$ is measurable. As $G_n$ increases to $G$, we can conclude that $G$ is measurable. Now we can use Tonelli’s theorem to conclude that $$\begin{split}
\int_{\mathbf{R}}f \,d\operatorname{\mathbf{E}}\mu &= \int_0^\infty (\operatorname{\mathbf{E}}\mu)(U_t) \,dt
= \int_0^{\infty} \operatorname{\mathbf{E}}[\mu(U_t)] \,dt \\
&= \operatorname{\mathbf{E}}\biggl[\int_0^\infty \mu(U_t)\,dt\biggr]
= \operatorname{\mathbf{E}}\biggl[\int_{\mathbf{R}}f\,d\mu\biggr].
\end{split}$$
Spectra of Hermitian matrices {#sec:spectra}
=============================
Basic facts {#subsec:spectra_basic}
-----------
Recall the following version of the spectral theorem from linear algebra.
\[thm:fin\_spec\_thm\] Let $V$ be an $n$-dimensional complex inner product space. For any self-adjoint linear operator $T:V \to V$, there exists an orthonormal basis of $V$ consisting of eigenvectors of $V$.
See [@Tao12 Theorem 1.3.1] or [@HK71 Theorem 9 in Section 9.5].
Also recall the following.
\[prop:self\_adj\_ev\_real\] Any eigenvalue of a self-adjoint linear operator on a complex inner product space is real.
Let $\lambda$ be an eigenvalue of a self-adjoint linear operator $T$ on a complex inner product space $V$. If $Tv = \lambda v$ and $v \ne 0$, then $$\bar{\lambda} {\lVertv\rVert}^2 = {\langlev,\lambda v\rangle}
= {\langlev,Tv\rangle} = {\langleTv,v\rangle} = {\langle\lambda v,v\rangle}
= \lambda {\lVertv\rVert}^2,$$ and so $\bar{\lambda} = \lambda$.
The following naturally follows from Theorem \[thm:fin\_spec\_thm\] and Proposition \[prop:self\_adj\_ev\_real\].
\[cor:unit\_diag\] For any $n \times n$ Hermitian matrix $A$, there exists an $n \times n$ diagonal matrix $D$ with real entries and an $n \times n$ unitary matrix $U$ such that $A = UDU^\ast$.
If $A$ is an $n \times n$ Hermitian matrix, then we denote the eigenvalues of $A$ counted with multiplicities as $$\lambda_1(A) \ge \cdots \ge \lambda_n(A).$$
If $A$ is an $n \times n$ Hermitian matrix, then the *spectral distribution* of $A$ is the Borel probability measure on ${\mathbf{R}}$ defined by $$\mu_A := \frac{1}{n} \sum_{i=1}^n \delta_{\lambda_i(A)}.$$ We write $F_A$ as a shorthand for $F_{\mu_A}$.
\[thm:ev\_minimax\] Let $A$ be an $n \times n$ Hermitian matrix. For each $1 \le i \le n$, we have $$\lambda_i(A) = \sup_{\dim V = i}
\inf_{v \in V : {\lVertv\rVert}=1} v^\ast Av$$ and $$\lambda_i(A) = \inf_{\dim V = n-i+1}
\sup_{v \in V : {\lVertv\rVert}=1} v^\ast Av,$$ where $V$ ranges over the subspaces of ${\mathbf{C}}^n$.
We only need to show the first equality, since the second follows by applying the first to $-A$. By the spectral theorem, we may assume that $A$ is diagonal with $\lambda_i(A)$ at the $(i,i)$-entry. Note that if $v = \begin{pmatrix} v_1 & \cdots & v_n \end{pmatrix}^T$, then we have $v^\ast Av = \sum_{i=1}^n \lambda_i(A)|v_i|^2$. If $V$ is the subspace spanned by $e_1,\ldots,e_i$, then $\inf_{v \in V : {\lVertv\rVert}=1} v^\ast Av = \lambda_i(A)$. To show the other direction, let $V$ be any $i$-dimensional subspace of ${\mathbf{C}}^n$. If $W$ is the subspace spanned by $e_i,\ldots,e_n$, then $V \cap W \ne \{\mathbf{0}\}$ follows from $$\dim V + \dim W = \dim (V \cap W) + \dim (V + W).$$ If we choose any $w \in V \cap W$ with ${\lVertw\rVert}=1$, then $$\inf_{v \in V:{\lVertv\rVert}=1} v^\ast Av \le w^\ast Aw \le \lambda_i(A).$$
\[thm:cauchy\_inter\] If $A_n$ is an $n \times n$ Hermitian matrix and $A_{n-1}$ is the top left $(n-1) \times (n-1)$ minor of $A_n$, then $$\lambda_{i+1}(A_n) \le \lambda_i(A_{n-1}) \le \lambda_i(A_n)$$ for any $1 \le i \le n-1$.
For any $v \in {\mathbf{C}}^{n-1}$, we have $$v^\ast A_{n-1}v = \begin{pmatrix} v & 0 \end{pmatrix}
A_{n} \begin{pmatrix} v & 0\end{pmatrix}^T.$$ So, by Theorem \[thm:ev\_minimax\], we have $$\begin{aligned}
\lambda_i(A_{n-1}) &= \sup_{V \subset {\mathbf{C}}^{n-1}:\dim V = i}
\inf_{v \in V:{\lVertv\rVert}=1} v^\ast A_{n-1}v \\
&\le \sup_{V \subset {\mathbf{C}}^n:\dim V = i} \inf_{v \in V:{\lVertv\rVert}=1}
v^\ast A_nv = \lambda_i(A_n).
\end{aligned}$$ By applying this result to $-A_n$, we have $$\lambda_{i+1}(A_n) = -\lambda_{n-i}(-A_n)
\le -\lambda_{n-i}(-A_{n-1}) = \lambda_i(A_{n-1}).$$
Perturbations by small Frobenius norms
--------------------------------------
We will show that spectral distributions are stable under two types of perturbations. The first can be described using the following norm.
\[def:fnorm\] If $A = (a_{ij})_{i,j=1}^n$ is an $n \times n$ complex matrix, then the *Frobenius norm* of $A$ is given by $${\lVertA\rVert_{F}} := \biggl( \sum_{i,j=1}^n |a_{j}|^2 \biggr)^{1/2}.$$
Note that the Frobenius norm is just the $L^2$-norm on ${\mathbf{C}}^{n^2}$. If $A$ is a Hermitian matrix, then ${\lVertA\rVert_{F}}^2 = \operatorname{tr}(A^2)$. The following inequality tells us that the ordered tuple of eigenvalues is stable under perturbations with small Frobenius norms.
\[thm:hof-wie\_ineq\] If $A$ and $B$ are $n \times n$ Hermitian matrices, then $$\sum_{i=1}^n (\lambda_i(A) - \lambda_i(B))^2 \le {\lVertA-B\rVert_{F}}^2.$$
Recall that eigenvalues and traces are similarity invariant. So, by the spectral theorem, we have $$\sum_{i=1}^n\lambda_i(A)^2 = \operatorname{tr}A^2 \qquad\text{and}\qquad
\sum_{i=1}^n\lambda_i(B)^2 = \operatorname{tr}B^2.$$ Thus, it is enough to show $$\operatorname{tr}(AB) \le \sum_{i=1}^n \lambda_i(A)\lambda_i(B).$$ (Recall that the right side of the desired inequality is equal to $\operatorname{tr}(A-B)^2$.) Again by the spectral theorem, we may assume that $A$ is diagonal with $\lambda_i(A)$ at its $i$-th entry, and write $B = UDU^\ast$ for some unitary $U$ where $D$ is the diagonal matrix with $\lambda_i(B)$ as its $i$-th entry. If $u_{ij}$ denotes the $(i,j)$-entry of $U$, we have $$\operatorname{tr}(AB) = \operatorname{tr}(AUDU^\ast)
= \sum_{i,j=1}^n |u_{ij}|^2\lambda_i(A)\lambda_j(B).$$ It is enough to show that if $a_1 \ge \cdots \ge a_n$ and $b_1 \ge \cdots \ge b_n$, then the maximum of $\sum_{i,j=1}^n v_{ij} a_ib_j$ where $v_{ij} = v_{ji} \ge 0$ and $\sum_{j=1}^n v_{ij} = 1$ is obtained when $(v_{ij})_{i,j=1}^n = I$; that is, $v_{11} = \cdots = v_{nn} = 1$ and $v_{ij} = 0$ whenever $i \ne j$. Let $(v_{ij})_{i,j=1}^n$ where $v_{ij} = v_{ji} \ge 0$ and $\sum_{j=1}^n v_{ij} = 1$ is given. If $(v_{ij}) \ne I$, we have $v_{kk} < 1$ for some $k$. Since $\sum_{j=1}^n v_{kj} = 1$, we have $v_{k\ell} > 0$ for some $\ell \ne k$. Let $w_{kk} = v_{kk} + v_{k\ell}$, $w_{\ell \ell} = v_{\ell \ell} + v_{k\ell} $, $w_{k\ell} = w_{\ell k} = 0$, and $w_{ij} = v_{ij}$ for all other $(i,j)$’s. Then we have $w_{ij} = w_{ji} \ge 0$ and $\sum_{j=1}^n w_{ij} = 1$. Also, $$\begin{split}
\sum_{i,j=1}^n w_{ij}a_ib_j - \sum_{i,j=1}^n v_{ij}a_ib_j
&= v_{k\ell}(a_kb_k + a_\ell b_\ell - a_kb_\ell - a_\ell b_k) \\
&= v_{k\ell}(a_k-a_\ell)(b_k-b_\ell) \ge 0.
\end{split}$$ Note that $(w_{ij})$ has more $1$’s on the diagonal than $(v_{ij})$. If we repeat this procedure, we will arrive at $I$, and this shows our claim.
From the Hoffman-Wielandt inequality (Theorem \[thm:hof-wie\_ineq\]), it follows that the spectral distribution is also stable under perturbations of small Frobenius norms.
\[cor:perturb\_frob\_norm\] If $A$ and $B$ are $n \times n$ Hermitian matrices, then $$[L(F_A,F_B)]^3 \le \frac{1}{n}{\lVertA-B\rVert_{F}}^2.$$ (For the definition of ${\lVert\cdot\rVert_{F}}$, see Definition \[def:fnorm\].)
For any $x \in {\mathbf{R}}$ and ${\epsilon}> 0$, we will show $$\label{eq:perturb_frob_1}
F_A(x) \le F_B(x + {\epsilon}) + \frac{1}{{\epsilon}^2n}{\lVertA-B\rVert_{F}}^2$$ and $$\label{eq:perturb_frob_2}
F_A(x) \ge F_B(x-{\epsilon}) - \frac{1}{{\epsilon}^2n}{\lVertA-B\rVert_{F}}^2.$$ Let $i := \bigl|\{\,\ell\mid\lambda_\ell(A) > x\,\}\bigr|$ and $j := \bigl|\{\,\ell\mid\lambda_\ell(B) > x+{\epsilon}\,\}\bigr|$. Since $\lambda_k(A) \le x$ and $\lambda_k(B) > x + {\epsilon}$ for each $i < k \le j$, we have $$(j-i){\epsilon}^2 \le \sum_{i < k \le j} |\lambda_k(A) - \lambda_k(B)|^2
\le {\lVertA-B\rVert_{F}}^2.$$ As $j-i = n(F_A(x) - F_B(x+{\epsilon}))$, follows. Now let $i' := \bigl|\{\,\ell\mid\lambda_\ell(A) > x\,\}\bigr|$ and $j' := \bigl|\{\,\ell\mid\lambda_\ell(B) > x-{\epsilon}\,\}\bigr|$. Then, $$(i'-j'){\epsilon}^2 \le \sum_{j' < k \le i'}|\lambda_k(A) - \lambda_k(B)|^2
\le {\lVertA-B\rVert_{F}}^2,$$ and so follows.
Now let ${\epsilon}> 0$ be such that ${\epsilon}^3 := \frac{1}{n}{\lVertA-B\rVert_{F}}^2$. Then, $\frac{1}{{\epsilon}^2n}{\lVertA-B\rVert_{F}}^2 = {\epsilon}$. Since $$F_B(x-{\epsilon}) - {\epsilon}\le F_A(x) \le F_B(x+{\epsilon}) + {\epsilon}$$ for all $x \in {\mathbf{R}}$, we have $L(F_A,F_B) \le {\epsilon}$, and thus the desired claim follows.
Perturbations by small ranks
----------------------------
The second type of perturbation is the low-rank perturbation.
\[thm:rank\_ineq\] If $A$ and $B$ are $n \times n$ Hermitian matrices, then $${\lVertF_{A} - F_{B}\rVert_\infty} \le \frac{\operatorname{rank}(A - B)}{n}$$ where ${\lVertf\rVert_\infty} := \sup_{x \in {\mathbf{R}}} |f(x)|$.
Note that $L(F_A,F_B) \le {\lVertF_A - F_B\rVert_\infty}$.
Let $k := \operatorname{rank}(A-B)$. Note that replacing $A$ and $B$ with $U AU^\ast$ and $UBU^\ast$ for some unitary $U$ doesn’t change each side of the desired inequality. So, using Corollary \[cor:unit\_diag\], we may assume that $A-B$ is diagonal. By swapping rows and columns, we can further assume that $$A =
\begin{pmatrix}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{pmatrix}
\text{ and }
B =
\begin{pmatrix}
B_{11} & A_{12} \\
A_{21} & A_{22}
\end{pmatrix},$$ where $A_{22}$ is a $(n-k) \times (n-k)$ matrix. If $x \in [\lambda_{i+1}(A_{22}),\lambda_i(A_{22}))$, then $\lambda_{k+i+1}(A) \le \lambda_{i+1}(A_{22})$ and $\lambda_i(A_{22}) \le \lambda_i(A)$ by the Cauchy interlacing law (Theorem \[thm:cauchy\_inter\]), and so $$\frac{n-k-i}{n} \le F_A(x) \le \frac{n-i}{n}.$$ By the same reasoning, we also have $$\frac{n-k-i}{n} \le F_B(x) \le \frac{n-i}{n},$$ and so $$|F_A(x) - F_B(x)| \le \frac{k}{n} = \frac{\operatorname{rank}(A-B)}{n}.$$ Even if $x < \lambda_{n-k}(A_{22})$ or $x \ge \lambda_1(A_{22})$, this inequality can be proved by a similar argument. Now the desired inequality follows since $x$ is arbitrary.
The following is a generalization of Theorem \[thm:rank\_ineq\].
\[cor:bv\_rank\_ineq\] If $A$ and $B$ are $n \times n$ Hermitian matrices and $f:{\mathbf{R}}\to {\mathbf{R}}$ satisfies ${\lVertf\rVert_{TV}} \le 1$, then $$\left|\int_{\mathbf{R}}f \,d\mu_A - \int_{\mathbf{R}}f \,d\mu_B\right|
\le \frac{\operatorname{rank}(A-B)}{n}.$$
Let $-\infty < t_1 < \cdots < t_m < \infty$ be such that $$\{t_1,\ldots,t_m\} = \{\lambda_1(A),\ldots,\lambda_n(A),
\lambda_1(B),\ldots,\lambda_n(B)\}.$$ Define $g:{\mathbf{R}}\to {\mathbf{R}}$ by letting $g(t_i)=f(t_i)$ for each $i=1,\ldots,m$, extending linearly between $t_i$ and $t_{i+1}$ for each $i=1,\ldots,m-1$, and setting to constants on $(-\infty,t_1]$ and $[t_m,\infty)$. Note that $g':{\mathbf{R}}\to {\mathbf{R}}$ exists as an integrable function, and we have $$f(t_i) = f(t_m) - \int_{t_i}^\infty g'(t)\,dt$$ for any $i=1,\ldots,m$. Also, as ${\lVertf\rVert_{TV}} \le 1$, we have $$\int_{\mathbf{R}}|g'(t)| \,dt = \sum_{i=1}^{n-1} |f(t_{i+1}) - f(t_i)| \le 1.$$ Observe that $$\begin{split}
\int_{\mathbf{R}}f \,d\mu_A &= \sum_{i=1}^n f(\lambda_i(A)) \\
&= nf(t_m) - \sum_{i=1}^n \int_{\lambda_i(A)}^\infty g'(t)\,dt \\
&= nf(t_m) - \int_{\mathbf{R}}g'(t)F_A(t) \,dt.
\end{split}$$ Similarly we have $$\int_{\mathbf{R}}f \,d\mu_B = nf(t_m) - \int_{\mathbf{R}}g'(t)F_B(t)\,dt,$$ and so $$\begin{split}
\left| \int_{\mathbf{R}}f\,d\mu_A - \int_{\mathbf{R}}f\,d\mu_B \right|
&\le \int_{\mathbf{R}}|g'(t)|\,dt {\lVertF_A - F_B\rVert}_\infty \\
&\le \frac{\operatorname{rank}(A-B)}{n}
\end{split}$$ by Theorem \[thm:rank\_ineq\].
Random Hermitian matrices {#subsec:rand_hermitian}
-------------------------
\[def:rand\_hermitian\] Let $H_n$ denote the space of all $n \times n$ Hermitian matrices. We equip $H_n$ with the standard Euclidean metric (and so the metric topology and the Borel $\sigma$-algebra) by identifying $H_n$ with $C^{n(n-1)/2}\times{\mathbf{R}}^n$, which is thought to represent the lower triangle of an $n \times n$ Hermitian matrix. A random element of $H_n$ is called a *random $n \times n$ Hermitian matrix*.
From Hoffman-Wielandt inequality (Theorem \[thm:hof-wie\_ineq\]), it follows that the map $\lambda: H_n \to {\mathbf{R}}^n$ given by $\lambda(A) := (\lambda_1(A),\ldots,\lambda_n(A))$ is continuous. This fact combined with the following lemma shows that the spectral distribution of a random Hermitian matrix is measurable.
\[lem:esd\_conti\] The map ${\mathbf{R}}^n \to \Pr({\mathbf{R}})$ given by $$(x_1,\ldots,x_n) \mapsto \frac{1}{n}\sum_{i=1}^n \delta_{x_i}$$ is continuous (and so is measurable).
For any continuous bounded $f:{\mathbf{R}}\to {\mathbf{R}}$, the map $(x_1,\ldots,x_n) \in {\mathbf{R}}^n \mapsto \int_{\mathbf{R}}f \,d(\frac{1}{n}
\sum_{i=1}^n \delta_{x_i}) = \frac{1}{n}\sum_{i=1}^n f(x_i)$ is continuous. Thus the given map is continuous by the definition of the topology of weak convergence.
The following is a “random version" of Corollary \[cor:perturb\_frob\_norm\]. If $X$ is a random $n \times n$ Hermitian matrix, we let $\operatorname{\mathbf{E}}F_X$ be the distribution function of $\operatorname{\mathbf{E}}\mu_X$, i.e. $(\operatorname{\mathbf{E}}F_X)(x) = \operatorname{\mathbf{E}}[F_X(x)]$.
\[cor:pertrub\_frob\_norm\_exp\] If $X$ and $Y$ are random $n \times n$ Hermitian matrices, then $$[L(\operatorname{\mathbf{E}}F_X, \operatorname{\mathbf{E}}F_Y)]^3
\le \frac{1}{n}\operatorname{\mathbf{E}}\bigl[{\lVertX - Y\rVert_{F}}^2\bigr].$$ (For the definition of ${\lVert\cdot\rVert_{F}}$, see Definition \[def:fnorm\].)
If $\operatorname{\mathbf{E}}\bigl[{\lVertX-Y\rVert_{F}}^2\bigr] = \infty$, there is nothing to prove; so we may assume $\operatorname{\mathbf{E}}\bigl[{\lVertX-Y\rVert_{F}}^2\bigr] < \infty$. By applying and to $X$ and $Y$, and taking the expectation, we have $$\begin{gathered}
\operatorname{\mathbf{E}}[F_Y(x-{\epsilon})] - \frac{1}{{\epsilon}^2 n} \operatorname{\mathbf{E}}\bigl[{\lVertX-Y\rVert_{F}}^2\bigr]
\le \operatorname{\mathbf{E}}[F_X(x)] \\
\le \operatorname{\mathbf{E}}[F_Y(x+{\epsilon})] + \frac{1}{{\epsilon}^2 n} \operatorname{\mathbf{E}}\bigl[{\lVertX-Y\rVert_{F}}^2\bigr]
\end{gathered}$$ for all $x \in {\mathbf{R}}$ and ${\epsilon}> 0$. As in the proof of Corollary \[cor:perturb\_frob\_norm\], let ${\epsilon}> 0$ be such that ${\epsilon}^3 := \frac{1}{n}\operatorname{\mathbf{E}}\bigl[{\lVertX-Y\rVert_{F}}^2\bigr]$. Since $$\operatorname{\mathbf{E}}[F_Y(x-{\epsilon})]-{\epsilon}\le \operatorname{\mathbf{E}}[F_X(x)] \le \operatorname{\mathbf{E}}[F_Y(x+{\epsilon})]+{\epsilon}$$ for all $x \in {\mathbf{R}}$, we have $L(\operatorname{\mathbf{E}}F_X, \operatorname{\mathbf{E}}F_Y) \le {\epsilon}$, and thus the desired inequality holds.
Concentration of measure {#sec:concen_meas}
========================
\[lem:hoeffding\_lemma\] Let $-\infty < a < b < \infty$. If $X$ is a $[a,b]$-valued random variable, then $$\operatorname{\mathbf{E}}\bigl[e^{X-\operatorname{\mathbf{E}}[X]}\bigr] \le \exp\bigl(2(b-a)^2\bigr).$$
We may assume $\operatorname{\mathbf{E}}[X]=0$. Note that $a \le 0 \le b$. For any $x \in [a,b]$, $$e^x = 1 + x + (e^c/2)x^2 \qquad \text{for some $c \in [a,b]$,}$$ and so $$e^x \le 1+x+(e^{b-a}/2)x^2.$$ Since $\operatorname{\mathbf{E}}[X]=0$ and $\operatorname{\mathbf{E}}[X^2]\le(b-a)^2$, we have $$\operatorname{\mathbf{E}}[e^X] \le 1+\operatorname{\mathbf{E}}[X]+(e^{b-a}/2)\operatorname{\mathbf{E}}[X^2]
\le 1+\bigl((b-a)^2/2\bigr) e^{b-a}.$$ If $b-a < 1$, then $$\operatorname{\mathbf{E}}[e^X] \le 1+(e/2)(b-a)^2 \le \exp\bigl((e/2)(b-a)^2\bigr).$$ If $b-a \ge 1$, then $$\begin{gathered}
\operatorname{\mathbf{E}}[e^X]\le\bigl(1+(b-a)^2/2\bigr)e^{b-a} \\
\le \exp\bigl((b-a)^2/2\bigr)e^{(b-a)^2} = \exp\bigl((3/2)(b-a)^2\bigr).
\end{gathered}$$ In any case, we have $\operatorname{\mathbf{E}}[e^X]\le\exp\bigl(2(b-a)^2\bigr)$.
\[thm:mcdiarmid\_ineq\] Let $S_1,\ldots,S_n$ be measurable spaces, and $F \colon S_1\times\cdots\times S_n \to {\mathbf{R}}$ be a bounded measurable function. Assume that $$\bigl|F(x_1,\ldots,x_n) - F(x_1,\ldots,x_{i-1},x'_i,x_{i+1},\ldots,x_n)
\bigr| \le c_i$$ for any $i \in \{1,\ldots,n\}$, $x_j \in S_j$ for each $j\in\{1,\ldots,n\}$, and $x'_i \in S_i$, where $c_i > 0$ doesn’t depend on $x_1,\ldots,x_n,x'_i$. If $X_1,\ldots,X_n$ are independent random elements of $S_1,\ldots,S_n$, then $$\operatorname{\mathbf{P}}\bigl(\bigl|F(X_1,\ldots,X_n)-\operatorname{\mathbf{E}}\bigl[F(X_1,\ldots,X_n)\bigr]\bigr|
\ge \lambda\sigma\bigr) \le 2\exp(-\lambda^2/8)$$ for any $\lambda > 0$ where $\sigma = \sqrt{c_1^2+\cdots+c_n^2}$.
Let us first show $$\label{eq:mcdiarmid_first}
\operatorname{\mathbf{E}}\Bigl[\exp\bigl(tF(X_1,\ldots,X_n)\bigr)\Bigr]
\le \exp\Bigl(2t^2\sigma^2 + t\operatorname{\mathbf{E}}\bigl[F(X_1,\ldots,X_n)\bigr]\Bigr)$$ for any $t > 0$ by induction on $n$. If $n = 0$, in which case $S_1 \times \cdots \times S_n$ is a singleton, $F$ is essentially a constant, and $\sigma = 0$, there is nothing to prove. We now proceed by induction on $n$.
Note that we may assume that each $X_i$ is the projection $\pi_i\colon S_1 \times \cdots \times S_n \to S_i$. Let $\mu$ and $\mu_n$ be the distributions of $(X_1,\ldots,X_{n-1})$ and $X_n$. Let $G:S_1 \times \cdots \times S_{n-1} \to {\mathbf{R}}$ be defined by $$G(x_1,\ldots,x_{n-1}):=\int_{S_n} F(x_1,\ldots,x_{n-1},y) \,\mu_n(dy).$$ For any $i \in \{1,\ldots,n-1\}$, $x_j \in S_j$ for each $j \in \{1,\ldots,n-1\}$, and $x'_i \in S_i$, we have $$\begin{split}
\bigl|G(x_1,\ldots,x_{n-1})&-G(x_1,\ldots,x_{i-1},x'_i,x_{i+1},
\ldots,x_{n-1})\bigr| \\
&\le \int_{S_n}\bigl|F(x_1,\ldots,x_{n-1},y)\\
&\qquad -F(x_1,\ldots,x_{i-1},x'_i,x_{i+1},\ldots,x_{n-1},y)\bigr|
\,\mu_n(dy) \\
&\le c_i.
\end{split}$$ Since $$\begin{split}
\operatorname{\mathbf{E}}[&G(X_1,\ldots,X_{n-1})] \\
&= \int_{S_1\times\cdots\times S_{n-1}}\int_{S_n}
F(x_1,\ldots,x_{n-1},y)\,\mu_n(dy)\mu(d(x_1,\ldots,x_{n-1})) \\
&= \operatorname{\mathbf{E}}[F(X_1,\ldots,X_n)],
\end{split}$$ the induction hypothesis implies $$\label{eq:mcdiarmid_Gbdd}
\begin{split}
\operatorname{\mathbf{E}}\Bigl[&\exp\bigl(tG(X_1,\ldots,X_{n-1})\bigr)\Bigr] \\
&\le \exp\Bigl(2t^2(c_1^2+\cdots+c_{n-1}^2)+
t\operatorname{\mathbf{E}}\bigl[F(X_1,\ldots,X_n)\bigr]\Bigr).
\end{split}$$ Define $H_t:S_1 \times \cdots \times S_{n-1} \to {\mathbf{R}}$ by $$\begin{gathered}
H_t(x_1,\ldots,x_{n-1}) := \\
\int_{S_n} \exp\bigl(t\bigl(
F(x_1,\ldots,x_{n-1},y)-G(x_1,\ldots,x_{n-1})\bigr)\bigr)\,\mu_n(dy).
\end{gathered}$$ Whenever $x_i \in S_i$ is fixed for each $i \in \{1,\ldots,n-1\}$, we have $$\label{eq:mcdiarmid_Hbdd}
H_t(x_1,\ldots,x_{n-1}) \le \exp(2t^2c_n^2)$$ by Hoeffding’s lemma (Lemma \[lem:hoeffding\_lemma\]). Thus, and yield $$\begin{split}
\operatorname{\mathbf{E}}\Bigl[&\exp\bigl(tF(X_1,\ldots,X_n)\bigr)\Bigr]\\
&=\int_{S_1\times\cdots\times S_{n-1}}
\exp\bigl(tG(x_1,\ldots,x_{n-1})\bigr) \\
&\qquad H_t(x_1,\ldots,x_{n-1})
\,\mu(d(x_1,\ldots,x_{n-1}))\\
&\le \exp(2t^2c_n^2) \int_{S_1\times\cdots\times S_{n-1}}
\exp\bigl(tG(x_1,\ldots,x_{n-1})\bigr)\,\mu(d(x_1,\ldots,x_{n-1}))\\
&\le \exp\Bigl(2t^2\sigma^2+t\operatorname{\mathbf{E}}\bigl[F(X_1,\ldots,X_n)\bigr]\Bigr),
\end{split}$$ finishing the proof of .
We can now finish the proof. Observe that $$\begin{split}
\operatorname{\mathbf{P}}\bigl(F(X_1,&\ldots,X_n)-\operatorname{\mathbf{E}}\bigl[F(X_1,\ldots,X_n)\bigr]
\ge \lambda\sigma\bigr)\\
&\le e^{-t\lambda\sigma} \operatorname{\mathbf{E}}\Bigl[\exp\bigl(tF(X_1,\ldots,X_n)-
t\operatorname{\mathbf{E}}\bigl[F(X_1,\ldots,X_n)\bigr]\bigr)\Bigr]\\
&\le \exp(2t^2\sigma^2-t\lambda\sigma).
\end{split}$$ By some calculus one can find that $t = \lambda/4\sigma$ minimizes the right side, yielding $$\operatorname{\mathbf{P}}\bigl(F(X_1,\ldots,X_n)-\operatorname{\mathbf{E}}\bigl[F(X_1,\ldots,X_n)\bigr]
\ge \lambda\sigma\bigr) \le \exp(-\lambda^2/8).$$ Applying this result to $-F$, we obtain $$\operatorname{\mathbf{P}}\bigl(\bigl|F(X_1,\ldots,X_n)-\operatorname{\mathbf{E}}\bigl[F(X_1,\ldots,X_n)\bigr]\bigr|
\ge \lambda\sigma\bigr) \le 2\exp(-\lambda^2/8).$$
The following inequality was found independently by Guntuboyina and Leeb [@GL09], and Bordenave, Caputo, and Chafaï [@BCC11].
\[thm:concen\_spec\] Let $X$ be a random $n \times n$ Hermitian matrix whose rows of the lower triangle are jointly independent. If $f:{\mathbf{R}}\to {\mathbf{R}}$ satisfies ${\lVertf\rVert_{TV}} \le 1$, and $t > 0$, then $$\operatorname{\mathbf{P}}\biggl( \biggl| \int_{\mathbf{R}}f\,d\mu_X - \operatorname{\mathbf{E}}\int_{\mathbf{R}}f\,d\mu_X \biggr| \ge t
\biggr) \le 2\exp(-nt^2/32).$$
Let $S_i := {\mathbf{C}}^i$ and $X_i := (X_{i1},\ldots,X_{ii})$ for each $i=1,\ldots,n$. Given $(x_1,\ldots,x_n) \in S_1 \times \cdots \times S_n$, let $H(x_1,\ldots,x_n)$ be the Hermitian matrix whose $i$th row of the lower triangle is $x_i$ for each $i=1,\ldots,n$. Let $(x_1,\ldots,x_n) \in S_1 \times \cdots \times S_n$ and $x_i' \in S_i$. If we change a row or a column of a matrix, then the change in rank is at most $1$. Since $H(x_1,\ldots,x_{i-1},x_i',x_{i+1},\ldots,x_n)$ can be obtained from $H(x_1,\ldots,x_n)$ by changing a row and then changing a column, the rank of $$H(x_1,\ldots,x_n) - H(x_1,\ldots,x_{i-1},x_i',x_{i+1},\ldots,x_n)$$ is at most $2$. Thus, Corollary \[cor:bv\_rank\_ineq\] tells us that $$\left|\int_{\mathbf{R}}f\,d\mu_{H(x_1,\ldots,x_n)} -
\int_{\mathbf{R}}f\,d\mu_{H(x_1,\ldots,x_{i-1},x_i',x_{i+1},\ldots,x_n)}
\right| \le \frac{2}{n}.$$ Let $X_i$ be the $i$th row of the lower triangle of $X$. Then, $X_1,\ldots,X_n$ are independent, and $X = H(X_1,\ldots,X_n)$. By applying Theorem \[thm:mcdiarmid\_ineq\] to $F:S_1 \times \cdots \times S_n \to {\mathbf{R}}$ given by $F(x_1,\ldots,x_n) := \int_{\mathbf{R}}f \,\mu_{H(x_1,\ldots,x_n)}$ and $X_1,\ldots,X_n$, we obtain $$\operatorname{\mathbf{P}}\left(\left|\int_{\mathbf{R}}f\,d\mu_X - \operatorname{\mathbf{E}}\int_{\mathbf{R}}f\,d\mu_X\right|
\ge \frac{2\lambda}{\sqrt{n}}\right) \le 2\exp(-\lambda^2/8)$$ for any $\lambda > 0$. Our desired result follows by letting $\lambda = t\sqrt{n}/2$.
Reduction to unit variance case {#sec:unit_reduction}
===============================
The Stieltjes transform method, which is the topic of the next section, is able to prove a semicircular law (Theorem \[thm:unit\_sclaw\]) which assumes that every entry of $W_n$ has variance excatly $1/n$. However, it seems not so easy to reduce Theorem \[thm:sclaw\] itself to the case the Stieltjes transform can handle. This section provides an alternative semicircular law, which is somewhat weaker than \[thm:sclaw\], that can still be reduced to what the Stieltjes transform can handle. If you’re satisfied by the reduced version (Theorem \[thm:unit\_sclaw\]), feel free to skip to Section \[sec:stieltjes\]. Otherwise, the following is the alternative semicirular law. It was pointed out in the Remark following Theorem \[thm:sclaw\] as a special case of Theorem \[thm:sclaw\].
\[thm:alter\_sclaw\] For each $n \in {\mathbf{N}}$, let $W_n=({w_{ij}^{(n)}})_{i,j=1}^n$ be a random $n \times n$ Hermitian matrix whose upper triangular entries are jointly independent, have mean zero, and have finite variances. We assume that $W_1, W_2, \ldots$ are defined on the same probability space. If $$\label{eqn:alter_var}
{\lim_{n\to\infty}\frac{1}{n} \sum_{i,j=1}^n \biggl| {\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}- \frac{1}{n} \biggr| =0}$$ and $${\lim_{n\to\infty}\frac{1}{n} \sum_{i,j=1}^n {\operatorname{\mathbf{E}}\bigl[\,|{w_{ij}^{(n)}}|^2{\mathrel{;}}|{w_{ij}^{(n)}}|>{\epsilon}\,\bigr]}=0} \qquad \text{for every ${\epsilon}> 0$,}$$ then ${\mu_{W_n}}{\Rightarrow}{\mu_{\mathrm{sc}}}$ as $n \to \infty$ a.s.
Extension of the underlying probability space
---------------------------------------------
Let $(\Omega,\mathcal{B},\operatorname{\mathbf{P}})$ be the probability space on which $W_1, W_2, \ldots$ are defined. If $(\Omega',\mathcal{B}',\operatorname{\mathbf{P}}')$ is another probability space, and $T\colon\Omega'\to\Omega$ is a measurable map such that $\operatorname{\mathbf{P}}(A) = \operatorname{\mathbf{P}}'(T^{-1}(A))$ for all $A \in \mathcal{B}$, then the random matrices $W_n \circ T$ satisfy all conditions of Theorem \[thm:alter\_sclaw\]. Assume that we proved $\mu_{W_n\circ T} {\Rightarrow}{\mu_{\mathrm{sc}}}$ $\operatorname{\mathbf{P}}'$-a.s. Since $$\{\,\omega'\in\Omega' \mid \mu_{W_n\circ T(\omega')}{\Rightarrow}{\mu_{\mathrm{sc}}}\,\}
= T^{-1}\bigl(\{\,\omega\in\Omega \mid \mu_{W_n(\omega)}{\Rightarrow}{\mu_{\mathrm{sc}}}\,\}\bigr),$$ we will have $\mu_{W_n} {\Rightarrow}{\mu_{\mathrm{sc}}}$ $\operatorname{\mathbf{P}}$-a.s. if we can show that $$\label{eqn:red_unit_meas}
\{\,\omega\in\Omega \mid \mu_{W_n(\omega)}{\Rightarrow}{\mu_{\mathrm{sc}}}\,\} \in \mathcal{B}.$$ For any $p,q \in {\mathbf{Q}}$ with $p < q$, let $f_{p,q}$ be defined as in Theorem \[thm:ch\_weak\_conv\]. Since $\int_{\mathbf{R}}f_{p,q}\,d\mu_{X}$ is a real-valued random variable for any random Hermitian matrix $X$, the event $$\biggl\{ \lim_{n\to\infty} \int_{\mathbf{R}}f_{p,q}\,d\mu_{W_n}
= \int_{\mathbf{R}}f_{p,q}\,d{\mu_{\mathrm{sc}}}\biggr\}$$ is measurable. So, follows from Theorem \[thm:ch\_weak\_conv\], and thus $\mu_{W_n} {\Rightarrow}{\mu_{\mathrm{sc}}}$ $\operatorname{\mathbf{P}}$-a.s. follows. This shows that we can think that $W_n \circ T$’s and $\Omega'$ are the given random matrices and the underlying space. By considering $\Omega'=\Omega\times\{0,1\}^{\mathbf{N}}$, we may assume that we have i.i.d. random variables $\xi_{ij}^{(n)}$’s, where $n \in {\mathbf{N}}$ and $1\le i,j\le n$, independent from $W_1,W_2,\ldots$, which satisfy $$\operatorname{\mathbf{P}}(\xi_{ij}^{(n)}=1/\sqrt{n}) = \operatorname{\mathbf{P}}(\xi_{ij}^{(n)}=-1/\sqrt{n}) = 1/2.$$
Repeating what we already know
------------------------------
The first three steps of Section \[sec:prelim\_reductions\] (that is, until centralization) works for our case with a slight change. Applying those steps, we can now assume the following, and need to prove $\operatorname{\mathbf{E}}{\mu_{W_n}}{\Rightarrow}{\mu_{\mathrm{sc}}}$.
1. The upper triangular entries of $W_n$ are jointly independent and have mean zero.
2. We have .
3. There are $\eta_1,\eta_2,\ldots>0$ such that $|{w_{ij}^{(n)}}|\le\eta_n$ and $\eta_n \to 0$ as $n \to \infty$.
Replacing and rescaling
-----------------------
Let $$E^{(n)} := \bigl\{ (i,j) \bigm| 1 \le i,j \le n, i \ne j,
{\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}\le \tfrac{1}{2n} \bigr\}$$ and define $W' = (v_{ij}^{(n)})_{i,j=1}^n$ by $$v_{ij}^{(n)} := \begin{cases}
\frac{1}{\bigl(n{\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}\bigr)^{1/2}}{w_{ij}^{(n)}}& \text{if $(i,j) \notin E^{(n)}$} \\
\xi_{ij}^{(n)} & \text{if $(i,j) \in E^{(n)}$} \\
\end{cases}.$$ Note that $$\begin{split}
\frac{1}{n} \sum_{(i,j)\in E^{(n)}}
\operatorname{\mathbf{E}}\bigl[\bigl|{w_{ij}^{(n)}}- v_{ij}^{(n)}\bigr|^2\bigr]
&= \frac{1}{n} \sum_{(i,j)\in E^{(n)}} \Bigl( {\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}+ \frac{1}{n} \Bigr)\\
&\le \frac{3}{2n^2} \bigl|E^{(n)}\bigr|\\
&\le \frac{3}{n} \sum_{i,j=1}^n \Bigl| {\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}- \frac{1}{n} \Bigr|
\to 0
\end{split}$$ as $n \to \infty$. Since $(a-b)^2 \le |a-b|(a+b) = |a^2-b^2|$ for any $a,b \ge 0$, we also have $$\begin{split}
\frac{1}{n} \sum_{(i,j)\notin E^{(n)}}
\operatorname{\mathbf{E}}\bigl[\bigl|&{w_{ij}^{(n)}}- v_{ij}^{(n)}\bigr|^2\bigr]\\
&= \frac{1}{n} \sum_{(i,j)\notin E^{(n)}}
\biggl( 1 - \frac{1}{\bigl(n{\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}\bigr)^{1/2}} \biggr)^2 {\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}\\
&\le \frac{1}{n} \sum_{i,j=1}^n
\biggl( \bigl({\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}\bigr)^{1/2} - \frac{1}{\sqrt{n}} \biggr)^2\\
&\le \frac{1}{n} \sum_{i,j=1}^n
\biggl| {\operatorname{\mathbf{Var}}\bigl[{w_{ij}^{(n)}}\bigr]}- \frac{1}{n} \biggr| \to 0
\qquad \text{as $n\to\infty$.}
\end{split}$$ Combining previous two displays, we obtain $${\lim_{n\to\infty}\frac{1}{n} \sum_{i,j=1}^{n} \operatorname{\mathbf{E}}\bigl[\bigl|{w_{ij}^{(n)}}- v_{ij}^{(n)}\bigr|^2\bigr] =0},$$ and so it is enough to show $\operatorname{\mathbf{E}}\mu_{W'_n} {\Rightarrow}{\mu_{\mathrm{sc}}}$ by Corollary \[cor:pertrub\_frob\_norm\_exp\] and Theorem \[thm:ch\_weak\_conv\].
If $i,j \notin E^{(n)}$, then $\operatorname{\mathbf{Var}}\bigl[v_{ij}^{(n)}\bigr] \ge \tfrac{1}{2n}$, and so $$|v_{ij}^{(n)}| \le \sqrt{2}|{w_{ij}^{(n)}}| \le \sqrt{2}\eta_n.$$ As $|v_{ij}^{(n)}| \le 1/\sqrt{n}$ for any $(i,j) \in E^{(n)}$, by letting $\eta'_n := \eta_n \vee (1/\sqrt{n})$ we have $\eta'_n \to 0$ and $|v_{ij}^{(n)}| \le \eta'_n$. Since the upper triangular entries of $W'_n$ are jointly independent random variables with mean zero and variance $1/n$, we can now assume that the upper triangular entries of $W_n$ have variance $1/n$, and there are $\eta_1,\eta_2,\ldots > 0$ such that $\eta_n \to 0$ and $|{w_{ij}^{(n)}}|\le\eta_n$.
The Stieltjes transform method {#sec:stieltjes}
==============================
\[thm:unit\_sclaw\] For each $n \in {\mathbf{N}}$, let $W_n = ({w_{ij}^{(n)}})_{i,j=1}^n$ be a random $n \times n$ Hermitian matrix whose upper triangular entries are jointly independent random variables with mean zero and variance $1/n$. We assume that $W_1, W_2, \ldots$ are defined on the same probability space. If there are $\eta_1, \eta_2 , \ldots > 0$ with $|{w_{ij}^{(n)}}| \le \eta_n$ and $\eta_n \to 0$ as $n \to \infty$, then $\operatorname{\mathbf{E}}{\mu_{W_n}}{\Rightarrow}{\mu_{\mathrm{sc}}}$ as $n \to \infty$ a.s.
For the readers who skipped Section \[sec:unit\_reduction\]: note that the first step in Section \[sec:prelim\_reductions\] lets us upgrade the conclusion of Theorem \[thm:unit\_sclaw\] to ${\mu_{W_n}}{\Rightarrow}{\mu_{\mathrm{sc}}}$ as $n \to \infty$ a.s.
Stieltjes transform
-------------------
Let ${\mathbf{C}}_+ := \{z \in {\mathbf{C}}\mid \Im z > 0\}$. Weak convergence of probability measures on ${\mathbf{R}}$ can be coded in terms of Stieltjes transforms.
Let $\mu$ be a positive, finite Borel measure on ${\mathbf{R}}$. The *Stieltjes transform* $s_\mu:{\mathbf{C}}_+ \to {\mathbf{C}}$ of $\mu$ is given by $$s_\mu(z) := \int_{\mathbf{R}}\frac{1}{x-z} \,\mu(dx).$$
Note that $|1/(x-z)| \le \Im z$ for all $x \in {\mathbf{R}}$. So $x \mapsto 1/(x-z)$ is a continuous and bounded function, and it also follows that $|s_\mu(z)| \le \Im z$.
\[thm:stie\_inv\] If $\mu$ is a positive, finite Borel measure on ${\mathbf{R}}$, then the following hold:
1. for any $b > 0$, $a \in {\mathbf{R}}\mapsto \frac{1}{\pi}\Im s_\mu(a + ib)$ is a nonnegative function with $\int_{\mathbf{R}}\frac{1}{\pi}
\Im s_\mu(a+ib)\,da = \mu({\mathbf{R}})$;
2. $\frac{1}{\pi}\Im s_\mu(a+ib) \,da {\Rightarrow}\mu$ as $b \downarrow 0$.
3. $b\Im s_\mu(ib) \to \mu({\mathbf{R}})$ as $b \to \infty$;
If $\mu = 0$, then there is nothing to prove. By renormalization, we may, and will, assume $\mu({\mathbf{R}}) = 1$. Note that $$\label{eq:stieltjes_inversion_1}
\frac{1}{\pi}\Im s_\mu(a+ib)
= \int_{\mathbf{R}}\frac{1}{\pi}\frac{b}{(x-a)^2 + b^2} \,\mu(dx).$$ Let $X$ be a real-valued random variable with distribution $\mu$, and $C$ be a standard Cauchy random variable (i.e. the law of $C$ has density $\frac{1}{\pi(x^2+1)}$) independent of $X$. Then $X+bC \to X$ as $b \downarrow 0$ a.s., and thus in distribution. Since the right side of is the density of the law of $X + bC$, both (i) and (ii) are proved. As $$bs_\mu(ib) = \int_{\mathbf{R}}\frac{b^2}{x^2+b^2}\,\mu(dx) \to \mu({\mathbf{R}})
\qquad \text{as $b \to \infty$}$$ by dominated convergence, (iii) is also proved.
\[thm:stie\_conv\] If $\mu,\mu_1,\mu_2,\ldots$ are Borel probability measures on ${\mathbf{R}}$, then $\mu_n {\Rightarrow}\mu$ if and only if $s_{\mu_n}(z) \to s_\mu(z)$ for all $z \in {\mathbf{C}}_+$.
The “only if" direction follows immediately from the definition of weak convergence. To show the “if" direction, assume that $s_{\mu_n}(z) \to s_\mu(z)$ for all $z \in {\mathbf{C}}_+$. Whenever $n_1 < n_2 < \cdots$ and $\mu_{n_k} \to \nu$ vaguely for some finite $\nu$, we have $s_{\mu_{n_k}}(z) \to s_\nu(z)$ for all $z \in {\mathbf{C}}_+$ since $x \mapsto 1/(x-z)$ vanishes at infinity. This implies $s_\nu(z) = s_\mu(z)$ for all $z \in {\mathbf{C}}_+$, and thus $\nu = \mu$ by Theorem \[thm:stie\_inv\]. As any subsequence of $(\mu_n)_{n \in {\mathbf{N}}}$ has a vaguely convergent further subsequence, it follows that any subsequence of $(\mu_n)_{n \in {\mathbf{N}}}$ has a further subsequence converging vaguely to $\mu$. This shows $\mu_n \to \mu$ vaguely, and so $\mu_n {\Rightarrow}\mu$.
Predecessor comparison
----------------------
In the remainder of this section, we will show that $s_{\operatorname{\mathbf{E}}\mu_{W_n}}(z) \to s_{{\mu_{\mathrm{sc}}}}(z)$ for all $z \in {\mathbf{C}}_+$. To do so, we will first express $s_{\operatorname{\mathbf{E}}\mu_{W_n}}(z)$ in terms of the *resolvent* $(W_n-zI)^{-1}$. By the spectral theorem, for any Hermitian matrix $A$ the matrix $A - zI$ is invertible for any $z \in {\mathbf{C}}_+$. Let $S_A(z) := (A - zI)^{-1}$ for any Hermitian $A$ and $z \in {\mathbf{C}}_+$. Using the spectral theorem, we can also see that $$s_{\mu_A}(z) = \frac{1}{n} \operatorname{tr}S_A(z).$$ Thus, we have $$s_{\operatorname{\mathbf{E}}\mu_{W_n}}(z) = \operatorname{\mathbf{E}}s_{\mu_{W_n}}(z) = \frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z),$$ and so it is enough to show that $$\frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z) \to s_{{\mu_{\mathrm{sc}}}}(z)$$ for all $z \in {\mathbf{C}}_+$.
To understand the limiting behavior of $\frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z)$, we relate it with the $(n-1) \times (n-1)$ minors of $W_n$ using the Schur complement formula, which will be presented below. For each $i \in \{1,\ldots,n\}$, let $W^{(i)}$ be the $(n-1) \times (n-1)$ matrix obtained by removing the $i$-th row and column from $W_n$. Also, let $w_i$ denote the $i$-th column of $W_n$ with $w_{ii}$ removed. (So, $w_i$ is an $(n-1)$-dimensional column vector.) Let us denote the $(i,j)$-entry of a matrix $A$ by $A(i,j)$. Recall that if $A$ is an invertible matrix, then $$A^{-1}(i,j) = \frac{1}{\det A} C_{ji}(A)$$ where $C_{ji}(A)$ is the $(i,j)$-cofactor of $A$. So we have $$S_{W_n}(z)(i,i) = \frac{\det (W^{(i)} - zI_{n-1})}{\det (W_n - zI)}$$ where $I_{n-1}$ is the $(n-1) \times (n-1)$ identity matrix.
\[prop:schur\_complement\] Consider a matrix $$\begin{pmatrix} A & B \\ C & D\end{pmatrix}$$ with complex entries, where $A$ and $D$ are square matrices and $A$ is invertible. Then we have $$\det \begin{pmatrix} A & B \\ C & D\end{pmatrix}
= \det(A)\det(-CA^{-1}B+D).$$
Note that $$\begin{pmatrix} I & 0 \\ -CA^{-1} & I\end{pmatrix}
\begin{pmatrix} A & B \\ C & D\end{pmatrix}
= \begin{pmatrix} A & B \\ 0 & -CA^{-1}B +D\end{pmatrix}.$$ Since $$\det \begin{pmatrix} I & 0 \\ -CA^{-1} & I\end{pmatrix} = 1$$ and $$\det \begin{pmatrix} A & B \\ 0 & -CA^{-1}B +D\end{pmatrix} = \det(A)\det(-CA^{-1}B+D),$$ we have $$\det \begin{pmatrix} A & B \\ C & D\end{pmatrix} = \det(A)\det(-CA^{-1}B+D)$$ by the multiplicativity of determinant.
By Proposition \[prop:schur\_complement\], we have $$\det (W_n - zI) = (-z-w_i^\ast S_{W^{(i)}}(z)w_i) \det(W^{(i)} - zI_{n-1}),$$ and so $$S_{W_n}(z)(i,i) = \frac{-1}{z+w_i^\ast S_{W^{(i)}}(z)w_i}.$$ Summing over $i=1,\ldots,n$ and taking the expectation, we obtain $$\label{eq:stieltjes_reduction_1}
\frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z) = \frac{1}{n} \sum_{i=1}^n
\operatorname{\mathbf{E}}\frac{-1}{z+w_i^\ast S_{W^{(i)}}(z)w_i}.$$ (The fact that the expectation on the right side is well-defined follows from Lemma \[lem:resolvent\_im\_pos\] below.) We will show that the right side of gets close to $$\frac{-1}{z + \frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z)}$$ as $n$ grows, and obtain a recursive relation involving the limit of $\frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z)$.
Derivation of a recurrence relation
-----------------------------------
The following fact will be used repeatedly. In particular, it will guarantee that many denominators we face in the computation below are nonzero.
\[lem:resolvent\_im\_pos\] If $A$ is an $n \times n$ Hermitian matrix and $z \in {\mathbf{C}}_+$, then the following hold:
1. $\operatorname{tr}S_A(z) \in {\mathbf{C}}_+$;
2. $\operatorname{tr}\bigl((A-zI)(A-\bar{z}I)\bigr)^{-1} \le n/(\Im z)^2$;
3. $u^\ast S_A(z) u \in {\mathbf{C}}_+$ for any $u \in {\mathbf{C}}^n$.
Let $A = U D U^\ast$ where $U$ is unitary and $D$ is real diagonal with diagonal entries $d_1,\ldots,d_n$. (i) Since trace is similarity invariant, we have $$\operatorname{tr}S_A(z) = \operatorname{tr}U(D-zI)^{-1}U^\ast = \operatorname{tr}(D-zI)^{-1}
= \sum_{i=1}^n \frac{1}{d_i-z} \in {\mathbf{C}}_+.$$ (ii) Also, $$\begin{aligned}
\operatorname{tr}\bigl((A-zI)(A-\bar{z}I)\bigr)^{-1}
&= \operatorname{tr}\bigl((D-zI)^{-1}(D-\bar{z}I)^{-1}\bigr) \\
&= \sum_{i=1}^n \frac{1}{(d_i-z)(d_i-\bar{z})}
= \sum_{i=1}^n \frac{1}{|d_i-z|^2} \le \frac{n}{(\Im z)^2}.
\end{aligned}$$ (iii) Finally, if we let $U^\ast u = (u_1,\ldots,u_n)$, then $$u^\ast S_A(z) u = (U^\ast u)^\ast (D-zI)^{-1} (U^\ast u)
= \sum_{i=1}^n \frac{|u_i|^2}{d_i-z} \in {\mathbf{C}}_+.$$
In the rest of this subsection, we fix $z \in {\mathbf{C}}_+$, and transform $$\frac{1}{n} \sum_{i=1}^n \operatorname{\mathbf{E}}\frac{-1}{z + w_i^\ast S_{W^{(i)}}(z) w_i}$$ step-by-step to obtain $$\frac{-1}{z + \frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z)}$$ (asymptotically) in the end.
### From $w_i^\ast S_{W^{(i)}}(z)w_i$ to $\frac{1}{n}\operatorname{tr}S_{W^{(i)}}(z)$
Instead of numbering the rows and columns of $W^{(i)}$ using $1,\ldots,n-1$, let us use $1,\ldots,\hat{i},\ldots,n$ as if $W^{(i)}$ still lies in $W_n$. For $j,k \ne i$, let $b_{jk}$ denote the $(j,k)$-entry of $S_{W^{(i)}}(z)$. Since $$\begin{aligned}
\sum_{j,k \ne i} |b_{jk}|^2
&= \operatorname{tr}\bigl((S_{W^{(i)}}(z))^\ast S_{W^{(i)}}(z)\bigr) \\
&= \operatorname{tr}\bigl([(W^{(i)}-zI_{n-1})^\ast]^{-1} (W^{(i)}-zI_{n-1})^{-1}\bigr) \\
&= \operatorname{tr}\bigl((W^{(i)}-zI_{n-1})(W^{(i)}-\bar{z}I_{n-1})\bigr)^{-1},\end{aligned}$$ we have $$\label{eq:stieltjes_resol_sq_bdd}
\sum_{j,k \ne i} \operatorname{\mathbf{E}}|b_{jk}|^2 \le \frac{n}{(\Im z)^2}$$ by Lemma \[lem:resolvent\_im\_pos\] (ii). The fact that $b_{jk}$’s are in $L^2$ will guarantee that all terms in the computation below are well-defined and finite. Using the fact that $W^{(i)}$ and $w_i$ are independent, and each entry of $W_n$ is of mean zero and variance $1/n$, we have $$\begin{split}
&\operatorname{\mathbf{E}}\Bigl[\Bigl|w_i^\ast S_{W^{(i)}}(z)w_i
- \frac{1}{n} \operatorname{tr}S_{W^{(i)}}(z)\Bigr|^2\Bigr] \\
&= \sum_{\substack{j,k \ne i \\ j \ne k}}
\Bigl[\operatorname{\mathbf{E}}\bigl[|w_{ji}|^2\bigr]
\operatorname{\mathbf{E}}\bigl[|w_{ki}|^2\bigr]\operatorname{\mathbf{E}}\bigl[|b_{jk}|^2\bigr]
+ \operatorname{\mathbf{E}}\bigl[\overline{w_{ji}}^2\bigr] \operatorname{\mathbf{E}}\bigl[w_{ki}^2\bigr]
\operatorname{\mathbf{E}}\bigl[b_{jk}\overline{b_{kj}}\bigr] \\
&\qquad + \operatorname{\mathbf{E}}\bigl[|w_{ji}|^2\bigr]\operatorname{\mathbf{E}}\bigr[|w_{ki}|^2\bigr]
\operatorname{\mathbf{E}}\bigl[b_{jj}\overline{b_{kk}}\bigr]\Bigr]
+ \sum_{j \ne i} \operatorname{\mathbf{E}}\bigl[|w_{ji}|^4\bigr]\operatorname{\mathbf{E}}\bigl[|b_{jj}|^2\bigr] \\
&\qquad - \sum_{j,k \ne i} \Bigl[ \frac{1}{n}
\operatorname{\mathbf{E}}\bigl[|w_{ji}|^2\bigr]\operatorname{\mathbf{E}}\bigl[b_{jj} \overline{b_{kk}}\bigr]
+ \frac{1}{n}\operatorname{\mathbf{E}}\bigl[|w_{ki}|^2\bigr]
\operatorname{\mathbf{E}}\bigl[b_{jj}\overline{b_{kk}}\bigr] \Bigr] \\
&\qquad + \frac{1}{n^2}\sum_{j,k \ne i}
\operatorname{\mathbf{E}}\bigl[b_{jj}\overline{b_{kk}}\bigr] \\
&= \frac{1}{n^2} \sum_{\substack{j,k \ne i \\ j \ne k}}
\operatorname{\mathbf{E}}\bigl[|b_{jk}|^2\bigr] + \sum_{j \ne i} \operatorname{\mathbf{E}}\bigl[|b_{jj}|^2\bigr]
\Bigl(\operatorname{\mathbf{E}}\bigl[|w_{ji}|^4\bigr] - \frac{1}{n^2}\Bigr) \\
&\qquad + \sum_{\substack{j,k \ne i \\ j \ne k}}
\operatorname{\mathbf{E}}\bigl[\overline{w_{ji}}^2\bigr] \operatorname{\mathbf{E}}\bigl[w_{ki}^2\bigr]
\operatorname{\mathbf{E}}\bigl[b_{jk}\overline{b_{kj}}\bigr].
\end{split}$$ Note that the last term in the last line must be real. Since $$\begin{aligned}
\Bigl|\sum_{\substack{j,k \ne i \\ j \ne k}}
\operatorname{\mathbf{E}}&\bigl[\overline{w_{ji}}^2\bigr]
\operatorname{\mathbf{E}}\bigl[w_{ki}^2\bigr] \operatorname{\mathbf{E}}\bigl[b_{jk}\overline{b_{kj}}\bigr] \Bigr|\\
&\le \frac{1}{n^2} \sum_{j,k \ne i} \operatorname{\mathbf{E}}\bigl|b_{jk}\overline{b_{kj}}\bigr|
\le \frac{1}{n^2} \sum_{j,k \ne i}
\bigl(\operatorname{\mathbf{E}}\bigl[|b_{jk}|^2\bigr]\bigr)^{1/2}
\bigl(\operatorname{\mathbf{E}}\bigl[|b_{kj}|^2\bigr]\bigr)^{1/2} \\
&\le \frac{1}{n^2} \Bigl( \sum_{j,k \ne i}
\operatorname{\mathbf{E}}\bigl[|b_{jk}|^2\bigr] \Bigr)^{1/2}
\Bigl( \sum_{j,k \ne i} \operatorname{\mathbf{E}}\bigl[|b_{jk}|^2\bigr] \Bigr)^{1/2} \\
&= \frac{1}{n^2} \sum_{j,k \ne i} \operatorname{\mathbf{E}}\bigl[|b_{jk}|^2\bigr]\end{aligned}$$ by the Cauchy-Schwarz inequality, we have $$\begin{split}
\operatorname{\mathbf{E}}\Bigl[\Bigl|w_i^\ast S_{W^{(i)}}(z)w_i
- \frac{1}{n}&\operatorname{tr}S_{W^{(i)}}(z)\Bigr|^2\Bigr] \\
&\le \frac{2}{n^2} \sum_{j,k \ne i} \operatorname{\mathbf{E}}\bigl[|b_{jk}|^2\bigr]
+ \frac{\eta_n^2}{n} \sum_{j \ne i} \operatorname{\mathbf{E}}\bigl[|b_{jj}|^2\bigr] \\
&\le \left(\frac{2}{n}+\eta_n^2\right) \frac{1}{n}\sum_{j,k \ne i}
\operatorname{\mathbf{E}}\bigl[|b_{jk}|^2\bigr] \\
&\le \left(\frac{2}{n}+\eta_n^2\right) \frac{1}{(\Im z)^2} \\
\end{split}$$ by . It follows that $$\begin{gathered}
\frac{1}{n}\sum_{i=1}^n \operatorname{\mathbf{E}}\Bigl[\Bigl|w_i^\ast S_{W^{(i)}}(z)w_i
- \frac{1}{n} \operatorname{tr}S_{W^{(i)}}(z)\Bigr|\Bigr] \\
\le \frac{1}{n}\sum_{i=1}^n \Bigl(\operatorname{\mathbf{E}}\Bigl[\Bigl|w_i^\ast S_{W^{(i)}}(z)w_i
- \frac{1}{n} \operatorname{tr}S_{W^{(i)}}(z)\Bigr|^2\Bigr]\Bigr)^{1/2} \\
\le \Bigl( \Bigl(\frac{2}{n} + \eta_n^2\Bigr)
\frac{1}{(\Im z)^2} \Bigr)^{1/2} \to 0 \qquad \text{as $n \to \infty$.}\end{gathered}$$ (We are fixing $z \in {\mathbf{C}}_+$.) Therefore, by \[lem:resolvent\_im\_pos\] (i) and (iii), we have $$\begin{split}
\biggl|\frac{1}{n} \sum_{i=1}^n &\operatorname{\mathbf{E}}\frac{-1}{z+w_i^\ast S_{W^{(i)}}(z)w_i}
- \frac{1}{n} \sum_{i=1}^n \operatorname{\mathbf{E}}\frac{-1}{z+\frac{1}{n}
\operatorname{tr}S_{W^{(i)}}(z)} \biggr| \\
&\le \frac{1}{n} \sum_{i=1}^{n} \operatorname{\mathbf{E}}\biggl|\frac{w_i^\ast S_{W^{(i)}}w_i
- \frac{1}{n}\operatorname{tr}S_{W^{(i)}}}{(z+w_i^\ast S_{W^{(i)}}(z)w_i)
(z+\frac{1}{n}\operatorname{tr}S_{W^{(i)}}(z))} \biggr| \\
&\le \frac{1}{n(\Im z)^2} \sum_{i=1}^n \operatorname{\mathbf{E}}\Bigl|w_i^\ast S_{W^{(i)}}(z)w_i
- \frac{1}{n}\operatorname{tr}S_{W^{(i)}}(z)\Bigr| \to 0
\end{split}$$ as $n \to \infty$.
### From $\frac{1}{n}\operatorname{tr}S_{W^{(i)}}(z)$ to $\frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}S_{W^{(i)}}(z)$
Since the maps $x \in {\mathbf{R}}\mapsto \Re(1/(x-z))$ and $x \in {\mathbf{R}}\mapsto \Im(1/(x-z))$ have bounded variations, Theorem \[thm:concen\_spec\] implies that there are $c, C > 0$ (depending on $z$, but we are fixing $z \in {\mathbf{C}}_+$) such that $$\begin{gathered}
\operatorname{\mathbf{P}}\left( \left|\frac{1}{n}\operatorname{tr}S_X(z) - \frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}S_X(z) \right|
\ge t \right) \\
= \operatorname{\mathbf{P}}\left( \left| \int_{\mathbf{R}}\frac{1}{x-z} \,\mu_{X}(dx)
- \operatorname{\mathbf{E}}\int_{\mathbf{R}}\frac{1}{x-z} \,\mu_{X}(dx) \right| \ge t \right)
\le c \exp(-Cnt^2)\end{gathered}$$ for any $n \in {\mathbf{N}}$, $t > 0$, and a random $n \times n$ Hermitian matrix $X$. So, we have $$\begin{split}
\operatorname{\mathbf{E}}\bigg|\frac{1}{n}&\operatorname{tr}S_{W^{(i)}}(z)
- \frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}S_{W^{(i)}}(z)\bigg| \\
&= \int_0^\infty \operatorname{\mathbf{P}}\left( \left|\frac{1}{n-1}\operatorname{tr}S_{W^{(i)}}(z)
- \frac{1}{n-1} \operatorname{\mathbf{E}}\operatorname{tr}S_{W^{(i)}}(z)\right|
\ge \frac{n}{n-1}t \right) \,dt \\
&\le \int_0^\infty c\exp\left(-\frac{Cn^2t^2}{(n-1)}\right) \,dt
\le \int_0^\infty c \exp(-Cnt^2)\,dt
\end{split}$$ for any $n \in {\mathbf{N}}$ and $i = 1,\ldots,n$. It follows that $$\begin{gathered}
\frac{1}{n}\sum_{i=1}^n \operatorname{\mathbf{E}}\left|\frac{1}{n}\operatorname{tr}S_{W^{(i)}}(z)
- \frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}S_{W^{(i)}}(z)\right| \\
\le \int_0^\infty c \exp(-Cnt^2)\,dt \to 0 \qquad \text{as $n \to \infty$}\end{gathered}$$ by dominated convergence. Therefore, $$\begin{split}
\bigg|\frac{1}{n} \sum_{i=1}^n
&\operatorname{\mathbf{E}}\frac{-1}{z+\frac{1}{n}\operatorname{tr}S_{W^{(i)}}(z)}
- \frac{1}{n} \sum_{i=1}^n \frac{-1}{z + \frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}S_{W^{(i)}}(z)}
\bigg| \\
&\le \frac{1}{n} \sum_{i=1}^n \operatorname{\mathbf{E}}\left| \frac{\frac{1}{n}
\operatorname{tr}S_{W^{(i)}}(z)- \frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}S_{W^{(i)}}(z)}{(z+\frac{1}{n}
\operatorname{tr}S_{W^{(i)}}(z)) (z + \frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}S_{W^{(i)}}(z))} \right| \\
&\le \frac{1}{n (\Im z)^2} \sum_{i=1}^n \operatorname{\mathbf{E}}\left|\frac{1}{n}
\operatorname{tr}S_{W^{(i)}}(z)- \frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}S_{W^{(i)}}(z)\right|
\to 0
\end{split}$$ as $n \to \infty$.
### From $\frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}S_{W^{(i)}}(z)$ to $\frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z)$
Let $\overline{W}^{(i)}$ be the $n \times n$ matrix obtained by replacing all the entries in the $i$-th row and column of $W$ by $0$. Since $\overline{W}^{(i)}$ has the same (multi)set of eigenvalues as $W^{(i)}$ except that it has one more zero eigenvalue, we have $$\left|\frac{1}{n} \operatorname{tr}S_{W^{(i)}}(z) - \frac{1}{n}
\operatorname{tr}S_{\overline{W}^{(i)}}(z) \right| \le \frac{1}{n \Im z}.$$ By the Hoffman-Wielandt inequality (Theorem \[thm:hof-wie\_ineq\]), we have $$\begin{split}
\bigg|\frac{1}{n} \operatorname{tr}S_{\overline{W}^{(i)}}(z)
- \frac{1}{n} \operatorname{tr}S_{W_n}&(z) \bigg|
\le \frac{1}{n} \sum_{j=1}^n \biggl|\frac{\lambda_j(\overline{W}^{(i)})
-\lambda_j(W_n)}{\bigl(\lambda_j(\overline{W}^{(i)}) - z\bigr)
\bigl(\lambda_j(W_n)-z\bigr)}\biggr| \\
&\le \frac{1}{n(\Im z)^2} \sum_{j=1}^n
\bigl|\lambda_j(\overline{W}^{(i)})-\lambda_j(W_n)\bigr| \\
&\le \frac{1}{(\Im z)^2} \biggl( \frac{1}{n} \sum_{j=1}^n
|\lambda_j(\overline{W}^{(i)})-\lambda_j(W_n)|^2 \biggr)^{1/2} \\
&\le \frac{1}{(\Im z)^2} \biggl( \frac{2}{n} \sum_{j=1}^n |w_{ij}|^2
\biggr)^{1/2}
\le \frac{\sqrt{2}}{(\Im z)^2} \eta_n.
\end{split}$$ Combining the results of the previous two displays, we obtain $$\operatorname{\mathbf{E}}\left| \frac{1}{n} \operatorname{tr}S_{W^{(i)}}(z) - \frac{1}{n} \operatorname{tr}S_{W_n}(z)
\right| \le \frac{1}{n\Im z} + \frac{\sqrt{2}}{(\Im z)^2}\eta_n$$ for all $n \in {\mathbf{N}}$ and $i = 1,\ldots,n$, and so $$\frac{1}{n} \sum_{i=1}^n \operatorname{\mathbf{E}}\left| \frac{1}{n} \operatorname{tr}S_{W^{(i)}}(z)
- \frac{1}{n} \operatorname{tr}S_{W_n}(z) \right| \le \frac{1}{n\Im z}
+ \frac{\sqrt{2}}{(\Im z)^2}\eta_n \to 0$$ as $n \to \infty$. Therefore, $$\begin{split}
\biggl|\frac{1}{n} \sum_{i=1}^n & \frac{-1}{z+\frac{1}{n}
\operatorname{\mathbf{E}}\operatorname{tr}S_{W^{(i)}}(z)} - \frac{-1}{z + \frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z)} \biggr|\\
&= \biggl|\frac{1}{n} \sum_{i=1}^n \frac{-1}{z+\frac{1}{n}
\operatorname{\mathbf{E}}\operatorname{tr}S_{W^{(i)}}(z)} - \frac{1}{n} \sum_{i=1}^n \frac{-1}{z
+ \frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z)} \biggr| \\
&\le \frac{1}{n} \sum_{i=1}^n \biggl| \frac{\frac{1}{n}
\operatorname{\mathbf{E}}\operatorname{tr}S_{W^{(i)}}(z)- \frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z)}{(z+\frac{1}{n}
\operatorname{\mathbf{E}}\operatorname{tr}S_{W^{(i)}}(z)) (z + \frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z))} \biggr| \\
&\le \frac{1}{n (\Im z)^2} \sum_{i=1}^n \operatorname{\mathbf{E}}\Bigl|\frac{1}{n}
\operatorname{tr}S_{W^{(i)}}(z)- \frac{1}{n}\operatorname{tr}S_{W_n}(z)\Bigr|
\to 0
\end{split}$$ as $n \to \infty$.
### The result
Combining and the final results of the previous three subsubsections, we obtain $$\label{eq:stieltjes_recurrence}
\left| \frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z) - \frac{-1}{z + \frac{1}{n}
\operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z)} \right| \to 0 \qquad \text{as $n \to \infty$}$$ for any $z \in {\mathbf{C}}_+$.
Convergence of the Stieltjes transform
--------------------------------------
Let $s_n(z) := \frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z)$. Fix $z \in {\mathbf{C}}_+$ for now, and let us write $s_n = s_n(z)$. If there are $n_1 < n_2 < \cdots$ such that $|s_{n_k}| \to \infty$ as $k \to \infty$, then we would have $$\left|s_{n_k} - \frac{-1}{z+s_{n_k}} \right| \to \infty
\qquad \text{as $k \to \infty$,}$$ which contradicts . Thus $\{s_n \mid n \in {\mathbf{N}}\}$ is bounded, and therefore any subsequence of $(s_n)_{n \in {\mathbf{N}}}$ has a convergent subsequence. If we show that any convergent subsequence of $(s_n)_{n \in {\mathbf{N}}}$ should converge to a number independent of the subsequence we choose, then we will have the convergence of $(s_n)_{n \in {\mathbf{N}}}$ to that number.
Assume $s_{n_k} \to s \in {\mathbf{C}}_+ \cup {\mathbf{R}}$ as $k \to \infty$. Since the left side of converges to $\bigl|s+1/(z+s)\bigr|$ along $n_1 < n_2 < \cdots$, we have $$s + \frac{1}{z + s} = 0.$$ Solving the quadratic equation, we obtain $$s = \frac{-z \pm \sqrt{z^2 - 4}}{2}.$$
We need to decide which branch of $\sqrt{z^2 - 4}$ we use. For simplicity, we will define $\sqrt{z^2-4}$ only for $z \in {\mathbf{C}}_+ \cup {\mathbf{R}}$, and it will suffice. Choose the branch of $\sqrt{z-2}$ and $\sqrt{z+2}$ defined on ${\mathbf{C}}_+ \cup {\mathbf{R}}$ which are continuous and have nonnegative imaginary part for all $z \in {\mathbf{C}}_+ \cup {\mathbf{R}}$. Then let $\sqrt{z^2-4} := \sqrt{z-2} \sqrt{z+2}$. This will make $\sqrt{z^2-4}$ continuous and have nonnegative imaginary part on ${\mathbf{C}}_+ \cup {\mathbf{R}}$.
Since $s_{n_k} = \frac{1}{n_k} \operatorname{\mathbf{E}}\operatorname{tr}S_{W_{n_k}}(z) \in {\mathbf{C}}_+$ for all $k \in {\mathbf{N}}$, we have $\Im s \ge 0$. On the other hand, $(-z-\sqrt{z^2-4})/2$ has a negative imaginary part. Thus we have $$s = \frac{-z + \sqrt{z^2 - 4}}{2}.$$ Since this is true for any subsequence $(s_{n_k})_{k \in {\mathbf{N}}}$ of $(s_n)_{n \in {\mathbf{N}}}$ converging to $s$, we have $$\label{eqn:stie_to_sc}
s_{\operatorname{\mathbf{E}}\mu_{W_n}}(z) = \frac{1}{n} \operatorname{\mathbf{E}}\operatorname{tr}S_{W_n}(z) = s_n \to
\frac{-z+\sqrt{z^2-4}}{2} \qquad \text{as $n \to \infty$.}$$
Computation of the limiting distribution
----------------------------------------
\[lem:sc\_stie\] If $\mu$ is a positive, finite Borel measure on ${\mathbf{R}}$ satisfying $s_\mu(z) = (-z + \sqrt{z^2 - 4})/2$ for all $z \in {\mathbf{C}}_+$, then $\mu(dx) = \sqrt{(4-x^2)_+}\,dx$.
As $$b \Im s_\mu(ib) = b\frac{-b+\sqrt{b^2+4}}{2}
= 2\frac{-1+\sqrt{1+4/b^2}}{4/b^2}\to1 \qquad \text{as $b \to \infty$,}$$ we have $\mu({\mathbf{R}}) = 1$ by Theorem \[thm:stie\_inv\] (iii). Since $z \mapsto (-z+\sqrt{z^2-4})/2$ is continuous on ${\mathbf{C}}_+ \cup {\mathbf{R}}$, we have $$\frac{1}{\pi}\Im s_\mu(a+ib) \to \Im \frac{-a+\sqrt{a^2-4}}{2\pi}
= \frac{1}{2\pi} \sqrt{(4-a^2)_+} \qquad \text{as $b \downarrow 0$}$$ for each fixed $a \in {\mathbf{R}}$. Since $\frac{1}{\pi} \Im s_\mu(a+ib)\,da$ is a probability density (by Theorem \[thm:stie\_inv\] (i)) converging pointwise to the probability density $\frac{1}{2\pi}\sqrt{(4-a^2)_+}$ as $b \downarrow 0$, we have $\frac{1}{\pi}\Im s_\mu(a+ib)\,da {\Rightarrow}\frac{1}{2\pi}\sqrt{(4-a^2)_+}\,da$ as $b \downarrow 0$ by Scheffé’s theorem ([@Bil12 Theorem 16.12]). Now $\mu(dx) = \frac{1}{2\pi}\sqrt{(4-x^2)_+}\,dx$ follows from Theorem \[thm:stie\_inv\] (ii).
The proof of Lemma \[lem:sc\_stie\] shows how one can figure out what the limiting spectral distribution should be in the first place. Now we finish our proof of the semicircular law. Since $\operatorname{\mathbf{E}}\mu_{W_1}, \operatorname{\mathbf{E}}\mu_{W_2}, \ldots$ are probability measures, there are integers $n_1 < n_2 < \cdots$ such that $\operatorname{\mathbf{E}}\mu_{W_{n_k}} \to \mu$ vaguely as $k \to \infty$ for some positive, finite measure $\mu$. Since $s_{\operatorname{\mathbf{E}}\mu_{W_{n_k}}}(z) \to s_\mu(z)$ as $k \to \infty$ for all $z \in {\mathbf{C}}_+$, implies $s_\mu(z) = (-z+\sqrt{z^2-4})/2$ for all $z \in {\mathbf{C}}_+$. So, we have $\mu = {\mu_{\mathrm{sc}}}$ by Lemma \[lem:sc\_stie\]. Now $\operatorname{\mathbf{E}}\mu_{W_n} {\Rightarrow}{\mu_{\mathrm{sc}}}$ follows from and Theorem \[thm:stie\_conv\]. Interestingly, we were able to avoid an actual computation of $s_{{\mu_{\mathrm{sc}}}}$, in which we might have used something like the residue theorem or the Cauchy integral formula.
|
---
abstract: 'We consider the currents formed by a heavy and a light quark within Quantum Chromodynamics and compute the matching to Heavy Quark Effective Theory to three-loop accuracy. As an application we obtain the third-order perturbative corrections to ratios of $B$-meson decay constants.'
address:
- 'Institut für Theoretische Teilchenphysik, , 76128 Karlsruhe, Germany'
- 'Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia'
- 'Department of Physics, University of Alberta, Edmonton, Alberta T6G 2G7, Canada'
author:
- 'S. Bekavac'
- 'A.G. Grozin'
- 'P. Marquard'
- 'J.H. Piclum'
- 'D. Seidel'
- 'M. Steinhauser'
title: 'Matching QCD and HQET heavy–light currents at three loops'
---
TTP09-41\
SFB/CPP-09-110\
Alberta Thy 16-09
,
,
,
,
,
Introduction {#S:Intro}
============
Quite often there are problems within Quantum Chromodynamics (QCD) involving a single heavy quark with momentum $$p = m v + k\,,
\label{Intro:p}$$ where $m$ is the on-shell heavy-quark mass and $v^2=1$. In situations when the characteristic residual momentum is small ($|k^\mu|\ll m$) and light quarks and gluons have small momenta ($|k_i^\mu|\ll m$) it is possible to use a simpler effective field theory — Heavy Quark Effective Theory (HQET) (see, e.g., the review [@N:94] and the textbooks [@MW:00; @G:04]). Its Lagrangian is a series in $1/m$. QCD operators are also given by series in $1/m$ in terms of HQET operators. Here we shall consider $\overline{\mbox{MS}}$ renormalized heavy–light QCD quark currents $$j(\mu) = Z_j^{-1}(\mu) j_0\,,\quad
j_0 = \bar{q}_0 \Gamma Q_0\,,
\label{Intro:j}$$ where $\Gamma$ is a Dirac matrix, and the index 0 means bare quantities. They can be expressed via operators in HQET $$j(\mu) = C_\Gamma(\mu) \tilde{\jmath}(\mu)
+ \frac{1}{2m} \sum_i B_i(\mu) O_i(\mu)
+ \mathcal{O}\left(\frac{1}{m^2}\right)\,,
\label{Intro:match}$$ where $$\tilde{\jmath}(\mu) = \tilde{Z}_j^{-1}(\mu) \tilde{\jmath}_0\,,\quad
\tilde{\jmath}_0 = \bar{q}_0 \Gamma Q_{v0}\,,
\label{Intro:jtilde}$$ $Q_{v0}$ is the bare HQET heavy-quark field satisfying $\rlap/v Q_{v0} = Q_{v0}$, and $O_i$ are dimension-4 HQET operators with appropriate quantum numbers. We shall not discuss $1/m$ corrections in this paper; our main subject is the matching coefficients $C_\Gamma(\mu)$.
The coefficients $C_\Gamma(\mu)$ have been calculated at one-loop order in the pioneering paper [@EH:90]. At two loops, they were calculated in Ref. [@BG:95], and corrected in Ref. [@G:98].[^1] All-order results in the large-$\beta_0$ limit were obtained in [@BG:95] (see also [@NS:95]). Asymptotics of perturbative series were investigated in a model-independent way in Ref. [@CGM:03].
In the present paper we calculate the matching coefficients $C_\Gamma$ up to three loops. These coefficients are useful for obtaining matrix elements of QCD currents (such as $f_B$) from results of lattice HQET simulations (see, e.g., Refs. [@lattice:09; @Sommer]) or HQET sum rules (see, e.g., Refs. [@SR; @N:94]).
Matching {#S:Match}
========
There are eight Dirac structures giving non-vanishing quark currents in four dimensions: $$\begin{aligned}
\Gamma &{}={}& 1\,,\quad
\rlap/v\,,\quad
\gamma_\bot^\alpha\,,\quad
\gamma_\bot^\alpha \rlap/v\,,
\nonumber\\
&&\gamma_\bot^{[\alpha} \gamma_\bot^{\beta]}\,,\quad
\gamma_\bot^{[\alpha} \gamma_\bot^{\beta]} \rlap/v\,,\quad
\gamma_\bot^{[\alpha} \gamma_\bot^{\beta} \gamma_\bot^{\gamma]}\,,\quad
\gamma_\bot^{[\alpha} \gamma_\bot^{\beta} \gamma_\bot^{\gamma]} \rlap/v\,,
\label{Match:Gamma}\end{aligned}$$ where $\gamma_\bot^\alpha = \gamma^\alpha - \rlap/v v^\alpha$, and square brackets mean antisymmetrization. The last four of them can be obtained from the first four by multiplying by the ’t Hooft–Veltman $\gamma_5^{\mbox{\scriptsize HV}}$. We are concerned with flavour non-singlet currents only, therefore, we may also use the anticommuting $\gamma_5^{\mbox{\scriptsize AC}}$ (there is no anomaly). The currents renormalized at a scale $\mu$ with different prescriptions for $\gamma_5$ are related by [@LV:91] $$\begin{aligned}
\left(\bar{q} \gamma_5^{\mbox{\scriptsize AC}} Q\right)_\mu &{}={}&
Z_P(\mu) \left(\bar{q} \gamma_5^{\mbox{\scriptsize HV}} Q\right)_\mu\,,
\nonumber\\
\left(\bar{q} \gamma_5^{\mbox{\scriptsize AC}} \gamma^\alpha Q\right)_\mu &{}={}&
Z_A(\mu) \left(\bar{q} \gamma_5^{\mbox{\scriptsize HV}} \gamma^\alpha Q\right)_\mu\,,
\nonumber\\
\left(\bar{q} \gamma_5^{\mbox{\scriptsize AC}} \gamma^{[\alpha} \gamma^{\beta]} Q\right)_\mu &{}={}&
Z_T(\mu) \left(\bar{q} \gamma_5^{\mbox{\scriptsize HV}} \gamma^{[\alpha}
\gamma^{\beta]} Q\right)_\mu\,,
\label{Match:Larin}\end{aligned}$$ where the finite renormalization constants $Z_{P,A,T}$ can be reconstructed from the differences of the anomalous dimensions of the currents. Multiplying $\Gamma$ by $\gamma_5^{\mbox{\scriptsize AC}}$ does not change the anomalous dimension. In the case of $\Gamma=\gamma^{[\alpha} \gamma^{\beta]}$, multiplying it by $\gamma_5^{\mbox{\scriptsize HV}}$ just permutes its components, and also does not change the anomalous dimension, therefore, $$Z_T(\mu) = 1\,;
\label{Match:ZT}$$ $Z_{P,A}(\mu)$ are known up to three loops [@LV:91].
The anomalous dimension of the HQET current (\[Intro:jtilde\]) does not depend on the Dirac structure $\Gamma$. Therefore, there are no factors similar to $Z_{P,A}$ in HQET. Multiplying $\Gamma$ by $\gamma_5^{\mbox{\scriptsize AC}}$ does not change the matching coefficient. Therefore, the matching coefficients for the currents in the second row of (\[Match:Gamma\]) are not independent: they can be obtained from those for the first row. In the $v$ rest frame, where $\rlap/v=\gamma^0$, we have $$\begin{aligned}
Z_P(\mu) &{}={}&
\frac{C_{\gamma_5^{\mbox{\scriptsize AC}}}(\mu)}{C_{\gamma_5^{\mbox{\scriptsize HV}}}(\mu)} =
\frac{C_1(\mu)}{C_{\gamma^0 \gamma^1 \gamma^2 \gamma^3}(\mu)}\,,
\nonumber\\
Z_A(\mu) &{}={}&
\frac{C_{\gamma_5^{\mbox{\scriptsize AC}} \gamma^0}(\mu)}{C_{\gamma_5^{\mbox{\scriptsize HV}} \gamma^0}(\mu)} =
\frac{C_{\gamma^0}(\mu)}{C_{\gamma^1 \gamma^2 \gamma^3}(\mu)}
\nonumber\\
&{}={}&
\frac{C_{\gamma_5^{\mbox{\scriptsize AC}} \gamma^3}(\mu)}{C_{\gamma_5^{\mbox{\scriptsize HV}} \gamma^3}(\mu)} =
\frac{C_{\gamma^3}(\mu)}{C_{\gamma^0 \gamma^1 \gamma^2}(\mu)}\,,
\nonumber\\
Z_T(\mu) &{}={}&
\frac{C_{\gamma_5^{\mbox{\scriptsize AC}} \gamma^0 \gamma^1}(\mu)}{C_{\gamma_5^{\mbox{\scriptsize HV}} \gamma^0 \gamma^1}(\mu)} =
\frac{C_{\gamma^0 \gamma^1}(\mu)}{C_{\gamma^2 \gamma^3}(\mu)}
\nonumber\\
&{}={}&
\frac{C_{\gamma_5^{\mbox{\scriptsize AC}} \gamma^2 \gamma^3}(\mu)}{C_{\gamma_5^{\mbox{\scriptsize HV}} \gamma^2 \gamma^3}(\mu)} =
\frac{C_{\gamma^2 \gamma^3}(\mu)}{C_{\gamma^0 \gamma^1}(\mu)} = 1\,.
\label{Match:Ratios}\end{aligned}$$ In particular, two matching coefficients are equal: $$C_{\gamma_\bot \rlap{\scriptsize/}v}(\mu) =
C_{\gamma_\bot^{[\alpha} \gamma_\bot^{\beta]}}(\mu)\,.
\label{Match:CT}$$ In the following we shall consider only the matching coefficients for the first four Dirac structures in (\[Match:Gamma\]).
In order to find the matching coefficients $C_\Gamma(\mu)$, we equate on-shell matrix elements of the left- and right-hand side of Eq. (\[Intro:match\]). They are obtained by considering transitions of the heavy quark with momentum $p=mv+k$ (\[Intro:p\]) to the light quark with momentum $k_q$: $${\langle}q(k_q)|j(\mu)|Q(mv+k){\rangle} =
C_\Gamma(\mu) {\langle}q(k_q)|\tilde{\jmath}(\mu)|Q_v(k){\rangle}
+ \mathcal{O}\left(\frac{k}{m},\frac{k_q}{m}\right)\,.
\label{Match:match}$$ The on-shell matrix elements are[^2] $$\begin{aligned}
{\langle}q(k_q)|j(\mu)|Q(p){\rangle} &{}={}&
\bar{u}_q(k_q) \Gamma(p,k_q) u(p)\,
Z_j^{-1}(\mu) Z_Q^{1/2} Z_q^{1/2}\,,
\nonumber\\
{\langle}q(k_q)|\tilde{\jmath}(\mu)|Q_v(k){\rangle} &{}={}&
\bar{u}_q(k_q) \tilde{\Gamma}(k,k_q) u_v(k)\,
\tilde{Z}_j^{-1}(\mu) \tilde{Z}_Q^{1/2} \tilde{Z}_q^{1/2}\,,
\label{Match:onshell}\end{aligned}$$ where $\Gamma(p,k_q)$ and $\tilde{\Gamma}(k,k_q)$ are the bare vertex functions, $Z_Q$ and $Z_q$ are the on-shell wave-function renormalization constants of the heavy and the light quark in QCD, $\tilde{Z}_Q$ is the on-shell wave-function renormalization constant of the HQET quark field $Q_v$, and $\tilde{Z}_q$ differs from $Z_q$ because there are no $Q$ loops in HQET. The difference between $u(mv+k)$ and $u_v(k)$ is of order $k/m$, and can be neglected. It is most convenient to use $k=k_q=0$, then the $\mathcal{O}(1/m)$ term is absent. The QCD vertex has two Dirac structures: $$\Gamma(mv,0) = \Gamma \cdot (A + B \rlap/v)\,.$$ This leads to $$\bar{u}(0) \Gamma(mv,0) u(mv) = \bar{\Gamma}(mv,0)\,\bar{u}(0) \Gamma u(mv)
\quad\mbox{with}\quad
\bar{\Gamma}(mv,0) = A + B\,.$$ The scalar vertex function $\bar{\Gamma}(mv,0)$ can be obtained by multiplying the diagrams by a projector and taking the trace. For the first two Dirac structures in (\[Match:Gamma\]), the projector $(\rlap/v+1)$ can be used; for the next two structures, $(\rlap/v+1)\gamma_\alpha$. The HQET vertex has just one Dirac structure. Therefore, $$C_\Gamma(\mu) =
\frac{\bar{\Gamma}(mv,0) Z_j^{-1}(\mu) Z_Q^{1/2} Z_q^{1/2}}{\tilde{\Gamma}(0,0) \tilde{Z}_j^{-1}(\mu) \tilde{Z}_Q^{1/2} \tilde{Z}_q^{1/2}}\,.
\label{Match:main}$$ Here $m$ is the on-shell mass of the heavy quark (because the external heavy quark with $p^2=m^2$ should be on its mass shell). Therefore, mass-counterterm vertices have to be taken into account on all $Q$ lines.
If all flavours except $Q$ are massless, all loop corrections to $\tilde{\Gamma}(0,0)$, $\tilde{Z}_Q$, and $\tilde{Z}_q$ contain no scale and hence vanish: $\tilde{\Gamma}(0,0)=1$, $\tilde{Z}_Q=1$, $\tilde{Z}_q=1$. If there is another massive flavour (charm in $b$-quark HQET), this is no longer so. The vertex $\tilde{\Gamma}(0,0)$ and $\tilde{Z}_Q$ have been calculated up to three loops in Ref. [@GSS:06]. The massless-quark on-shell wave-function renormalization constant in the presence of a massive flavour has been calculated up to three loops in Ref. [@CKS:98] (an explicit expression can be found in Appendix A of Ref. [@GSS:06]). The three-loop renormalization of the HQET current $\tilde{Z}_j$ has been calculated in Ref. [@CG:03].
If all flavours except $Q$ are massless, $\Gamma(mv,0)$, $Z_Q$, and $Z_q$ contain a single scale $m$. The on-shell heavy-quark wave-function renormalization constant $Z_Q$ has been calculated at three loops in [@MR:00] (and confirmed in Ref. [@MMPS:07]); $Z_q$ has been found in Ref. [@CKS:98]. The vertex $\Gamma(mv,0)$ is the subject of the present paper. If there is another flavour with a non-zero mass $m_c<m$, there are two scales, and calculations become more difficult. The renormalization constant $Z_Q$ has been calculated in this case, up to three loops, in Ref. [@BGSS:07] (the master integrals appearing in this case are discussed in Ref. [@BGSS:08]). The vertex $\Gamma(mv,0)$ and $Z_q$ with two masses are considered in Sect. \[S:Calc\] and \[S:Zq\]. The three-loop renormalization of the QCD currents with all Dirac structures $\Gamma$ has been obtained in Ref. [@G:00].
The QCD quantities $\bar{\Gamma}(mv,0)$, $Z_Q$, and $Z_q$ in the numerator of (\[Match:main\]) are calculated in terms of $\alpha_s^0=g_0^2/(4\pi)^{1-\varepsilon}$, the bare coupling of the $n_f$-flavour QCD. If there is another massive flavour, say charm, they also contain its bare mass $m_{c0}$; we re-express it via the on-shell mass $m_c$. These quantities do not involve $\mu$. The $\overline{\mbox{MS}}$ renormalization constant $Z_j^{-1}(\mu)$ is expressed in terms of $\alpha_s^{(n_f)}(\mu)$. The HQET quantities $\tilde{\Gamma}(0,0)$, $\tilde{Z}_Q$, and $\tilde{Z}_q$ in the denominator of (\[Match:main\]) are calculated via $\alpha_s^{0\prime}$ and $m_{c0}'$, the bare coupling and the bare $c$-quark mass in the $n_f'$-flavour QCD[^3] (we re-express $m_{c0}'$ via the on-shell mass $m_c$, which is the same in both theories). The $\overline{\mbox{MS}}$ renormalization constant $\tilde{Z}_j^{-1}(\mu)$ is expressed in terms of $\alpha_s^{(n_f')}(\mu)$. To combine all these quantities in (\[Match:main\]), we re-express them via the coupling $\alpha_s^{(n_f')}(\mu)$ using the decoupling relation [@CKS:98] (an explicit expression for $\alpha_s^{(n_f)}(\mu)$ via $\alpha_s^{(n_f')}(\mu)$, including the necessary terms with positive powers of $\varepsilon$, is given in Eq. (12) of Ref. [@GMPS:08]).
There exists an exact relation [@BG:95] between the matching coefficients $C_1(\mu)$ and $C_{\rlap{\scriptsize/}v}(\mu)$. Namely, the renormalized vector and scalar currents are related by $$i \partial_\alpha j^\alpha = m(\mu) j(\mu)\,,
\label{Match:div}$$ where $m(\mu)$ is the $\overline{\mbox{MS}}$ mass of the heavy quark $Q$. Taking the on-shell matrix element of this equality between the heavy quark with $p=mv$ and the light quark with $k_q=0$ and re-expressing both QCD matrix elements via the matrix element of the HQET current with $\Gamma=1$, we obtain $$m C_{\rlap{\scriptsize/}v}(\mu) = m(\mu) C_1(\mu)\,.
\label{Match:mm}$$ The ratio $m(\mu)/m$ has been calculated at three loops in Refs. [@CS:99; @MR:00a]. Comparing $C_{\rlap{\scriptsize/}v}(\mu)/C_1(\mu)$ with the analytical result [@MR:00a] for $m(\mu)/m$ provides a strong check of our calculations. For $m_c\neq0$, this ratio has been calculated in Ref. [@BGSS:07].
Bare vertex functions {#S:Calc}
=====================
The calculation of $\bar{\Gamma}(mv,0)$ at $m_c=0$ is a single-scale problem. The calculation is almost completely automated and similar to Refs. [@MMPS:07; @GMPS:08]. The Feynman diagrams were generated with `QGRAF` [@N:91] and classified into various topologies with the help of `q2e` and `exp` [@H:97]. Some sample diagrams are shown in Fig. \[fig:nm\_diagrams\] (the first four diagrams). Scalar Feynman integrals were reduced to master integrals using integration-by-parts identities [@IBP]. This was done in two independent ways: using the `Form` [@Vermaseren:2000nd] package `SHELL3` [@MR:00] and the `C++` program `Crusher` [@PMDS] implementing the Laporta algorithm [@Laporta]. The master integrals for the case of a massless charm quark are known from Ref. [@MR:00] (see also comments in Ref. [@MMPS:07]). As an independent check the automatic setup described in Ref. [@MPSS:09] has been used and the bare vertex functions have been checked numerically with the help of `FIESTA` [@Smirnov:2008py]. The results for the bare vertex functions with all four Dirac structures $\Gamma$ can be found on the web page [@web].
\
\
Next we study the influence of a non-zero $c$-quark mass, $m_c<m$, on the $b$-quark matching coefficients $C_\Gamma$. Then $c$-quark loops exist both in the full QCD and in the $b$-quark HQET. The full-theory quantities in the numerator of (\[Match:main\]) depend on the ratio of the on-shell quark masses, $$x = \frac{m_c}{m}\,.$$ Some sample diagrams contributing to $\bar{\Gamma}(mv,0)$ are shown in Fig. \[fig:nm\_diagrams\] (all diagrams except the first four ones depend on $x$). The HQET quantities in the denominator of (\[Match:main\]) contain a single scale $m_c$.
From the technical point of view the calculation is similar to Refs. [@BGSS:07; @BGSS:09]. We have used `Crusher` [@PMDS] for the reduction. The master integrals are known from Refs. [@BGSS:07; @BGSS:08], see also Ref. [@PHDStefan]. Most of the needed terms of their $\varepsilon$ expansions are known analytically, in terms of Harmonic Polylogarithms [@RV:00; @HPL] (HPLs) of $x$ (the status of these expansions is summarized in Tables 1–4 of Ref. [@BGSS:08]). From the requirement of cancellation of $1/\varepsilon$ poles in the matching coefficients (Sect. \[S:Coefs\]) we were able to find exact analytical expressions (in terms of HPLs) for the $\mathcal{O}(\varepsilon^0)$ terms of the master integrals 5.2 and 5.2a (Fig. 8 in [@BGSS:08]); formerly, they were known only as truncated series in $x$. The corresponding entry in Table 3 of Ref. [@BGSS:08] needs updating. We do not present these long expressions here, they can be found at [@web].
We have checked the $m_c$ dependent results by taking the limit $x\to0$ and reproducing the $n_l$ part of the results given in Sect. \[S:Coefs\]. Another check is taking the limit $x \to 1$. If we set $n_h=0$ and re-express the renormalized matching coefficients $C_\Gamma$ via $\alpha_s^{(n_l)}$, the results of Sect. \[S:Coefs\] with the substitution $n_m\to n_h$ are reproduced.
Wave-function renormalization of massless quarks {#S:Zq}
================================================
The last ingredient of Eq. (\[Match:main\]) which we had to calculate is the on-shell wave-function renormalization constant $Z_q$ of a massless quark. The result for can be extracted from Ref. [@CKS:98], however, we have performed an independent calculation. The $m_c$-dependent part is a new result. The calculation is similar to that of Ref. [@BGSS:07], where $Z_Q$ has been calculated. The integrals contributing to $Z_q$ reduce to tadpoles when the mass of the incoming particle is set to zero. There is only one new type of diagram which is shown in Fig. \[fig:zq\], all others can be reduced to known results.
For completeness we give here the result for an arbitrary gauge group. It reads $$\begin{aligned}
Z_q &{}={}& 1 + C_F T_F
\left(\frac{\alpha_s^0}{\pi}\Gamma(\varepsilon)\right)^2
\left(n_h m^{-4\varepsilon} + n_m m_c^{-4\varepsilon}\right)
\nonumber\\
&&\hphantom{{}+C_F T_F} \times
\frac{\varepsilon}{16}
\left(1
- \frac{5}{6} \varepsilon
+ \frac{89}{36} \varepsilon^2
+ \mathcal{O}(\varepsilon^3)
\right)
\nonumber\\
&&{} + C_F T_F
\left(\frac{\alpha_s^0}{\pi}\Gamma(\varepsilon)\right)^3
\Bigl(n_h Z_1(n_h) m^{-6\varepsilon} + n_m Z_1(n_m) m_c^{-6\varepsilon}
\nonumber\\
&&\hphantom{{}+C_F T_F}
+ T_F n_h n_m (m m_c)^{-3\varepsilon} Z_2(m_c/m) \Bigr)
+ \mathcal{O}(\alpha_s^4)\,,\end{aligned}$$ where $\alpha_s^0$ has the dimensionality $m^{2\varepsilon}$, and the single-scale contributions are [@CKS:98; @GSS:06] $$\begin{aligned}
Z_1(n) &{}={}&
C_F Z_F + C_A Z_A + T_F n_l Z_L + T_F n Z_H\,,
\nonumber\\
Z_F &{}={}& \frac{\varepsilon}{96}
\left[ 1 - \frac{3}{2} \varepsilon
+ \left(12 \zeta_3 + \frac{443}{12}\right) \varepsilon^2
\right]
+ \mathcal{O}(\varepsilon^4)\,,
\nonumber\\
Z_A &{}={}& \frac{1}{192} \biggl\{
1 + \frac{10}{3} \varepsilon + \frac{227}{9} \varepsilon^2
- \left(16 \zeta_3 + \frac{1879}{54}\right) \varepsilon^3
\nonumber\\
&&{} - \xi
\left[ 1 - 3 \varepsilon + \frac{35}{3} \varepsilon^2
+ \left(8 \zeta_3 - \frac{407}{9}\right) \varepsilon^3
\right] \biggr\}
+ \mathcal{O}(\varepsilon^4)\,,
\nonumber\\
Z_L &{}={}& - \frac{\varepsilon}{72}
\left[1 - \frac{5}{6} \varepsilon + \frac{337}{36} \varepsilon^2\right]
+ \mathcal{O}(\varepsilon^4)\,,
\nonumber\\
Z_H &{}={}& - \frac{\varepsilon}{36}
\left[1 - \frac{5}{6} \varepsilon + \frac{151}{36} \varepsilon^2\right]
+ \mathcal{O}(\varepsilon^4)\,,\end{aligned}$$ where $\xi=1-a_0$, $a_0$ is the bare gauge-fixing parameter.[^4] The two-scale contribution (Fig. \[fig:zq\]) is given by [@PHDStefan] $$Z_2(x) = Z_2(x^{-1}) = 2 Z_H - \frac{\varepsilon^3}{12} \ln^2 x
+ \mathcal{O}(\varepsilon^4)\,.$$
Matching coefficients {#S:Coefs}
=====================
In this Section, we present the results for the matching coefficients of the different heavy–light currents for the colour group SU(3) (results for a general colour group can be found at [@web]). For this purpose we decompose the coefficients as follows $$\begin{aligned}
C_\Gamma(\mu) &{}={}& 1 + \frac{\alpha_s^{(n_f')}(m)}{\pi}\, C_\Gamma^{(1)}
+ \left(\frac{\alpha_s^{(n_f')}(m)}{\pi}\right)^2\, C_\Gamma^{(2)}(x)
\nonumber\\
&&{} + \left(\frac{\alpha_s^{(n_f')}(m)}{\pi}\right)^3\, C_\Gamma^{(3)}(x)
+ \mathcal{O}(\alpha_s^4)\,,
\\
C_\Gamma^{(2)}(x) &{}={}& C_\Gamma^G
+ C_\Gamma^H n_h + C_\Gamma^L n_l + C_\Gamma^M(x) n_m\,,
\nonumber\\
C_\Gamma^{(3)}(x) &{}={}& C_\Gamma^{GG} + C_\Gamma^{GH} n_h + C_\Gamma^{GL} n_l
+ C_\Gamma^{HH} n_h^2 + C_\Gamma^{HL} n_h n_l + C_\Gamma^{LL} n_l^2
\nonumber\\
&&{} + C_\Gamma^{GM}(x) n_m +C_\Gamma^{HM}(x) n_h n_m + C_\Gamma^{LM}(x) n_l n_m
+ C_\Gamma^{MM}(x) n_m^2\,,
\nonumber\end{aligned}$$ where $\Gamma=1$, $\rlap/v$, $\gamma_\bot$, $\gamma_\bot\rlap/v$, and $n_f'=n_l+n_m$ is the number of active flavours in HQET. Furthermore, we use the abbreviation ${L}= \ln(\mu^2/m^2)$ ($m$ is the on-shell $b$-quark mass).
We first present the results for $m_c = 0$. The individual contributions read $$\begin{aligned}
C^{(1)}_1 &{}={}&
\frac{2}{3}
+ \frac{1}{2} {L}\,,\nonumber\\
C^{G}_1 &{}={}&
\frac{1843}{192}
+ \frac{11}{72} \pi^2
+ \frac{1}{18} \pi^2 \ln2
- \frac{17}{36} \zeta_3
+ \left(
\frac{527}{144}
- \frac{7}{108} \pi^2
\right) {L}- \frac{9}{16} {L}^2
\,,\nonumber\\
C^{H}_1 &{}={}&
\frac{149}{216}
- \frac{1}{18} \pi^2
- \frac{5}{36} {L}+ \frac{1}{12} {L}^2
\,,\nonumber\\
C^{L}_1 &{}={}&
- \frac{95}{288}
- \frac{1}{36} \pi^2
- \frac{5}{72} {L}+ \frac{1}{24} {L}^2
\,,\nonumber\\
C^{GG}_1 &{}={}&
\frac{8765231}{62208}
+ \frac{235073}{46656} \pi^2
+ \frac{917}{324} \pi^2 \ln2
- \frac{1}{81} \pi^2 \ln^22
+ \frac{3371}{1296} \zeta_3
\nonumber\\
&&{} + \frac{4733}{3888} \pi^2 \zeta_3
- \frac{50039}{116640} \pi^4
- \frac{28975}{2592} \zeta_5
- \frac{4}{27} \ln^42
- \frac{32}{9} a_4
\nonumber\\
&&{} + \left(
\frac{46123}{1728}
- \frac{25}{108} \pi^2
+ \frac{1}{36} \pi^2 \ln2
+ \frac{55}{144} \zeta_3
- \frac{95}{1944} \pi^4
\right) {L}\nonumber \\
&&{} + \left(
- \frac{605}{64}
+ \frac{7}{48} \pi^2
\right) {L}^2
+ \frac{15}{16} {L}^3
\,,\nonumber\end{aligned}$$ $$\begin{aligned}
C^{GH}_1 &{}={}&
\frac{3349}{1944}
- \frac{30917}{9720} \pi^2
+ \frac{443}{81} \pi^2 \ln2
+ \frac{1}{9} \pi^2 \ln^22
- \frac{27845}{5184} \zeta_3
+ \frac{29}{96} \pi^2 \zeta_3
\nonumber\\
&&{} - \frac{19}{2430} \pi^4
- \frac{45}{32} \zeta_5
- \frac{1}{9} \ln^42
- \frac{8}{3} a_4
\nonumber\\
&&{} + \left(
- \frac{319}{432}
- \frac{1}{36} \pi^2
- \frac{5}{6} \zeta_3
\right) {L}+ \frac{211}{144} {L}^2
- \frac{19}{72} {L}^3
\,,\nonumber\\
C^{GL}_1 &{}={}&
- \frac{528353}{46656}
- \frac{15553}{17496} \pi^2
- \frac{25}{324} \pi^2 \ln2
+ \frac{1}{81} \pi^2 \ln^22
- \frac{1591}{972} \zeta_3
\nonumber\\
&&{} + \frac{3281}{116640} \pi^4
+ \frac{1}{162} \ln^42
+ \frac{4}{27} a_4
+ \left(
- \frac{7399}{5184}
+ \frac{11}{972} \pi^2
- \frac{97}{216} \zeta_3
\right) {L}\nonumber\\
&&{}
+ \left(
\frac{809}{864}
- \frac{7}{648} \pi^2
\right) {L}^2
- \frac{19}{144} {L}^3
\,,\nonumber\\
C^{HH}_1 &{}={}&
- \frac{4045}{11664}
- \frac{1}{135} \pi^2
+ \frac{11}{27} \zeta_3
- \frac{35}{1296} {L}- \frac{5}{216} {L}^2
+ \frac{1}{108} {L}^3
\,,\nonumber\\
C^{HL}_1 &{}={}&
\frac{353}{5832}
+ \frac{1}{216} \pi^2
- \frac{2}{27} \zeta_3
- \frac{35}{648} {L}- \frac{5}{108} {L}^2
+ \frac{1}{54} {L}^3
\,,\nonumber\\
C^{LL}_1 &{}={}&
\frac{6457}{46656}
+ \frac{13}{648} \pi^2
+ \frac{7}{108} \zeta_3
- \frac{35}{2592} {L}- \frac{5}{432} {L}^2
+ \frac{1}{216} {L}^3
\,,\\
C^{(1)}_{\rlap{\scriptsize/}v} &{}={}&
- \frac{2}{3}
- \frac{1}{2} {L}\,,\nonumber\\
C^{G}_{\rlap{\scriptsize/}v} &{}={}&
- \frac{177}{64}
- \frac{5}{72} \pi^2
- \frac{1}{18} \pi^2 \ln2
- \frac{11}{36} \zeta_3
+ \left(
- \frac{79}{144}
- \frac{7}{108} \pi^2
\right) {L}+ \frac{13}{16} {L}^2
\,,\nonumber\\
C^{H}_{\rlap{\scriptsize/}v} &{}={}&
\frac{727}{432}
- \frac{1}{6} \pi^2
\,,\nonumber\\
C^{L}_{\rlap{\scriptsize/}v} &{}={}&
\frac{47}{288}
+ \frac{1}{36} \pi^2
+ \frac{5}{72} {L}- \frac{1}{24} {L}^2
\,,\nonumber\\
C^{GG}_{\rlap{\scriptsize/}v} &{}={}&
- \frac{62575}{62208}
- \frac{231253}{46656} \pi^2
- \frac{517}{324} \pi^2 \ln2
+ \frac{20}{81} \pi^2 \ln^22
+ \frac{5645}{1296} \zeta_3
\nonumber\\
&&{} + \frac{2089}{486} \pi^2 \zeta_3
- \frac{17347}{58320} \pi^4
- \frac{49435}{2592} \zeta_5
+ \frac{11}{54} \ln^42
+ \frac{44}{9} a_4
\nonumber\\
&&{}
+ \bigg(
\frac{115}{54}
- \frac{121}{648} \pi^2
+ \frac{1}{36} \pi^2 \ln2
+ \frac{37}{48} \zeta_3
- \frac{95}{1944} \pi^4
\bigg) {L}\nonumber\\
&&{}
+ \left(
\frac{2257}{576}
+ \frac{91}{432} \pi^2
\right) {L}^2
- \frac{13}{8} {L}^3
\,,\nonumber\\
C^{GH}_{\rlap{\scriptsize/}v} &{}={}&
\frac{2051}{96}
- \frac{24583}{2430} \pi^2
+ \frac{361}{27} \pi^2 \ln2
+ \frac{10}{81} \pi^2 \ln^22
- \frac{45869}{5184} \zeta_3
+ \frac{53}{96} \pi^2 \zeta_3
\nonumber\\
&&{}
- \frac{1}{20} \pi^4
- \frac{85}{32} \zeta_5
- \frac{10}{81} \ln^42
- \frac{80}{27} a_4
+ \left(
- \frac{727}{864}
+ \frac{1}{12} \pi^2
\right) {L}\,,\nonumber\\
C^{GL}_{\rlap{\scriptsize/}v} &{}={}&
\frac{24457}{46656}
+ \frac{5575}{8748} \pi^2
+ \frac{19}{324} \pi^2 \ln2
- \frac{1}{81} \pi^2 \ln^22
+ \frac{3181}{1944} \zeta_3
- \frac{379}{116640} \pi^4
\nonumber\\
&&{}
- \frac{1}{162} \ln^42
- \frac{4}{27} a_4
+ \left(
- \frac{319}{5184}
+ \frac{11}{972} \pi^2
+ \frac{83}{216} \zeta_3
\right) {L}\nonumber\\
&&{}
+ \left(
- \frac{469}{864}
- \frac{7}{648} \pi^2
\right) {L}^2
+ \frac{25}{144} {L}^3
\,,\nonumber\end{aligned}$$ $$\begin{aligned}
C^{HH}_{\rlap{\scriptsize/}v} &{}={}&
- \frac{5857}{7776}
+ \frac{1}{405} \pi^2
+ \frac{11}{18} \zeta_3
\,,\nonumber\\
C^{HL}_{\rlap{\scriptsize/}v} &{}={}&
- \frac{193}{432}
+ \frac{29}{648} \pi^2
\,,\nonumber\\
C^{LL}_{\rlap{\scriptsize/}v} &{}={}&
\frac{1751}{46656}
- \frac{13}{648} \pi^2
- \frac{7}{108} \zeta_3
+ \frac{35}{2592} {L}+ \frac{5}{432} {L}^2
- \frac{1}{216} {L}^3
\,,\\
C^{(1)}_{\gamma_\bot} &{}={}&
- \frac{4}{3}
- \frac{1}{2} {L}\,,\nonumber\\
C^{G}_{\gamma_\bot} &{}={}&
- \frac{14651}{1728}
- \frac{125}{648} \pi^2
- \frac{7}{54} \pi^2 \ln2
- \frac{7}{36} \zeta_3
+ \left(
- \frac{31}{144}
- \frac{7}{108} \pi^2
\right) {L}\nonumber\\
&&{} + \frac{13}{16} {L}^2
\,,\nonumber\\
C^{H}_{\gamma_\bot} &{}={}&
\frac{133}{144}
- \frac{5}{54} \pi^2
\,,\nonumber\\
C^{L}_{\gamma_\bot} &{}={}&
\frac{445}{864}
+ \frac{1}{36} \pi^2
+ \frac{5}{72} {L}- \frac{1}{24} {L}^2
\,,\nonumber\\
C^{GG}_{\gamma_\bot} &{}={}&
- \frac{5046967}{62208}
- \frac{361033}{46656} \pi^2
- \frac{1745}{324} \pi^2 \ln2
+ \frac{124}{243} \pi^2 \ln^22
+ \frac{3929}{1296} \zeta_3
\nonumber\\
&&{}
+ \frac{3463}{972} \pi^2 \zeta_3
- \frac{461}{3888} \pi^4
- \frac{43835}{2592} \zeta_5
+ \frac{163}{486} \ln^42
+ \frac{652}{81} a_4
\nonumber\\
&&{}
+ \left(
\frac{301}{54}
- \frac{53}{648} \pi^2
+ \frac{7}{108} \pi^2 \ln2
+ \frac{103}{144} \zeta_3
- \frac{95}{1944} \pi^4
\right) {L}\nonumber\\
&&{}
+ \left(
\frac{1945}{576}
+ \frac{91}{432} \pi^2
\right) {L}^2
- \frac{13}{8} {L}^3
\,,\nonumber\\
C^{GH}_{\gamma_\bot} &{}={}&
\frac{4133}{288}
- \frac{3385}{486} \pi^2
+ \frac{2069}{243} \pi^2 \ln2
+ \frac{26}{243} \pi^2 \ln^22
- \frac{10445}{5184} \zeta_3
\nonumber\\
&&{}
+ \frac{35}{288} \pi^2 \zeta_3
- \frac{233}{14580} \pi^4
- \frac{35}{96} \zeta_5
- \frac{26}{243} \ln^42
- \frac{208}{81} a_4
\nonumber\\
&&{}
+ \left(
- \frac{133}{288}
+ \frac{5}{108} \pi^2
\right) {L}\,,\nonumber\\
C^{GL}_{\gamma_\bot} &{}={}&
\frac{455461}{46656}
+ \frac{4937}{4374} \pi^2
+ \frac{169}{972} \pi^2 \ln2
- \frac{7}{243} \pi^2 \ln^22
+ \frac{5173}{1944} \zeta_3
\nonumber\\
&&{}
- \frac{2963}{116640} \pi^4
- \frac{7}{486} \ln^42
- \frac{28}{81} a_4
+ \left(
- \frac{1471}{5184}
+ \frac{11}{972} \pi^2
+ \frac{83}{216} \zeta_3
\right) {L}\nonumber\\
&&{}
+ \left(
- \frac{445}{864}
- \frac{7}{648} \pi^2
\right) {L}^2
+ \frac{25}{144} {L}^3
\,,\nonumber\\
C^{HH}_{\gamma_\bot} &{}={}&
- \frac{3641}{7776}
+ \frac{11}{1215} \pi^2
+ \frac{17}{54} \zeta_3
\,,\nonumber\\
C^{HL}_{\gamma_\bot} &{}={}&
- \frac{2545}{3888}
+ \frac{127}{1944} \pi^2
\,,\nonumber\\
C^{LL}_{\gamma_\bot} &{}={}&
- \frac{7993}{46656}
- \frac{7}{216} \pi^2
- \frac{7}{108} \zeta_3
+ \frac{35}{2592} {L}+ \frac{5}{432} {L}^2
- \frac{1}{216} {L}^3
\,,\\
C^{(1)}_{\gamma_\bot\rlap{\scriptsize/}v} &{}={}&
- \frac{4}{3}
- \frac{5}{6} {L}\,,\nonumber\\
C^{G}_{\gamma_\bot\rlap{\scriptsize/}v} &{}={}&
- \frac{20749}{1728}
- \frac{7}{24} \pi^2
- \frac{1}{6} \pi^2 \ln2
- \frac{5}{36} \zeta_3
+ \left(
- \frac{329}{144}
- \frac{7}{108} \pi^2
\right) {L}\nonumber\\
&&{}
+ \frac{215}{144} {L}^2
\,,\nonumber\end{aligned}$$ $$\begin{aligned}
C^{H}_{\gamma_\bot\rlap{\scriptsize/}v} &{}={}&
\frac{809}{648}
- \frac{7}{54} \pi^2
+ \frac{13}{108} {L}- \frac{1}{36} {L}^2
\,,\nonumber\\
C^{L}_{\gamma_\bot\rlap{\scriptsize/}v} &{}={}&
\frac{1745}{2592}
+ \frac{5}{108} \pi^2
+ \frac{41}{216} {L}- \frac{5}{72} {L}^2
\,,\nonumber\\
C^{GG}_{\gamma_\bot\rlap{\scriptsize/}v} &{}={}&
- \frac{21556403}{186624}
- \frac{488167}{46656} \pi^2
- \frac{757}{108} \pi^2 \ln2
+ \frac{142}{243} \pi^2 \ln^22
+ \frac{8357}{3888} \zeta_3
\nonumber\\
&&{}
+ \frac{16153}{3888} \pi^2 \zeta_3
- \frac{2447}{23328} \pi^4
- \frac{15925}{864} \zeta_5
+ \frac{113}{243} \ln^42
+ \frac{904}{81} a_4
\nonumber\\
&&{}
+ \left(
\frac{7871}{1728}
+ \frac{7}{108} \pi^2
+ \frac{5}{36} \pi^2 \ln2
+ \frac{61}{48} \zeta_3
- \frac{95}{1944} \pi^4
\right) {L}\nonumber\\
&&{}
+ \left(
\frac{22177}{1728}
+ \frac{301}{1296} \pi^2
\right) {L}^2
- \frac{4085}{1296} {L}^3
\,,\nonumber\\
C^{GH}_{\gamma_\bot\rlap{\scriptsize/}v} &{}={}&
\frac{125005}{5832}
- \frac{268333}{29160} \pi^2
+ \frac{301}{27} \pi^2 \ln2
+ \frac{1}{9} \pi^2 \ln^22
- \frac{22469}{5184} \zeta_3
\nonumber\\
&&{}
+ \frac{59}{288} \pi^2 \zeta_3
- \frac{73}{2430} \pi^4
- \frac{25}{32} \zeta_5
- \frac{1}{9} \ln^42
- \frac{8}{3} a_4
\nonumber\\
&&{}
+ \left(
- \frac{1117}{3888}
+ \frac{35}{324} \pi^2
+ \frac{5}{18} \zeta_3
\right) {L}- \frac{1225}{1296} {L}^2
+ \frac{1}{8} {L}^3
\,,\nonumber\\
C^{GL}_{\gamma_\bot\rlap{\scriptsize/}v} &{}={}&
\frac{211705}{15552}
+ \frac{28133}{17496} \pi^2
+ \frac{71}{324} \pi^2 \ln2
- \frac{1}{27} \pi^2 \ln^22
+ \frac{3347}{972} \zeta_3
\nonumber\\
&&{}
- \frac{4183}{116640} \pi^4
- \frac{1}{54} \ln^42
- \frac{4}{9} a_4
+ \left(
\frac{4091}{15552}
- \frac{13}{972} \pi^2
+ \frac{143}{216} \zeta_3
\right) {L}\nonumber\\
&&{}
+ \left(
- \frac{3845}{2592}
- \frac{7}{648} \pi^2
\right) {L}^2
+ \frac{5}{16} {L}^3
\,,\nonumber\\
C^{HH}_{\gamma_\bot\rlap{\scriptsize/}v} &{}={}&
- \frac{21281}{34992}
+ \frac{1}{81} \pi^2
+ \frac{31}{81} \zeta_3
+ \frac{1}{144} {L}+ \frac{13}{648} {L}^2
- \frac{1}{324} {L}^3
\,,\nonumber\\
C^{HL}_{\gamma_\bot\rlap{\scriptsize/}v} &{}={}&
- \frac{14567}{17496}
+ \frac{17}{216} \pi^2
+ \frac{2}{81} \zeta_3
+ \frac{1}{72} {L}+ \frac{13}{324} {L}^2
- \frac{1}{162} {L}^3
\,,\nonumber\\
C^{LL}_{\gamma_\bot\rlap{\scriptsize/}v} &{}={}&
- \frac{29309}{139968}
- \frac{89}{1944} \pi^2
- \frac{35}{324} \zeta_3
+ \frac{53}{2592} {L}+ \frac{41}{1296} {L}^2
- \frac{5}{648} {L}^3
\,,\end{aligned}$$ where $a_4=\mathop{\mathrm{Li}}\nolimits_4\left(\frac{1}{2}\right)$. The two-loop results, as well as the coefficients $C_\Gamma^{LL}$, are known from Ref. [@BG:95]; all remaining three-loop results are new.
It is instructive to re-write the matching coefficients for $m_c=0$ and $\mu=m$ via the leading $\beta$-function coefficient in HQET, $\beta_0'=11-\frac{2}{3}n_f'$. We obtain $$\begin{aligned}
C_1^{(2)}
&{}={}& 0.91 \beta_0' + 1.09 = 7.55 + 1.09 = 8.64\,,
\nonumber\\
C_{\rlap{\scriptsize/}v}^{(2)}
&{}={}& - 0.66 \beta_0' + 3.06 = - 5.47 + 3.06 = - 2.41\,,
\nonumber\\
C_{\gamma_\bot}^{(2)}
&{}={}& - 1.18 \beta_0' + 1.53 = - 9.87 + 1.53 = - 8.34\,,
\nonumber\\
C_{\gamma_\bot\rlap{\scriptsize/}v}^{(2)}
&{}={}& - 1.70 \beta_0' + 2.42 = - 14.13 + 2.42 = - 11.70
\label{nna2}\end{aligned}$$ at two loops, and the following results at three loops: $$\begin{aligned}
C_1^{(3)}
&{}={}& 0.93 \beta_0^{\prime2} + 9.04 \beta_0' - 38.16
= 64.74 + 75.34 - 38.16 = 101.92\,,
\nonumber\\
C_{\rlap{\scriptsize/}v}^{(3)}
&{}={}& - 0.54 \beta_0^{\prime2} - 1.29 \beta_0' + 29.74
= - 37.25 - 10.72 + 29.74 = - 18.23\,,
\nonumber\\
C_{\gamma_\bot}^{(3)}
&{}={}& - 1.28 \beta_0^{\prime2} - 5.56 \beta_0' + 45.34
= - 88.92 - 46.34 + 45.34 = - 89.92\,,
\nonumber\\
C_{\gamma_\bot\rlap{\scriptsize/}v}^{(3)}
&{}={}& - 1.78 \beta_0^{\prime2} - 7.63 \beta_0' + 63.22
= - 123.61 - 63.57 + 63.22
\nonumber\\
&{}={}& - 123.96\,.
\label{nna3}\end{aligned}$$ A method to estimate higher loop contributions called naive nonabelianization has been formulated in Ref. [@BG:95]. It is based on the fact that each polynomial in $n_f$ can be re-written as a polynomial in $\beta_0$. Usually it is relatively simple to calculate the term with the highest power of $n_f$. This means that we know the coefficient of the leading power of $\beta_0$. Neglecting subleading powers of $\beta_0$ we obtain an estimate of the full result. The two-loop corrections to the matching coefficients (\[nna2\]) were among the examples confirming naive nonabelianization [@BG:95]. At three loops we see that this prescription reproduces the correct signs and roughly the correct magnitude of the full results. In all cases we observe a compensation between the $\mathcal{O}(\beta_0')$ and $\mathcal{O}(1)$ terms. In the case of $C_{\gamma_\bot}^{(3)}$ and $C_{\gamma_\bot\rlap{\scriptsize/}v}^{(3)}$ this compensation is almost complete, and naive nonabelianization works surprisingly well.
Since the expressions for the charm-mass dependence are quite involved and not completely expressed in terms of HPLs of $x$ (the $\mathcal{O}(\varepsilon)$ terms of the master integrals 5.2, 5.2a, 5.3, 5.3a are known analytically only as truncated series in $x$, see Table 3 in Ref. [@BGSS:08]), we refrain from listing them here. Instead, we present an expansion of our results to the second order in $x$. The results in terms of the master integrals and expansions to higher orders can be obtained from the web page [@web].
Our results for the $m_c$-dependent coefficients read $$\begin{aligned}
C_1^{M}(x) &{}={}& C_1^{L}
+\frac{1}{8}\pi^2\, x
+\left(
\ln x
+\frac{1}{2}
\right) x^2
+\mathcal{O}(x^3)\,,
\nonumber\\
C_1^{GM}(x) &{}={}& C_1^{GL}
+\left(
\frac{4361}{1296} \pi^2
-\frac{119}{108} \pi^2 \ln 2
+\frac{13}{216} \pi^3
+\frac{1}{16} \pi^2 {L}-\frac{9}{8} \pi^2 \ln x
\right) x
\nonumber\\
&&{}
+\biggl[
\frac{48493}{5184}
+\frac{395}{432} \pi^2
+\frac{7}{36} \pi^2 \ln 2
+\frac{27}{16} \zeta_3
-\pi^2 \zeta_3
-\frac{49}{720} \pi^4
-\frac{5}{2} \zeta_5
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
+\frac{1}{4} {L}+\left(
\frac{6239}{432}
+\frac{1}{4} \pi^2
+\frac{15}{8} \zeta_3
-\frac{19}{180} \pi^4
+\frac{1}{2} {L}\right) \ln x
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
-\frac{37}{9} \ln^2 x
\biggr] x^2
+\mathcal{O}(x^3)\,,
\nonumber\end{aligned}$$ $$\begin{aligned}
C_1^{HM}(x) &{}={}& C_1^{HL}
+\left(
-\frac{517}{1350}
+\frac{1}{18} \pi^2
-\frac{1}{15} \ln x
\right) x^2
+ \mathcal{O}(x^3)\,,
\nonumber\\
C_1^{LM}(x) &{}={}& 2 C_1^{LL}
+\left(
-\frac{7}{36} \pi^2
+\frac{1}{6} \pi^2 \ln 2
+\frac{1}{12} \pi^2 \ln x
\right) x
\nonumber\\
&&{}
+\left(
-\frac{5}{18}
-\frac{1}{18} \pi^2
-\frac{2}{9} \ln x
+\frac{1}{3} \ln^2 x
\right) x^2
+ \mathcal{O}(x^3)\,,
\nonumber\\
C_1^{MM}(x) &{}={}& C_1^{LL}
+\left(
-\frac{1}{45} \pi^2
+\frac{1}{12} \pi^2 \ln x
\right) x
\nonumber\\
&&{}
+\left(
-\frac{19}{36}
-\frac{1}{18} \pi^2
-\frac{2}{9} \ln x
+\frac{1}{3}\ln^2 x
\right) x^2
+\mathcal{O}(x^3)\,,
\\
C_{\rlap{\scriptsize/}v}^{M}(x) &{}={}& C_{\rlap{\scriptsize/}v}^{L}
-\frac{1}{24} \pi^2\, x
+\left(
\frac{3}{2}
+\ln x
\right) x^2
+\mathcal{O}(x^3)\,,
\nonumber\\
C_{\rlap{\scriptsize/}v}^{GM}(x) &{}={}& C_{\rlap{\scriptsize/}v}^{GL}
+\biggl(
-\frac{35333}{3888} \pi^2
+\frac{4439}{324} \pi^2 \ln 2
-\frac{13}{648} \pi^3
+\frac{1}{48} \pi^2 {L}\nonumber\\
&&\hphantom{C_{\rlap{\scriptsize/}v}^{GL}+\biggl(\biggr.}
+\frac{17}{24} \pi^2 \ln x
\biggr) x
\nonumber\\
&&{}
+\biggl[
\frac{86509}{5184}
+\frac{863}{432} \pi^2
+\frac{7}{36} \pi^2 \ln 2
+\frac{115}{16} \zeta_3
-\frac{7}{4} \pi^2 \zeta_3
-\frac{49}{720} \pi^4
\nonumber\\
&&\hphantom{{}=\biggl[\biggr.}
-5 \zeta_5
-\frac{3}{4} {L}+\left(
\frac{7391}{432}
+\frac{7}{4} \pi^2
+\frac{15}{8} \zeta_3
-\frac{17}{90} \pi^4
-\frac{1}{2} {L}\right) \ln x
\nonumber\\
&&\hphantom{{}=\biggl[\biggr.}
-\frac{37}{9} \ln^2 x
\biggr] x^2
+\mathcal{O}(x^3)\,,
\nonumber\\
C_{\rlap{\scriptsize/}v}^{HM}(x) &{}={}& C_{\rlap{\scriptsize/}v}^{HL}
+\left(
-\frac{817}{1350}
+\frac{1}{18} \pi^2
-\frac{1}{15} \ln x
\right) x^2
+\mathcal{O}(x^3)\,,
\nonumber\\
C_{\rlap{\scriptsize/}v}^{LM}(x) &{}={}& 2 C_{\rlap{\scriptsize/}v}^{LL}
+\left(\frac{7}{108} \pi^2
-\frac{1}{18} \pi^2 \ln 2
-\frac{1}{36} \pi^2 \ln x
\right) x
\nonumber\\
&&{}
+\left(
-\frac{1}{2}
-\frac{1}{18} \pi^2
-\frac{2}{9} \ln x
+\frac{1}{3} \ln^2 x
\right) x^2
+\mathcal{O}(x^3)\,,
\nonumber\\
C_{\rlap{\scriptsize/}v}^{MM}(x) &{}={}& C_{\rlap{\scriptsize/}v}^{LL}
+\left(
\frac{1}{135} \pi^2
-\frac{1}{36} \pi^2 \ln x
\right) x
\nonumber\\
&&{}
+\left(
-\frac{3}{4}
-\frac{1}{18} \pi^2
-\frac{2}{9} \ln x
+\frac{1}{3} \ln^2 x
\right) x^2
+\mathcal{O}(x^3)\,,
\\
C_{\gamma_\bot}^{M}(x) &{}={}& C_{\gamma_\bot}^{L}
-\frac{1}{24} \pi^2\, x
+\left(
\frac{3}{2}
+\ln x
\right) x^2
+\mathcal{O}(x^3)\,,
\nonumber\\
C_{\gamma_\bot}^{GM}(x) &{}={}& C_{\gamma_\bot}^{GL}
+\biggl[
-\frac{130327}{11664} \pi^2
+\frac{14089}{972} \pi^2 \ln 2
-\frac{143}{1944} \pi^3
+\frac{11}{144} \pi^2 {L}\nonumber\\
&&\hphantom{C_{\gamma_\bot}^{GL}+\biggl[\biggr.}
+\frac{409}{216} \pi^2 \ln x
\biggr] x
\nonumber\end{aligned}$$ $$\begin{aligned}
&&{}
+\biggl[
\frac{24217}{5184}
+\frac{1517}{1296} \pi^2
-\frac{7}{108} \pi^2 \ln 2
+\frac{595}{144} \zeta_3
-\frac{7}{12} \pi^2 \zeta_3
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
-\frac{19}{2160} \pi^4
-\frac{5}{12} {L}\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
+\left(
-\frac{79}{144}
+\frac{595}{324} \pi^2
-\frac{1}{8} \zeta_3
-\frac{7}{135} \pi^4
+\frac{1}{6} {L}\right) \ln x
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
+\frac{7}{9} \ln^2 x
\biggr] x^2
+\mathcal{O}(x^3)\,,
\nonumber\\
C_{\gamma_\bot}^{HM}(x) &{}={}& C_{\gamma_\bot}^{HL}
+\left(
\frac{349}{4050}
-\frac{1}{54} \pi^2
-\frac{1}{15} \ln x
\right) x^2
+\mathcal{O}(x^3)\,,
\nonumber\\
C_{\gamma_\bot}^{LM}(x) &{}={}& 2 C_{\gamma_\bot}^{LL}
+\left(
\frac{77}{324} \pi^2
-\frac{11}{54} \pi^2 \ln 2
-\frac{11}{108} \pi^2 \ln x
\right) x
\nonumber\\
&&{}
+\left(
-\frac{19}{54}
+\frac{1}{54} \pi^2
+\frac{2}{27} \ln x
-\frac{1}{9} \ln^2 x
\right) x^2
+\mathcal{O}(x^3)\,,
\nonumber\\
C_{\gamma_\bot}^{MM}(x) &{}={}& C_{\gamma_\bot}^{LL}
+\left(
\frac{11}{405} \pi^2
-\frac{11}{108} \pi^2 \ln x
\right) x
\nonumber\\
&&{}
+\left(
-\frac{29}{108}
+\frac{1}{54} \pi^2
+\frac{2}{27} \ln x
-\frac{1}{9} \ln^2 x
\right) x^2
+\mathcal{O}(x^3)\,,
\\
C_{\gamma_\bot\rlap{\scriptsize/}v}^{M}(x) &{}={}& C_{\gamma_\bot\rlap{\scriptsize/}v}^{L}
-\frac{5}{24} \pi^2\, x
+\left(
\frac{7}{6}
-\frac{1}{3} \ln x
\right) x^2
+\mathcal{O}(x^3)\,,
\nonumber\\
C_{\gamma_\bot\rlap{\scriptsize/}v}^{GM}(x) &{}={}& C_{\gamma_\bot\rlap{\scriptsize/}v}^{GL}
+\biggl(
-\frac{59221}{3888} \pi^2
+\frac{6295}{324} \pi^2 \ln 2
-\frac{65}{648} \pi^3
+\frac{25}{144} \pi^2 {L}\nonumber\\
&&\hphantom{C_{\gamma_\bot\rlap{\scriptsize/}v}^{GL}+\biggl(\biggr.}
+\frac{541}{216} \pi^2 \ln x
\biggr) x
\nonumber\\
&&{}
+\biggl[
\frac{35653}{5184}
+\frac{631}{432} \pi^2
-\frac{7}{108} \pi^2 \ln 2
+\frac{313}{48} \zeta_3
-\frac{5}{6} \pi^2 \zeta_3
-\frac{19}{2160} \pi^4
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
-\frac{5}{6} \zeta_5
-\frac{35}{36} {L}\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
+\left(
\frac{575}{432}
+\frac{9}{4} \pi^2
-\frac{1}{8} \zeta_3
-\frac{43}{540} \pi^4
+\frac{5}{18} {L}\right) \ln x
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
+\frac{1}{2} \ln^2 x
\biggr] x^2
+\mathcal{O}(x^3)\,,
\nonumber\\
C_{\gamma_\bot\rlap{\scriptsize/}v}^{HM}(x) &{}={}& C_{\gamma_\bot\rlap{\scriptsize/}v}^{HL}
+\left(
\frac{49}{4050}
-\frac{1}{54} \pi^2
-\frac{1}{15} \ln x
\right) x^2
+\mathcal{O}(x^3)\,,
\nonumber\\
C_{\gamma_\bot\rlap{\scriptsize/}v}^{LM}(x) &{}={}& 2 C_{\gamma_\bot\rlap{\scriptsize/}v}^{LL}
+\left(
\frac{35}{108} \pi^2
-\frac{5}{18} \pi^2 \ln 2
-\frac{5}{36} \pi^2 \ln x
\right) x
\nonumber\\
&&{}
+\left(
-\frac{23}{54}
+\frac{1}{54} \pi^2
+\frac{2}{27} \ln x
-\frac{1}{9} \ln^2 x
\right) x^2
+\mathcal{O}(x^3)\,,
\nonumber\\
C_{\gamma_\bot\rlap{\scriptsize/}v}^{MM}(x) &{}={}& C_{\gamma_\bot\rlap{\scriptsize/}v}^{LL}
+\left(
\frac{1}{27} \pi^2
-\frac{5}{36} \pi^2 \ln x
\right) x
\nonumber\\
&&{}
+\left(
-\frac{37}{108}
+\frac{1}{54} \pi^2
+\frac{2}{27} \ln x
-\frac{1}{9} \ln^2 x
\right) x^2
+\mathcal{O}(x^3)\,.\end{aligned}$$
Meson matrix elements {#S:Meson}
=====================
We are now in the position to apply our results to the matrix elements between a $B$ or $B^*$ meson with momentum $p$ and the vacuum. They are defined through $$\begin{aligned}
{\langle}0| \left(\bar{q} \gamma_5^{\mbox{\scriptsize AC}} Q\right)_\mu |B{\rangle} &{}={}&
- i m_B f_B^P(\mu)\,,
\nonumber\\
{\langle}0| \bar{q} \gamma^\alpha \gamma_5^{\mbox{\scriptsize AC}} Q |B{\rangle} &{}={}&
i f_B p^\alpha\,,
\nonumber\\
{\langle}0| \bar{q} \gamma^\alpha Q |B^*{\rangle} &{}={}&
i m_{B^*} f_{B^*} e^\alpha\,,
\nonumber\\
{\langle}0| \left(\bar{q} \sigma^{\alpha\beta} Q\right)_\mu |B^*{\rangle} &{}={}&
f_{B^*}^{T}(\mu) (p^\alpha e^\beta - p^\beta e^\alpha)\,,
\label{Meson:matel}\end{aligned}$$ where $e^\alpha$ is the $B^*$ polarization vector. The corresponding HQET matrix elements (at $m\to\infty$) in the $v$ rest frame are $$\begin{aligned}
{\langle}0| \left(\bar{q}\gamma_5^{\mbox{\scriptsize AC}} Q_v\right)_\mu
|B(\vec{k}\,){\rangle}\strut_{\mbox{\scriptsize nr}} &{}={}&
- i F(\mu)\,,
\nonumber\\
{\langle}0| \left(\bar{q} \vec{\gamma} Q_v\right)_\mu
|B^*(\vec{k}\,){\rangle}\strut_{\mbox{\scriptsize nr}} &{}={}&
i F(\mu) \vec{e}\,,
\label{Meson:HQET}\end{aligned}$$ where the single-meson states are normalized by the non-relativistic condition $$\strut_{\mbox{\scriptsize nr}}{\langle}B(\vec{k}\,')|B(\vec{k}\,){\rangle}\strut_{\mbox{\scriptsize nr}}
= (2\pi)^3 \delta(\vec{k}\,'-\vec{k}\,)\,.$$ We also remind the reader that $\bar{q} \Gamma \rlap/v Q_v = \bar{q} \Gamma Q_v$, so that there are only two currents. These two matrix elements are characterized by a single hadronic parameter $F(\mu)$ due to the heavy-quark spin symmetry. From Eq. (\[Match:div\]) we have [@BG:95] $$\frac{f_B^P(\mu)}{f_B} = \frac{m_B}{m(\mu)}\,.
\label{Meson:mm}$$ Here $m_B=m+\bar{\Lambda} + \mathcal{O}(\Lambda_{\mbox{\scriptsize QCD}}^2/m)$ where $\bar{\Lambda}$ is the residual energy of the ground-state $B$ meson in the limit $m\to\infty$. Neglecting $1/m$ corrections, we see that this equation coincides with (\[Match:mm\]). We have checked that our results agree with the known formulas for $m(\mu)/m$ at $m_c=0$ [@MR:00a] and with $m_c$ corrections taken into account [@BGSS:07].
Using the expressions of Section \[S:Coefs\] we find the following results for the ratios of decay constants. Again, we present an expansion to the second order in $x$ for the charm mass dependent terms. For the numerical evaluations we use an expansion to the eighth order in $x$ and the values $\alpha_s^{(4)}(m_b) =
0.2163$ and $x=0.3$. $$\begin{aligned}
\frac{f_{B^*}}{f_B} &{}={}&
\frac{C_{\gamma_\bot}(m_b)}{C_{\rlap{\scriptsize/}v}(m_b)}
+ \mathcal{O}\left(\frac{\Lambda_{\mbox{\scriptsize QCD}}}{m_b}\right)
= 1 - \frac{2}{3} \frac{\alpha_s^{(4)}(m_b)}{\pi}
\nonumber\\
&&{} + \Bigl[ -7.749 - 0.028 n_h + 0.352 n_l
\nonumber\\
&&{} + \bigl(0.352 - 1.097 x -0.667 x^2 - 1.333 x^2 \ln x\bigr) n_m
\Bigr]
\left(\frac{\alpha_s^{(4)}(m_b)}{\pi}\right)^2
\nonumber\\
&&{}
+ \Bigl[ - 129.211 - 0.198 n_h + 14.294 n_l
- 0.006 n_h^2 - 0.005 n_h n_l
\nonumber\\
&&{} - 0.331 n_l^2
+ \bigl( 14.294 - 17.816 x + 11.697 x \ln x
\nonumber\\
&&\qquad{} - 0.273 x^2 -6.082 x^2 \ln x + 4.889 x^2 \ln^2 x \bigr) n_m
\nonumber\\
&&{}
+ \bigl( -0.005 - 0.040 x^2 \bigr) n_h n_m
\nonumber\\
&&{}
+ \bigl( -0.661 + 0.692 x -0.731 x \ln x
\nonumber\\
&&\qquad{} + 0.879 x^2 + 0.296 x^2 \ln x -0.444 x^2 \ln^2 x \bigr) n_l n_m
\nonumber\\
&&{}
+ \bigl( - 0.331 + 0.195 x - 0.731 x \ln x
\nonumber\\
&&\qquad{} + 1.213 x^2 + 0.296 x^2 \ln x - 0.444 x^2 \ln^2 x \bigr) n_m^2
\Bigr]
\left(\frac{\alpha_s^{(4)}(m_b)}{\pi}\right)^3
\nonumber\\
&&{} + \mathcal{O}\left(\alpha_s^4,\frac{\Lambda_{\mbox{\scriptsize QCD}}}{m_b}\right)
\nonumber\\
&{}={}& 1 - \frac{2}{3} \frac{\alpha_s^{(4)}(m_b)}{\pi}
- (6.370 + 0.189) \left(\frac{\alpha_s^{(4)}(m_b)}{\pi}\right)^2
\nonumber\\
&&{}
- (77.549 + 6.575) \left(\frac{\alpha_s^{(4)}(m_b)}{\pi}\right)^3
+ \mathcal{O}\left(\alpha_s^4,\frac{\Lambda_{\mbox{\scriptsize QCD}}}{m_b}\right)
\nonumber\\
&{}={}& 1 - 0.046 - (0.030 + 0.001) - (0.025 + 0.002)
\nonumber\\
&{}={}& 0.899 - 0.003 = 0.896
+ \mathcal{O}\left(\alpha_s^4,\Lambda_{\mbox{\scriptsize QCD}}/m_b\right)\,.\end{aligned}$$ In the second line from the bottom the corrections from tree level, first, second and third order in $\alpha_s$ are given separately. Also the contributions stemming from the finite charm mass are separated (the second number in the parentheses). In the first part of the last line, the $m_c$ correction is also separated. Power corrections $\mathcal{O}(\Lambda_{\mbox{\scriptsize QCD}}/m_b)$ are discussed in Refs. [@N:92; @CGM:03] and amount to several per cent.
For the second ratio we obtain for $\mu=m_b$ $$\begin{aligned}
\frac{f_{B^*}^T(m_b)}{f_{B^*}} &{}={}&
\frac{C_{\rlap{\scriptsize/}v\gamma_\bot}(m_b)}{C_{\gamma_\bot}(m_b)}
+ \mathcal{O}\left(\frac{\Lambda_{\mbox{\scriptsize QCD}}}{m_b}\right)
\nonumber\end{aligned}$$ $$\begin{aligned}
&{}={}& 1 +
\Bigl[ - 4.690 - 0.041 n_h + 0.341 n_l
\nonumber\\
&&{} + \bigl(0.341 - 0.548 x + 0.333 x^2 \bigr) n_m
\Bigr]
\left(\frac{\alpha_s^{(4)}(m_b)}{\pi}\right)^2
\nonumber\\
&&{}
+ \Bigl[ - 70.923 - 0.666 n_h + 9.175 n_l - 0.026 n_h^2 - 0.016 n_h n_l
\nonumber\\
&&{} - 0.222 n_l^2
+ \bigl(9.175 - 7.859 x + 6.031 x \ln x
\nonumber\\
&&\qquad{} + 4.555 x^2 + 3.256 x^2 \ln x - 0.278 x^2 \ln^2 x \bigr) n_m
\nonumber\\
&&{} + \bigl( - 0.016 - 0.074 x^2 \bigr) n_h n_m
\nonumber\\
&&{} + \bigl( - 0.444 + 0.346 x - 0.366 x \ln x - 0.074 x^2 \bigr) n_l n_m
\nonumber\\
&&{} + \bigl( - 0.222 + 0.097 x -0.366 x \ln x - 0.074 x^2 \bigr) n_m^2
\Bigr]
\left(\frac{\alpha_s^{(4)}(m_b)}{\pi}\right)^3
\nonumber\\
&&{} + \mathcal{O}\left(\alpha_s^4,\frac{\Lambda_{\mbox{\scriptsize QCD}}}{m_b}\right)
\nonumber\\
&{}={}& 1 - (3.367 + 0.142) \left(\frac{\alpha_s^{(4)}(m_b)}{\pi}\right)^2
\nonumber\\
&&{} - (38.530 + 3.973) \left(\frac{\alpha_s^{(4)}(m_b)}{\pi}\right)^3
+ \mathcal{O}\left(\alpha_s^4,\frac{\Lambda_{\mbox{\scriptsize QCD}}}{m_b}\right)
\nonumber\\
&{}={}& 1 - (0.016 + 0.001) - (0.013 + 0.001) \nonumber\\
&{}={}& 0.971 - 0.002 = 0.969
+ \mathcal{O}\left(\alpha_s^4,\Lambda_{\mbox{\scriptsize QCD}}/m_b\right)\,.\end{aligned}$$
The coefficients of these perturbative series (at $m_c=0$ and $\mu=m_b$) can be re-written via $\beta_0'$: $$\begin{aligned}
\left(\frac{f_{B^*}}{f_B}\right)^{(2)} &{}={}&
- 0.53 \beta_0' - 1.97 = - 4.40 - 1.97 = - 6.37\,,
\nonumber\\
\left(\frac{f_{B^*}^T(m_b)}{f_{B^*}}\right)^{(2)} &{}={}&
- 0.51 \beta_0' + 0.89 = - 4.26 + 0.89 = - 3.37\,,
\nonumber\\
\left(\frac{f_{B^*}}{f_B}\right)^{(3)} &{}={}&
- 0.74 \beta_0^{\prime2} - 5.06 \beta_0' + 16.33
\nonumber\\
&{}={}& - 51.67 - 42.21 + 16.33 = - 77.55\,,
\nonumber\\
\left(\frac{f_{B^*}^T(m_b)}{f_{B^*}}\right)^{(3)} &{}={}&
- 0.50 \beta_0^{\prime2} - 2.75 \beta_0' + 19.07
\nonumber\\
&{}={}& - 34.69 - 22.91 + 19.07 = - 38.53\,.\end{aligned}$$ Again, naive nonabelianization [@BG:95] works quite well for these ratios predicting the correct sign and order of magnitude.
Asymptotics of the perturbative coefficients for the matching coefficients at a large number of loops $l\gg1$ have been investigated in Ref. [@CGM:03] in a model-independent way. The results contain three unknown normalization constants $N_{0,1,2}\sim1$. Let us in the following assume that the number of loops $l=3$ is much larger than one and compare our results with Ref. [@CGM:03]. The asymptotics of the perturbative coefficients for $f_{B^*}/f_B$ contain $N_0$ and $N_2$ (see (5.6) in Ref. [@CGM:03]); in the case of $m/\hat{m}$ it contains only $N_0$ (see (5.9) in Ref. [@CGM:03])[^5]: $$\begin{aligned}
\left(\frac{f_{B^*}}{f_B}\right)^{(n+1)}_{L=-5/3} &{}={}&
- \frac{14}{27} \left\{ 1 + \mathcal{O}\left(\frac{1}{n}\right)
+ \frac{2}{7} \left(\frac{50}{3} n\right)^{-9/25}
\left[1 + \mathcal{O}\left(\frac{1}{n}\right) \right]
\frac{N_2}{N_0} \right\}
\nonumber\\
&&{}\times\left(\frac{m}{\hat{m}}\right)^{(n+1)}_{L=-5/3}\,.
\label{asympt}\end{aligned}$$ The coefficient of $N_2/N_0$ is about $0.08$ at $n=2$, and it seems reasonable to neglect this contribution. Neglecting also $1/n$ corrections, we obtain [@CGM:03] $$\left(\frac{f_{B^*}}{f_B}\right)^{(3)}_{L=-5/3} =
- \frac{14}{27} \cdot 56.37 = - 29.23\,.
\label{predict}$$ Our exact result $$\left(\frac{f_{B^*}}{f_B}\right)^{(3)}_{L=-5/3} = -37.787
\label{exact}$$ agrees with this prediction reasonably well, thus confirming the simple relation (5.14) in Ref. [@CGM:03]. However, $1/n$ corrections are large and tend to break this agreement. It is natural to expect that $1/n^2$ (and higher) corrections are also substantial at $n=2$.
Conclusion {#S:Conc}
==========
We have calculated the N$^3$LO corrections to the matching coefficients of heavy–light currents in HQET. Our result takes into account effects due to the mass of the charm quark. Strictly speaking, our results should be used together with the N$^3$LO $\beta$-function [@vanRitbergen:1997va; @Czakon:2004bu] and anomalous dimensions of both the QCD currents and the HQET one. The four-loop anomalous dimensions are known for some of the QCD currents (for $\Gamma=\rlap/v$, $\gamma_\bot$ the anomalous dimension is exactly zero; for $\Gamma=1$ it is just the anomalous dimension of the $\overline{\mbox{MS}}$ mass [@Chetyrkin:1997dh; @Vermaseren:1997fq] with a minus sign). However, the four-loop anomalous dimension of the HQET current is not known (this anomalous dimension does not depend on the Dirac structure $\Gamma$).[^6] The effect of this unknown anomalous dimension cancels in ratios of $B$-meson decay constants, $f_{B^*}/f_B$ and $f_{B^*}^T(\mu)/f_{B^*}$, discussed in Sect. \[S:Meson\].
All the previous experience shows that contributions from anomalous dimensions are numerically much smaller than from matching coefficients. Matching coefficients have renormalon singularities at the Borel parameter $u=1/2$, this is the position closest to the origin out of all possible ones; this means that the factorial growth of their perturbative coefficients is fastest among all possible variants. On the other hand, it is generally believed that anomalous dimensions have no renormalon singularities, and their perturbative series have finite convergence radii.
Only $f_B$ has been measured experimentally [@exp]. Our results can be used for predicting the $B^*$ decay constants. We find that the perturbative series for $f_{B^*}/f_B$ and $f_{B^*}^T/f_{B^*}$ converge very slowly at best. The effects due to the charm-quark mass are small and of the order of $10^{-3}$.
The matching coefficients $C_\Gamma$ can be used for extraction of $B$ (and $B^*$) decay constants from lattice HQET simulations (see Ref. [@lattice:09] for recent reviews), or from HQET sum rules.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to T. van Ritbergen for the package `SHELL3` [@MR:00]. We thank Rainer Sommer for communication. This work was supported by the DFG through SFB/TR9 and the Graduiertenkolleg “Hochenergiephysik und Teilchenastrophysik”, by NSERC and the Alberta Ingenuity Fund. The Feynman diagrams were drawn with the help of `Axodraw` [@Vermaseren:1994je] and `JaxoDraw` [@Binosi:2003yf].
[99]{}
M. Neubert, Phys. Rept. **245** (1994) 259 \[arXiv:hep-ph/9306320\].
A.V. Manohar, M.B. Wise, *Heavy Quark Physics*, Cambridge University Press (2000).
A.G. Grozin, *Heavy Quark Effective Theory*, Springer Tracts Mod. Phys. **201** (2004).
E. Eichten, B. Hill, Phys. Lett. **B 234** (1990) 511.
D.J. Broadhurst, A.G. Grozin, Phys. Rev. **D 52** (1995) 4082 \[arXiv:hep-ph/9410240\].
A.G. Grozin, Phys. Lett. **B 445** (1998) 165 \[arXiv:hep-ph/9810358\].
M. Neubert, C.T. Sachrajda, Nucl. Phys. **B 438** (1995) 235 \[arXiv:hep-ph/9407394\];\
M. Neubert, Phys. Rev. **D 51** (1995) 5924 \[arXiv:hep-ph/9412265\].
F. Campanario, A.G. Grozin, T. Mannel, Nucl. Phys. **B 663** (2003) 280; Erratum: ibid. **B 670** (2003) 331 \[arXiv:hep-ph/0303052\].
E. Gamiz, Proceedings of Science (Lattice 2008) 014 \[arXiv:0811.4146 \[hep-lat\]\];\
C. Aubin, arXiv:0909.2686 \[hep-lat\].
M. Della Morte *et al*., JHEP **02** (2008) 078 \[arXiv:0710.2201 \[hep-lat\]\];\
M. Della Morte *et al*., Proceedings of Science (Lattice 2008) 226 \[arXiv:0810.3166 \[hep-lat\]\];\
B. Blossier *et al*., arXiv:0911.1568 \[hep-lat\].
D.J. Broadhurst, A.G. Grozin, Phys. Lett. **B 274** (1992) 421 \[arXiv:hep-ph/9908363\];\
E. Bagan, P. Ball, V.M. Braun, H.G. Dosch, Phys. Lett. **B 278** (1992) 457.
S.A. Larin, J.A.M. Vermaseren, Phys. Lett. **B 259** (1991) 345;\
S.A. Larin, in *Quarks-92*, ed. D.Yu. Grigoriev, V.A. Matveev, V.A. Rubakov, P.G. Tinyakov, World Scientific (2003) p. 201 \[arXiv:hep-ph/9302240\].
A.G. Grozin, A.V. Smirnov, V.A. Smirnov, JHEP **11** (2006) 022 \[arXiv:hep-ph/0609280\].
K.G. Chetyrkin, B.A. Kniehl, M. Steinhauser, Nucl. Phys. **B 510** (1998) 61 \[arXiv:hep-ph/9708255\].
K.G. Chetyrkin, A.G. Grozin, Nucl. Phys. **B 666** (2003) 289 \[arXiv:hep-ph/0303113\].
K. Melnikov, T. van Ritbergen, Nucl. Phys. **B 591** (2000) 515 \[arXiv:hep-ph/0005131\].
P. Marquard, L. Mihaila, J.H. Piclum, M. Steinhauser, Nucl. Phys. **B 773** (2007) 1 \[arXiv:hep-ph/0702185\].
S. Bekavac, A.G. Grozin, D. Seidel, M. Steinhauser, JHEP **10** (2007) 006 \[arXiv:0708.1729 \[hep-ph\]\].
S. Bekavac, A.G. Grozin, D. Seidel, V.A. Smirnov, Nucl. Phys. **B 819** (2009) 183 \[arXiv:0903.4760 \[hep-ph\]\].
J.A. Gracey, Phys. Lett. **B 488** (2000) 175 \[arXiv:hep-ph/0007171\].
A.G. Grozin, P. Marquard, J.H. Piclum, M. Steinhauser, Nucl. Phys. **B 789** (2008) 277 \[arXiv:0707.1388 \[hep-ph\]\].
K.G. Chetyrkin, M. Steinhauser, Phys. Rev. Lett. **83** (1999) 4001 \[arXiv:hep-ph/9907509\]; Nucl. Phys. **B 573** (2000) 617 \[arXiv:hep-ph/9911434\].
K. Melnikov, T. van Ritbergen, Phys. Lett. **B 482** (2000) 99 \[arXiv:hep-ph/9912391\].
P. Nogueira, J. Comput. Phys. **105** (1993) 279.
R. Harlander, T. Seidensticker, M. Steinhauser, Phys. Lett. **B 426** (1998) 125 \[arXiv:hep-ph/9712228\];\
T. Seidensticker, arXiv:hep-ph/9905298.
K.G. Chetyrkin, F.V. Tkachov, Nucl. Phys. **B 192** (1981) 159. J.A.M. Vermaseren, arXiv:math-ph/0010025. P. Marquard, D. Seidel, unpublished.
S. Laporta, E. Remiddi, Phys. Lett. **B 379** (1996) 283 \[arXiv:hep-ph/9602417\];\
S. Laporta, Int. J. Mod. Phys. **A 15** (2000) 5087 \[arXiv:hep-ph/0102033\]. P. Marquard, J.H. Piclum, D. Seidel, M. Steinhauser, Phys. Lett. **B 678** (2009) 269 \[arXiv:0904.0920 \[hep-ph\]\].
A.V. Smirnov, M.N. Tentyukov, Comput. Phys. Commun. **180** (2009) 735 \[arXiv:0807.4129 \[hep-ph\]\]. http://www-ttp.particle.uni-karlsruhe.de/Progdata/ttp09/ttp09-41/.
S. Bekavac, A.G. Grozin, D. Seidel, M. Steinhauser, arXiv:0906.0130 \[hep-ph\].
S. Bekavac, *Charmmasseneffekte in den Renormierungskonstanten und den statischen Formfaktoren des Bottom-Quarks*, PhD-Thesis, Karlsruhe, January 2009; Shaker Verlag, Aachen, May 2009.
E. Remiddi, J.A.M. Vermaseren, Int. J. Mod. Phys. **A 15** (2000) 725 \[arXiv:hep-ph/9905237\]. D. Maître, Comput. Phys. Commun. **174** (2006) 222 \[arXiv:hep-ph/0507152\]; arXiv:hep-ph/0703052. M. Neubert, Phys. Rev. **D 46** (1992) 1076.
T. van Ritbergen, J.A.M. Vermaseren, S.A. Larin, Phys. Lett. **B 400** (1997) 379 \[arXiv:hep-ph/9701390\]. M. Czakon, Nucl. Phys. **B 710** (2005) 485 \[arXiv:hep-ph/0411261\]. K.G. Chetyrkin, Phys. Lett. **B 404** (1997) 161 \[arXiv:hep-ph/9703278\]. J.A.M. Vermaseren, S.A. Larin, T. van Ritbergen, Phys. Lett. **B 405** (1997) 327 \[arXiv:hep-ph/9703284\]. K. Ikado *et al.* (Belle Collaboration), Phys. Rev. Lett. **97** (2006) 251802 \[arXiv:hep-ex/0604018\];\
B. Aubert *et al.* (BaBar Collaboration), Phys. Rev. **D 77** (2008) 011107 \[arXiv:0708.2260 \[hep-ex\]\]. J.A.M. Vermaseren, Comput. Phys. Commun. **83** (1994) 45. D. Binosi, L. Theussl, Comput. Phys. Commun. **161** (2004) 76 \[arXiv:hep-ph/0309015\].
[^1]: The results incorporating this correction are given by formulae (5.65–5.68) and Table 5.1 in [@G:04]. Note a misprint in this table: in the row for $\Gamma=\gamma^1$, the term with $C_F$ in the square bracket, $-1453/48$, should be $+1453/48$.
[^2]: Both on-shell matrix elements of the renormalized currents are UV-finite. Both contain IR divergences, which are the same on the left- and right-hand sides of (\[Match:match\]), yielding a finite $C_\Gamma(\mu)$.
[^3]: To keep track of flavours in quark loops, we introduce the number $n_h$ of heavy flavours with mass $m$ and the number $n_m$ of massive flavours with mass $m_c$, so that $n_f=n_l+n_m+n_h$ and $n_f'=n_l+n_m$. In reality, $n_h=1$ and $n_m=1$. If the $c$ quark is considered massless, we can also include it in the number of light flavours $n_l$, and set $n_m=0$.
[^4]: The gauge-fixing term in the Lagrangian is $-\left(\partial_\mu A_0^{a\mu}\right)^2/(2 a_0)$, the free gluon propagator is $-(i/k^2) \left(g_{\mu\nu} - \xi k_\mu k_\nu/k^2\right)$.
[^5]: Note that for convenience the choice $\mu=m e^{-5/6}$ has been adopted in Ref. [@CGM:03]; $\hat{m}$ is the renormalization-group invariant mass (the exact definition used here is given in the unnumbered formula after (3.8) in Ref. [@CGM:03]).
[^6]: If a matrix element of the HQET current is extracted (from lattice simulations or sum rules) at a low scale $\mu<m_c$ (without dynamic $c$ quarks), then, in addition to HQET evolution without $c$ quark ($\mu<m_c$) and with it ($\mu>m_c$), one has to know the decoupling relation at $\mu=m_c$. It is known with three-loop (N$^3$LO) accuracy [@GSS:06].
|
---
---
(20,15)
(-10,31) (-10,31)[(4,- 3)[2]{}]{} (-10,28) (-10,28)[(4,3)[2]{}]{} (-8,29.5) (-8,29.5)[(3,-4)[1.6]{}]{} (-8.3,32.7) (-8.3,32.7)[(4,1)[2.5]{}]{} (-5.75,33.35) (-5.75,33.35)[(4,- 3)[2]{}]{} (-5.75,30.35) (-5.75,30.35)[(4,3)[2]{}]{} (-3.75,31.85) (-3.75,31.85)[(3,-2)[1.8]{}]{} (-12.24,29) (-12.24,29)[(2,-1)[2]{}]{} (-4.24,27.34) (-6.34,27.34) (-6.34,27.34)[(1,0)[2]{}]{} (-1.8,30.5) (-1.8,30.5)[(0,-1)[2.5]{}]{} (-1.8,28) (-1.8,28)[(-4,-1)[2.5]{}]{} (-4.24,27.34)[(0,-1)[2.8]{}]{}
(2.5,31) (2.5,31)[(-1,-4)[0.75]{}]{} (1.7,27.8) (1.7,27.8)[(-1,-4)[0.5]{}]{} (1.2,25.55) (1.2,25.6)[(-3,-4)[1.7]{}]{} (-0.6,26.1) (-0.6,26.1)[(0,-1)[2.6]{}]{} (-0.6,23.4) (-0.6,23.4)[(-4,1)[3]{}]{} (4.5,29,8) (4.5,29.8)[(-3, -2)[2.7]{}]{} (6.1,29.1) (6.1,29)[(-3, -4)[1.5]{}]{} (6.1,25) (6.1,25)[(-3, 4)[1.5]{}]{} (4.5,26.9) (4.5,27)[(-2, -1)[3]{}]{} (4.1,24) (4.1,24)[(-2, 1)[2.8]{}]{}
(-13,23) (-13,23)[(1,0)[2]{}]{} (-10.9,23) (-10.9,23)[(4,-1)[3]{}]{} (-7.7,22.2) (-7.7,22.2)[(2,1)[3.2]{}]{} (-12,21.5) (-12,21.5)[(2,3)[1]{}]{} (-13.2,19) (-13.2,19)[(4,-1)[2.5]{}]{} (-12.5,16) (-12.5,16)[(-1,4)[0.7]{}]{} (-9.7,21.2) (-9.7,21.2)[(2,1)[1.9]{}]{} (-10.5,18.3) (-10.5,18.3)[(1,4)[0.7]{}]{} (-10.5,15.2) (-10.5,15.2)[(0,1)[3]{}]{}
(-4,21) (-4,21)[(0,1)[2.7]{}]{} (-3.5,19) (-3.5,19)[(-1,4)[0.45]{}]{} (-6,18.4) (-6,18.4)[(4,1)[2.3]{}]{} (-4.5,14.4) (-4.5,14.4)[(1,4)[1.1]{}]{} (-2.4,14.9) (-2.4,14.9)[(-4,-1)[2]{}]{} (-1.4,20) (-1.4,20)[(-2,-1)[2]{}]{} (1,20.55) (1,20.55)[(-4,-1)[2.2]{}]{} (0.8,18.3) (0.8,18.3)[(-3,2)[2]{}]{} (3.4,19) (3.4,19)[(-4,-1)[2.3]{}]{} (1.6,15) (1.6,15)[(-1,4)[0.75]{}]{} (4.4,14.2) (4.4,14.2)[(-4,1)[2.6]{}]{}
(0.8,23.5)[(6,8)]{} (-4.1,24.1) (-3.7, 24.7)(1,1)[3]{} [(1,0)[0.1]{}]{} (-3.5, 24.9)(1,1)[3]{} [(1,0)[0.1]{}]{} (-3.3, 25.1)(1,1)[3]{} [(1,0)[0.1]{}]{} (-3.1, 25.3)(1,1)[3]{} [(1,0)[0.1]{}]{} (-2.9, 25.5)(1,1)[3]{} [(1,0)[0.1]{}]{}
(-0.6, 27.8)(0,1)[4]{} [(0,1)[0.1]{}]{} (-0.6, 28)(0,1)[4]{} [(0,1)[0.1]{}]{} (-0.6, 28.2)(0,1)[4]{} [(0,1)[0.1]{}]{} (-0.6, 28.4)(0,1)[4]{} [(0,1)[0.1]{}]{} (-0.6, 28.6)(0,1)[4]{} [(0,1)[0.1]{}]{}
(-4.8, 24.5)(-2,1)[1]{} [(1,0)[0.1]{}]{} (-5, 24.6)(-2,1)[1]{} [(1,0)[0.1]{}]{} (-5.2, 24.7)(-2,1)[1]{} [(1,0)[0.1]{}]{} (-5.4, 24.8)(-2,1)[1]{} [(1,0)[0.1]{}]{} (-5.6, 24.9)(-2,1)[1]{} [(1,0)[0.1]{}]{} (-5.8, 25)(-2,1)[1]{} [(1,0)[0.1]{}]{} (-6, 25.1)(-2,1)[1]{} [(1,0)[0.1]{}]{} (-6.2, 25.2)(-2,1)[1]{} [(1,0)[0.1]{}]{} (-6.4, 25.3)(-2,1)[1]{} [(1,0)[0.1]{}]{} (-6.6, 25.4)(-2,1)[1]{} [(1,0)[0.1]{}]{} (-6.8, 25.5)(-2,1)[1]{} [(1,0)[0.1]{}]{} (-7, 25.6)(-2,1)[1]{} [(1,0)[0.1]{}]{} (-7.2, 25.7)(-2,1)[1]{} [(1,0)[0.1]{}]{}
(-7.2, 25.7)(-1,0)[6]{} [(1,0)[0.1]{}]{} (-7.5, 25.7)(-1,0)[6]{} [(1,0)[0.1]{}]{} (-7.8, 25.7)(-1,0)[6]{} [(1,0)[0.1]{}]{}
(-13, 25.7)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-13.2, 25.5)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-13.4, 25.3)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-13.6, 25.1)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-13.8, 24.9)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-14, 24.7)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-14.2, 24.5)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-14.4, 24.3)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-14.6, 24.1)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-14.8, 23.9)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-15, 23.7)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-15.2, 23.5)(-2,-3)[1]{} [(1,0)[0.1]{}]{}
(-4.2, 23.7)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-4.3, 23.5)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-4.4, 23.3)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-4.5, 23.1)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-4.6, 22.9)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-4.7, 22.7)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-4.8, 22.5)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-4.9, 22.3)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-5, 22.1)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-5.1, 21.9)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-5.2, 21.7)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-5.3, 21.5)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-5.4, 21.3)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-5.5, 21.1)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-5.7, 20.9)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-5.9, 20.7)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-6.1, 20.5)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-6.3, 20.3)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-6.5, 20.1)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-6.7, 19.9)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-6.9, 19.7)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.1, 19.5)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.3, 19.3)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.5, 19.1)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.6, 18.8)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.6, 18.5)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.7, 18.2)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.7, 17.9)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.8, 17.6)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.8, 17.3)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.8, 17)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.8, 16.7)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.8, 16.4)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.8, 16.1)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.8, 15.8)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.8, 15.5)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.8, 15.2)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.8, 14.9)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.8, 14.6)(-2,-3)[1]{} [(1,0)[0.1]{}]{} (-7.8, 14.3)(-2,-3)[1]{} [(1,0)[0.1]{}]{}
(-3.6, 23.7)(2,-3)[1]{} [(1,0)[0.1]{}]{} (-3.4, 23.5)(2,-3)[1]{} [(1,0)[0.1]{}]{} (-3.2, 23.3)(2,-3)[1]{} [(1,0)[0.1]{}]{} (-3, 23.1)(2,-3)[1]{} [(1,0)[0.1]{}]{} (-2.8, 22.9)(2,-3)[1]{} [(1,0)[0.1]{}]{} (-2.6, 22.7)(2,-3)[1]{} [(1,0)[0.1]{}]{} (-2.4, 22.5)(2,-3)[1]{} [(1,0)[0.1]{}]{} (-2.2, 22.3)(2,-3)[1]{} [(1,0)[0.1]{}]{} (-2, 22.1)(2,-3)[1]{} [(1,0)[0.1]{}]{} (-1.8, 21.9)(2,-3)[1]{} [(1,0)[0.1]{}]{} (-1.5, 21.9)(1,0)[9]{} [(1,0)[0.1]{}]{} (-1.2, 21.9)(1,0)[9]{} [(1,0)[0.1]{}]{} (-0.9, 21.9)(1,0)[9]{} [(1,0)[0.1]{}]{}
(21.5,29.8)[$n_1$]{} (22,30)
(15.5,25.3)[$n_2$]{} (16,25.5) (16.7,26.2)[(4,3)[4.4]{}]{} (19.5,23.8)[$n_3$]{} (20,24) (20,25)[(1,2)[2]{}]{}
(27.5,25.3)[$n_6$]{} (28,25.5) (27.3,26.2)[(-4,3)[4.4]{}]{}
(21.5,17.3)[$n_5$]{} (22,17.5) (22,18.5)[(-1,3)[1.55]{}]{}
(16.5,18.3)[$n_4$]{} (17,18.5) (17.5,19.4)[(1,2)[1.9]{}]{}
(31,19.8)[$n_{8} $]{} (31.5,20) (30.8,20.7)[(-2,3)[2.55]{}]{}
(26,18.3)[$n_{7}$]{} (26.5,18.5) (26.5,19.5)[(1,4)[1.25]{}]{}
|
---
abstract: 'In this paper, we develop a patch reconstruction finite element method for the Stokes problem. The weak formulation of the interior penalty discontinuous Galerkin is employed. The proposed method has a great flexibility in velocity-pressure space pairs whose stability properties are confirmed by the inf-sup tests. Numerical examples show the applicability and efficiency of the proposed method.'
address:
- 'CAPT, LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China'
- 'School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China'
- 'School of Mathematics and Statistics, Wuhan University'
author:
- Ruo Li
- Zhiyuan Sun
- Fanyi Yang
- Zhijian Yang
title: A finite element method by patch reconstruction for the Stokes problem using mixed formulations
---
Stokes problem $\cdot$ Reconstructed basis function $\cdot$ Discontinuous Galerkin method $\cdot$ Inf-sup test 49N45,65N21
Introduction {#sec:introduction}
============
We are concerned in this paper with the incompressible Stokes problem, which has a wide range of applications on the approximation of low Reynolds number flows and the time discretizations of the Oseen equation or Naiver-Stokes equation. One of the major difficulties in finite element discretizations for the Stokes problem is the incompressible constraint, which leads to a saddle-point problem. The stability condition often referred as the inf-sup (LBB) condition requires the approximation spaces for velocity and pressure need to be carefully chosen [@boffi2013mixed]. We refer to [@girault1986finite; @taylor1973numerical] for some specific spaces used in the traditional finite element methods to solve the Stokes problem.
Most recently, the discontinuous Galerkin (DG) methods have achieved a great success in computational fluid dynamics, see the state of art survey [@cockburn2000development]. Hansbo and Larson propose and analyze an interior penalty DG method for incompressible and nearly incompressible linear elasticity on triangular meshes in [@hansbo2002discontinuous] where polynomial spaces of degree $k$ and $k-1$ are employed to approximate velocity and pressure, respectively. In [@toselli2002hp] Toselli considers the $hp$-approximations for the Stokes problem using piecewise polynomial spaces. The uniform divergence stability and error estimates with respect to $h$ and $p$ are proven for this DG formulation when velocity is approximated one or two degrees higher than pressure. Numerical results show that using equal order spaces for velocity and pressure can also work well. Sch[ö]{}tzau et al. improve the estimates on tensor product meshes in [@schotzau2002mixed]. A local discontinuous Galerkin method (LDG) for the Stokes problem is proposed in [@cockburn2002local]. The LDG method can be considered as a stabilized method when the approximation spaces for velocity and pressure are chosen with the same order. [ Hybrid discontinuous Galerkin methods are also of interest due to their capability of providing a superconvergent post processing, we refer to [@carrero2006hybridized; @Lederer2018Hybrid; @Nguyen2010Hybridizable; @Cockburn2009Hybridization] for more discussion. ]{}
Some special finite element spaces can be adopted to Stokes problem in DG framework. Karakashian and his coworkers [@baker1990piecewise; @karakashian1998nonconforming] propose a DG method with piecewise solenoidal vector fields which are locally divergence-free. Cockburn et al. [@cockburn2005incompressiblei; @cockburn2005incompressibleii; @carrero2006hybridized; @cockburn2007note] develop the LDG method with solenoidal vector fields. By introducing the hybrid pressure, the pressure and the globally divergence-free velocity can be obtained by a post-process of the LDG solution. While Montlaur et al.[@montlaur2008discontinuous] present two DG formulations for the incompressible flow, the first formulation is derived from an interior penalty method such that the computation of the velocity and the pressure is decoupled and the second formulation follows the methodology in [@baker1990piecewise]. With an inconsistent penalty, the velocity can be computed with absence of pressure terms. Liu [@liu2011penalty] presents a penalty-factor-free DG formulation for the Stokes problem with optimal error estimates.
However, one of the limitations of DG methods is the computational cost is higher than using continuous Galerkin method directly [@zienkiewicz2003discontinuous; @montlaur2009high] because of the duplication of the degrees of freedom at interelement boundaries especially in three-dimensional case. In this paper, we follow the methodology in [@li2012efficient; @2018arXiv180300378L] to apply the patch reconstruction finite element method to the Stokes problem. Piecewise polynomial spaces built by patch reconstruction procedure are taken to approximate velocity and pressure. The new space is a sub-space of the common approximation space used in DG framework, which allows us to employ the interior penalty formulation directly to solve the Stokes problem. As we mentioned before, it is important to verify the inf-sup condition for a mixed formulation to guarantee the stability, which is often severe for a specific discretization [@bathe2000inf]. We carry out a series of numerical inf-sup tests proposed in [@chapelle1993inf; @boffi2013mixed] to show this method is numerically stable.
The proposed method provides many merits. First, the DOFs of the system are totally decided by the mesh partition and have no relationship with the interpolation order. Then the method is easy to implement on arbitrary polygonal meshes because of the independence between the process of the construction of the space and the geometry structure of meshes. Third, we emphasize that the spaces to approximate velocity and pressure can be engaged with great flexibility. The results of numerical inf-sup tests exhibit the robustness of our method even in some extreme cases.
The outline of this paper is organized as follows. In Section \[sec:reconopreator\], we briefly introduce the patch reconstruction procedure and the finite element space. Then the scheme of the mixed interior penalty DG method and its error analysis for the Stokes problem are presented in Section \[sec:weakform\]. In Section \[sec:infsuptest\], we briefly review the inf-sup test and carry out a series of numerical inf-sup tests in several situations to show the proposed method satisfies the inf-sup condition. Finally, two-dimensional numerical examples are presented in Section \[sec:numericalresults\] to illustrate the accuracy and efficiency of the proposed approach, and verify our theoretical results.
Reconstruction operator {#sec:reconopreator}
=======================
In this section, we will introduce a reconstruction operator which can be constructed on any polygonal meshes and its corresponding approximation properties.
Let $\Omega\subset\mathbb R^d, d=2, 3$, be a convex polygonal domain with boundary $\partial\Omega$. We denote by $\mathcal{T}_h$ a subdivision that partitions $\Omega$ into polygonal elements. And let $\mathcal E_h$ be the set of $(d-1)$-dimensional interfaces (edges) of all elements in $\mathcal T_h$, $\mathcal E_h^i$ the set of interior faces and $\mathcal E_h^b$ the set of the faces on the domain boundary $\partial\Omega$. We set $$h=\max_{K\in\mathcal T_h} h_K,\quad h_K=\text{diam}(K),\quad
h_e=\text{diam}(e),$$ for $\forall K\in \mathcal T_h,\ \forall e \in \mathcal E_h$. Further, we assume that the partition $\MTh$ admits the following shape regularity conditions [@Brezzi:2009; @DaVeiga2014]:
1. There exists an integer number $N$ independent of $h$, that any element $K$ admits a sub-decomposition ${\widetilde{\mathcal
T}}_{h | K}$ made of at most $N$ triangles.
2. ${\widetilde{\MTh}}$ is a compatible sub-decomposition, that any triangle $T\in{\widetilde{\MTh}}$ is shape-regular in the sense of Ciarlet-Raviart [@ciarlet:1978]: there exists a real positive number $\sigma$ independent of $h$ such that $h_T/\rho_T\le\sigma$, where $\rho_T$ is the radius of the largest ball inscribed in $T$.
There many useful properties using for the analysis in finite difference schemes and DG framework can be derived from the above assumptions, such as Agmon inequality and inverse inequality [@DaVeiga2014; @antonietti2013hp; @2018arXiv180300378L]:
1. There exists $C$ that depends on $N$ and $\sigma$ but independent of $h_K$ such that $$\|v\|_{L^2(\partial K)}^2 \leq C\left( h_K^{-1}\|v\|_{L^2(K)}^2
+ h_K\|\nabla v\|_{L^2(K)}^2 \right), \quad \forall v \in
H^1(K).$$
2. There exists $C$ that depends on $N$ and $\sigma$ but independent of $h_K$ such that $$\|\nabla v\|_{L^2(K)} \leq Cm^2/h_K\|v\|_{L^2(K)}, \quad \forall
v \in \mathbb P_m(K).$$
[ Given the partition $\MTh$, we define the reconstruction operator as follows. First in each element $K \in \MTh$, we specify a point ${\boldsymbol}x_K \in K$ as the collocation point. Here we just let ${\boldsymbol}x_K$ be the barycenter of $K$. Then for each $K \in \MTh$ we construct an element patch $S(K)$, which is a set of $K$ itself and some elements around $K$. Specifically, we construct $S(K)$ in a recursive manner. For element $K$, we set $S(K) = \left\{ K \right\}$ first, and we enlarge $S(K)$ by adding all the von Neumann neighbours (adjacent edge-neighbouring elements) of $S(K)$ into $S(K)$ recursively until we have collected enough elements into the element patch. We denote by $\# S(K)$ the cardinality of $S(K)$ and an example of construction of $S(K)$ with $\# S(K) = 12$ is shown in Fig \[fig:buildpatch\]. ]{}
[.3]{}
[.3]{}
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[.3]{}
For element $K$, we collect all collocation points in a set $\mathcal
I_K$: $$\mathcal I_K\triangleq\left\{{\boldsymbol}x_{\widetilde K}\ |\ {\boldsymbol}x_{\widetilde K}\ \text{is the barycenter of}\ \widetilde
K,\ \forall\widetilde K\in S(K)\right\}.$$
Let $U_h$ be the space consisting of piecewise constant functions: $$U_h\triangleq\left\{ v\in L^2(\Omega)\ \big |\ v|_K \in \mathbb
P_0(K),\ \forall K\in \mathcal T_h\right\},$$ where $\mathbb P_n$ is the polynomial space of degree not greater than $n$. For any $v\in U_h$, we reconstruct a $m$th-order polynomial denoted by $\mathcal R_K^m v$ on $S(K)$ by the following least squares problem: $$\mathcal R_K^mv= \mathop{\arg\min}_{p\in\mathbb P_m(S(K))}
\ \sum_{{\boldsymbol}x\in \mathcal I_K}|v({\boldsymbol}x)-p({\boldsymbol}x)|^2.
\label{eq:lsproblem}$$
The uniqueness condition for the problem is provided by the condition $\# S(K)\geq\text{dim}(\mathbb P_m) $ and the following assumption [@li2012efficient; @li2016discontinuous]:
For $\forall K\in\mathcal T_h$ and $\forall p \in \mathbb
P_m(S(K))$, problem satisfies $$p|_{\mathcal I(K)}={\boldsymbol}0\quad\Longrightarrow\quad p|_{S(K)}\equiv0.$$ \[as:unique\]
Hereafter, we assume the uniqueness condition for always holds. For any $g \in U_h$, we restrict the definition domain of the polynomial $\mathcal R_K^m g$ on element $K$ to define a global reconstruction operator which is denoted by $\mathcal R^m$: $$\mathcal R^mg|_K = (\mathcal R_K^mg)|_K,\quad \forall K\in \mathcal
T_h.$$ Then we extend the reconstruction operator to an operator defined on $C^0(\Omega)$, still denoted as $\mathcal R^m$: $$\mathcal R^mu=\mathcal R^m\tilde u,\quad \tilde u\in U_h, \quad
\tilde u({\boldsymbol}x_K)=u({\boldsymbol}x_K), \quad \forall u\in C^0(\Omega).$$
We note that $\mathcal R^m$ is a linear operator whose image is actually a piecewise $m$th-order polynomial space which is denoted as $$V_{h}^{m} = \mathcal R^m U_h.$$
Further, we give a group of basis functions of the space $V_h^m$. We define $w_K({\boldsymbol}x) \in C^0(\Omega)$ such that $$w_K({\boldsymbol}x) = \begin{cases} 1, \quad &{\boldsymbol}x = {\boldsymbol}x_K, \\ 0, \quad
&{\boldsymbol}x \in \widetilde K, \quad \widetilde K \neq K.
\end{cases}$$ Then we denote $\left\{ \lambda_K\ |\ \lambda_K = {\mathcal{R}}^m w_K
\right\}$ as a group of basis functions. Given ${\lambda_K}$, we may write the reconstruction operator in an explicit way: $${\mathcal{R}}^m g = \sum_{K \in \MTh} g({\boldsymbol}x_K) \lambda_K(x), \quad \forall
g \in C^0(\Omega).
\label{eq:explicit}$$ From , it is clear that the degrees of freedom of ${\mathcal{R}}^m$ are the values of the unknown function at the collocation points of all elements in partition. We present a 2D example in Section \[sec:2dexample\] to demonstrate the reconstruction process and the implementation of basis functions.
We note that
$\mathcal R^m u(\forall u\in C^0(\Omega))$ may be discontinuous across the inter-element boundaries. The fact inspires us to share some well-developed theories of DG methods and enjoy its advantages.
We first introduce the traditional average and jump notations in DG method. Let $e$ be an interior edge shared by two adjacent elements $e=\partial K^{+} \cap \partial K^{-}$ with the unit outward normal vector ${\boldsymbol}{\mathrm n}^{+}$ and ${\boldsymbol}{\mathrm n}^{-}$, respectively. Let $v$ and ${\boldsymbol}{v}$ be the scalar-valued and vector-valued functions on $\mathcal T_h$, respectively, we define the $average$ operator $\{
\cdot \}$ as follows: $$\{v\}=\frac{1}{2}(v^{+} + v^{-}), \quad \{ {\boldsymbol}{v} \} =
\frac{1}{2}({\boldsymbol}{v}^{+} + {\boldsymbol}{v}^{-}) , \quad\text{on }\ e\in\mathcal
E_h^i,$$ with $v^+=v|_{K^+},\ v^-=v|_{K^-},\ {\boldsymbol}v^+={\boldsymbol}v|_{K^+},\ {\boldsymbol}v^-={\boldsymbol}v|_{K^-}$.
Further, we set the $jump$ operator ${[ \hspace{-2pt} [}\cdot {] \hspace{-2pt} ]}$ as $$\begin{aligned}
{[ \hspace{-2pt} [}v {] \hspace{-2pt} ]}=v^{+} {\boldsymbol}{\mathrm n}^{+} + v^{-} {\boldsymbol}{\mathrm n}^{-},
\quad {[ \hspace{-2pt} [}{\boldsymbol}{v} {] \hspace{-2pt} ]}={\boldsymbol}{v}^{+}\cdot {\boldsymbol}{\mathrm
n}^{+}+{\boldsymbol}{v}^{-}\cdot {\boldsymbol}{\mathrm n}^{-}, \\
{[ \hspace{-2pt} [}{\boldsymbol}{v} \otimes{\boldsymbol}{\mathrm n} {] \hspace{-2pt} ]}={\boldsymbol}{v}^{+}\otimes
{\boldsymbol}{\mathrm n}^{+}+{\boldsymbol}{v}^{-}\otimes {\boldsymbol}{\mathrm n}^{-},\quad
\text{on}\ e\in\mathcal E_h^i. \\
\end{aligned}$$
For $e \in \mathcal E^b_h$, we set $$\begin{aligned}
\{v\}=v&,\quad \{{\boldsymbol}v\}={\boldsymbol}v,\quad {[ \hspace{-2pt} [}v {] \hspace{-2pt} ]}=v{\boldsymbol}{\mathrm n},
\\ {[ \hspace{-2pt} [}{\boldsymbol}v {] \hspace{-2pt} ]}={\boldsymbol}v\cdot{\boldsymbol}{\mathrm n}&, \quad {[ \hspace{-2pt} [}{\boldsymbol}{v}\otimes {\boldsymbol}{\mathrm n} {] \hspace{-2pt} ]}= {\boldsymbol}{v}\otimes {\boldsymbol}{\mathrm n}
,\quad\text{on}\ e \in \mathcal E^b_h.\\
\end{aligned}$$
Now we will present the error analysis of $\mathcal R^m$. We begin by defining broken Sobolev spaces of composite order ${\boldsymbol}{\mathrm s}=\{
s_K\geq0: \forall K\in\mathcal T_h\}$: $$\begin{aligned}
H^{{\boldsymbol}{\mathrm s}}(\Omega,\mathcal T_h)\triangleq\{u\in
L^2(\Omega)&: u|_K \in H^{s_K}(K),\forall K\in\mathcal T_h\}, \\
\end{aligned}$$ where $H^{s_K}(K)$ is the standard Sobolev spaces on element $K$. The associated broken norm is defined as $$\begin{aligned}
\|u\|_{H^{{\boldsymbol}{\mathrm s}}(\Omega,\mathcal T_h)}^2=
\sum_{K\in\mathcal T_h}\|u\|_{H^{s_K}(K)}^2,
\end{aligned}$$ where $\|\cdot\|_{H^{s_K}(K)}$ is the standard Sobolev norm on element $K$. For ${\boldsymbol}u\in [H^{{\boldsymbol}{\mathrm s}}(\Omega, \mathcal T_h)]^d$, the norm is defined as $$\|{\boldsymbol}u\|_{H^{{\boldsymbol}{\mathrm s}}(\Omega,\mathcal T_h)}^2= \sum_{i=1}^d
\|{\boldsymbol}u_i\|_{H^{{\boldsymbol}{\mathrm s}}(\Omega, \mathcal T_h)}^2.$$ When $s_K=s$ for all elements in $\mathcal T_h$, we simply write $H^s(\Omega,\mathcal T_h)$ and $[H^s(\Omega, \mathcal T_h)]^d$.
Then we define a constant $\Lambda(m,\mathcal{I}_K)$ for $K\in
\mathcal T_h$: $$\label{eq:constant}
\Lambda(m, \mathcal{I}_K) \triangleq \max_{p\in \mathbb{P}_m(S(K))}
\frac{\max_{{\boldsymbol}{x} \in S(K)} |p({\boldsymbol}{x})|}{\max_{{\boldsymbol}{x} \in
\mathcal{I}_K} |p({\boldsymbol}{x})|},$$ the Assumption \[as:unique\] is equivalent to $$\Lambda(m, \mathcal{I}_K) < \infty.$$ The uniform upper bound of $\Lambda(m, \mathcal I_K)$ exists if element patches are convex and the triangulation is quasi-uniform [@li2012efficient]. We also refer to [@li2016discontinuous] for the estimate of $\Lambda(m, \mathcal I_K)$ in more general cases such as polygonal partition and non-convex element patch. We denote by $\Lambda_m$ the uniform upper bound of $\Lambda(m, \mathcal I_K)$.
With $\Lambda_m$, we have the following estimates.
Let $g\in H^{m+1}(\Omega)(m\geq0)$ and $ K\in\mathcal T_h$, then $$\|g-\mathcal{R}^m g\|_{L^2(K)}\lesssim \Lambda_{m}
h^{m+1}\|g\|_{H^{m+1}(S(K))}.
\label{eq:reconL2error}$$ \[le:reconL2error\]
For convenience, the symbol $\lesssim$ and $\gtrsim$ will be used in this paper. That $X_1\lesssim Y_1$ and $X_2\gtrsim Y_2$ mean that $X_1\leq C_1Y_1$ and $X_2\geq C_2Y_2$ for some positive constants $C_1$ and $C_2$ which are independent of mesh size $h$.
Let $g\in H^{m+1}(\Omega)(m\geq0)$ and $ K\in\mathcal T_h$, then $$\|g-\mathcal R^m g\|_{L^2(\partial K)}\lesssim \Lambda_{m}
h^{m+\frac12}\|g\|_{H^{m+1}(S(K))}.
\label{eq:recontraceinequality}$$ \[th:recontraceinequality\]
For the standard Sobolev norm, we have the following estimates:
Let $g\in H^{m+1}(\Omega)(m\geq0)$ and $K\in\mathcal T_h$, then $$\|g-\mathcal R^mg\|_{H^1(K)}\lesssim \Lambda_{m}
h^{m}\|g\|_{H^{m+1}(S(K))}.
\label{eq:reconSobleverror}$$ \[th:reconSobleverror\]
We refer to [@li2012efficient; @li2016discontinuous] for detailed proofs and more discuss about $S(K)$ and $\# S(K)$. Here we note that one of the conditions of guaranteeing the uniform upper bound $\Lambda_m$ is $\# S(K)$ should be much larger than $\text{dim}(\mathbb P_m)$. In Section \[sec:infsuptest\] we will list the values of $\# S(K)$ used in all numerical experiments.
Finally, we derive the estimate in DG energy norm. For the scalar-valued function, the DG energy norm is defined as: $$\begin{aligned}
\|u\|_{\mathrm{DG}}^2&\triangleq\sum_{K\in\mathcal
T_h}|u|_{H^1(K)}^2 + \sum_{e\in\mathcal E_h} \frac1{h_e}\|{[ \hspace{-2pt} [}u
{] \hspace{-2pt} ]}\|_{L^2(e)}^2,\quad \forall u\in H^1(\Omega, \mathcal T_h),\\
\end{aligned}$$
Let $g\in H^{m+1}(\Omega)(m\geq0)$, then $$\|g-\mathcal R^mg\|_{\mathrm{DG}}\lesssim \Lambda_{m} h^{m}
\|g\|_{H^{m+1}(\Omega)}.
\label{eq:interpolation}$$ \[th:interpolationerrorDG\]
From Lemma \[th:reconSobleverror\], we have $$\begin{aligned}
\sum_{K\in\mathcal T_h}|g-\mathcal R^mg|_{H^1(K)}&\lesssim
\sum_{K\in\mathcal T_h}\Lambda_m
h^{m}\|g\|_{H^{m+1}(S(K))}\\ &\lesssim \Lambda_m
h^{m}\|g\|_{H^{m+1}(\Omega)}.\\
\end{aligned}$$ For any $e\in\mathcal E_h^i$ shared by elements $K_1$ and $K_2$, we have $$\frac1{h_e}\|{[ \hspace{-2pt} [}g-\mathcal R^m g{] \hspace{-2pt} ]}\|_{L^2(e)}^2\leq
\frac{1}{h_e}\left( \|g-\mathcal R^m g\|_{L^2(\partial K_1)}^2
+\|g-\mathcal R^m g \|_{L^2(\partial K_2)}^2\right).$$ From Lemma \[th:recontraceinequality\], we get $$\begin{aligned}
\frac1{h_e}\|g-\mathcal R^m g\|_{L^2(\partial
K_1)}^2&\lesssim\Lambda_m
h^{2m}\|g\|_{H^{m+1}(K_1)}^2,\\ \frac1{h_e}\|g-\mathcal R^m
g\|_{L^2(\partial K_2)}^2&\lesssim\Lambda_m
h^{2m}\|g\|_{H^{m+1}(K_2)}^2.\\
\end{aligned}$$ For any $e\in\mathcal E_h^b$, assume $e$ is a face of element $K$, we have $$\begin{aligned}
\frac1{h_e}\|{[ \hspace{-2pt} [}g-\mathcal R^m g{] \hspace{-2pt} ]}\|_{L^2(e)}^2&\leq
\frac{1}{h_e}\|g-\mathcal R^m g\|_{L^2(\partial K)}^2\\ &\lesssim
\Lambda_m h^{2m}\|g\|_{H^{m+1}(K)}^2.\\
\end{aligned}$$ Combining the above inequalities gives the estimate , which completes the proof.
For the vector-valued function, the DG energy norm is defined as: $$\|{\boldsymbol}u\|_{\mathrm{DG}}^2\triangleq\sum_{i=1}^d\|{\boldsymbol}u_i\|_{
\mathrm{DG}}^2,\quad\forall {\boldsymbol}u\in [H^1(\Omega,\mathcal T_h)]^d,$$ and the reconstruction operator is defined component-wisely for $[U_h]^d$, still denoted by $\mathcal R^m$: $$\begin{aligned}
{\mathcal R}^m{\boldsymbol}v&=[\mathcal R^m{\boldsymbol}v_i]^d, \quad 1\leq i\leq
d,\quad \forall {\boldsymbol}v \in [U_h]^d.\\
\end{aligned}$$ Then the operator can be extended on $[C^0(\Omega)]^d$ and the corresponding estimate is written as:
let ${\boldsymbol}g\in [H^{m+1}(\Omega)]^d(m\geq0)$, then $$\|{\boldsymbol}g-{\mathcal R}^m{\boldsymbol}g\|_{\mathrm{DG}}\lesssim \Lambda_m
h^{m}\|{\boldsymbol}g\|_{H^{m+1}(\Omega)}.$$
It is a direct extension from Theorem \[th:interpolationerrorDG\].
The weak form of the stokes problem {#sec:weakform}
===================================
In this section, we consider the incompressible Stokes problem with Dirichlet boundary condition, which seeks the velocity field ${\boldsymbol}u$ and its associated pressure $p$ satisfying $$\begin{aligned}
-\Delta{\boldsymbol}u+\nabla p&={\boldsymbol}f \qquad \text{in}
\ \Omega,\\ \nabla\cdot{\boldsymbol}u&=0 \qquad\text{in}\ \Omega,\\ {\boldsymbol}u&={\boldsymbol}g \qquad\text{on}\ \partial\Omega,\\
\end{aligned}
\label{eq:stokes}$$ where ${\boldsymbol}f$ is the given source term and ${\boldsymbol}g$ is a Dirichlet boundary condition that satisfies the compatibility condition $$\int_{\partial \Omega} {\boldsymbol}g\cdot{\boldsymbol}{\mathrm n}\mathrm ds=0.$$
For positive integer $k,k'$, we define the following finite element spaces to approximate velocity and pressure: $$\begin{aligned}
{\boldsymbol}V_h^k=[V_h^k]^d,\quad Q_{h}^{k'}=V_{h}^{k'}.\\
\end{aligned}$$
We note that finite element spaces ${\boldsymbol}V_{h}^{k}$ and $Q_{h}^{k'}$ are the subspace of the common discontinuous Galerkin finite element spaces, which implies that the interior penalty discontinuous Galerkin method [@hansbo2002discontinuous; @montlaur2008discontinuous] can be directly applied to the Stokes problem .
For a vector ${\boldsymbol}u$, we define the second-order tensor $\nabla{\boldsymbol}u$ by $$(\nabla{\boldsymbol}u)_{i,j}=\frac{\partial{\boldsymbol}u_i}{\partial x_j}, \quad
1\leq i,j\leq d.$$
The discrete problem for the Stokes problem is as: find $({\boldsymbol}u_h, p_h)\in {\boldsymbol}V_{h}^{k} \times Q_{h}^{k'}$ such that $$\begin{aligned}
a({\boldsymbol}u_h, {\boldsymbol}v_h)+b({\boldsymbol}v_h, p_h)&=l({\boldsymbol}v_h),\quad \forall {\boldsymbol}v_h\in {\boldsymbol}V_{h}^{k},\\ b({\boldsymbol}u_h, q_h)&=(q_h,{\boldsymbol}{\mathrm
n}\cdot{\boldsymbol}g)_{\partial \Omega},\quad \forall q_h\in
Q_{h}^{k'},\\
\end{aligned}
\label{eq:weakform}$$ where symmetric bilinear form $a(\cdot, \cdot)$ is given by $$\begin{aligned}
a({\boldsymbol}{u}, {\boldsymbol}{v})&=\int_{\Omega}\nabla {\boldsymbol}{u} : \nabla {\boldsymbol}{v}
{\mathrm{d}\boldsymbol{x}}\\ &-\int_{\mathcal E_h} (\{\nabla {\boldsymbol}{u}\} : {[ \hspace{-2pt} [}{\boldsymbol}{v} \otimes
{\boldsymbol}{n} {] \hspace{-2pt} ]}+{[ \hspace{-2pt} [}{\boldsymbol}{u} \otimes {\boldsymbol}{n} {] \hspace{-2pt} ]}: \{\nabla
{\boldsymbol}{v}\})\mathrm ds \\ &+\int_{\mathcal E_h}\eta{[ \hspace{-2pt} [}{\boldsymbol}{u} \otimes
{\boldsymbol}{n} {] \hspace{-2pt} ]}: {[ \hspace{-2pt} [}{\boldsymbol}{v} \otimes {\boldsymbol}{n} {] \hspace{-2pt} ]}\mathrm ds, \quad
\forall {\boldsymbol}{u},{\boldsymbol}{v}\in [H^1(\Omega, \mathcal T_h)]^d.\\
\end{aligned}
\label{eq:ellipticform}$$
The term $\eta$ is referred to as the penalty parameter which is defined on $\mathcal E_h$ by $$\eta|_e = \eta_e,\quad \forall e\in \mathcal E_h,$$ and will be specified later. The bilinear form $b(\cdot, \cdot)$ and the linear form $l(\cdot)$ are defined as $$\begin{aligned}
b({\boldsymbol}v, p)&=-\int_{\Omega} p\nabla\cdot{\boldsymbol}v{\mathrm{d}\boldsymbol{x}}+ \int_{\mathcal
E_h} \{p\}{[ \hspace{-2pt} [}{\boldsymbol}{v} {] \hspace{-2pt} ]}\mathrm ds,\\
l({\boldsymbol}v)&=\int_{\Omega}{\boldsymbol}f
\cdot {\boldsymbol}v{\mathrm{d}\boldsymbol{x}}-\int_{\mathcal E_h^b}{\boldsymbol}g \cdot (\nabla
{\boldsymbol}v\cdot{\boldsymbol}{\mathrm n}) \mathrm ds +
\int_{\mathcal E_h^b}\eta {\boldsymbol}g\cdot {\boldsymbol}v\mathrm ds,\\
\end{aligned}
\label{eq:divergenceform}$$ for $\forall {\boldsymbol}v\in [H^1(\Omega,\mathcal T_h)]^d$ and $p\in
L^2(\Omega)$.
Now we present the standard continuity and coercivity properties of the bilinear form $a(\cdot. \cdot)$. Actually the bilinear form $a(\cdot, \cdot)$ is a direct extension from the interior penalty bilinear form used for solving the elliptic problems [@arnold1982interior]. It is easy to extend the theoretical results of solving the elliptic problems to $a(\cdot, \cdot)$.
The bilinear form $a(\cdot, \cdot)$, defined in , is continuous when $\eta\geq0$. The following inequality holds: $$|a({\boldsymbol}u, {\boldsymbol}v)|\lesssim\|{\boldsymbol}u\|_{\mathrm{\mathrm{DG}}}\|{\boldsymbol}v\|_{\mathrm{\mathrm{DG}}},\quad \forall {\boldsymbol}{u,v}\in [H^1(\Omega, \mathcal
T_h)]^d.$$ \[le:ellipticcontinuous\]
Let $$\eta|_e=\frac{\mu}{h_e},\quad \forall e\in \mathcal E_h,$$ where $\mu$ is a positive constant. With sufficiently large $\mu$, the following inequality holds: $$|a({\boldsymbol}u_h, {\boldsymbol}u_h)|\gtrsim \|{\boldsymbol}u_h\|_{\mathrm{\mathrm{DG}}}^2, \quad
\forall {\boldsymbol}u_h\in {\boldsymbol}V_{h}^{k}.$$ \[le:ellipticcoercivity\]
The detailed proofs of Lemma \[le:ellipticcontinuous\] and Lemma \[le:ellipticcoercivity\] could be found in [@arnold2002unified; @hansbo2002discontinuous; @montlaur2008discontinuous]. We also refer to [@arnold2002unified] where a unified method is employed to analyse the choices of the penalty parameter $\eta$.
For $b(\cdot,\cdot)$, we have the analogous continuity property.
The bilinear form $b(\cdot, \cdot)$, defined in , is continuous. The following inequality holds: $$|b({\boldsymbol}v, q)|\lesssim\|{\boldsymbol}v\|_{\mathrm{\mathrm{DG}}} \| q
\|_{L^2(\Omega)},\quad \forall {\boldsymbol}{v}\in [H^1(\Omega,
\MTh)]^d,\ \forall q\in L^2(\Omega).$$ \[le:divergencecontinuous\]
Besides the continuity of $a(\cdot, \cdot), b(\cdot,\cdot)$ and the coercivity of $a(\cdot, \cdot)$, [ the existence of a stable finite element approximation solution $({\boldsymbol}u_h, p_h)$ depends on choosing a pair of spaces ${\boldsymbol}V_{h}^{k}$ and $Q_h^{k'}$ such that the following inf-sup condition holds [@boffi2013mixed] ]{}: $$\mathop{\inf\qquad\sup}_{q_h\in Q_{h}^{k'}\ {\boldsymbol}v_h\in {\boldsymbol}V_{h}^{k}} \frac{b({\boldsymbol}v_h, q_h)}{ \|{\boldsymbol}v_h\|_{\mathrm{\mathrm{DG}}}\|q_h\|_{L^2(\Omega)}}\geq \beta,
\label{eq:infsupcondition}$$ where $\beta$ is a positive constant.
The finite element space we build depends on the collocation points and element patches, the theoretical verification of the inf-sup condition for the pair ${\boldsymbol}V_{h}^{k} \times Q_{h}^{k'}$ is very difficult in all situations. Chapelle and Bathe [@chapelle1993inf] propose a numerical test on whether the inf-sup condition is passed for a given finite element discretization. In next section, we will carry out a series of numerical evaluations for different $k$ and $k'$ to give an indication of the verification of the inf-sup condition.
Then if the inf-sup condition holds, we could state a standard priori error estimate of the mixed method .
Let the exact solution $({\boldsymbol}u, p)$ to the Stokes problem belong to $[H^{k+1}(\Omega)]^d
\times H^{k'+1}(\Omega)$ with $k\geq 1$ and $k'\geq
0$, and let $({\boldsymbol}u_h, p_h)$ be the numerical solution to , and assume that the inf-sup condition holds and the penalty parameter $\eta$ is set properly. Then the following estimate holds: $$\|{\boldsymbol}u- {\boldsymbol}u_h\|_{\mathrm{\mathrm{DG}}}+\|p -
p_h\|_{L^2(\Omega)}\lesssim h^s\left( \|{\boldsymbol}u\|_{H^{k+1}(
\Omega)}+\|p\|_{H^{k'+1}(\Omega)}\right),
\label{eq:prioriestimate}$$ \[th:prioriestimate\] where $s=\min(k,k'+1)$.
We define ${\boldsymbol}Z ({\boldsymbol}g)\subset {\boldsymbol}V_h^{k}$ by $$\label{eq:kernel_space}
{\boldsymbol}Z({\boldsymbol}g)=\{{\boldsymbol}v \in {\boldsymbol}V_h:b({\boldsymbol}v ,q )=
\int_{\mathcal E_h^b} {\boldsymbol}g\cdot {\boldsymbol}n q \ds, \ \forall q\in
Q_h^{k'} \}.$$
Consider ${\boldsymbol}w \in {\boldsymbol}Z ({\boldsymbol}g)$ and $q\in Q_h^{k'}$. Since Lemma \[le:ellipticcoercivity\], we have $$\begin{aligned}
\|{\boldsymbol}w -{\boldsymbol}u_h\|_{\mathrm{DG}}^2 &\lesssim a({\boldsymbol}w -{\boldsymbol}u_h, {\boldsymbol}w- {\boldsymbol}u_h)
\\ &\lesssim a({\boldsymbol}w -{\boldsymbol}u, {\boldsymbol}w- {\boldsymbol}u_h)+ a({\boldsymbol}u -{\boldsymbol}u_h, {\boldsymbol}w-
{\boldsymbol}u_h)\\ &=a({\boldsymbol}w -{\boldsymbol}u, {\boldsymbol}w- {\boldsymbol}u_h) - b({\boldsymbol}w -{\boldsymbol}u_h, p-
p_h).\end{aligned}$$ Since ${\boldsymbol}w -{\boldsymbol}u_h\in {\boldsymbol}Z(0)$, the $q_h$ can be replaced by any $q\in Q_h^{k'}$, we obtain $$\|{\boldsymbol}w -{\boldsymbol}u_h\|_{\mathrm{DG}}^2\lesssim a({\boldsymbol}w -{\boldsymbol}u, {\boldsymbol}w- {\boldsymbol}u_h) -
b({\boldsymbol}w -{\boldsymbol}u_h, p- q).$$ Using Lemma \[le:ellipticcontinuous\] and \[le:divergencecontinuous\] gives $$\label{eq:kernel_ineq}
\|{\boldsymbol}u -{\boldsymbol}u_h\|_{\mathrm{DG}} \lesssim \|{\boldsymbol}u- {\boldsymbol}w\|_{\mathrm{DG}} + \| p-
q\|_{L^2(\Omega)}, \ {\boldsymbol}w\in {\boldsymbol}Z({\boldsymbol}g), q\in Q_h^{k'}.$$ Then we deal with an arbitrary function in ${\boldsymbol}V_h^k$. For the fixed ${\boldsymbol}v \in {\boldsymbol}V_h^k$, we consider the problem of finding ${\boldsymbol}z({\boldsymbol}v) \in {\boldsymbol}V_h^k$, such that $$b({\boldsymbol}z({\boldsymbol}v) , q)= b({\boldsymbol}u-{\boldsymbol}u_h,q), \ q\in Q_h^{k'}.$$ Thanks to the inf-sup condition and [@boffi2013mixed Proposition 5.1.1,p.270]. We can find a solution ${\boldsymbol}z\in {\boldsymbol}V_h^k$, such that $$\label{eq:inf_sup_ineq}
\| {\boldsymbol}z({\boldsymbol}v) \|_{\mathrm{DG}} \lesssim \sup_{0\neq q\in Q_{h}^{k'}}
\frac{b({\boldsymbol}z({\boldsymbol}v), q)}{\|q\|_{L^2(\Omega)}}= \sup_{0 \neq q\in
Q_{h}^{k'}} \frac{b({\boldsymbol}u -{\boldsymbol}u_h, q)}{\|q\|_{L^2(\Omega)}}
\lesssim \|{\boldsymbol}u -{\boldsymbol}u_h\|_{\mathrm{DG}}.$$ Since $$b({\boldsymbol}z({\boldsymbol}v) +{\boldsymbol}v,q)=b ({\boldsymbol}u_h, q)=\int_{\mathcal E_h^b}
{\boldsymbol}g\cdot {\boldsymbol}n q \ds, \ \forall q\in Q_h^{k'},$$ we have ${\boldsymbol}z({\boldsymbol}v) +{\boldsymbol}v \in {\boldsymbol}Z ({\boldsymbol}g)$. Taking ${\boldsymbol}w={\boldsymbol}z({\boldsymbol}v) +{\boldsymbol}v$ in yields $$\label{eq:kernel_app}
\|{\boldsymbol}u -{\boldsymbol}u_h\|_{\mathrm{DG}} \lesssim \|{\boldsymbol}u -{\boldsymbol}v\|_{\mathrm{DG}} + \|{\boldsymbol}z({\boldsymbol}v)\|_{\mathrm{DG}} + \|p-q\|_{L^2(\Omega)}.$$ together with , $$\label{eq:velocity_app}
\begin{split}
\|{\boldsymbol}u -{\boldsymbol}u_h\|_{\mathrm{DG}} &\lesssim \inf_{{\boldsymbol}v \in {\boldsymbol}V_h^k} \|{\boldsymbol}u
-{\boldsymbol}v\|_{\mathrm{DG}} + \inf_{q\in Q_h^{k'}}\|p-q\|_{L^2(\Omega)}\\ &\lesssim
h^{k}\|{\boldsymbol}u\|_{H^{k+1}(\Omega)} + h^{k'+1}
\|p\|_{H^{k'+1}(\Omega)}.
\end{split}$$
Next we consider the pressure term, let $q\in Q_h^{k'}$. Using the inf-sup condition in we have $$\label{eq:pressure_ineq}
\begin{split}
\|q-p_h\|_{L^2(\Omega)}& \lesssim \sup_{0\neq {\boldsymbol}v \in {\boldsymbol}V_h^k}
\frac{b({\boldsymbol}v, q-p_h)}{\|{\boldsymbol}v\|_{\mathrm{DG}}}\\ &=\sup_{0\neq {\boldsymbol}v \in {\boldsymbol}V_h^k} \frac{b({\boldsymbol}v, q-p)+b({\boldsymbol}v, p-p_h)}{\|{\boldsymbol}v\|_{\mathrm{DG}}}\\ &=\sup_{0\neq {\boldsymbol}v \in {\boldsymbol}V_h^k} \frac{b({\boldsymbol}v, q-p)-
a({\boldsymbol}u -{\boldsymbol}u_h ,{\boldsymbol}v)}{\|{\boldsymbol}v\|_{\mathrm{DG}}} \\ &\lesssim
\|p-q\|_{L^2(\Omega)} + \|{\boldsymbol}u -{\boldsymbol}u_h\|_{\mathrm{DG}}.
\end{split}$$ From the triangle inequality and , we obtain $$\label{eq:pressure_app}
\begin{split}
\|p-p_h\|_{L^2(\Omega)} &\lesssim \|{\boldsymbol}u -{\boldsymbol}u_h\|_{\mathrm{DG}} + \inf_{q\in
Q_h^{k'}}\|p-q\|_{L^2(\Omega)} \\
&\lesssim h^{k}\|{\boldsymbol}u\|_{H^{k+1}(\Omega)} + h^{k'+1} \|p\|_{H^{k'+1}(\Omega)},
\end{split}$$ and the proof is concluded by combining and .
Inf-sup test {#sec:infsuptest}
============
In this section, we perform the inf-sup tests with some velocity-pressure finite element space pairs to validate the inf-sup condition numerically. After the discretization, the matrix form of the problem is obtained, $$\begin{bmatrix}
A & B^T\\ B & 0 \\
\end{bmatrix}
\begin{bmatrix}
{\boldsymbol}{\mathrm U}\\ {\boldsymbol}{\mathrm P}\\
\end{bmatrix}
= \begin{bmatrix} {\boldsymbol}{\mathrm F}\\ {\boldsymbol}{\mathrm G}\\
\end{bmatrix},$$ where the matrix $A$ and the matrix $B$ associate with the bilinear form $a(\cdot, \cdot)$ and $b(\cdot, \cdot)$, respectively. The vector ${\boldsymbol}U, {\boldsymbol}P$ is the solution vector corresponding to ${\boldsymbol}u_h, p_h$ and ${\boldsymbol}F, {\boldsymbol}G$ is the right hand side corresponding to ${\boldsymbol}f, {\boldsymbol}g$.
Then the numerical inf-sup test is based on the following lemma.
Let $S$ and $T$ be symmetric matrices of the norms $\|\cdot\|_{\mathrm{DG}}$ in ${\boldsymbol}V_h^k$ and $\|\cdot\|_{L^2(\Omega)}$ in $Q_{h}^{k'}$, respectively, and let $\mu_{\min}$ be the smallest nonzero eigenvalue defined by the following generalized eigenvalue problem: $$B^TS^{-1}B{\boldsymbol}{V}=\mu_{\min}^2T{\boldsymbol}{V},$$ then the value of $\beta$ is simply $\mu_{\min}$. \[le:numericalinfsup\]
The proof of this lemma can be found in [@boffi2013mixed; @malkus1981eigenproblems]. In numerical tests, we would consider a sequence of successive refined meshes and monitor $\mu_{\min}$ of each mesh. If a sharp decrease of $\mu_{\min}$ is observed while the mesh size approaches to zero, we could predict that the pair of approximation spaces violates the inf-sup condition. Otherwise, if $\mu_{\min}$ stabilizes as the mesh is refined, we can conclude that the inf-sup test is passed.
The numerical tests are conducted with following settings, let $\Omega$ be the unit square domain in two dimension and we consider two groups of quasi-uniform meshes which are generated by the software *gmsh* [@geuzaine2009gmsh]. The first ones are triangular meshes(see Fig \[fig:infsuptesttriangularmesh\]) and the second ones consist of triangular and quadrilateral elements(see Fig \[fig:infsuptestmixedmesh\]). In both cases, the mesh size $h$ is taken by $h=\frac1n,\ n=10,20,30,\cdots 80$.
![The triangular meshes, $h=\frac1{10}$(left)/$h=\frac1{
20}$(right).[]{data-label="fig:infsuptesttriangularmesh"}](./figure/tri1-crop.pdf "fig:"){width="40.00000%"} ![The triangular meshes, $h=\frac1{10}$(left)/$h=\frac1{
20}$(right).[]{data-label="fig:infsuptesttriangularmesh"}](./figure/tri2-crop.pdf "fig:"){width="40.00000%"}
![The mixed meshes, $h=\frac1{10}$(left)/$h=\frac1{2
0}$(right).[]{data-label="fig:infsuptestmixedmesh"}](./figure/mixed1-crop.pdf "fig:"){width="40.00000%"} ![The mixed meshes, $h=\frac1{10}$(left)/$h=\frac1{2
0}$(right).[]{data-label="fig:infsuptestmixedmesh"}](./figure/mixed2-crop.pdf "fig:"){width="40.00000%"}
With the given mesh partition, the finite element space can be constructed. As we mention before, for element $K$, $\# S(K)$ should be large enough to ensure the uniform upper bound $\Lambda_m$. For simplicity, $\# S(K)$ is taken uniformly and for different order $k$ we list a group of reference values of $\# S(K)$ for both meshes in Table \[tab:patchnumber2d\].
\[tab:patchnumber2d\]
We consider three choices of velocity-pressure pairs:
- **Method 1.** $ ({\boldsymbol}u_h, p_h)\in{\boldsymbol}V_{h}^{k}\times
Q_{h}^{k-1},\ 1\leq k \leq 5.$
- **Method 2.** $ ({\boldsymbol}u_h, p_h)\in{\boldsymbol}V_{h}^{k}\times
Q_{h}^{k},\ 1\leq k \leq 5.$
- **Method 3.** $ ({\boldsymbol}u_h, p_h)\in{\boldsymbol}V_{h}^{k}\times
Q_{h}^{0},\ 1\leq k \leq 5.$
Here the space $Q_h^0$ is just the piecewise constant space. These methods correspond to the choices $k'=k,\ k-1,\ 0$, respectively.
**Method 1.** The combination of polynomial degrees for the velocity and pressure approximation spaces is common in traditional FEM and DG while $k\geq 2$, known as Taylor-Hood elements. Numerical results for the method 1 are shown in Fig \[fig:m1infsuptest\]. $\mu_{\min}$ appears to be bounded in every case, which clearly indicates the method 1 has passed the inf-sup test. It is noticeable that ${\boldsymbol}V_{h}^{1}\times Q_{h}^{0}$ is a stable pair which will lead to the locking-phenomenon in traditional FEM.
![Inf-sup tests for method 1 on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m1infsuptest"}](./figure/infsup_tri_m_m-1.pdf "fig:"){width="48.00000%"} ![Inf-sup tests for method 1 on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m1infsuptest"}](./figure/infsup_mix_m_m-1.pdf "fig:"){width="48.00000%"}
**Method 2.** We consider equal polynomial degrees for both approximation spaces. This method is more efficient because the reconstruction procedure is carried out only once. Fig \[fig:m2infsuptest\] displays the history of $\mu_{\min}$. Similar with method 1, the values of $\mu_{\min}$ stabilize as $h$ decreases to zero. This method surprisingly keeps valid with ${\boldsymbol}V_{h}^{1}\times Q_{h}^{1}$ which is unstable due to the spurious pressure models in traditional FEM.
![Inf-sup tests for method 2 on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m2infsuptest"}](./figure/infsup_tri_m_m.pdf "fig:"){width="48.00000%"} ![Inf-sup tests for method 2 on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m2infsuptest"}](./figure/infsup_mix_m_m.pdf "fig:"){width="48.00000%"}
**Method 3.** We note that the number of DOFs of our finite element space, which are always equal to the number of elements in partition, has no concern to the order of approximation accuracy. In the sense of that, for all $k$, high order space $V_{h}^{k}$ is in the same size as the piecewise constant piece $Q_{k}^{0}$. Thus, we take ${\boldsymbol}V_{h}^{k}$ as the velocity approximation space while we select $Q_{h}^{0}$ for the pressure. Fig \[fig:m3infsuptest\] summarizes the results of this inf-sup test, which show that the inf-sup condition holds.
![Inf-sup tests for method 3 on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m3infsuptest"}](./figure/infsup_tri_m_0.pdf "fig:"){width="48.00000%"} ![Inf-sup tests for method 3 on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m3infsuptest"}](./figure/infsup_mix_m_0.pdf "fig:"){width="48.00000%"}
The satisfaction of the inf-sup condition has been checked in this section by the numerical tests. All experiments show that the inf-sup value $\mu_{\min}$ is bounded. In fact, the combination of two approximation spaces can be more flexible, such as ${\boldsymbol}V_{h}^{k}\times Q_{h}^{k+1}$ or ${\boldsymbol}V_{h}^{k}\times Q_{h}^{k+2}$, see Fig \[fig:m4infsuptest\] and Fig \[fig:m5infsuptest\]. Both cases could pass the inf-sup test. The numerical results demonstrate that our finite element space possesses more robust properties than the traditional finite element method. An analytical proof of the verification of the inf-sup condition is considered as the future work.
![Inf-sup tests for ${\boldsymbol}V_{h}^{k}\times Q_{h}^{k+1}$ on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m4infsuptest"}](./figure/infsup_tri_m_m+1.pdf "fig:"){width="48.00000%"} ![Inf-sup tests for ${\boldsymbol}V_{h}^{k}\times Q_{h}^{k+1}$ on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m4infsuptest"}](./figure/infsup_mix_m_m+1.pdf "fig:"){width="48.00000%"}
![Inf-sup tests for ${\boldsymbol}V_{h}^{k}\times Q_{h}^{k+2}$ on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m5infsuptest"}](./figure/infsup_tri_m_m+2.pdf "fig:"){width="48.00000%"} ![Inf-sup tests for ${\boldsymbol}V_{h}^{k}\times Q_{h}^{k+2}$ on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m5infsuptest"}](./figure/infsup_mix_m_m+2.pdf "fig:"){width="48.00000%"}
Numerical Results {#sec:numericalresults}
=================
In this section, we give [ some implementation details and]{} some numerical examples in two dimensions to verify the theoretical error estimates in Theorem \[th:prioriestimate\]. The numerical settings remain unchanged as in the previous section. For the resulting sparse system, a direct sparse solver is employed to solve it.
Implementation {#sec:2dexample}
--------------
We present a 2D example on the domain $[0, 1]\times[0, 1]$ to illustrate the implementation of our method. The key point is to calculate the basis functions. We consider a quasi-uniform triangular mesh, see Fig \[fig:2dquasi\_uniform\].
![The triangulation and the element patch $S(K_0)$ and the collocation points set ${\mathcal{I}}_{K_0}$ (left) / the basis function $\lambda_{K_0}$ (right)[]{data-label="fig:2dquasi_uniform"}](./figure/tri_patch.pdf "fig:"){width="48.00000%"} ![The triangulation and the element patch $S(K_0)$ and the collocation points set ${\mathcal{I}}_{K_0}$ (left) / the basis function $\lambda_{K_0}$ (right)[]{data-label="fig:2dquasi_uniform"}](./figure/K0_basis.pdf "fig:"){width="48.00000%" height="48.10000%"}
Here we consider a linear reconstruction. The barycenters of all elements are assigned as the collocation points. For any element $K$, we let $S(K)$ consist of $K$ itself and all edge-neighboring elements. Then we obtain the basis functions by solving the least squares problem on every element.
We take $K_0$ as an example (see Fig \[fig:2dquasi\_uniform\]), the element patch $S(K_0)$ is chosen as $$S(K_0) = \left\{ K_{0}, K_{1}, K_{2},K_{3} \right\},$$ and the corresponding collocation points are $${\mathcal{I}}_{K_0} = \left\{ (x_{K_{0}}, y_{K_{0}}),(x_{K_{1}},
y_{K_{1}}),(x_{K_{2}}, y_{K_{2}}),(x_{K_{3}}, y_{K_{3}}) \right\},$$ where $(x_{K_i}, y_{K_i})$ is the barycenter of $K_i$.
For a continuous function $g$, the least squares problem is $${\mathcal{R}}_{K_0} = \mathop{\arg \min}_{ (a, b, c) \in \mathbb R}
\sum_{(x_{K'},y_{K'}) \in {\mathcal{I}}_{K_0}} |g(x_{K'},y_{K'}) -
(a + bx_{K'} + cy_{K'})|^2.$$ By the Assumption 1, we obtain the unique solution $$[a, b, c]^T = (A^TA)^{-1}A^Tq,$$ where $$A = \begin{bmatrix} 1 & x_{K_{0}}& y_{K_{0}} \\ 1 & x_{K_{1}}&
y_{K_{1}} \\ 1 & x_{K_{2}} & y_{K_{2}}\\ 1 & x_{K_{3}} & y_{K_{3}}
\end{bmatrix}, \quad
q = \begin{bmatrix} g(x_{K_{0}},y_{K_{0}}) \\g(x_{K_{1}},y_{K_{1}})
\\g(x_{K_{2}},y_{K_{2}}) \\g(x_{K_{3}},y_{K_{3}})
\end{bmatrix}.$$ Thus the matrix $(A^TA)^{-1}A^T$ contains all necessary information of the basis functions $\lambda_{K_0}, \lambda_{K_1}, \lambda_{K_2},
\lambda_{K_3}$ on $K_0$ and we just store it to represent the basis functions. All the basis functions could be obtained by solving the least squares problem on every element. Besides, the basis function $\lambda_{K_0}$ is presented in Fig \[fig:2dquasi\_uniform\] and we shall point out that the support of the basis function is not always equal to the element patch, and vice versa.
2D smooth problem
-----------------
We first consider a 2D example on $\Omega=[0,1]^2$ with smooth analytical solution to investigate the convergence properties. The exact solution is taken as $${\boldsymbol}u(x,y)=\begin{bmatrix} \sin(2\pi x)\cos(2\pi y) \\ -\cos(2\pi
x)\sin(2\pi y) \\
\end{bmatrix}, \quad
p(x,y)=x^2+y^2,$$ and the source term ${\boldsymbol}f$ and the boundary condition ${\boldsymbol}g$ are chosen accordingly. We consider three methods in Section \[sec:infsuptest\] and solve the Stokes problem on the given triangular meshes and mixed meshes, respectively, with mesh size $h=\frac1n, n=10, 20, 40, 80$.
In Fig \[fig:m1L2error\] and Fig \[fig:m1DGerror\], we present the $L^2$ norm and the DG energy norm of the error in the approximation to the exact velocity on both meshes when using method 1. And Fig \[fig:m1Pressureerror\] shows the pressure error in $L^2$ norm. Here we observe that the optimal convergence rates for $\|{\boldsymbol}u - {\boldsymbol}u_h\|_{L^2( \Omega)}$, $\|{\boldsymbol}u - {\boldsymbol}u_h\|_{\mathrm{DG}}$ and $\|p -
p_h\|_{L^2(\Omega)}$ are obtained, which are $O(h^{k+1})$, $O(h^{k})$ and $O(h^k)$, respectively. The numerical results confirm the estimate .
![Velocity $L^2$ norm error with method 1 for the smooth case on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m1L2error"}](./figure/VelocityL2_tri_m_m-1.pdf "fig:"){width="48.00000%"} ![Velocity $L^2$ norm error with method 1 for the smooth case on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m1L2error"}](./figure/VelocityL2_mix_m_m-1.pdf "fig:"){width="48.00000%"}
![Velocity DG energy norm error with method 1 for the smooth case on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m1DGerror"}](./figure/VelocityDG_tri_m_m-1.pdf "fig:"){width="48.00000%"} ![Velocity DG energy norm error with method 1 for the smooth case on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m1DGerror"}](./figure/VelocityDG_mix_m_m-1.pdf "fig:"){width="48.00000%"}
![Pressure $L^2$ norm error with method 1 for the smooth case on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m1Pressureerror"}](./figure/PressureL2_tri_m_m-1.pdf "fig:"){width="48.00000%"} ![Pressure $L^2$ norm error with method 1 for the smooth case on triangular meshes (left) / mixed meshes (right)[]{data-label="fig:m1Pressureerror"}](./figure/PressureL2_mix_m_m-1.pdf "fig:"){width="48.00000%"}
Now we consider the method 2, the convergence rates are displayed in Fig \[fig:m2L2error\], \[fig:m2DGerror\] and \[fig:m2Pressureerror\]. All convergence orders are identical to the results in method , which agrees with the developed theory. For this method, the approximation to the pressure converges in a suboptimal way, but we build the approximation space only once which makes the method more effective.
![Velocity $L^2$ norm error with method 2 for the smooth case on triangular meshes(left) /mixed meshes(right)[]{data-label="fig:m2L2error"}](./figure/VelocityL2_tri_m_m.pdf "fig:"){width="48.00000%"} ![Velocity $L^2$ norm error with method 2 for the smooth case on triangular meshes(left) /mixed meshes(right)[]{data-label="fig:m2L2error"}](./figure/VelocityL2_mix_m_m.pdf "fig:"){width="48.00000%"}
![Velocity DG energy norm error with method 2 for the smooth case on triangular meshes(left)/mixed meshes(right)[]{data-label="fig:m2DGerror"}](./figure/VelocityDG_tri_m_m.pdf "fig:"){width="48.00000%"} ![Velocity DG energy norm error with method 2 for the smooth case on triangular meshes(left)/mixed meshes(right)[]{data-label="fig:m2DGerror"}](./figure/VelocityDG_mix_m_m.pdf "fig:"){width="48.00000%"}
![Pressure $L^2$ norm error with method 2 for the smooth case on triangular meshes (left)/mixed meshes(right)[]{data-label="fig:m2Pressureerror"}](./figure/PressureL2_tri_m_m.pdf "fig:"){width="48.00000%"} ![Pressure $L^2$ norm error with method 2 for the smooth case on triangular meshes (left)/mixed meshes(right)[]{data-label="fig:m2Pressureerror"}](./figure/PressureL2_mix_m_m.pdf "fig:"){width="48.00000%"}
Finally, we investigate the numerical performance of the method 3. The errors are plotted in Fig \[fig:m3L2error\], \[fig:m3DGerror\] and \[fig:m3Pressureerror\]. Here the theoretical convergence rates under norm $\|{\boldsymbol}u - {\boldsymbol}u_h\|_{\mathrm{DG}}$ and $\|p-p_h\|_{L^2(\Omega)}$ are $O(h^1)$. We observe that the numerical results do not coincide with the theory exactly which results from the numerical error in approximation to pressure is much larger than the interpolation error. The super convergence is spurious and the convergence orders will drop to the expected values as the mesh size $h$ approaches to zero. But it does not imply that the high order is not preferred in method , [the results show that using ${\boldsymbol}V_{h}^{k}$ with larger $k$ could give a more accurate approximation to the velocity and the pressure.]{}
![Velocity $L^2$ norm error on with method 3 for the smooth case triangular meshes(left)/mixed meshes(right)[]{data-label="fig:m3L2error"}](./figure/VelocityL2_tri_m_0.pdf "fig:"){width="48.00000%"} ![Velocity $L^2$ norm error on with method 3 for the smooth case triangular meshes(left)/mixed meshes(right)[]{data-label="fig:m3L2error"}](./figure/VelocityL2_mix_m_0.pdf "fig:"){width="48.00000%"}
![Velocity DG energy norm error with method 3 for the smooth case on triangular meshes(left)/mixed meshes(right)[]{data-label="fig:m3DGerror"}](./figure/VelocityDG_tri_m_0.pdf "fig:"){width="48.00000%"} ![Velocity DG energy norm error with method 3 for the smooth case on triangular meshes(left)/mixed meshes(right)[]{data-label="fig:m3DGerror"}](./figure/VelocityDG_mix_m_0.pdf "fig:"){width="48.00000%"}
![Pressure $L^2$ norm error on with method 3 for the smooth case triangular meshes (left)/mixed meshes(right)[]{data-label="fig:m3Pressureerror"}](./figure/PressureL2_tri_m_0.pdf "fig:"){width="48.00000%"} ![Pressure $L^2$ norm error on with method 3 for the smooth case triangular meshes (left)/mixed meshes(right)[]{data-label="fig:m3Pressureerror"}](./figure/PressureL2_mix_m_0.pdf "fig:"){width="48.00000%"}
Driven cavity problem
---------------------
The driven cavity problem is a standard benchmark test for the incompressible flow. It models a plane flow of an isothermal fluid in a unit square lid-driven cavity. The domain $\Omega$ is $[0,1]^2$ and the boundary condition and the source term are given by $${\boldsymbol}g(x,y)=\begin{cases} (1,0)^T,\quad
0<x<1,\ y=1,\\ (0,0)^T,\quad\text{otherwise},\\
\end{cases}\quad {\boldsymbol}f(x,y)=\begin{bmatrix}
0\\0\\
\end{bmatrix}.$$
The domain is partitioned by triangular mesh with mesh size $h=\frac1{60}$. Fig \[fig:m1liddriven\] shows the velocity vectors and the streamline of the flow for the discretization of ${\boldsymbol}V_{h}^3\times Q_{h}^2$. Fig \[fig:m2liddriven\] and \[fig:m3liddriven\] present the results for the pair of ${\boldsymbol}V_{h}^3\times Q_{h}^3$ and ${\boldsymbol}V_h^3\times Q_h^0$, respectively.
![Velocity vectors (left) and the streamline of the flow (right) for ${\boldsymbol}V_h^3\times Q_h^2$[]{data-label="fig:m1liddriven"}](./figure/lid_v_3-2.pdf "fig:"){width="48.00000%"} ![Velocity vectors (left) and the streamline of the flow (right) for ${\boldsymbol}V_h^3\times Q_h^2$[]{data-label="fig:m1liddriven"}](./figure/lid_streamline_3-2.pdf "fig:"){width="48.00000%"}
![Velocity vectors(left) and the streamline of the flow(right) for ${\boldsymbol}V_h^3\times Q_{h}^3$[]{data-label="fig:m2liddriven"}](./figure/lid_v_3-3.pdf "fig:"){width="48.00000%"} ![Velocity vectors(left) and the streamline of the flow(right) for ${\boldsymbol}V_h^3\times Q_{h}^3$[]{data-label="fig:m2liddriven"}](./figure/lid_streamline_3-3.pdf "fig:"){width="48.00000%"}
![Velocity vectors(left) and the streamline of the flow(right) for ${\boldsymbol}V_{h}^3\times Q_{h}^0$[]{data-label="fig:m3liddriven"}](./figure/lid_v_3-0.pdf "fig:"){width="48.00000%"} ![Velocity vectors(left) and the streamline of the flow(right) for ${\boldsymbol}V_{h}^3\times Q_{h}^0$[]{data-label="fig:m3liddriven"}](./figure/lid_streamline_3-0.pdf "fig:"){width="48.00000%"}
Non-smooth problem
------------------
In this example, we investigate the performance of our method dealing with the Stokes problem with a corner singularity in the analytical solution. Let $\Omega$ be the L-shaped domain $[-1,1]\times[-1,1]\backslash [0,1)\times(-1,0]$ and the meshes we use, which are generated by *gmsh*, are refinements of a triangular mesh of 250 triangles(see Fig \[fig:Lshape\]). The exact solution(from [@verfurth1996review; @hansbo2008piecewise]) is given by $${\boldsymbol}u(r, \theta)=r^\lambda\begin{bmatrix}
(1+\lambda)\sin(\theta)\psi(\theta)+\cos(\theta)\psi'(\theta)
\\ \sin(\theta)\psi'(\theta)-(1+\lambda)\cos(\theta)\psi(\theta) \\
\end{bmatrix},$$ in polar coordinates, where $$\begin{aligned}
\psi(\theta)=&\frac1{1+\lambda}\sin( (1+\lambda)\theta)
\cos(\lambda\omega)-\cos(
(1+\lambda)\theta)\\ &-\frac1{1-\lambda}\sin( (1-\lambda)\theta)
\cos(\lambda\omega) + \cos( (1-\lambda)\theta),\\
\end{aligned}$$ with $\omega=\frac32\pi$ and $\lambda\approx0.5444837$ as the smallest positive root to $$\sin(\lambda\omega) + \lambda\sin\omega=0.$$ At the corner $(0,0)$, the exact solution contains a singularity which indicates ${\boldsymbol}u(r, \theta)$ does not belong to $H^{2}(\Omega)$.
![The triangular meshes of L-shaped domain, 250 elements (left)/ 1000 elements (right)[]{data-label="fig:Lshape"}](./figure/L1-crop.pdf "fig:"){width="40.00000%"} ![The triangular meshes of L-shaped domain, 250 elements (left)/ 1000 elements (right)[]{data-label="fig:Lshape"}](./figure/L2-crop.pdf "fig:"){width="40.00000%"}
The $\# S(K)$ is chosen also as the Tab \[tab:patchnumber2d\] shows. In Tab \[tab:Lshapeerror\] we list the $L^2$ norm error of the velocity against the degrees of freedom for different pairs of approximation spaces. We observe that all convergence orders are about 1, which are consistent with the results in [@hansbo2008piecewise] where a piece divergence-free discontinuous Galerkin method is developed to solve this problem.
\[tab:Lshapeerror\]
Conclusion
==========
In this paper, we have introduced a new discontinuous Galerkin method to solve the Stokes problem. A novelty of this method is the new piecewise polynomial space that is reconstructed by solving local least squares problem. A variety of numerical inf-sup tests demonstrate the stability of this method. The optimal error estimates in $L^2$ norm and DG energy norm are presented and the numerical results are reported to show good agreement with the theoretical predictions.
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D. Boffi, F. Brezzi, M. Fortin, [[Mixed Finite Element Methods and Applications]{}](https://doi.org/10.1007/978-3-642-36519-5), Vol. 44 of Springer Series in Computational Mathematics, Springer, Heidelberg, 2013. <https://doi.org/10.1007/978-3-642-36519-5>
V. Girault, P. A. Raviart, [Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms](http://dx.doi.org/10.1007/978-3-642-61623-5), Springer-Verlag, 1986. <http://dx.doi.org/10.1007/978-3-642-61623-5>
C. Taylor, P. Hood, [A numerical solution of the [N]{}avier-[S]{}tokes equations using the finite element technique](https://doi.org/10.1016/0045-7930(73)90027-3), Internat. J. Comput. & Fluids 1 (1) (1973) 73–100. <https://doi.org/10.1016/0045-7930(73)90027-3>
B. Cockburn, G. E. Karniadakis, C. W. Shu, [The development of discontinuous [G]{}alerkin methods](https://doi.org/10.1007/978-3-642-59721-3_1), in: Discontinuous [G]{}alerkin methods ([N]{}ewport, [RI]{}, 1999), Vol. 11 of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2000, pp. 3–50. <https://doi.org/10.1007/978-3-642-59721-3_1>
P. Hansbo, M. G. Larson, [Discontinuous [G]{}alerkin methods for incompressible and nearly incompressible elasticity by [N]{}itsche’s method](https://doi.org/10.1016/S0045-7825(01)00358-9), Comput. Methods Appl. Mech. Engrg. 191 (17-18) (2002) 1895–1908. <https://doi.org/10.1016/S0045-7825(01)00358-9>
A. Toselli, [[$hp$]{} discontinuous [G]{}alerkin approximations for the [S]{}tokes problem](https://doi.org/10.1142/S0218202502002240), Math. Models Methods Appl. Sci. 12 (11) (2002) 1565–1597. <https://doi.org/10.1142/S0218202502002240>
D. Schötzau, C. Schwab, A. Toselli, [Mixed [$hp$]{}-[DGFEM]{} for incompressible flows](https://doi.org/10.1137/S0036142901399124), SIAM J. Numer. Anal. 40 (6) (2002) 2171–2194 (2003). <https://doi.org/10.1137/S0036142901399124>
B. Cockburn, G. Kanschat, D. Schötzau, C. Schwab, [Local discontinuous [G]{}alerkin methods for the [S]{}tokes system](https://doi.org/10.1137/S0036142900380121), SIAM J. Numer. Anal. 40 (1) (2002) 319–343. <https://doi.org/10.1137/S0036142900380121>
J. Carrero, B. Cockburn, D. Schötzau, [Hybridized globally divergence-free [LDG]{} methods. [I]{}. [T]{}he [S]{}tokes problem](https://doi.org/10.1090/S0025-5718-05-01804-1), Math. Comp. 75 (254) (2006) 533–563. <https://doi.org/10.1090/S0025-5718-05-01804-1>
P. Lederer, C. Lehrenfeld, J. Schöberl, [Hybrid discontinuous [G]{}alerkin methods with relaxed h(div)-conformity for incompressible flows. part [I]{}](https://doi.org/10.1137/17M1138078), SIAM Journal on Numerical Analysis 56 (4) (2018) 2070–2094. <https://doi.org/10.1137/17M1138078>
N. Nguyen, J. Peraire, B. Cockburn, [A hybridizable discontinuous [G]{}alerkin method for stokes flow](https://doi.org/10.1016/j.cma.2009.10.007), Computer Methods in Applied Mechanics and Engineering 199 (9) (2010) 582 – 597. <https://doi.org/10.1016/j.cma.2009.10.007>
B. Cockburn, J. Gopalakrishnan, R. Lazarov, [Unified hybridization of discontinuous galerkin, mixed, and continuous galerkin methods for second order elliptic problems](https://doi.org/10.1137/070706616), SIAM Journal on Numerical Analysis 47 (2) (2009) 1319–1365. <https://doi.org/10.1137/070706616>
G. A. Baker, W. N. Jureidini, O. A. Karakashian, [Piecewise solenoidal vector fields and the [S]{}tokes problem](https://doi.org/10.1137/0727085), SIAM J. Numer. Anal. 27 (6) (1990) 1466–1485. <https://doi.org/10.1137/0727085>
O. A. Karakashian, W. N. Jureidini, [A nonconforming finite element method for the stationary [N]{}avier-[S]{}tokes equations](https://doi.org/10.1137/S0036142996297199), SIAM J. Numer. Anal. 35 (1) (1998) 93–120. <https://doi.org/10.1137/S0036142996297199>
B. Cockburn, J. Gopalakrishnan, [Incompressible finite elements via hybridization. [I]{}. [T]{}he [S]{}tokes system in two space dimensions](https://doi.org/10.1137/04061060X), SIAM J. Numer. Anal. 43 (4) (2005) 1627–1650. <https://doi.org/10.1137/04061060X>
B. Cockburn, J. Gopalakrishnan, [Incompressible finite elements via hybridization. [II]{}. [T]{}he [S]{}tokes system in three space dimensions](https://doi.org/10.1137/040610659), SIAM J. Numer. Anal. 43 (4) (2005) 1651–1672. <https://doi.org/10.1137/040610659>
B. Cockburn, G. Kanschat, D. Schötzau, [A note on discontinuous [G]{}alerkin divergence-free solutions of the [N]{}avier-[S]{}tokes equations](https://doi.org/10.1007/s10915-006-9107-7), J. Sci. Comput. 31 (1-2) (2007) 61–73. <https://doi.org/10.1007/s10915-006-9107-7>
A. Montlaur, S. Fernandez-Mendez, A. Huerta, [Discontinuous [G]{}alerkin methods for the [S]{}tokes equations using divergence-free approximations](https://doi.org/10.1002/fld.1716), Internat. J. Numer. Methods Fluids 57 (9) (2008) 1071–1092. <https://doi.org/10.1002/fld.1716>
J. Liu, [Penalty-factor-free discontinuous [G]{}alerkin methods for 2-dim [S]{}tokes problems](https://doi.org/10.1137/10079094X), SIAM J. Numer. Anal. 49 (5) (2011) 2165–2181. <https://doi.org/10.1137/10079094X>
O. C. Zienkiewicz, R. L. Taylor, S. J. Sherwin, J. Peiró, [On discontinuous [G]{}alerkin methods](https://doi.org/10.1002/nme.884), Internat. J. Numer. Methods Engrg. 58 (8) (2003) 1119–1148. <https://doi.org/10.1002/nme.884>
A. D. V. D. Montlaur, High-order discontinuous galerkin methods for incompressible flows, Ph.D. thesis, Universitat Politècnica de Catalunya (2009).
R. Li, P. B. Ming, F. Tang, [An efficient high order heterogeneous multiscale method for elliptic problems](https://doi.org/10.1137/110836626), Multiscale Model. Simul. 10 (1) (2012) 259–283. <https://doi.org/10.1137/110836626>
R. [Li]{}, P. [Ming]{}, Z. [Sun]{}, Z. [Yang]{}, [An Arbitrary-Order Discontinuous Galerkin Method with One Unknown Per Element]{}, ArXiv e-prints[](http://arxiv.org/abs/1803.00378).
K.-J. Bathe, A. Iosilevich, D. Chapelle, [An inf-sup test for shell finite elements](http://dx.doi.org/10.1016/S0045-7949(99)00213-8), Computers & Structures 75 (5) (2000) 439–456. <http://dx.doi.org/10.1016/S0045-7949(99)00213-8>
D. Chapelle, K.-J. Bathe, [The inf-sup test](https://doi.org/10.1016/0045-7949(93)90340-J), Comput. & Structures 47 (4-5) (1993) 537–545. <https://doi.org/10.1016/0045-7949(93)90340-J>
F. Brezzi, A. Buffa, K. Lipnikov, [Mimetic finite differences for elliptic problems](http://dx.doi.org/10.1051/m2an:2008046), ESAIM Numer. Anal. 43 (2009) 277–295. <http://dx.doi.org/10.1051/m2an:2008046>
L. Beirão da Veiga, K. Lipnikov, G. Manzini, [The [M]{}imetic [F]{}inite [D]{}ifference [M]{}ethod for [E]{}lliptic [P]{}roblems](https://doi.org/10.1007/978-3-319-02663-3), Vol. 11 of MS&A. Modeling, Simulation and Applications, Springer, Cham, 2014. <https://doi.org/10.1007/978-3-319-02663-3>
P. G. Ciarlet, [The [F]{}inite [E]{}lement [M]{}ethod for [E]{}lliptic [P]{}roblems](http://dx.doi.org/10.1115/1.3424474), North-Holland, Amsterdam, 1978. <http://dx.doi.org/10.1115/1.3424474>
P. F. Antonietti, S. Giani, P. Houston, [[$hp$]{}-version composite discontinuous [G]{}alerkin methods for elliptic problems on complicated domains](https://doi.org/10.1137/120877246), SIAM J. Sci. Comput. 35 (3) (2013) A1417–A1439. <https://doi.org/10.1137/120877246>
R. Li, P. B. Ming, Z. Y. Sun, Z. J. Yang, [An arbitrary-order discontinuous [G]{}alerkin method with one unknown per element](http://adsabs.harvard.edu/abs/2018arXiv180300378L), arXiv:1803.00378[](http://arxiv.org/abs/1803.00378). <http://adsabs.harvard.edu/abs/2018arXiv180300378L>
D. N. Arnold, [An interior penalty finite element method with discontinuous elements](https://doi.org/10.1137/0719052), SIAM J. Numer. Anal. 19 (4) (1982) 742–760. <https://doi.org/10.1137/0719052>
D. N. Arnold, F. Brezzi, B. Cockburn, L. D. Marini, [Unified analysis of discontinuous [G]{}alerkin methods for elliptic problems](https://doi.org/10.1137/S0036142901384162), SIAM J. Numer. Anal. 39 (5) (2001/02) 1749–1779. <https://doi.org/10.1137/S0036142901384162>
D. S. Malkus, [Eigenproblems associated with the discrete [LBB]{} condition for incompressible finite elements](https://doi.org/10.1016/0020-7225(81)90013-6), Internat. J. Engrg. Sci. 19 (10) (1981) 1299–1310. <https://doi.org/10.1016/0020-7225(81)90013-6>
C. Geuzaine, J. F. Remacle, [Gmsh: [A]{} 3-[D]{} finite element mesh generator with built-in pre- and post-processing facilities](https://doi.org/10.1002/nme.2579), Internat. J. Numer. Methods Engrg. 79 (11) (2009) 1309–1331. <https://doi.org/10.1002/nme.2579>
R. Verf[ü]{}rth, [A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques]{}, Wiley-Teubner, 1996.
P. Hansbo, M. G. Larson, [Piecewise divergence-free discontinuous [G]{}alerkin methods for [S]{}tokes flow](https://doi.org/10.1002/cnm.975), Comm. Numer. Methods Engrg. 24 (5) (2008) 355–366. <https://doi.org/10.1002/cnm.975>
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---
abstract: 'This letter advances the non-orthogonal multiple access (NOMA) technique for cellular downlink co-channel interference mitigation, via exploiting the (limited) cooperation among base stations (BSs). Specifically, we consider a simplified but practically relevant scenario of two co-channel cells with [***asymmetric interference***]{}, i.e., only the user in one cell receives the strong interference from the BS in the other cell. To mitigate such interference, we propose a new [***cooperative NOMA***]{} scheme, where the interfered user’s serving BS sends a superposed signal comprising both the desired message and the co-channel user’s message (shared by the interfering BS). The co-channel user’s signal is aimed to add constructively with the interfering BS’s signal at the interfered user’s receiver so that the combined interference with enhanced power can be effectively decoded and cancelled. This thus leads to a new problem on how to optimally allocate the transmit power for the two superposed messages. We provide the closed-form solution to this problem and investigate the conditions under which the performance of the proposed scheme is superior over the existing schemes.'
author:
- 'Weidong Mei and Rui Zhang, [^1] [^2] [^3]'
bibliography:
- 'Coop\_NOMA.bib'
title: Cooperative NOMA for Downlink Asymmetric Interference Cancellation
---
Cooperative NOMA, cellular downlink, asymmetric interference cancellation, power allocation.
Introduction
============
Thanks to its ability to realize massive connectivity, low latency and high spectral efficiency in wireless communications, non-orthogonal multiple access (NOMA) technique has been recognized as a key enabler for future cellular networks. As such, NOMA has drawn a great deal of attention from both academia and industry (see, e.g., [@islam2017power; @ding2017survey; @3GPP36859] and the references therein). However, most of the existing studies on NOMA are limited to the single-cell setup, while only a handful of works have recently addressed the more challenging multi-cell scenario (see e.g., [@choi2014non; @tian2016performance; @ali2018downlink; @shin2016coordinated]). For multi-cell NOMA, inter-cell interference (ICI) is a major issue as it makes the successive interference cancellation (SIC) design far more complicated as compared to the single-cell case without ICI, especially for cell-edge user equipments (UEs) that generally suffer strong co-channel interference from other cells.
To resolve the above issue, NOMA has been combined with various interference mitigation techniques such as ICI coordination (ICIC) and cooperative multi-point (CoMP), generally referred to as cooperative NOMA, to reap its benefits as in the single-cell system. Specifically, cooperative NOMA involves multiple BSs to serve the cell-edge UEs at the same time by leveraging the message sharing among cooperating BSs. For example, in [@choi2014non], the author proposed a coordinated superposition coding scheme in a two-cell downlink network, where a cell-edge UE is served by two BSs via Alamouti code. To reduce the complexity of multi-user NOMA, an opportunistic NOMA scheme was proposed in [@tian2016performance], where each cell-edge UE is allowed to select its own preferred set of serving BSs. Furthermore, the work [@ali2018downlink] developed a general cooperation model with coexisting CoMP and non-CoMP UEs, and NOMA is applied at each BS to schedule their communications over the same resource block (RB). The authors in [@shin2016coordinated] proposed two interference alignment-based cooperative NOMA schemes so as to completely eliminate the ICI suffered by cell-edge UEs.
As shown in Fig.\[down\], we consider in this letter a simplified two-cell system with [*asymmetric interference*]{}, where only the downlink transmission from BS1 to UE1 is strongly interfered by that from BS2 to UE2. This scenario can occur in practice, e.g., when UE1 is at the cell edges of both BS1 and BS2 and the distance between BS1 and UE2 is much longer than that between BS1/BS2 and UE1 (see Fig.\[down\](a)). Alternatively, if UE1 is an unmanned aerial vehicle (UAV)[@cooperative2019mei], then it suffers much stronger ground-to-air interference from the co-channel BS2 as compared to the terrestrial interference caused by BS1 to UE2, even if UE1 (the UAV) is much closer to its serving BS (BS1) than the interfering BS (BS2), as shown in Fig.\[down\](b). This is because for high-altitude UAVs, their channels with ground BSs are dominated by line-of-sight (LoS) propagation[@3GPP36777], and thus suffer less path-loss, shadowing and multi-path fading as compared to the typical terrestrial channels (e.g., that between BS1 and UE2 in Fig.\[down\]) with rich scatterers[^4]. In both the above scenarios, despite that UE1 can perform SIC to subtract the interference from BS2, its achievable rate can be limited for satisfying the rate demand of UE2 due to the comparable signal and interference links from BS1 and BS2, respectively. To improve the rate performance of UE1 without affecting the co-channel transmission for UE2, we propose a new cooperation scheme by exploiting only one-sided message sharing from BS2 to BS1 (as opposed to the two-sided message sharing between the two BSs required by CoMP-based NOMA as in [@choi2014non; @tian2016performance; @ali2018downlink; @shin2016coordinated]). Specifically, BS2 via the backhaul link shares the data symbol for UE2 with BS1, which then transmits a superposed signal comprising both UE1’s and UE2’s data symbols. As a result, UE2’s signal can be added destructively or constructively with BS2’s interference at UE1’s receiver to suppress it (for decoding UE1’s signal directly) or enhance it (for decoding and cancelling UE2’s signal first before decoding UE1’s signal), respectively. In both cases, BS1’s transmit power allocations for UE1’s and UE2’s signals need to be optimized accordingly to maximize the achievable rate of UE1. For the former case, the optimal power allocation has been derived in our prior work [@cooperative2019mei], while in this letter we address this problem in the latter case. Note that in this case, the proposed cooperative NOMA resembles the conventional NOMA [@islam2017power; @ding2017survey; @3GPP36859] in the sense that BS1 sends a superposed signal of both UEs, and the near UE (UE1) applies SIC to decode and cancel the interference due to the far UE (UE2). However, a key difference between them lies in that the transmitted UE2’s signal in our scheme is not intended for UE2 (as in conventional NOMA), but instead for enhancing the combined interference at UE1’s receiver to facilitate its SIC. We compare the proposed cooperative NOMA scheme with other existing schemes both analytically and numerically to characterize the conditions under which it gives superior rate performance.
[*Notations:*]{} For a complex number $s$, $\lvert s \rvert$ denotes its amplitude, $\angle s$ denotes its phase, and $s \sim \mathcal{CN}(\mu,\sigma^2)$ means that it is a circularly symmetric complex Gaussian (CSCG) random variable with mean $\mu$ and variance $\sigma^2$. ${\mathbb E}[\cdot]$ denotes the expected value of random variables.
System Model and Problem Formulation
====================================
As shown in Fig.\[down\], we consider a two-cell setting[^5], where two BSs (BS1 and BS2) serve their respective UEs (UE1 and UE2) in the downlink over the same time-frequency RB. We assume that each BS employs an antenna array with fixed directional gain pattern, while each UE has a single antenna. We consider the asymmetric interference scenario as explained in Section [slowromancap1@]{}, where UE1 receives the strong co-channel interference from BS2, while the interference from BS1 to UE2 is negligible and thus ignored. It is assumed that the downlink transmission from BS2 to UE2 has already started over the considered RB before BS1 serves UE1 using the same RB. As such, we consider that BS2 cannot change its transmission to UE2 and thus the CoMP-based cooperative NOMA[@choi2014non; @tian2016performance; @ali2018downlink; @shin2016coordinated] is not applicable. Nonetheless, BS2 can help BS1 mitigate its interference to UE1 by sharing some useful information via their backhaul link (e.g., the X2 interface in Long Term Evolution (LTE)[@dahlman20134g]), such as UE2’s data symbol and the channel gain from it to UE1, which thus enables our proposed cooperative NOMA scheme. For ease of reference, the main symbols used in this letter are listed in Table \[variable\].
Symbol Description
------------------------------ --------------------------------------------------------
$P$ Maximum transmit power of BS1
$Q$ Maximum transmit power of BS2
$h_1$ Baseband equivalent channel from BS1 to UE1
$h_2$ Baseband equivalent channel from BS2 to UE1
$x_1$ Complex data symbol for UE1
$x_2$ Complex data symbol for UE2
$\sigma^2$ UE1’s receiver noise power
$\gamma^\star_{1,\text{S}i}$ UE1’s maximum SINR achievable by scheme $i, i=1,2,3,4$
$w_1$ Complex weight for transmitting $x_1$ by BS1
$w_2$ Complex weight for transmitting $x_2$ by BS1
$v_1$ Amplitude of of $w_1$, i.e., $\lvert w_1 \rvert$
$v_2$ Amplitude of of $w_2$, i.e., $\lvert w_2 \rvert$
$\rho$ Minimum SINR required for decoding UE2’s message
: List of Main Symbols[]{data-label="variable"}
Conventional Schemes {#conv.sc}
--------------------
First, we characterize the achievable signal-to-interference-plus-noise ratio (SINR) of UE1 via the conventional single-cell schemes by BS1 without BS2’s cooperation. Two schemes are considered, with or without SIC applied at UE1. Let $h_1$ be the complex-valued baseband equivalent channel gain from BS1 to UE1, and $h_2$ be that from BS2 to UE1. Let $P$ and $Q$ denote the maximum transmit power of BS1 and BS2, respectively. Then, the received signal at UE1 can be expressed as $$y_1=\sqrt{P_1}h_1x_1+\sqrt{Q}h_2x_2+z_1,$$ where $P_1 \le P$ denotes the transmit power of BS1, $x_1$ and $x_2$ denote the complex-valued data symbols for UE1 and UE2 with ${\mathbb E}[{\lvert{x_1}\rvert}^2] = 1$ and ${\mathbb E}[{\lvert{x_2}\rvert}^2] = 1$, respectively, and $z_1 \sim \mathcal{CN}(0,\sigma^2)$ denotes UE1’s receiver noise with $\sigma^2$ denoting the power.
**:** If SIC is not implemented at UE1, then the co-channel interference $\sqrt{Q}h_2x_2$ is treated as Gaussian noise at its receiver. In this case, BS1 should transmit at its full power, i.e., $P_1=P$, to maximize UE1’s receive SINR, which can be expressed as $$\label{sinr1}
\gamma^\star_{1, \text{S1}} = \frac{P\lvert h_1 \rvert^2}{\sigma^2 + Q\lvert h_2 \rvert^2}.$$
**:** On the other hand, if UE1 first decodes UE2’s message and then subtracts it, UE1 will be free of co-channel interference. As a result, its receive SINR is given by $$\label{sinr2}
\gamma_{1,\text{S2}} = \frac{P_1\lvert h_1 \rvert^2}{\sigma^2}.$$ Note that to successfully cancel UE2’s signal, its receive SINR at UE1 is given by $$\label{sinr2}
\gamma_{2,\text{S2}} = \frac{Q\lvert h_2 \rvert^2}{\sigma^2 + P_1\lvert h_1 \rvert^2}.$$
As a result, the maximum receive SINR of UE1 under scheme 2 can be obtained by solving the following optimization problem $$\label{op1}
\text{(P-S2)}\;\mathop {\max}\limits_{0 \le P_1 \le P}\; \gamma_{1,\text{S2}},\quad\text{s.t.}\; \gamma_{2,\text{S2}} \ge \rho,$$ where $\rho$ denotes the minimum SINR required for decoding UE2’s message (say, at UE2’ receiver), which is assumed to be given and fixed. Obviously, (P-S2) is feasible if and only if $\frac{Q\lvert h_2 \rvert^2}{\sigma^2} \ge \rho$. Assuming that (P-S2) is feasible, its optimal solution, denoted as $P_1^\star$, can be easily shown to be $$P_1^\star=
\begin{cases}
\frac{Q{\lvert h_2 \rvert}^2}{\rho{\lvert h_1 \rvert}^2} - \frac{\sigma ^2}{{\lvert h_1 \rvert}^2}, &\text{if}\;\frac{Q\lvert h_2 \rvert^2}{\sigma^2 + P\lvert h_1 \rvert^2} \le \rho\\
P, &\text{otherwise.}
\end{cases}$$ Accordingly, its optimal value, denoted as $\gamma^\star_{1,\text{S2}}$, is given by $$\label{sinr3}
\gamma^\star_{1,\text{S2}}=\frac{P_1^\star\lvert h_1 \rvert^2}{\sigma^2}=
\begin{cases}
\frac{Q{\lvert h_2 \rvert}^2}{\rho\sigma ^2} - 1, &\text{if}\;\frac{Q\lvert h_2 \rvert^2}{\sigma^2 + P\lvert h_1 \rvert^2} \le \rho\\
\frac{P\lvert h_1 \rvert^2}{\sigma^2}, &\text{otherwise.}
\end{cases}$$
Cooperation Schemes
-------------------
Next, we consider the case where BS2 cooperatively sends $h_2$ and $x_2$ to BS1 to facilitate the interference cancellation at UE1. With $x_2$ available at BS1, it transmits the superposition of $x_1$ and $x_2$, i.e., $w_1x_1 + w_2x_2$, where $w_1$ and $w_2$ denote the complex weights. To satisfy the power constraint at BS1, it must hold that $\lvert w_1 \rvert^2 + \lvert w_2 \rvert^2 \le P$. Then, the received signal at UE1 becomes $$\label{superpose}
y_1=h_1w_1x_1 + (h_1w_2+\sqrt{Q}h_2)x_2+z_1.$$
Based on (\[superpose\]), we introduce two interference cancellation schemes for UE1, depending on whether $h_1w_2$ is designed to be in- or out-of-phase with the interference channel gain $h_2$.
**[@cooperative2019mei]:** If $h_1w_2$ is designed to be opposite to $h_2$, i.e., $\angle w_2= \angle h_2-\angle h_1 + \pi$, the interference due to $x_2$ at UE1 can be suppressed, and its receive SINR can be improved as compared to (\[sinr1\]), which is given by $$\label{sinr5}
\gamma_{1,\text{S3}}=\frac{\lvert h_1 \rvert^2\lvert w_1 \rvert^2}{\sigma ^2 + (\lvert h_2 \rvert\sqrt{Q} -\lvert h_1 \rvert \lvert w_2 \rvert)^2}.$$ For convenience, let $v_1 \triangleq \lvert w_1 \rvert$ and $v_2 \triangleq \lvert w_2 \rvert$ be the amplitude of the complex weights $w_1$ and $w_2$, respectively. Then the problem for maximizing (\[sinr5\]) can be formulated as $$\begin{aligned}
\text{(P-S3)}\mathop {\max}\limits_{v_1,v_2 \ge 0}&\; \frac{\lvert h_1 \rvert^2v^2_1}{\sigma ^2 + (\lvert h_2 \rvert\sqrt{Q} -\lvert h_1 \rvert v_2)^2}\nonumber\\
\text{s.t.}\;\;&v_1^2 + v_2^2 \le P.\label{op2}\end{aligned}$$ Notice that with $v_1 = 0$ in (P-S3), scheme 3 reduces to scheme 1. Consequently, the solution to (P-S3) should generally yield a higher receive SINR for UE1 than scheme 1 without BS2’s cooperation.
From [@cooperative2019mei], the optimal solution to (P-S3), denoted by $(v_1^\star,v_2^\star)$, is given by $$\begin{split}
v_2^\star&=\frac{X - \sqrt {X^2 - 4{\lvert h_1 \rvert}^2{\lvert h_2 \rvert}^2PQ}}{2{\lvert h_1 \rvert}{\lvert h_2 \rvert}\sqrt Q},\\
\;v_1^\star&=\sqrt {P - v_2^{\star 2}},
\end{split}$$ where $X \triangleq \sigma^2 + {\lvert h_1 \rvert}^2P + {\lvert h_2 \rvert}^2Q$. Moreover, UE1’s maximum receive SINR, denoted as $\gamma^\star_{1,\text{S3}}$, is given by $$\label{itc}
\gamma^\star_{1,\text{S3}}=\frac{-Y + \sqrt {Y^2 + 4\sigma^2P{{\lvert h_1 \rvert}^2}}}{2\sigma^2},$$ where $Y \triangleq \sigma ^2 + Q{\lvert h_2 \rvert}^2 - P{\lvert h_1 \rvert}^2$. It is worth noting that SIC is not applied at UE1’s receiver in this scheme. Moreover, it is shown in [@cooperative2019mei] that $\gamma^\star_{1,\text{S3}}$ monotonically increases with $P{\lvert h_1 \rvert}^2 - Q{\lvert h_2 \rvert}^2$. This implies that if the interference power $Q{\lvert h_2 \rvert}^2$ becomes stronger (relative to the desired signal power $P{\lvert h_1 \rvert}^2$), scheme 3 achieves lower SINR and thus becomes less effective.
**:** Note that in scheme 2, the use of SIC for canceling UE2’s signal at UE1 limits the achievable rate of UE1, especially when the desired signal power $P{\lvert h_1 \rvert}^2$ becomes comparable with the interference power $Q{\lvert h_2 \rvert}^2$ or $\rho$ is large (i.e., when the first case in (\[sinr3\]) is likely to be true). To improve over scheme 2, a new cooperative NOMA scheme, referred to as scheme 4, is proposed in this letter, where $h_1w_2$ is designed to be in-phase with the interference channel gain $h_2$, i.e., $\angle w_2 = \angle h_2 - \angle h_1$, for enhancing the combined interference due to UE2’s signal so as to cancel it more effectively by SIC at UE1’s receiver.
As a result, with scheme 4, UE1’s and UE2’s achievable SINRs with SIC can be expressed as $$\begin{split}
\gamma_{1,\text{S4}}&=\frac{\lvert h_1 \rvert^2v_1^2}{\sigma ^2},\\
\gamma_{2,\text{S4}}&=\frac{(\lvert h_1 \rvert v_2 + \sqrt{Q}\lvert h_2 \rvert)^2}{\sigma ^2 + \lvert h_1 \rvert^2 v_1^2},
\end{split}$$ respectively. To ensure that UE2’s signal can be decoded, the following inequality should be met, i.e., $\gamma_{2,\text{S4}} \ge \rho$.
The new power allocation problem for maximizing $\gamma_{1,\text{S4}}$ is thus formulated as
\[op3\] $$\begin{aligned}
\text{(P-S4)} \mathop {\max}\limits_{v_1,v_2 \ge 0}\; &\frac{\lvert h_1 \rvert^2v_1^2}{\sigma ^2} \nonumber\\
\text{s.t.}\;\;&\frac{(\lvert h_1 \rvert v_2 + \sqrt{Q}\lvert h_2 \rvert)^2}{\sigma ^2 + \lvert h_1 \rvert^2 v_1^2} \ge \rho, \label{op3a}\\
&v_1^2 + v_2^2 \le P.\label{op3b}\end{aligned}$$
Notice that with $v_2 = 0$ in problem (P-S4), (P-S4) reduces to (P-S2). Consequently, the proposed cooperative NOMA scheme generally yields a higher SINR or achievable rate for UE1 than the conventional NOMA (scheme 2) without BS2’s cooperation.
Optimal Solution and Performance Comparison
===========================================
In this section, we first derive the optimal solution to (P-S4) which achieves the maximum receive SINR of UE1 by our proposed scheme (scheme 4). Then, we compare the performance of the proposed scheme with that of scheme 3 to reveal the conditions under which the proposed scheme achieves superior performance.
Optimal Solution to (P-S4) {#opt.sol}
--------------------------
It is easy to show, by contradiction, that the constraint (\[op3b\]) must hold with equality at the optimality of (P-S4), i.e., $v_1^2+v_2^2=P$. Otherwise, we can construct a new solution $(\hat v_1, \hat v_2)=(\sqrt{v_1^2+\delta},\sqrt{v_2^2+\rho\delta})$ with $\delta=(P-v_1^2-v_2^2)/(\rho+1)$. Obviously, we have $\hat v_1^2+\hat v_2^2=P$ and $$\label{ineq1}
\frac{(\lvert h_1 \rvert \hat v_2 + \sqrt{Q}\lvert h_2 \rvert)^2}{\sigma ^2 + \lvert h_1 \rvert^2 \hat v_1^2}=\frac{(\lvert h_1 \rvert v_2 + \sqrt{Q}\lvert h_2 \rvert)^2+\lvert h_1 \rvert^2\rho\delta+2Z}{\sigma^2+\lvert h_1 \rvert^2 v_1^2+\lvert h_1 \rvert^2 \delta},$$ with $Z\triangleq\sqrt{Q}\lvert h_1 \rvert\lvert h_1 \rvert(\hat v_2-v_2)>0$. Since $(v_1, v_2)$ is a feasible solution to (P-S4), it should satisfy the constraint (\[op3a\]), i.e., $(\lvert h_1 \rvert v_2 + \sqrt{Q}\lvert h_2 \rvert)^2 \ge \rho(\sigma^2+\lvert h_1 \rvert^2 v_1^2)$. Moreover, as $\lvert h_1 \rvert^2\rho\delta+2Z>\lvert h_1 \rvert^2\rho\delta$, it follows that the right-hand side (RHS) of (\[ineq1\]) is greater than $\rho$. Hence, $(\hat v_1, \hat v_2)$ is a feasible solution to (P-S4). However, since $\hat v_1 > v_2$, this new solution yields a larger objective value of (P-S4) than $(v_1,v_2)$. This contradicts the presumption, and thus $v_1^2+v_2^2=P$ must hold at the optimality of (P-S4). By substituting $v_1^2=P-v_2^2$ into (P-S4), we obtain the following equivalent problem with only a single variable $v_2$, i.e., $$\label{op4}
\mathop {\max}\limits_{0 \le v_2 \le \sqrt{P}}\; P-v_2^2, \quad\text{s.t.}\;\;\frac{(\lvert h_1 \rvert v_2 + \sqrt{Q}\lvert h_2 \rvert)^2}{\sigma ^2 + \lvert h_1 \rvert^2(P-v_2^2)} \ge \rho,$$ where the constant term $\lvert h_1 \rvert^2/\sigma^2$ is omitted in the objective function. Since $v_2 \ge 0$, the above problem is equivalent to $$\label{op5}
\mathop {\min}\limits_{0 \le v_2 \le \sqrt{P}}\; v_2, \quad \text{s.t.}\;\; F(v_2) \ge \rho,$$ where $F(v_2) \triangleq \frac{(\lvert h_1 \rvert v_2 + \sqrt{Q}\lvert h_2 \rvert)^2}{\sigma ^2 + \lvert h_1 \rvert^2(P-v_2^2)}$.
It is easy to verify that as $v_2$ increases, the numerator and the denominator of $F(v_2)$ increase and decrease, respectively. As such, $F(v_2)$ is a monotonically increasing function of $v_2$. It then follows that problem (\[op5\]) is feasible if and only if $F(\sqrt{P}) \ge \rho$, which can be shown equivalent to $$\label{cond1}
\frac{(\sqrt{P}\lvert h_1 \rvert + \sqrt{Q}\lvert h_2 \rvert)^2}{\sigma^2} \ge \rho.$$ Moreover, if $F(0)=\frac{Q\lvert h_2 \rvert^2}{\sigma^2 + P\lvert h_1 \rvert^2} \ge \rho$, i.e., the optimal solution to problem (\[op5\]) is $v_2=0$, scheme 4 becomes equivalent to scheme 2. Finally, if $F(0) < \rho \le F(\sqrt{P})$, the optimal solution to (P2) should be the solution to the equation $F(v_2) = \rho$, or equivalently, the quadratic equation $G(v_2)=0$, where $$G(v_2) \!=\! (1+\rho)\lvert h_1 \rvert^2v_2^2+2\sqrt{Q}\lvert h_1 \rvert\lvert h_2 \rvert v_2+Q\lvert h_2 \rvert^2\!-\!\rho\sigma^2\!-\!P\rho\lvert h_1 \rvert^2\!.$$ Since $Q\lvert h_2 \rvert^2-\rho\sigma^2-P\rho\lvert h_1 \rvert^2 = (F(0)-\rho)(\sigma^2+P\lvert h_1 \rvert^2)<0$, the quadratic equation $G(v_2)=0$ only has a single positive root, which is the optimal solution to problem (\[op4\]) and given by $$\label{optSol}
v_2^*=\frac{\sqrt A - \sqrt Q \lvert h_2 \rvert}{\lvert h_1 \rvert(1 + \rho)},$$ where $A \triangleq P\rho(1+\rho)\lvert h_1 \rvert^2+\rho(\rho+1)\sigma^2-Q\rho\lvert h_2 \rvert^2$.
Correspondingly, if scheme 4 is feasible, i.e., $\rho \le F(\sqrt{P})$, UE1’s maximum receive SINR can be expressed as $$\label{sinr6}
\gamma^\star_{1,{\text{S4}}}=
\begin{cases}
\frac{\lvert h_1 \rvert^2}{\sigma ^2}(P-v_2^{*2}), &\text{if}\;\rho > F(0)\\
\frac{P\lvert h_1 \rvert^2}{\sigma^2}, &\text{otherwise}.
\end{cases}$$
It follows from (\[optSol\]) that when the interference power $Q\lvert h_2 \rvert^2$ increases, the numerator of $v_2^*$ decreases. As such, $v_2^*$ monotonically decreases with $Q\lvert h_2 \rvert^2$. This implies that UE1’s maximum receive SINR, as given in (\[sinr6\]), is non-decreasing with $Q\lvert h_2 \rvert^2$. This is in a sharp contrast to scheme 3 for which UE1’s maximum receive SINR, i.e., $\gamma^\star_{1,{\text{S3}}}$ in (\[itc\]), decreases with $Q\lvert h_2 \rvert^2$. The above observations imply that our proposed scheme (scheme 4) is more advantageous over scheme 2 or 3 when $\rho$ is larger or the interference power $Q\lvert h_2 \rvert^2$ is larger, respectively.
Performance Comparison {#perf.comp}
----------------------
Since scheme 2 is a special case of scheme 4, while scheme 1 is a special case of scheme 3, it suffices to compare the performance of our proposed scheme 4 with that of scheme 3 analytically, as pursued in this subsection. For convenience, we define $\alpha \triangleq \frac{P\lvert h_1 \rvert^2}{\sigma^2}$ and $\beta \triangleq \frac{Q\lvert h_2 \rvert^2}{\sigma^2}$.
To this end, we compare (\[itc\]) with (\[sinr6\]). Firstly, if $\rho \le F(0)$, scheme 4 (or scheme 2) outperforms scheme 3 as $\gamma^\star_{1,\text{S4}}=\frac{P\lvert h_1 \rvert^2}{\sigma^2} \ge \gamma^\star_{1,\text{S3}}$. Secondly, if $F(0) \le \rho \le F(\sqrt{P})$, it can be shown that $\gamma^\star_{1,\text{S4}} \ge \gamma^\star_{1,\text{S3}}$ if $v_2^* \le \sqrt {P\xi}$, where $\xi = \frac{(1 +\alpha + \beta)-\sqrt{{(1 - \alpha + \beta )}^2 + 4\alpha}}{2\alpha}$. Since $v_2^*$ is the unique positive root of the quadratic equation $G(v_2) = 0$, the above inequality holds if and only if $G\left(\sqrt {P\xi}\right) \ge 0$, which, after some manipulations, can be shown equivalent to $$\label{cond6}
\rho \le \frac{(1+\alpha+3\beta)-\sqrt{(1-\alpha+\beta)^2+4\alpha}+2\sqrt W}{1+\alpha-\beta+\sqrt{(1-\alpha+\beta)^2+4\alpha}},$$ where $W=2\beta(1+\alpha+\beta-\sqrt{(1-\alpha+\beta)^2 + 4\alpha})$.
By combining the results in the above two cases, it follows that scheme 4 yields a better performance than scheme 3 if the condition (\[cond6\]) is met. Since $\gamma^\star_{1,\text{S4}}$ and $\gamma^\star_{1,\text{S3}}$ monotonically increase and decrease with the interference power $Q\lvert h_2 \rvert^2$, respectively, the threshold given in the RHS of (\[cond6\]) must monotonically increase with $\beta$ or the interference power at UE1, $Q\lvert h_2 \rvert^2$, which is in accordance with our previous discussion at the end of Section \[opt.sol\], as will be also shown via numerical results in the next section.
Numerical Results
=================
In this section, numerical results are provided to evaluate the performance of the proposed cooperative NOMA scheme (scheme 4), as compared to the benchmark schemes 1, 2 and 3. We consider a cellular-connected UAV for UE1, while UE2 is a terrestrial user. Unless otherwise specified, the simulation settings are as follows. The bandwidth is set to 180 kHz, which is equal to the width of a time-frequency RB in LTE[@dahlman20134g]. The carrier frequency $f_c$ is $2$ GHz, and the noise power spectrum density at UE1’s receiver is $-164$ dBm/Hz. The height of BSs is set to be 25 in meter (m). The altitude of the UAV is fixed as 200 m. The horizontal distance between the UAV and BS1 (BS2) is 0.92 km (2.88 km). The BS antenna elements are placed vertically with half-wavelength spacing and electrically steered with 10-degree downtilt angle. The UAV-BS channels follow the probabilistic LoS channel model based on the urban macro scenario in [@3GPP36777]. The transmit power of BS1 is set to be $P=20$ dBm.
![UE1’s achievable rate versus UE2’s given rate.[]{data-label="IERate_QoS"}](Mei_WCL2019-1495_fig3.eps "fig:"){width="3.2in"} .
Fig.\[IERate\_QoS\] shows UE1’s achievable rate (defined as $\log_2(1+\text{SINR})$ in bits per second per Hertz (bps/Hz), where SINR denotes the maximum achievable SINR in each scheme) by different schemes versus UE2’s given rate, $\log_2(1+\rho)$. The transmit power of BS2 is assumed to be identical to that of BS1, i.e., $P=Q=20$ dBm. It is observed that the performance of schemes 2 and 4 decreases with increasing UE2’s rate or $\rho$, since more transmit power needs to be allocated for transmitting UE2’s message by BS1 in order to cancel its (combined) interference at UE1 by SIC. In contrast, without the need of applying SIC at UE1’s receiver to cancel UE2’s interference, the performance of schemes 1 and 3 is observed to be unaffected by UE2’s rate. In addition, it is observed that the proposed scheme 4 significantly outperforms schemes 1 and 3 when UE2’s rate is not high. Moreover, the performance gap between schemes 2 and 4 is observed to be enlarged as UE2’s rate increases, which is consistent with our discussion at the end of Section \[opt.sol\].
![UE1’s achievable rate versus BS2’s transmit power.[]{data-label="IERate_Intf"}](Mei_WCL2019-1495_fig4.eps "fig:"){width="3.2in"} .
Next, we plot UE1’s achievable rate versus BS2’s transmit power $Q$ in Fig.\[IERate\_Intf\] by fixing $\log_2(1+\rho)=5$ bps/Hz. It is observed that UE1’s achievable rates by schemes 1 and 3 quickly diminish as $Q$ increases, due to the increasing (residual) co-channel interference. In contrast, UE1’s achievable rates by schemes 2 and 4 increase with $Q$ or the interference power and finally converge to the same maximum value when UE2’s rate can be satisfied even without BS2’s cooperation, i.e., $\log_2(1+\frac{P\lvert h_1 \rvert^2}{\sigma^2})$, corresponding to the second case of (\[sinr3\]) and (\[sinr6\]). It is also observed that scheme 4 outperforms scheme 3 when $Q$ or the interference power at UE1 is sufficiently large, as analytically shown in Section \[perf.comp\].
Conclusions
===========
This letter proposes a new cooperative NOMA scheme for cellular downlink to resolve the strong asymmetric interference issue. The key difference from the conventional NOMA lies in the new superposition signal design for the purpose of enhancing the co-channel interference at the receiver to facilitate SIC. It is shown both analytically and numerically that the proposed scheme significantly outperforms the conventional NOMA with SIC (scheme 2) when the co-channel interference is comparable to the desired signal in power, as well as the existing interference transmission and cancellation (ITC) scheme without SIC (scheme 3)[@cooperative2019mei] when the co-channel interference is strong. Both scenarios may practically occur in cellular networks (e.g., for cellular-connected UAVs).
[^1]: Manuscript received November 27, 2019; accepted February 11, 2020. The associate editor coordinating the review of this article and approving it for publication was W. Hamouda. [*(Corresponding author: Weidong Mei.)*]{}
[^2]: W. Mei is with the NUS Graduate School for Integrative Sciences and Engineering, National University of Singapore, Singapore 119077, and also with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583 (e-mail: wmei@u.nus.edu).
[^3]: R. Zhang is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583 (e-mail: elezhang@nus.edu.sg).
[^4]: Please refer to [@cooperative2019mei] and [@mei2019uplink; @liu2018multi; @cellular2018mei] for more details on the aerial-ground interference mitigation techniques for the downlink and uplink cellular UAV communications, respectively.
[^5]: This is for the convenience of illustrating the proposed scheme, while we will leave the extension to the general multi-cell system for our future work.
|
---
abstract: |
Let $G=(V,E)$ be a simple graph of maximum degree $\Delta$. The edges of $G$ can be colored with at most $\Delta +1$ colors by Vizing’s theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\Delta$ colors.
Vizing’s Theorem gives a bound of $\frac{\Delta}{\Delta+1}|E|$. This is known to be tight for cliques $K_{\Delta+1}$ when $\Delta$ is even. However, for $\Delta=3$ it was improved to $\frac{26}{31}|E|$ by Albertson and Haas \[*Parsimonious edge colorings*, Disc. Math. 148, 1996\] and later to $\frac{6}7|E|$ by Rizzi \[*Approximating the maximum 3-edge-colorable subgraph problem*, Disc. Math. 309, 2009\]. It is tight for $B_3$, the graph isomorphic to a $K_4$ with one edge subdivided.
We improve previously known bounds for $\Delta\in\{3,\ldots,7\}$, under the assumption that for $\Delta=3,4,6$ graph $G$ is not isomorphic to $B_3$, $K_5$ and $K_7$, respectively. For $\Delta \geq 4$ these are the first results which improve over the Vizing’s bound. We also show a new bound for subcubic multigraphs not isomorphic to $K_3$ with one edge doubled.
In the second part, we give approximation algorithms for the Maximum $k$-Edge-Colorable Subgraph problem, where given a graph $G$ (without any bound on its maximum degree or other restrictions) one has to find a $k$-edge-colorable subgraph with maximum number of edges. In particular, when $G$ is simple for $k=3,4,5,6,7$ we obtain approximation ratios of $\frac{13}{15},\frac{9}{11}$, $\frac{19}{22}$, $\frac{23}{27}$ and $\frac{22}{25}$, respectively. We also present a $\frac{7}{9}$-approximation for $k=3$ when $G$ is a multigraph. The approximation algorithms follow from a new general framework that can be used for any value of $k$.
author:
- 'Marcin Kamiński[^1]'
- 'Łukasz Kowalik[^2]'
title: 'Beyond the Vizing’s bound for at most seven colors[^3]'
---
Introduction
============
A graph is said to be $k$-edge-colorable if there exists an assignment of colors from the set $\{1,\ldots,k\}$ to the edges of the graph, such that every two incident edges receive different colors. For a graph $G$, let $\Delta(G)$ denote the maximum degree of $G$. Clearly, we need at least $\Delta(G)$ colors to color all edges of graph $G$. On the other hand, the celebrated Vizing’s Theorem [@vizing] states that for simple graphs $\Delta+1$ colors always suffice. However, if $k<\Delta+1$ it is an interesting question how many edges of $G$ can be colored in $k$ colors. The maximum $k$-edge-colorable subgraph of $G$ (maximum $k$-ECS in short) is a $k$-edge-colorable subgraph $H$ of $G$ with maximum number of edges. By $\gamma_k(G)$ we denote the ratio $|E(H)|/|E(G)|$; when $|E(G)|=0$ we define $\gamma_k(G)=1$. The [Maximum $k$-Edge-Colorable Subgraph]{} problem (aka Maximum Edge $k$-coloring [@FOW02]) is to compute a maximum $k$-ECS of a given graph. It is known to be APX-hard when $k \geq 2$ [@CP80; @H81; @LG83].
The research on approximation algorithms for max $k$-ECS problem was initiated by Feige, Ofek and Wieder [@FOW02]. Among other results, they suggested the following simple strategy. Begin with finding a maximum $k$-matching $F$ of the input graph, i.e. a subgraph of maximum degree $k$ which has maximum number of edges. This can be done in polynomial time (see e.g. [@schrijver]). Since a $k$-ECS is a $k$-matching itself, $F$ has at least as many edges as the maximum $k$-ECS. Hence, if we color $\rho|E(F)|$ edges of $F$ we get a $\rho$-approximation. It follows that studying large $k$-edge-colorable subgraphs of graphs of maximum degree $k$ is particularly interesting. Let us conclude this paragraph by the following proposition.
\[prop:approx\] If every graph $G=(V,E)$ of maximum degree $k$ has a $k$-edge-colorable subgraph with at least $\rho|E|$ edges, and such a subgraph can be found in polynomial-time, then there is a $\rho$-approximation algorithm for the maximum $k$-ECS problem.
Large $\Delta$-edge-colorable subgraphs of graphs of maximum degree $\Delta$ {#intro:combin}
----------------------------------------------------------------------------
As observed in [@FOW02], if we have a simple graph $G$ of maximum degree $\Delta(G)$, and we find its $(\Delta+1)$-edge-coloring by the algorithm which follows from the proof of Vizing’s Theorem, we can simply choose the $\Delta$ largest color classes to $\Delta$-color at least $\frac{\Delta}{\Delta+1}$ edges of $G$. Can we do better? In general we cannot, and the tight examples are the graphs $K_{\Delta+1}$, for even values of $\Delta$ (see Lemma \[lem:even-clique\]). However, for odd values of $\Delta$ the best upper bound is $\frac{\Delta+1}{\Delta+2-1/\Delta}$ which is attained by graph $B_\Delta$ (see Lemma \[lem:upper-odd\]). This raises two natural questions.
\[q-odd\] When $\Delta$ is odd, can we obtain a better lower bound than $\frac{\Delta}{\Delta+1}$ for simple graphs?
\[q-even\] When $\Delta$ is even and $G\ne K_{\Delta+1}$, can we obtain a better lower bound than $\frac{\Delta}{\Delta+1}$ for simple graphs?
[**Previous Work**. ]{} Question \[q-odd\] has been answered in affirmative for $\Delta=3$ by Albertson and Haas [@AH96], namely they showed that $\gamma_3(G)\ge\frac{26}{31}$ for simple graphs. They also showed that $\gamma_3(G)\ge\frac{13}{15}$ when $G$ is cubic simple graph. Recently, Rizzi [@R09] showed that $\gamma_3(G)\ge\frac{6}{7}$ when $G$ is a simple subcubic graph. The bound is tight by a $K_4$ with an arbitrary edge subdivided (we denote it by $B_3$). Rizzi also showed that when $G$ is a multigraph with no cycles of length 3, then $\gamma_3(G)\ge\frac{13}{15}$, which is tight by the Petersen graph. We are not aware on any results for $\Delta$ bigger than $3$.
[**Our Contribution**. ]{} In the view of the result of Rizzi it is natural to ask whether $B_3$ is the only subcubic simple graph $G$ with $\gamma_3(G)=\frac{6}{7}$. We answer this question in affirmative, namely we show that $\gamma_3(G)\ge\frac{13}{15}$ when $G$ is a simple subcubic graph different from $B_3$. This generalizes both the bound of Rizzi for triangle-free graphs and the bound of Albertson and Haas [@AH96] for cubic graphs, and is tight by the Petersen graph. For a subcubic multigraph, the bound $\gamma_3(G)\ge\frac{3}{4}$ (Vizing’s Theorem holds for subcubic multigraphs) is tight by the $K_3$ with an arbitrary edge doubled (we denote it by $G_3$). Again, we show that $G_3$ is the only tight example: $\gamma_3(G)\ge\frac{7}{9}$ when $G$ is a subcubic multigraph different from $G_3$.
The two results mentioned above follow relatively fast from the work of Rizzi [@R09]. Our main technical contribution is the positive answer to Questions \[q-odd\] and \[q-even\] for $\Delta\in\{4,\ldots,7\}$. Namely, we show that
- $\gamma_4(G)\ge\frac{5}{6}$ when $G$ is a simple graph of maximum degree 4 different from $K_5$,
- $\gamma_5(G)\ge\frac{23}{27}$ when $G$ is a simple graph of maximum degree 5,
- $\gamma_6(G)\ge\frac{19}{22}$ when $G$ is a simple graph of maximum degree 6 different from $K_7$,
- $\gamma_7(G)\ge\frac{22}{25}$ when $G$ is a simple graph of maximum degree 7.
In order to achieve the above bounds we develop a mini-theory describing the structure of maximum $\Delta$-edge-colorable subgraphs and their colorings, which may be useful for further research.
Very recently Mkrtchyan and Steffen [@steffen] showed that every simple graph $G$ has a maximum $\Delta(G)$-edge-colorable subgraph $H$ such that $E(G)\setminus E(H)$ is a matching. Hence, our bounds combined with this result can be seen as a strengthening of Vizing’s theorem: e.g. we show that every graph of maximum degree 4 distinct from $K_5$ has a $5$-edge-coloring such that the $4$ largest color classes contain at least $\frac{5}{6}|E|$ edges.
Approximation algorithms for the max $k$-ECS problem
----------------------------------------------------
**Previous work**. As observed in [@FOW02], the $k$-matching technique mentioned in the beginning of this section together with the bound $\gamma_k(G)\ge\frac{k}{k+1}$ of Vizing’s Theorem gives a $\frac{k}{k+1}$-approximation algorithm for simple graphs and every $k\ge 2$. Note that the approximation ratio approaches 1 as $k$ approaches $\infty$. For multigraphs, we get a $\frac{k}{k+\mu(G)}$-approximation by Vizing’s Theorem and a $k/\lfloor\frac{3}{2}k\rfloor$-approximation by the Shannon’s Theorem on edge-colorings [@shannon].
Feige et al. [@FOW02] show a polynomial-time algorithm which, for a given multigraph and an integer $k$, finds a subgraph $H$ such that $|E(H)|\ge {\rm OPT}$, $\Delta(H)\le k+1$ and $\Gamma(H)\le k+\sqrt{k+1}+2$, where ${\rm OPT}$ is the number of edges in the maximum $k$-edge colorable sugraph of $G$, and $\Gamma(H)$ is the odd density of $H$, defined as $\Gamma(H)=\max_{S\subseteq V(H), |S|\ge 2}\frac{|E(S)|}{\lfloor |S|/2\rfloor}$. The subgraph $H$ can be edge-colored with at most $\max\{\Delta+\sqrt{\Delta/2},\lceil \Gamma(H)\rceil\}\le\lceil k + \sqrt{k+1} + 2 \rceil$ colors in $n^{O(\sqrt{k})}$-time by an algorithm of Chen, Yu and Zang [@chen-jco]. By choosing the $k$ largest color classes as a solution this gives a $k/\lceil k + \sqrt{k+1} + 2 \rceil$-approximation. One can get a slightly worse $k/(k+(1+3/\sqrt{2})\sqrt{k} + o(\sqrt{k}))$-approximation by replacing the algorithm of Chen et al. by an algorithm of Sanders and Steurer [@ss] which takes only $O(nk(n+k))$-time. Note that in both cases the approximation ratio approaches 1 when $k$ approaches $\infty$, similarly as in the case of simple graphs.
The results above work for all values of $k$. However, for small values of $k$ tailor-made algorithms are known, with much better approximation ratios. The most intensively studied case is $k=2$. The research of this basic variant was initiated by Feige et al. [@FOW02], who proposed an algorithm for multigraphs based on an LP relaxation with an approximation ratio of $\frac{10}{13} \approx 0.7692$. They also pointed out a simple $\frac{4}{5}$-approximation for simple graphs. This was later improved several times [@CTW08; @CT09]. In 2009 Kosowski [@K09] achieved a $\frac{5}{6}$-approximation by a very interesting extension of the $k$-matching technique (see Section \[sec:OTW\]). Finally, Chen, Konno and Matsushita [@chen-0-842] got a $0.842$-approximation, essentially by a very careful analysis of the structure of the $k=2$ case.
Kosowski [@K09] studied also the case of $k=3$ and obtained a $\frac{4}{5}$-approximation for simple graphs, which was later improved by a $\frac{6}{7}$-approximation resulting from the mentioned result of Rizzi [@R09].
Finally, there is a simple greedy algorithm by Feige et al. [@FOW02] with approximation ratio $1-(1-\frac{1}k)^k$, which is still the best result for the case $k=4$ in multigraphs.
$k$ simple graphs reference multigraphs reference
----------------- ----------------- ------------------- ----------------------------------------------------------------- ------------------------------
2 $0.842$ [@chen-0-842] $\frac{10}{13}$ [@FOW02]
3 $\frac{13}{15}$ [**this work**]{} $\frac{7}{9}$ [**this work**]{}
4 $\frac{9}{11}$ [**this work**]{} $1-(\frac{3}{4})^4> 0.683$ [@FOW02]
5 $\frac{23}{27}$ [**this work**]{} $\frac{5}{7}$ [@shannon; @FOW02]
6 $\frac{19}{22}$ [**this work**]{} $\max\{\frac{2}{3},\frac{6}{6+\mu}\}$ [@shannon; @vizing; @FOW02]
7 $\frac{22}{25}$ [**this work**]{} $\max\{\frac{7}{10},\frac{7}{7+\mu}\}$ [@shannon; @vizing; @FOW02]
$8, \ldots, 13$ $\frac{k}{k+1}$ [@vizing; @FOW02] $\max\{\frac{k}{\lfloor 3k/2\rfloor},\frac{k}{k+\mu}\}$ [@shannon; @vizing; @FOW02]
$\ge 14$ $\frac{k}{k+1}$ [@vizing; @FOW02] $\max\{\frac{k}{\lceil k+\sqrt{k+1}+2\rceil},\frac{k}{k+\mu}\}$ [@chen-jco; @vizing; @FOW02]
: \[piekna-tabelka\]Best approximation ratios for the Maximum $k$-Edge-Colorable Subgraph problem
[**Our contribution**. ]{} We generalize the technique that Kosowski used in his algorithm for the max 2-ECS problem so that it may be applied for an arbitrary number of colors. Roughly, we deal with the situation when for a graph $G$ of maximum degree $k$ one can find in polynomial time a $k$-edge colorable subgraph $H$ with at least $\alpha|E(G)|$ edges, unless $G$ belongs to a family ${\mathcal{F}}$ of “exception graphs”, i.e. $\gamma(G)<\alpha$. As we have seen in the case of $k=3,4,6$ the set of exception graphs is small and in the case of $k=2$ the exceptions form a very simple family of graphs (odd cycles). The exception graphs are the only obstacles which prevent us from obtaining an $\alpha$-approximation algorithm (for general graphs) by using the $k$-matching approach. In such situation we provide a general framework, which allows to obtain approximation algorithms with approximation ratio better than $\min_{A\in{\mathcal{F}}}\gamma_k(A)$. See Theorem \[th:meta-algorithm\] for the precise description of our general framework.
By combining the framework and our combinatorial results described in Section \[intro:combin\] we get the following new results (see Table \[piekna-tabelka\]): a $\frac{7}{9}$-approximation of the max-$3$-ECS problem for multigraphs, a $\frac{13}{15}$-approximation of the max-$3$-ECS problem for simple graphs, a $\frac{9}{11}$-approximation of the max-$4$-ECS problem for simple graphs, a $\frac{23}{27}$-approximation of the max-$5$-ECS problem for simple graphs, a $\frac{19}{22}$-approximation of the max-$6$-ECS problem for simple graphs, and a $\frac{22}{25}$-approximation of the max-$7$-ECS problem for simple graphs. Note that for 4 up to 7 colors our algorithms are the first which break the barrier of Vizing’s Theorem. Although we were able to get improved approximation ratios only for at most seven colors, note that these are the most important cases, since the approximation ratio of the algorithm based on Vizing’s theorem is very close to 1 for large number of colors.
Notation
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We use standard terminology; for notions not defined here, we refer the reader to [@Diestel]. Let $G=(V,E)$ be an undirected graph. For a vertex $x$ by $N_G(x)$ we denote the set of neighbors of $x$ and $N_G[x]=N_G(x)\cup\{x\}$. For a set of vertices $S$ we denote $N_G(S)=\bigcup_{x\in S}N_G(x)\setminus S$ and $N_G[S]=\bigcup_{x\in S}N_G[x]$. Moreover, $d_G(S)=\{uv \in E\ :\ u\in S, v\not\in S\}$. For two sets $X,Y\subseteq V$ we define $E_G(X,Y)=\{xy \in E\ :\ x\in X\setminus Y, y\in Y\setminus X\}$. In all of the above denotations we omit the subscripts when it is clear what graph we refer to. A graph with maximum degree 3 is called *subcubic*. Following [@AH96], let $c_k(G)$ be the maximum number of edges of a $k$-edge-colorable subgraph of $G$. We also denote $\overline{c}_k(G)=|E(G)|-c_k(G)$, $c(G)=c_{\Delta(G)}(G)$ and $\overline{c}(G)=\overline{c}_{\Delta(G)}(G)$.
Large 3-edge-colorable subgraphs of graphs maximum degree 3
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In this section we will work with multigraphs. We will also need the following result on triangle-free multigraphs from Rizzi [@R09].
\[lemma-Rizzi\] Every subcubic, triangle-free multigraph $G$ has a 3-edge-colorable subgraph with at least $\frac{13}{15}|E(G)|$ edges. Moreover, this subgraph can be found in polynomial time.
We need one more definition. Let $G^*_5$ be the multigraph on 5 vertices obtained from the four-vertex cycle by doubling one edge of the cycle and adding a vertex of degree two adjacent to the two vertices of the cycle not incident with the double edge.
\[theorem-13-15\] Let $G$ be a biconnected subcubic multigraph different from $G_3$, $B_3$ and $G^*_5$. There exists a 3-edge-colorable subgraph of $G$ with at least $\frac{13}{15} |E(G)|$ edges. Moreover, this subgraph and its coloring can be found in polynomial time.
We will prove the theorem by induction on the number of vertices of the multigraph. We introduce the operation of *triangle contraction* which is to contract the three edges of a triangle (order of contracting is inessential) keeping multiple edges that appear. Note that since $G$ is biconnected and $G\ne G_3$, no triangle in $G$ has a double edge, so loops do not appear after the triangle contraction operation. If a multigraph is subcubic, then it will be subcubic after a triangle contraction. Notice that if a multigraph has at least five vertices, the operation of triangle contraction in subcubic multigraphs preserves biconnectivity. It is easy to check that that all subcubic multigraphs on at most 4 vertices, different from $G_3$, have a 3-edge-colorable subgraph with least $\frac{13}{15} |E(G)|$ edges.
Let $G$ be a biconnected subcubic multigraph with at least 5 vertices and different from $B_3$. If $G$ is triangle-free, then the theorem follows from Lemma \[lemma-Rizzi\]. Let us assume that $G$ has at least one triangle $T$ and let $G'$ be the multigraph obtained from $G$ by contracting $T$.
We can assume that $G'$ is subcubic and biconnected. First, let us assume that $G'$ is not isomorphic to $G_3$, $B_3$, or $G^*_5$. $G'$ has less vertices than $G$ so by the induction hypothesis it has a 3-edge-colorable subgraph with at least $\frac{13}{15} |E(G')|$. Notice that it can always be extended to contain all three edges of $T$. Hence, $G$ has a 3-edge-colorable subgraph with at least $\frac{13}{15} |E(G')| + 3 \geq \frac{13}{15} |E(G)|$ edges.
Now we consider the case when $G'$ is isomorphic to $G_3$, $B_3$ or $G^*_5$. In fact, $G'$ cannot be isomorphic to $G_3$, because then $G$ would be $B_3$ or $G^*_5$. There are only three multigraphs from which $B_3$ can be obtained after triangle contraction; they all have 10 edges and a [$3$-edge-colorable]{} subgraph with $9 > \frac{13}{15} \cdot 10$ edges. Similarly, there are only three multigraphs from which $G^*_5$ can be obtained after triangle contraction; they all have 10 edges and a [$3$-edge-colorable]{} subgraph with $9 > \frac{13}{15} \cdot 10$ edges.
\[corollary-13-15\] Let $G$ be a connected subcubic multigraph not containing $G_3$ as a subgraph and different from $B_3$ and $G^*_5$. There exists a 3-edge-colorable subgraph of $G$ with at least $\frac{13}{15} |E(G)|$ edges. Moreover, this subgraph and its coloring can be found in polynomial time.
Suppose that the theorem is not true. Let $G$ be a counter-example with the least number of vertices.
It is easy to check that if every biconnected component of $G$ has a 3-edge-colorable subgraph with at least $\frac{13}{15}$ of its edges, then so does $G$. Thus, by Theorem \[theorem-13-15\] we can assume that there exists a biconnected component $C$ of $G$ which is isomorphic to $B_3$ or $G^*_5$. Since $C$ is not the whole multigraph, there is an edge $vw$ with $v \in V(C)$ and $w\not\in V(C)$. If $C \cup vw$ is the whole multigraph, it does have a 3-edge-colorable subgraph with at least $\frac{13}{15} |E(G)|$ edges. Hence, $H := G[V \setminus (V(C) \cup \{w\})]$ is not empty.
Notice that $vw$ is a bridge. Since $C \cup \{vw\}$ has a 3-edge-colorable subgraph with at least $\frac{13}{15}$ of its edges, and $w$ is a cut-vertex, then – by a similar reasoning as above – $G[V(H) \cup \{w\}]$ does not have a 3-edge-colorable subgraph with at least $\frac{13}{15}$ of its edges. By minimality of $G$, $G[V(H) \cup \{w\}]$ is isomorphic to $B_3$ or $G^*_5$. However, then, $G$ is a cubic multigraph with 15 edges and it has a 3-edge-colorable subgraph with at least 13 edges; a contradiction.
\[corollary-7-9\] Every connected subcubic multigraph $G$ different from $G_3$ has a 3-edge-colorable subgraph with at least $\frac{7}{9} |E(G)|$ edges. Moreover, this subgraph and its coloring can be found in polynomial time.
Let $G$ be a connected multigraph different from $G_3$. We use induction on $|V(G)|$. First, assume that $G$ is also biconnected. If $G$ is isomorphic to $B_3$ or $G^*_5$, then it has a 3-edge-colorable subgraph with at least $\frac{6}{7} |E(G)| \geq \frac{7}{9} |E(G)|$ edges. Otherwise, from Theorem \[theorem-13-15\], it has a 3-edge-colorable subgraph with at least $\frac{13}{15} |E(G)| \geq \frac{7}{9} |E(G)|$ edges.
Now, let us assume that $G$ has a cut-vertex $v$. Since $G$ is subcubic, it has also a cut-edge $vw$. Let $C'$ and $C''$ be the connected components of $G-vw$. If both $C'$ and $C''$ have 3-edge-colorable subgraphs with at least $\frac{7}{9}$ of its edges, then so does $G$. Hence, by the induction hypothesis we can assume that at least one component (say, $C'$) is isomorphic to $G_3$. If $C''$ is not isomorphic to $G_3$, then by the induction hypothesis we can color $\frac{7}9$ of the edges of $C''$. Next, we can color four out of the five edges of $C'\cup\{vw\}$ and thus we obtain a 3-edge-colorable subgraph of $G$ with more than $\frac{7}{9} |E(G)|$ edges. In the remaining case the whole multigraph consists of two copies of $G_3$ with the degree 2 vertices connected by the edge $vw$. It has 9 edges and a 3-edge-colorable subgraph with 7 edges.
Large $\Delta$-edge-colorable subgraphs in simple graphs with maximum degree $\Delta$ from four to seven
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In this section by a graph we mean a simple graph. We prove the following theorem.
\[thm:main\] Let $G$ be a connected simple graph of maximum degree $\Delta\in\{4, 5, 6, 7\}$. Then $G$ has a $\Delta$-edge-colorable subgraph with at least
a) $\frac{5}{6}|E|$ edges when $\Delta=4$ and $G\ne K_5$,
b) $\frac{23}{27}|E|$ edges when $\Delta=5$,
c) $\frac{19}{22}|E|$ edges when $\Delta=6$ and $G\ne K_7$,
d) $\frac{22}{25}|E|$ edges when $\Delta=7$.
Moreover, the subgraph can be found in polynomial time.
We will work with partially colored graphs. A [*partial $k$-coloring*]{} of a graph $G=(V,E)$ is a function $\pi : E \rightarrow \{1, \ldots, k\} \cup \{\bot\}$ such that if two edges $e_1, e_2\in E$ are incident then $\pi(e_1) \ne \pi(e_2)$, or $\pi(e_1) = \bot$, or $\pi(e_2) = \bot$. We will call the pair $(G,\pi)$ a [*colored graph*]{}. We say an edge $e$ is [*uncolored*]{} if $\pi(e)=\bot$; otherwise, we say that $e$ is [*colored*]{}. For a vertex $v$, ${\pi}(v)$ is the set of colors of edges incident with $v$, i.e. $\pi(v)=\{\pi(e)\ :\ \text{$e$ is incident with $v$}\}\setminus\{\bot\}$, while $\overline{\pi}(v)=\{1, \ldots, k\}\setminus \pi(v)$ is the set of free colors at $v$.
Our plan for proving Theorem \[thm:main\] is the following. We introduce a notion of the potential function $\Psi$, which measures “the quality” of a partial $\Delta$-coloring $\pi$ of a given graph $G$. It turns out that if we are unable to improve the potential of a partial coloring $\pi$ then the pair $(G,\pi)$ exhibits certain structure. We are going to determine this structure in a series of lemmas so that we are able to show that $\pi$ has few uncolored edges. In the proofs of the structural lemmas we show that if the claim of the lemma does not hold, one can find in polynomial time a new coloring so that the potential increases. Hence, in order to find a partial coloring which satisfies the claimed lower bound on the number of colored edges it suffices to start with an empty coloring and then, as long as the claim of some of the structural lemmas does not hold, find a new coloring with improved potential, as described in the relevant proof. Since, as we will see, the potential can be increased only polynomial number of times, the whole procedure works in polynomial time.
The structure of maximum $\Delta$-edge-colorable subgraphs {#sec:structure}
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Let $G$ be an arbitrary connected graph and let $\Delta$ denote its maximum degree. In this section we study the structure of a partial edge-coloring $\pi$ of $G$, such that the number of colored edges cannot be increased. We defer the choice of our potential $\Psi$ until we show the full motivation for its definition. However, the potential $\Psi$ grows with the number of colored edges, so the structure of $(G, \pi)$ described in this section applies also when $\Psi$ cannot be increased. Another reason for deferring its full description is that we prefer to state the claims of this section under weaker assumptions since we believe they might be useful in further research.
Let $a$ and $b$ be two distinct colors and $x$ and $y$ be two distinct vertices. An [*$(ab, xy)$-path*]{} is a path $P=x_1x_2\ldots x_t$ for some $t>0$, such that:
- $x=x_1$ and $y=x_t$,
- the edges of $P$ are colored alternately with $a$ and $b$, i.e. $\pi(x_{i}x_{i+1})\in\{a,b\}$ and if $\pi(x_{i}x_{i+1})=a$ and $\pi(x_{j}x_{j+1})=b$ then $i \not\equiv j \bmod 2$,
- $P$ is maximal, i.e. $|{\overline{\pi}}(x)\cap\{a,b\}|=|{\overline{\pi}}(y)\cap\{a,b\}|=1$.
We also say that $P$ is an alternating path, $(ab,\cdot)$-path, $(ab,x)$-path, $(\cdot, xy)$-path or $(a,xy)$-path.
The idea of alternating paths dates back to Kempe [@kempe1879geographical] and his first attempts to prove the Four Color Theorem. The basic property of an alternating path $P$ is that we can recolor the graph along $P$ so that all edges of $P$ colored with $a$ get color $b$ and vice versa. Note that as a result, if $a$ (resp. $b$) was free in one end of the path $P$, say in $x$ then in ${\overline{\pi}}(x)$ the color $a$ is replaced by $b$ (resp. $b$ is replaced by $a$), and for every vertex $v\not\in\{ x,y\}$ the set of free colors ${\overline{\pi}}(v)$ stays the same. We will often use this operation, called [*swapping*]{}.
Let $V_{\bot} = \{v \in V\ :\ {\overline{\pi}}(v)\ne \emptyset\}$. In what follows, $\bot(G,\pi)=(V_{\bot},\pi^{-1}(\bot))$ is called [*the graph of free edges*]{}. Every connected component of the graph $\bot(G,\pi)$ is called a [*free component*]{}. If a free component has only one vertex, it is called [*trivial*]{}. The set of all nontrivial free components of colored graph $(G,\pi)$ is denoted by ${\textnormal{nfc}}(G,\pi)$.
\[lem:distinct-free\] Let $G$ be a graph and let $\pi$ be a partial coloring of $G$ which maximizes the number of colored edges. For any free component $Q$ of $(G,\pi)$ and for every two distinct vertices $v,w\in V(Q)$
1. ${\overline{\pi}}(v) \cap {\overline{\pi}}(w) = \emptyset$,
2. for every $a\in {\overline{\pi}}(v)$, $b\in{\overline{\pi}}(w)$ there is an $(ab, vw)$-path.
First we prove $(i)$ and we use induction on the length $d$ of the shortest path $P$ in $Q$ from $v$ to $w$. The proof is by contradiction, i.e. we show that if ${\overline{\pi}}(v)\cap{\overline{\pi}}(w)\ne\emptyset$ then one can increase the number of colored edges. If $d=1$ just color $vw$ with a color from ${\overline{\pi}}(v)\cap{\overline{\pi}}(w)$. Now we consider $d>1$. Assume there is a color $a \in {\overline{\pi}}(v) \cap {\overline{\pi}}(w)$. Let $x$ be the second to last vertex on $P$, i.e. $xw\in E(P)$. Since $x$ is incident with an uncolored edge, there is a free color at $x$, say $b$. Since we have already proved the claim for $d=1$, we infer that $a\ne b$ and $b\not\in{\overline{\pi}}(w)$. Let $R$ be the $(ab,w)$-path. We swap $R$. If $x$ is not incident with $R$ then $b$ is free at both $x$ and $w$ and we just color $xw$ with $b$ and we increase the number of colored edges; a contradiction. If $x$ is incident with $R$ it means that $R$ is an $(ab, wx)$-path. Hence after swapping, $a\in{\overline{\pi}}(v)\cap {\overline{\pi}}(x)$. Since $v$ and $x$ are at distance $d-1$ in $Q$ we get a contradiction with the induction hypothesis.
To see $(ii)$, just consider the $(ab,v)$-path and note that by $b\not\in{\overline{\pi}}(v)$ by $(i)$ so the path has length at least one. If this path does not end in $w$ we can swap it and get $b\in{\overline{\pi}}(v)\cap {\overline{\pi}}(w)$, contradicting $(i)$. Also, by $(i)$, we have $v\ne w$.
For a free component $Q$, by ${\overline{\pi}}(Q)$ we denote the set of free colors at the vertices of $Q$, i.e. ${\overline{\pi}}(Q)=\bigcup_{v\in V(Q)}{\overline{\pi}}(v)$.
\[cor:num-uncolored-0\] Let $G$ be a graph and let $\pi$ be a partial coloring of $G$ which maximizes the number of colored edges. For any free component $Q$ of $(G,\pi)$ we have $|{\overline{\pi}}(Q)|\ge 2|E(Q)|$.
We have $|{\overline{\pi}}(Q)| = \sum_{v\in V(Q)}|{\overline{\pi}}(v)| \ge \sum_{v\in V(Q)}\deg_Q(v) \ge 2|E(Q)|$, where the first equality follows from Lemma \[lem:distinct-free\]$(i)$.
Since $|{\overline{\pi}}(Q)| \le \Delta$ we immediately get the following.
\[cor:num-uncolored\] Let $G$ be a graph and let $\pi$ be a partial coloring of $G$ which maximizes the number of colored edges. Every free component $Q$ of $(G,\pi)$ has at most $\lfloor\frac{\Delta}{2}\rfloor$ edges.
Let $Q_1,Q_2$ be two distinct free components of $(G,\pi)$ and assume that for some pair of vertices $x\in V(Q_1)$ and $y\in V(Q_2)$, there is an edge $xy\in E$ such that $\pi(xy)\in{\overline{\pi}}(Q_1)$. Then we say that [*$Q_1$ sees $Q_2$ with $xy$*]{}, or shortly [*$Q_1$ sees $Q_2$*]{}.
\[lem:A\] Let $G$ be a graph and let $\pi$ be a partial coloring of $G$ which maximizes the number of colored edges. If $Q_1,Q_2$ are two distinct free components of $(G,\pi)$ such that $Q_1$ sees $Q_2$ then ${\overline{\pi}}(Q_1) \cap {\overline{\pi}}(Q_2) = \emptyset$.
Let $x\in V(Q_1)$, $y\in V(Q_2)$ be vertices such that $Q_1$ sees $Q_2$ with $xy$. Denote $a=\pi(xy)$. Let $v$ be a vertex of $Q_1$ such that $a\in{\overline{\pi}}(v)$. The proof is by contradiction.
First assume $a\in {\overline{\pi}}(Q_2)$. Since $a\not\in {\overline{\pi}}(y)$ it follows that $|E(Q_2)|>0$ and, in particular, $y$ has a neighbor in $Q_2$, say $y'$. By Lemma \[lem:distinct-free\]$(i)$ there is exactly one vertex $z\in V(Q_2)$ such that $a\in{\overline{\pi}}(z)$. Now we use induction on the length $d$ of the shortest path $P$ in $Q_2$ from $y'$ to $z$. If $d=0$, i.e. $z=y'$ we uncolor $xy$ and we color $yy'$ with $a$. As a result, the number of colored edges has not changed and we get a free component in which two vertices (namely, $v$ and $x$) share the same free color $a$, which is a contradiction with Lemma \[lem:distinct-free\]$(i)$. Now assume $d>0$ and let $z'$ be the second to last vertex on $P$, i.e. $z'z\in E(P)$. Let $c$ be any color of ${\overline{\pi}}(z')$. Consider the $(ac,zz')$ path $R$ described in Lemma \[lem:distinct-free\]$(ii)$. If $R$ does not contain $xy$, we just swap $R$ (note that after the swapping we still have $a\in{\overline{\pi}}(Q_1)$) and proceed by induction hypothesis. Otherwise let $R'$ be the maximal subpath of $R$ which starts in $z$ and does not contain $xy$. We uncolor $xy$, swap $R'$ and color $zz'$ with $c$. Again, the number of colored edges has not changed and we get a free component with two vertices (namely, $v$ and the endpoint of $xy$ which is not incident with $R'$) that share the same free color $a$.
Now assume that for some color $b\ne a$ we have $b\in{\overline{\pi}}(x')\cap{\overline{\pi}}(y')$ for some $x'\in V(Q_1)$ and $y'\in V(Q_2)$. If $x'\ne x$, choose any color $c \in {\overline{\pi}}(x)$ and swap the $(bc,x'x)$-path described in Lemma \[lem:distinct-free\]$(ii)$. We proceed analogously when $y'\ne y$. Hence we can assume that $b\in{\overline{\pi}}(x)\cap{\overline{\pi}}(y)$. Then we recolor $xy$ to $b$. As a result, $a\in {\overline{\pi}}(v)\cap{\overline{\pi}}(x)$ and $v$ and $x$ still belong to the same free component, which is a contradiction with Lemma \[lem:distinct-free\]$(i)$.
\[lem:C\] Let $G$ be a graph and let $\pi$ be a partial coloring of $G$ which maximizes the number of colored edges. Let $P$, $Q$ and $R$ be free components of $(G,\pi)$, $P\ne Q$ and $P\ne R$. Assume that for some $x\in P$ and $y\in Q$ there is an edge $xy\in E(G)$ and for some $u\in P$ and $v\in R$ there is an edge $uv\in E(G)$, $xy\ne uv$. If $\pi(xy)=\pi(uv)$ then there are no two distinct colors $a,b\in {\overline{\pi}}(P)$ such that $a\in {\overline{\pi}}(Q)$ and $b\in {\overline{\pi}}(R)$.
The proof is by contradiction.
Let $x'$ be the vertex of $P$ such that $a\in{\overline{\pi}}(x')$ and let $c$ be any color of ${\overline{\pi}}(x)$. By Lemma \[lem:distinct-free\]$(i)$, $a\ne c$. Note that by Lemma \[lem:A\] we have $\pi(xy)\ne a$. In particular, $\pi(xy)=\pi(uv)\ne a,c$. If $x'\ne x$ we swap the $(ac,xx')$ path described in Lemma \[lem:distinct-free\]$(ii)$. Note that the colors of $xy$ and $uv$ do not change. Similarly, let $y'$ be the vertex of $Q$ such that $a\in{\overline{\pi}}(y')$ and let $d$ be any color of ${\overline{\pi}}(y)$. Again, $\pi(xy)=\pi(uv)\ne a,d$. If $y'\ne y$ we swap the $(ad,yy')$ path described in Lemma \[lem:distinct-free\]$(ii)$ and again this does not change the colors of $xy$ and $uv$. Observe also that the sets of free colors of $P$ and $Q$ have not changed. Then we recolor $xy$ to $a$. After this operation, $\pi(uv)$ becomes free in $P$ and $P$ sees $R$ with $uv$. However, $b \in {\overline{\pi}}(P)\cap{\overline{\pi}}(R)$; a contradiction with Lemma \[lem:A\].
\[cor:C1\] Let $G$ be a graph and let $\pi$ be a partial coloring of $G$ which maximizes the number of colored edges. Let $Q$ be a free component of $(G,\pi)$ such that $\Delta-1 \le |{\overline{\pi}}(Q)| \le \Delta$. Then there are at most $\Delta-|{\overline{\pi}}(Q)|$ edges incident both with $Q$ and other nontrivial free components. Moreover, each such an edge is colored with a color from $\{1,\ldots,\Delta\}\setminus {\overline{\pi}}(Q)$.
We can assume that $|{\overline{\pi}}(Q)|=\Delta-1$ for otherwise by Lemma \[lem:A\] there are no edges incident with $Q$ and other free components and the claim follows. We infer that there is exactly one color $c\not\in {\overline{\pi}}(Q)$.
Assume to the contrary, that there are two edges $xy$ and $uv$ with the property described in the statement, with $x,u\in V(Q)$. Let $P$ and $R$ be the nontrivial free components such that $y\in V(P)$ and $v\in V(R)$, possibly $P=R$. Any nontrivial free component has at least two free colors by Lemma \[lem:distinct-free\]$(i)$, so in particular it has a color from ${\overline{\pi}}(Q)$, and hence by Lemma \[lem:A\] both $xy$ and $uv$ are colored with $c$ (this, in particular, proves the second part of the claim). Then $c\not\in {\overline{\pi}}(P)\cup {\overline{\pi}}(R)$ for otherwise $P$ or $R$ sees $Q$; a contradiction with Lemma \[lem:A\]. It follows that both ${\overline{\pi}}(P)$ and ${\overline{\pi}}(R)$ are subsets of ${\overline{\pi}}(Q)$, both of cardinality at least 2, which is a contradiction with Lemma \[lem:C\].
Now we need another classical notion in the area of edge-colorings: the notion of a fan. We use a somewhat relaxed definition, due to Favrholdt, Stiebitz and Toft [@stiebitz], adapted to our setting of partially colored graphs. Let $(G,\pi)$ be a partially edge-colored graph and let $xy$ be an uncolored edge of $G$. An [*$(x,y)$-fan*]{} is a sequence of edges $F =
(xy_1, \ldots, xy_{\ell})$, where $y_1=y$ and for each $i=2,\ldots,\ell$ there is an index ${\textnormal{pred}}_F(i)<i$ such that the edge $xy_i$ is colored with a color $\pi(xy_i)\in {\overline{\pi}}(y_{{\textnormal{pred}}_F(i)})$. We say that a fan is [*maximal*]{} when it is not a proper prefix of another fan. The vertices $y_2,\ldots,y_{\ell}$ are called [*ends*]{} of $F$. A proof of the following fact can be found in [@stiebitz]; see Theorem 2.1: point (a) below appears explicitly while point (b) can be found in the proof.
\[lem:fan1\] Let $F = (xy_1, \ldots, xy_{\ell})$ be a maximal fan in a partial $\Delta$-edge-coloring $(G,\pi)$ such that the number of colored edges cannot be increased. Then
(a) if $1\le i < j \le \ell$ then ${\overline{\pi}}(y_i) \cap {\overline{\pi}}(y_j)=\emptyset$ and
(b) $\{\pi(xy_2),\ldots,\pi(xy_{\ell})\} = \bigcup_{i=1}^{\ell}{\overline{\pi}}(y_i)$,
For a fan $F = (xy_1, \ldots, xy_{\ell})$, if ${\overline{\pi}}(y_i)=\emptyset$ then we say $y_i$ is a [*full vertex*]{} and $xy_i$ is a [*full edge*]{}.
\[cor:num-full\] Let $G$ be a graph and let $\pi$ be a partial coloring of $G$ which maximizes the number of colored edges. Any maximal $(x,y)$-fan $F$ in $(G,\pi)$ has at least $|{\overline{\pi}}(y)|$ full edges.
By Lemma \[lem:fan1\], $\sum_{i=1}^{\ell}|{\overline{\pi}}(y_i)| = \ell - 1$. Let $f$ be the number of full edges of $F$. Then clearly, $\sum_{i=1}^{\ell}|{\overline{\pi}}(y_i)|\ge |{\overline{\pi}}(y)| + \ell - 1 - f$. Hence, $f \ge |{\overline{\pi}}(y)|$, as required.
Let $F=(xy_1,\ldots,xy_{\ell})$ be a fan in a partially colored graph $(G,\pi)$. Fix a vertex $y_i$, $i>1$. Define ${\textnormal{pred}}_F(1)=1$. Consider the following sequence of indices: $a_1=i$, and for every $j>1$, $a_j={\textnormal{pred}}_F(a_{j-1})$. Let $d=\min\{j\ :\ a_j = 1\}$. Consider the following recoloring procedure which transforms the coloring $\pi$ into a new coloring $\pi'$: begin with $\pi'=\pi$ and for every $j=2,\ldots,d$ put $\pi'(xy_{a_j})=\pi(xy_{a_{j-1}})$. Finally, uncolor $xy_i$. Note that $\pi'$ is a proper partial coloring with the same number of colored edges as $\pi$. This procedure is called [*rotating $F$ at $y_i$*]{}.
\[lem:fans-edge-disjoint\] Let $G$ be a graph and let $\pi$ be a partial coloring of $G$ which maximizes the number of colored edges. Let $F_1$ be an $(x,y)$-fan in $(G,\pi)$ and let $F_2$ be an $(x,z)$-fan in $(G,\pi)$, for some $y\ne z$. Then $F_1$ and $F_2$ do not share an edge.
Let $F_1=(xy_1,\ldots,xy_{\ell})$ and $F_2=(xz_1,\ldots,xz_t)$. Assume $(i,j)$ is the lexicographically first pair of indices such that $y_i = z_j$. Let $c=\pi(xy_i)$. Rotate $F_1$ at $y_{{\textnormal{pred}}_{F_1}(i)}$ and rotate $F_2$ at $z_{{\textnormal{pred}}_{F_2}(j)}$. Note that by our choice of $(i,j)$ it is possible to perform both rotations. As a result, the number of colored edges does not change and we get a free component with color $c$ free at two vertices, namely $y_{{\textnormal{pred}}_{F_1}(i)}$ and $z_{{\textnormal{pred}}_{F_2}(j)}$; a contradiction with Lemma \[lem:distinct-free\].
The structure of $\Psi_0$-maximal partial $\Delta$-edge-colorings {#sec:struct-psi0}
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Now we are ready to define a potential function $\Psi_0$ for a partial coloring $(G,\pi)$. Let $c$ be the number of colored edges, i.e. $c=|\pi^{-1}(\{1,\ldots,\Delta\})|$. For every $i=1,\ldots,{\ensuremath{\lfloor{\Delta/2}\rfloor}}$, let $n_i$ be the number of free components with $i$ edges. Then $$\Psi_0(G,\pi)=(c,n_{{\ensuremath{\lfloor{\Delta/2}\rfloor}}},n_{{\ensuremath{\lfloor{\Delta/2}\rfloor}}-1},\ldots,n_1).$$
We use the lexicographic order on tuples to compare values of $\Psi_0$. In what follows we study the structure of a partial $\Delta$-edge-coloring $\pi$ of a graph $G$ which is $\Psi_0$-maximal, i.e. there is no partial $\Delta$-edge-coloring $\pi'$ with $\Psi_0(G,\pi') > \Psi_0(G,\pi)$. Note that the claims of the lemmas in Section \[sec:structure\] also hold for $(G,\pi)$.
The intuition behind the choice of the potential $\Psi_0$ is as follows. Our goal is to find a partial coloring so that we can injectively assign many colored edges to every uncolored edge. As we will see, to maintain the injectiveness of the assignment, edges of a free component $Q$ are assigned only edges that are [*close*]{} to $Q$ (mostly edges incident with $Q$). In particular, if a colored edge is incident with two free components, we assign [*half*]{} of it to each of them. Assume $\Delta$ is even and consider a free component with $\Delta/2$ edges. Such a component will be called [*maximal*]{}. Observe that Corollary \[cor:num-uncolored-0\] and Lemma \[lem:A\] imply that a colored edge is incident with at most one maximal component. Hence it seems that maximal components are good for us: they get assigned [*the whole*]{} incident edges, not just halves. This is why if we increase the number of maximal free components, our potential will increase, even if the number of colored edges stays the same. Our choice of $\Psi_0$ will also help when considering smaller components: for a smaller free component we will be able to argue that some (but not all) edges incident with it cannot be incident with another free component for otherwise by fan rotations we can “merge” the two components to form a bigger one. However, a rotation can increase the number of free components, and in particular it can decrease the potential. Hence we use rotations only for very special fans. Consider an $(x,y)$-fan $F$ and let $Q$ be the free component that contains $xy$. We say that $F$ is [*stable*]{}, if $Q-xy$ has no edges or $Q-xy$ has exactly one nontrivial (i.e. with at least one edge) connected component and this component contains $x$. (Note that even if every $(x,y)$-fan is stable it does not mean that a $(y,x)$-fan is stable).
\[prop:rotate\] Rotating a stable fan does not decrease $\Psi_0$.
\[lem:usually-fans-do-not-meet\] Let $G$ be a graph and let $\pi$ be a partial coloring of $G$ which maximizes the potential $\Psi_0$. Let $P$ and $Q$ be two distinct free components of $(G,\pi)$ and let $xy \in E(P)$, $zu \in E(Q)$. Assume $F_1=(xy_1,\ldots,xy_{\ell})$ is a stable $(x,y)$-fan. Let $F_2=(zu_1,\ldots,zu_t)$ be a $(z,u)$-fan. If $|E(Q)|\le|E(P)|$ or $F_2$ is stable then the ends of $F_1$ and $F_2$ are distinct, i.e. for every $i=1,\ldots,\ell$ and $j=1,\ldots,t$ we have $y_i \ne u_j$.
Assume $(i,j)$ is the lexicographically first pair of indices such that $y_i = u_j$. Then we rotate $F_1$ at $y_i$ and we rotate $F_2$ at $u_j$ (note that because of the choice of $i$ and $j$, the free colors at $u_1,\ldots,u_j$ do not change during the rotation of $F_1$ so the rotation of $F_2$ is still possible). In the graph $\bot(G,\pi)$ it corresponds to removing edges $xy$ and $zu$ and adding edges $xy_i$ and $zu_j$ (note that $zu_j=zy_i$). Both when $|E(Q)|\le|E(P)|$ and when $F_2$ is stable the potential $\Psi_0$ increases (we get a new component of size at least $|E(P)|+1$ in the former case and of size exactly $|E(P)|+|E(Q)|$ in the latter case); a contradiction.
Let $Q$ be a free component. Then $S_1(Q)$ is the set of all vertices $v$ such that for some edge $xy\in E(Q)$ there is a stable $(x,y)$-fan which contains $xv$ as a full edge. For any $v\in S_1(Q)$ the stable fan from the definition above is denoted by $F(v)$; if there are many such fans then we choose an arbitrary one as $F(v)$. We also define $S(Q)=V(Q)\cup S_1(Q)$. Note that by Lemma \[lem:usually-fans-do-not-meet\] for two distinct free components $Q$ and $R$ the sets $S(Q)$ and $S(R)$ are disjoint. For two sets $A$ and $B$, any edge $ab$ with $a\in A$ and $b\in B$ will be called an [*$AB$-edge*]{}.
\[lem:QR-edges-matching\] Let $G$ be a graph of maximum degree $\Delta$ and let $\pi$ be a partial coloring of $G$ which maximizes the potential $\Psi_0$. Assume $\Delta$ is odd and let $Q$ be a free component of $(G,\pi)$ such that $|E(Q)|=(\Delta-1)/2$ and $|{\overline{\pi}}(Q)| = \Delta - 1$. Let $R$ be a free component, $R\ne Q$. Then the set of all $S(Q)S(R)$-edges is a matching.
Let $c$ be the only color in $\{1,\ldots,\Delta\}\setminus{\overline{\pi}}(Q)$. Consider an arbitrary $S(Q)S(R)$-edge $vw$, $v\in S(Q)$. We can assume that $v\in V(Q)$ for otherwise we rotate the stable fan $F(v)$ at $v$; note that then the component which replaces $Q$ has also $(\Delta-1)/2$ edges so by Corollary \[cor:num-uncolored-0\] it has at least $\Delta-1$ free colors. Then $\pi(vw)=c$, because if $w\in V(R)$ this follows from Corollary \[cor:C1\] and otherwise, i.e. when $w\in S_1(R)$, we can rotate the fan $F(w)$ at $w$ and get $w\in V(R)$. (Note that rotating both $F(v)$ and $F(w)$ is possible because they are disjoint by Lemma \[lem:usually-fans-do-not-meet\].) We have just proved that an arbitrary $S(Q)S(R)$-edge is colored by $c$, so the claim follows.
\[lem:QR-edges\] Let $G$ be a graph of maximum degree $\Delta$ and let $\pi$ be a partial coloring of $G$ which maximizes the potential $\Psi_0$. Assume $\Delta$ is odd and let $Q$ be a free component of $(G,\pi)$ such that $|E(Q)|=(\Delta-1)/2$ and $|{\overline{\pi}}(Q)| = \Delta - 1$. Let $R$ be a free component, $R\ne Q$. If there are at least two $V(Q)S(R)$-edges and at least two $S(Q)V(R)$-edges then there is no $V(Q)V(R)$-edge.
In this proof we use the following definition. Let $v_1\in S_1(P)$ and $v_2\in V(P)$ for some free component $P$. We say that $v_1$ is [*safe for $v_2$*]{} if after rotating $F(v_1)$ at $v_1$ the vertices $v_1$ and $v_2$ are in the same free component.
Now we proceed with the proof. Assume on the contrary that there is an edge $qr$ such that $q\in V(Q)$ and $r\in V(R)$. Let $q'r'$ be another $V(Q)S(R)$-edge, $q'\in V(Q)$, and let $q''r''$ be another $S(Q)V(R)$-edge, $r''\in V(R)$; both edges exist by our assumption. Note that $r' \in S_1(R)$ and $q'' \in S_1(Q)$ for otherwise we get a contradiction with Corollary \[cor:C1\], so in particular $q'r'\ne q''r''$. By Lemma \[lem:QR-edges-matching\] we see that $q$, $q'$ and $q''$ are pairwise distinct, and so are $r$, $r'$ and $r''$.
If $r'$ is safe for $r$ then we rotate $F(r')$ at $r'$ and we get a new component $R'$ with two $V(Q)V(R')$-edges; a contradiction with Corollary \[cor:C1\].
If $q''$ is safe for $q$ then we rotate $F(q'')$ at $q''$ and we get a new component $Q'$. Since $|E(Q')|=|E(Q)|$ by Corollary \[cor:num-uncolored-0\] we have $|{\overline{\pi}}(Q')|\ge\Delta-1$. However, there are two $V(Q')V(R)$-edges; a contradiction with Corollary \[cor:C1\].
Now assume that $r'$ is not safe for $r$ and $q''$ is not safe for $q$. Observe that any vertex $v\in S_1(P)$ can be not safe for at most one vertex, namely if $F(v)$ is a $(x,y)$-fan then $v$ can be not safe only for $y$. Hence $r'$ is safe for $r''$ and $q''$ is safe for $q'$. We rotate both $F(r')$ at $r'$ and $F(q'')$ at $q''$. As a result we get two new components $Q'$ and $R'$ where $q'r'$ and $q''r''$ are $V(Q')V(R')$-edges. By the same argument as before, $|{\overline{\pi}}(Q')|\ge\Delta-1$ so we get a contradiction with Corollary \[cor:C1\].
The structure of a $\Psi$-maximal partial $\Delta$-edge-coloring
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Now we define our final potential function $\Psi$ for a partial coloring $(G,\pi)$. Let $\#_c$ be the number of cycles in all free components. Recall that ${\textnormal{nfc}}(G,\pi)$ denotes the set of nontrivial (i.e., with at least one edge) free components of $(G,\pi)$. Then $$\Psi(G,\pi)=(\Psi_0(G,\pi), \#_c, \Delta|V| - \sum_{Q\in{\textnormal{nfc}}(G,\pi)}|{\overline{\pi}}(Q)|).$$
Again assume that $(G,\pi)$ maximizes $\Psi$. Note that all the results from Sections \[sec:structure\] and \[sec:struct-psi0\] apply.
\[lem:cycles\] Let $G$ be a graph and let $\pi$ be a partial coloring of $G$ which maximizes the potential $\Psi$. Let $F_1=(xy_1,\ldots,xy_{\ell})$ be a stable $(x,y)$-fan and $F_2=(zu_1,\ldots,zu_t)$ be a stable $(z,u)$-fan, where $xy$ and $zu$ are distinct edges of the same free component $Q$ of $(G,\pi)$. If $Q$ is a tree, then the ends of $F_1$ and $F_2$ are distinct, i.e. for every $i=1,\ldots,\ell$ and $j=1,\ldots,t$ we have $y_i \ne u_j$.
Assume $y_i=u_j$ for some $i=1,\ldots,\ell$ and $j=1,\ldots,t$. Since $F_1$ and $F_2$ are stable, $y$ and $u$ are leaves of $Q$. Hence if $xy$ and $zu$ are incident then $x=z$ and the claim follows from Lemma \[lem:fans-edge-disjoint\]. Otherwise we perform the two rotations described in the proof of Lemma \[lem:usually-fans-do-not-meet\]. As a result we get a new component $Q'=Q-\{xy,zu\}\cup\{xy_i,y_iz\}$. Then not only $\Psi_0$ does not decrease but also $\#_c$ increases, so $\Psi$ increases; a contradiction.
\[prop:free-colors\] Let $G$ be a graph and let $\pi$ be a partial coloring of $G$ which maximizes the potential $\Psi$. Let $Q$ be a free component of $(G,\pi)$ and let $xy$ be an edge of $Q$. Let $F$ be a stable $(x,y)$-fan and let $Q'$ be the free component that replaces $Q$ after rotating $F$. Then $|{\overline{\pi}}(Q')|\ge|{\overline{\pi}}(Q)|$.
Assume $|{\overline{\pi}}(Q')|<|{\overline{\pi}}(Q)|$. By Proposition \[prop:rotate\], $\Psi_0$ does not decrease. Let $F=(xy_1,\ldots,xy_{\ell})$ and assume $F$ is rotated at $y_i$. Assume $xy$ belongs to a cycle in $Q$. Then $V(Q')=V(Q)\cup\{y_{i}\}$. Since the number of free colors in every vertex from $\{y_2,\ldots,y_{i-1}\}$ does not change after the rotation, the only vertex for which which the number of free colors decreases is $y$, but it stays in the component. Hence by Lemma \[lem:distinct-free\] we have $|{\overline{\pi}}(Q')|\ge|{\overline{\pi}}(Q)|$, a contradiction. Since $xy$ does not belong to a cycle in $Q$ we infer that $\#_c$ does not decrease. Observe that $y_2,\ldots,y_{i}$ do not belong to nontrivial free components different from $Q$, for otherwise in the process of rotating $F$ we can merge two components and $\Psi_0$ increases. Hence, after the rotation all nontrivial components different from $Q'$ do not change their free colors. Then $|{\overline{\pi}}(Q')|<|{\overline{\pi}}(Q)|$ implies that $\sum_{R\in{\textnormal{nfc}}(G,\pi)}|{\overline{\pi}}(R)|$ decreases, so $\Psi$ increases; a contradiction.
\[lem:full-component-no-QR-edges\] Let $G$ be a graph of maximum degree $\Delta$ and let $\pi$ be a partial coloring of $G$ which maximizes the potential $\Psi$. Let $Q$ be a free component of $(G,\pi)$ such that $|{\overline{\pi}}(Q)|=\Delta$. Then for any other free component $R$ there are no $S(Q)S(R)$-edges.
The proof is by contradiction. Assume there is an edge $uv$, such that $u\in S(Q)$ and $v \in S(R)$. First assume that $v\in S_1(R)$. Then we rotate the fan $F(v)$ at $v$. Note that the number of colored edges does not change. Note that by Lemma \[lem:usually-fans-do-not-meet\] rotating $F(v)$ does not affect stable fans of $Q$, so in particular $S_1(Q)$ does not change after the rotation. Hence we can assume that $v\in V(R)$. Now assume that $u\in S_1(Q)$. Then we rotate $F(u)$ at $u$; again the number of colored edges does not change and moreover the new free component also has $\Delta$ free colors by Proposition \[prop:free-colors\]. Hence we can assume that $u\in V(Q)$, i.e. $uv$ is incident with both $Q$ and $R$. Since the number of colored edges is maximal this is a contradiction with Corollary \[cor:C1\].
Bounding the number of uncolored edges
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In this section we assume that $(G,\pi)$ is a partially colored graph such that $\pi$ maximizes the potential $\Psi$ and our goal is to give a bound on the number of uncolored edges. Here is our plan: We put a [*charge*]{}, equal to 1 to every colored edge of graph $G$. Next, every colored edge sends its charge to its endpoints following carefully selected rules. Finally, we assign disjoint sets of vertices to nontrivial free components. Then, we show a lower bound on the total charge at vertices assigned to a nontrivial free component divided by the number of edges in this component. This gives the desired bound. Let us be more precise now. The lemma below will be used in describing the sets of vertices assigned to free components.
\[lem:A\_1\] Assume $4\le\Delta\le 7$. For every free component $Q$ there is a set $A_1(Q)\subseteq S_1(Q)$ such that
1. if $Q$ is a tree, $z_1, z_2 \in A_1(Q)$, $F(z_1)$ is an $(x_1,y_1)$-fan and $F(z_2)$ is an $(x_2,y_2)$-fan then $\{x_1,y_1\}\ne \{x_2,y_2\}$,
2. if $|E(Q)|\le 2$ then $|A_1(Q)|=|E(Q)|$,
3. if $|E(Q)|=3$ then $|A_1(Q)|=2$ if $Q$ is a tree and $|A_1(Q)|=3$ otherwise.
First assume $|E(Q)|=1$ and let $E(Q)=\{xy\}$. Pick any maximal $(x,y)$-fan $F$. Then $F$ is stable and by Corollary \[cor:num-full\] fan $F$ has at least one full edge $xz$. We put $A_1(Q)=\{z\}$.
Now assume $|E(Q)|\ge 2$ and $Q$ is a tree. Consider an arbitrary leaf $\ell$ of $Q$ and let $x\ell$ be the edge of $Q$ incident with $\ell$. Pick any maximal $(x,\ell)$-fan $F_{\ell}$. Since $\ell$ is a leaf $F_{\ell}$ is stable. By Corollary \[cor:num-full\], $F_{\ell}$ has at least $|{\overline{\pi}}(\ell)|\ge 1$ full edges. Pick any such edge $x v_\ell$. Since $|E(Q)|\ge 2$ there are at least two leaves. We pick an arbitrary pair of leaves $\ell_1, \ell_2$ and we put $A_1(Q)=\{v_{\ell_1},v_{\ell_2}\}$. By Lemma \[lem:cycles\] the fans $F_{\ell_1}$ and $F_{\ell_2}$ are disjoint (note that we can apply the lemma since $\ell_1$ and $\ell_2$ are not the endpoints of the same edge), so $|A_1(Q)|=2$.
Finally assume $|E(Q)|=3$ and $Q$ is a cycle. Pick any vertex $v\in V(Q)$. Observe that for any $w\in V(Q)$ we have $|{\overline{\pi}}(w)|\ge 2$. Hence, by Corollary \[cor:num-full\] and Lemma \[lem:fans-edge-disjoint\] there are at least 4 full fan edges incident with $v$. Moreover, since $Q$ is a cycle, for any $xy\in E(Q)$ all $(x,y)$-fans are stable. Let $vu_1$, $vu_2$, $vu_3$ be three of the at least four full fan edges incident with $v$. We put $A_1(Q)=\{u_1,u_2,u_3\}$.
For every nontrivial free component $Q$ the set of vertices assigned to $Q$ is defined as $A(Q)=V(Q)\cup A_1(Q)$. Note that $A(Q)\subseteq S(Q)$. It follows that for any two distinct free components $P$ and $Q$ the sets $A(P)$ and $A(Q)$ are disjoint, since $S(P)$ and $S(Q)$ are disjoint. Observe also that some vertices of $G$ may not be assigned to any of the free components. Let us denote $A_0 (Q) = V(Q)$, $A = \bigcup_{Q\in {\textnormal{nfc}}(G,\pi)} A(Q)$, $A_0 = \bigcup_{Q\in {\textnormal{nfc}}(G,\pi)} A_0(Q)$, and $A_1 = \bigcup_{Q\in {\textnormal{nfc}}(G,\pi)} A_1(Q)$.
Our rules for moving the charge are the following. Let $xy$ be an arbitrary colored edge. By symmetry we can assume that if one of its endpoints is in $A_0$ then $x \in A_0$.
1. $xy$ divides its charge equally between its endpoints in $A$, i.e. it sends $\frac{1}{|\{x,y\}\cap A|}$ to each of its endpoints from $A$, unless (R2) applies.
2. If $x\in A(P)$, $y \in A_1(Q)$ for two distinct free components $P$ and $Q$ such that $|E(P)|\ge 2$ and $|E(Q)|=1$, then $xy$ sends $(1-\epsilon_\Delta)$ to $x$ and $\epsilon_\Delta$ to $y$, where $$\epsilon_4 = \tfrac{1}{2}, \quad \epsilon_5 = \tfrac{1}{4}, \quad \epsilon_6 = \tfrac{1}{12}, \quad \epsilon_7 = \tfrac{3}{28}.$$
Let ${\textnormal{ch}}(v)$ denote the amount of charge received by a vertex $v$. For a set $S\subseteq V$ we denote ${\textnormal{ch}}(S)=\sum_{v\in S}{\textnormal{ch}}(v)$. The disjointness of the sets $A(Q)$ immediately gives the following.
$$\gamma_{\Delta}(G) \ge \min_{Q\in{\textnormal{nfc}}(G,\pi)}\frac{{\textnormal{ch}}(A(Q))}{{\textnormal{ch}}(A(Q))+|E(Q)|}.$$
In what follows we give lower bounds for the ratio $\frac{{\textnormal{ch}}(A(Q))}{{\textnormal{ch}}(A(Q))+|E(Q)|}$ for $\Delta=4,\ldots,7$ and $|E(Q)|=1,\ldots,{\ensuremath{\lfloor{\Delta/2}\rfloor}}$, which is sufficient by Corollary \[cor:num-uncolored\]. We begin with some simple cases.
\[lem:incident\] Let $e$ be a colored edge incident with a free component $Q$. Then the charge $e$ sends to $A(Q)$ is
1. at least $\frac{1}2$,
2. at least $1-\epsilon_\Delta$ if $e$ is a full edge of a non-stable fan,
3. 1 if $e$ is a full edge of a stable fan.
The discharging rules easily imply $(i)$. Let $e=vw$ for $v\in V(Q)$ and $w\not\in V(Q)$. If $e$ is a full edge, then ${\overline{\pi}}(w)=\emptyset$, so $w \not\in A_0$ and hence the rules imply $(ii)$. Finally, if $e$ is a full edge of a stable fan then by Lemma \[lem:usually-fans-do-not-meet\] there is no free component $P\ne Q$ such that $w\in A_1(P)$. It follows that if $w \in A$ then $w\in A_1(Q)$, so by (R1) $e$ sends 1 to $A(Q)$ and $(iii)$ follows.
\[lem:bound-incident-1-edge\] Let $Q$ be a free component consisting of exactly one edge. Then, the edges incident with $Q$ send the charge of at least $\Delta$ to $A(Q)$.
By Lemma \[lem:incident\] every colored edge incident with $Q$ sends at least $1/2$ to $A(Q)$. Since for every vertex $v\in V(Q)$ there are exactly $\Delta-{\overline{\pi}}(v)$ such edges, they send at least $\frac{1}2\sum_{v\in Q}(\Delta-|{\overline{\pi}}(v)|)$ to $A(Q)$.
Let $E(Q)=\{xy\}$. Then we choose a maximal $(x,y)$-fan $F_1$ and a maximal $(y,x)$-fan $F_2$. Note that both $F_1$ and $F_2$ are stable, since $|E(Q)|=1$. The fan $F_1$ (resp. $F_2$) has at least $|{\overline{\pi}}(y)|$ (resp. $|{\overline{\pi}}(x)|$) full edges by Corollary \[cor:num-full\]. Hence there are at least $\sum_{v\in Q}|{\overline{\pi}}(v)|$ full fan edges incident with $Q$ and by Lemma \[lem:incident\] each of them sends $1$ to $A(Q)$. It follows that the total charge $A(Q)$ receives from the incident edges is at least $\frac{1}2\sum_{v\in Q}(\Delta-{\overline{\pi}}(v))+\frac{1}{2}\sum_{v\in Q}|{\overline{\pi}}(v)| = \Delta$.
\[prop:bound-full-end\] Let $F$ be a stable $(x,y)$-fan and let $xz$ be a full edge of $F$. If $z\in A_1(Q)$ for some free component $Q$, then the charge received by $z$ from edges not incident with $Q$ is at least $\eta_\Delta(\Delta-|V(Q)|)$, where $$\eta_\Delta = \begin{cases}
\frac{1}{2} & \text{when $|E(Q)|\ge 2$} \\
\epsilon_\Delta & \text{when $|E(Q)|=1$.}
\end{cases}$$
\[cor:bound-1-edge\] Let $Q$ be a one-edge free component of $(G,\pi)$. Then, $$\frac{{\textnormal{ch}}(A(Q))}{{\textnormal{ch}}(A(Q))+|E(Q)|} \ge \begin{cases}
\frac{5}6 & \text{when $\Delta=4$,}\\
\frac{23}{27} & \text{when $\Delta=5$,}\\
\frac{19}{22} & \text{when $\Delta=6$,}\\
\frac{211}{239}>\frac{22}{25} & \text{when $\Delta=7$.}
\end{cases}$$
By Lemma \[lem:A\_1\] we have $|A_1(Q)|=1$. Hence, by Lemma \[lem:bound-incident-1-edge\] and Proposition \[prop:bound-full-end\] we have ${\textnormal{ch}}(A(Q)) \ge \Delta + \eta_\Delta(\Delta-2)$, which is equal to $5$, $\frac{23}{4}$, $\frac{19}3$ and $\frac{211}{28}$ when $\Delta=4,5,6,7$, respectively. The claim follows.
From our charge moving rules and Lemma \[lem:full-component-no-QR-edges\] we immediately get the following.
\[prop:bound-full-component\] For every free component $Q$ such that $|{\overline{\pi}}(Q)|=\Delta$ every edge incident with $A(Q)$ sends 1 to $A(Q)$.
\[lem:degrees-full-component\] Let $Q$ be a free component such that $|E(Q)|={\ensuremath{\lfloor{\Delta/2}\rfloor}}$. Then, $A(Q)$ contains exactly $|{\overline{\pi}}(Q)|-2{\ensuremath{\lfloor{\Delta/2}\rfloor}}$ vertices of degree $\Delta-1$ in $G$ and all the remaining vertices of $A(Q)$ are of degree $\Delta$.
Clearly, for every $v\in V(Q)$ we have $|{\overline{\pi}}(v)| = \Delta - |\pi(v)| = \Delta - (\deg_G(v) - \deg_Q(v))$. By Lemma \[lem:A\], $|{\overline{\pi}}(Q)|=\sum_{v\in V(Q)}{\overline{\pi}}(v)$, so $$|{\overline{\pi}}(Q)| = \sum_{v\in V(Q)}(\Delta-\deg_G(v)) + 2|E(Q)|.$$ By plugging in our assumptions and rearranging the formula, we get $$\sum_{v\in V(Q)}(\Delta-\deg_G(v)) = |{\overline{\pi}}(Q)| - 2{\ensuremath{\lfloor{\Delta/2}\rfloor}}.$$ By Corollary \[cor:num-uncolored-0\] we have $|{\overline{\pi}}(Q)|\ge 2|E(Q)|\ge \Delta-1$. Hence, $|{\overline{\pi}}(Q)| - 2{\ensuremath{\lfloor{\Delta/2}\rfloor}} \le 1$. It follows that $Q$ has exactly $|{\overline{\pi}}(Q)|-2{\ensuremath{\lfloor{\Delta/2}\rfloor}}$ vertices of degree $\Delta-1$ in $G$ and all the remaining vertices of $Q$ are of degree $\Delta$. Moreover, since the vertices of $A_1(Q)$ are ends of [*full*]{} fan edges, each of them is of degree $\Delta$. The claim follows.
\[cor:bound-full-even-component\] When $\Delta$ is even, for every free component $Q$ such that $|E(Q)|=\Delta/2$, $${\textnormal{ch}}(A(Q)) \ge \Delta|A(Q)| - |E(G[A(Q)])| - |E(Q)|.$$
By Corollary \[cor:num-uncolored-0\] we have $|{\overline{\pi}}(Q)|=\Delta$. Hence by Lemma \[lem:degrees-full-component\] there are exactly $\Delta|A(Q)| - |E(G[A(Q)])|$ edges incident with $A(Q)$, and $|E(Q)|$ of them are uncolored. This, together with Proposition \[prop:bound-full-component\], gives the claim.
Now we are very close to establishing our bound for $\Delta=4$. We will need just one more auxiliary claim (Lemma \[lem:K\_(k+1)-e\] below).
\[lem:odd-clique\] For every odd $k$, the clique $K_{k+1}$ is $k$-edge colorable.
\[lem:even-clique\] For every even $k$ we have $c(K_{k+1})=k^2/2$. Moreover, there is a partial $k$-edge-coloring $\pi$ of $K_{k+1}$ with $k^2/2$ colored edges such that the uncolored edges form a matching, and for each pair of distinct vertices $x$ and $y$, ${\overline{\pi}}(x)\ne{\overline{\pi}}(y)$.
Since every color class covers at most ${\ensuremath{\lfloor{(k+1)/2}\rfloor}}=k/2$ edges, we have $c(K_{k+1})\le k^2/2$.
Now we show that $k^2/2$ edges of $K_{k+1}$ can be colored with $k$ colors. Begin by a $(k+1)$-edge-coloring of $K_{k+2}$, which exists by Lemma \[lem:odd-clique\]. Remove one vertex to get a $(k+1)$-colored $K_{k+1}$. Uncolor the edges colored with the color $k+1$. There are at most $k/2$ of them, so the the number of colored edges is at least ${k+1\choose 2} - k/2 = k^2/2$. The coloring satisfies the desired property because in the $(k+1)$-coloring of $K_{k+2}$ every vertex, including the removed one, is incident with all $k+1$ colors.
Let ${\mathcal{G}}^\Delta_d$ be the family of all simple graphs which (i) have at least one edge, (ii) are of maximum degree at most $\Delta$ and (iii) such that any subset of vertices of size $(\Delta+1)$ induces a subgraph with at most ${\Delta+1\choose 2} - d$ edges.
\[lem:K\_(k+1)-e\] Assume $\Delta\ge 4$ and $\Delta$ is even. If for every graph $G\in{\mathcal{G}}^\Delta_2$ we have $\gamma_\Delta(G)\ge \alpha$ for some constant $\alpha\in[0,1]$, then for every graph $G\in{\mathcal{G}}^\Delta_1$ we have $\gamma_\Delta(G)\ge \min\{\alpha,\frac{\Delta^2}{\Delta^2+\Delta-2}\}$.
Let $G$ be an arbitrary graph from $G\in{\mathcal{G}}^\Delta_1$. We use induction on $|V(G)|$. For the base case when $|V(G)|=2$, i.e. $G$ consists of a single edge, $\gamma_\Delta(G)=1$ so the claim follows. Let $|V(G)|>3$. We can assume that $V(G)$ contains a subset $S$ of size $(\Delta+1)$ such that $|G[S]| = {\Delta+1 \choose 2}-1$ for otherwise the claim follows from the assumed property of ${\mathcal{G}}^\Delta_2$. If there are no edges leaving $S$, then we just color $G-S$ inductively and we color $\Delta^2/2$ edges of $G[S]$ according to Lemma \[lem:even-clique\]. Then $\gamma_\Delta(G)\ge\min\{\gamma_\Delta(G-S),(\Delta^2/2)/({\Delta+1\choose 2}-1)\}=\min\{\gamma_\Delta(G-S),\frac{\Delta^2}{\Delta^2+\Delta-2}\}\ge \min\{\alpha,\frac{\Delta^2}{\Delta^2+\Delta-2}\}$. We can also assume that $G$ has no cutvertex for otherwise it is easy to get the claim from the induction hypothesis. It follows that there are exactly two edges leaving $S$, say $xx'$ and $yy'$, with $x,y\in S$, and $x,x',y,y'$ distinct. Then we remove $S$ and add a new vertex $q$ and two new edges $x'q$, $y'q$. Denote the resulting graph by $G'$. Find the partial coloring of $G'$ corresponding to the largest $\Delta$-colorable subgraph of $G'$. Then in the partially colored $G'$ we remove $q$ and put back the set $S$ with incident edges. Color $xx'$ and $yy'$ with the colors of $x'q$ and $y'q$, respectively (and if one of the edges $x'q$, $y'q$ is uncolored, then the corresponding edge is also uncolored; note that $x'q$ and $y'q$ do not get the same color). By Lemma \[lem:even-clique\] we can color $\Delta^2/2$ edges of $G[S]$ so that the edges of $G[S]$ incident with $x$ do not get the color of $xx'$ and the edges of $G[S]$ incident with $y$ do not get the color of $yy'$. Then again $\gamma_\Delta(G)\ge\min\{\gamma_\Delta(G'),(\Delta^2/2)/({\Delta+1\choose 2}-1)\}\ge \min\{\alpha,\frac{\Delta^2}{\Delta^2+\Delta-2}\}$.
\[lem:bound-4-colors-2-edges\] Let $\Delta=4$ and let $Q$ be a two-edge free component of $(G,\pi)$. If $G\in {\mathcal{G}}^\Delta_2$ then $\frac{{\textnormal{ch}}(A(Q))}{{\textnormal{ch}}(A(Q))+|E(Q)|} \ge \frac{5}{6}$.
By Lemma \[lem:A\_1\], $|A_1(Q)|= 2$. By Corollary \[cor:bound-full-even-component\] we have ${\textnormal{ch}}(A(Q))\ge 4\cdot 5 - ({5 \choose 2} -2) - 2 = 10$ and we get ${\textnormal{ch}}(A(Q)) / ({\textnormal{ch}}(A(Q))+|E(Q)|) \ge \frac{5}{6}$, as required.
\[cor:main-4\] Every connected simple graph $G$ of maximum degree $4$ has a $4$-edge-colorable subgraph with at least $\frac{5}{6}|E|$ edges, unless $G = K_5$.
By Corollary \[cor:num-uncolored\] every free component of a partially 4-edge-colored graph which maximizes the potential $\Psi$ has at most two edges. Hence, by Corollary \[cor:bound-1-edge\] and Lemma \[lem:bound-4-colors-2-edges\] for every $G\in {\mathcal{G}}^4_2$ we have $\gamma_4(G)\ge\frac{5}6$. By Lemma \[lem:K\_(k+1)-e\] the same bound holds also for graphs in ${\mathcal{G}}^4_1$, which is equivalent to our claim.
\[lem:bound-almost-full-component\] Assume $\Delta\in \{5,7\}$ and let $Q$ be a free component such that $|E(Q)|=(\Delta-1)/2$, $|{\overline{\pi}}(Q)| = \Delta-1$. Let $D$ be the set of colored edges incident with $A(Q)$. Then, the charge sent from $D$ to $A(Q)$ is at least $|D|-\frac{|A(Q)|-2}2$.
Call an edge $e\in D$ [*bad*]{} if it sends less than 1 to $A(Q)$. Note that every bad edge sends either $\frac{1}2$ or $1-\epsilon_\Delta\ge\frac{1}2$ to $A(Q)$. Hence in what follows we assume that there are at least $|A(Q)|-1$ bad edges, for otherwise we get the claim immediately.
Clearly, every bad edge has only one endpoint in $A(Q)$ and the other endpoint is in $A(P)$ for some $P\ne Q$. We prove the following two auxiliary claims:
[**Claim 1:**]{} There is a free component $P\ne Q$ such that every bad edge has an endpoint in $A(P)$.
[*Proof of Claim 1.*]{} The proof is by contradiction, i.e. we assume that there are two edges $uv$ and $xy$ such that $u,x \in A(Q)$ and $v \in A(P)$ and $y \in A(R)$ for some distinct free components $P,R\ne Q$. We consider two cases.
[[**CASE A: **]{}]{} $u,x\in V(Q)$. If $v \in A_1$ then we rotate $F(v)$ at $v$. Similarly, if $y \in A_1$ then we rotate $F(y)$ at $y$. Note that if both $v\in A_1$ and $y\in A_1$ then the fans $F(v)$ and $F(y)$ are distinct by Lemma \[lem:usually-fans-do-not-meet\]. It follows that if both $v\in A_1$ and $y\in A_1$ then rotating $F(v)$ does not destroy $F(y)$ and we can indeed perform both rotations. As a result, $v,y\in A_0$, which is a contradiction with Corollary \[cor:C1\].
[[**CASE B: **]{}]{} case A does not apply. However, since there are at least $|A(Q)|-1$ bad edges, and each vertex of $A(Q)$ is incident with at most one of them by Lemma \[lem:QR-edges-matching\], we infer that at most one vertex of $V(Q)$ is not incident with a bad edge. Since Case A does not apply, for some free component $P\ne Q$ each bad edge incident with $V(Q)$ has the other endpoint in $A(P)$. If Claim 1 does not hold, there is a bad edge $uv$, $u\in A_1(Q)$ and $v\in A(R)$, for some $R\ne Q,P$. Then we rotate $F(u)$ at $u$ and the component $Q$ is replaced by a new component $Q'$ with $u\in V(Q')$. Since $|V(Q)|\ge 3$, $|V(Q)\setminus V(Q')|\le 1$ and at most one vertex of $V(Q)$ is not incident with a bad edge it means that at least one vertex of $V(Q)\cap V(Q')$ is incident with a bad edge in the new colored graph, and every such bad edge has an endpoint in $A(P)$. However, then we proceed as in Case A (note that by Lemma \[lem:usually-fans-do-not-meet\] rotating the fan $F(u)$ does not affect the fans $F(v)$ and $F(y)$). This finishes the proof of Claim 1.
Let $P$ be the free component from Claim 1.
[**Claim 2:**]{} There is at most one bad edge incident both with $A(Q)$ and $V(P)$.
[*Proof of Claim 2.*]{} Assume there are two such edges, say $q_1p_1$ and $q_2p_2$ with $q_1,q_2\in A(Q)$. Since there are at least $|A(Q)|-1$ bad edges and $|V(Q)|\ge 3$, there are also at least two bad edges incident both with $V(Q)$ and with $A(P)$, say $q_3p_3$ and $q_4p_4$ with $q_3,q_4\in V(Q)$. By Lemma \[lem:QR-edges\], $q_1,q_2\in A_1(Q)$ and $p_3,p_4\in A_1(P)$. Then we rotate $F(q_1)$ at $q_1$ and let $Q'$ be the component that replaces $Q$. Note that at least one of $q_3$, $q_4$ is in $V(Q')$ and $|{\overline{\pi}}(Q')|\ge\Delta-1$ by Proposition \[prop:free-colors\]. By symmetry assume $q_3\in V(Q')$. First assume $Q$ is a tree. Then by Lemma \[lem:A\_1\]$(i)$ and Lemma \[lem:cycles\] rotating $F(q_1)$ does not affect $F(q_2)$ so in particular $q_2\in A_1(Q')$. We see that $q_1p_1$, $q_2p_2$ and $q_3p_3$ are $S(Q')S(P)$-edges, $q_1,q_3\in V(Q')$ and $p_1,p_2\in V(P)$. This is a contradiction with Lemma \[lem:QR-edges\]. Now assume $Q$ is not a tree. By Corollary \[cor:num-uncolored\] we have $|E(Q)|\le{\ensuremath{\lfloor{\Delta/2}\rfloor}}$, so $Q$ is a 3-cycle. Then $q_3,q_4\in V(Q')$. If after rotating $F(p_3)$ at $p_3$ the component $P'$ that replaces $P$ contains $p_1$, we do rotate $F(p_3)$ at $p_3$. As a result, we get two $V(Q)V(P')$-edges, namely $q_1p_1$ and $q_3p_3$; a contradiction with Corollary \[cor:C1\]. Hence we can assume that after rotating $F(p_3)$ at $p_3$ the component that replaces $P$ does not contain $p_1$. Hence $P$ is a tree and $F(p_3)$ is a $(v,p_1)$-fan for some $v\in V(P)$. By Lemma \[lem:A\_1\]$(i)$ we see that $F(p_4)$ is not a $(w,p_1)$-fan for any $w\in V(P)$. It follows that after rotating $F(p_4)$ at $p_4$ the component $P'$ that replaces $P$ contains $p_1$. We get a contradiction with Corollary \[cor:C1\] as before. This finishes the proof of Claim 2.
The value of $\Delta$ is odd, so by Corollary \[cor:num-uncolored\] we have $|E(P)|\le (\Delta-1)/2$. Hence by Lemma \[lem:A\_1\] we have $|A_1(P)|\le (\Delta-1)/2$, so by Lemma \[lem:QR-edges-matching\] there are at most $(\Delta-1)/2$ bad edges not incident with $V(P)$. This, together with Claim 2 implies that the total number of bad edges is at most $(\Delta+1)/2$, which is at most $3$ when $\Delta=5$ and at most $4$ when $\Delta=7$. This is a contradiction with our assumption that there are at least $|A(Q)|-1$ bad edges. Indeed, if $\Delta=5$ then by Lemma \[lem:A\_1\] we have $|A(Q)|-1=4$, and if $\Delta=7$ then by Lemma \[lem:A\_1\] we have $|A(Q)|-1= 5$.
\[lem:bound-5-colors-2-edges\] Let $\Delta=5$ and let $Q$ be a 2-edge free component. Then, $\frac{{\textnormal{ch}}(A(Q))}{{\textnormal{ch}}(A(Q))+|E(Q)|} \ge \frac{23}{27}$.
We show that ${\textnormal{ch}}(Q) \ge \frac{23}2$, which implies the claim.
By Corollary \[cor:num-uncolored-0\], $|{\overline{\pi}}(Q)|\ge 4$ and by Lemma \[lem:A\_1\], $|A(Q)|= 5$.
Assume $|{\overline{\pi}}(Q)|=5$. Then by Proposition \[prop:bound-full-component\] and Lemma \[lem:degrees-full-component\] , ${\textnormal{ch}}(A(Q)) \ge 5|A(Q)| - 1 - |E(G[A(Q)])| - |E(Q)|$. Since $|E(G[A(Q)])|\le {5 \choose 2}$ we get ${\textnormal{ch}}(A(Q))\ge 12$, as required.
Finally assume $|{\overline{\pi}}(Q)|=4$. Let $D$ be the set of colored edges incident with $A(Q)$. By Lemma \[lem:degrees-full-component\], all vertices of $A(Q)$ are of degree $\Delta$ in $G$. Hence, $|D|=5|A(Q)|-|E(G[A(Q)])|-2$ and by Lemma \[lem:bound-almost-full-component\] we get ${\textnormal{ch}}(A(Q))\ge \frac{9}2|A(Q)|-|E(G[A(Q)])|-1$. Since $|E(G[A(Q)])|\le {5 \choose 2}$ we get ${\textnormal{ch}}(A(Q))\ge \frac{23}2$, as required.
\[cor:main-5\] Every simple graph $G$ of maximum degree $5$ has a $5$-edge-colorable subgraph with at least $\frac{23}{27}|E|$ edges.
By Corollary \[cor:num-uncolored\] every free component of a partially 5-edge-colored graph which maximizes the potential $\Psi$ has at most two edges. Hence, by Corollary \[cor:bound-1-edge\] and Lemma \[lem:bound-5-colors-2-edges\] the claim follows.
\[lem:full-fan-edges-0\] For every free component $Q$, if $|E(Q)|\ge 2$ and $Q$ is a tree then the number of edges incident with $Q$ which are full edges of some stable fan is at least $\displaystyle\sum_{\substack{\ell \in V(Q)\\ \deg_Q(\ell)=1}}|{\overline{\pi}}(\ell)|$.
Consider an arbitrary leaf $\ell$ of $Q$ and let $x\ell$ be the edge of $Q$ incident with $\ell$. Pick any maximal $(x,\ell)$-fan $F_{\ell}$. Since $\ell$ is a leaf $F_{\ell}$ is stable. By Corollary \[cor:num-full\], $F_{\ell}$ has at least $|{\overline{\pi}}(\ell)|$ full edges. By Lemma \[lem:cycles\], for two different leaves $\ell$ and $\ell'$ the ends of $F_{\ell}$ and $F_{\ell'}$ are disjoint (note that we can apply the lemma since $\ell$ and $\ell'$ are not the endpoints of the same edge). Hence the claim follows.
\[lem:bound-incident\] Let $Q$ be a 2-edge free component and assume that $\Delta\in\{6,7\}$. Then, the charge received by $A(Q)$ from the edges incident with $Q$ is at least $\frac{3}2\Delta+1-4\epsilon_\Delta$.
Similarly as in the proof of Lemma \[lem:bound-incident-1-edge\], there are exactly $\sum_{v\in Q}(\Delta-{\overline{\pi}}(v))$ colored edges incident with $Q$ and each of them sends at least $\frac{1}{2}$ to $Q$ by Lemma \[lem:incident\].
For every vertex $y\in V(Q)$, for every incident uncolored edge $xy \in E(Q)$ we choose a maximal $(x,y)$-fan and it has at least $|{\overline{\pi}}(y)|$ full edges by Corollary \[cor:num-full\]. It follows that there are at least $\sum_{v\in V(Q)}\deg_Q(v)|{\overline{\pi}}(v)|$ full fan edges incident with $V(Q)$, and by Lemma \[lem:incident\] each of them sends at least $1-\epsilon_\Delta$ to $Q$,
The component $Q$ is a 2-path, say $pqr$, where $|{\overline{\pi}}(p)|,|{\overline{\pi}}(r)|\ge 1$ , and $|{\overline{\pi}}(q)|\ge 2$. By Lemma \[lem:full-fan-edges-0\] there are at least $|{\overline{\pi}}(p)|+|{\overline{\pi}}(r)|$ edges incident with $Q$ that are full edges of stable fans, and by Lemma \[lem:incident\] each of them sends $1$ to $Q$.
The charge sent from the edges incident with $Q$ to $A(Q)$ is at least $$\begin{aligned}
&&\tfrac{1}{2}\sum_{v\in Q}(\Delta-{\overline{\pi}}(v)) + (\tfrac{1}{2}-\epsilon_\Delta)\sum_{v\in Q}\deg_Q(v)|{\overline{\pi}}(v)| + \epsilon_\Delta(|{\overline{\pi}}(p)|+|{\overline{\pi}}(r)|) = \\
&&\frac{\Delta|V(Q)|}2 + \tfrac{1}{2}\underbrace{\sum_{v\in Q}|{\overline{\pi}}(v)|(\deg_Q(v)-1)}_{|{\overline{\pi}}(q)|} - \epsilon_\Delta\underbrace{\sum_{v\in Q}\deg_Q(v)|{\overline{\pi}}(v)|}_{|{\overline{\pi}}(p)|+2|{\overline{\pi}}(q)|+|{\overline{\pi}}(r)|} + \epsilon_\Delta(|{\overline{\pi}}(p)|+|{\overline{\pi}}(r)|) = \\
&&\tfrac{3}2\Delta + (\tfrac{1}{2}-2\epsilon_\Delta)|{\overline{\pi}}(q)| \ge \tfrac{3}2\Delta + 1 - 4\epsilon_\Delta.
\end{aligned}$$
\[cor:6-7-colors-2-edges\] Let $Q$ be a free component of $(G,\pi)$. If $|E(Q)|=2$ and $\Delta\in\{6,7\}$, then $$\frac{{\textnormal{ch}}(A(Q))}{{\textnormal{ch}}(A(Q))+|E(Q)|} \ge \begin{cases}
\frac{19}{22} & \text{when $\Delta=6$,}\\
\frac{211}{239}>\frac{22}{25} & \text{when $\Delta=7$.}
\end{cases}$$
By Lemma \[lem:A\_1\] we have $|A_1(Q)|= 2$. Hence, by Lemma \[lem:bound-incident\] and Proposition \[prop:bound-full-end\] we have ${\textnormal{ch}}(A(Q)) \ge \tfrac{3}2\Delta + 1 - 4\epsilon_\Delta + 2\cdot {\frac{1}2}\cdot (\Delta-3)$. Hence, ${\textnormal{ch}}(A(Q))\ge 12\frac{2}3$ if $\Delta=6$ and ${\textnormal{ch}}(A(Q))\ge 15\frac{1}{14}$ if $\Delta=7$. Then ${\textnormal{ch}}(A(Q))/({\textnormal{ch}}(A(Q))+|E(Q)|)$ is at least $\frac{19}{22}$ for $\Delta=6$ and $\frac{211}{239}$ for $\Delta=7$, as required.
\[lem:6-colors-3-edges\] Let $\Delta=6$ and let $Q$ be a 3-edge free component. Assume that $G$ does not contain a set of 6 vertices $S$ such that $G[S]$ induces a clique and exactly 6 edges leave $S$. Then, $\frac{{\textnormal{ch}}(A(Q))}{{\textnormal{ch}}(A(Q))+|E(Q)|} \ge \frac{19}{22}$.
By Corollary \[cor:num-uncolored-0\] we have $|{\overline{\pi}}(Q)|= 6$ and by Lemma \[lem:A\_1\] we have $|A(Q)|=6$. By Corollary \[cor:bound-full-even-component\] we have ${\textnormal{ch}}(A(Q)) \ge 6|A(Q)| - |E(G[A(Q)])| - 3$. If $G[A(Q)]$ does not induce a clique then $|E(G[A(Q)])|\le {6 \choose 2} - 1$ and hence ${\textnormal{ch}}(A(Q)) \ge 19$. Finally, if $G[A(Q)]$ induces a clique then since every vertex in $A(Q)$ is of degree 6 there are exactly 6 edges leaving $A(Q)$; a contradiction. It follows that $\frac{{\textnormal{ch}}(A(Q))}{{\textnormal{ch}}(A(Q))+|E(Q)|}\ge\frac{19}{22}$, as required.
In the following lemma by [*extending*]{} a partial coloring $\pi$ we mean finding a new coloring which matches $\pi$ at the edges already colored in $\pi$.
\[lem:6-colors-extend\] Let $G$ be a graph of maximum degree 6 that contains a subgraph $H$ isomorphic to a 6-clique. Let $\pi$ be an arbitrary partial 6-edge-coloring of $G$ such that the edges of $H$ are uncolored. Assume there are two vertices $v,w$ of $H$ such that $|{\overline{\pi}}(v)\cap{\overline{\pi}}(w)|\ge 5$. Then, $\pi$ can be extended so that at most 2 edges of $H$ are left uncolored.
Let $V(H)=\{v,w,x_1,\ldots,x_4\}$. Note that for every $i=1,\ldots,4$ we have ${\overline{\pi}}(x_i)\ge 5$. Assume w.l.o.g. that $\{1,\ldots,5\}\subseteq {\overline{\pi}}(v)\cap{\overline{\pi}}(w)$. If among $x_1,\ldots,x_4$ there are at most two vertices incident with an edge colored with a color from $\{1,\ldots,5\}$ then we just color $H$ with colors $1,\ldots,5$ using Lemma \[lem:odd-clique\]. As a result, at most two edges of $H$ get the same color as an incident edge so we can uncolor these two edges and get the claim. Otherwise, by Lemma \[lem:odd-clique\] and by the symmetry we can color $E(H)$ with colors $1,\ldots,5$ so that $\pi(x_1x_2) \in (\pi(x_1)\cup\pi(x_2))\setminus\{6\}$ and $\pi(x_3x_4) \in (\pi(x_3)\cup\pi(x_4))\setminus\{6\}$. Next we recolor $x_1x_2$ and $x_3x_4$ to color 6. Clearly, then at most two edges of $H$ still have the same color as an incident edge so we can uncolor these two edges and get the claim.
Now we are ready to finish the proof of our bound for 6 colors. Similarly as in the case of 4 colors we need to exclude some special case when $G$ contains a dense structure, which unfortunately turns out to be quite technical this time.
\[lem:main-6\] Every connected simple graph $G$ of maximum degree at most $6$ has a $6$-edge-colorable subgraph with at least $\frac{19}{22}|E|$ edges, unless $G=K_7$.
We use the induction on $|E(G)|$. For the base case observe that the claim holds for the empty graph. Now we proceed with the induction step.
First assume that $G$ does not contain a set of 6 vertices $S$ such that $G[S]$ induces a clique and exactly 6 edges leave $S$. By Corollary \[cor:num-uncolored\] every free component of a partially 6-edge-colored graph which maximizes the potential $\Psi$ has at most three edges. Hence, by Corollary \[cor:bound-1-edge\], Corollary \[cor:6-7-colors-2-edges\] and Lemma \[lem:6-colors-3-edges\] the claim follows.
Hence in what follows we assume that there is a set $S\subset V(G)$ such that $G[S]$ induces a $K_6$ and exactly 6 edges leave $S$ (each vertex of $S$ is incident with one of them).
Now assume that there are two edges leaving $S$, say $vx$ and $wy$ with $v,w\in S$, such that $x\ne y$ and $xy\not\in E(G)$. Then we remove $S$ from $G$ and add edge $xy$. Next we apply the induction hypothesis to the resulting graph $G'$, getting a partial coloring $\pi'$. We color $E(G)\cap E(G')$ according to $\pi'$, and we color $vx$ and $wy$ with $\pi'(xy)$. Next we color the remaining 4 edges leaving $S$ with free colors and we color $E(G[S])$ using Lemma \[lem:6-colors-extend\] so that at most two edges are left uncolored. As a result we get a partial coloring where the number of colored edges is at least $\frac{19}{22}|E(G')| + 18 = \frac{19}{22}|E(G')| + \frac{18}{20}(|E(G)|-|E(G')|) > \frac{19}{22}|E(G)|$, as required.
Hence we can assume that $N(S)$ induces a clique. Since $G\ne K_7$, $|N(S)|>1$. Let $N(S)=\{v_1,\ldots,v_{|N(S)|}\}$. We remove the edges of $E(N[S])$ and we (partially) color the resulting graph $G'$ inductively. In what follows we show (for each value of $|N(S)|$ separately) that the coloring $\pi'$ of $G'$ can be extended to a coloring $\pi''$ so that (1) at most one edge of $E(G[N(S)])$ is uncolored and (2) there are two vertices $v,w\in N(S)$ such that ${\overline{\pi}}(v)\cap {\overline{\pi}}(w) \ne \emptyset$. Having that, we extend the coloring further. We pick an edge $vx$ for $x\in S$ and we color it with a color $a\in{\overline{\pi}}(v)\cap {\overline{\pi}}(w)$. Next we pick an edge $wy$ for $y\in S$ and we color it with the same color $a$. The remaining edges of $E(N(S),S)$ are colored with free colors. Note that $|{\overline{\pi}}(x)\cap{\overline{\pi}}(y)|=5$. Finally we partially color $G[S]$ using Lemma \[lem:6-colors-extend\] so that at most 2 edges remain uncolored. As a result we get a partial coloring of $G$ where the number of colored edges is at least $\frac{19}{22}|E(G')| + {|N(S)| \choose 2} - 1 + 6 + {6\choose 2} - 2 \ge \frac{19}{22}|E(G')| + \frac{{|N(S)| \choose 2} + 18}{{|N(S)| \choose 2} + 21}(|E(G)|-|E(G')|) \ge \frac{19}{22}|E(G)|$, as required.
[[**CASE 1: **]{}]{} $|N(S)|=6$. Then $E(N(S),V\setminus N[S])=\emptyset$. We color $E(G[N(S)])$ with colors $1,\ldots,5$ according to Lemma \[lem:odd-clique\]. Then for every $v,w\in N(S)$ we have ${\overline{\pi}}(v)\cap {\overline{\pi}}(w) =\{6\}$.
[[**CASE 2: **]{}]{} $|N(S)|=5$. W.l.o.g. we can assume that for every $i=1,\ldots,4$ we have $|E(\{v_i\},S)|=1$ and $|E(\{v_5\},S)|=2$. Then, for every $i=1,\ldots,4$ we have $|E(\{v_i\},V\setminus N[S])|\le 1$ and $E(\{v_5\},V\setminus N[S])|=0$
First assume that there is a color, say color 1, which appears at all the four edges of $E(N(S),V\setminus N[S])$. Then by Lemma \[lem:even-clique\] we can color $G[N(S)]$ with colors $2,\ldots,5$ so that only $v_1v_2$ and $v_3v_4$ are uncolored. Then we color $v_1v_2$ with 6 and in the resulting coloring $\pi$ we have $6\in{\overline{\pi}}(v_3)\cap{\overline{\pi}}(v_4)$, so ${\overline{\pi}}(v_3)\cap{\overline{\pi}}(v_4)\ne \emptyset$, as required.
Now w.l.o.g. we can assume that $\pi'(v_1)=\{1\}$, $\pi'(v_2)=\{2\}$ and $\pi'(v_3),\pi'(v_4)\subset\{1,\ldots,4\}$ (recall that $|\pi'(v_3)|=|\pi'(v_4)|=1$). Then by Lemma \[lem:even-clique\] we can color $G[N(S)]$ with colors $1,\ldots,4$ so that exactly two edges are uncolored and they form a matching, color 1 is not used at the edges of $G[N(S)]$ incident with $v_1$ and color 2 is not used at the edges of $G[N(S)]$ incident with $v_2$. Then there are at most two edges in $G[N(S)]$ which are colored with the same color as an incident edge (each of $v_3$, $v_4$ is incident with at most one such edge). We uncolor these edges. Hence $G[N(S)]$ has at most four uncolored edges and two of them form a matching. We color these two edges with 5 and one of the remaining two (if any) with 6. Hence we get a proper partial coloring with at most one edge of $G[N(S)]$ uncolored and such that 6 is free in at least two of vertices $v_1,\ldots,v_4$, as required.
[[**CASE 3: **]{}]{} $|N(S)|=4$. Note that for every $i=1,\ldots,4$ we have $1\le |E(\{v_i\},S)| \le 3$. Since $|E(\{v_1,v_2,v_3,v_4\},S)|=6$ there are two subcases to consider.
[[**CASE 3.1: **]{}]{} $N(S)$ has a vertex, say $v_4$, such that $|E(\{v_4\},S)| = 3$. Then for every $i=1,\ldots,3$ we have $|E(\{v_i\},S)|=1$ and $|E(\{v_i\},V\setminus N[S])|\le 2$. Moreover, $|E(\{v_4\},V\setminus N[S])|=0$.
First assume that the edges of $E(N(S),V\setminus N[S])$ use at most 5 colors (say, colors $1,\ldots,5$). Then at least one pair of vertices from $\{v_1,v_2,v_3\}$ is incident with edges of at most 3 colors, by symmetry we can assume $v_1,v_2$ is such a pair. We color $v_1v_3$ and $v_2v_4$ with 6. Then $|{\overline{\pi}}(v_2v_3)|\ge 1$, $|{\overline{\pi}}(v_1v_2)|\ge 2$, $|{\overline{\pi}}(v_1v_4)|\ge 3$ and $|{\overline{\pi}}(v_3v_4)|\ge 3$, so we can color $v_2v_3$, $v_1v_2$, and $v_1v_4$ in this order, always using a free color. We see that $v_3v_4$ still has at least one free color so ${\overline{\pi}}(v_3)\cap{\overline{\pi}}(v_4)\ne\emptyset$ as required.
Now assume that the edges of $E(N(S),V\setminus N[S])$ use all 6 colors. W.l.o.g. $\pi(v_1)=\{1,2\}$, $\pi(v_2)=\{3,4\}$, $\pi(v_3)=\{5,6\}$. Then we color $v_2v_3$ with 1, $v_2v_4$ with 2, $v_1v_3$ with 3, $v_3v_4$ with 4, $v_1v_2$ with 5, and we get the color 6 free at $v_1$ and $v_4$, as required.
[[**CASE 3.2: **]{}]{} $|E(\{v_1\},S)| = |E(\{v_2\},S)| = 1$ and $|E(\{v_3\},S)| = |E(\{v_4\},S)| = 2$ (if Case 3.1 does not apply all the other cases are symmetric). Then $|E(\{v_1\},V\setminus N[S])|\le 2$, $|E(\{v_2\},V\setminus N[S])|\le 2$, $|E(\{v_3\},V\setminus N[S])|\le 1$ and $|E(\{v_4\},V\setminus N[S])|\le 1$.
First assume that the edges of $E(N(S),V\setminus N[S])$ use at most 5 colors (say, colors $1,\ldots,5$). We color $v_1v_2$ and $v_3v_4$ with 6. Then we are left with coloring of the 4-cycle $v_1v_4v_2v_3v_1$ and each of its edges has at least two free colors. Since even cycles are 2-edge-choosable [@erdos1979choosability], we can color it. Finally we uncolor one edge, say $v_1v_2$ and we get ${\overline{\pi}}(v_1)\cap{\overline{\pi}}(v_2)\ne\emptyset$, as required.
Now assume that the edges of $E(N(S),V\setminus N[S])$ use all 6 colors. W.l.o.g. $\pi(v_1)=\{1,2\}$, $\pi(v_2)=\{3,4\}$, $\pi(v_3)=\{5\}$ and $\pi(v_4)=\{6\}$. Then we color $v_2v_3$ with 1, $v_2v_4$ with 2, $v_3v_4$ with 3, $v_1v_4$ with 4, $v_1v_2$ with 5, and we get the color 6 free at $v_1$ and $v_3$, as required.
[[**CASE 4: **]{}]{} $|N(S)|=3$. For every $i=1,2,3$ we have $1\le |E(\{v_i\},S)| \le 4$. Assume w.l.o.g. that $|E(\{v_1\},S)|\le|E(\{v_2\},S)|\le|E(\{v_3\},S)|$. There are two subcases to consider.
[[**CASE 4.1: **]{}]{} $|E(\{v_1\},S)|=1$. Then $|E(\{v_1\},V\setminus N[S])|\le 3$. We have either $|E(\{v_2\},S)|=1$ and $|E(\{v_3\},S)|=4$ or $|E(\{v_2\},S)|=2$ and $|E(\{v_3\},S)|=3$. In the former case we have $|E(\{v_2\},V\setminus N[S])|\le 3$ and $|E(\{v_3\},V\setminus N[S])|=0$. In the latter case we have $|E(\{v_2\},V\setminus N[S])|\le 2$ and $|E(\{v_3\},V\setminus N[S])|\le 1$. Hence in both cases $|{\overline{\pi}}(v_2v_3)|\ge 3$ and $|{\overline{\pi}}(v_1v_3)|\ge 2$. We color $v_2v_3$ and $v_1v_3$ with free colors and we still have at least one free color at $v_2v_3$, so ${\overline{\pi}}(v_2)\cap{\overline{\pi}}(v_3)\ne\emptyset$, as required.
[[**CASE 4.2: **]{}]{} $|E(\{v_1\},S)|\ge 2$. Then $|E(\{v_1\},S)|=|E(\{v_2\},S)|=|E(\{v_3\},S)|=2$. Hence, for every $i=1,2,3$ we have $|E(\{v_i\},V\setminus N[S])|\le 2$. Hence $|{\overline{\pi}}(v_1v_2)|,|{\overline{\pi}}(v_2v_3)|,|{\overline{\pi}}(v_1v_3)|\ge 2$. If $|{\overline{\pi}}(v_1v_2)|=|{\overline{\pi}}(v_2v_3)|=|{\overline{\pi}}(v_1v_3)|=2$ then the sets ${\overline{\pi}}(v_1v_2)$, ${\overline{\pi}}(v_2v_3)$ and ${\overline{\pi}}(v_1v_3)$ are pairwise disjoint so we just color $v_1v_2$ and $v_2v_3$ with free colors and $|{\overline{\pi}}(v_1)\cap{\overline{\pi}}(v_3)|\ge 2$. Otherwise one of these sets, say ${\overline{\pi}}(v_1v_2)$, has cardinality at least 3. Then we color $v_2v_3$ and $v_1v_3$ with free colors and $v_1v_2$ still has a free color so ${\overline{\pi}}(v_1)\cap{\overline{\pi}}(v_2)\ne\emptyset$.
[[**CASE 5: **]{}]{} $|N(S)|=2$. We just put $\pi=\pi'$. Note that $|E(N(S),V\setminus N[S])|\le 2\cdot 6 - 2 - 6 = 4$. It follows that $|{\overline{\pi}}(v_1v_2)|\ge 2$, so ${\overline{\pi}}(v_1)\cap{\overline{\pi}}(v_2)\ne\emptyset$, as required.
\[lem:7-colors-3-edges\] Let $\Delta=7$ and let $Q$ be a 3-edge free component. Then, $\frac{{\textnormal{ch}}(A(Q))}{{\textnormal{ch}}(A(Q))+|E(Q)|} \ge \frac{22}{25}$.
By Corollary \[cor:num-uncolored-0\] we have $|{\overline{\pi}}(Q)|\ge 6$ and by Lemma \[lem:A\_1\] we have $|A(Q)|= 6$. Let $D$ be the set of colored edges incident with $A(Q)$.
Assume $|{\overline{\pi}}(Q)|=7$. By Lemma \[lem:degrees-full-component\] there are at least $7|A(Q)| -1 - {|A(Q)|\choose 2}$ edges incident with $A(Q)$, so $|D|\ge 7|A(Q)| -1 - {|A(Q)|\choose 2} -3= 23$. This, together with Proposition \[prop:bound-full-component\] gives the claim.
Finally assume $|{\overline{\pi}}(Q)|=6$. By Lemma \[lem:degrees-full-component\] there are at least $7|A(Q)| - {|A(Q)|\choose 2}$ edges incident with $A(Q)$, so $|D|\ge 7|A(Q)| - {|A(Q)|\choose 2} -3$ and by Lemma \[lem:bound-almost-full-component\] we have ${\textnormal{ch}}(Q)\ge \frac{13}2|A(Q)|- {|A(Q)|\choose 2} -2= 22$. This gives the claim.
\[cor:main-7\] Every simple graph $G$ of maximum degree $7$ has a $7$-edge-colorable subgraph with at least $\frac{22}{25}|E|$ edges.
By Corollary \[cor:num-uncolored\] every free component of a partially 7-edge-colored graph which maximizes the potential $\Psi$ has at most three edges. Hence, by Corollary \[cor:bound-1-edge\], Corollary \[cor:6-7-colors-2-edges\] and Lemma \[lem:7-colors-3-edges\] the claim follows.
Approximation Algorithms {#sec:OTW}
========================
In this section we describe a meta-algorithm for the maximum $k$-edge-colorable subgraph problem. It is inspired by a method of Kosowski [@K09] developed originally for $k=2$. In the end of the section we show that the meta-algorithm yields new approximation algorithms for $k=3$ in the case of multigraphs and for $k=3,\ldots,7$ in the case of simple graphs.
Throughout this section $G=(V,E)$ is the input graph from a family of graphs ${\mathcal{G}}$ (later on, we will use $\mathcal{G}$ as the family of all simple graphs or of all multigraphs). We fix a maximum $k$-edge-colorable subgraph ${{\rm OPT}}$ of $G$.
As many previous algorithms, our method begins with finding a maximum $k$-matching $F$ of $G$ in polynomial time. Clearly, $|E({{\rm OPT}})| \le |E(F)|$. Now, if we manage to color $\rho|E(F)|$ edges of $F$, we get a $\rho$-approximation. Unfortunately, this way we can get a low upper bound on the approximation ratio. Consider for instance the case of $k=3$ and ${\mathcal{G}}$ being the family of multigraphs. Then, if a connected component $Q$ of $F$ is isomorphic to $G_3$, we get $\rho \le \frac{3}{4}$. In the view of Corollary \[corollary-7-9\] this is very annoying, since $G_3$ is the only graph which prevents us from obtaining the $\frac{7}{9}$ ratio there. However, we can take a closer look at the relation of $Q$ and ${{\rm OPT}}$. Observe that if ${{\rm OPT}}$ does not leave $Q$, i.e. ${{\rm OPT}}$ contains no edge with exactly one endpoint in $Q$ then $|E({{\rm OPT}})|=|E({{\rm OPT}}[V\setminus V(Q)])|+|E({{\rm OPT}}[V(Q)])|$. Note also that $|E({{\rm OPT}}[V(Q)])|=3$, so if we take only three of the four edges of $Q$ to our solution we do not lose anything — locally our approximation ratio is 1. It follows that if there are many components of this kind, the approximation ratio is better than $3/4$. What can we do if there are many components isomorphic to $G_3$ with an incident edge of ${{\rm OPT}}$? The problem is that we do not know ${{\rm OPT}}$. However, then there are many components isomorphic to $G_3$ with an incident edge of the input graph $G$. The idea is to add some of these edges in order to form bigger components (possibly with maximum degree bigger than $k$) which have larger $k$-colorable subgraphs than the original components.
In the general setting, we consider a family graphs ${\mathcal{F}}\subset {\mathcal{G}}$ such that for every graph $A\in {\mathcal{F}}$,
1. $\Delta(A)=k$ and $A$ has at most one vertex of degree smaller than $k$,
2. for every graph $G\in {\mathcal{G}}$, for every subgraph $H$ of $G$ with $|V(A)|$ vertices, $c_k(H)\le c_k(A)$,
3. a maximum $k$-edge colorable subgraph of $A$ (together with its $k$-edge-coloring) can be found in polynomial time; similarly, for every edge $uv\in E(A)$ a maximum $k$-ECS of $A-uv$ (together with its $k$-edge-coloring) can be found in polynomial time,
4. for a given graph $B$ one can check whether $A$ is isomorphic to $B$ in polynomial time,
5. $A$ is 2-edge-connected,
6. for every edge $uv\in A$, we have $c_k(A-uv)=c_k(A)$.
A family that satisfies the above properties will be called a [*$k$-normal family*]{}. We assume there is a number $\alpha \in (0,1]$ and a polynomial-time algorithm $\mathcal{A}$ such that for every $k$-matching $H\not\in{\mathcal{F}}$ of a graph in ${\mathcal{G}}$, the algorithm ${\mathcal A}$ finds a $k$-edge-colorable subgraph of $H$ with at least $\alpha |E(H)|$ edges. Intuitively, ${\mathcal{F}}$ is a family of “bad exceptions” meaning that for every graph $A$ in ${\mathcal{F}}$, there is $c(A) < \alpha |E(A)|$, e.g. in the above example of subcubic multigraphs ${\mathcal{F}}=\{G_3\}$. We note that the family ${\mathcal{F}}$ needs not to be finite, e.g. in the work [@K09] of Kosowski ${\mathcal{F}}$ contains all odd cycles. We also denote $$\beta = \min_{A,B\in{\mathcal{F}}\atop\text{$A$ is not $k$-regular}}\frac{c_k(A)+c_k(B)+1}{|E(A)|+|E(B)|+1}\text{,\quad } \gamma=\min_{A\in{\mathcal{F}}}\frac{c_k(A)+1}{|E(A)|+1}.$$ As we will see, the approximation ratio of our algorithm is $\min\{\alpha,\beta,\gamma\}$.
Let $\Gamma$ be the set of all connected components of $F$ that are isomorphic to a graph in ${\mathcal{F}}$.
\[obs-degree-k\] Without loss of generality, there is no edge $xy\in E(G)$ such that for some $Q\in\Gamma$, $x\in V(Q)$, $y\not\in V(Q)$ and $\deg(y)<k$.
If such an edge exists, we replace in $F$ any edge of $Q$ incident with $x$ with the edge $xy$. The new $F$ is still a maximum $k$-matching in $G$. By (F5) the number of connected components of $F$ increases, so the procedure eventually stops with a $k$-matching having the desired property.
When $H$ is a subgraph of $G$ we denote $\Gamma(H)$ as the set of components $Q$ in $\Gamma$ such that $H$ contains an edge $xy$ with $x\in V(Q)$ and $y\not\in V(Q)$. We denote $\overline{\Gamma}(H)=\Gamma\setminus\Gamma(H)$. The following lemma, a generalization of Lemma 2.1 from [@K09], motivates the whole approach.
\[lem-approx-lower-bound\] $\displaystyle|E({{\rm OPT}})| \le |E(F)|-\sum_{Q\in\overline{\Gamma}({{\rm OPT}})}{\overline{c}}_k(Q)$.
Since for every component $Q\in{\overline{\Gamma}}({{\rm OPT}})$ the graph ${{\rm OPT}}$ has no edges with exactly one endpoint in $Q$, $$\label{eq-zuzia}
|E({{\rm OPT}})| = |E({{\rm OPT}}[V'])| + \sum_{Q\in{\overline{\Gamma}}({{\rm OPT}})}|E({{\rm OPT}}[V(Q)])|,$$ where $V'=V \setminus \bigcup_{Q\in{\overline{\Gamma}}({{\rm OPT}})}V(Q)$. By (F2), we get $$\label{eq-fredzia}
|E({{\rm OPT}}[V(Q)])|\le c_k(Q).$$ Since ${{\rm OPT}}$ is $k$-edge-colorable, $E({{\rm OPT}}[V'])$ is a $k$-matching. Clearly $|E({{\rm OPT}}[V'])| \le |E(F[V'])|$ for otherwise $F$ is not maximal. This, together with and gives the desired inequality as follows. $$|E({{\rm OPT}})| \le |E(F [V'])| + \sum_{Q\in{\overline{\Gamma}}({{\rm OPT}})}c_k(Q)=|E(F)|-\sum_{Q\in\overline{\Gamma}({{\rm OPT}})}{\overline{c}}_k(Q).$$
The above lemma allows us to leave up to $\sum_{Q\in\overline{\Gamma}({{\rm OPT}})}{\overline{c}}_k(Q)$ edges of components in $\Gamma$ uncolored for free, i.e. without obtaining approximation factor worse than $\alpha$. In what follows we “cure” some components in $\Gamma$ by joining them with other components by edges of $G$. We want to do it in such a way that the remaining, “ill”, components have a partial $k$-edge-coloring with no more than $\sum_{Q\in\overline{\Gamma}({{\rm OPT}})}{\overline{c}}_k(Q)$ uncolored edges. To this end, we find a $k$-matching $R\subseteq G$ which satisfies the following conditions:
1. for each edge $xy\in R$ there is a component $Q\in \Gamma$ such that $x\in V(Q)$ and $y\not\in V(Q)$,
2. $R$ maximizes $\sum_{Q\in\Gamma(R)}{\overline{c}}_k(Q)$,
3. $R$ is inclusion-wise minimal $k$-matching subject to (M1) and (M2).
\[lem:R-polynomial\] $R$ can be found in polynomial time.
We use a slightly modified algorithm from the proof of Proposition 2.2 in [@K09]. We define a graph $G'=(V',E')$ as follows. Let $V'=V\cup\{u_Q,w_Q\ :\ Q\in\Gamma\}$. Then, for each $Q\in\Gamma$, the set $E'$ contains three types of edges:
- all edges $xy\in E(G)$ such that $x\in V(Q)$ and $y\not\in V(Q)$,
- an edge $vu_Q$ for every vertex $v\in V(Q)$, and
- an edge $u_Qw_Q$.
Next we define functions $f,g:V'\rightarrow\mathbb{N}\cup\{0\}$ as follows: for every $v\in \bigcup_{Q\in\Gamma}V(Q)$ we set $f(v)=1$, $g(v)=k$; for every $v\in V\setminus\bigcup_{Q\in\Gamma}V(Q)$ we set $f(v)=0$, $g(v)=k$; for every $Q\in\Gamma$ we set $f(u_Q)=0$, $g(u_Q)=|V(Q)|$ and $f(w_Q)=0$, $g(w_Q)=1$. An [*$[f,g]$-factor*]{} $R'$ in $G'$ is a subgraph $R'\subseteq G'$ such that for every $v\in V(R')$ there is $f(v)\le\deg_{R'}(v)\le g(v)$. All edges $u_Qw_Q$ have weight ${\overline{c}}_k(Q)$ while all the other edges have weight 0. Then we find a maximum weight $[f,g]$-factor $R'$ in $G'$, which can be done in polynomial time (see e.g. [@schrijver]). It is easy to see that $R=E(R')\cap E(G)$ satisfies (M1) and (M2). Next, as long as $R$ contains an edge $xy$ such that $R-xy$ still satisfies (M1) and (M2), we replace $R$ by $R-xy$.
Assuming that the components from $\Gamma(R)$ will be “cured” by joining them to other components, the following lemma shows that we do not need to care about the remaining components, i.e. the components from ${\overline{\Gamma}}(R)$. Informally, the lemma says that the number of uncolored edges in such components is bounded by the the number of uncolored edges in components in ${\overline{\Gamma}}({{\rm OPT}})$, which will turn out to be optimal thanks to property (F2).
\[lem-good-matching\] $\displaystyle \sum_{Q\in{\overline{\Gamma}}(R)}{\overline{c}}_k(Q)\le\sum_{Q\in{\overline{\Gamma}}({{\rm OPT}})}{\overline{c}}_k(Q)$.
Let $R_{{{\rm OPT}}}=\{xy\in E({{\rm OPT}}) : \text{for some $Q\in\Gamma$, } x\in Q \text{ and } y\not\in Q\}$. Since ${{\rm OPT}}$ is $k$-edge-colorable, $R_{{{\rm OPT}}}$ is a $k$-matching. By (M2) it follows that $$\label{eq-rysia}
\sum_{Q\in\Gamma(R)}{\overline{c}}_k(Q) \ge^{\rm (M2)} \sum_{Q\in\Gamma(R_{{{\rm OPT}}})}{\overline{c}}_k(Q)=\sum_{Q\in\Gamma({{\rm OPT}})}{\overline{c}}_k(Q),$$ and next $$\sum_{Q\in{\overline{\Gamma}}(R)}{\overline{c}}_k(Q) = \sum_{Q\in\Gamma}{\overline{c}}_k(Q) - \sum_{Q\in\Gamma(R)}{\overline{c}}_k(Q) \le^{\eqref{eq-rysia}} \sum_{Q\in\Gamma}{\overline{c}}_k(Q) - \sum_{Q\in\Gamma({{\rm OPT}})}{\overline{c}}_k(Q) = \sum_{Q\in{\overline{\Gamma}}({{\rm OPT}})}{\overline{c}}_k(Q).$$
The following observation is immediate from the minimality of $R$, i.e. from condition (M3).
\[obs:stars\] Let $H_F$ be a graph with vertex set $\{Q\ :\ Q \text{ is a connected component of $F$}\}$ and the edge set $\{PQ\ :\ \text{there is an edge $xy\in R$ incident with both $P$ and $Q$}\}$. Then $H_F$ is a forest, and every connected component of $H_F$ is a star.
In what follows, the components of $F$ corresponding to leaves in $H_F$ are called [*leaf components*]{}. Now we proceed with finding a $k$-edge-colorable subgraph $S$ of $G$ together with its coloring, using the algorithm described below. In the course of the algorithm, we maintain the following invariants:
\[inv-degrees\] For every $v\in V$, $\deg_R(v) \le \deg_F(v)$.
\[inv-no-new-gammas\] If $F$ contains a connected component $Q$ isomorphic to a graph in ${\mathcal{F}}$, then $Q\in\Gamma$, in other words a new component isomorphic to a graph in ${\mathcal{F}}$ cannot appear.
By Observation \[obs:stars\], each edge of $R$ connects a vertex $x$ of a leaf component and a vertex $y$ of another component. Hence $\deg_R(x)=1\le\deg_F(x)$. By Observation \[obs-degree-k\], initially $\deg_F(y)=k$, so also $\deg_R(y)\le\deg_F(y)$. It follows that Invariant \[inv-degrees\] holds at the beginning, as well as Invariant \[inv-no-new-gammas\], the latter being trivial. Now we describe the coloring algorithm.
1. Begin with the graph with no edges $S=(V,\emptyset)$.
2. \[step-jadzia\] As long as $F$ contains a leaf component $Q\in\Gamma$ and a component $P$, such that
- there is an edge $xy\in R$ with $x\in Q$ and $y\in P$,
- there is an edge $yz\in E(P)$ such that no connected component of $P-yz$ is isomorphic to a graph in ${\mathcal{F}}$,
then we remove $xy$ from $R$ and both $Q$ and $yz$ from $F$. Notice that if $z$ was incident with an edge $zw\in R$ then by Observation \[obs:stars\], $w$ belongs to another leaf component $Q'$. Then we also remove $zw$ from $R$ and $Q'$ from $F$ (if there are many such edges $zw$ we perform this operation only for [*one*]{} of them). It follows that Invariants \[inv-degrees\] and \[inv-no-new-gammas\] hold.
3. \[step-ula\] As long as there is a leaf component $Q\in\Gamma(R)$ we do the following. Let $P$ be the component of $F$ such that there is an edge $xy\in R$ with $x\in Q$ and $y\in P$. Then, by Step \[step-jadzia\], for each edge $yz\in E(P)$ in graph $P-yz$ there is a connected component isomorphic to a graph in ${\mathcal{F}}$. In particular, by (F1) every edge $yz\in E(P)$ is a bridge in $P$. By (F5), $P\not\in\Gamma$. Let $yz$ be any edge incident with $y$ in $P$, which exists by Invariant \[inv-degrees\]. Note that if $P-yz$ has a connected component $C$ isomorphic to a graph in ${\mathcal{F}}$ and containing $y$ then every edge of $C$ incident with $y$ is a bridge in $C$; a contradiction with (F5). Hence $P-yz$ has exactly one connected component isomorphic to a graph in ${\mathcal{F}}$, call it $P_{yz}$, and $V(P_{yz})$ contains $z$. Assume $P_{yz}$ is incident with an edge of $R$, i.e. there is an edge $x'y'$ with $x'\in V(Q')$ for some leaf component $Q'\in\Gamma(R)$ and $y'\in P_{yz}$. By the same argument, $y'$ is incident with a bridge $y'z'$ in $P$ and $P-y'z'$ contains a connected component $P'_{y'z'}$ from ${\mathcal{F}}$, such that $z'\in V(P'_{y'z'})$. But since $P_{yz}$ has no bridges, $y'=z$ and $z'=y$, which implies that $P-yz$ has [*two*]{} connected components isomorphic to a graph in ${\mathcal{F}}$, a contradiction. Hence $P_{yz}$ is not incident with an edge of $R$. Then we remove $Q$, $yz$ and $P_{yz}$ from $F$ and $xy$ from $R$. The above discussion shows that Invariants \[inv-degrees\] and \[inv-no-new-gammas\] hold.
4. Process each of the remaining components $Q$ of $F$, depending on its kind.
1. If $Q\in\Gamma$, it means that $Q\in{\overline{\Gamma}}(R)$, because otherwise there are leaf components in $\Gamma(R)$, which contradicts Step \[step-ula\]. Then we find a maximum $k$-edge-colorable subgraph $S_Q\subseteq Q$, which is possible in polynomial time by (F3), and add it to $S$ with the relevant $k$-edge-coloring.
2. If $Q\not\in\Gamma$ we use the algorithm $\mathcal{A}$ to color at least $\alpha|E(Q)|$ edges of $Q$ and we add the colored edges to $S$.
3. For every $Q$, $yz$ and $P_{yz}$ deleted in Step \[step-ula\], we find the maximum $k$-edge-colorable subgraph $Q^*$ of $Q$ and $P^*$ of $P_{yz}$. Note that the coloring of $P^*$ can be extended to $P^*+yz$ since $\deg_{P^*}(z)<k$. Next we add $Q^*$, $P^*$ and $yz$ to $S$ (clearly we can rename the colors of $P^*+yz$ so that we avoid conflicts with the already colored edges incident with $y$). To sum up, we added $c_k(Q)+c_k(P_{yz})+1$ edges to $S$, which is $\frac{c_k(Q)+c_k(P_{yz})+1}{|E(Q)|+|E(P_{yz})|+1}\ge\beta$ of the edges of $F$ deleted in Step \[step-ula\].
4. For every $xy$ and $Q$ deleted in Step \[step-jadzia\], let $zw$ be any edge of $Q$ incident with $x$ and then we find the maximum $k$-edge-colorable subgraph $Q^*$ of $Q-zw$ using the algorithm guaranteed by (F3). Next we add $Q^*$ and $xy$ to $S$ (similarly as before, we can rename the colors of $Q^*+xy$ so that we avoid conflicts with the already colored edges incident with $y$). By (F6), $c_k(Q-zw)=c_k(Q)$. Recall that in Step \[step-jadzia\] two cases might happen: either we deleted only $Q$ and $yz$ from $F$, or we deleted $Q$, $yz$ and $Q'$. In the former case we add $c_k(Q)+1$ edges to $S$, which is $\frac{c_k(Q)+1}{|E(Q)|+1}\ge\gamma$ of the edges removed from $F$. In the latter case we add $c_k(Q)+c_k(Q')+2$ edges to $S$, which is $\frac{c_k(Q)+c_k(Q')+2}{|E(Q)|+|E(Q')|+1}>\gamma$ of the edges removed from $F$.
Our algorithm has approximation ratio of $\min\{\alpha,\beta,\gamma\}$.
Let $\rho = \min\{\alpha,\beta,\gamma\}$. $$\begin{aligned}
|S| & \ge & \rho (|E(F)| - \sum_{Q\in{\overline{\Gamma}}(R)}|E(Q)|) + \sum_{Q\in{\overline{\Gamma}}(R)}c_k(Q) \ge \\
& & \rho (|E(F)| - \sum_{Q\in{\overline{\Gamma}}(R)}|E(Q)| + \sum_{Q\in{\overline{\Gamma}}(R)}c_k(Q)) = \\
& & \rho (|E(F)| - \sum_{Q\in{\overline{\Gamma}}(R)}{\overline{c}}_k(Q)) \ge^{\text{(Lemma~\ref{lem-good-matching})}} \\
& & \rho (|E(F)| - \sum_{Q\in{\overline{\Gamma}}({{\rm OPT}})}{\overline{c}}_k(Q)) \ge^{\text{(Lemma~\ref{lem-approx-lower-bound})}} \rho |E(OPT)|.
\end{aligned}$$
\[th:meta-algorithm\] Let ${\mathcal{G}}$ be a family of graphs and let ${\mathcal{F}}$ be a $k$-normal family of graphs. Assume there is a polynomial-time algorithm such that for every connected $k$-matching $H \not \in {\mathcal F}$ of a graph in ${\mathcal G}$, the algorithm finds a $k$-edge-colorable subgraph of $H$ with at least $\alpha|E(H)|$ edges and a $k$-edge-coloring of it. Moreover, let $$\beta = \min_{A,B\in{\mathcal{F}}\atop\text{$A$ is not $k$-regular}}\frac{c_k(A)+c_k(B)+1}{|E(A)|+|E(B)|+1}\text{,\quad } \gamma=\min_{A\in{\mathcal{F}}}\frac{c_k(A)+1}{|E(A)|+1}.$$ Then, there is an approximation algorithm for the maximum $k$-ECS problem for graphs in ${\mathcal{G}}$ with approximation ratio $\min\{\alpha,\beta,\gamma\}$.
The above theorem summarizes our discussion in this section. Now we apply it to particular cases.
The maximum $3$-ECS problem has a $\frac{7}{9}$-approximation algorithm for multigraphs.
Let ${\mathcal{F}}=\{G_3\}$. It is easy to check that ${\mathcal{F}}$ is $3$-normal. Now we give the values of parameters $\alpha, \beta$ and $\gamma$ from Theorem \[th:meta-algorithm\]. By Corollary \[corollary-7-9\], $\alpha=\frac{7}{9}$. Notice that $c_3(G_3)=3$ and $|E(G_3)|=4$. Hence, $\beta=\frac{7}{9}$ and $\gamma=\frac{4}{5}$. By Theorem \[th:meta-algorithm\] the claim follows.
The maximum $3$-ECS problem has a $\frac{13}{15}$-approximation algorithm for simple graphs.
Let ${\mathcal{F}}=\{B_3\}$. It is easy to check that ${\mathcal{F}}$ is $3$-normal. Now we give the values of parameters $\alpha, \beta$ and $\gamma$ from Theorem \[th:meta-algorithm\]. By Corollary \[corollary-13-15\], $\alpha=\frac{13}{15}$. Notice that $c_3(B_3)=6$ and $|E(B_3)|=7$. Hence, $\beta=\frac{13}{15}$ and $\gamma=\frac{7}{8}$. By Theorem \[th:meta-algorithm\] the claim follows.
The maximum $4$-ECS problem has a $\frac{9}{11}$-approximation algorithm for simple graphs.
Let ${\mathcal{F}}=\{K_5\}$. It is easy to check that ${\mathcal{F}}$ is $4$-normal. Now we give the values of parameters $\alpha, \beta$ and $\gamma$ from Theorem \[th:meta-algorithm\]. By Theorem \[thm:main\], $\alpha=\frac{5}{6}$. Observe that $\beta=\infty$, since ${\mathcal{F}}$ contains only $K_5$ which is $4$-regular. Notice that $c_4(K_5)=8$ and $|E(K_5)|=10$. Hence, $\gamma=\frac{9}{11}$. By Theorem \[th:meta-algorithm\] the claim follows.
The maximum $6$-ECS problem has a $\frac{19}{22}$-approximation algorithm for simple graphs.
Let ${\mathcal{F}}=\{K_7\}$. It is easy to check that ${\mathcal{F}}$ is $6$-normal. Now we give the values of parameters $\alpha, \beta$ and $\gamma$ from Theorem \[th:meta-algorithm\]. By Theorem \[thm:main\], $\alpha=\frac{19}{22}$. Observe that $\beta=\infty$, since ${\mathcal{F}}$ contains only $K_7$ which is $6$-regular. Notice that by Lemma \[lem:even-clique\], $c_6(K_7)=18$ and $|E(K_7)|=21$. Hence, $\gamma=\frac{19}{22}$. By Theorem \[th:meta-algorithm\] the claim follows.
Directly from Proposition \[prop:approx\] and from Theorem \[thm:main\] we get the following corollaries.
The maximum $5$-ECS problem has a $\frac{23}{27}$-approximation algorithm for simple graphs.
The maximum $7$-ECS problem has a $\frac{22}{25}$-approximation algorithm for simple graphs.
Further Work
============
The most important open problem seems to be to provide answers to Questions \[q-odd\] and \[q-even\] from Section \[intro:combin\] for all $\Delta\ge 8$. We think that although our techniques (with some hard work) might be sufficient to improve the Vizing bound when $\Delta=9$ or $\Delta=10$, for large values of $\Delta$ some new ideas are needed.
It would be also interesting to improve our bounds for $\Delta\le 7$. In particular the best upper bound for even $\Delta$, and for $G\ne K_{\Delta+1}$ we are aware of is $\frac{\Delta}{\Delta+1-2/\Delta}$, attained by $K_{\Delta+1}-e$, i.e. the $K_{\Delta+1}$ with one edge removed. The lemma below provides an upper bound for odd values of $\Delta$.
\[lem:upper-odd\] For every odd value of $\Delta$ there is a graph of maximum degree $\Delta$ such that $$\gamma_\Delta(G)=\frac{\Delta+1}{\Delta+2-\tfrac{1}\Delta}.$$
Let $\Delta = 2\ell + 1$. Begin with $K_{\Delta+1}$. Remove a matching $M$ of size $\ell$. Add a new vertex $v$ and add edges between $v$ and $V(M)$. Denote the resulting graph by $B_\Delta$. Observe that the maximum degree of $B_\Delta$ is $\Delta$. We see that $|E(B_\Delta)|={\Delta+1\choose 2} + \ell$. Consider a maximum $\Delta$-edge-colorable subgraph $H$ of $B_\Delta$. Since each of the $\Delta$ color classes has at most $(\Delta+1)/2$ edges, $|E(H)| \le \Delta\cdot (\Delta+1)/2 = {\Delta\choose 2}$. It is easy to see that actually $|E(H)|={\Delta\choose 2}$; just consider the coloring of $K_\Delta$ from Lemma \[lem:odd-clique\], and for each of the removed edges, say $xy$, copy its color to one of the new edges incident with $xy$, say $vx$. It follows that $\gamma_\Delta(B_\Delta)={\Delta+1\choose 2}/({\Delta+1\choose 2}+\ell)=\frac{\Delta+1}{\Delta+2-1/\Delta}$.
Another interesting question is the following.
\[q:13-15\] Does $\gamma_3(G)\ge\frac{13}{15}+\varepsilon$ for some $\varepsilon>0$ when $G$ is a simple graph isomorphic neither to $B_3$ nor to the Petersen graph?
Acknowledgments {#acknowledgments .unnumbered}
===============
We are very grateful to Adrian Kosowski for helpful remarks regarding the state-of-art of the $k$-ECS problem in multigraphs. We also thank anonymous reviewers for careful reading and many helpful remarks.
[10]{}
M. O. Albertson and R. Haas. Parsimonious edge coloring. , 148(1-3):1–7, 1996.
V. Bryant. . Cambridge University Press, 1993.
G. Chen, X. Yu, and W. Zang. Approximating the chromatic index of multigraphs. , 21(2):219–246, 2011.
Z. Chen, S. Konno, and Y. Matsushita. Approximating maximum edge 2-coloring in simple graphs. , 158(17):1894–1901, 2010.
Z.-Z. Chen and R. Tanahashi. Approximating maximum edge 2-coloring in simple graphs via local improvement. , 410(45):4543–4553, 2009.
Z.-Z. Chen, R. Tanahashi, and L. Wang. An improved approximation algorithm for maximum edge 2-coloring in simple graphs. , 6(2):205–215, 2008.
G. Cornuéjols and W. Pulleyblank. A matching problem with side conditions. , 29:135–159, 1980.
Z. G. D. Leven. -completeness of finding the chromatic index of regular graphs. , 4:35–44, 1983.
R. Diestel. . Springer-Verlag, Electronic Edition, 2005.
P. Erdős, A. Rubin, and H. Taylor. Choosability in graphs. , 26:125–157, 1979.
L. M. Favrholdt, M. Stiebitz, and B. Toft. Graph edge colouring: [Vizing]{}’s theorem and [Goldberg]{}’s conjecture. Technical Report 20, IMADA, The University of Southern Denmark, 2006.
U. Feige, E. Ofek, and U. Wieder. Approximating maximum edge coloring in multigraphs. In K. Jansen, S. Leonardi, and V. V. Vazirani, editors, [*APPROX*]{}, volume 2462 of [*Lecture Notes in Computer Science*]{}, pages 108–121. Springer, 2002.
I. Holyer. The [NP]{}-completeness of edge coloring. , 10:718–720, 1981.
M. Kamiński and [Ł]{}. Kowalik. Approximating the maximum 3- and 4-edge-colorable subgraph. In H. Kaplan, editor, [*SWAT*]{}, volume 6139 of [*Lecture Notes in Computer Science*]{}, pages 395–407. Springer, 2010.
A. Kempe. On the geographical problem of the four colours. , 2(3):193–200, 1879.
A. Kosowski. Approximating the maximum $2-$ and $3-$edge-colorable subgraph problems. , 157:3593–3600, 2009.
V. V. Mkrtchyan and E. Steffen. Maximum [$\Delta$]{}-edge-colorable subgraphs of class [II]{} graphs. , 70(4):473–482, 2012.
R. Rizzi. Approximating the maximum 3-edge-colorable subgraph problem. , 309:4166–4170, 2009.
P. Sanders and D. Steurer. An asymptotic approximation scheme for multigraph edge coloring. , 4(2):1–24, 2008.
A. Schrijver. . Springer, 2003.
C. E. Shannon. A theorem on coloring the lines of a network. , 28:148–151, 1949.
V. G. Vizing. On the estimate of the chromatic class of a $p$-graph. , 3:25–30, 1964.
[^1]: Département d’Informatique, Université Libre de Bruxelles and Institute of Informatics, University of Warsaw. Email: [mjk@mimuw.edu.pl]{}
[^2]: Institute of Informatics, University of Warsaw. Email: [kowalik@mimuw.edu.pl]{}. Supported by ERC StG project PAAl no. 259515.
[^3]: A preliminary version of this work (with a proper subset of the results) was presented at 12th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2010) and is published as an extended abstract [@swat].
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1.4cm 1.4cm = 15truecm = 21truecm = -1.3truecm = -2truecm 0.3cm
[**Matrix oscillator and Calogero-type models**]{}
S.Meljanac [[^1]]{} and A. Samsarov[ [^2]]{}\
Rudjer Bošković Institute, Bijenička c.54, HR-10002 Zagreb, Croatia\
**Abstract**
We study a single matrix oscillator with the quadratic Hamiltonian and deformed commutation relations. It is equivalent to the multispecies Calogero model in one dimension, with inverse-square two-body and three-body interactions. Specially, we have constructed a new matrix realization of the Calogero model for identical particles, without using exchange operators. The critical points at which singular behaviour occurs are briefly discussed.
PACS number(s): 03.65.Fd, 03.65.Sq, 05.30.Pr\
Keywords: Matrix oscillator, Multispecies Calogero model, Fock space .
Introduction
============
A class of integrable many-body systems in one spatial dimension is known, referred to as Calogero systems \[1\]. These systems are formed of $ N $ identical particles on the line which interact through an inverse-square two-body interaction and are subjected to a common confining harmonic force. These models are completely integrable in both the classical and the quantum case \[2\] and are related to a number of mathematical and physical problems, ranging from random matrices \[3,4\] to gravity, black hole physics \[5\] and two-dimensional strings \[6\]. The algebraic structure of the Calogero model has recently been reconsidered by a number of authors in the framework of the exchange operator formalism \[7,8\] based on a symmetric group. An advantage of this approach is the possibility of an explicit construction of wave functions for an arbitrary number of particles. This approach also emphasizes the interpretation in terms of generalized statistics \[9\] that allows for the possibility of having particles of different species with a mutual coupling parameter depending on the species coupled.
It is known that a random matrix theory provides a simple relation between the quantum mechanics of the harmonic oscillator and the Calogero model. However, this connection is known to be true only for three special values of the coupling parameter $ \nu $ : $ \nu = \frac{1}{2}, 1, 2 $. In this approach the Calogero model appears through the calculation of averages in the Gaussian ensembles \[3,4\]. Also, another remarkable connection between the matrix models and the Calogero models was established in \[2,10,11,12\]. In Ref’s.\[2,11,12\] a classical matrix system without the quadratic potential was considered by the technique of the Hamiltonian reduction. The quantization is performed through path integral methods \[12\].
In the present Letter we introduce matrices whose matrix elements are operators and define the matrix Hamiltonian of the quadratic type. We show that this matrix formulation is in one-to-one correspondence with the Calogero model for an arbitrary value of the coupling parameter that can even depend on the particles coupled. In addition, exchange operators do not appear in our formulation.
In section $ 2 $, by introducing a pair of $ N \times N $ matrices $ \bf{X}$ , $ \bf{P}$, we define the quadratic Hamiltonian $ \mathcal{H} $ for a single quantum matrix oscillator that is required to satisfy deformed commutation relations. This is a generalization of a single oscillator with the quadratic Hamiltonian and the deformed commutation relation \[13,14\]. After finding the representation that solves these commutation relations, we show that the single matrix oscillator with the quadratic Hamiltonian and deformed commutation relations is equivalent to a multispecies Calogero model \[15-18\] with inverse-square two-body and three-body interactions. Generalization of $ SU(1,1) $ generators is proposed and their form is used to construct matrix ladder operators. By applying the Fock space analysis, a class of the excited states of the matrix oscillator Hamiltonian $ \mathcal{H} $ has been found. In section $ 3 $ we specialize our considerations to the case where all masses and coupling constants are equal. After stating some non-trivial identities, the matrix oscillator considered in this paper turns out to be an alternative simple formulation of the Calogero model that avoids the necessity of using exchange operators. Finally, brief inspection of the Fock space that corresponds to the relative motion of particles reveals the existence of the critical points at which the system exhibits singular behaviour.
Matrix oscillator and a multispecies Calogero model
====================================================
Let us consider $ N \times N $ matrices $ \bf{X}$ , $ \bf{P}$ with operator-valued matrix elements and a non-singular mass matrix $ \; \mathcal{ M}$. The matrix Hamiltonian, generally non-Hermitean, is given by $${\mathcal{H}} = \frac{1}{2} ( {\bf{P}} {\mathcal{ M}}^{-1} {\bf{P}} + {{\omega}^{2}}
{\bf{X}} {\mathcal{ M}} {\bf{X}} ),$$ with $ \hbar = 1 $. It represents a matrix generalization of a single harmonic oscillator.\
We assume the following matrix commutation relations: $$[ \bf {X} , \bf {P}] = \imath \mathcal{V},$$ where $ \mathcal{V} $ is a Hermitean $ N \times N $ matrix with constant, real and symmetric matrix elements $ \nu_{ij} = \nu_{ji} , \;\; i,j = 1,2,...,N $. We set the diagonal elements to be equal to unity, reflecting the quantum nature of the system. We further assume that the matrix $\bf{X}$ is Hermitean and can be represented as a diagonal matrix with real elements $ x_{i} , \;\; i = 1,2,...,N $ : $${\bf{X}}_{ij} = x_{i}\delta_{ij}$$ and $ {\mathcal{ M}}_{ij} = m_{i}\delta_{ij}, \;\; m_{i} > 0 $ . By introducing the matrix operator $ \mathcal{ D}$ such that the relation $ \; \bf{ P} = -\imath \mathcal{ D} \; $ holds, we can rewrite Eq. (2) in the following way : $$\begin{array}{c}
{\mathcal{D}}_{ij} x_{j} - x_{i}{\mathcal{D}}_{ij} = \nu_{ij} \;\;\;\; \forall i,j ; \;\; i \neq j \\
{\mathcal{D}}_{ii} x_{i} - x_{i}{\mathcal{D}}_{ii} = 1 \;\;\;\; \forall i .
\end{array}$$ There are many solutions of equations (4) since the addition of a diagonal piece to $ {\mathcal{D}}_{ij} $ depending only on the coordinates does not affect these equations. To express this fact in a more explicit maneer, we can consider a transformation $ \; {\mathcal{ D}}^{(f)} = f^{-1} {\mathcal{ D}} f \; $ of the operator $ \; \mathcal{ D} \; $ by the arbitrary function $ \; f \; $ of the coordinates. The corresponding Hamiltonians are connected by non-unitary gauge transformations, i.e. by similarity transformations of the form $ \; {\mathcal{ H}} ^{(f)} = f^{-1} {\mathcal{ H}} f \; $. We shall restrict ourselves to gauge transformations defined by $ \; f = \prod_{i < j}(x_{i} - x_{j})^{\lambda_{ij}}, \;\;\; \lambda_{ij} = \lambda_{ji} $. A corresponding class of solutions of equations (4) is given by $${\mathcal{D}}_{ij} = \delta_{ij} ( \frac{\partial}{\partial x_{i} } + \sum_{k \neq i}\frac{\lambda_{ik}}{x_{i} - x_{k}} )
- \frac{\nu_{ij}(1 - \delta_{ij})}{x_{i} - x_{j}},$$ where $ \; \lambda_{ij} \; $ are gauge parameters. Note that the dependence of the Hamiltonian $ \; \mathcal{ H} \; $ on the gauge parameters $ \; \lambda_{ij} \; $ enters through the operator $ \mathcal{ D}$.
The matrix Schroedinger equation is $${\mathcal{H}} \ast \Psi ({\bf{X}}) \equiv {\mathcal{H} \mathcal{J}} \Psi({\bf{X}})\;
= {\mathcal{E}} \Psi({\bf{X}}),$$ where $ \Psi(\bf{X})$ is a column wave function $ ( \psi_{i}(\bf{X})), \;\;$ $ i = 1,...,N $, $ \mathcal{J} $ is an $ N \times N $ matrix with units at all positions, and multiplication $ \ast $ is defined in the above equation. For example, the ground state in the gauge $ \; \lambda_{ij} = \nu_{ij}, \;\;\; \forall i,j \; $ is described by the column matrix $$\| 0 \rangle \sim \left ( e^{-\frac{\omega}{2}\sum_{i=1}^{N} m_{i}x_{i}^{2}}\right ) \; C .$$ Here $ C $ is the column matrix with all elements equal to unity. Analogously, one can introduce the left action of the Hamiltonian $ \mathcal{H}$ on the row wave function. In this case (in the gauge $ \; \lambda_{ij} = \nu_{ij}, \;\;\; \forall i,j \; $), the ground state is represented by $ \; \langle 0 \| \sim R e^{-\frac{\omega}{2}\sum_{i=1}^{N} m_{i}x_{i}^{2}} \; $, where $ R $ is a transpose of $\; C $. Note that $ \; RC = N $, $ \; CR = {\mathcal{J}}, \; {\mathcal{J}}\| 0 \rangle = N \| 0 \rangle \; $ and $ \; \langle 0 \|0 \rangle = 1 $. At this point it should be emphasized that $ \;\; \Psi ({\bf{X}}) \;\; $ is not the eigenstate of the Hamiltonian $ \; {\mathcal{H}} \; $ in the usual sense, but rather it satisfies the “eigenvalue” equation in which the “eigenvalue” $ \; {\mathcal{E}}\; $ is a matrix. We can give a well-defined meaning to this equation after performing multiplication of both sides by the row matrix $ \; R \; $ from the left. In this case, Eq. (6) is reduced to the eigenvalue equation $ \; H\psi = E \psi \; $, where $ \; H \; $ is the Hamiltonian corresponding to the matrix Hamiltonian $ \; {\mathcal{H}} \; $ by $ \; H = R {\mathcal{H}} C = tr( {\mathcal{H}} {\mathcal{J}}) \; $ . We point out that in the special case where the gauge $ \; \lambda_{ij} = \nu_{ij}, \;\;\; \forall i,j \; $ is chosen, we get the familiar eigenvalue equation $ \; \tilde{H} \psi = E \psi, \; $ where $ \; \tilde{H} \; $ stands for the transformed Hamiltonian for a multispecies Calogero model \[17\]. In this case $ \; {\mathcal{H}} \;$ is related to $ \; \tilde{H} \; $ as $$\tilde{H} = R {\mathcal{H}} C = tr (\mathcal{H} \mathcal{J})$$ $$= -\frac{1}{2}\sum_{i=1}^{N}\frac{1}{m_{i}}\frac{{\partial}^{2}}
{\partial x_{i}^{2}} + \frac{{\omega}^{2}}{2}\sum_{i=1}^{N }m_{i} x_{i}^{2}$$ $$- \frac{1}{2} \sum_{i \neq j }\frac{{\nu}_{ij}}
{(x_{i}-x_{j})}(\frac{1}{m_{i}} \frac{\partial}{{\partial} x_{i}}
- \frac{1}{m_{j}} \frac{\partial}{{\partial} x_{j}})$$ $$\equiv - T_{-} + {{\omega}^{2}} T_{+},$$ where $ \; T_{\pm} \; $ are $ \; SU(1,1) \; $ generators. After performing the similarity transformation with the inverse Yastrow factor $ \; \prod_{i < j}(x_{i} - x_{j})^{ - {\nu}_{ij}} \; $, we obtain an original Hamiltonian $ H_{cal} $ for the multispecies Calogero model, with inverse-square two-body and three-body interactions \[17\]. This original Hamiltonian $ H_{cal} $ can also be reproduced directly, $ H_{cal} = tr( {\mathcal{H}} {\mathcal{J}}) $ in the gauge $ \; \lambda_{ij} = 0, \;\;\; \forall i,j $. The procedure outlined here differs significantly from that followed in Ref.\[12\] in the way how the Calogero Hamiltonian appears. Namely, in this paper the Hamiltonian is not just the trace of the matrix Hamiltonian $ \mathcal{H}, $ but is given by $ \; H = tr( {\mathcal{H}} {\mathcal{J}}) , \; $ see Eq.(8). In the rest of the paper we restrict to the gauge $ \; \lambda_{ij} = \nu_{ij}, \;\;\; \forall i,j \; $.
In the same way as we have introduced the matrix Hamiltonian $ \mathcal{H} $, we introduce matrix generators with operator- valued matrix elements $$\begin{array}{l}
{\mathcal{T}}_{+} = \frac{1}{2} \bf{X} \mathcal{ M} \bf{X}, \\
{\mathcal{T}}_{-} = \frac{1}{2} {\mathcal{D}} {\mathcal{ M}}^{-1} {\mathcal{D}}, \\
{\mathcal{T}}_{0} = \frac{1}{4}( \bf{X}{\mathcal{ D}} + {\mathcal{ D}}\bf{X})
= \frac{1}{2} ( \bf{X}{\mathcal{ D}} + \frac{1}{2} {\mathcal{ V}}).
\end{array}$$ They satisfy the relations $$R [{\mathcal{T}}_{-}, {\mathcal{T}}_{+}]_{J}C = 2 R {\mathcal{T}}_{0}C ,$$ $$R [{\mathcal{T}}_{0}, {\mathcal{T}}_{\pm}]_{J}C = \pm R {\mathcal{T}}_{\pm}C ,$$ where the $ J - \mbox{commutator} $ is defined by $$[A,B ]_{J} = A {\mathcal{J}} B - B {\mathcal{J}} A .$$ $ {\mathcal{T}}_{\pm}, {\mathcal{T}}_{0} \;\; $ are related to the generators $ \;\; T_{\pm}, T_{0} \;\;$ \[17\] of $ \; SU(1,1) \;$ algebra in the following way : $$\begin{array}{l}
T_{\pm} = R {\mathcal{T}}_{\pm} C = tr ({\mathcal{T}}_{\pm}{\mathcal{J}}), \\
T_{0} = R {\mathcal{T}}_{0} C = tr ({\mathcal{T}}_{0}{\mathcal{J}}) .
\end{array}$$
The wave functions $ \psi $ of the Hamiltonian (8) $ ( \tilde{H}\psi = E \psi ) $ are related to the column wave functions defined in (6). As it is readily seen, the connection is simply $ \psi ( x_{1},..., x_{N}) \sim R \Psi(\bf{X}) \; $ and $ \; {\psi ^{\ast}} ( x_{1},..., x_{N}) \sim {\Psi}^{\dagger} ({\bf{X}}) C \; $.
The model described by the Hamiltonian (8) was partially solved in \[17\]. Corresponding solutions in the matrix formulation are obtained in terms of the following pairs of creation and annihilation operators : $${\mathcal{A}}_{1}^{\pm} = \frac{1}{\sqrt{2 tr {\mathcal{M}}}}( \sqrt{\omega}{\bf {X}}{\mathcal{M}} \mp
\frac{1}{\sqrt{\omega}}{\mathcal{D}} ),$$ $${\mathcal{A}}_{2}^{\pm} = \frac{1}{2}
( \omega {\mathcal{T}}_{+} + \frac{1}{\omega} {\mathcal{T}}_{-}) \mp {\mathcal{T}}_{0} .$$ Note that for the case in which all masses $\; m_{i}\;$ are equal, there is a simple relation between these two sets of operators, namely $ \;\; N R {{\mathcal{A}}_{1}^{\pm}}^{2}C = R {\mathcal{A}}_{2}^{\pm}C \;\;$ .
The generators $ \; {\mathcal{T}}_{\pm} \; $ are defined in (9) and play the role of collective radial variables corresponding to dilatation modes. The first pair of operators (14) describes center-of-mass ($ CM $) modes, while the second pair describes collective radial modes. These operators satisfy the following commutation relations : $$R [ {\mathcal{A}}_{1}^{-}, {\mathcal{A}}_{1}^{+}]_{J}C = 1 , \;\;\;\;\;
R [ {\mathcal{A}}_{2}^{-}, {\mathcal{A}}_{2}^{+}]_{J}C = \frac{1}{\omega} R {\mathcal{H}}C,$$ $$R [ {\mathcal{A}}_{1}^{\mp}, {\mathcal{A}}_{2}^{\mp}]_{J}C = 0, \;\;\;\;\;
R [ {\mathcal{A}}_{1}^{\mp}, {\mathcal{A}}_{2}^{\pm}]_{J}C = \pm R {\mathcal{A}}_{1}^{\pm}C ,$$ $$R [ {\mathcal{H}}, {\mathcal{A}}_{1}^{\pm}]_{J}C = \pm \omega R {\mathcal{A}}_{1}^{\pm}C , \;\;\;\;\;
R [ {\mathcal{H}}, {\mathcal{A}}_{2}^{\pm}]_{J}C = \pm 2 \omega R {\mathcal{A}}_{2}^{\pm}C .$$
The partial matrix Fock space corresponding to $ CM $ modes and collective radial modes is spanned by the states of the form $$\| n_{1},n_{2} \rangle \sim ({\mathcal{A}}_{1}^{+} {\mathcal{J}})^{n_{1}}({\mathcal{A}}_{2}^{+} {\mathcal{J}})^{n_{2}}
\| 0 \rangle,$$ which are governed by the matrix Schroedinger equation $${\mathcal{H}}{\mathcal{J}} \| n_{1},n_{2} \rangle = {\mathcal{E}}_{n_{1},n_{2}}\| n_{1},n_{2} \rangle .$$ Here $ \;\; {\mathcal{E}}_{n_{1},n_{2}} \;\; $ is the matrix that satisfies the relation $$\frac{1}{\sqrt{N}} R {\mathcal{E}}_{n_{1},n_{2}}\| n_{1},n_{2} \rangle =
\frac{1}{\sqrt{N}} E_{n_{1},n_{2}} R \| n_{1},n_{2} \rangle$$ $$= E_{n_{1},n_{2}} | n_{1},n_{2} \rangle = ( E_{0} + \omega (n_{1} + 2n_{2})) | n_{1},n_{2} \rangle ,$$ where $ \; E_{0} = \omega ( \frac{N}{2} + \frac{1}{2}\sum_{i \neq j} \nu_{ij})\; $ is the energy of the ground state. The state $ \; \| n_{1},n_{2} \rangle \; $ is the column Fock state and $ \; \| 0 \rangle \; $ is the vacuum state defined by (7), while $ \;\; E_{n_{1},n_{2}} \;\; $ and $ \;\; | n_{1},n_{2} \rangle \;\; $ are eigenvalues and eigenstates of the partial Fock space of the corresponding multispecies Calogero problem. The ground state is well defined if $ \; E_{0} > \frac{1}{2} \; $ \[17-19\].
Note that the correspondence between the matrix ladder operators (14) and the analogous operators $ \;A_{1}^{\pm}, A_{2}^{\pm} \; $ \[17\] that define the partial Fock space in the related multispecies Calogero model is simply given by $$A_{1}^{\pm} = R {\mathcal{A}}_{1}^{\pm} C = tr ( {\mathcal{A}}_{1}^{\pm}{\mathcal{J}} ),$$ $$A_{2}^{\pm} =
R {\mathcal{A}}_{2}^{\pm} C = tr ( {\mathcal{A}}_{2}^{\pm}{\mathcal{J}} ).$$
Special case : the Calogero model
==================================
Specially, for $ \; \nu_{ij} = \nu \; $, $ \; m_{i} = m $ for all $ \; i,j = 1,...,N \; $, we recover the original Calogero model ( cf. Eq. (8)).
In this case, we introduce the matrix creation and annihilation operators $ {\mathcal{A}}^{\pm} $ defined by $${\mathcal{A}}^{\pm} = \frac{1}{\sqrt{2 m \omega}} ( m \omega {\bf{X}} \mp {\mathcal{D}}) =
\frac{1}{\sqrt{2 m \omega}} (m \omega {\bf{X}} \mp \imath \bf{ P} ).$$
Then the following relations hold : $$[{\mathcal{A}}^{-}, {\mathcal{A}}^{+}] = (1 - \nu){\bf {1}} + \nu \mathcal{J}$$ and $${\mathcal{H}} = \frac{\omega}{2} \{{\mathcal{A}}^{-}, {\mathcal{A}}^{+}\}, \;\;\;\;
R {\mathcal{H}}C = \omega R [{{\mathcal{A}}^{-}}^{2} , {{\mathcal{A}}^{+}}^{2} ]_{J}C,$$ $$R [{{\mathcal{A}}^{\mp}}^{k}, {{\mathcal{A}}^{\mp}}^{l} ]_{J}C = 0, \;\;\;\;\;
R [{\mathcal{H}}, {{\mathcal{A}}^{\pm}}^{k} ]_{J}C = \pm \omega k R {{\mathcal{A}}^{\pm}}^{k}C ,$$ where $ \; k,l \; $ can be arbitrary non-negative integers. In the above relations, $ [ , ] $ and $ \{ , \} $ denote the ordinary commutators and anticommutators, respectively. Note that the set of operators $ \{ {{\mathcal{A}}^{-}}^{2} ,{{\mathcal{A}}^{+}}^{2} , {\mathcal{H}} \} $ define $ \; SU(1,1) \; $ algebra and that the $ \; {\mathcal{A}} \; $ operators (20) coincide (up to the factor $ \; \sqrt{N} $ ) with the operators $ \; {\mathcal{A}}_{1}^{\pm} \; $ defined by (14), for $ \; m_{i} = m \;\; \mbox{and} \;\; \nu_{ij} = \nu $, namely $ \; {\mathcal{A}}^{\pm} = \sqrt{N} {\mathcal{A}}_{1}^{\pm} \; $.
Starting from the column vacuum state (7) ( now with all $ m_{i} $ equal to $ m $ ), which satisfies the condition $ \;\; {\mathcal{A}}^{-} \| 0 \rangle = 0 \;\; $, the complete matrix Fock space can be constructed. The general state of the full Fock space is generated by the operators (20) and can be written as $$\| n_{1},...,n_{N} \rangle \sim \prod_{k=1}^{N}( {{\mathcal{A}}^{+}}^{k}{\mathcal{J}})^{n_{k}}
\| 0 \rangle ,$$ where $ \;\; n_{k} = 0,1,2,... \;\; $ and $${\mathcal{A}}^{-}{\mathcal{J}}{{\mathcal{A}}^{+}}^{n}\| 0 \rangle =
n {{\mathcal{A}}^{+}}^{n - 1}\| 0 \rangle .$$
The general state, Eq(24), satisfies the equation $${\mathcal{H}} \ast \| n_{1},...,n_{N} \rangle \equiv {\mathcal{H}}{\mathcal{J}}\| n_{1},...,n_{N} \rangle =
{\mathcal{E}}_{\{n\}} \| n_{1},...,n_{N} \rangle .$$ Here again, $ \;\; {\mathcal{E}}_{\{n\}} \;\; $ is the matrix that satisfies the relation $$\frac{1}{\sqrt{N}} R {\mathcal{E}}_{\{n\}} \| n_{1},...,n_{N} \rangle =
\frac{1}{\sqrt{N}} E_{\{n\}} R \| n_{1},...,n_{N} \rangle$$ $$= E_{\{n\}} | n_{1},...,n_{N} \rangle =
((\sum_{k=1}^{N} k n_{k} + \varepsilon_{0} ) \omega ) | n_{1},...,n_{N} \rangle ,$$ where $ \;\; \varepsilon_{0} = \frac{1}{2}[ 1+ \nu (N - 1)] N \; $ and $ \;\; E_{\{n\}} \;\; $ and $ \;\; | n_{1},...,n_{N} \rangle \;\; $ are eigenvalues and eigenstates of the corresponding Calogero problem. For example, we explicitly show how the Hamiltonian $\; {\mathcal{H}} \; $ acts on the state $ \;\; {{\mathcal{A}}^{+}}^{n}\| 0 \rangle \;\; $ : $${\mathcal{H}} \ast {{\mathcal{A}}^{+}}^{n}\| 0 \rangle =
\omega (n{\bf {1}} + \frac{1}{2}{\mathcal{V}} {\mathcal{J}}){{\mathcal{A}}^{+}}^{n}\| 0 \rangle .$$ Note that after multiplying both sides of this equation by $ R $ from the left, we get the familiar result. Now we make complete correspondence with the ordinary Calogero model for identical particles.
The Fock space for the ordinary single-species Calogero model is generated by totally symmetric combinations $ \;\; A_{k}^{\pm} = \sum_{i=1}^{N} ( a_{i}^{\pm} )^{k} \;\;,(k = 1,2,...,N) \;\; $ of auxiliary creation and annihilation operators $ \;\; a_{i}^{\pm} = \frac{1}{\sqrt{2 \omega}}( \omega x_{i} \mp \frac{\partial}{\partial x_{i}} \mp \nu \sum_{i \neq j}
\frac{1}{x_{i}-x_{j}} ( 1 - K_{ij} )) \;\; $, where the operators $\; K_{ij}\; $ exchange labels $ \; i \; $ and $ \; j \; $ in all quantities \[7,8\].
To be more precise, the complete Fock space is spanned by the states of the form $ \;\; \prod_{k=1}^{N} ( A_{k}^{+})^{n_{k}} |0 \rangle \;\; $, where $ \;\; |0 \rangle \sim
e^{-\frac{ m \omega}{2}\sum_{i=1}^{N} x_{i}^{2}} \;\; $ is the ordinary vacuum, $ \;\; a_{i}^{-}|0 \rangle = 0 \;\;$ for all $ \;\; i = 1,...,N $.
At this point we emphasize that the following crucial relations hold: $${(a_{i}^{+})}^{k}_{/_{symm}} = {( {{\mathcal{A}}^{+}}^{k} C )}_{i}, \;\;\;\;\;
{(a_{i}^{-})}^{k}_{/_{symm}} = {( R {{\mathcal{A}}^{-}}^{k} )}_{i}$$ for every $ \;\; k = 1,2,...,N \;\; $, which can be proved by induction. From this relation we immediately obtain $$A_{k}^{\pm} = \sum_{i=1}^{N}{(a_{i}^{\pm})}^{k}_{/_{symm}} = R {{\mathcal{A}}^{\pm}}^{k} C =
tr ( {{\mathcal{A}}^{\pm}}^{k} {\mathcal{J}}).$$ The label $ \; symm \; $ designates that the action is restricted to the symmetric states only. In this case, the exchange operators $ \; K_{ij} \; $ coincide with the identity operator $ ( K_{ij} \psi = \psi ) $. Equations (29) provide one-to-one correspondence between our matrix oscillator and the ordinary single-species Calogero model. Note that in the matrix oscillator approach the exchange operators $ \; K_{ij} \; $ do not appear. The approach considered in this paper can be repeated in an arbitrary gauge, particularly in the gauge $ \lambda = 0 $.
After removing the $ CM $ mode, one is left with $ \;\; {\mathcal{V}}_{rel} = ( 1 - \frac{1}{N} - \nu){\bf {1}}
+ \nu {\mathcal{J}} \;\;$. The critical points are determined by $ \;\; det {\mathcal{V}}_{rel} = 0 \;\; $. They are $ \; \nu = - \frac{1}{N} \; $ and $ \; \nu = 1 - \frac{1}{N} \; $. The first critical point corresponds to the existence of the ground state for $ \; \nu > - \frac{1}{N} \; $ \[17-19\], whereas the second critical point should be investigated more carefully.
Conclusion
============
In conclusion, by introducing a pair of $ N \times N $ matrices, we have constructed a Hamiltonian for a single quadratic matrix oscillator that is required to obey deformed commutation relations. This is a generalization of a single deformed oscillator. Straightforward analysis has shown that this oscillator is equivalent to the multispecies Calogero model in one dimension. Specially, in the case of identical particles, an alternative formulation of ladder operators has been proposed in which exchange operators do not appear. In terms of these operators, the complete Fock space has been constructed. We point out that our matrix formulation can be extended to all Calogero-like models, such as models in arbitrary dimensions,the Sutherland model (on a circle) and models defined on root systems and the supersymmetric Calogero model.
[**Acknowledgment**]{}\
We would like to thank K.S.Gupta and M.Mileković for useful discussions. This work was supported by the Ministry of Science and Technology of the Republic of Croatia under contract No. 0098003.
[99]{}
F.Calogero, J.Math.Phys. [**10**]{} (1969) 2191, 2197;\
F.Calogero, J.Math.Phys. [**12**]{} (1971) 419\
B.Sutherland, J.Math.Phys. [**12**]{} (1971) 246. M.A.Olshanetsky, A.M.Perelomov, Phys.Rept.[**71**]{}(1981)313. M.Caselle, U.Magnea, Phys.Rept[**394**]{} (2004)41;\
T.Guhr, A.Muller-Groeling, H.A.Weidenmuller, Phys.Rept.[**299**]{}(1998)189;\
M.L.Mehta,[*[Random matrices]{}*]{} (Academic Press, 1991). T.Guhr, H.Kohler, J.Math.Phys.[**43**]{},(2002)2741;\
H.J.Stoeckmann, J.Phys.A Math.Gen.[**37**]{} (2004)137;\
S.Jain, Mod.Phys.Lett. A[**11**]{} (1996)1201;\
B.D.Simons, P.A.Lee, B.L.Altshuler, Phys.Rev.Lett. [**70**]{} (1993)4122. D.Birmingham, K.S.Gupta, S.Sen, Phys.Lett. B [**505**]{} (2001) 191;\
G.W.Gibbons, P.K.Townsend, Phys.Lett. B[**454**]{}, 187(1999). J.Gomis, A.Kapustin, JHEP 0406(2004)002. A.P.Polychronakos, Phys.Rev.Lett. [**69**]{} (1992) 703. L.Brink, T.H.Hansson, M.Vasiliev, Phys.Lett. B [**286**]{} (1992) 109;\
L.Brink, T.H.Hansson, S.E.Konstein, M.Vasiliev, Nucl.Phys. B [**384**]{} (1993) 591. A.P.Polychronakos,in “Topological Aspects of Low-Dimensional Systems”, Proceedings of the Les Houches session LXIX Summer school, Springer 1998. A.P.Polychronakos, Phys.Lett.B[**277**]{}(1992)102;\
A.P.Polychronakos, Phys.Lett.B[**403**]{}(1997)239;\
A.P.Polychronakos, Nucl.Phys.B[**546**]{}(1999)495. D.Kazhdan, B.Kostant, S.Sternberg, Comm.Pure Appl. Math.vol.31 (1978) 481. A.Gorsky, N.Nekrasov, Nucl.Phys. B414 (1994) 213. A.Kempf, G.Mangano, R.B.Mann, Phys.Rev.D [**52**]{} (1995) 1108;\
L.N.Chang, D.Minic, N.Okamura, T.Takeuchi, Phys.Rev.D [**65**]{} (2002) 125027;\
I.Dadić , L.Jonke, S.Meljanac, Phys.Rev.D [**67**]{} (2003) 087701. V.P.Nair, A.P.Polychronakos,Phys.Lett.B [**505**]{} (2001) 267;\
L.Jonke, S.Meljanac, Eur.Phys.J.C [**29**]{} (2003) 433. P.Forrester, J. Phys. A: Math. Gen. [**25**]{} (1992) L607. C.Furtlehner, S.Ouvry, Mod.Phys.Lett. B[**9**]{} (1995) 503;\
D.Sen, Nucl. Phys. B [**479**]{} (1996) 554;\
A.Dasniers de Veigy, Nucl. Phys. B [**483**]{} (1997) 580;\
S.Maskhevich, Phys. Lett. A [**233**]{} (1997) 30. S.Meljanac, M.Mileković , A.Samsarov, Phys. Lett.B[**573**]{}(2003) 202;\
S.Meljanac, M.Mileković , A.Samsarov, M. Stojić , Mod. Phys. Lett. B, Vol.[**18**]{}, (2004) 603. S.Meljanac, M.Mileković , A.Samsarov, hep-th/0405131, to appear in Phys. Lett. B. S.Meljanac, M.Mileković , M.Stojić , Eur. Phys. J. C [**24**]{} (2002) 331;\
V.Bardek, L.Jonke, S.Meljanac, M.Milekovi, Phys.Lett. B [**531**]{} (2002) 311.
[^1]: e-mail: meljanac@irb.hr
[^2]: e-mail:asamsarov@irb.hr
|
---
author:
- |
Kyaw Zaw Lin\
National University of Singapore\
[kyawzl@comp.nus.edu.sg]{}
- |
Weipeng Xu\
Max Planck Institute for Informatics\
[wxu@mpi-inf.mpg.de]{}
- |
Qianru Sun\
Max Planck Institute for Informatics\
[qsun@mpi-inf.mpg.de]{}
- |
Christian Theobalt\
Max Planck Institute for Informatics\
[theobalt@mpi-inf.mpg.de]{}
- |
Tat-Seng Chua\
National University of Singapore\
[dcscts@nus.edu.sg]{}
- |
Kyaw Zaw Lin$^{1}$ Weipeng Xu$^{2}$ Qianru Sun$^{1,2}$ Christian Theobalt$^{2}$ Tat-Seng Chua$^{1}$\
\
$^{1}$National University of Singapore\
$^{2}$Max Planck Institute for Informatics, Saarland Informatics Campus\
\
bibliography:
- 'egbib.bib'
title: Learning a Disentangled Embedding for Monocular 3D Shape Retrieval and Pose Estimation
---
Acknowledgement
===============
This research is part of NExT research which is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its IRC@SG Funding Initiative. It is partially supported by German Research Foundation (DFG CRC 1223).
|
---
author:
- 'L. Grassitelli [^1] , N. Langer , N.J. Grin , J. Mackey , J.M. Bestenlehner and G. Gräfener'
bibliography:
- 'sonic.bib'
date: 'Received //, 2017'
title: |
Subsonic structure and optically thick winds\
from Wolf–Rayet stars
---
Introduction
============
A star loses energy from its surface not only in the form of radiation, but also in the form of a stream of particles, the stellar wind [@1958Parker]. Stellar winds are outflows of material from a star which play a major role especially in massive star evolution. Stellar winds can in fact lead massive stars to lose a significant fraction of their initial mass, thus enriching the interstellar medium and influencing the structure of the outer layers and the spectroscopic appearance of these stars [@1986Chiosi; @1991Lamers; @2012Langer].
The proximity of massive stars to the Eddington limit in the upper part of the Hertszprung–Russell (HR) diagram can lead to dense outflows. High mass-loss rates can veil the hydrostatic layers and result in the appearance of emission lines in the spectra of luminous stars [e.g. @2006Hamann; @2008Grafener; @2014Bestenlehner]. In this case part of the radiation is scattered back from the radially expanding atmosphere, giving feedback in the form of a back-warming of the outer hydrostatic layers. This feedback is often described as proportional to the frequency independent mean optical depth $\tau$, which is the average number of photon mean-free paths along the line of sight [@1978Mihalas]. For optically thin winds the photosphere arises in the subsonic part of the outflow. More massive stars have winds that can become optically thick, where the continuum – or part of it – is produced in the supersonic part of the outflow.
The highest steady mass-loss rates by stellar wind are found in the late stages of the evolution of massive stars, in what is known as the Wolf–Rayet phenomenon. Wolf–Rayet (WR) stars are very luminous stars close to the Eddington limit and characterized by highly supersonic dense winds which can shroud the hydrostatic layers from direct observation [@1988Langer; @1989Langer; @2008Crowther]. The mass loss is thought to be driven by radiation pressure, i.e. momentum transfer via absorption and scattering of photons in the partially optically thick wind. High momentum transfer efficiency is necessary to explain the high mass-loss rates observed in WR stars, implying that multiple scattering events and enhanced opacities due to line-blanketing are necessary to accelerate these stellar winds up to the observed terminal wind velocities of the order of thousands of kms$^{-1}$ [@1987Abbott; @1999LamersCassinelli; @2013Owocki; @2014Bestenlehner].
Empirical determinations of the hydrostatic radii of Galactic WR stars in recent decades has revealed a disagreement between the spectral analysis based on wind models [@2006Hamann; @2012Sander] and stellar structure calculations of mostly helium burning stars [@1989Langer; @1996Heger; @2006Petrovic; @2012Grafener; @2012Georgy; @2016GrassitelliWR]. The difficulties in reconciling this disagreement, where there is a difference of up to a factor 10 in the derived hydrostatic radii (known as the ‘WR radius problem’), has been ascribed to the location of the wavelength-dependent photosphere [@1987Hillier; @2015Hillier; @1988Langer; @1992Kato; @1996Heger] which forms in the supersonic part of the wind. Consequently, the detailed dynamics and density structure in the expanding envelope and supersonic wind are uncertain, which relates also to the complex opacities and their interplay with the accelerating outflow [@1996Schaerer; @1996Heger] and to the presence of density inhomogeneities in the wind [@1988Moffat; @2009StLouis; @2011CheneCIR; @2014Michaux].
There appears to be a lack of radiative force to drive the high mass-loss rates observed in WR stars [e.g. @2005Grafener; @2015Sander]. This is true even considering the line opacity contribution of the absorption lines (especially UV) exposed to unattenuated radiative flux by the progressive Doppler shift of the stellar wind, which allows for the absorption of the unattenuated continuum, reducing thus the effect of line self-shadowing [@1970Lucy; @1975CAK; @1985Abbott; @1993Lucy; @1997Owocki]. Consequently, an approximate relation is commonly adopted to describe the dynamics of the winds (i.e. the velocity and density profiles), namely a beta-velocity law [@1975CAK; @1989Langer; @2006Hamann; @2015Hillier; @2017Sander]. This approximation contributes to the uncertainty in the determination of the stellar radius and the detailed temperature stratification within the supersonic outflow.
Hydrostatic models of helium stars above $\sim 10 \Msun$ with Galactic metallicity show the formation of a low-density inflated envelope having a density inversion below their plane parallel grey atmosphere [@2006Petrovic; @2012Grafener; @2016GrassitelliWR; @2016Eldridge]. Inflation and density inversion arise as these models approach a local Eddington factor of unity in the proximity of the iron opacity bump at a temperature of $\approx 200$kK. This hydrostatic limit leads to the flattening of the temperature and density gradients in the outer layers, up to building a positive gas pressure gradient to counterbalance the strong radiative force [@1973Joss; @1997Langer; @2015Owocki; @2015Sanyal].
@1996Heger, @1996Schaerer, @2006Petrovic, and more recently @2016Ro and @2018Nakauchi have investigated the envelope configuration of massive stars while considering the effects of mass loss by stellar winds on the structure of the outer layers. The need for a strong radiative acceleration to launch these dense winds points towards the role of the iron opacity bump as a source of momentum for the flow already from the subsonic quasi-hydrostatic part of the star. @2006Petrovic showed that sufficiently high mass-loss rates can lead to configurations in which the inflation of the envelope is inhibited, and where the flow is accelerated by the radiative forces in the proximity of the iron opacity bump.
In this paper we investigate the interplay between an optically thick wind and the subsonic structure of massive helium stars, focusing especially on the conditions at the sonic point. In Sect.2 we explore some theoretical considerations about optically thick winds and the importance of the sonic point. In Sect.3 we apply our considerations to computing hydrodynamic stellar structure models, in Sect.4 we show selected results and applications, in Sect.5 we develop our results to constrain predicted mass-loss rates by introducing new useful tools, in Sect.6 these results are discussed, and in Sect.7 we draw our conclusions.
Optically thick winds {#Sect.optthickwinds}
=====================
The requirement of a smooth, radiation-pressure-driven, steady-state, transonic flow sets specific conditions that the physical quantities and their gradients must satisfy at the sonic point for optically thick winds. Based on the reasonable assumption of a radiation-driven wind [analogous to the O stars, @2015Vink], @2002Nugis estimated that the sonic point of hot WR stars is located at high optical depth (i.e. $\tau \approx$3–30) and that it has to occur within the temperature ranges where the Rosseland opacity $\kappa$ increases as a function of radius, i.e. where the flow accelerates as a result of the increase in the local opacity ($\mathrm{d}\kappa/\mathrm{d}r>0$), because energy and momentum input are necessary at the sonic point to accelerate the outflow.
From the momentum equation, the velocity gradient in Eulerian coordinates describing the hydrodynamics of a steady-state stellar wind can be written as [@1995Bjorkman; @1999LamersCassinelli; @2002Nugis; @2014Hubeny; @2017Sander] $$\label{criticalpoint}
\frac{1}{\varv}\frac{\mathrm{d}\varv}{\mathrm{d}r} = -\left(g - g_\mathrm{ rad} - 2 \frac{c_\mathrm{ s}^2}{r} + \frac{\mathrm{d}c_\mathrm{ s}^2}{\mathrm{d}r}\right)/(\varv^2-c_\mathrm{ s}^2) \quad ,$$ where $g$ and $g_{\rm rad}$ are the gravitational and radiative accelerations, $c_\mathrm{ s}$ the local isothermal sound speed, $\varv$ the flow velocity, and $r$ the radial coordinate. Equation \[criticalpoint\], known as the ‘Bondi equation’ [@1952Bondi], implies that when $\varv=c_\mathrm{ s}$, $\mathrm{d}\varv/\mathrm{d}r$ diverges unless the numerator is null as well. The requirement of a null numerator at the sonic point implies that $$\label{numsonic}
g_\mathrm{ rad} = g - 2 \frac{c_\mathrm{ s}^2}{r} + \frac{\mathrm{d}c_\mathrm{ s}^2}{\mathrm{d}r} \quad ,$$ where $g=Gm/r^2$ with $m$ being the mass coordinate. In this equation the radiative acceleration can be expressed as $$\label{grad}
g_\mathrm{ rad} = \frac{\kappa L}{4 \pi r^2 c} \quad ,$$ with $\kappa$ the Rosseland mean opacity from Thompson scattering and from bound-bound and bound-free absorption, which is assumed in this context to be consistent with the flux-weighted mean opacity, and with the constants $G$ and $c$ holding their usual meaning. Assuming that the opacities are not affected by the velocity and the velocity gradient, Eq.\[criticalpoint\] shows the presence of a critical point for the momentum equation at the sonic point (see Appendix \[appendixsoniccritical\] for further discussion).
Equation \[numsonic\] can be written in terms of the Eddington factor $\Gamma = g_\mathrm{ rad}/g$, and becomes $$\label{gammasonic}
\Gamma = 1- \frac{2 \frac{c_\mathrm{ s}^2}{r} - \frac{\mathrm{d}c_\mathrm{ s}^2}{\mathrm{d}r} }{g}
,$$ which shows that the condition $\frac{\mathrm{d}c_\mathrm{ s}^2}{\mathrm{d}r} - 2 \frac{c_\mathrm{ s}^2}{r} \ll g$ leads to $\Gamma \approx 1$ (or equivalently to a local effective gravity $g_\mathrm{ eff} \approx 0$) at the sonic point. This has a crucial implication: for $\frac{\mathrm{d}c_\mathrm{ s}^2}{\mathrm{d}r} - 2 \frac{c_\mathrm{ s}^2}{r} \ll g$, a radiation-driven, smooth, steady-state transonic outflow from a star needs to have its sonic point located at $\Gamma\approx 1$. A transonic radiation-pressure-driven stellar wind has to self-adjust its structure to meet this requirement.
Temperature stratification at the sonic point {#temperature-stratification-at-the-sonic-point .unnumbered}
----------------------------------------------
Formal studies of the dynamics of stellar winds and their stability in spherical symmetry were first conducted by @1958Parker [@1966Parker]. However, from a mathematical and conceptual point of view, solving the physical problem of the dynamics of a gas outflow is analogous to studying spherically symmetric mass accretion [@1952Bondi; @1975Tamazawa; @1980Moncrief; @1981Thorne; @1995Bjorkman; @1998Visser]. The topology of the steady-state solutions for the stellar wind equation shows two characteristic solutions: one that starts subsonic and reaches supersonic finite velocity (known as the wind solution) and one that starts supersonic and becomes subsonic for increasing distance (the accretion solution). The problem of stellar wind and accretion are conceptually analogous and, [*mutatis mutandis*]{}, the results obtained in one context can be applied to the other.
Accretion onto a black hole can also be divided into optically thin and optically thick accretion depending on the location of the sonic point radius with respect to the photosphere [@1975Tamazawa; @1981Thorne; @1982Meier; @1984Flammang; @1991Nobili]. In the optically thin case the hydrodynamics of the flow and the radiative transfer are effectively decoupled [@1978Mihalas; @1981ThorneB; @1982Flammang]. The non-local coupling of the radiation field to the structure of the atmosphere as a whole implies only a weak dependence of the energy transfer problem on the local thermodynamical conditions [@1978Mihalas]. However, as the photon mean free path $\lambda$, defined via $$\lambda = (\rho \kappa)^{-1} \quad$$ becomes smaller than the length scale on which the macroscopic quantities change, the system enters into the photon diffusion regime [@1978Mihalas; @1981Thorne; @1990Kippenhahn]. The radiative transport equation in the asymptotic limiting case of large optical depth and small mean free paths yields the diffusive approximation, which can be expressed as [@1978Mihalas; @1990Kippenhahn] $$\label{diffapprox}
\frac{\mathrm{d}P_\mathrm{ rad}}{\mathrm{d}\tau}=\frac{F}{c} \quad ,$$ with $P_\mathrm{ rad}$ the radiation pressure, $F$ the flux, and $\mathrm{d}\tau = - \kappa \rho \, \mathrm{d}r$.
Similarly, the stellar wind problem can be considered in the optically thin and optically thick regime. In Appendix \[SH\] we investigate in more detail the behaviour of the radiation field in the proximity of the sonic point at the conditions typically met in stellar models of WNE stars. Backed by these considerations, we assume the conditions at the sonic point of WNE stars to closely match local thermodynamical equilibrium (LTE).
The LTE assumption implies that the temperature gradient can be defined by [local]{} quantities only, with the energy flux set by the need to transport the stellar luminosity outwards. At the high optical depth assumed for the sonic point, in fact, the temperature gradient becomes independent of the detailed global structure of the atmosphere, losing its explicit dependence on the optical depth characteristic of the optically thin situation (see Appendix B).
For a given mass-loss rate, the sonic point conditions of an optically thick wind (density and temperature) are uniquely defined, and thus the subsonic structure becomes independent from the detailed conditions above it, which only define the velocity profile and the terminal wind velocity of the outflow. Setting the mass-loss rate, in fact, sets the temperature at the sonic point. This can be seen combining the condition $$\label{sonicpointiso}
\varv = \sqrt{\frac{k_\mathrm{ B} T}{\mu m_\mathrm{ H}}} = c_\mathrm{ s}
,$$ where $\mu$ is the mean molecular weight, $m_\mathrm{ H}$ the mass of a proton, and $k_\mathrm{ B}$ the Boltzman constant, with the definition of steady-state mass-loss rate $$\label{steadymdot}
\dot{M} = 4 \pi r^2 \rho \varv
,$$ which leads to $$\label{tsonic}
T_\mathrm{ S}= \frac{\mu m_\mathrm{ H}}{k_\mathrm{ B}}\left(\frac{\dot{M}}{4 \pi r^2 \rho}\right)^2 \quad .$$ Therefore, having the mass-loss rate imposing the blanketed sonic point temperature while coupled to the subsonic stellar structure, hydrodynamic models can be computed independently from wind calculations without the need of a global computation (as we show in Sect.\[sectrisults\]).
Envelope inflation {#envelope-inflation .unnumbered}
------------------
Both hydrostatic and hydrodynamic stellar models with high luminosity-to-mass ratios close to the Eddington limit show diluted and extended envelopes, often characterized by very inefficient convective energy transport and the presence of a density inversion [@1997Langer; @1999Ishii; @2012Grafener; @2015Kohler; @2015Sanyal; @2017Sanyal; @2015GrassitelliA; @2015Owocki]. These stellar models are said to be [inflated]{}, having a sub-photospheric envelope configuration with large pressure and density scale heights and the local Eddington factor close to unity. This happens in the temperature range where an opacity bump arises, most notably the iron opacity bump.
Assuming that the temperature stratification is given only by radiative transport in the diffusive approximation, thus neglecting the contribution from convective energy transport [appropriate for the envelopes of massive helium star models, @2016GrassitelliWR], for chemically homogeneous layers with constant luminosity, using Eq.\[diffapprox\] the temperature profile can be written as $$\label{dTdr}
\frac{\mathrm{d}T}{\mathrm{d}r}= -\frac{\rho T}{4 P_\mathrm{ rad}}\frac{\kappa L}{4 \pi r^2 c} \quad ,$$ from which $$\label{dcsdr}
\frac{\mathrm{d}c_\mathrm{ s}^2}{\mathrm{d}r}= c_\mathrm{ s}^2\frac{\mathrm{d}\,\ln(T)}{\mathrm{d}r} =-\frac{\beta}{4(1-\beta)}\frac{\kappa L}{4 \pi r^2 c}\quad ,$$ with $\beta$ the ratio of gas to total pressure. Combining then Eq.\[dTdr\] with Eq.\[criticalpoint\] and with the differential form of the steady-state continuity equation $$\mathrm{d}\,{\ln}(\varv) + \mathrm{d}\,{\ln} (\rho) + 2\mathrm{d}\,{\ln} (r)=0\quad,$$ the slope of the density profile follows $$\label{densder}
\frac{\mathrm{d}\,\ln(\rho)}{\mathrm{d}r}=\left(g-\frac{\kappa L}{4\pi r^2 c}\left(1+\frac{P_{\rm gas}}{4P_\mathrm{ rad}}\right)+\frac{2\varv^2}{r}\right)/(\varv^2-c_\mathrm{ s}^2) \quad .$$ Here we can distinguish two regimes: the subsonic and the supersonic. In the subsonic, sub-Eddington region, an increase in opacity or generally any outward-directed force (e.g. the centrifugal force) tends to reduce the slope of the density profile, thus increasing the density scale height and giving rise to the [envelope inflation]{} encountered in the helium star models of e.g. @2006Petrovic, @2012Grafener, and @2016GrassitelliWR. Consequently, although counter-intuitive, from Eq.\[densder\] an increase in the radiative force in the subsonic region implies a less steep increase in flow velocity, while making the density profile more and more flat; the layers aim to preserve hydrostatic equilibrium counterbalancing the local gravitational force, and the radiative acceleration effectively acts as a reduction in the local gravity. As the gas pressure gradient needed to counterbalance this reduced effective gravity is smaller, the increased opacity can lead, together with the acceleration of the flow, to an almost flat density profile when the local $\Gamma$ approaches unity [@2017Sanyal].
However, reaching and exceeding the local Eddington limit does not necessarily guarantee that the outflow reaches supersonic velocities. In case of a (still) subsonic steady-state flow, if the numerator on the right-hand side of Eq.\[densder\] becomes negative, a density inversion has to arise associated with a decrease in velocity, whereas a further increase in radiative force leads to a steep decrease in the velocity and an increase in the gas to total pressure. This hydrodynamic envelope inflation solution is similar to the ‘breeze’ solution for winds which do not asymptotically reach supersonic velocities [@1999LamersCassinelli; @2014Hubeny].
A clear definition, and therefore a criterion, of envelope inflation was not available prior to publication of this work [@2017Sanyal]. In Appendix \[app.inflation\] we discuss more extensively the appearance of inflated envelopes. We do so by performing hydrostatic stellar structure calculations of massive helium star models using different opacities and thus comparing how the outer layers react to radiative forces of different intensity. Non-inflated hydrostatic massive helium star models show density and gas pressure scale height which rapidly decrease as they approach the surface. This is due to the gas pressure gradient being the leading force counterbalancing gravity and preserving hydrostatic equilibrium. However, in the presence of high Eddington factors in stars with high $L/M$ ratios, an inflection point in density and gas pressure can appear, and thus density and gas pressure scale heights start to increase.
We show that an inflection point can appear as a consequence of a rapid increase in opacity (or equivalently in the Eddington factor) when the radiation pressure gradient becomes the primary agent counterbalancing gravity. Analytically, an inflection point in the gas pressure profile is found when (see Eq.\[critinf\]) $$\frac{\mathrm{d}\,\ln(\kappa)}{\mathrm{d}\,\ln(T)} \lesssim 1 -\frac{1}{\Gamma} \quad,$$ showing both the need for an Eddington factor close to unity and for a steep increase in the opacity with temperature. In view of this discussion and of Appendix \[app.inflation\], we can thus understand envelope inflation as [the appearance of large gas pressure scale heights in response to a steep increase in the radiative opacity in radiation-pressure-dominated layers close to the Eddington limit]{}. The low gas pressure gradient prevents the density from decreasing steeply for extended regions of space, and implies that the surface is found at significantly larger radii compared to non-inflated stellar models.
Model assumptions {#model}
=================
Computing the stellar structure is a task that requires solving a hyperbolic set of differential equations simultaneously with well-defined boundary conditions [@1990Kippenhahn]. We adopt the Bonn evolutionary code (BEC), a Lagrangian [*hydrodynamic*]{} one-dimensional stellar evolution code [@1988Langer; @1994Langer; @1996Heger; @1998Heger; @2000Heger; @2006Petrovic; @2006Yoon; @2014Kozyreva]. It solves the set of coupled non-linear partial differential stellar structure equations in the form [@1990Kippenhahn]
$$\label{consmass}
\left(\frac{\partial m}{\partial r}\right)_t = 4 \pi r^2 \rho
,$$
$$\label{defvel}
\left(\frac{\partial r}{\partial t}\right)_m = \varv\\
,$$
$$\label{momentumeq}
\left(\frac{a}{4 \pi r^2}\right)_t = \frac{G m}{4 \pi r^4} + \frac{\partial P}{\partial m}\\
,$$
$$\label{energytransp}
\left(\frac{\partial T}{\partial m}\right)_t = - \frac{G m }{4 \pi r^4 }\frac{T}{P}\nabla \left( 1+ \frac{r^2}{G m}\frac{ \partial \varv}{\partial t} \right)_m
,$$
$$\label{energycons}
\left(\frac{\partial L}{\partial m}\right)_t=\epsilon_n- \epsilon_g - \epsilon_\nu
,$$
where $m$ and $t$, the mass coordinate and time, are the two independent variables; $\rho$ is the density; $\varv$ is the velocity; $a$ is the acceleration; $r$ is the radial coordinate; $T$ is the temperature; $L$ is the luminosity; $\nabla:=\frac{\mathrm{d}\,\log(T)}{\mathrm{d}\,\log(P)}$ is the temperature gradient; $P$ is the total pressure given by the sum of gas and radiation pressure; $G$ is the gravitational constant; and $\epsilon$ is the energy production/loss per unit mass and unit time related to nuclear processes (subscript n), to gravitational contraction/expansion (subscript g), and neutrino loss (subscript $\nu$). These equations express: the conservation of mass (Eq.\[consmass\]), the definition of velocity (Eq.\[defvel\]), the conservation of momentum (Eq.\[momentumeq\]), the energy transport (Eq.\[energytransp\]), and the energy conservation (Eq.\[energycons\]). These equations together with the network of nuclear reaction rates, the set of equations of the mixing length theory [@1958Vitense], and the OPAL opacity tables [@1996Iglesias], define the structure and evolution of a stellar model. In addition, mass loss by stellar wind can also be applied.
The set of non-linear coupled partial differential equations above has five dependent variables (namely $\rho$, $L$, $\varv$, $T$, and $r$) and requires a number of boundary conditions equal to the degrees of freedom of the system, i.e. five. The central boundary conditions are trivially set by the requirement of zero $r$, $L$, and $\varv$ in the centre of the stellar models, while the outer boundary conditions are usually determined by the assumption of plane parallel grey atmosphere [@1989Langer; @1998Heger].
The boundary conditions set constraints on the family of possible solutions (if any) of the set of partial differential equations describing a star. We modify BEC adopting as surface boundary conditions $$\label{boundmdot}
\dot{M} = 4 \pi r^2 \rho \varv$$ and $$\label{sonicpoint}
\varv = \sqrt{\frac{k_\mathrm{ B} T}{\mu m_\mathrm{ H}}} = c_\mathrm{ s}
,$$ i.e. we impose the boundary of the stellar model at the sonic point while preserving continuity. The mass-loss (or mass-accretion) rate is a free parameter in our calculations, i.e. it can be chosen either manually or according to mass loss by stellar wind prescriptions [such as those in e.g. @2000Nugis; @2001Vink]. Mass loss is treated adopting a pseudo-Lagrangian scheme in the outer part of the stellar model, i.e. the independent variable becomes $q=m/M_\mathrm{ tot}$ (the relative mass coordinate) and consequently the Lagrangian operator time derivative for a steady state becomes $\frac{\mathrm{d}}{\mathrm{d}t}= - \frac{q \dot{M}}{M_\mathrm{ tot}}\frac{\partial}{\partial q} $ [@1977Neo; @1998Heger].
We adopt BEC to compute the subsonic structure of massive helium stars. Models are computed with a chemical composition as in @2000Heger, with metallicity Z=0.02 and helium fraction 0.98, and stationarity is achieved with large time steps (i.e. greater than $10^3$ seconds) while inhibiting chemical evolution. The velocity profile is computed self-consistently with the radiative acceleration derived via the velocity-independent Rosseland opacity and the imposed mass-loss rate. Convection is included in the calculations, but as shown by @2016GrassitelliWR, the convective flux in the outer layers is several orders of magnitude smaller than the radiative flux. We compute models for chemically homogeneous massive helium zero age main sequence stars of mass 10, 15, and 20 at solar metallicity and under various applied mass-loss rates by stellar wind. Rotation and magnetic field are not included in the calculations; however, they are not expected to affect the general theoretical considerations in Sect.2. We also neglect the structural effects from the turbulent pressure on the stellar models. For comparison purposes, we also compute models with classical plane parallel grey atmosphere boundary conditions [cf. @1998Heger; @2006Petrovic].
Results {#sectrisults}
=======
$\log\, (\dot{M})$ $R_\mathrm{ S}$ $\log\, (L) $ $\kappa$ $\log\, (\rho_\mathrm{ S})$ $\log\, (T_\mathrm{ S})$ $\log\, (\lambda) $ $\varv$ $\Gamma$ $t$ $\log\,(H_P)$ $\log\, (T_\mathrm{ eff})$ $\tau_\mathrm{ S}$
-------------------- ----------------- --------------- ---------- ----------------------------- -------------------------- --------------------- --------- ---------- ------- --------------- ---------------------------- --------------------
$M_\odot/$yr $R_\odot$ $L_{\odot}$ cm$^2$/g g/cm$^3$ K cm km/s $ $ $ $ cm K $ $
-4.3 1.152 5.464 0.658 -7.971 5.331 8.125 36.55 0.9901 8.097 9.218 5.099 33
-4.5 1.158 5.466 0.659 -8.165 5.312 8.350 35.73 0.9918 6.689 9.335 5.097 20
-4.7 1.165 5.467 0.660 -8.359 5.291 8.521 34.91 0.9931 5.754 9.451 5.096 13
-5.0 1.182 5.469 0.661 -8.656 5.259 8.827 33.62 0.9946 4.708 9.628 5.093 8
-5.1 1.190 5.468 0.661 -8.753 5.246 8.921 33.14 0.9952 4.787 9.680 5.091 6
-5.3 1.216 5.469 0.661 -8.959 5.212 9.133 31.87 0.9964 5.790 9.768 5.087 3
-5.4 1.260 5.470 0.626 -8.843 4.726 8.985 18.20 0.8978 0.138 7.827 5.079 2
Same as above, but for our set of models with a mass of 10
$\log\, (\dot{M})$ $R_\mathrm{ S}$ $\log\, (L) $ $\kappa$ $\log\, (\rho_\mathrm{ S})$ $\log\, (T_\mathrm{ S})$ $\log\, (\lambda) $ $\varv$ $\Gamma$ $t$ $\log\,(H_P)$ $\log\, (T_\mathrm{ eff})$ $\tau_\mathrm{ S}$
-------------------- ----------------- --------------- ---------- ----------------------------- -------------------------- --------------------- --------- ---------- ------- --------------- ---------------------------- --------------------
$M_\odot/$yr $R_\odot$ $L_{\odot}$ cm$^2$/g g/cm$^3$ K cm km/s $ $ $ $ cm K $ $
-4.3 0.910 5.115 0.986 -7.748 5.295 7.748 35.03 0.9828 9.766 8.832 5.061 49
-4.5 0.914 5.120 0.979 -7.937 5.267 7.937 33.93 0.9861 8.418 8.908 5.061 28
-4.7 0.920 5.123 0.971 -8.118 5.221 8.118 32.17 0.9883 13.29 8.909 5.061 16
-5.0 0.930 5.136 0.842 -8.225 4.814 8.225 20.13 0.8173 0.625 7.565 5.061 8
-5.1 0.930 5.136 0.846 -8.311 4.791 8.311 19.62 0.8082 0.523 7.556 5.061 6
-5.3 0.931 5.136 0.855 -8.493 4.747 8.493 18.64 0.7866 0.377 7.544 5.061 3
Same as above, but for our set of models with a mass of 20
$\log\, (\dot{M})$ $R_\mathrm{ S}$ $\log\, (L) $ $\kappa$ $\log\, (\rho_\mathrm{ S})$ $\log\, (T_\mathrm{ S})$ $\log\, (\lambda) $ $\varv$ $\Gamma$ $t$ $\log\,(H_\mathrm{ P})$ $\log\, (T_\mathrm{ eff})$ $\tau_\mathrm{ S}$
-------------------- ----------------- --------------- ---------- ----------------------------- -------------------------- --------------------- --------- ---------- ------- ------------------------- ---------------------------- --------------------
$M_\odot/$yr $R_\odot$ $L_{\odot}$ cm$^2$/g g/cm$^3$ K cm km/s $ $ $ $ cm K $ $
-4.3 1.362 5.678 0.545 -8.123 5.344 8.376 37.09 0.9934 5.158 9.438 5.114 27
-4.5 1.371 5.679 0.544 -8.320 5.327 8.597 36.37 0.9945 4.213 9.571 5.113 18
-4.7 1.383 5.680 0.543 -8.517 5.310 8.775 35.68 0.9954 3.635 9.707 5.111 11
-5.0 1.409 5.681 0.542 -8.821 5.283 9.081 34.58 0.9963 2.931 9.917 5.108 6
-5.1 1.421 5.681 0.542 -8.921 5.274 9.182 34.23 0.9965 2.731 9.988 5.106 2
-5.3 1.453 5.682 0.542 -9.135 5.255 9.398 33.48 0.9971 2.487 10.14 5.101 1
Figure\[env\] shows the velocity profiles of the outflow in the outer subsonic part of a set of 15stellar models computed with sonic point boundary conditions and with different adopted mass-loss rates. For comparison the velocity profile of a plane parallel grey atmosphere BEC model is also shown.
All the models presented in Fig.\[env\] show a rapid increase in the radial velocity starting from $R \approx 1.13\, R_\odot$. This increase is associated with the increase in radiative force in correspondence with the iron opacity bump (Fe-bump) in the temperature range $5.2\lesssim {\log(T)}\lesssim 5.5$. The more compact, black models in Fig.\[env\] show that the velocity profile is steeper for the higher mass-loss rate applied, with a flow velocity monotonically increasing until it reaches the sound speed in the temperature range of the hot rising part of the Fe-bump. The model with the highest mass-loss rate (5$\times 10^{-5}\Msun\,{\rm yr}^{-1}$) achieves a transonic flow from the smallest radius, while the other stellar models show larger sonic point radii as the adopted mass-loss rates decrease (see Table \[tab\] for the parameters of the models). All the monotonic solutions are thus solutions where the outflow is accelerated by the iron opacity (see also Fig.\[den\]). Compared to the electron scattering opacity, this increase in opacity provides a stronger outward-directed radiative force, leading to higher local Eddington factors and flows that can reach supersonic velocities already at 200kK. The higher the mass-loss rate, the steeper the velocity profile, the sooner (in terms of radius) the flow meets the sound speed, and the hotter and more compact is the resulting model. However, solutions with non-monotonic velocity profiles are also present. While the group of more compact monotonic solutions has mass-loss rates of $\Mdot\geq 5\times10^{-6}\Msun\,{\rm yr}^{-1}$ (see Table \[tab\]), the pink model with a mass-loss rate of $\Mdot = 4\times10^{-6}\Msun\,{\rm yr}^{-1}$ shows an outflow that does not become supersonic in the proximity of the iron opacity bump. In this case the flow slows down to velocities of the order of a few km/s after being accelerated by the increased opacity of the Fe-bump. It then re-accelerates and finds the sonic point in the hot rising part of the helium opacity bump (He-bump) when $\log(T)\approx 4.6-5$. This kind of extended solution implies the formation of a density inversion following the decrease in flow velocity. This solution approaches and exceeds $\Gamma=1$ in the envelope (in correspondence with the positive density gradient), but shows an outflow that does not exceed the local sound speed before the opacity peak of the Fe-bump (${\log(T)}\approx 5.2$). For the 15model, however, even in these more extended solutions, the sonic point radius does not increase by more than 10% compared to the more compact solutions.
For comparison, a stellar model with plane parallel grey atmosphere boundary conditions and a mass-loss rate of $10^{-5}\Msun\,{\rm yr}^{-1}$ is plotted in Fig.\[env\], which shows an outflow accelerating to supersonic velocities near the Fe-bump, but then decelerating to give rise to a density inversion below the peak temperature of the iron opacity bump. This model also shows that the subsonic structure of the model computed with plane parallel grey atmosphere boundary conditions and the one with sonic point boundary conditions do not differ. This is the case even if the first has layers above the sonic point while the second has no information concerning the conditions above the sonic point other than the prescribed mass-loss rate.
Figure \[den\] shows the density inversion in the plane parallel and He-bump solutions, with a peak density one order of magnitude higher than the underlying layers. The inflection point in the density profile (i.e. $\mathrm{d}^2\rho/\mathrm{d}r^2=0$) is also visible at $R\approx 1.13 \Rsun$, indicating the location where inflation begins (see Sect.\[Sect.optthickwinds\]). This radius is defined as the core radius. For the compact, Fe-bump solutions, the larger the radius, the lower the density of the sonic point, and the larger the density scale height. Figure \[den\] also shows the steep increase in opacity due to the recombination of iron encountered at the base of envelope. This Fe-bump increases the opacity with respect to the electron scattering opacity by almost a factor of 3, leading to the appearance of the inflection point and providing the momentum to accelerate the flows up to the sonic point from temperatures as high as a few hundred thousand kelvins.
As a consequence of the need for a $\mathrm{d}\kappa/\mathrm{d}r>0$ [@2002Nugis], for the 15helium star models we find two types of configurations depending on the applied mass-loss rate:
- hot, Fe-bump stellar models having corresponding sonic temperatures which are higher than $\log(T_\mathrm{ S}) \gtrsim 5.2$ and radii similar to the core-size;
- cooler, slightly more extended He-bump models with sonic points in the range $\log(T_\mathrm{ S})\approx 4.6-4.8$.
Similarly, in Fig.\[env10\] we show how a 10helium star model readjusts to the different mass-loss rates. As in Fig.\[env\], higher mass-loss rates lead to more compact solutions, with steeper velocity profiles. Differently from the 15model, the more compact and less luminous 10helium star models find the sonic point in the helium opacity bump already starting from $ \Mdot = 1\times10^{-5}\Msun\,{\rm yr}^{-1}$ (see Table \[tab\]). This is due to the lower luminosities and higher densities in the envelopes of these models.
The opposite is true for the 20model in Fig.\[env20\], for which the solutions reach the sonic point in the hot part of the Fe-bump at $\approx 1.4 R_{\odot}$ for all the applied mass-loss rates. In this case the high luminosities (and low densities in the envelope) of the models allow a transonic flow, even for relatively low mass-loss rates. Figure \[env20\] also shows how the model with a plane parallel grey atmosphere behaves in the supersonic part of the outflow (for $\rm \Mdot = 8\times10^{-6}\Msun\,{\rm yr}^{-1}$), which rapidly reaches flow velocities of more than 100 km/s, to then slow down again following the density inversion due to the decrease in opacity between the Fe- and He-bump.
Sonic point conditions {#sonic-point-conditions .unnumbered}
----------------------
In Table \[tab\], the physical quantities computed at the sonic point are reported for our stellar models. The mean free paths estimated at the sonic point are small, of the order of 0.01–1% of the sonic radius and an order of magnitude smaller than the pressure scale height at the sonic point. They tend to increase for the lower mass loss by stellar wind applied and for the He-bump solutions, mostly due to the lower sonic point densities. From Appendix \[SH\] and the analysis concerning the radiation field at the sonic point, mean free paths smaller than the scale at which the thermodynamic quantities change suggest that local conditions are not far from LTE.
For the diffusive approximation to be valid, the mean free path should also be considered together with the optical depth of the sonic point. We estimate the optical depth of the wind via the integral $$\tau_\mathrm{ S}=\int_{R_\mathrm{ S}}^{\infty} \mathrm{d}r \,\, \kappa(\rho,T) \,\rho$$ from the sonic point radius $\rm R_\mathrm{ S}$ to infinity. In the supersonic region $\rho$ is computed via the continuity equation (Eq.\[boundmdot\]) while adopting a beta-velocity law [@1989Langer Eq.11] connected continuously on top of the stellar model, having the exponent unity and a typical velocity at infinity of 1600 km/s. The opacity $\kappa$ is instead computed via the OPAL opacity tables throughout the wind. To evaluate the opacity, the OPAL tables need as input not only the local density, but also the temperature, approximated in this case via the differential form of the $T-\tau$ relation from @2002Nugis with the sonic point temperature as reference temperature. These values, which should be considered only a rough estimate of the actual optical depth, are shown in Table \[tab\] and indicate that the optical depth of the sonic point is of the order of 10 in case of mass-loss rates greater than $\log(\Mdot)\gtrsim -5$. Instead, for $\log(\Mdot)\lesssim -5$ the optical depth becomes close to unity and the mean free path at the sonic point becomes quite large, indicating that the assumption of diffusive energy transport and LTE might not be valid for these mass-loss rates.
Another key difference between the He-bump and the Fe-bump models can be seen considering the optical depth parameter [@1975CAK; @1978Mihalas; @1999LamersCassinelli] $$t = \frac{\varv_\mathrm{ th}}{\lambda}/\frac{\mathrm{d}\varv}{\mathrm{d}r}
,$$ with $\varv_\mathrm{ th}$ being the thermal velocity of the particles [@1999LamersCassinelli]. The optical depth parameter tends to be of the order of 10 in the hot Fe-bump solutions, while it drops to less than 1 in the He-bump case. Considering that the optical depth parameter indicates how well the velocity independent Rosseland mean opacity can reproduce the flux-weighted opacity also in the presence of a velocity gradient, low values of $t$ point to the inadequacy of the use of the OPAL opacity tables in this context. For low $t$, the radiative acceleration due to the progressive line-deshadowing associated with the velocity gradient could instead already be important for launching these winds.
A key aspect that emerges from Table \[tab\] is that the sonic point coincides to a very good approximation with an Eddington factor of unity. In general the Eddington limit can be rewritten in the form of an Eddington opacity $\kappa_\mathrm{ EDD}$, $$\label{kedd}
\kappa_\mathrm{ EDD}:= 4 \pi c G M / L \quad,$$ only proportional to the mass-to-luminosity ratio; this is the opacity necessary for a star in hydrostatic equilibrium to reach the Eddington limit.
Figure \[tro\] shows the Rosseland opacity from the OPAL tables [@1996Iglesias] as a function of temperature and density, together with the structure of the outer layers of the stellar models shown in Fig.\[env\]. Associating an Eddington opacity with the opacity contours in Fig.\[env\], we can define the contour of $\Gamma=1$ relative to a given $L/M$. This is thus the contour where we expect to find the sonic points of our stellar models. In Fig.\[tro\] the Eddington opacities relative to our 10, 15, and 20models are plotted. They follow the increase in opacity around $\log(T)\approx 5.2$, i.e. the Fe-bump, and at $\log(T)\approx 4.6$, i.e. the HeII opacity bump. The more compact stellar models in Figs. \[env\], \[env10\], and \[env20\] have their sonic points located almost exactly on their respective opacity contours and in the hot part of the iron opacity bump, i.e. $5.2<\log(T)<5.5$. The two more extended models showing a density inversion, i.e. the He-bump sonic point model and the plane parallel grey atmosphere model, have their outer layers follow closely the contour of $\Gamma \approx 1$ for $ \log(T) \gtrsim 5.2$ (see Sect.\[Sect.optthickwinds\]), to then cross this limit within their density inversions in the proximity of the peak temperature of the Fe-bump. The plane parallel model finds its surface back into the sub-Eddington region, while the He-bump models continues to cooler temperatures to then find its sonic point close to its Eddington opacity contour at $ \log(T) \approx 4.7$. The grey shaded regions in Fig.\[tro\] mark instead combinations of parameters for which we do not expect to find the sonic point of radiation-driven stellar winds, due to the decrease in opacity as a function of temperature.
Mass-loss rate constraints {#sect.masslosspred}
==========================
We can make use of Eq. \[tsonic\], Eq. \[sonicpoint\], and Eq. \[kedd\] to obtain the stellar wind mass-loss rates of WR stars as a function of their sonic point temperatures and luminosity-to-mass ratios. This can be done via the opacity tables, assuming as before that the value of $\Gamma$ is exactly one at the sonic point (see Fig.\[tro\] and Table \[tab\]). From this assumption it follows that the sonic point opacity is equal to the Eddington opacity, which is only proportional to the ratio $M/L$ (Eq.\[kedd\]). Manipulating the OPAL tables we can unequivocally associate a sonic point density $\rho_\mathrm{ S}$ with a sonic point temperature and a given Eddington opacity (as in Fig.\[tro\]). Combining the sonic point density and the isothermal sound speed (which is a function of the temperature only) the mass flux at the sonic point can be obtained. Once the mass flux is defined, a radius still needs to be specified to obtain a mass-loss rate (Eq.\[boundmdot\]). In Fig.\[mdot\] we assume a typical sonic point radius of 1 $R_{\odot}$ for all luminosity-to-mass ratios, obtaining a Sonic HR diagram[^2] from which the mass-loss rate can be derived as a function of the sonic point temperature and the luminosity-to-mass ratio.
Two assumptions are used to construct this diagram: the validity of the velocity independent OPAL opacities and the adoption of a specific radius to derive mass-loss rates from mass-fluxes. However, as previously discussed, the sonic point Rosseland opacities are not expected to differ significantly due to the effects of a velocity gradient, at least in the case of the Fe-bump-driven flows. Concerning the adopted radius, as shown in Sect.\[sectrisults\], solutions with the sonic point in the Fe-bump are compact, due to the lack of a strongly inflated envelope. The core and sonic point radii in the Fe-bump tend to be very similar and are not expected to differ by more than a factor of 2 from 1 $\Rsun$, at least not in the mass range of most H-free WR stars, i.e. 10–20 $\Msun$[^3].
In Fig.\[mdot\] it can be seen how the mass-loss rate contours follow the opacity profile of the iron opacity bump, as in Fig. \[tro\]. Above $\log(T_\mathrm{ S})\approx5.2$, the higher the sonic point temperature, the higher the expected mass-loss rate at fixed $L/M$ (see Fig.\[env\]). Fixing instead the sonic point temperature, the higher the $L/M$, the lower the mass-loss rate, which indicates that as the $L/M$ ratio increases, an Eddington factor of unity can be more easily reached, and thus flows can be accelerated to supersonic velocities already for lower mass-loss rates. Equivalently, given $\Mdot$ (thus a contour in the diagram), the flow can be accelerated until it reaches the Eddington limit, and therefore transonic velocities, already from higher temperatures for higher $L/M$ ratios.
In contrast, below $\log(T_\mathrm{ S})\approx5.2$ the predictions tend to be much more uncertain, mostly because sonic point temperatures lower than the iron bump peak imply that the stellar wind is not accelerated to supersonic velocities by the iron opacity bump, but rather the opacity bump leads to a low-density inflated envelope configuration with a density inversion. In this configuration the assumption of a typical radius starts to break down together with the assumption of LTE in the subsonic part of the outflow (see Table \[tab\]).
Similarly, we can derive the $L/M$ ratio as a function of the sonic point temperature and the mass-loss rate. This is done in Fig.\[lmts\], where (thanks again to Eq.\[kedd\]) from the sonic point temperature and $\log(\Mdot$) we can obtain $\kappa_\mathrm{ EDD}$ and thus the $L/M$ ratio. For $\log(T_\mathrm{ S})\gtrsim 5.2$, higher sonic point temperatures lead to higher mass-loss rates at fixed $L/M$, or at fixed $\Mdot$ a higher luminosity-to-mass ratio requires a higher sonic point temperature. The same trend can also be seen in the temperature range of the raising helium opacity bump, i.e. $4.6 \lesssim \log(T_\mathrm{ S})\lesssim 5.0$, but with $L/M$ ratios which are lower at constant $\Mdot$ compared to the temperatures in the hot part of the iron opacity bump. This can be interpreted as stars needing to have higher $L/M$ in order to drive a certain $\Mdot$ while having the sonic point located in the iron bump compared to stars having their winds driven by the helium opacity bump[^4]. Moreover, in the temperature range of the helium opacity bump, flows can be accelerated to supersonic velocities already in stars with significantly lower $L/M$ than the values necessary in the iron opacity bump.
Comparison to observations {#comparison-to-observations .unnumbered}
--------------------------
Figure \[mdot\] shows that a low-mass, low-luminosity stellar model does not have an outflow that reaches transonic velocities within the iron opacity bump unless a certain minimum mass-loss rate is applied. Lower mass-loss rates lead instead to inflated envelope configurations such as the low mass-loss rate cases in Fig.\[env\] and Fig.\[env10\].
A minimum mass-loss rate as a function of the luminosity-to-mass ratio can therefore be derived, and indicates which mass-loss rate implies a radiation-driven supersonic flow starting from the iron opacity bump (minimum $\Mdot_\mathrm{ Fe}$). This minimum mass-loss rate marks the separation between compact and extended solutions in Fig.\[env\] and \[env10\]. It is shown in Fig.\[minmass\], where the derived minimum $L/M$ above which the flow becomes transonic in the hot part of the iron opacity bump is plotted as a function of the mass-loss rate. The minimum $\Mdot_\mathrm{ Fe}$ decreases at higher luminosities, and it can be closely approximated by a parabola $$\label{eqminmasslossrate}
\log\left(\frac{L}{M}\right) = 1.69 - 0.80\, \log\left(\Mdot\right) - 0.06\, \log\left(\Mdot\right)^2 \quad ,$$ or equivalently in terms of luminosity $$\label{eqminmasslossL}
\log\left(L\right) = 0.44 - 1.35\, \log\left(\Mdot\right) - 0.097\, \log\left(\Mdot\right)^2 \quad .$$
In Fig.\[minmass\] this minimum mass-loss rate as a function of $L/M$ is compared to the sample of observed Galactic WNE analysed by @2006Hamann. We collected $L/M$ and $\Mdot$ exclusively for the H-free WNE stars, as even a small mass fraction of hydrogen can significantly affect the structure of the envelopes [@2017Schootemeijer]. The vast majority of the observed stars exceed the minimum mass-loss rate. They are therefore consistent with models having outflows that are radiation pressure driven to supersonic velocities by the iron bump. Assuming the mass-luminosity relation from @1989Langer and, as before, a typical radius of 1, we can construct another Sonic HR diagram relating luminosity and sonic point temperature. This is done in Fig.\[mdotL\], where regions are colour-coded according to the expected mass-loss rate. In this diagram we locate the observed WNE stars from Fig.\[minmass\]. Their location clusters at sonic point temperatures around $\log(T_\mathrm{ S})\approx 5.3$. Therefore, based solely on their observed luminosities and mass-loss rates, all the WNE stars of this sample can be understood as having their outflows accelerated to supersonic velocities by the radiation pressure consequence of the high number of transitions associated with iron and iron-group elements at around $\log(T_\mathrm{ S})\approx 5.2$. The sonic point temperature as a function of luminosity for the best lintear fit of the observed WNE stars in Fig.\[mdotL\] follows the relation $$\label{eqfitLTsonic}
\log(L)= 3.18\,\log(T_\mathrm{ S}) -11.36 \quad ,$$ suggesting that, in general, WNE stars with higher luminosities have higher sonic point temperatures. A similar relation as in Eq.\[eqfitLTsonic\] can be derived also in terms of mass-loss rate and sonic point temperature, $$\label{eqfitMDOTTsonic}
\log(\Mdot) = 5.66 \,\log(T_\mathrm{ S}) - 34.65 \quad .$$ This time Eq.\[eqfitMDOTTsonic\] suggests that higher mass-loss rates lead to higher sonic point temperatures, or in other words that stars with high mass-loss rates have their sonic point located deeper inside their atmospheres (see also Table \[tab\]).
Discussion
==========
WR radius problem
-----------------
For typical Galactic WNE stars the continuum originates in the dense outflow at a significant fraction of the terminal velocities, i.e. well beyond the hydrostatic domain [@2008Crowther]. Therefore, the velocity law at the base of the wind is not well constrained by observations [@1992Schmutz; @2015Hillier].
As shown in Sect. \[sectrisults\], the subsonic structure of helium star models suggests that the location of the sonic point of WNE stars is in the range 0.9–1.5$\Rsun$, with supersonic flows originating from temperatures around 200kK (see Fig.\[mdotL\]). For the supersonic layers, @2017Sander has recently outlined how for hydrodynamically consistent models of hot, radiation-driven stellar winds the velocity profile is shaped by the different ionization levels of the elements present in the wind. This suggests that the adoption of a single beta-velocity law for the supersonic wind, independently of the detailed temperature stratification, available opacities, and subsonic structure is probably an oversimplification [see also @2005Grafener]. Hydrodynamic models for WR winds might bring the hydrostatic radii in the range predicted by our stellar structure calculation [see e.g. Fig.2 in @2015Sander]. A non-constant velocity gradient was found in the supersonic part of the flow of WR111, a WC5 star, for which a hydrodynamic model for the optically thick wind was available [@2005Grafener]. We were able to continuously connect the subsonic structure of one of our sonic point boundary conditions models and the supersonic wind model at the sonic point (see Appendix \[AWR111\]).
Still, the apparent homogeneity in the location of the observed WNE stars in the Sonic HR diagram from Fig.\[mdotL\] does not yet explain the scattered distribution found in the classical HR diagram [cf. Fig.8 in @2006Hamann]. From @1989Langer, at a given metallicity the properties of H-free post-main sequence stars are expected to be mostly defined by one single parameter, the luminosity of the star. Instead WNE stars with comparable luminosities are found to have different spectroscopic subclasses. In this respect in Fig.\[mdotL\] we note how the predicted mass-loss rate depends sensitively on the sonic point temperature. As such, small differences in the metallicity or in the subsonic force balance (e.g. due to rotation or magnetic fields), or the exact evolutionary phase and chemical composition of the core, might result in significantly different mass-loss rates, which consequently could lead to differences in the photospheric stellar parameters, which might reconcile observations with theoretical expectations.
A decelerating flow above the sonic point could possibly lead to configurations with multiple critical points and the appearance of a density inversion similar to that observed in inflated envelopes. The presence in massive main sequence stars of an increase in density shortly after the peak temperature of the iron opacity bump has recently been confirmed via 3D simulations by @2015Jiang, implying that the density inversion is not necessarily erased, especially not by subsonic flows. The stagnation could then possibly imply the appearance of instabilities such as photon bubbles [@2003Dessart; @2003Blaes; @2015Owocki], line-deshadowing instabilities [@1984Owocki; @2013SundqvistLDI], or stream-like lateral flows possibly connected with the co-rotating interaction regions [@1984Mullan; @2009StLouis]. Hydrodynamic and time-dependent multi-dimensional simulations are necessary to assess the dynamical stability of such configurations and might shed light on what ultimately sets the mass-loss rate of these objects.
Optical depth of the sonic point
--------------------------------
Integrating the diffusive temperature gradient (Eq. \[dTdr\]) from the sonic point to infinity $$\int_\infty^{r_\mathrm{ S}} \kappa \rho\,\mathrm{d}r = \int_{T_\infty}^{T_\mathrm{ S}}\frac{4 ac}{3 F}\,\mathrm{d}T^4
,$$ with $a$ the radiation density constant, and assuming that the sonic point temperature $T_\mathrm{ S}^4 \gg T_\mathrm{ eff}^4 > T_\infty^4$, valid in the case of optically thick winds, leads to a tautological relation between optical depth and sonic point temperature $$\tau_\mathrm{ S} \approx \frac{a c}{ 3 F} T_\mathrm{ S}^4 \quad .$$ This, combined with Eq. \[tsonic\], sets a relation which needs to be fulfilled at the sonic point for a given mass-loss rate, i.e. $$\label{taumdot}
\left(\frac{ 3 F}{a c}\tau_\mathrm{ S}\right)^{1/4} \approx \frac{\mu m_\mathrm{ H}}{k_\mathrm{ B}}\left(\frac{\dot{M}}{4 \pi r^2 \rho}\right)^2$$ The mass-loss rate is a free parameter in our hydrodynamic stellar models. Equation \[taumdot\] can then be used to constrain the only mass-loss rate which is consistent with the expected optical depth if combined with a prescribed velocity law for the wind [as in @2017Grafener], or assuming an optical depth for the sonic point.
Turbulence
----------
Atmosphere calculations of hot stellar winds often include an extra turbulent broadening of the lines, associated with the turbulent motion in the atmosphere [e.g. @2005Dessart; @2017Sander]. This turbulent velocity contributes to the equation of state of stellar matter and, via its gradients, to the structure of the outer layers[^5]. From the stellar structure calculations point of view, the structural effects of the inclusion of the turbulent pressure terms in the convective zones are marginal [@2015GrassitelliA]. Moreover, subsurface convection in the subsonic layers might be inhibited in helium star models [@2016Ro]. Therefore, we neglect the turbulent terms from our analysis, expecting at most a difference of a fraction of the gas pressure in the equation of state [@2015GrassitelliA; @2016Grassitelli]. If present, this turbulent motion might alter our results, and introduce a systematic difference between our analysis and stellar wind models.
Previous works
--------------
Compact stellar structure models were also favoured by wind models from @2016Ro. Considering the sonic point at the Fe-bump as a boundary condition for their wind models, these authors show that while using velocity independent Rosseland opacities, winds fail to accelerate up to the terminal wind velocities. However, the dynamics in the supersonic part of the flow show the presence of more compact, strong wind solutions and extended, weak stagnating solutions. The bifurcation between these solutions that takes place depends on the ratio between the temperature scale height (computed in the diffusive limit within the supersonic part of the wind) and the local radius. In particular, their more compact wind solutions are expected to be more prominently affected by line-force amplification due to Doppler enhancement from flow velocities of the order 150–200 km/s, suggesting that the Rosseland mean opacities become rapidly inadequate above the sonic point. The question is whether such an increase in opacity is sufficient to accelerate the outflow monotonically up to the escape speed of the star, or whether stagnation and a complex velocity profile might appear.
After this manuscript was submitted, @2018Nakauchi constructed hydrostatic He-star models connected to a single beta-velocity law, optically thick stellar wind models. They confirm the crucial role played by the iron opacity bump in launching the winds of WNE stars, in agreement with our fully hydrodynamic models. Less satisfactory is the comparison between the observed photospheric temperature and the temperature values estimated by their wind models, further pointing towards the inadequacy of a prescribed velocity law with a single beta exponent in reproducing the initial acceleration of these stellar winds. Similarly, @2017Grafener made use of the proximity to $\Gamma \approx 1$ at the sonic point (see Sect.\[Sect.optthickwinds\]) while adopting a prescribed stellar wind dynamics to investigate WR mass-loss rates and properties. Their $\Mdot-L$ trend at Galactic metallicity resembles the observed one, favouring therefore more compact solutions with outflows driven by the iron opacity bump. Both @2018Nakauchi and @2017Grafener make use of a prescribed single beta-velocity law for the supersonic layers to constrain the mass-loss rate of WR stars with optically thick winds, while $\dot{M}$ is a free parameter in our hydrodynamic models.
In the supersonic part of WR outflows, the presence of instabilities and inhomogeneities should be taken into account. Invoking density inhomogeneities, or ‘clumping’, in stellar models has an influence on the mean opacity, due to the enhanced density in clumps [@1988Moffat; @1998Hamann]. However, a porous structure could also imply lowered mean opacity, counteracting the effect of small-scale clumping [@1998Shaviv; @2007Oskinova]. Assuming that the material near the Fe-opacity peak is clumped, @2012Grafener were able to extend the surface radii of their hydrostatic helium star models, in practical terms, by significantly enhancing the iron bump opacity (see also Appendix \[app.inflation\]). The surface temperatures of such hydrostatic, strongly inflated models computed with plane parallel grey atmospheres are then compared with the fictitious effective temperatures at $\tau=20$ of the atmosphere calculations performed by @2006Hamann. This effective temperature at $\tau=20$ should not be confused with the actual blanketed temperature at the base of WNE wind models, which is about a factor of 2 larger (see e.g. Appendix \[AWR111\]). As we show in Sect.\[sectrisults\], strongly inflated hydrodynamic solutions imply supersonic flows already at the base of the inflated envelopes for the typical mass-loss rates of WNE stars (cf. the plane parallel atmosphere model in Fig.\[env20\]). Moreover, it would be inconsistent to think that the iron opacity bump is simultaneously responsible for both the inflation of the envelope and the acceleration of the flow. While the detailed effects of clumping and porosity remain a subject for future research, we concentrated in this work on the homogeneous case as those instabilities are expected to initiate above the sonic point [@2013Sundqvist; @2015Owocki]. If this is not the case and clumping is already present at the relatively high densities and optical depth of the sonic point of massive helium star models, it might affect to some extent the local opacity.
Standing waves in helium stars [@1993Glatzel] are also not expected. As already pointed out by @2016GrassitelliWR, while investigating pulsations in massive helium star models, mass loss has an inhibitory effect on pulsations already in the presence of inflated envelopes.
Conclusions
===========
We investigated the conditions at the sonic point and the subsonic structure of hydrodynamic models for chemically homogeneous massive helium stars, thought to be representative of WR stars in the WNE phase. For typical mass-loss rates we find that the outflows of our stellar models, computed with newly implemented boundary conditions at the sonic point, are accelerated to transonic velocities by the momentum provided by the increase in opacity associated with the recombination of the iron and iron-group elements, the iron opacity bump, in the temperature range 160–220kK or by the helium opacity bump in the range 40–60kK.
The Eddington factor at the sonic point of our models is very close to unity. This allows us to build a Sonic HR diagram relating the luminosity-to-mass ratio of stars with optically thick winds to their sonic point temperature. Knowing the sonic point temperature and relying on available opacity tables, it is possible to predict the mass-loss rates of stars with optically thick winds from this diagram. We use this to derive a simple relation for the minimum mass-loss rate for WNE stars necessary to have the outflow accelerated to supersonic velocities by the iron opacity bump only as a function of the luminosity-to-mass ratio. The vast majority of the observed Galactic WNE stars exceed this minimum mass-loss rate, suggesting therefore that the radiative acceleration in the hot part of the iron opacity bump is responsible for the launch of the outflows of these stars to supersonic velocities. This leads to sonic point radii of the order of one solar radius for WNE stars.
In this paper we shed new light on the WR radius problem and show that the iron opacity bump is key to driving outflows through the sonic point in WNE stars, and hence to determining the mass-loss rate of the star. Our models may provide a useful reference while adopting the Sonic HR diagram to compare stellar structure and atmosphere models of optically thick outflows with observations, and may provide solid inner boundary conditions for stellar atmosphere codes, reducing the complexity of the calculations with simple physical arguments and providing ab initio subsonic structure models.
We have also derived analytic relations to better understand the mechanisms at work when extended low-density envelopes appear in the stellar models, characterized by the proximity to the Eddington limit, the so-called envelope inflation. In the envelope of a stellar model in hydrostatic equilibrium the emergence of a large gas pressure scale height is the response to the rapid increase in radiative opacity when approaching an opacity bump in radiation-pressure-dominated layers close to the Eddington limit.
L.G. thanks Andreas Sander, Stan Owocki, Alina Istrate, Luca Fossati, and Jean-Claude Passy for the constructive discussions. J.M. acknowledges funding from a Royal Society–Science Foundation Ireland University Research Fellowship.
Envelope inflation criterion {#app.inflation}
============================
Here we attempt to characterize the structure of inflated envelopes. With BEC we compute three versions of the same 15$\Msun$ helium star model, but adopting three different OPAL opacity tables corresponding to the opacity of stellar matter at three different metallicities. For illustrative purposes, we impose hydrostatic equilibrium and no mass loss by stellar wind. For this computation we also recover the plane parallel grey atmosphere boundary conditions [@1998Heger; @2006Yoon]. The advantage of studying a helium zero age main sequence massive helium star model is that such a model is hot enough that only the iron opacity bump is present below its surface. This allows us to more clearly investigate the response of the outer layers subject only to the radiative force arising by this opacity bump.
The density profiles of these newly computed stellar models are shown in Fig.\[infpro\] where the blue model is computed with the opacity table for a metallicity Z=0, while the orange and the black models are computed for Z=0.02 and Z=0.05, respectively. The Z=0 blue model clearly shows a density profile that monotonically decreases as a function of radius, showing moreover a density scale height $H_\rho$ increasingly smaller as one approaches the surface. This model shows how the density profile behaves in absence of a pronounced increase in opacity in the outer subsurface layers, showing no sign of envelope inflation and no core-halo structure below its surface. It is considered the reference model.
This is not the case for the helium star models with opacities corresponding to the metallicity Z=0.02 and Z=0.05. Their density profiles in Fig.\[infpro\] show the characteristic envelope inflation configuration encountered by e.g. @2012Grafener, @2015Sanyal, and @2016GrassitelliWR. In these models the higher iron bump opacity brings the outer layers to the Eddington limit; this leads to the formation of an extended, low-density envelope which, for the Z=0.02, increases the surface radius by approximately 10%, while for the Z=0.05 case, the increase in radius is more than a factor of 2 compared to the non-inflated case. Consequently these two models also find their surfaces at lower effective temperatures, $T_\mathrm{eff}\approx{}$130kK, 120kK, 80kK for the metallicities $Z={0.00, 0.02, 0.05}$, respectively. At first the mildly and the more inflated models in Fig.\[infpro\] both show density profiles with density scale heights decreasing in the core region. However, around 1$\Rsun$, the two inflated models start to differ from the non-inflated model. Their density scale heights start to increase due to the increase in the outward-directed radiative force in the proximity of the iron opacity bump. This can be understood by writing Eq.\[densder\] in the general hydrostatic case ($\varv=0$) $$\label{densderhydrostatic}
c^2_\mathrm{ s}\frac{\mathrm{d}\,\ln(\rho)}{\mathrm{d}r}= -g + g_\mathrm{ rad} -c^2_\mathrm{ s}\frac{\mathrm{d}\,\ln(T)}{\mathrm{d}r} \quad .$$ In the force balance an increase of $g_\mathrm{ rad}$ leads to a larger density scale height, and thus the high-opacity models have more shallow density gradients. As the opacity keeps rising in the hot part of the iron opacity bump, the two inflated models flatten their density profiles and reach a minimum, followed by a rapid rise in density. From Eq.\[densderhydrostatic\] (or Eq.\[densder\] in the case of $\varv^2 \ll g r $ ), a positive density gradient (i.e. a density inversion) appears when $$\label{gammabeta}
\Gamma \gtrsim \frac{1-\beta}{1-\frac{3}{4}\beta}$$ or equivalently $$\frac{P_\mathrm{ gas}}{P_\mathrm{ rad}} \gtrsim 4 \frac{1-\Gamma}{\Gamma} \quad .$$ Equaling the sides in the inequality \[gammabeta\] gives $$\label{betagamma}
\beta \simeq \frac{1-\Gamma}{1-\frac{3}{4}\Gamma} \quad ,$$ which defines the condition when either $\frac{\mathrm{d}\rho}{\mathrm{d}r}=0$ (and the flow stays subsonic) or, in the hydrodynamic case, the sonic point $\beta$ of transonic outflows (see Sect.\[Sect.optthickwinds\]). Moreover, the inequality \[gammabeta\] can be compared with the criterion for convection [@1997Langer], $$\label{convectioncriterion}
\Gamma \geq (1-\beta)\frac{32-24\beta}{32-24\beta-3\beta^2}
,$$ showing that a density inversion is necessarily convective [see also @1973Joss].
Equation \[betagamma\] shows that a null density gradient can appear only for specific sets of $\beta$ and $\Gamma$, with lower $\beta$ (and thus densities) for higher $\Gamma$, as we can see in Fig.\[infpro\]. A modified Eddington factor could then be introduced, namely $$\label{gammagrassitelli}
\Gamma_\mathrm{ S}=\frac{\kappa L}{4 \pi c G M}\left(1+\frac{P_\mathrm{ gas}}{4 P_\mathrm{ rad}}\right)
,$$ which includes the outward-directed force associated with the gas temperature gradient, and which defines (when equal to unity) either the beginning of a density inversion or the condition at the sonic point of stars for which convection is inefficient (see Sect.\[Sect.optthickwinds\] and Eq.\[gammasonic\]). This equation also shows that the local Eddington factor cannot be exactly unity either at the sonic point or when $\frac{\mathrm{d}\rho}{\mathrm{d}r}=0$. In either case, these two quantities tend to differ by only few percent in the radiation-pressure-dominated outer layers of massive stars, which is in fact equivalent to the approximation $\frac{\mathrm{d}c_\mathrm{ s}^2}{\mathrm{d}r} - 2 \frac{c_\mathrm{ s}^2}{r} \ll g$ made in Sect.\[Sect.optthickwinds\] and below.
From Fig.\[infpro\] it is evident that the inflated stellar models differ markedly from the non-inflated model when the profiles show an inflection point, i.e. a minimum in $\mathrm{d}\,\ln(P_\mathrm{ gas})/\mathrm{d}r$, or almost equivalently $\mathrm{d}\,\ln(\rho)/\mathrm{d}r$. An increase in the density scale height follows the inflection point, contrary to the monotonically decreasing density scale height of stellar models which do not inflate. This is due to an increasing contribution from $g_\mathrm{ rad}$ in the momentum equation, affecting the balance between the gravitational force and the gas pressure gradient that defines the condition of hydrostatic equilibrium. Analytically, the second derivative of the gas pressure profile writes $$\label{seconder}
\begin{aligned}
\frac{\mathrm{d}^2\ln(P_\mathrm{ gas})}{\mathrm{d}r^2}={}&\frac{\mathrm{d}^2\ln(\rho)}{\mathrm{d}r^2}+\frac{\mathrm{d}^2\ln(T)}{\mathrm{d}r^2} \\ {}&= \frac{(g-g_\mathrm{ rad})}{c_\mathrm{ s}^2}\frac{\mathrm{d}\,\ln(T)}{\mathrm{d}r} +\frac{g_\mathrm{ rad}}{c_\mathrm{ s}^2} \frac{\mathrm{d}\,\ln(\kappa)}{\mathrm{d}r}
\end{aligned}
,$$ where for simplicity we have neglected the geometrical term as $2/r \ll \mathrm{d}{\rm ln}(T)/\mathrm{d}r$ (which might not be always possible for main sequence stars). In writing Eq.\[seconder\] we have assumed a fixed chemical composition and no energy source or sink in the outer layers (strictly true for hydrostatic models in thermal equilibrium, and true to a high degree for steady-state hydrodynamic models).
In Eq.\[seconder\] the first term on the right-hand side is always negative in sub-Eddington regions. Instead, the second term on the right-hand side depends on both the radiative acceleration and the opacity profile, and it is positive when the opacity increases with increasing radius. This is the term that becomes dominant at and above the inflection point. For a constant opacity, an inflection point would be present only when $\Gamma = 1$. However, in the case of more complex, non-constant opacity profiles, a rapid increase in opacity at temperatures where the recombination of some elements takes place can lead to an increase in the gas pressure scale height, and thus density scale height, as the second derivative of gas pressure changes sign.
From Eq.\[seconder\] a criterion for the appearance of the inflection point, i.e. for the beginning of an envelope inflation can be derived: $$\label{critinf}
\frac{\mathrm{d}\,\rm{ln}(\kappa)}{\mathrm{d}\rm{\, ln}(T)} \equiv \frac{4 \, \mathrm{d}\rm{\, ln}(\Gamma)}{\mathrm{d}\rm{\, ln}(P_\mathrm{ rad})} \lesssim 1 -\frac{1}{\Gamma}$$ We can now understand envelope inflation as [the appearance of low gas pressure gradients in response to a steep increase in the radiative opacity in radiation-pressure-dominated layers close to the Eddington limit]{}. Above the inflection point, the increase in opacity increases the gas pressure scale height $H_{P_\mathrm{ gas}}$ following the increase in $g_\mathrm{ rad}$ as $\Gamma \rightarrow 1$ [@2017Sanyal]: $$\label{gasscaleheight}
\frac{1}{H_{P_\mathrm{ gas}}} := \frac{\mathrm{d}\rm{ln}(P_\mathrm{ gas})}{\mathrm{d}r}=\frac{g-g_\mathrm{ rad}}{c_\mathrm{s}}= \frac{g}{c_\mathrm{s}} (1-\Gamma) .$$ From here it follows that as the gas pressure scale height increases, the location of the photosphere is found at larger radii. Analytical expressions for the latter in the case of envelope inflation originating from the iron bump in very massive helium stars and neglecting convection have been given in @2012Grafener. We note here that stellar models can be inflated, i.e. have extended low-density envelopes close to the Eddington limit, but do not show a density inversion [or a gas pressure inversion; see e.g. the 85$\Msun$ main sequence model in @2015Sanyal]. On the other hand, a density inversion appearing when $\Gamma\approx 1$ is always an indication of envelope inflation.
Effects of convection on inflated envelopes {#effects-of-convection-on-inflated-envelopes .unnumbered}
-------------------------------------------
The presence of convection can affect the structure of the inflated envelope [@2015Sanyal]. It effectively reduces the radial extent of the inflated layers by affecting the temperature stratification without contributing directly to the force balance. In convective layers, the ratio of the luminosity transported by radiation to the total luminosity can be expressed as $$\label{nablaL}
\frac{L_\mathrm{ rad}}{L}=\frac{\nabla}{\nabla_\mathrm{ rad}}
,$$ where $\nabla_\mathrm{ rad}$ is the temperature gradient required to transport all the luminosity by radiation, while $\nabla$ is the actual temperature gradient which includes the contribution from convection. In a convective layer, according to the Schwartzschild criterion, $\nabla_\mathrm{ rad}>\nabla$ [@1990Kippenhahn]. Together with Eq.\[nablaL\], the radiative acceleration from Eq.\[grad\] writes as $$g^{\rm conv}_\mathrm{ rad} = \frac{\kappa L }{4 \pi r^2 c}\frac{\nabla}{\nabla_\mathrm{ rad}} < g_\mathrm{ rad} \quad ,$$ where the inequality arises from $\nabla_\mathrm{ rad}>\nabla$. Therefore, convection effectively reduces the radiative acceleration by not contributing to the force balance or, equivalently, has the same effect of a reduction of the local opacity as $$\kappa_\mathrm{ conv}=\frac{\nabla}{\nabla_\mathrm{ rad}} \kappa \quad,$$ leading to small gas pressure scale height (Eq.\[gasscaleheight\]). The criterion in Eq.\[critinf\] is still valid, as far as the opacity gradient takes into account the effect of convection.
Radiation field at the sonic point {#SH}
==================================
In general, the temperature stratification in a stellar atmosphere is a global problem due to the intrinsic coupling of the radiation field to different regions of the atmosphere. However, we discuss here the limiting case in which the radiative energy transport turns from being a global problem, to a local one, and apply it to our analysis of the conditions at the sonic point of WNE stars. The intensity of radiation $I$ at the sonic point optical depth $\tau_\mathrm{ S}>1$ in the Rosseland approximation can be expressed in terms of a Taylor–McLaurin series around the equilibrium Planckian value $B(T)$ of the source function. The source function $S$ can be written as [@1978Mihalas; @1994Hansen] $$\label{sourcefunct}
S(t)=\sum\limits_{n=0}^\infty \frac{(t-\tau_\mathrm{ S})^n}{n!} \frac{\mathrm{d}^n B(\tau_\mathrm{ S})}{\mathrm{d}\tau^n} \quad ,$$ with $B(\tau_\mathrm{ S}):=B(T[\tau_\mathrm{ S}])$ the Planck function at the temperature of the sonic point and $t$ a dummy variable.\
Integration of the equation of radiative transfer $$\frac{\partial I e^{-\tau/\mu}}{\partial \tau}=-\frac{1}{\mu} S(t) e^{-\tau/\mu}$$ for the outgoing radiation $I^+$, i.e. from $\tau=\infty$ to $\tau=\tau_\mathrm{ S}$, leads to $$\label{radtransfequation+}
I^+(\tau_\mathrm{ S}, \mu\geq 0) = \sum\limits_{n=0}^\infty \mu^n \frac{\mathrm{d}^n B(\tau_\mathrm{ S})}{\mathrm{d}\tau^n}
,$$ with $\mu:=cos(\theta)$ indicating the direction of the beam. The expression for the incoming radiation $I^-$ from $\tau=0$ to $\tau=\tau_\mathrm{ S}$ writes instead $$\label{radtransfequation-}
I^-(\tau_\mathrm{ S}, \mu\leq 0) = \sum\limits_{n=0}^\infty \mu^n \frac{\mathrm{d}^n B(\tau_\mathrm{ S})}{\mathrm{d}\tau^n}\left( 1 - e^\frac{\tau_\mathrm{ S}}{\mu} \sum\limits_{k=0}^n \left(\frac{-\tau_\mathrm{ S}}{\mu}\right)^k\frac{1}{k!} \right) \quad.$$ The inward-directed radiation differs from the outward-directed radiation by terms of the order of $e^\frac{\tau}{\mu}$.
We can now more quantitatively refer to WNE stars. At first we can say that, assuming as reference a typical optical depth for the sonic point of the order of $\tau_\mathrm{ S} \approx 5$ [see e.g. Table \[tab\] and @2005Grafener], the exponential term in Eq. \[radtransfequation-\] contributes for less than 1$\%$. For low $n$ therefore, the term $e^\frac{\tau}{\mu} \sum\limits_{k=0}^n \left(\frac{-\tau}{\mu}\right)^k\frac{1}{k!}$ is negligible, as usually assumed in order to derive the diffusive approximation [@1978Mihalas]. Considering the momenta of the radiative energy transport equation, namely the energy density, the energy flux, and the radiation pressure, an approximation of the derivatives by appropriate differences shows that the ratio of successive terms in the series is of order O(1/$\tau_\mathrm{ S}^2$) (see @1978Mihalas). As such, we can study the convergence of the series via $$O\left(\frac{1}{\tau_\mathrm{ S}^2}\right)\approx O\left(\frac{\lambda^2}{H_\mathrm{ P}^2}\right) \quad ,$$ where we have taken the pressure scale height $H_\mathrm{ P}$ as the characteristic scale of the system. From Table \[tab\] the pressure scale heights are an order of magnitude larger than the mean free path at the sonic point, assuring the rapid convergence of the series to the Planckian value, with the high-order terms contributing less than 1$\%$. We can thus assert that the conditions at the sonic point of WNE stars, especially those with stellar winds driven by the iron opacity bump, closely approach the LTE conditions.
Consequently, the radiation field at the sonic point of these stars is close to being isotropic, with the radiative transfer losing the explicit dependence on the optical depth in Eq. \[radtransfequation-\]. We can further estimate the level of anisotropy of the radiation field at the sonic point in case of outflows driven by the iron opacity bump as [@1978Mihalas] $$\frac{\textrm{Anisotropic term}}{\textrm{Isotropic term}}\propto\left(\frac{T_\mathrm{ eff}|_\mathrm{ S}}{T_\mathrm{ S}}\right)^4\approx 5\%
,$$ with the $T_\mathrm{ eff}|_\mathrm{ S}$ fictitious effective temperature at the sonic point derived from the Stefan–Boltzmann law, having adopted a radius of 1$\Rsun$ and a typical luminosity of $10^5 \Lsun$. We can also estimate the proximity to the LTE conditions and the validity of the diffusive approximation by comparing the radiation timescale associated with the photon thermalization $$\tau_{t}=\frac{\lambda}{c}$$ to the other macroscopic timescales, i.e. the dynamical and thermal timescales of the envelopes [@1981Thorne; @1981ThorneB]. The timescale for photon thermalization, i.e. the time needed by the radiation field to approach its LTE value, is of the order of seconds, while the dynamical and thermal timescales were estimated to be of the order of hundreds of seconds for massive helium stars [@2016GrassitelliWR].
All these considerations lead us to conclude that the states of the gas and radiation are close to their LTE conditions. In LTE, it is assumed that the local T and $\rho$ are sufficient to determine the ionization state of the gas, all microscopic processes being close to detailed balance. This approximation is justified as far as the thermalization depth of radiation, i.e. the depth at which the source function approaches its equilibrium value, is smaller than the sonic point optical depth [@1978Mihalas; @1995Pistinner].
Nozzle analogy and critical point {#appendixsoniccritical}
=================================
The [critical solution]{} describing the kinematics of stellar winds can be understood in analogy with an ideal rocket (or de Laval) nozzle, i.e. a nozzle with walls that narrow at first, reach a throat, and then rapidly expand [@1980Abbott; @1995Bjorkman; @1999LamersCassinelli; @2007Shore]. Given a compressible steady-state flow of gas, the subsonic flow accelerates in the converging part of the nozzle due to mass conservation. If the initial conditions are such that the flow can reach supersonic velocities before passing the throat, the sonic point is always located exactly at the throat of the nozzle. This system, common to almost all modern rockets [@1989Raga], shows how a smooth steady-state flow can self-adjust such that a transonic flow can be established via information that travels at the speed of sound. Once the transonic flow is established, the conditions in the diverging supersonic part define the local kinematics of the flow, such as its velocity and temperature, but they can influence neither the subsonic part of the flow nor the mass-flow rate [@1999LamersCassinelli; @2007Shore; @2014Maciel].
The analogy with a stellar wind lies in the form of the momentum equation of a de Laval nozzle, that is [@1999LamersCassinelli] $$\varv^2 \frac{\mathrm{d}{\rm ln} \varv}{\mathrm{d}x} = -c_\mathrm{ s}^2\frac{\mathrm{d}{\rm ln} \rho}{\mathrm{d}x} -\frac{\mathrm{d} c_\mathrm{ s}^2}{\mathrm{d}x}$$ for an ideal gas flow, with $x$ spatial coordinate. Combining this with the continuity equation $\mathrm{d}{\rm ln}(\varv) + \mathrm{d}{\rm ln} (\rho) + \mathrm{d}{\rm ln} (S)=0$, where S corresponds to the area of a section of the nozzle, the force balance is given by $$\label{nozzleeq}
\left(\varv^2 - c_\mathrm{ s}^2\right)\frac{\mathrm{d}{\rm ln} \varv}{\mathrm{d}x} = -c_\mathrm{ s}^2\frac{\mathrm{d}{\rm ln} S }{\mathrm{d}x} -\frac{d c_\mathrm{ s}^2}{\mathrm{d}x} \quad .$$ Comparing Eq.\[nozzleeq\] with Eq.\[criticalpoint\], the pseudo-nozzle term in the context of radiation-pressure-driven winds is $$c_\mathrm{ s}^2 \frac{\mathrm{d}{\rm ln} S }{\mathrm{d}x} = g \left(1-\Gamma + \frac{2 c_\mathrm{ s}^2}{g r}\right)
,$$ which shows how, in the limit of $2c_\mathrm{ s}^2\ll gr$, the Eddington limit acts similarly to a nozzle term for a radiation-driven stellar outflow.
The common property between the hydrodynamic of a flow through a pipe and a stellar wind is the readjustment to the characteristic solution. It is the solution that starts subsonic, goes through the critical point of the momentum equation [where the mass-flow is defined, @1999LamersCassinelli], and has a finite velocity at infinity [@1980Abbott; @2002Nugis; @2007Shore]. In the nozzle the critical point also defines the last point of the system which can still communicate with all the regions of the flow. By reducing the perturbative analysis in e.g. @1980Abbott to a case where the line opacity and acceleration are independent from the velocity profile, we can easily show that the characteristic speed of the hyperbolic system of stellar wind equations is the local sound speed[^6]. However, as noticed by @1981Thorne while investigating the event horizon, the diffusive approximation for radiative transfer is notoriously acausal (due to the parabolic nature of the partial differential equation governing the phenomenon). This shortcoming implies that information can be transmitted instantaneously. Hence a relativistic formulation governing diffusion together with the introduction of the second sound speed[^7] would be more appropriate [@1982Flammang; @1984Mihalas; @2001Mandelis; @2005Ali]. If the speed of heat were, in this context, the same as or smaller than the speed of sound, a [sonic horizon]{} would arise in this class of stars, with the sonic point being the last point that could communicate with all the regions of the flow.
Effects of velocity gradients on the critical point {#effects-of-velocity-gradients-on-the-critical-point .unnumbered}
---------------------------------------------------
Equation \[criticalpoint\] shows that the sonic point is the critical point of a stellar wind, but this is true only if no other term with a dependency on $\mathrm{d}\varv/\mathrm{d}r$ contributes to the right-hand side of Eq.\[criticalpoint\]. In the context of line-driven winds, @1975CAK and later applications of their theory (the CAK theory) showed the importance of lines unsaturated due to Doppler shift in driving the winds of massive stars. In fact, in the presence of steep velocity gradients the rest wavelength of each accelerating element in the outflow is Doppler shifted such that unattenuated continuum can be absorbed and momentum transfer is very efficient. This in turn allows massive stars to have winds with very high terminal wind velocities. In this context the opacity arising from the spectral lines includes a dependency on the velocity gradient, i.e. $\kappa(\rho,T,\mathrm{d}\varv/\mathrm{d}r,...)$, and therefore @1980Abbott argue that the sonic point is no longer, strictly speaking, the critical point of the outflow.
However, @2002Nugis suggest that the contribution to the opacity by the Doppler shifted lines in hot WR stars is usually less than 1% at typical sonic point conditions. This can be quantified by investigating the inverse of the optical depth parameter (Table \[tab\]). This quantity compares the Doppler shift of the lines as being due to the velocity gradient through a mean free path to the thermal velocity of the protons [@2002Nugis; @2016Ro]. They estimate this quantity to be typically of the order of 100 for WR stars. Independently of this, @2007LucyB [@2007Lucy] also suggested that, in contrast to the standard CAK [*ansatz*]{}, the sonic point retains its role of critical point for the system of differential equations describing the system [see also @1975Lucy; @1998Lucy; @1990Poe; @2008Muller; @2017Sander]. The importance of the unsaturated lines in the highly supersonic outflow is instead clear [@1975CAK; @1980Abbott; @1985Abbott; @1989Kudritzki; @1995Gayley; @2005Grafener; @2005Vink; @2008Puls; @2008Grafener].
WR111 {#AWR111}
=====
We adopt the hydrodynamically self-consistent wind structure derived for WR111, a WC5 WR star, by @2005Grafener and connect it smoothly to our hydrodynamic model. This is done in order to show how our calculations of the subsonic structure can be smoothly matched to the sonic point conditions of a hydrodynamical wind model of a post-main sequence WR star and also to give a picture of what the internal profile of such a star looks like. This is done in Fig.\[WR111\], where the velocity and temperature profiles of a 12$\Msun$ stellar model with sonic point boundary conditions are plotted together with the supersonic structure of the wind [@2005Grafener]. The stellar model has been built such that its sonic point conditions, namely sonic point temperature, density, radius, luminosity, and mass-loss rate, match those derived for WR111. Luminosity and mass loss for the stellar model are, following @2005Grafener, $\log(L/\Lsun)=5.45$ and $\log(\Mdot)\approx -5.1$, respectively. Homogeneous zero age main sequence helium star models with such luminosity were found to have a core radius that was too large to reproduce the radius of this wind model. Therefore, given that WR111 is of the WC class and it is consequently thought to be in a more advanced stage of its evolution, we used a more evolved model (with approximately 0.5 carbon mass fraction in its centre) instead of a homogeneous helium star. The wind model by @2005Grafener found its sonic point at an optical depth of $\tau=5.4$ with temperature $T_\mathrm{ S}\approx 200\,kK$ and $\log(\rho_\mathrm{ S})\approx 8.5$. For this star @2005Grafener finds a $T_*$, i.e. the effective temperature at $\tau=20$, equal to 140kK. Instead, for the same object, spectral analysis conducted by @2002Grafener and @2012Sander assuming a beta-velocity law led to $T_*$ of the order of 85kK. The corresponding radius derived via the Stefan–Boltzmann law would thus be $\approx 2.5 \Rsun$, different by more than a factor of 2 from the actual quasi-hydrostatic radius in Fig.\[WR111\]. This shows how misleading it might be to use the Stefan–Boltzmann law at $\tau=20$ and, more importantly, how the adoption of a beta-velocity law to describe the dynamics of the optically thick winds of WR stars might lead to a significantly different radius estimate compared to hydrodynamic calculations.
[^1]: e-mail: luca@astro.uni-bonn.de
[^2]: Similar to the spectroscopic HR diagram [@2014Langer].
[^3]: This simplifying assumption could be replaced by the direct computation of the sonic point radii (see our models in Sect.\[sectrisults\]), but would introduce, at this point, an unnecessary complexity to Fig.\[mdot\].
[^4]: WN stars cooler than $\log(T_\mathrm{ S})\lesssim 5.0$ tend to show spectra with hydrogen spectral lines, which implies non-negligible amounts of hydrogen at their surface. Thus, the H-free opacity tables might not be adequate in that temperature range.
[^5]: The turbulent velocity is usually assumed constant in stellar atmosphere calculation, implying that the only contribution to the force balance would arise from the density gradient.
[^6]: @1980Abbott introduced the concept of effective sound speed and radiative-acoustic waves while treating the CAK critical point.
[^7]: The [second sound speed]{} is the speed at which a perturbation in the heat flux travels, i.e. the speed of heat. It arises when the Fourier equation for diffusive transport gets a hyperbolic form via a relativistic description of the phenomena [@2005Ali]
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abstract: 'We continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in [@AJP]. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes. Among our main results, we obtain the characterization of completion morphisms as pseudo-closed immersions that are flat. Also, the local structure of smooth and étale morphisms between locally noetherian formal schemes is described: the former factors locally as a completion morphism followed by a smooth adic morphism and the latter as a completion morphism followed by an étale adic morphism.'
address:
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Departamento de Álxebra\
Facultade de Matemáticas\
Universidade de Santiago de Compostela\
E-15782 Santiago de Compostela, SPAIN
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Departamento de Álxebra\
Facultade de Matemáticas\
Universidade de Santiago de Compostela\
E-15782 Santiago de Compostela, SPAIN
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Departamento de Matemáticas\
Escola Superior de Enxeñería Informática\
Campus de Ourense, Univ. de Vigo\
E-32004 Ourense, Spain
author:
- Leovigildo Alonso Tarrío
- Ana Jeremías López
- Marta Pérez Rodríguez
title: Local structure theorems for smooth maps of formal schemes
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[^1]
Introduction {#introduction .unnumbered}
============
Formal schemes have always been present in the backstage of algebraic geometry but they were rarely studied in a systematic way after the foundational [@EGA1 §10]. It has become more and more clear that the wide applicability of formal schemes in several areas of mathematics require such study. Let us cite a few of this applications. The construction of De Rham cohomology for a scheme $X$ of zero characteristic embeddable in a smooth scheme $P$, studied by Hartshorne [@ha2] (and, independently, by Deligne), is defined as the hypercohomology of the completion of the De Rham complex of the formal completion of $P$ along $X$. Formal schemes play a key role in $p$-adic cohomologies (crystalline, rigid …) and are also algebraic models of rigid analytic spaces. These developments go back to Grothendieck with further elaborations by Raynaud, in collaboration with Bosch and Lütkebohmert, and later work by Berthelot and de Jong. In a different vein, Strickland [@st] has pointed out the importance of formal schemes in the context of (stable) homotopy theory.
A particular assumption that it is almost always present in most earlier works on formal schemes is that morphisms are adic, [[*i.e.*]{} ]{}that the topology of the sheaf of rings of the initial scheme is induced by the topology of the base formal scheme. This hypothesis on a morphism of formal schemes guarantees that its fibers are usual schemes, therefore an adic morphism between formal schemes is, in the terminology of Grothendieck’s school, a relative scheme over a base that is a formal scheme. But there are important examples of maps of formal schemes that do not correspond to this situation. The first example that comes into mind is the natural map $\operatorname{Spf}(A[[X]]) \to \operatorname{Spf}(A)$ for an adic ring $A$. This morphism has a finiteness property that had not been made explicit until [@AJL1] (and independently, in [@y]). This property is called *pseudo-finite type*[^2]. The fact that pseudo-finite type morphisms need not be adic allows fibers that are not usual schemes, and the structure of these maps is, therefore, more complex than the structure of adic maps. The study of smoothness and, more generally, infinitesimal lifting properties in the context of noetherian formal schemes together with this hypothesis of finiteness was embraced in general in our previous work [@AJP]. We should mention a preceding study of smooth morphisms under the restriction that the base is a usual scheme in [@y] and also the overlap of several results in [@AJP] and a set of results in [@LNS §2], based on Nayak’s 1998 thesis. In [@AJP] we studied the good properties of these definitions and the agreement of their properties with the corresponding behavior for usual noetherian schemes, obtaining the corresponding statement of Zariski’s Jacobian criterion for smoothness. Now we concentrate on studying properties which make sense specifically in the formal context getting information about the infinitesimal lifting properties from information present in the structure of a formal scheme. This study continues by the third author in [@P] where a deformation theory for smooth morphisms is developed.
This paper can be structured roughly into three parts. The first, formed by sections 1, 2 and 3, includes preliminaries, introduces the notion of quasi-covering and the study of completion morphisms. We know of no previous reference about these matters, so we include all the needed details. They will be indispensable to state our results. The second part encompasses three sections (\[sec3\], \[sec4\] and \[sec5\]). We show that there exists a close relationship between the infinitesimal lifting properties of an adic morphism and the infinitesimal lifting properties of the underlying morphism of ordinary schemes $f_{0}$. The third part (section \[sec6\]) treats the structure theorems, which are the main results of this work. We characterize open immersions and completion morphisms in terms of the étale property. We classify étale adic coverings of a noetherian formal scheme. Finally, we give local structure theorems for unramified, étale and smooth maps, that show that it is possible to factor them locally into simpler maps.
Let us discuss in greater detail the contents of every section. Our framework is the category of locally noetherian formal schemes. In this category a morphism $f\colon{\mathfrak X}\to {\mathfrak Y}$ can be expressed as a direct limit $$f={\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} f_{n}$$ of a family of maps of ordinary schemes using appropriate ideals of definition. The first section sets the basic notations and recalls some definitions that will be used throughout the paper. The second section deals with morphisms between locally noetherian formal schemes expressed as before as a limit in which every map $f_{n}$ is a closed immersion of usual schemes. It is a true closed immersion of formal schemes when $f$ is adic. We treat radicial maps of formal schemes and see that the main results are completely similar to the case of usual schemes. On usual schemes, quasi-finite maps play a very important role in the understanding of the structure of étale maps. In the context of formal schemes there are two natural generalizations of this notion. The simplest one is *pseudo-quasi-finite* (Definition \[defncuasifin\]) — in a few words: “of pseudo-finite type with finite fibers". The key notion though is that of quasi-covering (Definition \[defncuasireves\]). While both are equivalent in the context of usual schemes, the latter is a basic property of unramified and, therefore, étale maps between formal schemes (*cf.* Corollaries \[corpnrimplcr\] and \[corcaractlocalpe\]). In section 3 we discuss flat morphisms in the context of locally noetherian formal schemes. Next, we study morphisms of completion in this setting. They form a class of flat morphisms that are closed immersions as topological maps. Such maps will be essential for the results of the last section.
Expressing a morphism $f\colon{\mathfrak X}\to {\mathfrak Y}$ between locally noetherian formal schemes as a limit as before, it is sensible to ask about the relation that exists between the infinitesimal lifting properties of $f$ and the infinitesimal lifting properties of the underlying morphisms of usual schemes $\{f_{n}\}_{n \in {\mathbb N}}$. This is one of the main themes of the next three sections. The case of unramified morphisms is simple: $f$ is unramified if and only if $f_n$ are unramified $\forall n \in {\mathbb N}$ (Proposition \[nrfn\]). Another characterization is that $f$ is unramified if and only if $f_0$ is *and* the fibers of $f$ and of $f_0$ agree (Corollary \[corf0imfpnr\]). A consequence of this result is a useful characterization of pseudo-closed immersion as those unramified morphisms such that $f_0$ is a closed immersion (Corollary \[pecigf0ecnr\]). Smooth morphisms are somewhat more difficult to characterize. An *adic* morphism $f$ is smooth if and only if $f_0$ is and $f$ is flat (Corollary \[flf0l\]). For a non adic morphism, one *cannot* expect that the maps $f_n$ are going to be smooth when $f$ is smooth as it is shown by example \[exf0lisonofliso\]. On the positive side, there is a nice characterization of smooth closed subschemes (Proposition \[ecppl\]). Also, the matrix jacobian criterion holds for formal schemes, see Corollary \[criteriojacobiano\] for a precise statement. In section \[sec5\] we combine these results to obtain properties of étale morphisms. It is noteworthy to point out that a smooth pseudo-quasi-finite map need not be étale (Example \[pcf+plnope\]).
The last section contains our main results. First we recover in our framework the classical fact for usual schemes [@EGA44 (17.9.1)] that an open immersion is a map that is étale and radicial (Theorem \[caractencab\]). We also characterize completion morphisms as those pseudo-closed immersions that are flat. This and other characterizations are given in Proposition \[caracmorfcompl\]. Writing a locally noetherian formal scheme ${\mathfrak Y}$ as $${\mathfrak Y}= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} Y_{n}$$ with respect to an ideal of definition, Proposition \[teorequivet\] says that there is an equivalence of categories between étale adic ${\mathfrak Y}$-formal schemes and étale $Y_{0}$-schemes. A special case already appears in [@y Proposition 2.4]. In fact, this result is a reinterpretation of [@EGA44 (18.1.2)]. The factorization theorems are based on Theorem \[tppalnr\] that says that an unramified morphism can be factored locally into a pseudo-closed immersion followed by an étale adic map. As consequences we obtain Theorem \[tppalet\] and Theorem \[tppall\]. They state that every smooth morphism and every étale morphism factor locally as a completion morphism followed by a smooth adic morphism and an étale adic morphism, respectively. These results explain the local structure of smooth and étale morphisms of formal schemes. It has been remarked by Lipman, Nayak and Sastry in [@LNS p. 132] that this observation may simplify some developments related to Cousin complexes and duality on formal schemes.
Preliminaries {#sec1}
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We denote by ${\mathsf {NFS}}$ the category of locally noetherian formal schemes and by ${{\mathsf {NFS}}_{\mathsf {af}}}$ the subcategory of locally noetherian affine formal schemes. We write ${\mathsf {Sch}}$ for the category of ordinary schemes.
We assume that the reader is familiar with the basic theory of formal schemes as is explained in [@EGA1 §10]: formal spectrum, ideal of definition of a formal scheme, fiber product of formal schemes, functor $M \leadsto M^{\triangle}$ for modules over adic rings, completion of a usual scheme along a closed subscheme, adic morphisms, separated morphisms, etc.
From now on and, except otherwise indicated, every formal scheme will belong to ${\mathsf {NFS}}$. Every ring under consideration will be assumed to be noetherian. So, every complete ring and every complete module will be separated under the corresponding adic topology.
\[lim\] Henceforth, the following notation [@EGA1 §10.6] will be used:
1. Given ${\mathfrak X}\in {\mathsf {NFS}}$ and ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ an ideal of definition for each $n\in {\mathbb N}$ we put $X_{n}:=({\mathfrak X},{\mathcal O}_{{\mathfrak X}}/{\mathcal J}^{n+1})$ and we indicate that ${\mathfrak X}$ is the direct limit of the schemes $X_{n}$ by $${\mathfrak X}={\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} X_{n}.$$ The ringed spaces ${\mathfrak X}$ and $X_{n}$ have the same underlying topological space, so we will not distinguish between a point in ${\mathfrak X}$ or $X_{n}$.
2. If $f\colon{\mathfrak X}\to {\mathfrak Y}$ is in ${\mathsf {NFS}}$, ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ are ideals of definition such that $f^{*}({\mathcal K}){\mathcal O}_{{\mathfrak X}} \subset {\mathcal J}$ and $f_{n}\colon X_{n}:=({\mathfrak X},{\mathcal O}_{{\mathfrak X}}/{\mathcal J}^{n+1}) \to Y_{n}:=({\mathfrak Y},{\mathcal O}_{{\mathfrak Y}}/{\mathcal K}^{n+1})$ is the morphism induced by $f$, for each $n \in {\mathbb N}$, then $f$ is expressed as $$f = {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n\in {\mathbb N}}} \end{array}} f_{n}.$$
3. Furthermore, given $f\colon{\mathfrak X}\to {\mathfrak Y}$ a morphism in ${\mathsf {NFS}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ an ideal of definition, there exist ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ an ideal of definition such that $f^*({\mathcal K}){\mathcal O}_{{\mathfrak X}} \subset {\mathcal J}$. Such a pair of ideals of definition will be called $f$-*compatible*.
Let $f \colon {\mathfrak X}\to {\mathfrak Y}$ be a morphism in ${\mathsf {NFS}}$ and let ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ be $f$-compatible ideals of definition. The morphism $f$ is of *pseudo-finite type (pseudo-finite)* [@AJL1 p.7] if $f_{0}$ (and in fact any $f_{n}$) is of finite type (finite, respectively). Moreover, if $f$ is adic we say that $f$ is of *finite type (finite)* [@EGA1 10.13.3] ([@EGA31 (4.8.2)], respectively). Note that these definitions do not depend on the choice of ideals of definition.
[@AJP Definition 2.1 and Definition 2.6] A morphism $f\colon{\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$ is *smooth (unramified, étale)* if it is of pseudo-finite type and satisfies the following lifting condition:
> For all affine ${\mathfrak Y}$-schemes $Z$ and for each closed subscheme $T{\hookrightarrow}Z$ given by a square zero ideal ${\mathcal I}\subset {\mathcal O}_{Z}$ the induced map $$\operatorname{Hom}_{{\mathfrak Y}}(Z,{\mathfrak X}) {\longrightarrow}\operatorname{Hom}_{{\mathfrak Y}}(T,{\mathfrak X})$$ is surjective (injective or bijective, respectively).
Moreover, if $f$ is in addition adic we say that $f$ is *smooth adic (unramified adic or étale adic, respectively)*.
We say that $f$ is *smooth (unramified or étale) at $x$* if there exists an open subset ${\mathfrak U}\subset {\mathfrak X}$ with $x \in {\mathfrak U}$ such that $f|_{\mathfrak U}$ is smooth (unramified or étale, respectively). It holds that $f$ is smooth (unramified or étale) if and only if $f$ is smooth (unramified or étale, respectively) at $x, \forall x \in {\mathfrak X}$ (*cf.* [@AJP Proposition 4.3, 4.1]).
(*cf.* [@AJP §3]) Given $f \colon {\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$ the *differential pair of ${\mathfrak X}$ over ${\mathfrak Y}$*, $({\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}}, {\widehat{d}}_{{\mathfrak X}/{\mathfrak Y}})$, is locally given by $ ({\widehat{\Omega}}^{1}_{A/B},{\widehat{d}}_{A/B})
$ for all open sets ${\mathfrak U}=\operatorname{Spf}(A) \subset {\mathfrak X}$ and ${\mathfrak V}=\operatorname{Spf}(B) \subset {\mathfrak Y}$ with $f({\mathfrak U}) \subset {\mathfrak V}$. The ${\mathcal O}_{{\mathfrak X}}$-Module ${\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}}$ is called the *module of $1$-differentials of ${\mathfrak X}$ over ${\mathfrak Y}$* and the continuous ${\mathfrak Y}$-derivation ${\widehat{d}}_{{\mathfrak X}/{\mathfrak Y}}$ is called the *canonical derivation of ${\mathfrak X}$ over ${\mathfrak Y}$*.
\[ec\] [@EGA1 p. 442] A morphism $f \colon {\mathfrak Z}\to {\mathfrak X}$ in ${\mathsf {NFS}}$ is a *closed immersion* if it factors as ${\mathfrak Z}{\xrightarrow}{g} {\mathfrak X}' \overset{j}{\hookrightarrow}{\mathfrak X}$ where $g$ is an isomorphism of ${\mathfrak Z}$ into a closed subscheme ${\mathfrak X}' {\hookrightarrow}{\mathfrak X}$ of the formal scheme ${\mathfrak X}$ ([@EGA1 (10.14.2)]). Recall from [@EGA31 (4.8.10)] that a morphism $f \colon {\mathfrak Z}\to {\mathfrak X}$ in ${\mathsf {NFS}}$ is a closed immersion if it is adic and, given ${\mathcal K}\subset {\mathcal O}_{{\mathfrak X}}$ an ideal of definition of ${\mathfrak X}$ and ${\mathcal J}=f^{*}({\mathcal K}){\mathcal O}_{{\mathfrak Z}}$ the corresponding ideal of definition of ${\mathfrak Z}$, the induced morphism $f_{0} \colon Z_{0} \to X_{0}$ is a closed immersion, equivalently, the induced morphisms $f_{n} \colon Z_{n} \to X_{n}$ are closed immersions for all $n \in {\mathbb N}$.
A morphism $f: {\mathfrak Z}\to {\mathfrak X}$ in ${\mathsf {NFS}}$ is an *open immersion* if it factors as ${\mathfrak Z}{\xrightarrow}{g} {\mathfrak X}' {\hookrightarrow}{\mathfrak X}$ where $g$ is an isomorphism of ${\mathfrak Z}$ into an open subscheme ${\mathfrak X}' {\hookrightarrow}{\mathfrak X}$.
Let ${\mathfrak X}$ be in ${\mathsf {NFS}}$, let ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ be an ideal of definition and $x \in {\mathfrak X}$. We define the *topological dimension of ${\mathfrak X}$ at $x$* as $$\operatorname{dimtop}_{x}{\mathfrak X}= \dim_{x}X_{0}.$$ It is easy to see that the definition does not depend on the chosen ideal of definition of ${\mathfrak X}$. We define the *topological dimension of ${\mathfrak X}$* as $$\operatorname{dimtop}{\mathfrak X}= \sup_{x \in {\mathfrak X}} \operatorname{dimtop}_{x} {\mathfrak X}= \sup_{x \in {\mathfrak X}} \dim_{x}X_{0} = \dim X_{0}.$$
Given $A$ an $I$-adic noetherian ring, put $X = \operatorname{Spec}(A)$ and ${\mathfrak X}= \operatorname{Spf}(A)$, then $ \operatorname{dimtop}{\mathfrak X}= \dim A / I $. While the only “visible part” of ${\mathfrak X}$ in $X = \operatorname{Spec}(A)$ is $V(I)$, it happens that $X \setminus V(I)$ has a deep effect on the behavior of ${\mathfrak X}$ as we will see along this work. So apart from the topological dimension of ${\mathfrak X}$, it is necessary to consider another notion of dimension that expresses part of the “hidden" information: the algebraic dimension.
Let ${\mathfrak X}$ be in ${\mathsf {NFS}}$ and let $x \in {\mathfrak X}$. We define the *algebraic dimension of ${\mathfrak X}$ at $x$* as $$\dim_{x}{\mathfrak X}= \dim {\mathcal O}_{{\mathfrak X},x}.$$ The *algebraic dimension of ${\mathfrak X}$* is $$\dim{\mathfrak X}= \sup_{x \in {\mathfrak X}} \dim_{x} {\mathfrak X}.$$
\[diaf\] If ${\mathfrak X}= \operatorname{Spf}(A)$ with $A$ an $I$-adic noetherian ring then $\dim {\mathfrak X}= \dim A.$
For each $x \in {\mathfrak X}$, if ${\mathfrak p}_{x}$ is the corresponding open prime ideal in $A$ we have that $\dim_{x} {\mathfrak X}= \dim A_{\{{\mathfrak p}_{x}\}} = \dim A_{{\mathfrak p}_{x}}$ since $ A_{{\mathfrak p}_{x}} {\hookrightarrow}A_{\{{\mathfrak p}_{x}\}} $ is a flat extension of local rings with the same residue field (*cf.* [@ma1 (24.D)]). Since $I \subset A$ is in the Jacobson radical, it holds that $\dim A = \sup_{x \in {\mathfrak X}} \dim A_{{\mathfrak p}_{x}}$, from which it follows the equality.
\[exdimafdis\] Given $A$ an $I$-adic noetherian ring and $\mathbf{T}= T_{1},\, T_{2},\, \ldots,\, T_{r}$ a finite number of indeterminates, the *affine formal space of dimension $r$ over $A$* is $ {\mathbb A}_{\operatorname{Spf}(A)}^{r}=\operatorname{Spf}(A\{\mathbf{T}\} )$ and the *formal disc of dimension $r$ over $A$* is $ {\mathbb D}_{\operatorname{Spf}(A)}^{r}=\operatorname{Spf}(A[[\mathbf{T}]])$ (see [@AJP Example 1.6]). It holds that $$\begin{array}{ccccc}
\operatorname{dimtop}{\mathbb A}_{\operatorname{Spf}(A)}^{r} &= &\dim {\mathbb A}_{\operatorname{Spec}(A/I)}^{r} &= &\dim A / I + r \\
\operatorname{dimtop}{\mathbb D}_{\operatorname{Spf}(A)}^{r} &= &\dim \operatorname{Spec}(A/I) &= &\dim A / I
\end{array}$$ and $$\begin{array}{ccccccc}
\dim {\mathbb A}_{\operatorname{Spf}(A)}^{r} & \underset{\textrm{\ref{diaf}}} = &\dim A\{\mathbf{T}\} &=& \dim A + r &\underset{\textrm{\ref{diaf}}} = &\dim \operatorname{Spf}(A) + r\\
\dim {\mathbb D}_{\operatorname{Spf}(A)}^{r} &\underset{\textrm{\ref{diaf}}} = & \dim A[[\mathbf{T}]] &=& \dim A + r &\underset{\textrm{\ref{diaf}}} = &\dim \operatorname{Spf}(A) + r.
\end{array}$$
From these examples, we see that the algebraic dimension of a formal scheme does not measure the dimension of the underlying topological space. In general, for ${\mathfrak X}$ in ${\mathsf {NFS}}$, $\dim_{x} {\mathfrak X}\ge \operatorname{dimtop}_{x} {\mathfrak X}$, for any $x \in {\mathfrak X}$ and, therefore $$\dim {\mathfrak X}\ge \operatorname{dimtop}{\mathfrak X}.$$ Moreover, if ${\mathfrak X}= \operatorname{Spf}(A)$ with $A$ an $I$-adic ring then $\dim {\mathfrak X}\ge \operatorname{dimtop}{\mathfrak X}+ \operatorname{ht}(I)$.
Let $f:{\mathfrak X}\to {\mathfrak Y}$ be in ${\mathsf {NFS}}$ and $y \in {\mathfrak Y}$. The *fiber of $f$ at the point $y$* is the formal scheme $$f^{-1} (y) = {\mathfrak X}\times_{{\mathfrak Y}} \operatorname{Spec}(k(y)).$$ For example, if $f:{\mathfrak X}= \operatorname{Spf}(B) \to {\mathfrak Y}= \operatorname{Spf}(A)$ is in ${{\mathsf {NFS}}_{\mathsf {af}}}$ we have that $f^{-1} (y) = \operatorname{Spf}(B {\widehat{\otimes}}_{A} k(y))$.
Let ${\mathfrak Y}= \operatorname{Spf}(A)$ be in ${{\mathsf {NFS}}_{\mathsf {af}}}$ and let $\mathbf{T}= T_{1},\, T_{2},\, \ldots,\, T_{r}$ be a set of indeterminates. If $p: {\mathbb A}_{{\mathfrak Y}}^{r} \to {\mathfrak Y}$ is the canonical projection of the affine formal $r$-space over ${\mathfrak Y}$, then for all $y \in {\mathfrak Y}$ we have that $$p^{-1}(y) = \operatorname{Spf}(A\{\mathbf{T}\}{\widehat{\otimes}}_{A}k(y)) = \operatorname{Spec}(k(y)[\mathbf{T}]) = {\mathbb A}_{\operatorname{Spec}(k(y))}^{r}.$$ If $q: {\mathbb D}_{{\mathfrak Y}}^{r}\to{\mathfrak Y}$ is the canonical projection of the formal $r$-disc over ${\mathfrak Y}$, given $y \in {\mathfrak Y}$, it holds that $$q^{-1}(y) = \operatorname{Spf}(A[[\mathbf{T}]]{\widehat{\otimes}}_{A}k(y)) = \operatorname{Spf}(k(y)[[\mathbf{T}]]) = {\mathbb D}_{\operatorname{Spec}(k(y))}^{r}.$$
\[fibra\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be in ${\mathsf {NFS}}$ and let us consider ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ $f$-compatible ideals of definition. According to \[lim\], $$f = {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} (f_{n}: X_{n} \to Y_{n}).$$ Then, by [@EGA1 (10.7.4)], it holds that $$f^{-1} (y) = {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}}f_{n}^{-1} (y)$$ where $f_{n}^{-1} (y) = X_{n} \times_{Y_{n}} \operatorname{Spec}(k(y))$, for each $n \in {\mathbb N}$.
If $f$ is adic, by base-change (*cf.* [@AJP 1.3]) we deduce that $f^{-1}(y) \to \operatorname{Spec}(k(y))$ is adic so, $f^{-1} (y)$ is an ordinary scheme and $f^{-1} (y) =f_{n}^{-1} (y)$, for all $n \in {\mathbb N}$.
We establish the following convention. Let $f:{\mathfrak X}\to {\mathfrak Y}$ be in ${\mathsf {NFS}}$, $x \in {\mathfrak X}$ and $y= f(x)$ and assume that ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ are $f$-compatible ideals of definition. From now and, except otherwise indicated, whenever we consider the rings ${\mathcal O}_{{\mathfrak X},x}$ and ${\mathcal O}_{{\mathfrak Y},y}$ we will associate them the ${\mathcal J}{\mathcal O}_{{\mathfrak X},x}$ and $ {\mathcal K}{\mathcal O}_{{\mathfrak Y},y}$-adic topologies, respectively. And we will denote by $\widehat{{\mathcal O}_{{\mathfrak X},x}}$ and $\widehat{{\mathcal O}_{{\mathfrak Y},y}}$ the completion of ${\mathcal O}_{{\mathfrak X},x}$ and ${\mathcal O}_{{\mathfrak Y},y}$ with respect to the ${\mathcal J}{\mathcal O}_{{\mathfrak X},x}$ and $ {\mathcal K}{\mathcal O}_{{\mathfrak Y},y}$-adic topologies, respectively. Note that these topologies do not depend on the choice of ideals of definition of ${\mathfrak X}$ and ${\mathfrak Y}$.
Let $f:{\mathfrak X}\to {\mathfrak Y}$ be in ${\mathsf {NFS}}$. Given $x \in {\mathfrak X}$ and $y= f(x)$, we define the *relative algebraic dimension of $f$ at $x$* as $$\dim_{x} f = \dim_{x} f^{-1}(y).$$ If ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ are $f$-compatible ideals of definition, then $$\dim_{x} f= \dim {\mathcal O}_{f^{-1}(y),x} = \dim {\mathcal O}_{{\mathfrak X},x} \otimes_{{\mathcal O}_{{\mathfrak Y},y}} k(y) = \dim \widehat{{\mathcal O}_{{\mathfrak X},x} }\otimes_{\widehat{{\mathcal O}_{{\mathfrak Y},y} }} k(y).$$ If the topology in $\widehat{{\mathcal O}_{{\mathfrak X},x} }\otimes_{\widehat{{\mathcal O}_{{\mathfrak Y},y} }} k(y)$ is the ${\mathcal J}\widehat{{\mathcal O}_{{\mathfrak X},x}}$-adic then $\widehat{{\mathcal O}_{{\mathfrak X},x} }{\widehat{\otimes}}_{\widehat{{\mathcal O}_{{\mathfrak Y},y} }} k(y) = \widehat{{\mathcal O}_{{\mathfrak X},x} }\otimes_{\widehat{{\mathcal O}_{{\mathfrak Y},y} }} k(y)$.
\[exdimalg\] Given an adic morphism $f:{\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$ and $f$-compatible ideals of definition ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$, then $\dim_{x} f = \dim _{x} f_{0}$ for every $x \in {\mathfrak X}$. For example:
1. \[exdimalg1\] If $p: {\mathbb A}_{{\mathfrak Y}}^{r}:={\mathbb A}_{{\mathbb Z}}^{r} \times_{{\mathbb Z}} {\mathfrak Y}\to {\mathfrak Y}$ is the canonical projection of the affine formal $r$-space over ${\mathfrak Y}$, given $x \in {\mathbb A}_{{\mathfrak Y}}^{r}$ we have that $$\dim_{x} p = \dim k(y)[\mathbf{T}] = r,$$ where $y=p(x)$. In contrast, if $q: {\mathbb D}_{{\mathfrak Y}}^{r}:={\mathbb D}_{{\mathbb Z}}^{r} \times_{{\mathbb Z}} {\mathfrak Y}\to{\mathfrak Y}$ is the canonical projection of the formal $r$-disc over ${\mathfrak Y}$, $x \in {\mathbb D}_{{\mathfrak Y}}^{r}$ and $y=q(x)$ it holds that $$\dim_{x} q = \dim k(y)[[\mathbf{T}]] \underset{\textrm{\ref{exdimafdis}}} = r > \dim k(y) = 0.$$
2. \[exdimalg2\] If $X$ is a usual noetherian scheme and $X'$ is a closed subscheme of $X$, recall that the morphism of completion of $X$ along $X' $, $\kappa: X_{/X' } \to X$ ([@EGA1 (10.8.5)]) is not adic, in general. Note however that $$\dim_{x} \kappa = \dim k(x) = 0$$ for all $x \in X_{/X' }$.
Pseudo-closed immersions and quasi-coverings
============================================
A morphism $f:{\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$ is a *pseudo-closed immersion* if there exists ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ $f$-compatible ideals of definition such that the induced morphisms of schemes $\{f_{n}:X_{n} \to Y_{n}\}_{n \in {\mathbb N}}$ are closed immersions.
Note that if $f:{\mathfrak X}\to {\mathfrak Y}$ is a pseudo-closed immersion, $f({\mathfrak X})$ is a closed subset of ${\mathfrak Y}$.
Let us show that this definition does not depend on the choice of ideals of definition. Being a local question, we can assume that $f: {\mathfrak X}= \operatorname{Spf}(A) \to {\mathfrak Y}= \operatorname{Spf}(B)$ is in ${{\mathsf {NFS}}_{\mathsf {af}}}$ and that $ {\mathcal J}= J^{{\triangle}},\, {\mathcal K}= K^{{\triangle}}$ for ideals of definition $J \subset A$ and $K \subset B$ such that $KA \subset J$. Then, given another pair of ideals of definition $J' \subset A$ and $K' \subset B$ such that ${\mathcal J}' = J'^{{\triangle}}\subset {\mathcal O}_{{\mathfrak X}},\, {\mathcal K}' = K'^{{\triangle}}\subset {\mathcal O}_{{\mathfrak Y}}$ satisfying that $f^{*}({\mathcal K}'){\mathcal O}_{{\mathfrak X}} \subset {\mathcal J}'$, there exists $n_{0} > 0$ such that $J^{n_{0}} \subset J'$ and $K^{n_{0}} \subset K'$. The morphism $B \to A$ induces the following commutative diagrams
B/K\^[n\_[0]{}(n+1)]{} & & A/J\^[n\_[0]{}(n+1)]{}\
& &\
B/K’\^[n+1]{} & & A/J’\^[n+1]{}\
and it follows that $B/K'^{n+1}\to A/J'^{n+1}$ is surjective, for all $n \in {\mathbb N}$. Then, using \[ec\], it follows that the morphism $ ({\mathfrak X}, {\mathcal O}_{{\mathfrak X}}/{\mathcal J}'^{n+1}) \to ({\mathfrak Y},{\mathcal O}_{{\mathfrak Y}}/{\mathcal K}'^{n+1})$ is a closed immersion, for all $n \in {\mathbb N}$.
Let $X$ be a noetherian scheme and let $X' \subset X$ be a closed subscheme defined by an ideal ${\mathcal I}\subset {\mathcal O}_{X}$. The morphism of completion $X_{/X'} {\xrightarrow}{\kappa} X$ of $X$ along $X'$ ([@EGA1 (10.8.5)]) is expressed as $${\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} \left((X' ,{\mathcal O}_{X}/{\mathcal I}^{n+1}) {\xrightarrow}{ \kappa_{n}} (X,{\mathcal O}_{X})\right),$$ therefore, it is a pseudo-closed immersion.
Notice that an adic pseudo-closed immersion is a closed immersion (*cf.* \[ec\]). However, to be a pseudo-closed immersion is not a topological property:
Given $K$ a field, let $p: {\mathbb D}_{\operatorname{Spec}(K)}^{1} \to \operatorname{Spec}(K)$ be the canonical projection. If we consider the ideal of definition $\langle T\rangle^{{\triangle}}$, of ${\mathbb D}_{\operatorname{Spec}(K)}^{1}$ then $p_{0} = 1_{\operatorname{Spec}(K)}$ is a closed immersion. However, the morphisms $$p_{n}: \operatorname{Spec}(K[T] / \langle T \rangle ^{n+1}) \to \operatorname{Spec}(K)$$ are not closed immersions, for all $ n > 0$ and, thus, $p$ is not a pseudo-closed immersion.
Let $f:{\mathfrak X}\to {\mathfrak Y}$ and $g:{\mathfrak Y}\to {\mathfrak S}$ be two morphisms in ${\mathsf {NFS}}$. It holds that:
1. If $f$ and $g$ are pseudo-closed immersions then $g \circ f$ is a pseudo-closed immersion.
2. If $f$ is a pseudo-closed immersion, given $h:{\mathfrak Y}' \to {\mathfrak Y}$ in ${\mathsf {NFS}}$ we have that ${\mathfrak X}_{{\mathfrak Y}'} = {\mathfrak X}\times_{{\mathfrak Y}} {\mathfrak Y}'$ is in ${\mathsf {NFS}}$ and that $f':{\mathfrak X}_{{\mathfrak Y}'} \to {\mathfrak Y}'$ is a pseudo-closed immersion.
As for (1) let ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$, ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ and ${\mathcal L}\subset {\mathcal O}_{{\mathfrak S}}$ be ideals of definition such that ${\mathcal J}$ and ${\mathcal K}$ are $f$-compatible, ${\mathcal K}$ and ${\mathcal L}$ are $g$-compatible and consider the corresponding expressions for $f$ and $g$ as direct limits: $$f ={\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} (X_{n} {\xrightarrow}{f_{n}} Y_{n}) \qquad g= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} (Y_{n} {\xrightarrow}{g_{n}}S_{n})$$ Since $$g \circ f = {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} g_{n} \circ f _{n}$$ the assertion follows from the stability under composition of closed immersions in ${\mathsf {Sch}}$. Let us show (2). Take ${\mathcal K}' \subset {\mathcal O}_{{\mathfrak Y}'}$ an ideal of definition with $h^{*}({\mathcal K}){\mathcal O}_{{\mathfrak Y}'} \subset {\mathcal K}'$ and such that, by \[lim\], $$h= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} (h_{n}: Y'_{n} \to Y_{n}).$$ Then by [@EGA1 (10.7.4)] we have that $$\begin{matrix}
\begin{diagram}[height=2.5em,w=2.5em,labelstyle=\scriptstyle]
{\mathfrak X}_{{\mathfrak Y}'} & \rTto^{f'} & {\mathfrak Y}' & \\
\dTto & &\dTto^h & \\
{\mathfrak X}&\rTto^f & {\mathfrak Y}& \\
\end{diagram}
& = \qquad & {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}}&
\left(
\begin{diagram}[height=2.5em,w=2.5em,labelstyle=\scriptstyle]
X_{n} \times_{Y_{n}} Y'_{n} & \rTto^{f'_{n}}&Y'_{n}\\
\dTto & &\dTto^{h_{n}}\\
X_{n} & \rTto^{f_{n}} & Y_{n}\\
\end{diagram}
\right)
\end{matrix}$$ By hypothesis, $f_{n}$ is a closed immersion and since closed immersions in ${\mathsf {Sch}}$ are stable under base-change we have that $f'_{n}$ is a closed immersion of noetherian schemes, $\forall n\in {\mathbb N}$. Finally, since $f$ is a morphism of pseudo-finite type, from [@AJP Proposition 1.8.(2)] we have that ${\mathfrak X}_{{\mathfrak Y}'}$ is in ${\mathsf {NFS}}$.
Next we turn to the study of radicial morphisms in the context of formal schemes. This notion will allow us later (Theorem \[caractencab\]) to give a characterization of open immersions in terms of étale morphisms.
\[rad\] A morphism $f:{\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$ is *radicial* if given ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ $f$-compatible ideals of definition the induced morphism of schemes $f_{0}:X_{0} \to Y_{0}$ is radicial.
Given $x \in {\mathfrak X}$, the residue fields of the local rings ${\mathcal O}_{{\mathfrak X},x}$ and ${\mathcal O}_{X_{0},x}$ agree and analogously for ${\mathcal O}_{{\mathfrak Y},f(x)}$ and ${\mathcal O}_{Y_{0},f_0(x)}$. Therefore the definition of radicial morphisms does not depend on the chosen ideals of definition of ${\mathfrak X}$ and ${\mathfrak Y}$.
\[prad\] From the sorites of radicial morphisms in ${\mathsf {Sch}}$ it follows that:
1. \[prad1\] Radicial morphisms are stable under composition and noetherian base-change.
2. \[prad2\] Every monomorphism is radicial. So, open immersions, closed immersions and pseudo-closed immersions are radicial morphisms.
The notion of quasi-finite morphism of usual schemes (see [@EGA1 Definition (6.11.3)]) is based on the equivalence between several conditions for morphisms between schemes (Corollaire (6.11.2) in *loc. cit.*) that are no longer equivalent in the full context of formal schemes. Specifically, we study two notions that generalize that of quasi-finite morphism of usual schemes. They will play a basic role in understanding the structure of unramified and étale morphisms in ${\mathsf {NFS}}$.
\[defncuasifin\] Let $f\colon{\mathfrak X}\to {\mathfrak Y}$ be a pseudo-finite type morphism in ${\mathsf {NFS}}$. We say that $f$ is *pseudo-quasi-finite* if there exist ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ $f$-compatible ideals of definition such that $f_{0}$ is quasi-finite. And $f$ is *pseudo-quasi-finite at $x \in {\mathfrak X}$* if there exists an open neighborhood $x \in{\mathfrak U}\subset {\mathfrak X}$ such that $f|_{{\mathfrak U}}$ is pseudo-quasi-finite.
Notice that if $f:{\mathfrak X}\to {\mathfrak Y}$ is a pseudo-quasi-finite morphism (in ${\mathsf {NFS}}$) then, *for all $f$-compatible ideals of definition* ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$, the induced morphism of schemes $f_{0}:X_{0} \to Y_{0}$ is quasi-finite.
As an immediate consequence of the analogous properties in ${\mathsf {Sch}}$ we have that:
1. The underlying sets of the fibers of a pseudo-quasi-finite morphism are finite.
2. \[soritpcf1\] Closed immersions, pseudo-closed immersions and open immersions are pseudo-quasi-finite.
3. \[soritpcf1bis\] Pseudo-finite and finite morphisms are pseudo-quasi-finite.
4. \[soritpcf2\] If $f:{\mathfrak X}\to {\mathfrak Y}$ and $g:{\mathfrak Y}\to {\mathfrak S}$ are pseudo-quasi-finite morphisms, then so is $g \circ f$.
5. \[soritpcf3\] If $f:{\mathfrak X}\to {\mathfrak Y}$ is pseudo-quasi-finite, given $h:{\mathfrak Y}' \to {\mathfrak Y}$ a morphism in ${\mathsf {NFS}}$ we have that $f':{\mathfrak X}_{{\mathfrak Y}'} \to {\mathfrak Y}'$ is pseudo-quasi-finite.
In ${\mathsf {Sch}}$ it is the case that a morphism is étale if and only if it is smooth and quasi-finite. However, we will show that in ${\mathsf {NFS}}$ not every smooth and pseudo-quasi-finite morphism is étale (see Example \[pcf+plnope\]). That is why we introduce a stronger notion than pseudo-quasi-finite morphism and that also generalizes quasi-finite morphisms in ${\mathsf {Sch}}$: the quasi-coverings.
\[defncuasireves\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a pseudo-finite type morphism in ${\mathsf {NFS}}$. The morphism $f$ is a *quasi-covering* if ${\mathcal O}_{{\mathfrak X},x} {\widehat{\otimes}}_{{\mathcal O}_{{\mathfrak Y},f(x)}} k(f(x))$ is a finite type $k(f(x))$-module, for all $x \in {\mathfrak X}$. We say that $f$ is a *quasi-covering at $x \in {\mathfrak X}$* if there exists an open ${\mathfrak U}\subset {\mathfrak X}$ with $x \in {\mathfrak U}$ such that $f|_{{\mathfrak U}}$ is a quasi-covering.
We reserve the word *covering* for a dominant ([[*i.e.*]{} ]{}with dense image) quasi-covering. These kind of maps will play no role in the present work but they are important, for instance, in the study of finite group actions on formal schemes.
If $X$ is a locally noetherian scheme and $X' \subset X $ is a closed subscheme the morphism of completion $\kappa: {\mathfrak X}=X_{/X'} \to X$ is a quasi-covering. In fact, for all $x \in {\mathfrak X}$ we have that $${\mathcal O}_{{\mathfrak X},x} {\widehat{\otimes}}_{{\mathcal O}_{X,\kappa(x)}} k(\kappa(x)) = k(\kappa(x)).$$
\[soritcr\] We have the following:
1. \[soritcr1\] Closed immersions, pseudo-closed immersions and open immersions are quasi-coverings.
2. \[soritcr2\] If $f:{\mathfrak X}\to {\mathfrak Y}$ and $g:{\mathfrak Y}\to {\mathfrak S}$ are quasi-coverings, the morphism $g \circ f$ is a quasi-covering.
3. \[soritcr3\] If $f:{\mathfrak X}\to {\mathfrak Y}$ is a quasi-covering, and $h:{\mathfrak Y}' \to {\mathfrak Y}$ a morphism in ${\mathsf {NFS}}$, then $f':{\mathfrak X}_{{\mathfrak Y}'} \to {\mathfrak Y}'$ is a quasi-covering.
Immediate.
\[cuairevdim0\] If $f:{\mathfrak X}\to {\mathfrak Y}$ is a quasi-covering in $x \in {\mathfrak X}$ then: $$\dim_{x} f =0$$
It is a consequence of the fact that ${\mathcal O}_{{\mathfrak X},x} {\widehat{\otimes}}_{{\mathcal O}_{{\mathfrak Y},f(x)}} k(f(x))$ is an artinian ring.
Observe that given ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ ideals of definition such that $f^{*}({\mathcal J}) {\mathcal O}_{{\mathfrak X}}\subset {\mathcal K}$, for all $x \in {\mathfrak X}$ it holds that $${\mathcal O}_{{\mathfrak X},x} {\widehat{\otimes}}_{{\mathcal O}_{{\mathfrak Y},f(x)}} k(f(x)) = {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longleftarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} {\mathcal O}_{X_{n},x} \otimes_{{\mathcal O}_{Y_{n},f_n(x)}} k(f(x)).$$ Over usual schemes quasi-coverings and pseudo-quasi-finite morphisms are equivalent notions. More generally we have the following.
Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a morphism in ${\mathsf {NFS}}$. If $f$ is a quasi-covering, then it is pseudo-quasi-finite. Furthermore, if $f$ is adic then the converse holds.
Suppose that $f$ is a quasi-covering and let ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ be $f$-compatible ideals of definition. Given $x \in {\mathfrak X}$ and $y=f(x)$, ${\mathcal O}_{{\mathfrak X},x} {\widehat{\otimes}}_{{\mathcal O}_{{\mathfrak Y},y}} k(y)$ is a finite $k(y)$-module and, therefore, $$\frac{{\mathcal O}_{X_{0},x}}{{\mathfrak m}_{Y_{0},y}{\mathcal O}_{X_{0},x}} = \frac{{\mathcal O}_{{\mathfrak X},x}}{ {\mathcal J}{\mathcal O}_{{\mathfrak X},x}} \otimes_{{\mathcal O}_{Y_{0},y}} k(y)$$ is $k(y)$-finite, so it follows that $f$ is pseudo-quasi-finite.
If $f$ is an adic morphism, $f^{-1}(y)=f_{0}^{-1}(y)$ for each $y \in {\mathfrak Y}$, so $${\mathcal O}_{X_{0},x}/{\mathfrak m}_{Y_{0},y}{\mathcal O}_{X_{0},x} = {\mathcal O}_{{\mathfrak X},x} {\widehat{\otimes}}_{{\mathcal O}_{{\mathfrak Y},y}} k(y)$$ for all $x \in {\mathfrak X}$ with $y=f(x)$. If $f$ is moreover pseudo-quasi-finite, it follows from [@EGA1 Corollaire (6.11.2)] that $f$ is a quasi-covering.
Every finite morphism $f:{\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$ is a quasi-covering.
Finite morphisms are adic and pseudo-quasi-finite. Therefore the result is consequence of the last proposition.
Nevertheless, by the next example, not every pseudo-finite morphism is a quasi-covering and, therefore, *pseudo-quasi-finite* does not imply *quasi-covering* for morphisms in ${\mathsf {NFS}}$.
For $r > 0$, the canonical projection $p: {\mathbb D}_{{\mathfrak X}}^{r} \to {\mathfrak X}$ is not a quasi-covering since $$\dim_{x} p \underset{\textrm{\ref{exdimalg}.(\ref{exdimalg1})}} = r > 0 \quad
\forall x \in {\mathfrak X}.$$ But considering an appropriate pair of ideals of definition, the scheme map $p_{0}= 1_{X_{0}}$ is finite.
In short, we have the following diagram of strict implications (with the conditions that imply adic morphism in italics): $$\begin{matrix}\textrm{\emph{closed immersion}}&{\Rightarrow}&\textrm{\emph{finite}}&{\Rightarrow}&
\textrm{quasi-covering}\\
\Downarrow & &\Downarrow & &\Downarrow\\
\textrm{pseudo-closed immersion}&{\Rightarrow}&\textrm{pseudo-finite}&{\Rightarrow}& \textrm{pseudo-quasi-finite}\\
\end{matrix}$$
Flat morphisms and completion morphisms {#sec2}
=======================================
In the first part of this section we discuss flat morphisms in ${\mathsf {NFS}}$. Whenever a morphism $$f = {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} f_{n}$$ is adic, the local criterion of flatness for formal schemes (Proposition \[clp\]) relates the flat character of $f$ and that of the morphisms $f_{n}$, for all $n \in {\mathbb N}$. In absence of the adic hypothesis this relation does not hold, though (Example \[excompl1\]). In the second part, we study the morphisms of completion in ${\mathsf {NFS}}$, a class of flat morphisms that are pseudo-closed immersions (so, they are closed immersions as topological maps). Even though the construction of the completion of a formal scheme along a closed formal subscheme is a natural one, it has not been systematically developed in the basic references about formal schemes. Morphisms of completion will be an essential ingredient in the main theorems of Section \[sec6\], namely, Theorems \[tppalnr\], \[tppalet\] and \[tppall\].
\[caracterizlocalplanos\] A morphism $f:{\mathfrak X}\to {\mathfrak Y}$ is *flat at $x \in {\mathfrak X}$* if ${\mathcal O}_{{\mathfrak X},x}$ is a flat ${\mathcal O}_{{\mathfrak Y},f(x)}$-module. We say that *$f$ is flat* if it is flat at $x$, for all $x \in {\mathfrak X}$.
Given ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ $f$-compatible ideals of definition, by [@b III, §5.4, Proposition 4] the following are equivalent:
1. $f$ is flat at $x \in {\mathfrak X}$.
2. ${\mathcal O}_{{\mathfrak X},x}$ is a flat ${\mathcal O}_{{\mathfrak Y},f(x)}$-module.
3. $\widehat{{\mathcal O}_{{\mathfrak X},x}}$ is a flat ${\mathcal O}_{{\mathfrak Y},f(x)}$-module.
4. $\widehat{{\mathcal O}_{{\mathfrak X},x}}$ is a flat $\widehat{{\mathcal O}_{{\mathfrak Y},f(x)}}$-module.
\[excompl1\] Let $K$ be a field let ${\mathbb A}_{K}^{1}= \operatorname{Spec}(K[T])$ and consider the closed subset $X' =V( \langle T \rangle) \subset {\mathbb A}_{K}^{1}$. The canonical morphism of completion of ${\mathbb A}_{K}^{1}$ along $X'$ $${\mathbb D}_{K}^{1} {\xrightarrow}{\kappa} {\mathbb A}_{K}^{1}$$ is flat but, the morphisms $$\operatorname{Spec}(K[T]/ \langle T \rangle ^{n+1}) {\xrightarrow}{\kappa_{n}} {\mathbb A}_{K}^{1}$$ are not flat, for every $n \in {\mathbb N}$.
\[clp\] (Local flatness criterion for formal schemes.) Given an *adic* morphism $f\colon{\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$, and an ideal of definition ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$, then ${\mathcal J}= f^{*}({\mathcal K}){\mathcal O}_{{\mathfrak X}} \subset {\mathcal O}_{{\mathfrak X}}$ is an ideal of definition. Let $\{f_{n}\colon X_{n} \to Y_{n}\}_{n \in {\mathbb N}}$ be the morphisms induced by $f$ through ${\mathcal K}$ and ${\mathcal J}$. The following assertions are equivalent:
1. The morphism $f$ is flat.
2. The morphism $f_{n}$ is flat, for all $n \in {\mathbb N}$.
3. The morphism $f_{0}$ is flat.
We may suppose that $f: {\mathfrak X}=\operatorname{Spf}(A) \to {\mathfrak Y}=\operatorname{Spf}(B)$ is in ${\mathsf {NFS}}$. Then if ${\mathcal K}=K^{{\triangle}}$ for an ideal of definition $K \subset B$, we have that ${\mathcal J}= (KA)^{{\triangle}}$ and the proposition is a consequence of [@AJL1 Lemma 7.1.1] and of the local flatness criterion for rings (*cf.* [@ma2 Theorem 22.3]).
Associated to a (usual) locally noetherian scheme $X$ and a closed subscheme of $X' \subset X$ there is a locally noetherian formal scheme $X_{/X'}$, called completion of $X$ along $X'$ and, a canonical morphism $\kappa: X_{/X'} \to X$ ([@EGA1 (10.8.3) and (10.8.5)]). Next, we define the completion of a formal scheme ${\mathfrak X}$ along a closed formal subscheme ${\mathfrak X}'\subset {\mathfrak X}$.
\[defcom\] Let ${\mathfrak X}$ be in ${\mathsf {NFS}}$ and let ${\mathfrak X}' \subset {\mathfrak X}$ be a closed formal subscheme defined by a coherent ideal ${\mathcal I}$ of ${\mathcal O}_{{\mathfrak X}}$. Given an ideal of definition ${\mathcal J}$ of ${\mathfrak X}$ we define the completion of a sheaf ${\mathcal F}$ on ${\mathfrak X}$ over ${\mathfrak X}'$, denoted by ${\mathcal F}_{/ {\mathfrak X}'}$, as the restriction to ${\mathfrak X}'$ of the sheaf $${\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longleftarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} \frac{{\mathcal F}}{({\mathcal J}+{\mathcal I})^{n+1}{\mathcal F}}.$$ The definition does not depend neither on the chosen ideal of definition ${\mathcal J}$ of ${\mathfrak X}$ nor on the coherent ideal ${\mathcal I}$ that defines ${\mathfrak X}'$.
We define the *completion of ${\mathfrak X}$ along ${\mathfrak X}'$*, and it will be denoted ${\mathfrak X}_{/ {\mathfrak X}'}$, as the topological ringed space whose underlying topological space is ${\mathfrak X}'$ and whose sheaf of topological rings is ${\mathcal O}_{{\mathfrak X}_{/{\mathfrak X}'}}$.
It is easy to check that ${\mathfrak X}_{/ {\mathfrak X}'}$ satisfies the hypothesis of [@EGA1 (10.6.3) and (10.6.4)], from which we deduce that:
1. The formal scheme ${\mathfrak X}_{/ {\mathfrak X}'}$ is locally noetherian.
2. The ideal $({\mathcal I}+{\mathcal J})_{/{\mathfrak X}'} \subset {\mathcal O}_{{\mathfrak X}_{/{\mathfrak X}'}}$ defined by the restriction to ${\mathfrak X}'$ of the sheaf $${\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longleftarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} \frac{{\mathcal J}+{\mathcal I}}{({\mathcal J}+{\mathcal I})^{n+1}}$$ is an ideal of definition of ${\mathfrak X}_{/{\mathfrak X}'}$.
3. It holds that ${\mathcal O}_{{\mathfrak X}_{/{\mathfrak X}'}}/ (({\mathcal I}+{\mathcal J})_{/{\mathfrak X}'})^{n+1}$ agrees with the restriction to ${\mathfrak X}'$ of the sheaf ${\mathcal O}_{{\mathfrak X}}/ ({\mathcal J}+{\mathcal I})^{n+1}$ for every $n \in {\mathbb N}$.
\[limcomple\] With the above notations, if $Z_{n} = ({\mathfrak X}', {\mathcal O}_{{\mathfrak X}}/ ({\mathcal J}+{\mathcal I})^{n+1})$ for all $n \in {\mathbb N}$, by \[lim\] we have that $${\mathfrak X}_{/ {\mathfrak X}'} = {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}}Z_{n}$$ For each $n \in {\mathbb N}$, let $X_{n} =({\mathfrak X}, {\mathcal O}_{{\mathfrak X}}/{\mathcal J}^{n+1})$ and $X'_{n} =({\mathfrak X}', {\mathcal O}_{{\mathfrak X}}/({\mathcal J}^{n+1}+{\mathcal I}))$. The canonical morphisms $$\frac{{\mathcal O}_{{\mathfrak X}}}{{\mathcal J}^{n+1}} {\twoheadrightarrow}\frac{{\mathcal O}_{{\mathfrak X}}}{({\mathcal J}+{\mathcal I})^{n+1}} {\twoheadrightarrow}\frac{{\mathcal O}_{{\mathfrak X}}}{{\mathcal J}^{n+1} + {\mathcal I}}$$ provide the closed immersions of schemes $X'_{n} {\xrightarrow}{j_{n}} Z_{n} {\xrightarrow}{\kappa_{n}} X_{n}$, such that the diagram, whose vertical maps are the obvious closed immersions,
X’\_[m]{} & \^[j\_[m]{}]{} & Z\_[m]{} & \^[\_[m]{}]{} & X\_[m]{}\
& & & &\
X’\_[n]{} & \^[j\_[n]{}]{} & Z\_[n]{} & \^[\_[n]{}]{} & X\_[n]{}\
commutes, for all $m \ge n \ge 0$. Then by \[lim\] we have the canonical morphisms in ${\mathsf {NFS}}$ $${\mathfrak X}' {\xrightarrow}{j} {\mathfrak X}_{/ {\mathfrak X}'} {\xrightarrow}{\kappa} {\mathfrak X}$$ where $j$ is a closed immersion (see \[ec\]). The morphism $\kappa$ as topological map is the inclusion and it is called *morphism of completion of ${\mathfrak X}$ along ${\mathfrak X}'$*.
Observe that $\kappa$ is adic only if ${\mathcal I}$ is contained in a ideal of definition of ${\mathfrak X}$, in which case ${\mathfrak X}= {\mathfrak X}_{/ {\mathfrak X}'}$ and $\kappa=1_{{\mathfrak X}}$.
\[compl\] If ${\mathfrak X}= \operatorname{Spf}(A)$ is in ${{\mathsf {NFS}}_{\mathsf {af}}}$ with $A$ a $J$-adic noetherian ring, and ${\mathfrak X}'= \operatorname{Spf}(A/I)$ is a closed formal subscheme of ${\mathfrak X}$, then $$\operatorname{\Gamma}({\mathfrak X}_{/ {\mathfrak X}'},{\mathcal O}_{{\mathfrak X}_{/ {\mathfrak X}'}}) = {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longleftarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} \frac{A}{(J+I)^{n+1}}=: {\widehat{A}}$$ and from [@EGA1 (10.2.2) and (10.4.6)] we have that ${\mathfrak X}_{/ {\mathfrak X}'}=\operatorname{Spf}({\widehat{A}})$ and the morphisms ${\mathfrak X}' {\xrightarrow}{j} {\mathfrak X}_{/ {\mathfrak X}'} {\xrightarrow}{\kappa} {\mathfrak X}$ correspond to the natural continuous morphisms $A \to {\widehat{A}}\to A /I $.
\[caractcom\] Given ${\mathfrak X}$ in ${\mathsf {NFS}}$ and ${\mathfrak X}'$ a closed formal subscheme of ${\mathfrak X}$, the morphism of completion $\kappa: {\mathfrak X}_{/ {\mathfrak X}'} \to {\mathfrak X}$ is a pseudo-closed immersion and étale (and therefore, from [@AJP Proposition 4.8], it is flat).
With the notations of \[limcomple\] we have that $$\kappa= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} \kappa_{n}.$$ Since $\kappa_{n}$ is a closed immersion for all $n \in {\mathbb N}$, it follows that $\kappa$ is a pseudo-closed immersion. In order to prove that $\kappa$ is an étale morphism we may suppose that ${\mathfrak X}=\operatorname{Spf}(A)$ and ${\mathfrak X}'= \operatorname{Spf}(A/I)$, where $A$ is a $J$-adic noetherian ring. Note that ${\mathfrak X}_{/ {\mathfrak X}'}=\operatorname{Spf}({\widehat{A}})$ where ${\widehat{A}}$ is the completion of $A$ for the $(J+I)$-adic topology and, therefore, is étale over $A$. By [@AJP 2.2], $\kappa$ is an étale morphism.
In Theorem \[caracmorfcompl\] we will see that the converse holds: every flat pseudo-closed immersion is a morphism of completion.
\[complmorf\] Given $f:{\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$, let ${\mathfrak X}' \subset {\mathfrak X}$ and ${\mathfrak Y}' \subset {\mathfrak Y}$ be closed formal subschemes given by ideals ${\mathcal I}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal L}\subset {\mathcal O}_{{\mathfrak Y}}$ such that $f^{*}({\mathcal L}){\mathcal O}_{{\mathfrak X}} \subset {\mathcal I}$, that is, $f({\mathfrak X}')\subset {\mathfrak Y}'$. If ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ are $f$-compatible ideals of definition, let us denote for all $n \in {\mathbb N}$ $$\begin{aligned}
X_{n} &= ({\mathfrak X}, {\mathcal O}_{{\mathfrak X}}/{\mathcal J}^{n+1}), & \quad
Y_{n} &= ({\mathfrak Y}, {\mathcal O}_{{\mathfrak Y}}/{\mathcal K}^{n+1}), \\
Z_{n} &= ({\mathfrak X}', {\mathcal O}_{{\mathfrak X}}/({\mathcal J}+{\mathcal I})^{n+1}), & \quad
W_{n} &= ({\mathfrak Y}', {\mathcal O}_{{\mathfrak Y}}/({\mathcal K}+{\mathcal L})^{n+1}) \\
X'_{n} &= ({\mathfrak X}', {\mathcal O}_{{\mathfrak X}}/({\mathcal J}^{n+1}+{\mathcal I})) & \quad
Y'_{n} &= ({\mathfrak Y}', {\mathcal O}_{{\mathfrak X}}/({\mathcal K}^{n+1}+{\mathcal L})).\end{aligned}$$ Then the morphism $f$ induces the following commutative diagram of locally noetherian schemes where the oblique maps are the canonical immersions:
X\_[n]{}&&\^[f\_[n]{}]{}&&Y\_[n]{}&&&\
&&&&&&&\
\^[\_[n]{}]{}&&X\_[m]{}&\^[f\_[m]{}]{}&&&Y\_[m]{}&\
&&\^[\_[m]{}]{}&&\^[’\_[n]{}]{}&&&\
Z\_[n]{}&&&\^[\_[n]{}]{}&W\_[n]{}&&\_[’\_[m]{}]{}&\
&&&&&&&\
\^[j\_[n]{}]{}&&Z\_[m]{}&\^[\_[m]{}]{}&&&W\_[m]{} &\
&&\^[j\_[m]{}]{}&&\^[j’\_[n]{}]{}&&&\
X’\_[n]{}&&&\^[f’\_[n]{}]{}& Y’\_[n]{}&&\_[j’\_[m]{}]{}&\
&&&&&&&\
&&X’\_[m]{}&&\^[f’\_[m]{}]{}&& Y’\_[m]{}&\
for all $m \ge n \ge 0$. Note that $$f' = {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} f'_{n}$$ is the restriction $f|_{{\mathfrak X}'} \colon {\mathfrak X}' \to {\mathfrak Y}'$. Applying the direct limit over ${n \in {\mathbb N}}$ we obtain a morphism $${\widehat{f}}: {\mathfrak X}_{/ {\mathfrak X}'} \to {\mathfrak Y}_{/ {\mathfrak Y}'}$$ in ${\mathsf {NFS}}$, such that the following diagram is commutative: $$\label{diagrcom}
\begin{diagram}[height=2em,w=2em,labelstyle=\scriptstyle]
{\mathfrak X}& \rTto^{f} & {\mathfrak Y}\\
\uTto^{\kappa} & & \uTto^{\kappa'} \\
{\mathfrak X}_{/ {\mathfrak X}'} & \rTto^{{\widehat{f}}} & {\mathfrak Y}_{/ {\mathfrak Y}'} \\
\uTto & & \uTto \\
{\mathfrak X}' & \rTto^{f|_{{\mathfrak X}'}} & {\mathfrak Y}'
\end{diagram}$$ We will call ${\widehat{f}}$ the *completion of $f$ along ${\mathfrak X}'$ and ${\mathfrak Y}'$*.
\[complafin\] Suppose that $f: {\mathfrak X}=\operatorname{Spf}(A) \to {\mathfrak Y}=\operatorname{Spf}(B)$ is in ${{\mathsf {NFS}}_{\mathsf {af}}}$ and that ${\mathfrak X}'= \operatorname{Spf}(A/I)$ and ${\mathfrak Y}'= \operatorname{Spf}(B/L)$ with $LA \subset I$. If $J \subset A$ and $K \subset B$ are ideals of definition such that $KA \subset J$, the morphism ${\widehat{f}}: {\mathfrak X}_{/ {\mathfrak X}'} \to {\mathfrak Y}_{/ {\mathfrak Y}'}$ corresponds to the morphism induced by $B \to A$ $$\widehat{B}\to {\widehat{A}}$$ (*cf.* [@EGA1 (10.4.6)]) where ${\widehat{A}}$ is the completion of $A$ for the $(I+J)$-adic topology and $\widehat{B}$ denotes the completion of $B$ for the $(K+L)$-adic topology.
\[cccom\] Given $f:{\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$, let ${\mathfrak Y}' \subset {\mathfrak Y}$ be a closed formal subscheme and ${\mathfrak X}' = f^{-1}({\mathfrak Y}')$. Then, $${\mathfrak X}_{/ {\mathfrak X}'} = {\mathfrak Y}_{/ {\mathfrak Y}'} \times_{{\mathfrak Y}} {\mathfrak X}.$$
We may restrict to the case in which ${\mathfrak X}=\operatorname{Spf}(A)$, ${\mathfrak Y}=\operatorname{Spf}(B)$ and ${\mathfrak Y}'= \operatorname{Spf}(B/L)$ are affine formal schemes and $J \subset A$ and $K \subset B$ are ideals of definition such that $KA \subset J$. By hypothesis, ${\mathfrak X}'= \operatorname{Spf}(A/LA)$, so ${\mathfrak X}_{/ {\mathfrak X}'} = \operatorname{Spf}({\widehat{A}})$ where ${\widehat{A}}$ is the completion of $A$ for the $(J+LA)$-adic topology. On the other hand, ${\mathfrak Y}_{/ {\mathfrak Y}'} = \operatorname{Spf}({\widehat{B}})$ where ${\widehat{B}}$ denotes the completion of $B$ for the $(K+L)$-adic topology and it holds that $${\widehat{B}}{\widehat{\otimes}}_{B} A= B{\widehat{\otimes}}_{B} A={\widehat{A}},$$ since $J+(K+L)A = J+KA+LA= J+LA$, the result follows.
\[complsorit\] Given $f:{\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$, let us consider closed formal subschemes ${\mathfrak X}' \subset {\mathfrak X}$ and ${\mathfrak Y}' \subset {\mathfrak Y}$ such that $f({\mathfrak X}')\subset {\mathfrak Y}'$.
1. Let ${\mathcal P}$ be one of the following properties of morphisms in ${\mathsf {NFS}}$:
*pseudo-finite type, pseudo-finite, pseudo-closed immersion, pseudo-quasi-finite, quasi-covering, flat, separated, radicial, smooth, unramified, étale.*
If $f$ satisfies ${\mathcal P}$, then so does ${\widehat{f}}$.
2. Moreover, if ${\mathfrak X}' = f^{-1}({\mathfrak Y}')$, let ${\mathcal Q}$ be one of the following properties of morphisms in ${\mathsf {NFS}}$:
*adic, finite type, finite, closed immersion, smooth adic, unramified adic, étale adic.*
Then, if $f$ satisfies ${\mathcal Q}$, then so does ${\widehat{f}}$.
Suppose that $f$ is flat and let us prove that ${\widehat{f}}$ is flat. The question is local so we may assume $f: {\mathfrak X}=\operatorname{Spf}(A) \to {\mathfrak Y}=\operatorname{Spf}(B)$ in ${{\mathsf {NFS}}_{\mathsf {af}}}$, ${\mathfrak X}'= \operatorname{Spf}(A/I)$ and ${\mathfrak Y}'= \operatorname{Spf}(B/L)$ with $LA \subset I$. Let $J \subset A$ and $ K \subset B$ be ideals of definition such that $KA \subset J$ and, ${\widehat{A}}$ and $\widehat{B}$ the completions of $A$ and $B$ for the topologies given by $(I+J)\subset A$ and $(K+L) \subset B$, respectively. By [@b III, §5.4, Proposition 4] we have that the morphism $\widehat{B}\to {\widehat{A}}$ is flat and, from \[complafin\] and [@AJL1 Lemma 7.1.1] it follows that ${\widehat{f}}$ is flat.
Suppose that $f$ satisfies any of the other properties ${\mathcal P}$ and let us prove that ${\widehat{f}}$ inherits them using the commutativity of the diagram
[X]{}& \^[f]{} & [Y]{}\
\^[ ]{} & & \^[ ’]{}\
[X]{}\_[/ [X]{}’]{} & \^ & [Y]{}\_[/ [Y]{}’]{}\
where the vertical arrows are morphisms of completion. Since all of these properties ${\mathcal P}$ are stable under composition and a morphism of completion satisfies ${\mathcal P}$ (Proposition \[caractcom\]) we have that ${\mathcal P}$ holds for $f \circ \kappa = \kappa' \circ {\widehat{f}}$. If ${\mathcal P}$ is smooth, unramified or étale the result is immediate from [@AJP Proposition 2.13].
If ${\mathcal P}$ is any of the other properties, then closed immersions verify ${\mathcal P}$ and ${\mathcal P}$ is stable under composition and under base-change in ${\mathsf {NFS}}$. Therefore, since $\kappa' \circ {\widehat{f}}$ has ${\mathcal P}$ and $\kappa'$ is separated (Proposition \[caractcom\]), by the analogous argument in ${\mathsf {NFS}}$ to the one in ${\mathsf {Sch}}$ [@EGA1 (5.2.7), $i),\, ii) \Rightarrow iii)$] we get that ${\widehat{f}}$ also satisfies ${\mathcal P}$.
Finally, if $f$ is adic, from Proposition \[cccom\] and from [@AJP 1.3], we deduce that ${\widehat{f}}$ is adic. Then, if ${\mathcal Q}$ is any of the properties in statement (2) and $f$ satisfies ${\mathcal Q}$, by (1) so does ${\widehat{f}}$.
Unramified morphisms {#sec3}
====================
Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a morphism of locally noetherian formal schemes. Given ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ $f$-compatible ideals of definition, express $f$ as a limit $$f:{\mathfrak X}\to {\mathfrak Y}= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} (f_{n}:X_{n} \to Y_{n}).$$ We begin relating the unramified character of $f:{\mathfrak X}\to {\mathfrak Y}$ and that of the underlying ordinary scheme morphisms $\{f_{n}\}_{n \in {\mathbb N}}$.
\[nrfn\] With the previous notations, the morphism $f$ is unramified if and only if $f_{n}:X_{n} \to Y_{n}$ is unramified, for all $n \in {\mathbb N}$.
Notice that both conditions in the statement imply that $f$ is a pseudo-finite type morphism. Applying [@AJP Proposition 4.6] we have to show that ${\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}} = 0$ is equivalent to $\Omega^{1}_{X_{n}/Y_{n}}=0$, for all $n \in {\mathbb N}$. If ${\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}} =0$, by the Second Fundamental Exact Sequence ([@AJP Proposition 3.13]) for the morphisms $$X_{n} {\hookrightarrow}{\mathfrak X}{\xrightarrow}{f} {\mathfrak Y},$$ we have that $\Omega^{1}_{X_{n}/ {\mathfrak Y}}=0$, for all $n \in {\mathbb N}$. From the First Fundamental Exact Sequence ([@AJP Proposition 3.10]) associated to the morphisms $$X_{n} {\xrightarrow}{f_{n}} Y_{n} {\hookrightarrow}{\mathfrak Y},$$ it follows that $\Omega^{1}_{X_{n}/Y_{n}}=0$. The converse follows from the identification $${\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}} ={\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longleftarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} \Omega ^{1}_{X_{n}/Y_{n}}$$ (*cf*. [@AJP §1.9] ).
\[pecimplnoram\] With the previous notations, if the morphisms $f_{n}:X_{n} \to Y_{n}$ are immersions for all $n \in {\mathbb N}$, then $f$ is unramified.
In the class of adic morphisms in ${\mathsf {NFS}}$ the following proposition provides a criterion, stronger than the last result, to determine when a morphism $f$ is unramified.
\[fnrf0nr\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be an *adic* morphism in ${\mathsf {NFS}}$ and let ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ be an ideal of definition. Write $$f={\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} f_{n}$$ by taking ideals of definition ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ and ${\mathcal J}=f^{*}({\mathcal K}){\mathcal O}_{{\mathfrak X}} \subset {\mathcal O}_{{\mathfrak X}}$. The morphism $f$ is unramified if and only if the induced morphism $f_{0}:X_{0} \to Y_{0}$ is unramified.
If $f$ is unramified by Proposition \[nrfn\] we have that $f_{0}$ is unramified. Conversely, suppose that $f_{0}$ is unramified and let us prove that ${\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}}=0$. The question is local so we may assume that $f:{\mathfrak X}=\operatorname{Spf}(A) \to {\mathfrak Y}= \operatorname{Spf}(B)$ is in ${{\mathsf {NFS}}_{\mathsf {af}}}$ and that ${\mathcal J}=J^{{\triangle}},\, $ with $J \subset A$ an ideal of definition. By hypothesis $\Omega^{1}_{X_{0}/Y_{0}}=0$ and thus, since $f$ is adic it holds that $$\label{eqmodifceromodiff0cero}
{\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}} \otimes_{{\mathcal O}_{{\mathfrak X}}} {\mathcal O}_{X_{0}}\underset{\textrm{\cite[3.8]{AJP}}} = \Omega^{1}_{X_{0}/Y_{0}}=0.$$ Then by the equivalence of categories [@EGA1 (10.10.2)], the last equality says that ${\widehat{\Omega}}^{1}_{A/B}/J {\widehat{\Omega}}^{1}_{A/B}=0$. Since $A$ is a $J$-adic ring it holds that $J$ is contained in the Jacobson radical of $A$. Moreover, [@AJP Proposition 3.3] implies that ${\widehat{\Omega}}^{1}_{A/B}$ is a finite type $A$-module. From Nakayama’s lemma we deduce that ${\widehat{\Omega}}^{1}_{A/B}=0$ and therefore, ${\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}}= ({\widehat{\Omega}}^{1}_{A/B})^{{\triangle}}= 0$. Applying [@AJP Proposition 4.6] it follows that $f$ is unramified.
The following example illustrates that in the non adic case the analogous of the last proposition does not hold.
\[framfonram\] Let $K$ be a field and $p:{\mathbb D}^{1}_{K} \to \operatorname{Spec}(K)$ be the projection morphism of the formal disc of dimension $1$ over $\operatorname{Spec}(K)$. By [@AJP Example 3.14] we have that ${\widehat{\Omega}}^{1}_{p} =(K[[T]] {\widehat{d}}T)^{{\triangle}}$ and therefore, ${\mathbb D}^{1}_{K}$ is ramified over $K$ ([@AJP Proposition 4.6]). However, given the ideal of definition $\langle T \rangle \subset K[[T]]$ the induced morphism $p_{0} = 1_{\operatorname{Spec}(K)}$ is unramified.
Let us consider for a morphism $f\colon{\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$ the notation established at the beginning of the section. In view of the example, our next goal will be to determine when the morphism $f$ such that $f_{0}$ is unramified but $f$ itself is not necessarily adic, is unramified (Corollary \[corf0imfpnr\]). In order to do that, we will need some results that describe the local behavior of unramified morphisms. Next, we provide local characterizations of unramified morphisms in ${\mathsf {NFS}}$, generalizing the analogous properties in the category of schemes (*cf.* [@EGA44 (17.4.1)]).
\[caraclocalpnr\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a morphism in ${\mathsf {NFS}}$ of pseudo-finite type. For $x \in {\mathfrak X}$ and $y=f(x)$ the following conditions are equivalent:
1. $f$ is unramified at $x$.
2. $f^{-1}(y)$ is an unramified $k(y)$-formal scheme at $x$.
3. ${\mathfrak m}_{{\mathfrak X},x}\widehat{{\mathcal O}_{{\mathfrak X},x}} = {\mathfrak m}_{{\mathfrak Y},y}\widehat{{\mathcal O}_{{\mathfrak X},x}}$ and $k(x)|k(y)$ is a finite separable extension.
4. ${\widehat{\Omega}}^{1}_{{\mathcal O}_{{\mathfrak X},x}/ {\mathcal O}_{{\mathfrak Y},y}} =0$.
5. $({\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}})_{x}=0$.
6. ${\mathcal O}_{{\mathfrak X},x}$ is a formally unramified ${\mathcal O}_{{\mathfrak Y},y}$-algebra for the adic topologies.
7. $\widehat{{\mathcal O}_{{\mathfrak X},x}}$ is a formally unramified $\widehat{{\mathcal O}_{{\mathfrak Y},y}}$-algebra for the adic topologies.
Keep the notation from the beginning of this section and write $$f:{\mathfrak X}\to {\mathfrak Y}= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} (f_{n}:X_{n} \to Y_{n})$$ (1) ${\Leftrightarrow}$ (2) By Proposition \[nrfn\], $f$ is unramified at $x$ if and only if all the morphisms $f_{n}:X_{n} \to Y_{n}$ are unramified at $x$. Applying [@EGA44 (17.4.1)], this is equivalent to $f_{n}^{-1}(y)$ being an unramified $k(y)$-scheme at $x$, for all $n \in {\mathbb N}$, which is also equivalent to $$f^{-1} (y) \underset{\textrm{\ref{fibra}}}= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} f_{n}^{-1}(y)$$ being an unramified $k(y)$-formal scheme at $x$.
\(1) ${\Rightarrow}$ (3) The assertion (1) is equivalent to $f_{n}:X_{n} \to Y_{n}$ being unramified at $x$, for all $n \in {\mathbb N}$, and from [@EGA44 *loc. cit.*] it follows that $k(x)|k(y)$ is a finite separable extension, and that ${\mathfrak m}_{X_{n},x} = {\mathfrak m}_{Y_{n},y} {\mathcal O}_{X_{n},x}$, for all $n \in {\mathbb N}$. Hence, $${\mathfrak m}_{{\mathfrak X},x} \widehat{{\mathcal O}_{{\mathfrak X},x}} = {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longleftarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} {\mathfrak m}_{X_{n},x} = {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longleftarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} {\mathfrak m}_{Y_{n},y} {\mathcal O}_{X_{n},x} = {\mathfrak m}_{{\mathfrak Y},y}\widehat{{\mathcal O}_{{\mathfrak X},x}}.$$
\(4) ${\Leftrightarrow}$ ($4'$) By [@AJP Proposition 3.3] it holds that $({\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}})_{x}$ is a finite type ${\mathcal O}_{{\mathfrak X},x}$-module and therefore, $${\widehat{\Omega}}^{1}_{ {\mathcal O}_{{\mathfrak X},x}/ {\mathcal O}_{{\mathfrak Y},y}} = \widehat{({\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}})_{x}} = ({\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}})_{x} \otimes_{{\mathcal O}_{{\mathfrak X},x}} \widehat{{\mathcal O}_{{\mathfrak X},x}}.$$ Then, since $\widehat{{\mathcal O}_{{\mathfrak X},x}}$ is a faithfully flat ${\mathcal O}_{{\mathfrak X},x}$-algebra, ${\widehat{\Omega}}^{1}_{ {\mathcal O}_{{\mathfrak X},x}/ {\mathcal O}_{{\mathfrak Y},y}}=0$ if and only if $({\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}})_{x}=0$.
\(3) ${\Rightarrow}$ (4) Since $k(x)|k(y)$ is a finite separable extension we have that $\Omega^{1}_{k(x)/k(y)}=0$ and from [@AJP Proposition 3.3] $ {\widehat{\Omega}}^{1}_{{\mathcal O}_{{\mathfrak X},x}/ {\mathcal O}_{{\mathfrak Y},y}}= \widehat{({\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}})_{x}}$ is a finite type $\widehat{{\mathcal O}_{{\mathfrak X},x}}$-module. Therefore, it holds that $${\widehat{\Omega}}^{1}_{{\mathcal O}_{{\mathfrak X},x}/ {\mathcal O}_{{\mathfrak Y},y}} \otimes_{\widehat{{\mathcal O}_{{\mathfrak X},x}}} k(x)= {\widehat{\Omega}}^{1}_{({\mathcal O}_{{\mathfrak X},x} \otimes_{{\mathcal O}_{{\mathfrak Y},y}} k(y))/k(y)} =\Omega^{1}_{k(x)/k(y)}=0.$$ By Nakayama’s lemma, ${\widehat{\Omega}}^{1}_{ {\mathcal O}_{{\mathfrak X},x}/ {\mathcal O}_{{\mathfrak Y},y}} =0$.
\(4) ${\Leftrightarrow}$ (5) It is straightforward from [@EGA41 (**0**, 20.7.4)].
\(5) ${\Leftrightarrow}$ ($5'$) Immediate.
($4'$) ${\Rightarrow}$ (1) Since ${\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}} \in \operatorname{\mathsf{Coh}}({\mathfrak X})$ ([@AJP Proposition 3.3]), assertion $(4')$ implies that there exists an open subset ${\mathfrak U}\subset {\mathfrak X}$ with $x \in {\mathfrak U}$ such that $({\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}})|_{{\mathfrak U}}=0$ and therefore, by [@AJP Proposition 4.6] we have that $f$ is unramified at $x$.
\[corcaraclocalpnr\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a pseudo-finite type morphism in ${\mathsf {NFS}}$. The following conditions are equivalent:
1. $f$ is unramified.
2. For all $x \in {\mathfrak X}$, $f^{-1}(f(x))$ is an unramified $k(f(x))$-formal scheme at $x$.
3. For all $x \in {\mathfrak X}$, ${\mathfrak m}_{{\mathfrak X},x}\widehat{{\mathcal O}_{{\mathfrak X},x}} = {\mathfrak m}_{{\mathfrak Y},f(x)}\widehat{{\mathcal O}_{{\mathfrak X},x}}$ and $k(x)|k(f(x))$ is a finite separable extension.
4. ${\widehat{\Omega}}^{1}_{{\mathcal O}_{{\mathfrak X},x}/ {\mathcal O}_{{\mathfrak Y},f(x)}} =0$, for all $x \in {\mathfrak X}$.
5. For all $x \in {\mathfrak X}$, $({\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}})_{x} =0$.
6. For all $x \in {\mathfrak X}$, ${\mathcal O}_{{\mathfrak X},x}$ is a formally unramified ${\mathcal O}_{{\mathfrak Y},f(x)}$-algebra for the adic topologies.
7. For all $x \in {\mathfrak X}$, $\widehat{{\mathcal O}_{{\mathfrak X},x}}$ is a formally unramified $\widehat{{\mathcal O}_{{\mathfrak Y},f(x)}}$-algebra for the adic topologies.
\[corpnrimplcr\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a pseudo-finite type morphism in ${\mathsf {NFS}}$. If $f$ is unramified at $x \in {\mathfrak X}$, then $f$ is a quasi-covering at $x$.
By assertion (3) of Proposition \[caraclocalpnr\] we have that $${\mathcal O}_{{\mathfrak X},x} {\widehat{\otimes}}_{{\mathcal O}_{{\mathfrak Y},f(x)}} k(f(x))=k(x)$$ with $k(x)|k(f(x))$ a finite extension and therefore, $f$ is a quasi-covering at $x$ (see Definition \[defncuasireves\]).
\[pnrdim0\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a pseudo-finite type morphism in ${\mathsf {NFS}}$. If $f$ is unramified at $x \in {\mathfrak X}$, then $\dim_{x} f =0$.
It is straightforward from the previous Corollary and Proposition \[cuairevdim0\].
\[fonrydislocal\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a pseudo-finite type morphism in ${\mathsf {NFS}}$. Given $x \in {\mathfrak X}$ and $y=f(x)$ the following conditions are equivalent:
1. $f$ is unramified at $x$.
2. $f_{0}:X_{0} \to Y_{0}$ is unramified at $x$ and $\widehat{{\mathcal O}_{{\mathfrak X},x}} \otimes_{\widehat{{\mathcal O}_{{\mathfrak Y},y}}} k(y) = k(x)$.
If $f$ is unramified at $x$, then $f_{0}$ is unramified at $x$ (Proposition \[nrfn\]). Moreover, assertion (3) of Proposition \[caraclocalpnr\] implies that $\widehat{{\mathcal O}_{{\mathfrak X},x}} \otimes_{\widehat{{\mathcal O}_{{\mathfrak Y},y}}} k(y) = k(x)$ so (1) $\Rightarrow$ (2) holds. Let us prove that (2) $\Rightarrow$ (1). Since $f_{0}$ is unramified at $x$ we have that $k(x)|k(y)$ is a finite separable extension (*cf.* [@EGA44 (17.4.1)]). From the equality $\widehat{{\mathcal O}_{{\mathfrak X},x}} \otimes_{\widehat{{\mathcal O}_{{\mathfrak Y},y}}} k(y) = k(x)$ we deduce that ${\mathfrak m}_{{\mathfrak X},x}\widehat{{\mathcal O}_{{\mathfrak X},x}} = {\mathfrak m}_{{\mathfrak Y},y}\widehat{{\mathcal O}_{{\mathfrak X},x}}$. Thus, the morphism $f$ and the point $x$ satisfy assertion (3) of Proposition \[caraclocalpnr\] and it follows that $f$ is unramified at $x$.
Now we are ready to state the non adic version of Proposition \[fnrf0nr\]:
\[corf0imfpnr\] Given $f:{\mathfrak X}\to {\mathfrak Y}$ a morphism in ${\mathsf {NFS}}$ of pseudo-finite type let ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ be $f$-compatible ideals of definition and let $f_0 \colon X_0 \to Y_0$ be the induced morphism. The following conditions are equivalent:
1. The morphism $f$ is unramified.
2. The morphism $f_{0}$ is unramified and, for all $x \in {\mathfrak X}$, $f^{-1}(y)=f_{0}^{-1}(y)$ with $y=f(x)$.
Suppose that $f$ is unramified and fix $x \in {\mathfrak X}$ and $y=f(x)$. By Proposition \[fonrydislocal\] we have that $f_{0}$ is unramified and that $\widehat{{\mathcal O}_{{\mathfrak X},x}} \otimes_{\widehat{{\mathcal O}_{{\mathfrak Y},y}}} k(y) = k(x)$. Therefore, ${\mathcal J}(\widehat{{\mathcal O}_{{\mathfrak X},x}} \otimes_{\widehat{{\mathcal O}_{{\mathfrak Y},y}}} k(y) )= 0$ and applying Lemma \[fibradiscr\] we deduce that $f^{-1}(y)=f_{0}^{-1}(y)$. Conversely, suppose that (2) holds and let us show that given $x \in {\mathfrak X}$, the morphism $f$ is unramified at $x$. If $y=f(x)$, we have that $f_{0}^{-1}(y)$ is an unramified $k(y)$-scheme at $x$ (*cf.* [@EGA44 (17.4.1)]) and since $f^{-1}(y)=f_{0}^{-1}(y)$, from Proposition \[caraclocalpnr\] it follows that $f$ is unramified at $x$.
\[fibradiscr\] Let $A$ be a $J$-adic noetherian ring such that for all open prime ideals ${\mathfrak p}\subset A$, $J_{{\mathfrak p}}=0$. Then $J=0$ and therefore, the $J$-adic topology in $A$ is the discrete topology.
Since every maximal ideal ${\mathfrak m}\subset A$ is open for the $J$-adic topology, we have that $J_{{\mathfrak m}}=0$, for all maximal ideal ${\mathfrak m}\subset A$, so $J=0$.
As a consequence of Corollary \[corf0imfpnr\] it holds that:
- If $f:{\mathfrak X}\to {\mathfrak Y}$ is an unramified morphism in ${\mathsf {NFS}}$ then $f^{-1}(y)$ is a usual scheme for all $x \in {\mathfrak X}$ where $y=f(x)$.
- In Corollary \[corcaraclocalpnr\] assertion (2) may be written:
- *For all $x \in {\mathfrak X},\, y=f(x)$, $f^{-1}(y)$ is a unramified $k(y)$-scheme at $x$.*
From Proposition \[caraclocalpnr\] we obtain the following result, in which we provide a description of pseudo-closed immersions that will be used in the characterization of completion morphisms (Theorem \[caracmorfcompl\]).
\[pecigf0ecnr\] Given $f:{\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$, let ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ be $f$-compatible ideals of definition and express $$f={\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n\in {\mathbb N}}} \end{array}} f_{n}.$$ The morphism $f$ is a pseudo-closed immersion if and only if $f$ is unramified and $f_{0}:X_{0} \to Y_{0}$ is a closed immersion.
If $f$ is a pseudo-closed immersion, by Corollary \[pecimplnoram\] it follows that $f$ is unramified. Conversely, suppose that $f$ is unramified and that $f_{0}$ is a closed immersion and let us show that $f_{n}:X_{n} \to Y_{n}$ is a closed immersion, for each $n \in {\mathbb N}$. By [@EGA1 (4.2.2.(ii))] it suffices to prove that, for all $x \in {\mathfrak X}$ with $y=f(x)$, the morphism ${\mathcal O}_{Y_{n},y} \to {\mathcal O}_{X_{n},x}$ is surjective, for all $n \in {\mathbb N}$. Fix $x \in {\mathfrak X}$, $ y=f(x) \in {\mathfrak Y}$ and $n \in {\mathbb N}$. Since $f_{0}$ is a closed immersion, by [@EGA1 *loc. cit.*], we have that ${\mathcal O}_{Y_{0},y} \to {\mathcal O}_{X_{0},x}$ is surjective and therefore, $\operatorname{Spf}(\widehat{{\mathcal O}_{{\mathfrak X}, x}}) \to \operatorname{Spf}(\widehat{{\mathcal O}_{{\mathfrak Y}, y}})$ is a pseudo-finite morphism, so, the morphism ${\mathcal O}_{Y_{n},y} \to {\mathcal O}_{X_{n},x}$ is finite. On the other hand, the morphism $f$ is unramified therefore by Proposition \[nrfn\] we get that $f_{n}$ is unramified and applying Proposition \[caraclocalpnr\] we obtain that ${\mathfrak m}_{Y_{n},y} {\mathcal O}_{X_{n},x}= {\mathfrak m}_{X_{n},x}$. Then by Nakayama’s lemma we conclude that ${\mathcal O}_{Y_{n},y} \to {\mathcal O}_{X_{n},x}$ is a surjective morphism.
Smooth morphisms {#sec4}
================
The contents of this section can be structured in two parts. In the first part we study the relationship between the smoothness of a morphism $$f= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} f_{n}$$ in ${\mathsf {NFS}}$ and the smoothness of the ordinary scheme morphisms $\{f_{n}\}_{n \in {\mathbb N}}$. In the second part, we provide a local factorization for smooth morphisms (Proposition \[factpl\]). In this section we also prove in Corollary \[criteriojacobiano\] the matrix Jacobian criterion, that is a useful explicit condition in terms of a matrix rank for determining whether a closed subscheme of the affine formal space or of the affine formal disc is smooth or not.
\[lfn\] Given $f:{\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$, let ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ be $f$-compatible ideals of definition and write $$f={\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n\in {\mathbb N}}} \end{array}} f_{n}.$$ If $f_{n}:X_{n} \to Y_{n}$ is smooth, for all $n \in {\mathbb N}$, then $f$ is smooth.
By [@AJP Proposition 4.1] we may assume that $f$ is in ${{\mathsf {NFS}}_{\mathsf {af}}}$. Let $Z$ be an affine scheme, consider a morphism $w: Z \to {\mathfrak Y}$, a closed ${\mathfrak Y}$-subscheme $T {\hookrightarrow}Z$ given by a square zero ideal and a ${\mathfrak Y}$-morphism $u: T \to {\mathfrak X}$. Since $f$ and $w$ are morphisms of affine formal schemes we find an integer $m \ge 0$ such that $w^{*}({\mathcal K}^{m+1}) {\mathcal O}_{Z} =0$ and $u^{*}({\mathcal J}^{m+1}){\mathcal O}_{T}=0$ and therefore $u$ and $w$ factors as $T {\xrightarrow}{u_{m}} X_{m} {\xrightarrow}{i_{m}} {\mathfrak X}$ and $Z {\xrightarrow}{w_{m}} Y_{m} {\xrightarrow}{i_{m}} {\mathfrak Y}$, respectively. Since $f_{m}$ is formally smooth, there exists a $Y_{m}$-morphism $v_{m}: Z \to X_{m}$ such that the following diagram is commutative
T & & & Z\
\^[u\_[m]{}]{} & & (3,2)\^[v\_[m]{}]{} & \^[w\_[m]{}]{}\
X\_[m]{} & \^[f\_[m]{}]{} & & Y\_[m]{}\
\^[i\_[m]{}]{} & & &\
[X]{}& \^[f]{} & & [Y]{}\
Thus the ${\mathfrak Y}$-morphism $v:=i_{m} \circ v_{m}$ satisfies that $v|_{T}=u$ and then, $f$ is formally smooth. Moreover, since $f_{0}$ is a finite type morphism, it holds that $f$ is of pseudo-finite type and therefore, $f$ is smooth.
\[ladfn\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be an *adic* morphism in ${\mathsf {NFS}}$ and consider ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ an ideal of definition. The morphism $f$ is smooth if and only if all the scheme morphisms $\{f_{n}:X_{n} \to Y_{n}\}_{n \in {\mathbb N}}$, determined by the ideals of definition ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ and ${\mathcal J}=f^{*}({\mathcal K}){\mathcal O}_{{\mathfrak X}}$, are smooth.
If $f$ is adic, by [@EGA1 (10.12.2)], we have that for each $n \in {\mathbb N}$, the diagram
[X]{}& \^[f]{} & [Y]{}\
& &\
X\_[n]{} & \^[f\_[n]{}]{} & Y\_[n]{}\
is a cartesian square. Then by base-change ([@AJP Proposition 2.9 (2)]) we have that $f_{n}$ is smooth, for all $n \in {\mathbb N}$. The converse follows from the previous proposition.
Next example shows us that the converse of Proposition \[lfn\] does not hold in general.
\[peynopen\] Let $K$ be a field and $ {\mathbb A}_{K}^{1}= \operatorname{Spec}(K[T])$. For the closed subset $X =V( \langle T \rangle) \subset {\mathbb A}_{K}^{1}$, Proposition \[caractcom\] implies that the canonical completion morphism $${\mathbb D}_{K}^{1} {\xrightarrow}{\kappa} {\mathbb A}_{K}^{1}$$ of ${\mathbb A}_{K}^{1}$ along $X$ is étale. However, picking in ${\mathbb A}_{K}^{1}$ the ideal of definition $0$, the morphisms $$\operatorname{Spec}(K[T]/ \langle T \rangle ^{n+1}) {\xrightarrow}{\kappa_{n}} {\mathbb A}_{K}^{1}$$ are not flat, whence it follows that $\kappa_{n}$ can not be smooth for all $n \in {\mathbb N}$ (see [@AJP Proposition 4.8]).
Our next goal will be to determine the relation between smoothness of a morphism $$f= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} f_{n}$$ and that of $f_{0}$ (Corollaries \[flf0l\] and \[corf0imfpl\]). In order to do that, we need to characterize smoothness locally.
\[pligplafibr\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a pseudo-finite type morphism in ${\mathsf {NFS}}$. Given $x \in {\mathfrak X}$ and $y = f(x)$ the following conditions are equivalent:
1. The morphism $f$ is smooth at $x$.
2. ${\mathcal O}_{{\mathfrak X},x}$ is a formally smooth ${\mathcal O}_{{\mathfrak Y},y}$-algebra for the adic topologies.
3. $\widehat{{\mathcal O}_{{\mathfrak X},x}}$ is a formally smooth $\widehat{{\mathcal O}_{{\mathfrak Y},y}}$-algebra for the adic topologies.
4. The morphism $f$ is flat at $x$ and $f^{-1}(y)$ is a $k(y)$-formal scheme smooth at $x$.
The question is local and $f$ is of pseudo-finite type, so we may assume that $f: {\mathfrak X}= \operatorname{Spf}(A) \to {\mathfrak Y}= \operatorname{Spf}(B)$ is in ${{\mathsf {NFS}}_{\mathsf {af}}}$, with $A = B\{T_{1},\, \ldots ,T_{r}\}[[Z_{1},\ldots, Z_{s}]]/I$ and $I \subset B':= B\{T_{1},\, \ldots ,T_{r}\}[[Z_{1},\ldots, Z_{s}]]$ an ideal ([@AJP Proposition 1.7]). Let ${\mathfrak p}\subset A$ be the open prime ideal corresponding to $x$, let ${\mathfrak q}\subset B'$ be the open prime such that ${\mathfrak p}= {\mathfrak q}/I$ and let ${\mathfrak r}\subset B$ be the open prime ideal corresponding to $y$.
\(1) ${\Rightarrow}$ (3) Replacing ${\mathfrak X}$ by a sufficiently small open neighborhood of $x$ we may suppose that $A$ is a formally smooth $B$-algebra. Then, by [@EGA41 (**0**, 19.3.5)] we have that $A_{{\mathfrak p}}$ is a formally smooth $B_{{\mathfrak r}}$-algebra and [@EGA41 (**0**, 19.3.6)] implies that $\widehat{{\mathcal O}_{{\mathfrak X},x}}=\widehat{A_{{\mathfrak p}}}$ is a formally smooth $\widehat{{\mathcal O}_{{\mathfrak Y},y}}=\widehat{B_{{\mathfrak r}}}$-algebra.
\(2) ${\Leftrightarrow}$ (3) It is a consequence of [@EGA41 (**0**, 19.3.6)].
\(3) ${\Rightarrow}$ (1) By [@EGA41 (**0**, 19.3.6)], assertion (3) is equivalent to $A_{{\mathfrak p}}$ being a formally smooth $B_{{\mathfrak r}}$-algebra. Then Zariski’s Jacobian criterion ([@AJP Proposition 4.14] implies that the morphism of $\widehat{A_{{\mathfrak p}}}$-modules $$\widehat{\frac{I_{{\mathfrak q}}}{I_{{\mathfrak q}}^{2}}} \to \Omega^{1}_{B'_{{\mathfrak q}}/B_{{\mathfrak r}}} {\widehat{\otimes}}_{B'_{{\mathfrak q}}} A_{{\mathfrak p}}$$ is right invertible. Since $\widehat{A_{{\mathfrak p}}}$ is a faithfully flat $A_{\{{\mathfrak p}\}}$-algebra and the $A_{\{{\mathfrak p}\}}$-module$({\widehat{\Omega}}^{1}_{B'/B} \otimes_{B'} A)_{\{{\mathfrak p}\}}$ is projective (see [@AJP Proposition 4.8]), it holds that the morphism $$\left(\frac{I}{I^{2}}\right)_{\{{\mathfrak p}\}} \to
({\widehat{\Omega}}^{1}_{B'/B} \otimes_{B'} A)_{\{{\mathfrak p}\}}$$ is right invertible by [@EGA41 (**0**, 19.1.14.(ii))]. From the equivalence of categories [@EGA1 (10.10.2)] we find an open subset ${\mathfrak U}\subset {\mathfrak X}$ with $x \in {\mathfrak U}$ such that the morphism $$\left(\frac{I}{I^{2}}\right)^{{\triangle}} \to {\widehat{\Omega}}^{1}_{{\mathbb D}^{s}_{{\mathbb A}^{r}_{{\mathfrak Y}}}/{\mathfrak Y}} \otimes _{{\mathcal O}_{{\mathbb D}^{s}_{{\mathbb A}^{r}_{{\mathfrak Y}}}}} {\mathcal O}_{{\mathfrak X}}$$ is right invertible over ${\mathfrak U}$. Now, by Zariski’s Jacobian criterion for formal schemes ([@AJP Corollary 4.15]) it follows that $f$ is smooth in ${\mathfrak U}$.
\(3) ${\Rightarrow}$ (4) By [@EGA41 (**0**, 19.3.8)] we have that $\widehat{{\mathcal O}_{{\mathfrak X},x}}$ is a formally smooth $\widehat{{\mathcal O}_{{\mathfrak Y},y}}$-algebra for the topologies given by the maximal ideals. Then it follows from [@EGA41 (**0**, 19.7.1)] that $\widehat{{\mathcal O}_{{\mathfrak X},x}}$ is $\widehat{{\mathcal O}_{{\mathfrak Y},y}}$-flat and by \[caracterizlocalplanos\], $f$ is flat at $x$. Moreover from [@EGA41 (**0**, 19.3.5)] we deduce that $\widehat{{\mathcal O}_{{\mathfrak X},x}}\otimes_{\widehat{{\mathcal O}_{{\mathfrak Y},y}}} k(y)$ is a formally smooth $k(y)$-algebra for the adic topologies or, equivalently, by (3) ${\Leftrightarrow}$ (1), $f^{-1}(y)$ is a $k(y)$-formal scheme smooth at $x$.
\(4) ${\Rightarrow}$ (3) By \[caracterizlocalplanos\] we have that $A_{{\mathfrak p}}$ is a flat $B_{{\mathfrak r}}$-module and therefore, it holds that $$\label{ecuacioncita}
0 \to \frac{I_{{\mathfrak q}}}{{\mathfrak r}I_{{\mathfrak q}}} \to \frac{B'_{{\mathfrak q}}}{{\mathfrak r}B'_{{\mathfrak q}}} \to \frac{A_{{\mathfrak p}} }{{\mathfrak r}A_{{\mathfrak p}}} \to 0$$ is an exact sequence. On the other hand, since $f^{-1}(y)$ is a $k(y)$-formal scheme smooth at $x$, from (1) $\Rightarrow$ (2) we deduce that $\widehat{{\mathcal O}_{{\mathfrak X},x}}\otimes_{\widehat{{\mathcal O}_{{\mathfrak Y},y}}} k(y)$ is a formally smooth $k(y)$-algebra for the adic topologies or, equivalently by [@EGA41 (**0**, 19.3.6)], $A_{{\mathfrak p}}/{\mathfrak r}A_{{\mathfrak p}}$ is a formally smooth $k({\mathfrak r})$-algebra for the adic topologies. Applying Zariski’s Jacobian criterion ([@AJP Proposition 4.14]), we have that the morphism $$\widehat{\frac{I_{{\mathfrak q}}}{I_{{\mathfrak q}}^{2}}} \otimes_{B_{{\mathfrak r}}} k({\mathfrak r}) \to ({\widehat{\Omega}}^{1}_{B'/B})_{{\mathfrak q}} {\widehat{\otimes}}_{B'_{{\mathfrak q}}} A_{{\mathfrak p}} \otimes_{B_{{\mathfrak r}}} k({\mathfrak r})$$ is right invertible. Now, since $({\widehat{\Omega}}^{1}_{B'/B})_{{\mathfrak q}}$ is a projective $B'_{{\mathfrak q}}$-module (see [@AJP Proposition 4.8]) by [@EGA1 (**0**, 6.7.2)] we obtain that $$\widehat{\frac{I_{{\mathfrak q}}}{I_{{\mathfrak q}}^{2}}} \to {\widehat{\Omega}}^{1}_{B'_{{\mathfrak q}}/B_{{\mathfrak r}}} {\widehat{\otimes}}_{\widehat{B'_{{\mathfrak q}}}} \widehat{A_{{\mathfrak p}}}$$ is right invertible. Again, by the Zariski’s Jacobian criterion, $A_{{\mathfrak p}}$ is a formally smooth $B_{{\mathfrak r}}$-algebra for the adic topologies or, equivalently by [@EGA41 (**0**, 19.3.6)], $\widehat{A_{{\mathfrak p}}}$ is a formally smooth $\widehat{B_{{\mathfrak r}}}$-algebra.
\[corpligplafibr\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a pseudo-finite type morphism in ${\mathsf {NFS}}$. The following conditions are equivalent:
1. The morphism $f$ is smooth.
2. For all $x \in {\mathfrak X}$, ${\mathcal O}_{{\mathfrak X},x}$ is a formally smooth ${\mathcal O}_{{\mathfrak Y},f(x)}$-algebra for the adic topologies.
3. For all $x \in {\mathfrak X}$, $\widehat{{\mathcal O}_{{\mathfrak X},x}}$ is a formally smooth $\widehat{{\mathcal O}_{{\mathfrak Y},f(x)}}$-algebra for the adic topologies.
4. The morphism $f$ is flat and $f^{-1}(f(x))$ is a $k(f(x))$-formal scheme smooth at $x$, for all $x \in {\mathfrak X}$.
\[flf0l\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be an *adic* morphism in ${\mathsf {NFS}}$ and let ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ be an ideal of definition. Put $$f={\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} f_{n}$$ using the ideals of definition ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ and ${\mathcal J}=f^{*}({\mathcal K}){\mathcal O}_{{\mathfrak X}} \subset {\mathcal O}_{{\mathfrak X}}$. Then, the morphism $f$ is smooth if and only if it is flat and the morphism $f_{0}:X_{0} \to Y_{0}$ is smooth.
Since $f$ is adic, the diagram
[X]{}& \^[f]{} & [Y]{}\
& &\
X\_[0]{} & \^[f\_[0]{}]{} & Y\_[0]{}\
is a cartesian square ([@EGA1 (10.12.2)]). If $f$ is smooth, by base-change it follows that $f_{0}$ is smooth. Moreover by [@AJP Proposition 4.8] we have that $f$ is flat. Conversely, if $f$ is adic, by \[fibra\], we have that $f^{-1}(f(x))=f_{0}^{-1}(f(x))$, for all $x \in {\mathfrak X}$. Therefore, since $f_{0}$ is smooth, by base-change it holds that $f^{-1}(f(x))$ is a $k(f(x))$-scheme smooth at $x$, for all $x \in {\mathfrak X}$ and applying Corollary \[corpligplafibr\] we conclude that $f$ is smooth.
The upcoming example shows that the last result is not true without assuming the *adic* hypothesis for the morphism $f$.
\[exf0lisonofliso\] Given $K$ a field, let $\mathbb{P}^{n}_{K}$ be the $n$-dimensional projective space and $X \subset \mathbb{P}^{n}_{K}$ a closed subscheme that is not smooth over $K$. If we denote by $(\mathbb{P}^{n}_{K})_{/ X}$ the completion of $\mathbb{P}^{n}_{K}$ along $X$, by Proposition \[complsorit\] we have that the morphism $$(\mathbb{P}^{n}_{K})_{/ X} {\xrightarrow}{f} \operatorname{Spec}(K)$$ is smooth but $f_{0} : X \to \operatorname{Spec}(K)$ is not smooth.
\[corf0imfpl\] Given $f:{\mathfrak X}\to {\mathfrak Y}$ a morphism in ${\mathsf {NFS}}$ let ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ be $f$-compatible ideals of definition. Write $$f={\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} f_{n}.$$ If $f$ is flat, $f_{0}: X_{0} \to Y_{0}$ is a smooth morphism and $f^{-1}(f(x))=f_{0}^{-1}(f(x))$, for all $x \in {\mathfrak X}$, then $f$ is smooth.
Since $f_{0}$ is smooth and $f^{-1}(y)=f_{0}^{-1}(y)$ for all $y = f(x)$ with $x \in {\mathfrak X}$, we deduce that $f^{-1}(y)$ is a smooth $k(y)$-scheme. Besides, by hypothesis $f$ is flat and Corollary \[corpligplafibr\] implies that $f$ is smooth.
Example \[exf0lisonofliso\] illustrates that the converse of the last corollary does not hold.
Every smooth morphism $f: X \to Y$ in ${\mathsf {Sch}}$ is locally a composition of an étale morphism $U \to {\mathbb A}^{r}_{Y}$ and a projection ${\mathbb A}^{r}_{Y} \to Y$. Proposition \[factpl\] generalizes this fact for smooth morphisms in ${\mathsf {NFS}}$. The same result has already appeared stated in local form in [@y Proposition 1.11]. We include it here for completeness.
\[factpl\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a pseudo-finite type morphism in ${\mathsf {NFS}}$. The morphism $f$ is smooth at $x \in {\mathfrak X}$ if and only if there exists an open subset ${\mathfrak U}\subset {\mathfrak X}$ with $x \in {\mathfrak U}$ such that $f|_{{\mathfrak U}}$ factors as $${\mathfrak U}{\xrightarrow}{g} {\mathbb A}^{n}_{{\mathfrak Y}} {\xrightarrow}{p} {\mathfrak Y}$$ where $g$ is étale, $p$ is the canonical projection and $n = \operatorname{rg}({\widehat{\Omega}}^{1}_{{\mathcal O}_{{\mathfrak X},x}/{\mathcal O}_{{\mathfrak Y},f(x)}})$.
As this is a local question, we may assume that $f: {\mathfrak X}= \operatorname{Spf}(A) \to {\mathfrak Y}= \operatorname{Spf}(B)$ is a smooth morphism in ${{\mathsf {NFS}}_{\mathsf {af}}}$. By [@AJP Proposition 4.8] and by [@EGA1 (10.10.8.6)] we have that ${\widehat{\Omega}}^{1}_{A/B}$ is a projective $A$-module of finite type and therefore, if ${\mathfrak p}\subset A$ is the open prime ideal corresponding to $x$, there exists $h\in A \setminus {\mathfrak p}$ such that $\operatorname{\Gamma}(\operatorname{\mathfrak{D}}(h), {\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}}) = {\widehat{\Omega}}^{1}_{A_{\{h\}}/B}$ is a free $A_{\{h\}}$-module of finite type. Put ${\mathfrak U}= \operatorname{Spf}(A_{\{h\}})$. Given $\{{\widehat{d}}a_{1},{\widehat{d}}a_{2},\ldots, {\widehat{d}}a_{n}\}$ a basis of ${\widehat{\Omega}}^{1}_{A_{\{h\}}/B}$ consider the morphism of ${\mathfrak Y}$-formal schemes $${\mathfrak U}{\xrightarrow}{g} {\mathbb A}^{n}_{{\mathfrak Y}}=\operatorname{Spf}(B\{T_{1},T_{2},\ldots,T_{n}\} )$$ defined by the continuous morphism of topological $B$-algebras $$\begin{array}{ccc}
B\{T_{1},T_{2},\ldots,T_{n}\} & \to & A_{\{h\}}\\
T_{i} &\leadsto &a_{i}
\end{array}$$ See [@EGA1 (10.2.2) and (10.4.6)]. The morphism $g$ satisfies that $f|_{{\mathfrak U}} = p \circ g$. Moreover, we deduce that $g^{*} {\widehat{\Omega}}^{1}_{{\mathbb A}^{n}_{{\mathfrak Y}}/{\mathfrak Y}}\cong {\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}}$ (see the definition of $g$) and by [@AJP Corollary 4.13] we have that $g$ is étale.
\[dimrango\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a smooth morphism at $x \in {\mathfrak X}$ and $y = f(x)$. Then $$\dim_{x} f = \operatorname{rg}({\widehat{\Omega}}^{1}_{{\mathcal O}_{{\mathfrak X},x}/{\mathcal O}_{{\mathfrak Y},y}}).$$
Put $n= \operatorname{rg}({\widehat{\Omega}}^{1}_{{\mathcal O}_{{\mathfrak X},x}/{\mathcal O}_{{\mathfrak Y},y}})$. By Proposition \[factpl\] there exists ${\mathfrak U}\subset {\mathfrak X}$ with $x \in {\mathfrak U}$ such that $f|_{{\mathfrak U}}$ factors as ${\mathfrak U}{\xrightarrow}{g} {\mathbb A}^{n}_{{\mathfrak Y}} {\xrightarrow}{p} {\mathfrak Y}$ where $g$ is an étale morphism and $p$ is the canonical projection. Applying [@AJP Proposition 4.8] we have that $f|_{{\mathfrak U}}$ and $g$ are flat morphisms and therefore, $$\begin{array}{ccccc}
\dim_{x} f & = &\dim \widehat{{\mathcal O}_{{\mathfrak X},x} }\otimes_{\widehat{{\mathcal O}_{{\mathfrak Y},y} }} k(y) &= &\dim \widehat{{\mathcal O}_{{\mathfrak X},x} } - \dim\widehat{{\mathcal O}_{{\mathfrak Y},y} }\\
\dim_{x} g &= &\dim \widehat{{\mathcal O}_{{\mathfrak X},x} }\otimes_{\widehat{{\mathcal O}_{{\mathbb A}^{n}_{{\mathfrak Y}},g(x)}}} k(g(x))&= &\dim \widehat{{\mathcal O}_{{\mathfrak X},x} } - \dim\widehat{{\mathcal O}_{{\mathbb A}^{n}_{{\mathfrak Y}},g(x)}}.
\end{array}$$ Now, since $g$ is unramified by Corollary \[pnrdim0\] we have that $\dim_{x} g =0$ and therefore $\dim_{x} f = \dim\widehat{{\mathcal O}_{{\mathbb A}^{n}_{{\mathfrak Y}},g(x)}}- \dim\widehat{{\mathcal O}_{{\mathfrak Y},y} }=n$.
\[ecppl\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a morphism of pseudo-finite type and let ${\mathfrak X}' {\hookrightarrow}{\mathfrak X}$ be a closed immersion given by the ideal ${\mathcal I}\subset {\mathcal O}_{{\mathfrak X}}$ and put $f'=f|_{{\mathfrak X}'}$. If $f$ is smooth at $x \in {\mathfrak X}'$, $n = \dim_{x} f$ and $y=f(x)$ the following conditions are equivalent:
1. The morphism $f'$ is smooth at $x$ and $\dim_{x} f'^{-1}(y)=n-m$.
2. The natural sequence of ${\mathcal O}_{{\mathfrak X}}$-modules $$0 \to \frac{{\mathcal I}}{{\mathcal I}^{2}} \to {\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}} \otimes_{{\mathcal O}_{{\mathfrak X}}} {\mathcal O}_{{\mathfrak X}'} \to {\widehat{\Omega}}^{1}_{{\mathfrak X}'/{\mathfrak Y}} \to 0$$ is exact[^3] at $x$ and, on a neighborhood of $x$, the displayed ${\mathcal O}_{{\mathfrak X}'}$-Modules are locally free of ranks $m,\, n$ and $n-m$, respectively.
Since $f:{\mathfrak X}\to {\mathfrak Y}$ is a smooth morphism at $x$, replacing ${\mathfrak X}$, if necessary, by a smaller neighborhood of $x$, we may assume that $f: {\mathfrak X}=\operatorname{Spf}(A) \to {\mathfrak Y}=\operatorname{Spf}(B)$ is a morphism in ${{\mathsf {NFS}}_{\mathsf {af}}}$ smooth at $x$ and that ${\mathfrak X}'= \operatorname{Spf}(A/I)$. Therefore, applying [@AJP Proposition 4.8] and Corollary \[dimrango\] we have that ${\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}}$ is a locally free ${\mathcal O}_{{\mathfrak X}}$-Module of rank $n$.
Let us prove that (1) $\Rightarrow $ (2). Replacing ${\mathfrak X}'$ with a smaller neighborhood of $x$ if necessary, we may also assume that $f':{\mathfrak X}' \to{\mathfrak Y}$ is a smooth morphism. Then, by an argument along the lines of the previous paragraph, it follows that ${\widehat{\Omega}}^{1}_{{\mathfrak X}'/{\mathfrak Y}}$ is a locally free ${\mathcal O}_{{\mathfrak X}'}$-Module of rank $n-m$. Zariski’s Jacobian criterion for formal schemes ([@AJP Corollary 4.15]) implies that the sequence $$0 \to \frac{{\mathcal I}}{{\mathcal I}^{2}} \to {\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}} \otimes_{{\mathcal O}_{{\mathfrak X}}} {\mathcal O}_{{\mathfrak X}'} \to {\widehat{\Omega}}^{1}_{{\mathfrak X}'/{\mathfrak Y}} \to 0$$ is exact and split, from which we deduce that ${\mathcal I}/{\mathcal I}^{2}$ is a locally free ${\mathcal O}_{{\mathfrak X}'}$-Module of rank $m$.
Conversely, applying [@EGA1 (**0**, 5.5.4)] and [@AJP Proposition 3.13] we deduce that there exists an open formal subscheme ${\mathfrak U}\subset {\mathfrak X}'$ with $x \in {\mathfrak U}$ such that $$0 \to \left(\frac{{\mathcal I}}{{\mathcal I}^{2}}\right)|_{{\mathfrak U}} \to ({\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}} \otimes_{{\mathcal O}_{{\mathfrak X}}} {\mathcal O}_{{\mathfrak X}'})|_{{\mathfrak U}} \to ({\widehat{\Omega}}^{1}_{{\mathfrak X}'/{\mathfrak Y}})|_{{\mathfrak U}} \to 0$$ is exact and split. From Zariski’s Jacobian criterion it follows that $f'|_{{\mathfrak U}}$ is smooth and therefore, $f'$ is smooth at $x$.
The natural sequence in Proposition \[ecppl\] is the Second Fundamental Exact Sequence associated to the morphisms ${\mathfrak X}' {\hookrightarrow}{\mathfrak X}{\xrightarrow}{f} {\mathfrak X}$ ([@AJP Proposition 3.13]).
Locally, a pseudo-finite type morphism $f: {\mathfrak X}\to {\mathfrak Y}$ factors as ${\mathfrak U}\overset{j} {\hookrightarrow}{\mathbb D}^{r}_{{\mathbb A}^{s}_{{\mathfrak Y}}} {\xrightarrow}{p} {\mathfrak Y}$ where $j$ is a closed immersion (see [@AJP Proposition 1.7]). In Corollary \[criteriojacobiano\] we provide a criterion in terms a matrix rank that tells whether ${\mathfrak U}$ is smooth over ${\mathfrak Y}$ or not.
Let ${\mathfrak Y}= \operatorname{Spf}(A) \in{{\mathsf {NFS}}_{\mathsf {af}}}$. Consider ${\mathfrak X}\subset {\mathbb D}^{s}_{{\mathbb A}^{r}_{{\mathfrak Y}}}$ a closed formal subscheme given by an ideal ${\mathcal I}=I^{{\triangle}}$, with $I=\langle g_{1}\,, g_{2},\, \ldots,\, g_{k} \rangle \subset A\{\mathbf{T}\}[[\mathbf{Z}]]$ where $\mathbf{T}= T_{1},\, T_{2},\, \ldots,\, T_{r}$ and $\mathbf{Z}= Z_{1},\, Z_{2},\, \ldots,\, Z_{s}$ are two sets of of indeterminates. From [@AJP 3.14] we have that $$\{{\widehat{d}}T_{1},\, \ldots ,{\widehat{d}}T_{r},\, {\widehat{d}}Z_{1},\,\ldots, {\widehat{d}}Z_{s}\}$$ is a basis of ${\widehat{\Omega}}^{1}_{A\{\mathbf{T}\}[[\mathbf{Z}]]/A}$ and also that given $g \in A\{\mathbf{T}\}[[\mathbf{Z}]]$ it holds that: $${\widehat{d}}g = \sum_{i=1}^{r} \frac{\partial g}{\partial T_{i}} {\widehat{d}}T_{i} + \sum_{j=1}^{s} \frac{\partial g}{\partial Z_{j}} {\widehat{d}}Z_{j},$$ where ${\widehat{d}}$ is the complete canonical derivation of $A\{\mathbf{T}\}[[\mathbf{Z}]]$ over $A$. For any $g \in A\{\mathbf{T}\}[[\mathbf{Z}]]$, $w \in \{{\widehat{d}}T_{1},\,\ldots ,{\widehat{d}}T_{r},\, {\widehat{d}}Z_{1},\,\ldots, {\widehat{d}}Z_{s} \}$ and $x \in {\mathfrak X}$, denote by $\frac{\partial g}{\partial w}(x) $ the image of $\frac{\partial g}{\partial w} \in A\{\mathbf{T}\}[[\mathbf{Z}]]$ in $k(x)$. We will call $$\operatorname{Jac}_{{\mathfrak X}/{\mathfrak Y}}(x)=
\begin{pmatrix}
\frac{\partial g_{1}}{\partial T_{1}}(x) & \ldots &
\frac{\partial g_{1}}{\partial T_{r}}(x) & \frac{\partial g_{1}}{\partial Z_{1}}(x) & \ldots & \frac{\partial g_{1}}{\partial Z_{s}}(x) \\
\frac{\partial g_{2}}{\partial T_{1}}(x) & \ldots &
\frac{\partial g_{2}}{\partial T_{r}}(x) & \frac{\partial g_{2}}{\partial Z_{1}}(x) & \ldots &
\frac{\partial g_{2}}{\partial Z_{s}}(x) \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\
\frac{\partial g_{k}}{\partial T_{1}}(x) & \ldots &
\frac{\partial g_{k}}{\partial T_{r}}(x) & \frac{\partial g_{k}}{\partial Z_{1}}(x) & \ldots & \frac{\partial g_{k}}{\partial Z_{s}}(x) \\
\end{pmatrix}$$ the *Jacobian matrix of ${\mathfrak X}$ over $ {\mathfrak Y}$ at $x$*. This matrix depends on the chosen generators of $I$ and therefore, the notation $\operatorname{Jac}_{{\mathfrak X}/{\mathfrak Y}}(x)$ is not completely accurate.
[(Jacobian criterion for the affine formal space and the affine formal disc.)]{} \[criteriojacobiano\] With the previous notations, the following assertions are equivalent:
1. The morphism $f: {\mathfrak X}\to{\mathfrak Y}$ is smooth at $x$ and $\dim_{x} f= r+s-l$.
2. There exists a subset $\{g_{1},\,, g_{2},\, \ldots,\, g_{l}\} \subset \{g_{1}\,, g_{2},\, \ldots,\, g_{k}\}$ such that ${\mathcal I}_{x}=\langle g_{1}\,, g_{2},\, \ldots,\, g_{l} \rangle{\mathcal O}_{{\mathfrak X},x}$ and $\operatorname{rg}(\operatorname{Jac}_{{\mathfrak X}/{\mathfrak Y}}(x)) = l$.
Assume (1). By Proposition \[ecppl\] we have that the sequence $$0 \to \frac{{\mathcal I}}{{\mathcal I}^{2}} \to {\widehat{\Omega}}^{1}_{ {\mathbb D}^{s}_{{\mathbb A}^{r}_{{\mathfrak Y}}}/{\mathfrak Y}} \otimes_{{\mathcal O}_{ {\mathbb D}^{s}_{{\mathbb A}^{r}_{{\mathfrak Y}}}}} \!\!\! {\mathcal O}_{{\mathfrak X}} \to {\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}} \to 0$$ is exact at $x$ and the corresponding ${\mathcal O}_{{\mathfrak X}}$-Modules are locally free, in a neighborhood of $x$, of ranks $l$, $r+s$ and $r+s-l$, respectively. Therefore, $$\label{sexcritjacobkev}
0 \to \frac{{\mathcal I}}{{\mathcal I}^{2}}\otimes_{{\mathcal O}_{{\mathfrak X}}} k(x) \to
{\widehat{\Omega}}^{1}_{ {\mathbb D}^{s}_{{\mathbb A}^{r}_{{\mathfrak Y}}}/{\mathfrak Y}} \otimes_{ {\mathcal O}_{{\mathbb D}^{s}_{{\mathbb A}^{r}_{{\mathfrak Y}}}}} \!\! k(x) \to {\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}} \otimes_{{\mathcal O}_{{\mathfrak X}}} k(x) \to 0$$ is an exact sequence of $k(x)$-vector spaces of dimension $l,\, r+s,\, r+s-l$, respectively. Thus, there exists a set $\{g_{1},\, g_{2},\, \ldots,\, g_{l}\} \subset \{g_{1},\, g_{2},\, \ldots,\, g_{k}\}$ such that $\{g_{1}(x),\, g_{2}(x),\, \ldots,\, g_{l}(x)\}$ provides a basis of ${\mathcal I}/{\mathcal I}^{2}\otimes_{{\mathcal O}_{{\mathfrak X}}} k(x)$ at $x$. By Nakayama’s lemma it holds that ${\mathcal I}_{x}=\langle g_{1}\,, g_{2},\, \ldots,\, g_{l} \rangle{\mathcal O}_{{\mathfrak X},x}$. Besides, from the exactness of the sequence (\[sexcritjacobkev\]) and from the equivalence of categories [@EGA1 (10.10.2)] we deduce that the set $$\{{\widehat{d}}g_{1}(x),\, {\widehat{d}}g_{2}(x),\, \ldots,\, {\widehat{d}}g_{l}(x)\} \subset {\widehat{\Omega}}^{1}_{ A\{\mathbf{T}\}[[\mathbf{Z}]]/A} \otimes_{A\{\mathbf{T}\} [[\mathbf{Z}]]} k(x)$$ is linearly independent. Therefore, $\operatorname{rg}(\operatorname{Jac}_{{\mathfrak X}/{\mathfrak Y}}(x)) = l$.
Conversely, from the Second Fundamental Exact Sequence associated to the morphisms ${\mathfrak X}{\hookrightarrow}{\mathbb D}^{s}_{{\mathbb A}^{r}_{{\mathfrak Y}}} \to {\mathfrak Y}$ [@AJP Proposition 3.13] we get the exact sequence $$\frac{{\mathcal I}}{{\mathcal I}^{2}}\otimes_{{\mathcal O}_{{\mathfrak X}}} k(x) \to
{\widehat{\Omega}}^{1}_{ {\mathbb D}^{s}_{{\mathbb A}^{r}_{{\mathfrak Y}}}/{\mathfrak Y}} \otimes_{ {\mathcal O}_{{\mathbb D}^{s}_{{\mathbb A}^{r}_{{\mathfrak Y}}}}} \!\! k(x) \to {\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}} \otimes_{{\mathcal O}_{{\mathfrak X}}} k(x) \to 0.$$ Since $\operatorname{rg}(\operatorname{Jac}_{{\mathfrak X}/{\mathfrak Y}}(x)) = l$, we have that $$\{{\widehat{d}}g_{1}(x),\,, {\widehat{d}}g_{2}(x),\, \ldots,\, {\widehat{d}}g_{l}(x)\} \subset {\widehat{\Omega}}^{1}_{ A\{\mathbf{T}\}[[\mathbf{Z}]]/A} \otimes_{A\{\mathbf{T}\} [[\mathbf{Z}]]} k(x)$$ is a linearly independent set. Extending this set to a basis of the vector space ${\widehat{\Omega}}^{1}_{ A\{\mathbf{T}\}[[\mathbf{Z}]]/A} \otimes_{A\{\mathbf{T}\} [[\mathbf{Z}]]} k(x)$, by Nakayama’s lemma we find a basis $\mathcal{B} \subset {\widehat{\Omega}}^{1}_{ A\{\mathbf{T}\}[[\mathbf{Z}]]/A}$ such that $\{{\widehat{d}}g_{1},\, {\widehat{d}}g_{2},\, \ldots, \, {\widehat{d}}g_{l}\} \subset \mathcal{B}$ and therefore $$\{{\widehat{d}}g_{1},\,, {\widehat{d}}g_{2},\, \ldots,\, {\widehat{d}}g_{l}\} \subset{\widehat{\Omega}}^{1}_{ A\{\mathbf{T}\}[[\mathbf{Z}]]/A} \otimes_{A\{\mathbf{T}\} [[\mathbf{Z}]]} A\{\mathbf{T}\}[[\mathbf{Z}]]/I$$ is a linearly independent set at $x$. Thus the set $\{g_{1}\,, g_{2,},\, \ldots,\, g_{l}\}$ provides a basis of ${\mathcal I}/{\mathcal I}^{2}$ at $x$ and by the equivalence of categories [@EGA1 (10.10.2)] we have that the sequence of ${\mathcal O}_{\mathfrak X}$-Modules $$0 \to \frac{{\mathcal I}}{{\mathcal I}^{2}} \to {\widehat{\Omega}}^{1}_{ {\mathbb D}^{s}_{{\mathbb A}^{r}_{{\mathfrak Y}}}/{\mathfrak Y}} \otimes_{{\mathcal O}_{ {\mathbb D}^{s}_{{\mathbb A}^{r}_{{\mathfrak Y}}}}} \!\!\! {\mathcal O}_{{\mathfrak X}} \to {\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}} \to 0$$ is split exact at $x$ of locally free Modules of ranks $l$, $r+s$ and $r+s-l$, respectively. Applying Proposition \[ecppl\] it follows that $f$ is smooth at $x$ and $\dim_{x} f= r+s-l$.
Notice that the matrix form of the Jacobian criterion for the affine formal space and the affine formal disc (Corollary \[criteriojacobiano\]) generalize the usual matrix form of the Jacobian criterion for the affine space in ${\mathsf {Sch}}$ ([@at Ch. VII, Theorem (5.14)]).
Étale morphisms {#sec5}
===============
The main results of this section are consequences of those obtained in Sections \[sec3\] and \[sec4\]. They will allow us to characterize in Section \[sec6\] two important classes of étale morphisms: open immersions and completion morphisms.
\[efn\] Given $f:{\mathfrak X}\to {\mathfrak Y}$ in ${\mathsf {NFS}}$ let ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ be $f$-compatible ideals of definition. Using these ideals, set $$f={\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} f_{n}.$$ If $f_{n}:X_{n} \to Y_{n}$ is étale, $\forall n \in {\mathbb N}$, then $f$ is étale.
The sum of Proposition \[nrfn\] and Proposition \[lfn\].
\[eadfn\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be an *adic* morphism in ${\mathsf {NFS}}$ and let ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ be an ideal of definition. Consider $\{f_{n}\}_{n \in {\mathbb N}}$ the direct system of morphisms of schemes associated to the ideals of definition ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ and ${\mathcal J}=f^{*}({\mathcal K}){\mathcal O}_{{\mathfrak X}} \subset {\mathcal O}_{{\mathfrak X}}$. The morphism $f$ is étale if and only if the morphisms $f_{n}:X_{n} \to Y_{n}$ are étale $\forall n \in {\mathbb N}$.
It follows from Proposition \[nrfn\] and Corollary \[ladfn\].
\[fetf0et\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be an *adic* morphism in ${\mathsf {NFS}}$ and let $f_{0}:X_{0} \to Y_{0}$ be the morphism of schemes associated to the ideals of definition ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ and ${\mathcal J}=f^{*}({\mathcal K}){\mathcal O}_{{\mathfrak X}} \subset {\mathcal O}_{{\mathfrak X}}$. Then, $f$ is étale if and only if $f$ is flat and $f_{0}$ is étale.
Put together Proposition \[fnrf0nr\] and Corollary \[flf0l\].
Note that Example \[peynopen\] shows that in the non adic case the last two results do not hold and also that, in general, the converse of Proposition \[efn\] is not true.
Let $f$ be a pseudo-finite type morphism in ${\mathsf {NFS}}$ and choose ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ $f$-compatible ideals of definition. Write $$f={\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} f_{n}.$$ If $f_{0}:X_{0} \to Y_{0}$ is étale, $f$ is flat and $f^{-1}(f(x))= f_{0}^{-1}(f(x))$, for all $x \in {\mathfrak X}$, then $f$ is étale.
It follows from Corollary \[corf0imfpnr\] and Corollary \[corf0imfpl\].
Example \[peynopen\] shows that the converse of the last result is not true. Next Proposition gives us a local characterization of étale morphisms.
\[caractlocalpe\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a morphism in ${\mathsf {NFS}}$ of pseudo-finite type, let $x \in {\mathfrak X}$ and $y = f(x)$, the following conditions are equivalent:
1. $f$ is étale at $x$.
2. ${\mathcal O}_{{\mathfrak X},x}$ is a formally étale ${\mathcal O}_{{\mathfrak Y},y}$-algebra for the adic topologies.
3. $\widehat{{\mathcal O}_{{\mathfrak X},x}}$ is a formally étale $\widehat{{\mathcal O}_{{\mathfrak Y},y}}$-algebra for the adic topologies.
4. $f$ is flat at $x$ and $f^{-1}(y)$ is a $k(y)$-formal scheme étale at $x$.
5. $f$ is flat and unramified at $x$.
6. $f$ is flat at $x$ and $({\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}})_{x}=0$.
7. $f$ is smooth at $x$ and a quasi-covering at $x$.
Applying Proposition \[caraclocalpnr\] and Proposition \[pligplafibr\] we have that $$(5) \Leftarrow (1) {\Leftrightarrow}(2) {\Leftrightarrow}(2') {\Leftrightarrow}(3) {\Rightarrow}(4) {\Leftrightarrow}(4').$$ Let $C := \widehat{{\mathcal O}_{{\mathfrak X},x}} \otimes_{\widehat{{\mathcal O}_{{\mathfrak Y},y}}} k(y)$. To show (4) ${\Rightarrow}$ (5), by Corollary \[corpnrimplcr\] it is only left to prove that $f$ is smooth at $x$. By hypothesis, we have that $f$ is unramified at $x$ and by Proposition \[caraclocalpnr\], it follows that $C = k(x)$ and $k(x)|k(y)$ is a finite separable extension, therefore, formally étale. Since $f$ is flat at $x$, by Proposition \[pligplafibr\] we conclude that $f$ is smooth at $x$.
To prove that (5) ${\Rightarrow}$ (1), it suffices to check that $f$ is unramified at $x$ or, equivalently by Proposition \[caraclocalpnr\], that $C = k(x)$ and that $k(x)|k(y)$ is a finite separable extension. As $f$ is smooth at $x$, applying Proposition \[pligplafibr\], we have that $\widehat{{\mathcal O}_{{\mathfrak X},x}}$ is a formally smooth $\widehat{{\mathcal O}_{{\mathfrak X},x}}$-algebra for the adic topologies. Then by base-change it holds that $C$ is a formally smooth $k(y)$-algebra. By [@EGA41 (**0**, 19.3.8)] we have that $C$ is a formally smooth $k(y)$-algebra for the topologies given by the maximal ideals and from [@ma2 Lemma 1, p. 216] it holds that $C$ is a regular local ring. Besides, by hypothesis we have that $C$ is a finite $k(y)$-module, therefore, an artinian ring, so $C = k(x)$. Since $k(x) = C$ is a formally smooth $k(y)$-algebra we have that $k(x)|k(y)$ is a separable extension (*cf.* [@EGA41 (**0**, 19.6.1)]).
\[corcaractlocalpe\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a pseudo-finite type morphism in ${\mathsf {NFS}}$. The following conditions are equivalent:
1. $f$ is étale.
2. For all $x \in {\mathfrak X}$, ${\mathcal O}_{{\mathfrak X},x}$ is a formally étale ${\mathcal O}_{{\mathfrak Y},f(x)}$-algebra for the adic topologies.
3. For all $x \in {\mathfrak X}$, $\widehat{{\mathcal O}_{{\mathfrak X},x}}$ is a formally étale $\widehat{{\mathcal O}_{{\mathfrak Y},f(x)}}$-algebra for the adic topologies.
4. For all $x \in {\mathfrak X}$, $f^{-1}(f(x))$ is a $k(f(x))$-formal scheme étale at $x$ and $f$ is flat.
5. $f$ is flat and unramified.
6. $f$ is flat and ${\widehat{\Omega}}^{1}_{{\mathfrak X}/{\mathfrak Y}}=0$.
7. $f$ is smooth and a quasi-covering.
\[pcf+plnope\] Given a field $K$, the canonical morphism ${\mathbb D}^{1}_{K} \to \operatorname{Spec}(K)$ is smooth, pseudo-quasi-finite but it is not étale.
In ${\mathsf {Sch}}$ a morphism is étale if and only if it is smooth and quasi-finite. The previous example shows that in ${\mathsf {NFS}}$ there are smooth and pseudo-quasi-finite morphisms that are not étale. That is why we consider quasi-coverings in ${\mathsf {NFS}}$ (see Definition \[defncuasireves\]) as the right generalization of quasi-finite morphisms in ${\mathsf {Sch}}$.
Structure theorems of the infinitesimal lifting properties {#sec6}
==========================================================
We begin with two results that will be used in the proof of the remainder results of this section.
\[levantpeusform\] Consider a formally étale morphism $f:{\mathfrak X}\to {\mathfrak Y}$ and a morphism $g\colon {\mathfrak S}\to {\mathfrak Y}$, both in ${\mathsf {NFS}}$. Take ${\mathcal L}\subset {\mathcal O}_{{\mathfrak S}}$ an ideal of definition of ${\mathfrak S}$ and write $${\mathfrak S}= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} S_{n}.$$ If $h_{0}: S_{0} \to {\mathfrak X}$ is a morphism in ${\mathsf {NFS}}$ that makes the diagram
S\_[0]{} & & [S]{}\
\^[h\_[0]{}]{} & & \_[g]{}\
[X]{}& \^[f]{} & [Y]{}\
commutative, where $S_{0} {\hookrightarrow}{\mathfrak S}$ is the canonical closed immersion, then there exists a unique ${\mathfrak Y}$-morphism $l: {\mathfrak S}\to {\mathfrak X}$ in ${\mathsf {NFS}}$ such that $l|_{S_{0}}=h_{0}$.
By induction on $n$ we are going to construct a collection of morphisms $\{h_{n}: S_{n} \to {\mathfrak X}\}_{n \in {\mathbb N}}$ such that the diagrams
S\_[n-1]{} & & & &\
&(4,2) (2,4)\_[h\_[n-1]{}]{} & & &\
& &S\_[n]{} && [S]{}\
& & \_[h\_[n]{}]{} & & \_[g]{}\
& & [X]{}&\^[f]{}& [Y]{}\
commute. For $n =1$, by [@AJP 2.4] there exists a unique morphism $h_{1}: S_{1} \to {\mathfrak X}$ such that $h_{1} |_{S_{0}}= h_{0}$ and $g|_{S_{1}}= f \circ h_{1}$. Now let $n \in {\mathbb N}, \,n > 1$ and suppose we already have for all $0< k <n$ morphisms $h_{k}: S_{k} \to {\mathfrak X}$ such that $h_{k} |_{S_{k-1}}= h_{k-1}$ and $g|_{S_{k}}= f \circ h_{k}$. Then by [@AJP *loc. cit.*] there exists a unique morphism $h_{n}: S_{n} \to {\mathfrak X}$ such that $h_{n} |_{S_{n-1}}= h_{n-1}$ and $g|_{S_{n}}= f \circ h_{n}$. It is straightforward that $$l:= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} h_{n}$$ is a morphism of formal schemes and is the unique one such that the diagram
S\_[0]{} & & [S]{}\
\^[h\_[0]{}]{} & \_[h]{} & \_[g]{}\
[X]{}& \_f & [Y]{}\
commutes.
\[iso0peis\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be an étale morphism in ${\mathsf {NFS}}$ and ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ $f$-compatible ideals of definition such that the corresponding morphism $f_{0}:X_{0} \to Y_{0}$ is an isomorphism. Then $f$ is an isomorphism.
By Proposition \[levantpeusform\] there exists a (unique) morphism $g: {\mathfrak Y}\to {\mathfrak X}$ such that the following diagram is commutative
Y\_[0]{} & & [Y]{}\
\^[f\_[0]{}\^[-1]{}]{} & (2,4)\^[g]{} &\
X\_[0]{} & & \_[1\_[[Y]{}]{}]{}\
& &\
[X]{}& \^[f]{} & [Y]{}\
Then, by [@AJP Proposition 2.13] it follows that $g$ is an étale morphism. Thus, applying Proposition \[levantpeusform\] we have that there exists a (unique) morphism $f': {\mathfrak X}\to {\mathfrak Y}$ such that the following diagram is commutative
X\_[0]{} & & [X]{}\
\^[f\_[0]{}]{} & (2,4)\^[f’]{} &\
Y\_[0]{} & & \_[1\_[[X]{}]{}]{}\
& &\
[Y]{}& \^[g]{} &[X]{}\
From $f \circ g=1_{{\mathfrak Y}}$ and $g \circ f'=1_{{\mathfrak X}}$ we deduce that $f=f'$ and therefore $f$ is an isomorphism.
In ${\mathsf {Sch}}$ open immersions are characterized as being those étale morphisms that are radicial (see [@EGA44 (17.9.1)]). In the following theorem we extend this characterization and relate open immersions in formal schemes with their counterparts in schemes.
\[caractencab\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a morphism in ${\mathsf {NFS}}$. The following conditions are equivalent:
1. $f$ is an open immersion.
2. $f$ is adic, flat and if ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ is an ideal of definition such that ${\mathcal J}= f^{*}({\mathcal K}) {\mathcal O}_{{\mathfrak X}} \subset {\mathcal O}_{{\mathfrak X}}$, the associated morphism of schemes $f_{0}:X_{0} \to Y_{0}$ is an open immersion.
3. $f$ is *adic* étale and radicial.
4. There are ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ $f$-compatible ideals of definition such that the morphisms $f_{n}:X_{n} \to Y_{n}$ are open immersions, for all $n \in {\mathbb N}$.
The implication (1) $\Rightarrow$ (2) is immediate. Given ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ an ideal of definition, assume (2) and let us show (3). Since $f_{0}$ is an open immersion, is radicial, so, $f$ is radicial (see Definition \[rad\] and its attached paragraph). Furthermore, $f$ is flat and $f_{0}$ is an étale morphism then $f$ is étale (see Proposition \[fetf0et\]). Let us prove that (3) $\Rightarrow$ (4). Given ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ an ideal of definition and ${\mathcal J}= f^{*}({\mathcal K}) {\mathcal O}_{{\mathfrak X}}$, by Corollary \[eadfn\] the morphisms $f_{n}:X_{n} \to Y_{n}$ are étale, for all $n \in {\mathbb N}$. The morphisms $f_{n}$ are also radicial for all $n \in {\mathbb N}$ (see Definition \[rad\]) and thus by [@EGA44 (17.9.1)] it follows that $f_{n}$ is an open immersion, for each $n \in {\mathbb N}$. Finally, suppose that (4) holds and let us see that $f$ is an open immersion. With the notations in (4), there exists an open subset $U_{0} \subset Y_{0}$ such that $f_{0}$ factors as $$X_{0} {\xrightarrow}{f'_{0}} U_{0} \overset{i_{0}} {\hookrightarrow}Y_{0}$$ where $f'_{0}$ is an isomorphism and $i_{0}$ is the canonical inclusion. Let ${\mathfrak U}\subset {\mathfrak Y}$ be the open formal subscheme with underlying topological space $U_{0}$. Since the open immersion $i: {\mathfrak U}\to {\mathfrak Y}$ is étale, then Proposition \[levantpeusform\] implies that there exists a morphism $f': {\mathfrak X}\to {\mathfrak U}$ of formal schemes such that the diagram
[X]{}& & \^[f]{}& & &[Y]{}\
& (3,2)\^[f’]{}& & &\^[i]{}&\
& & & [U]{}& &\
& & && &\
X\_[0]{} & & \^[f\_[0]{}]{}&&&Y\_[0]{}\
& (3,2)\_[f’\_[0]{}]{} & & &\^[i\_[0]{}]{} &\
& & & U\_[0]{} & &\
is commutative. Since the morphisms $f_{n}$ are étale, for all $n \in {\mathbb N}$, Proposition \[efn\] implies that $f$ is étale. By [@AJP Proposition 2.13] we have that $f'$ is étale and applying Corollary \[iso0peis\], $f'$ is an isomorphism and therefore, $f$ is an open immersion.
\[diagopen\] Let $f\colon{\mathfrak X}\to {\mathfrak Y}$ be a pseudo-finite type morphism in ${\mathsf {NFS}}$. Then $f$ is unramified if and only if the diagonal morphism $\Delta_f \colon {\mathfrak X}\to {\mathfrak X}\times_{\mathfrak Y}{\mathfrak X}$ is an open embedding.
Take ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ $f$-compatible ideals of definition such that $f$ can be expressed as the limit of maps of usual schemes $f_n \colon X_n \to Y_n$, $n \in {\mathbb N}$. The morphism $f\colon{\mathfrak X}\to {\mathfrak Y}$ is unramified if and only if $f_n$ is unramified for all $n \in {\mathbb N}$ by Proposition \[nrfn\]. By [@EGA44 Corollaire (17.4.2)] this is equivalent to $\Delta_{f_n}: X_n \to X_n \times_{Y_n} X_n$ being an open embedding for all $n \in {\mathbb N}$. But this, in turn, is equivalent to the fact that $\Delta_f \colon {\mathfrak X}\to {\mathfrak X}\times_{\mathfrak Y}{\mathfrak X}$ is an open embedding by Theorem \[caractencab\].
Every completion morphism is a pseudo-closed immersion that is flat (*cf.* Proposition \[caractcom\]). Next, we prove that this condition is also sufficient. Thus, we obtain a criterion to determine whether a ${\mathfrak Y}$-formal scheme ${\mathfrak X}$ is the completion of ${\mathfrak Y}$ along a closed formal subscheme.
\[caracmorfcompl\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a morphism in ${\mathsf {NFS}}$ and let ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ be $f$-compatible ideals of definition. Let $f_{0}:X_{0} \to Y_{0}$ be the corresponding morphism of ordinary schemes. The following conditions are equivalent:
1. There exists a closed formal subscheme ${\mathfrak Y}' \subset {\mathfrak Y}$ such that ${\mathfrak X}= {\mathfrak Y}_{/{\mathfrak Y}'}$ and $f$ is the morphism of completion of ${\mathfrak Y}$ along ${\mathfrak Y}'$.
2. The morphism $f$ is a flat pseudo-closed immersion.
3. The morphism $f$ is étale and $f_{0}:X_{0} \to Y_{0}$ is a closed immersion.
4. The morphism $f$ is a smooth pseudo-closed immersion.
The implication (1) ${\Rightarrow}$ (2) is Proposition \[caractcom\]. Let us show that (2) ${\Rightarrow}$ (3). Since $f$ is a pseudo-closed immersion, by Corollary \[pecigf0ecnr\] we have that $f$ is unramified. Then as $f$ is flat, Corollary \[corcaractlocalpe\] establishes that $f$ is étale. The equivalence (3) ${\Leftrightarrow}$ (4) is consequence of Corollary \[pecigf0ecnr\]. Finally, we show that (3) ${\Rightarrow}$ (1). By hypothesis, the morphism $f_{0}: X_{0} \to {\mathfrak Y}$ is a closed immersion. Consider $\kappa: {\mathfrak Y}_{/ X_{0}} \to {\mathfrak Y}$ the morphism of completion of ${\mathfrak Y}$ along $X_{0}$ and let us prove that ${\mathfrak X}$ and ${\mathfrak Y}_{/ X_{0}}$ are ${\mathfrak Y}$-isomorphic. By Proposition \[caractcom\] the morphism $\kappa$ is étale so, applying Proposition \[levantpeusform\], we have that there exists a ${\mathfrak Y}$-morphism $\varphi: {\mathfrak X}\to {\mathfrak Y}_{/ X_{0}}$ such that the following diagram is commutative
[X]{}& & \^[f]{} & & &[Y]{}\
& (3,2)\^& & &\^&\
& && [Y]{}\_[/ X\_[0]{}]{} & &\
& && & &\
X\_[0]{} & &\^[f\_[0]{}]{}& && Y\_[0]{}\
& (3,2)\_[\_[0]{}=1\_[X\_[0]{}]{}]{} && &\^[f\_[0]{}]{} &\
& && X\_[0]{} & &\
From [@AJP Proposition 2.13] it follows that $\varphi$ is étale and then by Corollary \[iso0peis\] we get that $\varphi$ is an isomorphism.
A consequence of the proof of (3) ${\Rightarrow}$ (1) is the following: Given ${\mathfrak Y}$ in ${\mathsf {NFS}}$ and a closed formal subscheme ${\mathfrak Y}' \subset {\mathfrak Y}$ defined by the ideal ${\mathcal I}\subset {\mathcal O}_{{\mathfrak Y}}$, then for every ideal of definition ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ of ${\mathfrak Y}$, it holds that $${\mathfrak Y}_{/{\mathfrak Y}'} = {\mathfrak Y}_{/Y'_{0}}$$ where $Y'_{0} = ({\mathfrak Y}',{\mathcal O}_{{\mathfrak Y}}/({\mathcal I}+{\mathcal K}))$.
Given a scheme $Y$ and a closed subscheme $Y_{0} \subset Y$ with the same topological space, the functor $X \leadsto X \times_{Y} Y_{0}$ defines an equivalence between the category of étale $Y$-schemes and the category of étale $Y_{0}$-schemes by [@EGA44 (18.1.2)]. In the next theorem we extend this equivalence to the category of locally noetherian formal schemes. A special case of this theorem, namely when ${\mathfrak Y}$ is smooth over a noetherian ordinary base scheme, appears in [@y Proposition 2.4].
\[teorequivet\] Let ${\mathfrak Y}$ be in ${\mathsf {NFS}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ an ideal of definition such that $${\mathfrak Y}= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} Y_{n}.$$ Then the functor $$\begin{matrix}\textrm { \'etale adic }{\mathfrak Y}\textrm{-formal schemes}& {\xrightarrow}{F} &\textrm{\'etale }Y_{0}\textrm{-schemes}\\
{\mathfrak X}& \leadsto& {\mathfrak X}\times_{{\mathfrak Y}} Y_{0}\\
\end{matrix}$$ is an equivalence of categories.
By [@mcl IV, §4, Theorem 1] it suffices to prove that: (a) $F$ is full and faithful; and (b) Given $X_{0}$ an étale $Y_{0}$-scheme there exists an étale adic ${\mathfrak Y}$-formal scheme ${\mathfrak X}$ such that $F({\mathfrak X})={\mathfrak X}\times_{{\mathfrak Y}}Y_{0} \cong X_{0}$.
The assertion (a) is an immediate consequence of Proposition \[levantpeusform\].
Let us show (b). Given $X_{0}$ an étale $Y_{0}$-scheme in ${\mathsf {Sch}}$ by [@EGA44 (18.1.2)] there exists $X_{1}$ a locally noetherian étale $Y_{1}$-scheme such that $X_{1}\times_{Y_{1}}Y_{0} \cong X_{0}$. Reasoning by induction on $n \in {\mathbb N}$ and using [@EGA44 *loc. cit.*], we get a family of schemes $\{X_{n}\}_{n \in {\mathbb N}}$ such that, for each $n \in {\mathbb N}$, $X_{n}$ is a locally noetherian étale $Y_{n}$-scheme and $X_{n}\times_{Y_{n}}Y_{n-1} \cong X_{n-1}$, for $n > 0$. Then $${\mathfrak X}:= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} X_{n}$$ is a locally noetherian adic ${\mathfrak Y}$-formal scheme (by [@EGA1 (10.12.3.1)]), $${\mathfrak X}\times_{{\mathfrak Y}}Y_{0} \underset{\textrm{\cite[(10.7.4)]{EGA1}} }= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} (X_{n} \times_{Y_{n}} Y_{0}) =X_{0}$$ and ${\mathfrak X}$ is an étale ${\mathfrak Y}$-formal scheme (see Proposition \[efn\]).
It seems plausible that there is a theory of an algebraic fundamental group for formal schemes that classifies *adic* étale surjective maps onto a noetherian formal scheme ${\mathfrak X}$. If this is the case, the previous theorem would imply that it agrees with the fundamental group of $X_0$. We also consider feasible the existence of a bigger fundamental group classifying arbitrary étale surjective maps onto a noetherian formal scheme ${\mathfrak X}$, that would give additional information on ${\mathfrak X}$.
Let $f:{\mathfrak X}\to {\mathfrak Y}$ be an étale morphism in ${\mathsf {NFS}}$. Given ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}},$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ $f$-compatible ideals of definition, if the induced morphism $f_{0}:X_{0} \to Y_{0}$ is étale, then $f$ is adic étale.
By Proposition \[teorequivet\] there is an adic étale morphism $f':{\mathfrak X}' \to{\mathfrak Y}$ in ${\mathsf {NFS}}$ such that ${\mathfrak X}' \times_{{\mathfrak Y}} Y_{0}= X_{0}$. Therefore by Proposition \[levantpeusform\] there exists a morphism of formal schemes $g: {\mathfrak X}\to {\mathfrak X}'$ such that the diagram
[X]{}& & & \^[f]{} & &[Y]{}\
& (3,2)\^[g]{} && & \^[f’]{} &\
& & & [X]{}’ & &\
& & & & &\
X\_[0]{} & & & &\^[f\_[0]{}]{} & Y\_[0]{}\
& (3,2)\_[g\_[0]{}=1\_[X\_[0]{}]{}]{} & & & \^[f’\_[0]{}]{} &\
& & & X\_[0]{} & &\
is commutative. Applying [@AJP Proposition 2.13] we have that $g$ is étale and from Corollary \[iso0peis\] we deduce that $g$ is an isomorphism and therefore, $f$ is adic étale.
\[fnetfet\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a morphism in ${\mathsf {NFS}}$. The morphism $f$ is adic étale if and only if there exist $f$-compatible ideals of definition ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ such that the induced morphisms $f_{n}:X_{n} \to Y_{n}$ are étale, for all $n \in {\mathbb N}$.
If $f$ is adic étale, given ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ an ideal of definition, take ${\mathcal J}=f^{*}({\mathcal K}) {\mathcal O}_{{\mathfrak X}}$ the corresponding ideal of definition of ${\mathfrak X}$. By base change, we have that the morphisms $f_{n}:X_{n} \to Y_{n}$ are étale, for all $n \in {\mathbb N}$. The converse is a consequence of Proposition \[efn\] and of the previous Corollary.
Proposition \[teorequivet\] says that given $${\mathfrak Y}= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} Y_{n}$$ in ${\mathsf {NFS}}$ and $X_{0}$ an étale $Y_{0}$-scheme there exists a unique (up to isomorphism) étale ${\mathfrak Y}$-formal scheme ${\mathfrak X}$ such that ${\mathfrak X}\times_{{\mathfrak Y}} Y_{0}=X_{0}$. But, what happens when $X_{0}$ is a smooth $Y_{0}$-scheme?
\[equivlocal\] Let ${\mathfrak Y}$ be in ${\mathsf {NFS}}$ and with respect to an ideal of definition ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ let us write $${\mathfrak Y}= {\begin{array}[t]{c} \mathrm{lim}\\[-7.5 pt]
{\longrightarrow} \\[-7.5 pt] {\scriptstyle {n \in {\mathbb N}}} \end{array}} Y_{n}.$$ Given $f_{0}:X_{0} \to Y_{0}$ a morphism in ${\mathsf {Sch}}$ smooth at $x \in X_{0}$, there exists an open subset $U_{0} \subset X_{0}$, with $x \in U_{0} $ and a smooth adic ${\mathfrak Y}$-formal scheme ${\mathfrak U}$ such that ${\mathfrak U}\times_{{\mathfrak Y}} Y_{0} \cong U_{0}$.
Since this is a local question in ${\mathfrak Y}$, we may assume that ${\mathfrak Y}= \operatorname{Spf}(B)$ is in ${{\mathsf {NFS}}_{\mathsf {af}}}$, ${\mathcal K}= K^{{\triangle}}$ with $K \subset B$ an ideal of definition of the adic ring $B$, $B_{0} = B/K$ and that $f_{0}:X_{0}= \operatorname{Spec}(A_{0}) \to Y_{0} = \operatorname{Spec}(B_{0})$ is a morphism in ${\mathsf {Sch}_{\mathsf {af}}}$ smooth at $x \in X_{0}$. By Proposition \[factpl\] there exists an open subset $U_{0} \subset X_{0}$ with $x \in U_{0}$ such that $f_{0} |_{U_{0}}$ factors as $$U_{0} {\xrightarrow}{f'_{0}} {\mathbb A}_{Y_{0}}^{n}= \operatorname{Spec}(B_{0}[\mathbf{T}]) {\xrightarrow}{p_{0}} Y_{0}$$ where $\mathbf{T}= T_{1},\, T_{2},\, \ldots,\, T_{r}$ is a set of indeterminates, $f'_{0}$ is an étale morphism and $p_{0}$ is the canonical projection. The morphism $p_{0}$ lifts to a projection morphism $p: {\mathbb A}_{{\mathfrak Y}}^{n}= \operatorname{Spf}(B\{\mathbf{T}\}) \to {\mathfrak Y}$ such that the square in the following diagram is cartesian
& & [A]{}\_[[Y]{}]{}\^[n]{} & \^[p]{} & [Y]{}\
& & & &\
U\_[0]{} & \^[f’\_[0]{}]{} & [A]{}\_[Y\_[0]{}]{}\^[n]{} & \^[p\_[0]{}]{} & Y\_[0]{}\
Applying Proposition \[teorequivet\], there exists a locally noetherian étale adic ${\mathbb A}_{{\mathfrak Y}}^{n}$-formal scheme ${\mathfrak U}$ such that $U_{0} \cong {\mathfrak U}\times_{{\mathbb A}_{{\mathfrak Y}}^{n}} {\mathbb A}_{Y_{0}}^{n}$. Then ${\mathfrak U}$ is an smooth adic ${\mathfrak Y}$-formal scheme such that $U_{0} \cong {\mathfrak U}\times_{{\mathfrak Y}} Y_{0}$.
The next theorem transfers the local description of unramified morphisms known in the case of schemes ([@EGA44 (18.4.7)]) to the framework of formal schemes.
\[tppalnr\] Let $f:{\mathfrak X}\to {\mathfrak Y}$ be a morphism in ${\mathsf {NFS}}$ unramified at $x \in {\mathfrak X}$. Then there exists an open subset ${\mathfrak U}\subset {\mathfrak X}$ with $x \in {\mathfrak U}$ such that $f|_{{\mathfrak U}}$ factors as $${\mathfrak U}{\xrightarrow}{\kappa} {\mathfrak X}' {\xrightarrow}{f'} {\mathfrak Y}$$ where $\kappa$ is a pseudo-closed immersion and $f'$ is an *adic* étale morphism.
Let ${\mathcal J}\subset {\mathcal O}_{{\mathfrak X}}$ and ${\mathcal K}\subset {\mathcal O}_{{\mathfrak Y}}$ be ideals of definition. The morphism of schemes $f_{0}$ associated to these ideals is unramified at $x$ (Proposition \[nrfn\]) and by [@EGA44 (18.4.7)] there exists an open set $U_{0}\subset X_{0}$ with $x \in U_{0}$ such that $f_{0}|_{U_{0}}$ factors as $$U_{0}\rTinc^{\kappa_{0}} X'_{0} \rTto^{f'_{0}} Y_{0}$$ where $\kappa_{0}$ is a closed immersion and $f'_{0}$ is an étale morphism. Proposition \[teorequivet\] implies that there exists an étale adic morphism $f':{\mathfrak X}' \to{\mathfrak Y}$ in ${\mathsf {NFS}}$ such that ${\mathfrak X}' \times_{{\mathfrak Y}} Y_{0} = X'_{0}$. Now, if ${\mathfrak U}\subset {\mathfrak X}$ is the open formal scheme with underlying topological space $U_{0}$, by Proposition \[levantpeusform\] there exists a morphism $\kappa: {\mathfrak U}\to {\mathfrak X}'$ such that the following diagram commutes
[U]{}& & \^[f|\_[[U]{}]{}]{} & & & [Y]{}\
& (3,2)\^ && &\^[f’]{}&\
& & & [X]{}’ & &\
& & && &\
U\_[0]{} & &\^[f\_[0]{}|\_[U\_[0]{}]{}]{}&&& Y\_[0]{}\
& (3,2)\_[\_[0]{}]{} &&& \^[f’\_[0]{}]{} &\
& & & X’\_[0]{} & &\
Since $f$ is unramified, by [@AJP Proposition 2.13] it holds that $\kappa$ is unramified. Furthermore, $\kappa_{0}$ is a closed immersion, then Corollary \[pecigf0ecnr\] shows us that $\kappa$ is a pseudo-closed immersion.
As a consequence of the last result we obtain the following local description for étale morphisms.
\[tppalet\] Let $f\colon{\mathfrak X}\to {\mathfrak Y}$ be a morphism in ${\mathsf {NFS}}$ étale at $x \in {\mathfrak X}$. Then there exists an open subset ${\mathfrak U}\subset {\mathfrak X}$ with $x \in {\mathfrak U}$ such that $f|_{{\mathfrak U}}$ factors as $${\mathfrak U}{\xrightarrow}{\kappa} {\mathfrak X}' {\xrightarrow}{f'} {\mathfrak Y}$$ where $\kappa$ is a completion morphism and $f'$ is an *adic* étale morphism.
By the last theorem we have that there exists an open formal subscheme ${\mathfrak U}\subset {\mathfrak X}$ with $x \in {\mathfrak U}$ such that $f|_{{\mathfrak U}}$ factors as $${\mathfrak U}{\xrightarrow}{\kappa} {\mathfrak X}' {\xrightarrow}{f'} {\mathfrak Y}$$ where $\kappa$ is a pseudo-closed immersion and $f'$ is an adic étale morphism. Since $f|_{{\mathfrak U}}$ is étale and $f'$ is an adic étale morphism we have that $\kappa$ is étale by [@AJP Proposition 2.13]. Now, applying Theorem \[caracmorfcompl\] it follows that $\kappa$ is a completion morphism.
\[tppall\] Let $f\colon{\mathfrak X}\to {\mathfrak Y}$ be a morphism in ${\mathsf {NFS}}$ smooth at $x \in {\mathfrak X}$. Then there exists an open subset ${\mathfrak U}\subset {\mathfrak X}$ with $x \in {\mathfrak U}$ such that $f|_{{\mathfrak U}}$ factors as $${\mathfrak U}{\xrightarrow}{\kappa} {\mathfrak X}' {\xrightarrow}{f'} {\mathfrak Y}$$ where $\kappa$ is a completion morphism and $f'$ is an *adic* smooth morphism.
By Proposition \[factpl\] there exists an open formal subscheme ${\mathfrak V}\subset {\mathfrak X}$ with $x \in {\mathfrak V}$ such that $f|_{{\mathfrak V}}$ factors as $${\mathfrak V}{\xrightarrow}{g} {\mathbb A}^{n}_{{\mathfrak Y}} {\xrightarrow}{p} {\mathfrak Y}$$ where $g$ is étale and $p$ is the canonical projection. Applying the last Theorem to the morphism $g$ we conclude that there exists an open subset ${\mathfrak U}\subset {\mathfrak X}$ with $x \in {\mathfrak U}$ such that $f|_{{\mathfrak U}}$ factors as $${\mathfrak U}{\xrightarrow}{\kappa} {\mathfrak X}' {\xrightarrow}{f''} {\mathbb A}_{{\mathfrak Y}}^{n} {\xrightarrow}{p}{\mathfrak Y}$$ where $\kappa$ is a completion morphism, $f''$ is an adic étale morphism and $p$ is the canonical projection, from where it follows that $f'= f'' \circ p$ is adic smooth.
Lipman, Nayak and Sastry note in [@LNS pag. 132] that this Theorem may simplify some developments related to Cousin complexes and duality on formal schemes. See the final part of Remark 10.3.10 of *loc. cit.*
We have benefited form conversations on these topics and also on terminology with Joe Lipman, Suresh Nayak and Pramath Sastry. The authors thank the Mathematics department of Purdue University for hospitality and support.
The diagrams were typeset with Paul Taylor’s `diagrams.sty`.
[AB]{}
Alonso Tarr[í]{}o, L.; Jerem[í]{}as López, A.; Lipman, J.: Duality and flat base-change on formal schemes, in *Studies in duality on noetherian formal schemes and non-noetherian ordinary schemes*. Providence, RI: American Mathematical Society. Contemp. Math. **244**, 3–90, 1999.
Alonso Tarr[í]{}o, L.; Jerem[í]{}as López, A.; Lipman, J.: Correction to the paper: “Duality and flat base change on formal schemes” *Proc. Amer. Math. Soc.* **131** (2003), no. 2, 351–357.
Alonso Tarr[í]{}o, L.; Jerem[í]{}as López, A.; Pérez Rodríguez, M.: Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes. *Comm. Alg*, **35**, 1341–1367, 2007.
Altman, A. B.; Kleiman, S.: [*Introduction to Grothendieck Duality Theory*]{}, Springer-Verlag, Heidelberg-Berlin-New York , 1970.
Bourbaki, N.: [*Commutative algebra, Chapters 1–7*]{}. Elements of Mathematics. Springer-Verlag, Berlin-New York, 1989.
Grothendieck, A.; Dieudonné, J. A.: [*Eléments of Géométrie Algébrique I*]{}, Grundlehren der math. Wissenschaften [**166**]{}, Springer-Verlag, Heidelberg, 1971.
Grothendieck, A.; Dieudonné, J. A.: [*Eléments of Géométrie Algébrique III, Étude cohomologique des faisceaux cohérents*]{}, Publications Mathématiques, [**11**]{}, Institut des Hautes Études Scientifiques, Paris, 1961.
Grothendieck, A.; Dieudonné, J. A.: [*Eléments of Géométrie Algébrique IV, Étude locale des schémas et des morphismes of schémas (première partie)*]{}, Publications Mathématiques, [**20**]{}, Institut des Hautes Études Scientifiques, Paris, 1964.
Grothendieck, A.; Dieudonné, J. A.: [*Eléments of Géométrie Algébrique IV, Étude locale des schémas et des morphismes of schémas (quatrième partie)*]{}, Publications Mathématiques, [**32**]{}, Institut des Hautes Études Scientifiques, Paris, 1967.
Hartshorne, R.: [*On the of Rham cohomology of algebraic varieties*]{}. [*Pub. Math. IHES*]{}, [**45**]{} (1975), 5–99.
Lipman, J.; Nayak, S.; Sastry P.: Variance and duality for Cousin complexes on formal schemes, pp. 3–133 in [*Pseudofunctorial behavior of Cousin complexes on formal schemes*]{}. Contemp. Math., **375**, Amer. Math. Soc., Providence, RI, 2005.
Matsumura, H.: [*Commutative algebra*]{}, W. A. Benjamin, New York, 1980.
Matsumura, H.: [*Commutative ring theory*]{}, Cambridge University Press, Cambridge, 1986.
MacLane, S.: [*Categories for the Working Mathematician*]{}, Springer-Verlag, New York Heidelberg Berlin, 1971.
Pérez Rodríguez, M.: Basic Deformation Theory of smooth formal schemes, to appear in *J. Pure Appl. Algebra*, `arXiv:0801.2846`.
Strickland, N. P.: Formal schemes and formal groups. *Homotopy invariant algebraic structures* (Baltimore, MD, 1998), 263–352, *Contemp. Math.*, **239**, Amer. Math. Soc., Providence, RI, 1999.
Yekutieli, A.: Smooth formal embeddings and the residue complex. [*Canad. J. Math.*]{} [**50**]{} (1998), no. 4, 863–896.
[^1]: This work was partially supported by Spain’s MCyT and E.U.’s FEDER research project MTM2005-05754.
[^2]: In [@y] the terminology *formally finite type* is used.
[^3]: Let $(X,{\mathcal O}_{X})$ be a ringed space. We say that the sequence of ${\mathcal O}_{X}$-Modules $0 \to {\mathcal F}\to {\mathcal G}\to {\mathcal H}\to 0$ is exact at $x \in X$ if and only if $0 \to {\mathcal F}_{x} \to {\mathcal G}_{x} \to {\mathcal H}_{x} \to 0$ is an exact sequence of ${\mathcal O}_{X,x}$-modules.
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---
abstract: 'Max-stable processes have been expanded to quantify extremal dependence in spatio-temporal data. Due to the interaction between space and time, spatio-temporal data are often complex to analyze. So, characterizing these dependencies is one of the crucial challenges in this field of statistics. This paper suggests a semiparametric inference methodology based on the spatio-temporal $F$-madogram for estimating the parameters of a space-time max-stable process using gridded data. The performance of the method is investigated through various simulation studies. Finally, we apply our inferential procedure to quantify the extremal behavior of radar rainfall data in a region in the State of Florida.'
author:
- |
Abdul-Fattah Abu-Awwad, abuawwad@math.univ-lyon1.fr &\
Véronique Maume-Deschamps, veronique.maume@univ-lyon1.fr &\
Pierre Ribereau, pierre.ribereau@univ-lyon1.fr\
Université de Lyon, Université Claude Bernard Lyon 1,\
Institut Camille Jordan ICJ UMR 5208 CNRS, France.
title: 'Semiparametric estimation for space-time max-stable processes: $F$-madogram-based estimation approach'
---
Introduction
============
Typically, extremes of environmental and climate processes like extreme wind speeds or heavy precipitation are modelled using extreme value theory. Max-stable processes are ideally suited for the statistical modeling of spatial extremes as they form the natural extension of multivariate extreme value distributions to infinite dimensions. Various families of max-stable models and estimation procedures have been proposed for extremal data. For a detailed overview of max-stable processes, we refer the reader to [@de2007extreme]. For statistical inference, it is then often assumed that the observations at spatial locations are independent in time, see, e.g., [@padoan2010likelihood; @davison2012statistical; @davison2013geostatistics]. However, many extreme environmental processes observations exhibit a spatial dependence structure, meaning that neighboring locations within some distance show similar patterns, as well as a temporal dependence, which can be seen from high values for two consecutive time moments (e.g., within hours). As an illustration, Figure \[Dailyint\] depicts the daily rainfall maxima for the wet seasons (June-September) from the years 2007-2012 at one fixed grid location in Florida. We observe that it is likely that a high value is followed by a value of a similar magnitude. So, the temporal dependence may be present. Accordingly, the temporal dependence structure should be considered in an appropriate way. More details on the rainfall data in Florida are given in Section \[sec:real\].
\[!htp\] ![Daily rainfall maxima in inches taken over hourly measurements from 2007-2012 for a fixed location in Florida, USA.[]{data-label="Dailyint"}](Dailyint.pdf "fig:")
Currently space-time models are still taking up little space in the literature. Only a few papers have introduced space-time max-stable models. For instance, [@davis2013max] extended the construction of spatial Brown-Resnick (BR) model [@brown1977extreme; @kabluchko2009stationary] and Smith’s storm profile model [@smith1990max] to the space-time domain, whereas [@buhl2016anisotropic] extended the space-time BR model [@davis2013max] to an anisotropic setting. Additionally, [@huser2014space] introduced an extension of spatial Schather model [@schlather2002models], which comprises a truncated Gaussian random process, so that storm shapes are stochastic, and includes a compact random set, that allows the process to be mixing in space as well as to exhibit a spatial diffusions, see also [@davison2012geostatistics]. A common feature for these models is that the major emphasis is in modeling asymptotic dependence treating the time just as additional dimension of the space. So, these models do not allow any interaction between the spatial components and temporal component in the underlying dependence function. However it seems reasonable to suppose that the spatial and temporal components behave asymptotically in a different way. Therefore, a new class of space-time max-stable models have been proposed by [@embrechts2016space] in which the influence of time and space are partly decoupled, where the time infuences space through a bijective operator on space.\
The inference on max-stable processes in both spatial and spatiotemporal contexts is an open field that is still in development. Many techniques have been proposed for parameter estimation in spatial extreme models. Each technique has its pros and cons. As with spatial max-stable processes, the pairwise likelihood estimation has been found useful to estimate the parameters of space-time max-stable processes due to its theoretical properties, see, e.g. [@davis2013statistical; @huser2014space; @embrechts2016space]. Recently, various semiparametric estimation approaches have been proposed to fit such processes. For instance [@buhl2016semiparametric] introduced a new semiparametric estimation procedure based on a closed form expression of the so-called extremogram [@davis2009extremogram] to estimate the parameters of space-time max-stable BR process. The extremogram has been estimated nonparametrically by its empirical version, where space and time are separated. A constrained weighted linear regression is then applied in order to produce parameter estimates. While in [@manaf] a semiparametric estimation procedure has been developed for spatial max-mixture processes [@doi:10.1093/biomet/asr080] based on the $F$-madogram [@cooley2006variograms]. A non-linear least squares (NLS) is then applied to minimize the squared difference between the empirical $F$-madogram and its model-based counterpart. A major advantage of the semiparametric methods is the substantial reduction of computation time compared to the pairwise likelihood estimation. Hence, these methods can be applied as an alternative or a prerequisite to the widely-adopted pairwise likelihood inference, which suffers from some defects; first, it can be onerous, since the computation and subsequent optimization of the objective function is time-consuming. Second, the choice of good initial values for optimization of the composite likelihood is essential.
An implicit difficulty in any extreme value analysis is the limited amount of data for model estimation, see, e.g. [@coles2001introduction]. Hence, inference based on the extremogram is difficult because few observations are available as the threshold increases. Consequently, the semiparametric estimates obtained by [@buhl2016semiparametric] showed a larger bias than the pairwise likelihood estimates and are sensible to the choice of the threshold used for the extremogram. Accordingly, the surrogates of existing estimation techniques should be welcomed.
In the present paper, we are interested in statistical inference for space-time max-stable processes. Motivated by deficiencies in existing inference approaches, we propose two novel and flexible semiparametric estimation schemes to fit space-time max-stable processes:
1. **Scheme 1:** we estimate spatial and temporal parameters separately. Based on NLS, we minimize the squared difference between the empirical estimates of spatial/temporal $F$-madograms and their model-based counterparts. Our inferential methodology is close to the one that has been proposed by [@buhl2016semiparametric] as an alternative or a preliminary analysis to the pairwise likelihood approach in [@davis2013statistical], where only isotropic space-time max-stable BR process has been fitted via the two approaches.
2. **Scheme 2:** we generalize the NLS to estimate spatial and temporal parameters simultaneously.
The remainder of the paper is organized as follows. Section \[sec:models\] defines the space-time max-stable models. The two semiparametric estimation schemes are described in Section \[sec:inference\]. Section \[sec:simulation\] illustrates the performance of our method through various simulation studies, where also a comparison with the semiparametric estimation [@buhl2016semiparametric] is performed. In Section \[sec:real\], we apply our method to radar rainfall data in a region in Florida by using spatial and temporal block maxima design. The concluding remarks in Section \[sec:conc\] address some remaining issues and perspectives.
Space-time max-stable models {#sec:models}
=============================
Throughout the paper, $X:=\left\{ X({\boldsymbol{s}},t): ({\boldsymbol{s}},t) \in \mathcal{S}\times \mathcal{T} \right\}$, $\mathcal{S}\times \mathcal{T} \subset {\ensuremath{\mathbb{R}}}^d \times {\ensuremath{\mathbb{R}}}^+$ (generally, $d=2$) is a spatiotemporal process, where the space $\mathcal{S}\times \mathcal{T}$ is the spatiotemporal domain. The points ${\boldsymbol{s}}$ denote the spatial coordinates and are called “sites” or “locations” or “stations” and the points $t$ denote the temporal coordinates and are called “times” or “moments”. The space index ${\boldsymbol{s}}$ and time index $t$ will respectively belong to the sets $\mathcal{S}$ and $\mathcal{T}$. In addition, we will denote by ${\boldsymbol{h}}={\boldsymbol{s}}_1-{\boldsymbol{s}}_2 \in {\ensuremath{\mathbb{R}}}^2$ (respectively $l=t_1-t_2 \in {\ensuremath{\mathbb{R}}}$) the spatial (respectively temporal) lag.
Space-time max-stable models without spectral separability
----------------------------------------------------------
According to [@de1984spectral], the simple space-time max-stable process $X$, where simple means that the margins are standard Fréchet, i.e., $F(x):=\mathbb{P} (X({\boldsymbol{s}},t)\leq x)= \exp \{-x^{-1}\}$, $x>0$, has the following spectral representation $$X(\boldsymbol{s},t)\stackrel{\mathcal{D}}{=} \bigvee_{i=1}^{\infty} {\xi_{i} U_i {(\boldsymbol{s},t)}}, \ (\boldsymbol{s},t) \in \mathcal {S}\times \mathcal{T}.
\label{spectral rep}$$
where $\bigvee$ denotes the max-operator, $\{\xi_{i}\}_{i \geq 1}$ are independent and identically distributed (i.i.d.) points of a Poisson process on $(0,\infty)$ with intensity $\xi^{-2} d\xi$ and $\{U_{i} (t,{\boldsymbol{s}})\}_{i \geq 1}$ is a sequence of independent replications of some space-time process $\{U({\boldsymbol{s}},t), ({\boldsymbol{s}},t) \in \mathcal {S}\times \mathcal{T}\}$ with ${\ensuremath{\mathbb{E}}}\{U({\boldsymbol{s}},t)\} < \infty$ for each $(t,\boldsymbol{s}) \in \mathcal {S}\times \mathcal{T}$, and $U({\boldsymbol{s}},t) \geq 0$, which are also independent of $\xi_{i}$.
For $D \in \mathbb{N}\setminus \{0\}$, $\boldsymbol{s}_{1},\ldots,\boldsymbol{s}_{D} \in \mathcal{S}$, $t_{1},\ldots,t_{D} \in \mathcal{T}$ and $x_{1},\ldots,x_{D}>0$, the finite $D$-dimensional distributions of the space-time max-stable process $X$ are given by $$\begin{aligned}
\mathbb{P}(X({\boldsymbol{s}}_1,t_1)\leq x_{1},\ldots,X({\boldsymbol{s}}_D,t_D)\leq x_{D})=& \mathbb{P} \left\{ \xi_i \bigvee_{j=1}^{D} \frac{U_i {({\boldsymbol{s}}_{j},t_j)}}{x_j} \leq 1, \forall i=1,2,\ldots \right\} \\ \nonumber
&=\exp \left\{ -\mathbb{E} \left(\bigvee_{j=1}^{D} \frac{U {(\boldsymbol{s}_{j},t_j)}}{x_j} \right) \right\}. \end{aligned}$$ Hence, all finite-dimensional distributions are multivariate extreme value distributions with unit Fréchet margins. In particular, for $x_1,x_2 > 0$, the bivariate cumulative distribution function (c.d.f.) $F_{{\boldsymbol{s}}_1,t_1,{\boldsymbol{s}}_2,t_2}$ of the space-time max-stable process $X({\boldsymbol{s}},t)$ in (\[spectral rep\]) can be expressed in terms of the underlying bivariate spatio-temporal exponent function $V_{{\boldsymbol{s}}_1,t_1,{\boldsymbol{s}}_2,t_2}$ as $$\begin{aligned}
\label{biv exponent measure}
- \log F_{{\boldsymbol{s}}_1,t_1;{\boldsymbol{s}}_2,t_2}(x_1,x_2)=&- \log \mathbb{P}\left[X(t_1,\boldsymbol{s}_1) \leq x_{1}, X( t_2,\boldsymbol{s}_2) \leq x_{2} \right]\\ \nonumber & =: V_{{\boldsymbol{s}}_1,t_1;{\boldsymbol{s}}_2,t_2}\left(x_1,x_2\right).\end{aligned}$$ Below, we will consider stationary space-time processes, so that $V_{{\boldsymbol{s}}_1,t_1;{\boldsymbol{s}}_2,t_2}$ depends only on ${\boldsymbol{h}}={\boldsymbol{s}}_1-{\boldsymbol{s}}_2$ and $l=t_1-t_2$. We will write $F_{{\boldsymbol{h}},l}$ for $F_{{\boldsymbol{s}}_1,t_1;{\boldsymbol{s}}_2,t_2}$ and $V_{{\boldsymbol{h}},l}$ for $V_{{\boldsymbol{s}}_1,t_1;{\boldsymbol{s}}_2,t_2}$.
### [Spatio-temporal extremal dependence summary measures]{}
In order to measure the spatio-temporal extremal dependence, we provide in the next Definition, extensions to the spatio-temporal setting of some quantities that have been introduced in the spatial context. For a stationary spatio-temporal max-stable process $X$ with univariate margin c.d.f. $F$, we have
1. **(Spatio-temporal extremal dependence function, originally due to [@schlather2003dependence])** $$\label{Def::spatiotemporal-theta}
\theta({\boldsymbol{h}},l)= - x\log {\mathbb P}\left( X({\boldsymbol{s}},t) \leq x, X({\boldsymbol{s}}+{\boldsymbol{h}},t+l) \leq x \right) \in [1,2], \ x>0.$$
2. **(Spatio-temporal upper tail dependence function, originally due to [@coles1999dependence])** $$\label{Def::spatiotemporal-chi}
\chi_u{({\boldsymbol{h}},l)}= 2- \frac{2 \log \mathbb{P}\{F(X({\boldsymbol{s}},t))<u , F(X({\boldsymbol{s}}+{\boldsymbol{h}},t+l))<u\}}{\log \mathbb{P}\{F(X({\boldsymbol{s}}+{\boldsymbol{h}},t+l))<u\}}$$ and $\chi{({\boldsymbol{h}},l)}=\lim_{u\rightarrow 1^-}\chi_u{({\boldsymbol{h}},l)}$, $u \in [0,1]$. Similarly to spatial setting, we have the simple link: $\chi{({\boldsymbol{h}},l)}=2-\theta({\boldsymbol{h}},l)$.
3. **(Spatio-temporal $F$-madogram, originally due to [@cooley2006variograms])** $$\nu_{F}({\boldsymbol{h}},l)= \frac{1}{2} \mathbb{E} \left[|F(X( {\boldsymbol{s}},t )) - F(X( {\boldsymbol{s}}+{\boldsymbol{h}},t+l ))|\right] \in [0,1/6].
\label{Def::spatiotemporal-mado}$$ Furthermore, the $F^\lambda$-madogram (originally due to [@bel2008assessing]) and $\lambda$-madogram (originally due to [@naveau2009modelling]) are defined analogously.
4. **(Spatio-temporal extremogram dependence function, originally due to [[@davis2009extremogram]]{})** $$\label{Space-time::Extremo}
\rho_{\mathscr{A}_1, \mathscr{A}_2}({\boldsymbol{h}},l)= \lim_{x \rightarrow \infty} \frac{ \mathbb{P}\left\{x^{-1} X({\boldsymbol{s}},t) \in \mathscr{A}_1 , x^{-1}X({\boldsymbol{s}}+{\boldsymbol{h}},t+l) \in \mathscr{A}_2\right\}}{ \mathbb{P}\left\{ x^{-1} X({\boldsymbol{s}},t) \in \mathscr{A}_1\right\}}.$$ Clearly, setting the Borel sets $\mathscr{A}_1=\mathscr{A}_2=(1,\infty)$ yields $\rho_{(1,\infty), (1,\infty)}({\boldsymbol{h}},l) =\chi({\boldsymbol{h}},l)$. The two cases $\chi({\boldsymbol{h}},l)=0$ and $\chi({\boldsymbol{h}},l)=1$ correspond to the boundary cases of asymptotic independence and complete dependence.
Both dependence functions $\theta({\boldsymbol{h}},l)$ and $\chi({\boldsymbol{h}},l)$ provide simple measures of extremal dependence within the class of asymptotic dependence distributions.
**(Stationary BR spatio-temporal process without spectral separability)** [@davis2013max] introduced the spatial BR model [@brown1977extreme; @kabluchko2009stationary] in space and time. A strictly stationary spatio-temporal BR process $X$ has the following spectral representation $$X({\boldsymbol{s}},t)= \bigvee_{i=1}^{\infty} \xi_i \exp \left\{\varepsilon_i({\boldsymbol{s}},t) - \gamma({\boldsymbol{s}},t) \right\}, \ ({\boldsymbol{s}},t) \in \mathcal {S}\times \mathcal{T},
\label{BR spectral fun}$$ where $ \left\{\xi_{i}\right\}_{i \geq 1}$ are points of a Poisson process on $(0,\infty)$ with intensity $\xi^{-2} d\xi$, the processes $\left\{\varepsilon_i({\boldsymbol{s}},t): {({\boldsymbol{s}},t)\in (\mathcal{S}\times \mathcal{T})} \right\}$ are independent replications of a Gaussian process $\{\varepsilon(t,{\boldsymbol{s}})\}$ with stationary increments, $\varepsilon(\boldsymbol{0},0)=0$, $\mathbb{E}[\varepsilon({\boldsymbol{s}},t)]=0$ and covariance function $$\mathbb{C}ov (\varepsilon({\boldsymbol{s}}_1,t_1),\varepsilon({\boldsymbol{s}}_2,t_2) ) = \gamma({\boldsymbol{s}}_1 ,t_1) + \gamma({\boldsymbol{s}}_2,t_2) - \gamma( {\boldsymbol{s}}_1-{\boldsymbol{s}}_2 , t_1-t_2),$$ for all $( {\boldsymbol{s}}_1,t_1),({\boldsymbol{s}}_2,t_2) \in\mathcal {S} \times \mathcal{T}$. The dependence function $\gamma$ which is termed the spatio-temporal semivariogram of the process $\{\varepsilon({\boldsymbol{s}},t)\}$, is non-negative and conditionally negative definite, that is, for any $k \in \mathbb{N}$, $({\boldsymbol{s}}_1,t_1),\ldots,({\boldsymbol{s}}_k,t_k) \in \mathcal{S}\times\mathcal{T} $ and $a_1, \ldots,a_k \in \mathbb{R}$, $$\sum_{i=1} ^{k} \sum_{j=1} ^{k} a_i a_j \gamma \left( {\boldsymbol{s}}_i -{\boldsymbol{s}}_j ,t_i-t_j \right) \leq 0,\ \sum_{i=1}^{k}a_i =0.$$
The process $X({\boldsymbol{s}},t)$ in (\[BR spectral fun\]) is fully characterized by the dependence function $\gamma$. In geostatistics, the function $\gamma$ is given by $$\gamma\left( {\boldsymbol{s}}_1-{\boldsymbol{s}}_2,t_1-t_2 \right)= \frac{1}{2} \mathbb{V}ar \left(\varepsilon({\boldsymbol{s}}_1,t_1)-\varepsilon({\boldsymbol{s}}_2,t_2) \right).$$ Let $\Phi$ denote the standard normal distribution function. For $x_1,x_2>0$, the bivariate c.d.f. ($F_{{\boldsymbol{h}},l}$) of $\left(X({\boldsymbol{s}}_1,t_1),X({\boldsymbol{s}}_2,t_2)\right)$ in the stationary case is given by $$\begin{aligned}
\label{Biv BR}
- \log F_{{\boldsymbol{h}},l}(x_1,x_2) = \frac{1}{x_{1}}\Phi\left(\sqrt{\frac{\gamma({{\boldsymbol{h}}}, l)}{2}}+\frac{\log\left(\frac{x_{2}}{x_{1}}\right)}{\sqrt{2\gamma({{\boldsymbol{h}}} ,l )}}\right )\\+ \nonumber
\frac{1}{x_{2}}\Phi\left(
\sqrt{\frac{\gamma( {{\boldsymbol{h}}},l)}{2}}+\frac{ \log\left(\frac{x_{1}}{x_{2}}\right)}{\sqrt{2\gamma( {{\boldsymbol{h}}},l)}}\right).\end{aligned}$$ Recall that if $\gamma$ is assumed to depend only on the norm of ${\boldsymbol{s}}_1-{\boldsymbol{s}}_2$, the associated process is spatially isotropic. The pairwise spatio-temporal extremal dependence function for this model is $\theta({{\boldsymbol{h}}},l)=2\Phi\left\{\sqrt{\gamma({{\boldsymbol{h}}},l)/2}\right\}$. This model has been used in [@buhl2016semiparametric] to quantify the extremal behavior of radar rainfall data in a region of Florida, where a new semiparametric procedure based on the extremogram is applied to estimate the model parameters.
Space-time max-stable models with spectral separability
-------------------------------------------------------
The fundamental advantages of the spectral representation in (\[spectral rep\]) are (i) the construction of spatio-temporal processes from widely studied max-stable processes (ii) the huge literature available on spatio-temporal correlation functions for Gaussian processes, allows for considerable diversity of spatio-temporal behavior. However, an important modeling issue is that they do not allow any interaction between the spatial and the temporal components in the underlying dependence function. Thus, the time has no specific role but is equivalent to an additional spatial dimension; the spatial and temporal distributions belong to a similar family of models. Hence, alternatively, a new class of space-time max-stable models with spectral separability has been suggested in [@embrechts2016space]. More precisely, $$X({\boldsymbol{s}},t)= \bigvee_{i=1}^{\infty} \xi_{i}U_ {t}(Q_i) U_ {\mathcal{R}(t,Q_i){{\boldsymbol{s}}}}(W_i),
\label{spectral rep1}$$ where $\{\xi_{i},Q_i,W_i\}_{i \geq 1}$ are the points of a Poisson process on $(0,\infty) \times E_{1} \times E_{2} $, and with intensity $\xi^{-2} d\xi \times \mu_{1} (dq) \times \mu_{2} (dw)$ for some Polish measure spaces $(E_{1},\mathcal{E}_{1},\mu_{1})$ and $(E_{2},\mathcal{E}_{2},\mu_{2})$. The spectral function $U_ {t}:E_{1} \rightarrow (0,\infty)$ is measurable such that $\int_{E_{1}} U_ {t}(q)\mu_{1} (dq) =1$ for each $t \in \mathcal {T}$ and contributes to the temporal dynamic of the process, whereas the spectral function $U_ {{\boldsymbol{s}}}:E_{2} \rightarrow (0,\infty)$ is measurable such that $\int_{E_{2}} U_ {{\boldsymbol{s}}}(w)\mu_{2} (dw) =1$ for each $\boldsymbol{s} \in \mathcal {S}$ and drives the shape of the main spatial patterns. The operators $\mathcal{R}(t,q)$ are bijective from $\mathcal{S}$ to $\mathcal{S}$ for each $(t,q) \in \mathcal {T}\times E_{1 }$ and describes how the spatial patterns move in space.
The construction (\[spectral rep1\]) allows one to deal with the temporal and spatial aspects separately. So, the estimation procedure can be simplified by estimating in a first step the spatial parameters independently from the temporal ones. Several examples of subclasses of the general class of space-time process $X$ (\[spectral rep1\]) were introduced by [@embrechts2016space], where the operator is either a translation or a rotation. The authors in that paper focused mainly on a special case of models where the function corresponding to the time in the spectral representation is the exponential density (continuous-time case) or the probability values of a geometric random variable (discrete-time case). So, the corresponding models become Markovian and have a useful max-autoregressive representation, i.e., $$\label{model space-time}
X({\boldsymbol{s}},t)=\max\left\{\delta X(\boldsymbol{s}-\boldsymbol{\tau},t-1),(1-\delta)H(\boldsymbol{s},t)\right\}, \ ({\boldsymbol{s}},t) \in \mathcal {S}\times \mathcal{T},$$ where the parameter $\delta \in (0,1)$ measures the influence of the past, the parameter $\boldsymbol{\tau} \in \mathbb{R}^2$ represents some kind of specific direction of propagation/contagion in space and $H=:\{H({\boldsymbol{s}},t), {\boldsymbol{s}}\in \mathcal{S}, t \in \mathcal{T}\}$ is a time-independent process and is derived from independent replications of a spatial max-stable process $\{H({\boldsymbol{s}}), {\boldsymbol{s}}\in \mathcal{S}\}$. This model can be seen as an extension of the real-valued max-autoregressive moving-average process MARMA(1,0) to the spatial context, see [@davis1989basic]. The value at location $\boldsymbol{s}$ and time $t$ is either related to the value at location $\boldsymbol{s}-\boldsymbol{\tau}$ at time $t-1$ or to the value of another process (the innovation), $H$, that characterizes a new event happening at location $\boldsymbol{s}$. This model may be useful for phenomena that propagate in space.
In the following, we will focus on the processes satisfying (\[model space-time\]). Let $V_{\boldsymbol{0},\boldsymbol{h}-l\boldsymbol{\tau}}$ denote the exponent function characterizing the spatial distribution of the process $H({\boldsymbol{s}},t)$, then the bivariate c.d.f. $F_{{\boldsymbol{h}},l}$ of $(X(\boldsymbol{0},0),X( {\boldsymbol{h}},l))$ can be expressed for $x_1,x_2>0$ as $$\label{Biv CDF}
-\log F_{{\boldsymbol{h}},l}(x_1,x_2) =V_{\boldsymbol{0},\boldsymbol{h}-l\boldsymbol{\tau}} \left(x_1,\frac{x_2}{\delta^{l}}\right)+\frac{1-\delta^{l}}{x_2}.$$ Moreover, the spatio-temporal extremal dependence function in (\[Def::spatiotemporal-theta\]) can be easily deduced in this case by setting $x_1=x_2=x$ in (\[Biv CDF\]), $$\label{spatiotemporal theta}
\theta({\boldsymbol{h}},l)=V_{\boldsymbol{0},{\boldsymbol{h}}-l\boldsymbol{\tau}}\left(1,\delta^{-l} \right) + 1-\delta^{l}.$$
Clearly, space and time are not fully separated in the extremal dependence function, even if $\boldsymbol{\tau}=\boldsymbol{0}$ (space and time are completely separated in the spectral representation). Asymptotic time independence is achieved when $\lim_{l \, \to \, \infty} \theta({\boldsymbol{h}},l)\rightarrow 2$. In the sequel, we give two examples of a bivariate space-time max-stable process satisfying (\[model space-time\]).
1. ***Spectrally separable space-time max-stable Smith process***
If the innovation process $H$ is derived from independent replications of a spatial Smith process [@smith1990max] with a covariance matrix $\boldsymbol{\Sigma}$. Then the bivariate c.d.f. $F_{{\boldsymbol{h}},l}$ of the resulting spatio-temporal model in (\[model space-time\]) has the form $$\begin{aligned}
\label{Smith::ERWAN}
-\log F_{{\boldsymbol{h}},l}(x_1,x_2)& =\frac{1}{x_{1}}\Phi\left( \frac{b ({\boldsymbol{h}},l)}{2}+\frac{1}{b({\boldsymbol{h}},l)}\log\left(\frac{x_{2}}{\delta^{l} x_{1}}\right)\right )\\ \nonumber&+ \frac{\delta^{l}}{x_{2}}\Phi\left( \frac{b({\boldsymbol{h}},l)}{2}+\frac{1}{b({\boldsymbol{h}},l)}\log\left(\frac{\delta^{l}x_{1}}{x_{2}}\right)\right)+\frac{1-\delta^{l}}{x_2},
\end{aligned}$$ where $b({\boldsymbol{h}},l)=\sqrt{(\boldsymbol{h}-l\boldsymbol{\tau})^{t}\boldsymbol{\Sigma}^{-1}(\boldsymbol{h}-l\boldsymbol{\tau})}$. The associated spatio-temporal extremal dependence function with this model is $$\begin{aligned}
\theta({\boldsymbol{h}},l)=& \Phi\left( \frac{b({\boldsymbol{h}},l)}{2}+\frac{1}{b({\boldsymbol{h}},l)}\log\left(\delta^{-l}\right)\right)+ \delta^{l}\Phi\left( \frac{b({\boldsymbol{h}},l)}{2}+\frac{1}{b({\boldsymbol{h}},l)}\log\left(\delta^{l}\right)\right) \\ \nonumber & + 1-\delta^{l}.
\end{aligned}$$
2. ***Spectrally separable space-time max-stable Schlather process***
The spatio-temporal model in (\[model space-time\]) with an innovation process $H$ derived from independent replications of a spatial Schlather process [@schlather2002models], has a bivariate c.d.f. $F_{{\boldsymbol{h}},l}$ of the form $$\begin{aligned}
\label{Schlather space time}
-\log F_{{\boldsymbol{h}},l}(x_1,x_2) &=\frac{1}{2}\left( \frac{1}{x_1}+\frac{\delta^{l}}{x_2}\right)\\ \times \nonumber &\left[\left(1+ \sqrt{1- \frac{2\delta^l (\rho({\boldsymbol{h}},l)+1)x_{1} x_{2}} {(\delta^{l}x_{1}+x_{2})^{2}}}\right)\right] + \frac{1-\delta^{l}}{x_2},
\end{aligned}$$ where $\rho({\boldsymbol{h}},l)$ is the spatio-temporal exponential correlation function related to this model. The associated spatio-temporal extremal coefficient with this model is $\theta({\boldsymbol{h}},l)=\frac{1}{2} (1+\delta^{l})\left[\left(1+ \sqrt{1- \frac{2\delta^l (\rho({\boldsymbol{h}},l)+1)} {(1+\delta^{l})^{2}}}\right)\right]+ 1-\delta^{l}.$
If the time lag $l=0$, the formulas in (\[Smith::ERWAN\]) and (\[Schlather space time\]) reduce to the bivariate distributions of the max-stable spatial fields.
Statistical inference for space-time max-stable processes {#sec:inference}
==========================================================
In what follows, we shall denote, respectively, by $h={\ensuremath{\lVert \boldsymbol{h} \rVert} }=: \lVert {\boldsymbol{s}}_1-{\boldsymbol{s}}_2 \rVert, \ {\boldsymbol{h}}\in {\ensuremath{\mathbb{R}}}^2$ and $l'=|l|=:|t_1-t_2|, \ l \in {\ensuremath{\mathbb{R}}}$ the Euclidean norm of spatial lag ${\boldsymbol{h}}$ and the absolute value of temporal lag $l$.
We now describe two semiparametric estimation schemes for space-time max-stable processes based on the spatio-temporal $F$-madogram in (\[Def::spatiotemporal-mado\]), which stems from a classical geostatistical tool; the madogram [@matheron1987suffit]. It has a clear link with extreme value theory throughout the spatio-temporal extremal dependence function $\theta(.)$, i.e., $$\nu_{F}({{\boldsymbol{h}}},l)=\frac{1}{2} -\frac{1}{\theta({\boldsymbol{h}},l)+1}.
\label{spatiotemporal madogram}$$
In practice, measurements are typically taken at various locations, sometimes on a grid, and at regularly spaced time intervals. In the following, the process $X:=\{X({\boldsymbol{s}},t): {({\boldsymbol{s}},t)\in \mathcal{S}\times \mathcal{T}} \}$ is assumed to be a stationary space-time max-stable process. It is observed on locations assumed to lie on a regular 2-dimensional (2D) grid, i.e., $$S_n = \left\{ {\boldsymbol{s}}_{i}: i=1,\ldots, n^2 \right\} = \left\{ (x,y),\ x,y \in \left\{1,\ldots,n\right\}\right\} ,$$ and at equidistant time moments, given by $\{t_1,\ldots,t_T\}=\{1,\ldots,T\}$. This sampling scheme has been adopted in various studies in the literature, see e.g., [@davis2013statistical; @buhl2016anisotropic; @buhl2016semiparametric]. For statistical inference on the process $X$, we develop the following two semiparametric estimation schemes.
Scheme 1 {#Sec::Scheme1}
--------
Let $\boldsymbol{\psi} =( {\boldsymbol{\psi}}^{(s)}, {\boldsymbol{\psi}}^{(t)})$ denotes the vector gathering the parameters of the process $X$ to be estimated, where ${\boldsymbol{\psi}}^{(s)}$ and ${\boldsymbol{\psi}}^{(t)}$ denote, respectively, the vectors gathering the spatial and temporal parameters. In this scheme, we consider how the process evolves at given time of reference (a merely spatial process), and its evolution over time at a given location (a merely temporal process). So, ${\boldsymbol{\psi}}^{(s)}$ and ${\boldsymbol{\psi}}^{(t)}$ can be estimated separately. More precisely, denote by $\mathcal{H} \subset [0,\infty)$ and $\mathcal{K} \subset [0,\infty)$ finite sets of spatial and temporal lags on which the estimation is performed. Let the set $\mathcal{B}_h$ summarizes all pairs of $ \mathcal{S}_n$ which give rise to the same spatial lag $h \in \mathcal{H}$, i.e., $$\mathcal{B}_{h} =\{(\ell,p) \in \{1,\ldots,n^2\}^2: \lVert \boldsymbol{s}_\ell-\boldsymbol{s}_p\rVert = {\ensuremath{\lVert \boldsymbol{h} \rVert} }= h\}.$$
The inferential methodology is summarized in the following steps:
1. As a first step, we estimate the purely spatial/temporal $F$-madogram nonparametrically by the empirical version. Denote by ${\widehat{\nu}}^{(t)}_{F} (\boldsymbol{h}), \ {\ensuremath{\lVert \boldsymbol{h} \rVert} }\in \mathcal{H}$ $\left( \text{respectively} \ {\widehat{\nu}}^{(\boldsymbol{{\boldsymbol{s}}})}_{F} (l'), \ l' \in \mathcal{K}\right)$ the nonparametric estimate of the purely spatial (respectively temporal) $F$-madogram. As is standard in geostatistics, we compute ${\widehat{\nu}}^{(t)}_{F} (\boldsymbol{h})$ from the empirical spatio-temporal $F$-madogram ${\widehat{\nu}}_F({\boldsymbol{h}},l)$ at spatio-temporal distances $({\boldsymbol{h}},0)$, that is for all $\{t_1,\ldots,t_T\}$, $${\widehat{\nu}}^{(t)}_{F} (\boldsymbol{h})={\widehat{\nu}}_F({\boldsymbol{h}},0)= \frac{1}{2 |\mathcal{B}_{h}|} \underset{\lVert \boldsymbol{s}_\ell-\boldsymbol{s}_p\rVert = \lVert \boldsymbol{h}\rVert= h}{\sum_{p=1}^{n^2}\sum_{\ell=1}^{n^2}} |F\{X (\boldsymbol{s}_\ell,t)\} - F\{X(\boldsymbol{s}_p,t)\}|\/,\ {h} \in \mathcal{H},$$ where $|.|$ denotes the cardinality of the set $B_{h}$ and $F$ is the standard Fréchet probability distribution function. Let us remark that, a similar estimator in the framework of $\lambda$-madogram has been adopted by [@naveau2009modelling] in an analysis of Bourgogne (France) annual maxima of daily rainfall measurements. On the other hand, ${\widehat{\nu}}^{(\boldsymbol{s})}_{F} (l')$ is computed from the empirical spatio-temporal $F$-madogram ${\widehat{\nu}}_F({\boldsymbol{h}},l')$ at spatio-temporal distances $(\boldsymbol{0},l')$, that is for all $\boldsymbol{s} \in \mathcal{S}_n $ $${\widehat{\nu}}^{(\boldsymbol{s})}_{F} (l')={\widehat{\nu}}_F(\boldsymbol{0},l')= \frac{1}{2 (T-l')} \sum_{k=1}^{T-l'} |F\{X ({\boldsymbol{s}},t_k)\} - F\{X({\boldsymbol{s}},t_{k +l'})\}|\/, \ l' \in \mathcal{K}.$$
2. Then, the overall purely spatial (respectively temporal) $F$-madogram estimates ${\widehat{\nu}}_{F} (\boldsymbol{h})$ (respectively ${\widehat{\nu}}_{F} (l')$) are computed from the means over the temporal moments (respectively the spatial locations). More precisely, $$\label{Spatial F}
{\widehat{\nu}}_{F} (\boldsymbol{h})= \frac{1}{T} \underset{\lVert \boldsymbol{h}\rVert = h }{\sum_{k=1}^{T}}{\widehat{\nu}}^{(t_k)}_{F}(\boldsymbol{h}), \ h \in \mathcal{H}.$$ $$\label{Temporal F}
{\widehat{\nu}}_{F} (l')= \frac{1}{n^2}\sum_{\ell=1}^{n^2} {\widehat{\nu}}^{(\boldsymbol{s}_\ell)}_{F} (l'), \ l' \in \mathcal{K}.$$
3. Finally, a NLS procedure is applied to estimate the parameters of interest. $$\label{eq:psi1}
\boldsymbol{{\widehat{\boldsymbol{\psi}}}}^{(s)}=\underset{ {\boldsymbol{\psi}}^{({\boldsymbol{s}})} \in {\boldsymbol{\Psi}}^{({\boldsymbol{s}})}} {\text{arg min}} \sum_{\Vert\boldsymbol{h}\Vert =h \in \mathcal{H}} \omega^{{\boldsymbol{h}}} \left({\widehat{\nu}}_{F} (\boldsymbol{h})- \nu_{F}^{({\boldsymbol{s}})} (\boldsymbol{h},\boldsymbol{\psi}^{(s)})
\right)^{2}, \ h \in \mathcal{H},$$ $$\label{eq:psi2}
\boldsymbol{{\widehat{\boldsymbol{\psi}}}}^{{(t)}}=\underset{ {\boldsymbol{\psi}}^{{(t)}} \in {\boldsymbol{\Psi}}^{{(t)}} } {\text{arg min}} \sum_{l' \in \mathcal{K}} \omega^{l'} \left({\widehat{\nu}}_{F} (l')- \nu_{F}^{(t)} (l',\boldsymbol{\psi}^{{(t)}})
\right)^{2}, \ l'\in \mathcal{K},$$ where $\nu_{F}^{({\boldsymbol{s}})} (\boldsymbol{h},\boldsymbol{\psi}^{(s)})=\nu_{F} ({\boldsymbol{h}},0,\boldsymbol{\psi}^{{(s)}})$ and $\nu_{F}^{(t)} (l',\boldsymbol{\psi}^{{(t)}})=\nu_{F} (\boldsymbol{0},l',\boldsymbol{\psi}^{{(t)}})$ denote, respectively, the spatial and temporal model-based $F$-madogram counterparts. $\omega^{{\boldsymbol{h}}} \geq 0 $ and $\omega^{l'} \geq 0$ denote, respectively, the spatial and temporal weights. Since it is expected that the spatio-temporal pairs which are far away in space or in time, have only little influence on the dependence parameters to be estimated, a simple choice for these weights is $\omega^{{\boldsymbol{h}}}=
\mathbbm{1}_{\{ {\ensuremath{\lVert \boldsymbol{h} \rVert} }\leq r\}}$, $\omega^{l'}=
\mathbbm{1}_{\{ l' \leq q\}}$, where $\mathbbm{1}(.)$ denotes the indicator function and $(r,q)$ is fixed.
Note that the setup of the inferential methodology in Scheme 1 is close to the one proposed in [@buhl2016semiparametric], in which the spatio-temporal extremogram in (\[Space-time::Extremo\]) was adopted.
Scheme 2
--------
We now generalize Scheme 1 in order to estimate temporal and spatial parameters simultaneously. Thus, we consider how the process $X$ evolves in both space and time. In the classical geostatistics, for a stationary spatio-temporal process $\left\{ X({\boldsymbol{s}},t): ({\boldsymbol{s}},t) \in \mathcal{S}\times \mathcal{T} \right\}$, the spatio-temporal empirical classical semivariogram is defined by $$\widehat{\gamma}({\boldsymbol{h}},l)= \frac{1}{2 \lvert \mathcal{B}_{({\boldsymbol{h}},l)}\rvert} \sum_{\mathcal{B}_{({\boldsymbol{h}},l)}} \left(X({\boldsymbol{s}}_i,t_i,) - X({\boldsymbol{s}}_j,t_j)\right)^2,$$ where $\mathcal{B}_{({\boldsymbol{h}},l)}=\left\{ ({\boldsymbol{s}}_i,t_i)({\boldsymbol{s}}_j,t_j): {\boldsymbol{s}}_i-{\boldsymbol{s}}_j={\boldsymbol{h}}\ \text{and} \ t_i-t_j=l \right\}$, see e.g., [@fernandez2015spatial]. By adapting this estimator to our framework, we consider the following estimation procedure:
1. First, the spatio-temporal $F$-madogram is estimated nonparametrically by its empirical version. Assume the set $\mathcal{B}_{(h,l')}$ summarizes all pairs of $ \mathcal{S}_n$ which give rise to the same spatial lag $h \in \mathcal{H} \subset [0,\infty)$ and the same temporal lag $l' \in \mathcal{K} \subset [0,\infty)$. In other words, combining the spatial and the temporal lags from Scheme 1, i.e., $$\mathcal{B}_{(h,l')} =\left\{\left({\boldsymbol{s}}_i,t_i),(\boldsymbol{s}_j,t_j)\right): \lVert \boldsymbol{s}_i-\boldsymbol{s}_j\rVert = h, |t_i-t_j|=l' \right\}.$$ We estimate $\nu_F({\boldsymbol{h}},l')$ by $${\widehat{\nu}}_F({\boldsymbol{h}},l')= \frac{1}{2 |\mathcal{B}_{(h,l')}|} {{\sum_{ \mathcal{B}_{(h,l')}}}} |F\{X (\boldsymbol{s}_i,t_i)\ - F\{X(\boldsymbol{s}_j,t_j)\}|\/,$$ where $|.|$ denotes the cardinality of the set $\mathcal{B}_{(h,l')}$ and $(h,l') \in \mathcal{H} \times \mathcal{K}$.
2. Then, we apply a NLS fitting to obtain the estimates of the process parameters; $\boldsymbol{\psi}$, i.e., $$\label{eq:psi}
\boldsymbol{{\widehat{\boldsymbol{\psi}}}}=\underset{{\boldsymbol{\psi}} \in {\boldsymbol{\Psi}}} {\text{arg min}} \underset{ \lVert\boldsymbol{h} \rVert=h} {\sum_{l' \in \mathcal{K}} \sum_{h \in \mathcal{H}}} \omega^{{\boldsymbol{h}},l'} \left({\widehat{\nu}}_{F} ({\boldsymbol{h}},l')- \nu_{F} ({\boldsymbol{h}},l',\boldsymbol{\psi})
\right)^{2}, \ (h,l') \in \mathcal{H} \times \mathcal{K},$$ where $\omega^{{\boldsymbol{h}},l'} \geq 0$ denotes the spatio-temporal weights and $\nu_{F} (\boldsymbol{h},l',\boldsymbol{\psi})$ is the model-based spatio-temporal $F$-madogram.
The idea underlying the construction of Scheme 2 is that when modeling and predicting a given phenomenon, significant benefits may be obtained by considering how it evolves in both space and time rather than only considering its spatial distribution at a given time of reference (a merely spatial process), or its evolution over time at a given location (a merely temporal process), such as those described in Scheme 1. Lastly, the establishment of the asymptotic properties of the resulting pairwise dependence estimates is deferred to future work. The derived asymptotic properties of the unbinned empirical $\lambda$-madogram in the spatial context, see [@naveau2009modelling] (Proposition 3 and 4) might provide a starting point. Nevertheless, this setting is more specialized. In the real data example of that study, a binned version of the empirical $\lambda$-madogram is adopted and deriving the convergence of this estimator as the cardinality of the distance class (i.e., $\mathcal{B}_h$) increases is still challenging. Therefore, we will provide some numerical indications for the asymptotic properties of our pairwise dependence estimates.
Illustration examples
---------------------
In order to illustrate how the proposed estimation schemes perform, we consider the following two examples, which we will revisit in Section \[sec:simulation\].
\[example1\] **(Estimation of isotropic space-time max-stable BR)** Let us consider the space-time max-stable BR process in (\[BR spectral fun\]) with bivariate c.d.f. (\[Biv BR\]), where the dependence structure is given by the following stationary isotropic fractional Brownian motion (FBM) spatio-temporal semivariogram $$\gamma({\boldsymbol{h}},l) :=\gamma(h,l')= 2 \phi_s h^{\kappa_s}+2 \phi_t {l'}^{\kappa_t} ,
\label{gamma}$$ where the scalar distance $h={\ensuremath{\lVert \boldsymbol{h} \rVert} }=\lVert {\boldsymbol{s}}_1-{\boldsymbol{s}}_2 \rVert$, $l' = \lvert l \rvert=\lvert t_1-t_2 \rvert$, $ \phi_s, \phi_t >0$ determine spatial and temporal scale parameters and $\kappa_s,\kappa_t \in (0,2]$ relate to the smoothness of the underlying Gaussian process in space and time. The associated spatio-temporal $F$-madogram with this process is $$\label{BR::F-mado}
\nu_F(h,l')= \frac{1}{2} - \frac{1}{2\Phi\left(\sqrt{\phi_s h^{\kappa_s} + \phi_t l'^{\kappa_t}}\right)+1},$$ where $\theta(h,l')=2\Phi\left(\sqrt{\phi_s h^{\kappa_s} + \phi_t l'^{\kappa_t}}\right)$ is the associated spatio-temporal extremal dependence function. Figure \[3DBR\] visualizes a 3D representation of the spatio-temporal FBM semivariogram in (\[gamma\]) and the associated dependence summary measures: the spatio-temporal extremal dependence function $\theta: \mathbb{R}^{2} \times \mathbb{R}^{+} \mapsto[1,2]$ and the spatio-temporal $F$-madogram $\nu_F: \mathbb{R}^{2} \times \mathbb{R}^{+} \mapsto[0,1/6]$. Complete dependence (respectively complete independence) is achieved at lower boundaries (respectively upper boundaries). Moreover, Figure \[Theoreticalbehavior\] displays the theoretical behaviors of the purely spatial FBM semivariogram $\gamma^{\boldsymbol{({\boldsymbol{s}})}}(h,\kappa_s)$ and the related purely spatial $F$-madogram $ \nu_{F}^{\boldsymbol{({\boldsymbol{s}})}} (h,\kappa_s)$. Obviously, depending on the value of the smoothness parameter $\kappa_s$, these measures exhibit a large variety of dependence behaviors.
\[!h\] ![Spatio-temporal FBM semivariogram $\gamma(h,l')=0.8 h^{1.5}+0.4l'$ (left panel). The associated spatio-temporal extremal dependence function (middle panel). The associated spatio-temporal $F$-madogram (right panel).[]{data-label="3DBR"}](3DBR.pdf "fig:")
\[!h\] ![The FBM semivariogram $\gamma^{({\boldsymbol{s}})}(h,\kappa_s)= 0.8 h^{\kappa_s}$ (left panel) and the related spatial $F$-madogram $ \nu_{F}^{({\boldsymbol{s}})} (h,\kappa_s)=0.5 - \left\{2\Phi\left(\sqrt{ 0.4 h^{\kappa_s}}\right)+1\right\}^{-1}$ (right panel) plotted as functions of space lag $h$, with different smoothness parameter $\kappa_s \in \{0.1,0.5,1,1.5,2\}$. []{data-label="Theoreticalbehavior"}](Theoreticalbehavior.pdf "fig:")
With this construction, based on Scheme 1, the NLS optimization problems in (\[eq:psi1\]) and (\[eq:psi2\]) can be expressed as $$\label{eq:BR1}
\begin{pmatrix}
{\widehat{\kappa}}_s\\ {\widehat{\phi}}_s
\end{pmatrix}=\underset{{\substack{\\\phi_s>0 \\ \kappa_s \in (0,2]}}} {\text{arg min}} \sum_{h \in \mathcal{H}} \omega^h \left({\widehat{\nu}}_{F} (h)- \left\{\frac{1}{2} - \frac{1}{2\Phi\left(\sqrt{\phi_s h^{\kappa_s}}\right)+1} \right\}
\right)^{2}, \ h \in \mathcal{H},$$ $$\label{eq:BR2}
\begin{pmatrix}
{\widehat{\kappa}}_t\\ {\widehat{\phi}}_t
\end{pmatrix}= \underset{{\substack{\\\phi_t>0 \\ \kappa_t \in (0,2]}}} {\text{arg min}} \sum_{l' \in \mathcal{K}} \omega^{l'} \left({\widehat{\nu}}_{F} (l')- \left\{\frac{1}{2} - \frac{1}{2\Phi\left(\sqrt{\phi_t l'^{\kappa_t}}\right)+1} \right\}
\right)^{2}, \ l' \in \mathcal{K}.$$
Lastly, with $(h,l') \in \mathcal{H} \times \mathcal{K}$ and on the basis of Scheme 2, the NLS estimation problem in ([\[eq:psi\]]{}) has the form $$\label{eq:BR}
\begin{pmatrix}
{\widehat{\kappa}}_s\\ {\widehat{\phi}}_s \\{\widehat{\kappa}}_t\\ {\widehat{\phi}}_t
\end{pmatrix}=\underset{{\substack{\\\phi_s,\phi_t>0 \\ \kappa_s,\kappa_t \in (0,2]}}} {\text{arg min}} \sum_{l' \in \mathcal{K}} \sum_{h \in \mathcal{H}} \omega^{h,l'} \left({\widehat{\nu}}_{F} (h,l')- \left\{\frac{1}{2} - \frac{1}{2\Phi\left(\sqrt{\phi_s h^{\kappa_s}+\phi_t {l'}^{\kappa_t} }\right)+1} \right\}
\right)^{2}.$$
\[example2\] **(Estimation of spectrally separable space-time max-stable Smith process)** We now describe the way to fit the spectrally separable space-time max-stable Smith process. Indeed, the estimation procedure can be simplified since the purely spatial parameters can be estimated independently of the purely temporal parameters. Formally, we consider the process in (\[model space-time\]), where the innovation process $H$ is derived from independent replications of a spatial Smith process with covariance matrix $$\label{Sigma}
\boldsymbol{\Sigma}=\begin{pmatrix}
\sigma_{11}&\sigma_{12} \\
\sigma_{12} &\sigma_{22}
\end{pmatrix}.$$
We donte by $\boldsymbol{\psi}$ the vector gathering the parameters to be estimated, i.e., $ \boldsymbol{\psi}=\left(\sigma_{11}, \sigma_{12}, \sigma_{22}, \boldsymbol{\tau}^{t},\delta \right)^{t}$. It is possible to separate the estimation. Firstly, the estimation of the spatial parameters ${\boldsymbol{\psi}}^{({s})}=\left(\sigma_{11}, \sigma_{12},\sigma_{22}\right)^{t}$ is carried out. Secondly, once ${\boldsymbol{\psi}}^{({s})}$ is known, it is held fixed and we estimate the temporal parameters ${\boldsymbol{\psi}}^{(t)}=\left( \boldsymbol{\tau}^{t},\delta\right)^{t} = \left( \tau_1, \tau_2, \delta\right)^t$. Subsequently, under Scheme 1, the NLS optimization problems in (\[eq:psi1\]) and (\[eq:psi2\]) can be expressed as $$\label{eq:Smith1}
\begin{pmatrix}
{\widehat{\sigma}}_{11}\\ {\widehat{\sigma}}_{12} \\ {\widehat{\sigma}}_{22}
\end{pmatrix}= \underset{{\substack{\\ \sigma_{11},\sigma_{22} >0 \\ \sigma_{12} \in {\ensuremath{\mathbb{R}}}}}} {\text{arg min}} \underset{\lVert\boldsymbol{h}\rVert=h} {\sum_{h\in \mathcal{H}}} \omega^{{\boldsymbol{h}}} \left({\widehat{\nu}}_{F} (\boldsymbol{h})- \left\{\frac{1}{2} - \frac{1}{2\Phi\left(\sqrt{\boldsymbol{h}^{t}\boldsymbol{\Sigma}^{-1}\boldsymbol{h}}/2\right)+1} \right\}
\right)^{2}, \ {h} \in \mathcal{H},$$ $$\label{eq:Smith2}
\begin{pmatrix}
{\widehat{\delta}}\\ {\widehat{\tau}}_1\\ {\widehat{\tau}}_2
\end{pmatrix}= \underset{{\substack{\\ a \in (0,1)\\ \tau_1,\tau_2 \in {\ensuremath{\mathbb{R}}}}}} {\text{arg min}} \sum_{l' \in \mathcal{K}} \omega^{l'}\left({\widehat{\nu}}_{F} (l')- \left\{\frac{1}{2} - \frac{1}{\theta(l')+1} \right\}
\right)^{2}, \ l' \in \mathcal{K},$$ where, $$\theta(l')= \Phi\left( \frac{b^{*}(l')}{2}+\frac{1}{b^{*} (l')}\log\left(\delta^{-l'}\right)\right)+ \delta^{l}\Phi\left( \frac{b^{*}(l')}{2}+\frac{1}{b^{*}(l')}\log\left(\delta^{l'}\right)\right)+ 1-\delta^{l'}$$ with $b^{*}(l')=\sqrt{( \boldsymbol{0}-l' \boldsymbol{\tau})^{t}\widehat{\boldsymbol{\Sigma}}^{-1} ( \boldsymbol{0}-l' \boldsymbol{\tau}) }.$
In order to figure out the role of the temporal parameter $\delta$ for this process. For a fixed site $\boldsymbol{s} \in \mathcal{S}$, Figure \[temporaltheta\] displays the temporal extremal function $\theta(l')$ and the associated temporal $F$-madogram $ \nu_{F}^{(t)}$ for $\delta \in \{0.1,0.3,0.5,0.7,0.9\}$. We set $\boldsymbol{\Sigma}=$ 10 Id$_2$ and $\boldsymbol{\tau}=(1,1)^{t}$ (translation to the top right). Clearly, as the value of $\delta$ increases, the independece (i.e., $\theta(l')\rightarrow2$) occurs at larger time lags $l'$.
\[!htp\] ![$\theta(l')$ and the associated $\nu_{F}^{(t)}(l')$ plotted as functions of time lag $l'$ for $\delta \in \{0.1,0.3,0.5,0.7,0.9\}$ based on the process (\[model space-time\]), where $H$ is a sequence of i.i.d. spatial Smith processes with covariance matrix $\boldsymbol{\Sigma}=$ 10 Id$_2$. []{data-label="temporaltheta"}](temporaltheta1.pdf "fig:")
Lastly, based on Scheme 2, the NLS estimator $\boldsymbol{\widehat{\psi}} = \left( \widehat{\sigma}_{11},\widehat{\sigma}_{12},\widehat{\sigma}_{22},\widehat{\tau}_1,\widehat{\tau}_2,\widehat{\delta}\right)^t$ is given by $$\label{eq:Smith}
{\widehat{\boldsymbol{\psi}}}= \underset{{\substack{\\ \boldsymbol{\psi} \in \boldsymbol{\Psi}}}} {\text{arg min}} \underset{{\substack{\\ \lVert\boldsymbol{h}\rVert = h}}}{\sum_{l' \in \mathcal{K}} \sum_{h \in \mathcal{H}}}\omega^{{\boldsymbol{h}},l'} \left({\widehat{\nu}}_{F} ({\boldsymbol{h}},l' )- \left\{\frac{1}{2} - \frac{1}{\theta({\boldsymbol{h}},l' )+1} \right\}
\right)^{2}, \ (h,l') \in \mathcal{H} \times \mathcal{K},$$ where $\theta({\boldsymbol{h}},l')= \Phi\left( \frac{b({\boldsymbol{h}},l')}{2}+\frac{1}{b({\boldsymbol{h}},l')}\log\left(\delta^{-l'}\right)\right)+ \delta^{l'}\Phi\left( \frac{b({\boldsymbol{h}},l')}{2}+ \frac{1}{b({\boldsymbol{h}},l')}\log\left(\delta^{l'}\right)\right)+ 1-\delta^{l'}$ $\text{with} \ b({\boldsymbol{h}},l')=\sqrt{({\boldsymbol{h}-l'\boldsymbol{\tau})}^{t}{\boldsymbol{\Sigma}}^{-1}{(\boldsymbol{h}-l'\boldsymbol{\tau}})}.$
Simulation study {#sec:simulation}
================
Throughout this section, we investigate the performance of the semiparametric estimation procedures introduced in Section \[sec:inference\] with three simulation studies.
Simulation study 1: Fitting space-time max-stable BR process {#Sec::sim1}
------------------------------------------------------------
In this study, we adopt the same experiment plan that has been proposed in [@buhl2016semiparametric] (Section 5), in order to make the results obtained there comparable with the results here.
### Setup for a simulation study {#Sec::setup1}
We simualte the space-time BR process with spectral representation (\[BR spectral fun\]) and dependence function $\gamma$ modeled as in (\[gamma\]). Namely, $$\label{FBM semi}
\gamma({\boldsymbol{h}},l) = 0.8 h^{3/2} +0.4 l'.$$ The simulations have been carried out using the function [*RFsimulate*]{} of the R package RandomFields [@schlatherRF] and based on the exact method proposed by [@dombry2016exact]. The space-time observation area is assumed to be on a $n \times n$ spatial grid and the time moments are equidistantly, i.e., $${\mathscr{A} } = \{(x,y): x,y \in\{1,\ldots,n\} \} \times \{1,\ldots,T\}.$$ Figure \[3DBRsim\] visualizes a realization simulated from space-time BR process with a spatio-temporal FBM semivariogram model (\[FBM semi\]) at six consecutive time points.
\[!htp\] ![Simulation from a space-time max-stable BR process with spatio-temporal FBM semivariogram $\gamma({\boldsymbol{h}},l)= 0.8 h^{1.5}+0.4l'$ at six consecutive time points (from left to right and top to bottom).[]{data-label="3DBRsim"}](3DBRsim3.pdf "fig:")
As in [@buhl2016semiparametric], we choose the sets $\mathcal{H} =\{1,\sqrt{2},2,\sqrt{5},\sqrt{8},3,\sqrt{10},\sqrt{13},4,\sqrt{17}\}$ and $\mathcal{K}=\{1,\ldots,10\}$, where permutation tests show that these lags are enough to capture the relevant extremal dependence structure, see Figure \[GRIDSPATIAL\]. Equal weights are assumed. We repeat this experiment 100 times to obtain summary plots of the resulting estimates and to compute performance metrics: the mean estimate, the root mean squared error (RMSE) and the mean absolute error (MAE).
\[htp!\] ![A regular $14 \times 14$ spatial grid. The distances between the peripheral locations (shown by red square symbols) and the central one (shown by blue square symbol) belong to the set $\mathcal{H}$. []{data-label="GRIDSPATIAL"}](GRIDSPATIAL.pdf "fig:")
### Estimation using Scheme 1 {#Sec::scheme1.1}
Simulation of space-time max-stable BR processes based on the exact method proposed in [@dombry2016exact] can be time-consuming. Hence, for the sake of time-saving and due to the fact that the estimation of the purely spatial (respectively purely temporal) parameters depends on a large number of spatial observations (respectively a large number of observed time instants), we examine the performance of the purely spatial (respectively purely temporal) estimates using two different space-time observation areas, i.e.,
- [$ {\mathscr{A} }_1 = \{(x,y): x,y \in\{1,\ldots,50\} \} \times \{1,\ldots,10\}.$]{}
- [$ {\mathscr{A} }_2 = \{(x,y): x,y \in\{1,\ldots,5\} \} \times \{1,\ldots,300\}.$]{}
We assess the quality of the fit between the theoretical values of spatial/temporal $F$-madograms and their estimates. Figure \[Empirical mado\] compares empirical estimates of purely spatial/temporal $F$-madograms with their asymptotic counterparts. Overall, both the purely spatial/temporal empirical versions are consistent, with a relatively higher variability for the temporal estimates. This is probably due to the fairly low number of time instants (300) used for the estimation of the purely temporal parameters compared to the number of spatial locations (2500) used for the estimation of the purely spatial parameters.
\[h!\] ![Scheme 1: (Top row) boxplots of purely spatial/temporal empirical $F$-madograms estimates at lags $(h,l') \in \mathcal{H} \times \mathcal{K}$ for 100 simulated BR processes (\[BR spectral fun\]) with FBM spatio-temporal semivariogram (\[FBM semi\]). The middle blue dotted/red solid lines show the overall mean of the estimates/true values. (Bottom row) boxplots of the corresponding estimation errors. []{data-label="Empirical mado"}](BR1.pdf "fig:")
Next, we present results for the semiparametric estimation with Scheme1. Figure \[Performance1\] displays the resulting estimates of the purely spatial parameters $(\phi_s,\alpha_s)$ and the purely temporal parameters $(\phi_t,\alpha_t)$. Generally, the estimation procedure appears to work well. Moreover, we observe that the estimation of the purely spatial parameters is more accurate (the RMSE and MAE are lower), see Table \[BR\]. Again this probably stems from the large number of spatial locations used in the estimation which is ($\approx 8.3$) times higher than the time points.
\[!h\] ![Scheme 1: Semiparametric estimates of $\boldsymbol{\widehat{\psi}} = \{ \widehat{\phi}_s, \widehat{\kappa}_s, \widehat{\phi}_t,\widehat{\kappa}_t\}$ for 100 simulated BR processes defined by (\[BR spectral fun\]) with FBM spatio-temporal semivariogram (\[FBM semi\]). The middle blue dotted/red solid lines show the overall mean of the estimates/true values.[]{data-label="Performance1"}](BR2.pdf "fig:")
As the last step in this study, we compare the statistical efficiency of our method and the one proposed in [@buhl2016semiparametric]. Table \[BR\] reports the performance metrics for both methods. Although in that study, the authors used a larger grid size $(n=70)$ to estimate the purely spatial parameters, clearly, the $F$-madogram semiparametric estimation outperforms their approach which based on the extremogram as an inferential tool (their semiparametric estimates show a larger bias than ours; the RMSE and MAE are higher). This is probably due to the fact that the estimates obtained in [@buhl2016semiparametric] are sensitive to the choice of the threshold used for computing (possibly bias corrected) empirical estimates of the extremogram.
### Estimation using Scheme 2 {#Sec::scheme1.2}
Based on Scheme 2, we estimate the parameters of the space-time max-stable BR process with a similar simulation setting which is previously described in Section \[Sec::setup1\]. We consider the space-time observation area where the spatial locations consisted of a $20 \times 20$ grid and equidistantly time points, $\{1,\ldots,200\}$. Figure \[3Dspatiomado\] compares the empirical spatio-temporal $F$-madogram estimates $\widehat{\nu}_F(h,l')$ with their model-based counterparts $\nu_F(h,l')$ over the spatio-temporal lags $(h,l') \in \mathcal{H} \times \mathcal{K}$. There is a good agreement overall. These diagnostic plots provide a satisfactory representation of the empirical spatio-temporal $F$-madogram estimates. Generally, the results lend support to the agreement between the empirical spatio-temporal $F$-madogram estimates and model-based counterparts, especially once sampling variability is taken into account.
\[h!\] ![Scheme 2: Diagnostic plots of the empirical spatio-temporal $F$-madogram estimates for 100 simulated BR processes defined by (\[BR spectral fun\]) with FBM spatio-temporal semivariogram (\[FBM semi\]). Histogram of the errors, $\widehat{\nu}_F(h,l')-\nu_F(h,l')$, $(h,l') \in \mathcal{H} \times \mathcal{K}$ (left panel). Blue/red cross symbols show the overall mean of the empirical spatio-temporal $F$-madogram estimates/model-based counterparts (right panel). []{data-label="3Dspatiomado"}](3Dspatiomado1.pdf "fig:")
\[h!\] ![Scheme 2: Semiparametric estimates of $\boldsymbol{\widehat{\psi}} = \{ \widehat{\phi}_s, \widehat{\kappa}_s,\widehat{\phi}_t,\widehat{\kappa}_t\}$ for 100 simulated BR processes defined by (\[BR spectral fun\]) with FBM spatio-temporal semivariogram (\[FBM semi\]). The middle blue dotted/red solid lines show overall mean of the estimates/true values. []{data-label="BRscheme2"}](BRscheme2.pdf "fig:")
Figure \[BRscheme2\] shows the estimation performance of the estimated parameters. Overall, the parameters are well estimated. Moreover, we observe that the estimation of the scale parameters $\{\phi_s$, $\phi_t\}$ is more accurate than the smoothness parameters $\{\kappa_s, \kappa_t\}$ (the RMSE and MAE are lower), see Table \[BR\].
To sum up, for both schemes, Table \[BR\] reports the mean estimate, RMSE, and MAE of the estimated parameters $\boldsymbol{\widehat{\psi}} = \{ \widehat{\phi}_s, \widehat{\kappa}_s,\widehat{\phi}_t,\widehat{\kappa}_t\}$. Let us remark that the comparison between the resulting parameter estimates from the two estimation schemes is not completely straightforward because we consider non-unified space-time observation areas due to the above-mentioned computational reasons. However, with the above sampling schemes, we observe that the estimation of the purely spatial parameters is more accurate when using Scheme 1 (the RMSE and MAE are lower). On the other hand, we notice a slight outperformance for Scheme 2 in estimating purely temporal parameters. Finally, the QQ-plots against a normal distribution in Figure \[QQBR\] provide an indication for asymptotic normality of the resulting estimates.
\[htp!\] ![QQ-plots of the estimates resulting from both estimation schemes for 100 simulated BR processes defined by (\[BR spectral fun\]) with the FBM spatio-temporal semivariogram (\[FBM semi\]) against the normal distribution. Scheme 1: purely spatial parameters (top row) and purely temporal parameters (second row). Scheme 2: purely spatial parameters (third row) and purely temporal parameters (bottom row). Dashed red lines correspond to 95% confidence intervals. []{data-label="QQBR"}](QQBR1.pdf "fig:")
Simulation study 2: Fitting spectrally separable space-time max-stable Smith process {#Sec::sim2}
------------------------------------------------------------------------------------
### Setup for a simulation study {#Sec::setup2}
We simulate data from the spatio-temporal Smith process considered in Example \[example2\], with parameter vector $\boldsymbol{\psi}=(1,0, 1, 1,1,0.7)^{t}$. As a reasonable compromise between accuracy and computation time, the locations are assumed to lie on a regular 2D grid of size $n=20$. The time points are equidistant, given by the set $\{1,\ldots,200\}$. The simulations have been carried out using R SpatialExtremes package with [*rmaxstab*]{} function, see [@ribatet2011spatialextremes]. The spatial lags set $\mathcal{H}$ and temporal lags set $\mathcal{K}$ are fixed as before, recall Section \[Sec::sim1\]. Equal weights are assumed. We repeat this experiment 100 times.
### Results for the two estimation schemes {#Sec::scheme2.1}
The top row of Figure \[Density\] displays the density of the errors between the empirical estimates of the purely spatial/temporal $F$-madograms and their model-based counterparts, whereas the bottom row displays the density of the errors between empirical spatio-temporal $F$-madogram estimates and model-based counterparts. Generally, all of the empirical versions are congruous with their asymptotic counterparts. Clearly, the density of the errors is close to a centered Gaussian distribution.
Figure \[Smithall\] displays boxplots the errors of the resulting estimates from both schemes: ($\boldsymbol{\widehat{\psi}}-\boldsymbol{{\psi}}$). The top row displays the estimation errors of purely spatial parameters $({\sigma}_{11},{\sigma}_{12},{\sigma}_{22})$ and purely temporal parameters $({\tau}_1,{\tau}_2,{\delta})$ resulting from Scheme 1, whereas the bottom row displays the estimation errors resulting form Scheme 2. Overall, the inference procedures perform well. Altogether, we observe that the estimates are close to the true values.
To sum up, for both schemes, Table \[ST Smith\] reports the mean estimate, RMSE, and MAE of the estimated parameters $\boldsymbol{\widehat{\psi}} = \{\widehat{\sigma}_{11},\widehat{\sigma}_{12},\widehat{\sigma}_{22},\widehat{\tau}_1,\widehat{\tau}_2,\widehat{\delta} \}$. Contrary to Scheme 2, we observe that the estimation of purely spatial parameters $\boldsymbol{\Sigma}$ is more accurate than the estimation of purely temporal parameters ($\boldsymbol{\tau}$ and $\delta$) when using Scheme 1 (RMSE and MAE are lower). This probably can be justified by the fact that in Scheme 1 the number of spatial locations used is higher than time moments. Additionally, there is probably an impact of the estimated covariance matrix $\widehat{\boldsymbol{\Sigma}}$ on the estimation efficiency of the purely temporal parameters, whereas, the purely temporal parameters are estimated independently of purely spatial parameters when using Scheme 2. Moreover, we notice that the estimation of purely spatial parameters is less accurate when using Scheme 2 (RMSE and MAE are higher). This is probably owing to the fact that in Scheme 2 the number of pairs used is higher than in Scheme 1, leading more variability. Whereas, both schemes seem to have the same performance order in estimating purely temporal parameters.
We also show QQ-plots against a normal distribution for all parameters in Figure \[qqplot\]. For both schemes, it seems that the semiparametric estimates are approximately normally distributed.
\[h!\] ![Density of the errors between the empirical versions of the $F$-madogram estimates and their model-based counterparts for 100 simulated spectrally separable space-time max-stable Smith processes with parameter $\boldsymbol{\psi}=(1,0, 1, 1,1,0.7)^{t}$. Scheme 1 (Top row): ${\widehat{\nu}}_{F} (\boldsymbol{h}) -{\nu}_{F} (\boldsymbol{h})$, $\lVert\boldsymbol{h}\rVert \in \mathcal{H}$ (left panel) ${\widehat{\nu}}_{F} (l') - {\nu}_{F} (l')$, $l' \in \mathcal{K}$ (right panel). Scheme 2 (Bottom row): $\widehat{\nu}_F({\boldsymbol{h}},l') - {\nu}_F({\boldsymbol{h}},l'),$ at spatio-temporal lags $({\ensuremath{\lVert \boldsymbol{h} \rVert} },l') \in \mathcal{H} \times \mathcal{K}$. []{data-label="Density"}](Density.pdf "fig:")
\[h!\] ![Boxplots of the errors $\boldsymbol{\widehat{\psi}}-\boldsymbol{{\psi}}$ resulting from both estimation schemes for 100 simulated spectrally separable space-time max-stable Smith processes with parameter $\boldsymbol{\psi}=(1,0, 1, 1,1,0.7)^{t}$. Scheme 1 (Top row): purely spatial parameters (left panel) and purely temporal parameters (right panel). Scheme 2 (Bottom row): all parameters. The middle blue dotted/red solid lines show the overall mean of errors estimates/zero value.[]{data-label="Smithall"}](Smithall1.pdf "fig:")
\[h!\] ![QQ-plots of the estimates from both estimation schemes for 100 simulated spectrally separable space-time max-stable Smith processes with parameter $\boldsymbol{\psi}=(1,0, 1, 1,1, 0.7)^{t}$ against the normal distribution. Scheme 1: purely spatial parameters (top row) and purely temporal parameters (second row). Scheme 2: purely spatial parameters (third row) and purely temporal parameters (bottom row). Dashed red lines correspond to 95% confidence intervals. []{data-label="qqplot"}](qqplot1.pdf "fig:")
Finally, let us remark that a simulation study has been carried out in [@embrechts2016space], where only the spectrally separable spatio-temporal Smith process has been fitted. Irregularly spaced locations have been considered. Two estimation schemes based on pairwise likelihood have been adopted (a two-step approach and a one-step approach). The obtained results have shown that, the estimation of purely spatial parameters is more accurate with a two-step approach.
Simulation study 3: Fitting spectrally separable STMS Schlather process {#app4}
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Finally, we perform a third simulation study to fit spectrally separable space-time max-stable Schlather process. The innovation process $H$ is derived from independent replications of a spatial Schlather process with correlation function of powered exponential type defined, for all ${\ensuremath{\lVert \boldsymbol{h} \rVert} }\geq0$, by $\rho({\boldsymbol{h}})=\exp [- ({{\ensuremath{\lVert \boldsymbol{h} \rVert} }} / \phi)^{\kappa}]$, $\phi>0$ and $0<\kappa<2$, where $\phi$ and $\kappa$ denote, respectively, the range and the smoothing parameters. We denote by $\boldsymbol{\psi}=(\phi,\kappa, \tau_1,\tau_2,\delta)^{t}$ the vector gathering the model parameters. We take $\phi=3$, $\kappa=3/2$, $\boldsymbol{\tau}=(1,0)^{t}$ and $\delta=0.3$ . As previously, we consider the same simulation setup used in Section \[Sec::setup2\]. The results are summarized in Figure \[Sch\] and Table \[Schlather\]. Generally, we obtain equally satisfying results.
\[h!\] ![Boxplots of errors $\boldsymbol{\widehat{\psi}}-\boldsymbol{{\psi}}$ from both estimation schemes for 100 simulated spectrally separable STMS Schlather processes with parameter $\boldsymbol{\psi}=(2,1.5, 1,0,0.3)^{t}$. Scheme 1 (Top row): purely spatial parameters (left panel) and purely temporal parameters (right panel). Scheme 2 (Bottom row): all parameters. The middle blue dotted/red solid lines show the overall mean of errors estimates/zero value. []{data-label="Sch"}](SCh1.pdf "fig:")
Real data analysis {#sec:real}
==================
In this section, we aim to quantify the extremal behavior of radar rainfall data in a region in the State of Florida. Our approach is to fit the data by different space-time max-stable classes based on a space-time block maxima design using the proposed semiparametric estimation procedure.
Description of the dataset
--------------------------
The dataset analyzed in this section is composed of radar rainfall values (in inches) measured on a square of 140 $\times$ 140 km region containing 4900 grid locations in the State of Florida. The database consists of radar hourly rainfall values measured on a regular grid with squared cells of size 2 km covering a region of 70 $\times$ 70 cells in the State of Florida. A map of the study area is shown in Figure \[Grid\]. We only consider the wet season (June-September) over the years 2007-2012. The data were collected by the Southwest Florida Water Management District (SWFWMD) and freely available on <ftp://ftp.swfwmd.state.fl.us/pub/radar_rainfall>. Moreover, the dataset is available in the Supplementary Material: <http://math.univ-lyon1.fr/homes-www/abuawwad/Florida_RadarRainfall/>.
\[h!\] ![Radar rainfall observation area in the State of Florida. Source: Southwest Florida Water Management District (SWFWMD).[]{data-label="Grid"}](Grid.jpg "fig:")
Data fitting
------------
We perform a block maxima design in space and time as follows: we take block maxima over 24 consecutive hours and over 10 km $\times$ 10 km areas (the daily maxima over 25 grid locations), resulting in $14 \times 14$ grid in space for all $6 \times 122$ days of the wet seasons. So, this gives a time series of dimension $14 \times 14$ and of length 732. For the sake of notational simplicity, we denote the set of resulting grid locations by $\mathbb{S} = \left\{(x,y): x,y \in\{1,\ldots,14\} \right\}$ and the spacetime realizations by $\left\{X({\boldsymbol{s}},t),\boldsymbol{s} \in \mathbb{S} , \ t\in \{t_1,\ldots,t_{732}\} \right\}$. This setup has been also considered in [@buhl2016semiparametric; @davis2013statistical] for analyzing radar rainfall measurements in a region in the State of Florida over the years 1999-2004, where only space-time max-stable BR process has been fitted to the data by a semiparametric approach in [@buhl2016semiparametric] and a pairwise likelihood approach in [@davis2013statistical]. Let us remark that both regions here and in the above-mentioned two studies are located in the central portion of Florida District, which would probably have the best square area of coverage. Having larger grid size will lead to some cells missing in the southwestern “corner” due to the coastline. Figure \[timeseries\] shows the obtained time series for daily maxima observations at four grid locations.
\[h!\] ![Plots of daily maximal rainfall in inches for four grid locations with simplified coordinates: (1,1), (6,5), (7,10) and (11,8).[]{data-label="timeseries"}](timeseries.pdf "fig:")
According to this sampling scheme of the process $X$, there are $196 \times 731= 143276$ spatio-temporal pairs of points at distance $(0,1)$, that is, $$\left\{ ({\boldsymbol{s}}_1,t_2), ({\boldsymbol{s}}_1,t_1) \right\}, \left\{ ({\boldsymbol{s}}_2,t_2), ({\boldsymbol{s}}_2,t_1) \right\}, \ldots, \left\{ ({\boldsymbol{s}}_{196},t_2), ({\boldsymbol{s}}_{196},t_1) \right\}$$ $$\vdots$$ $$\left\{ ({\boldsymbol{s}}_1,t_{732}), ({\boldsymbol{s}}_1,t_{731}) \right\}, \left\{ ({\boldsymbol{s}}_2,t_{732}), ({\boldsymbol{s}}_2,t_{731}) \right\}, \ldots, \left\{ ({\boldsymbol{s}}_{196},t_{732}), ({\boldsymbol{s}}_{196},t_{731}) \right\}.$$ Analogously, there are $196 \times 730= 143080$ spatio-temporal pairs of points at distance $(0,2)$, and so forth. Generally, for a set of spatio-temporal data measured in the time moments $t_1, \ldots, t_T$, on a regular $n \times n$ spatial grid, we have $n^2(t_T-l')$ spatio-temporal pairs of points at distance $(0,l')$. Computing the $F$-madogram values corresponding to the above spatio-temporal distances, we obtain the purely temporal empirical $F$-madogram. It is also easy to check that there are $364 \times 732 = 266448$ spatio-temporal pairs of points at distance $(1,0)$, $336 \times 732 = 245952$ at distance $(2,0)$, and so forth, see Table \[Lags\]. Computing the $F$-madogram values for the spatio-temporal distances $({\boldsymbol{h}},0)$, we obtain the purely spatial empirical $F$-madogram.
\[htp!\]
Since we are interested in modeling the joint occurrence of extremes over a region, then the dependence structure of a multivariate variable has to be explicitly stated. The usual modeling strategy consists of two steps: firstly, estimating the marginal distribution. Secondly, characterizing the dependence via a model issued by the multivariate extreme value theory, see e.g., [@beirlant2006statistics; @padoan2010likelihood]. For marginal modeling, we explain the procedure as follows:
1. We transform the data to stationarity by removing possible seasonal effects using a simple moving average with a period of 122 days (the number of days in the wet season considered in one particular year). More precisely, for each fixed location $\boldsymbol{s} \in \mathbb{S}$, we deseasonalize the time series $\left\{X({\boldsymbol{s}},t), t\in \{t_1,\ldots,t_{732}\} \right\}$ by computing for $i=1,\ldots,122$ $$\tilde{X}({\boldsymbol{s}},t_{i+122(j-1)}) = {X}({\boldsymbol{s}},t_{i+122(j-1)})- \frac{1}{6} \sum_{j=1}^{6} {X}({\boldsymbol{s}},t_{i+122(j-1)}),$$
2. For each fixed location $\boldsymbol{s} \in \mathbb{S}$, the deseasonalized observations are fitted to the generalized extreme value distribution, $$\text{GEV}_{\mu(\boldsymbol{s}),\sigma(\boldsymbol{s}),\xi(\boldsymbol{s})}(x)=\exp \left\{ - \left[1+\xi (\boldsymbol{s}) \left(\frac{x-\mu(\boldsymbol{s})}{\sigma{(\boldsymbol{s})}} \right) \right]^{-1/\xi (\boldsymbol{s})}\right\},$$ for some location $\mu(\boldsymbol{s}) \in \mathbb{R}$, scale $\sigma(\boldsymbol{s})>0$, and shape $\xi (\boldsymbol{s})\in \mathbb{R}$. Let us remark that the estimated shape parameters $\xi({\boldsymbol{s}})$ are sufficiently close to zero with confidence interval containing zero, see Figure \[RADARFIT\]. This suggests a Gumbel distribution (GEV with $\xi=0$) as appropriate model. Therefore, we fit directly a Gumbel distribution $$\text{GEV}_{\mu({\boldsymbol{s}}),\sigma({\boldsymbol{s}}),0}(x)= \left\{ \exp \left[ - \exp \left( - \frac{x-\mu({\boldsymbol{s}})}{\sigma({\boldsymbol{s}})} \right) \right] \right\}.$$ For each spatial location, we assess the goodness of the marginal fits by QQ-plots of deseasonalized rain series versus the fitted Gumbel distribution. The results at four spatial locations $(1,1), (6,5), (7,10) \ \text{and}\ (11,8)$ are summarized in Figure \[qqplotGumbel\]. All plots provide a reasonable fit.
3. The deseasonalized observations may be transformed either to standard Gumbel or standard Fréchet margins. More precisely, let $\widehat{\mu}({\boldsymbol{s}})$, $\widehat{\sigma}({\boldsymbol{s}})$ are the parameter estimates obtained from (ii), then we may use:
1. $\tilde{\tilde{X}}({\boldsymbol{s}},t) = \frac{{\tilde{X}}({\boldsymbol{s}},t)-\widehat{\mu}({\boldsymbol{s}})}{\widehat{\sigma}({\boldsymbol{s}})}, \ t \in \{1,\ldots,732\}$ to transform the deseasonalized observations to standard Gumbel margins;
2. $ \tilde{\tilde{X}}({\boldsymbol{s}},t) = -\frac{1}{\log\left\{ \text{GEV}_{\widehat{\mu}({\boldsymbol{s}}),\widehat{\sigma}({\boldsymbol{s}}),0}({\tilde{X}}({\boldsymbol{s}},t)) \right\}}, \ t \in \{1,\ldots,732\}$ to transform the deseasonalized observations to standard Fréchet margins. This transformation is called the probability integral transformation. In this study, we adopt this case.
\[htp!\] ![Estimated GEV shape parameter $\widehat{\xi}{({\boldsymbol{s}})}$ at all grid locations with 95% confidence intervals. []{data-label="RADARFIT"}](RADARFIT.pdf "fig:")
\[h!\] ![QQ-plots of deseasonalized rain series versus the fitted Gumbel distribution (GEV with $\widehat{\mu}({\boldsymbol{s}})$, $\widehat{\sigma}({\boldsymbol{s}})$ and 0) on the basis of the time series corresponding to the four grid locations shown in Figure \[timeseries\].[]{data-label="qqplotGumbel"}](QQplotGumbel.pdf "fig:")
In [@buhl2016semiparametric; @davis2013statistical], the authors assume that the observations $\tilde{\tilde{X}}({\boldsymbol{s}},t)$ are realizations from the space-time max-stable BR process. The contribution of the present section is to broaden the dependence structure by considering the spectrally separable space-time max-stable processes, that allow interactions between spatial and temporal components.
In the sequel, we estimate the extremal dependence structure for the daily maxima of rainfall measurements. Based on our findings in the simulation studies, we notice that Scheme 1 outperforms Scheme 2 generally. So, one may first estimate the extremal dependence parameters using Scheme 2. Afterward, re-estimating the parameters using Scheme 1, where the estimates resulting from Scheme 2 serve as starting values for the optimization routine used in Scheme 1. To that aim, we consider the following five spatio-temporal max-stable models:
1. Class A: consists of two non-spectrally separable models A$_1$ and A$_2$.
- A$_1$: a space-time max-stable BR model (\[BR spectral fun\]), with dependence function $\gamma({\boldsymbol{h}},l) = 2 \phi_s {\ensuremath{\lVert \boldsymbol{h} \rVert} }^{\kappa_s}+2 \phi_t {l'}^{\kappa_t}$, where $l'=\lvert l \rvert$, recall Example \[example1\].
- A$_2$: a space-time max-stable Schlather model. The space-time correlation function is chosen to be separable such that $$\rho({\boldsymbol{h}},l)=\exp\left\{- \left[ \left({\ensuremath{\lVert \boldsymbol{h} \rVert} }/\phi_s \right)^{\kappa_s} +\left(l'/\phi_t \right)^{\kappa_t} \right]\right\},$$ where the range parameters $\phi_t, \phi_s>0$ and the smoothing parameters $0<\kappa_t, \kappa_s<2$.
2. Class B: consists of spectrally separable models B$_1$, B$_2$ and B$_3$.
- B$_1$: a spectrally separable space-time max-stable model (\[model space-time\]), where the innovation process $H$ is derived from independent replications of a spatial BR process with semivariogram $\gamma({\boldsymbol{h}})= \left({\ensuremath{\lVert \boldsymbol{h} \rVert} }/\phi \right)^\kappa$, for some range parameter $\phi>0$ and smoothness parameter $\kappa \in (0,2]$. Obviously, models A$_1$ and B$_1$ are equivalent when the time lag $l'=0$.
- B$_2$: a spectrally separable space-time max-stable model (\[model space-time\]), where the innovation process $H$ is derived from independent replications of a spatial Smith process with covariance matrix $\boldsymbol{\Sigma}=\begin{pmatrix}
\sigma_{11}&\sigma_{12} \\
\sigma_{12} &\sigma_{22}
\end{pmatrix},$ recall Example \[example2\].
- B$_3$: a spectrally separable space-time max-stable model (\[model space-time\]), where the innovation process $H$ is derived from independent replications of a spatial extremal-$t$ process with degrees of freedom $\nu \geq 1$ and correlation function of type powered exponential defined, for all ${\ensuremath{\lVert \boldsymbol{h} \rVert} }\geq0$, by $\rho({\boldsymbol{h}})=\exp [- ({{\ensuremath{\lVert \boldsymbol{h} \rVert} }} / \phi)^{\kappa}]$, $\phi>0$ and $0<\kappa<2$, where $\phi$ and $\kappa$ denote, respectively, the range and the smoothing parameters.
To select the best-fitting model, we use the Akaike Information Criterion (AIC) which was first developed by [@akaike1974new] under the framework of maximum likelihood estimation. The AIC is one of the most widely used methods for selecting a best-fitting model from several competing models given a particular dataset. A concise formulation of the AIC under the framework of least squares estimation has been derived by [@banks2017aic]. The AIC under Scheme 1 is defined as $$\label{AIC}
\text{AIC}_{\text{NLS}}= \lvert\mathcal{H}\rvert \log \left( \frac{\mathcal{L}(\widehat{{\boldsymbol{\psi}}}^{(s)})}{ \lvert\mathcal{H} \rvert}\right)+2(k_s+1) + \lvert\mathcal{K}\rvert \log \left( \frac{\mathcal{L}(\widehat{{\boldsymbol{\psi}}}^{(t)})}{\lvert\mathcal{K} \rvert}\right)+2(k_t+1),$$ where $\mathcal{L}(\widehat{{\boldsymbol{\psi}}}^{(s)})$ and $\mathcal{L}(\widehat{{\boldsymbol{\psi}}}^{(t)})$ are the estimated objective functions in space and time with $\omega^{{\boldsymbol{h}}}=\omega^{l'}=1$, i.e., $$\mathcal{L}( \widehat{{\boldsymbol{\psi}}}^{(s)} )= \sum_{\Vert\boldsymbol{h}\Vert =h \in \mathcal{H}} \left({\widehat{\nu}}_{F} (\boldsymbol{h})- \nu_{F}^{({\boldsymbol{s}})} (\boldsymbol{h}, \widehat{{\boldsymbol{\psi}}}^{(s)})
\right)^{2}, \ h \in \mathcal{H},$$ $$\mathcal{L}( \widehat{{\boldsymbol{\psi}}}^{(t)} )= \sum_{l' \in \mathcal{K}} \left({\widehat{\nu}}_{F} (l')- \nu_{F}^{(t)} (l', \widehat{{\boldsymbol{\psi}}}^{(t)} )
\right)^{2}, \ l'\in \mathcal{K}.$$ $\lvert\mathcal{\mathcal{A}}\rvert$ denotes the cardinality of the set $\mathcal{A}$, and $k_s$ and $k_t$ are respectively the total number of purely spatial and purely temporal parameters in the underlying model. If $\lvert\mathcal{\mathcal{H}}\rvert / k_s+1 < 40$ and $\lvert\mathcal{\mathcal{K}}\rvert / k_t+1 < 40$, it is suggested to use an adjusted corrected version of $\text{AIC}_{\text{NLS}}$ (\[AIC\]), see [@banks2017aic], i.e., $$\label{AIC1}
\text{AIC}_{{\text{NLS}}_c}= \text{AIC}_{\text{NLS}} + \frac{2(k_s+1) (k_s+2)}{ \lvert\mathcal{\mathcal{H}}\rvert - k_s } + \frac{2(k_t+1) (k_t+2)}{ \lvert\mathcal{\mathcal{K}}\rvert - k_t }.$$
Our results are summarized in Table \[Fitted model\]. Model A$_1$ has the lowest AIC$_\text{NLS}$ value and therefore would be considered as the best candidate for this dataset, closely followed by model B$_3$. Obviously, the temporal estimates ($\widehat{\phi}{_t}$ and $\widehat{\kappa}{_t}$) in the best-fitting model A$_1$ indicate that there is a weak temporal extremal dependence. Recall that the purely temporal $F$-madogram for this model is given by $$\nu_{F}^{(t)} (l')=0.5 - \left\{2\Phi\left(\sqrt{ {\phi}{_t}l'^{\kappa_t}}\right)+1\right\}^{-1}, \ l' > 0.$$ Accordingly, $ \nu_{F}^{(t)} (l')$ is close to zero for large values of $\phi_t$, indicating asymptotic independence. On the other hand, $ \nu_{F}^{(t)} (l')$ is approximately constant when $\kappa_t$ is small, indicating that the extremal dependence is the same for all $l'$. So, both large $\phi_t$ and small $\kappa_t$ lead to temporal asymptotic independence.
For comparison, we present the semiparametric estimates obtained by [@buhl2016semiparametric]; $ \widehat{\phi}{_s}=0.3611, \widehat{\kappa}{_s}=0.9876, \widehat{\phi}{_t}=2.3650 \ \text{and} \ \widehat{\kappa}{_t}=0.0818$. On the other hand, the pairwise likelihood estimates obtained by [@davis2013statistical] are $ \widehat{\phi}{_s}=0.3485, \widehat{\kappa}{_s}=0.8858, \widehat{\phi}{_t}=2.4190 \ \text{and} \ \widehat{\kappa}{_t}=0.1973$. Obviously, these estimates are close to our estimates, except the temporal smoothness estimate $\widehat{\kappa}{_t}$ which is relatively large.
Figure \[Goodness\_of\_fit\] shows the empirical values of $\nu_{F}(h),\ h \in \mathcal{H}$ and $\nu_{F}(l'), \ l' \in \mathcal{K}$, and their model-based counterparts from the three best-fitting models according to the AIC$_\text{NLS}$. It seems that the three models give a quite reasonable fit with a little outperformance for model A$_1$. So, considering these plots and the AIC$_\text{NLS}$ values there is overall evidence in favor of model A$_1$.
\[htp!\] ![Red star symbols show the empirical values of $\nu_{F}(h)$ and $\nu_{F}(l')$ used for estimation. The curves show the fitted $\nu_{F}(h)$ and $\nu_{F}(l')$ from the three best-fitting models (A$_1$, B$_1$ and B$_3$). []{data-label="Goodness_of_fit"}](Goodness_of_fit1.pdf "fig:")
Lastly, permutation tests can be useful to determine the range of clear dependence. So, in order to examine whether the extremal dependence in space and time is significant, we perform a permutation test. We randomly permute the space-time data and compute the empirical spatial/temporal $F$-madograms. More precisely, to check how the extremal dependence lasts in space, for each fixed time point $t \in \{t_1,\ldots, t_{732}\}$ we permute the spatial locations. Afterward, the spatial $F$-madogram is computed and the procedure is repeated 1000 times. From the obtained spatial $F$-madogram sample, we compute 97.5% and 2.5% empirical quantiles which form a 95% confidence region for spatial extremal independence. On the other hand, to test the presence of temporal extremal independence, the analog procedure is done for the temporal $F$-madogram. In particular, for each fixed location ${\boldsymbol{s}}\in \mathbb{S} =\left\{(x,y): x,y \in\{1,\ldots,14\} \right\}$ we sample without replacement from the corresponding time series and compute the empirical temporal $F$-madogram. Our findings are shown in Figure \[Permutation\] together with the fitted values of spatial/temporal $F$-madograms derived from the best-fitting models A$_1$. Inspecting these plots, it appears that the spatial extremal dependence vanishes for spatial lags larger than four (the fitted values for the spatial $F$-madogram lies within the obtained independence confidence region), whereas the temporal extremal dependence vanishes for time lags larger than three. Let remark that the same conclusions are obtained in [@buhl2016semiparametric], where the permutation tests have been carried out based on the extremogram.
\[htp!\] ![Permutation test for extremal independence in space (left panel) and time (right panel). Upper/lower blue lines show 97.5% and 2.5% quantiles of empirical $F$-madograms for 1000 spatial (right) and temporal (left) permuations of the space-time observations. Red star symbols show the fitted values of $\nu_{F}(h),\ h \in \mathcal{H}$ and $\nu_{F}(l'), \ l' \in \mathcal{K}$ derived from the best-fitting model A$_1$. []{data-label="Permutation"}](Permutation.pdf "fig:")
Concluding remarks {#sec:conc}
==================
In summary, motivated by shortcomings in existing inferential methods, we proposed two novel and flexible semiparametric estimation schemes for space-time max-stable processes based on the spatio-temporal $F$-madogram, $\nu_{F}({\boldsymbol{h}},l)$. Working with the madogram has a few advantages. In addition to its simple definition and the computational facility, it has a clear link with extreme value theory throughout the extremal dependence function. The new estimation procedure may be considered as an alternative or a prerequisite to the widely used pairwise likelihood; the semiparametric estimates could serve as starting values for the optimization routine used to maximize the pairwise log-likelihood function to decrease the computational time and also improve the statistical efficiency, see [@castruccio2016high].
A simulation study has shown that the inference procedure performs well. Moreover, our estimation methodology outperforms the semiparametric estimation procedure suggested by [@buhl2016semiparametric] which was based on the dependence measure extremogram. The introduced method is applied to radar rainfall measurements in a region in the State of Florida (Section \[sec:real\]) in order to quantify the extremal properties of the space-time observations.
Our attention is concentrated on fitting space-time max-stable processes based on gridded datasets. In the future, we plan to generalize our method in order to fit space-time max-stable processes with extensions to irregularly spaced locations that may have a fundamental interest in practice. In addition, equally weighted inference approaches have been widely implemented. However, using non-constant weights seems appealing for at least two reasons. First from a computational point of view, for example discarding distant pairs, the CPU load for the evaluation might be smaller and the fitting procedure would be less time-consuming. On the other hand, as neighboring pairs are expected to be strongly dependent, thus providing valuable information for the estimation of dependence parameters, this may improve the statistical efficiency. Therefore, it could be interesting to investigate the gain in statistical efficiency of estimators as well as computational efficiency by adopting different weighting strategies. Since the number of spatial and temporal lags are limited, we could consider weights such that locations and time points which are further apart from each other have less influence on the estimation, i.e., $$\begin{aligned}
\omega^{{\boldsymbol{h}}} =& \exp \left\{ -c_1 {\ensuremath{\lVert \boldsymbol{h} \rVert} }\right\} \ \text{or}\ \exp \left\{ -c_1 {{\ensuremath{\lVert \boldsymbol{h} \rVert} }}^2 \right\} \ \text{or} \ {{\ensuremath{\lVert \boldsymbol{h} \rVert} }}^{-c_1} ,\\
\omega^{l'} =& \exp \left\{ -c_2 l' \right\} \ \text{or}\ \exp \left\{ -c_2 {l'} ^ 2\right\} \ \text{or}\ l'^{-c_2},\\
\omega^{{\boldsymbol{h}},l'} =& \exp \left\{ -c \left({\ensuremath{\lVert \boldsymbol{h} \rVert} }+l' \right) \right\} \ \text{or}\ \exp \left\{ -c \left({{\ensuremath{\lVert \boldsymbol{h} \rVert} }}^2+{l'}^2\right) \right\} \ \text{or} \ {\left({\ensuremath{\lVert \boldsymbol{h} \rVert} }+l' \right)}^{-c} ,\end{aligned}$$ where $c_1, \ c_2,\ c>0, \ {\ensuremath{\lVert \boldsymbol{h} \rVert} }\in \mathcal{H} \ \text{and} \ l' \in \mathcal{K}.$ Finally, it could be interesting to extend the spatial $\lambda$-madogram approach proposed by [@naveau2009modelling] to estimate the spatio-temporal extremal dependence function $V_{{\boldsymbol{h}},l}$. For example, in the case of (\[model space-time\]), it is easy to verify that for ${\boldsymbol{h}}\in {\ensuremath{\mathbb{R}}}^2$ and $l \in {\ensuremath{\mathbb{R}}}$, the spatio-temporal $\lambda$-madogram for any $\lambda \in (0,1)$ is given by $$\nu_{\lambda}({\boldsymbol{h}},l )=\frac{ (1-\lambda)\{V_{\boldsymbol{0},\boldsymbol{h}-l\boldsymbol{\tau}}\left(\lambda,(1-\lambda) \delta^{-l}\right)\} + {1-\delta^{l}}}{ (1-\lambda)\{1+V_{\boldsymbol{0},\boldsymbol{h}-l\boldsymbol{\tau}}\left(\lambda,(1-\lambda) \delta^{-l}\right)\} + 1-\delta^{l}} - c(\lambda),
\label{madogram formula1}$$ where $c(\lambda)=\frac{3}{2(1+\lambda)(2-\lambda)}$.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge the Southwest Florida Water Management District (SWFWMD) for providing the data. Especially, we would like to thank Margit L. Crowell for the help in finding the data and the related details. We also thank the authors of paper [@buhl2016semiparametric] for providing their space-time max-stable BR process simulation [R]{} code, that used to simulate data in Section \[sec:simulation\].
[9]{} <span style="font-variant:small-caps;">Ahmed, M., Maume-Deschamps, V., Ribereau, P. and Vial, C.</span> (2017). A semi-parametric estimation for max-mixture spatial processes. *submitted for publication- arxiv.org/abs/1710.08120.*
<span style="font-variant:small-caps;">Akaike, H.</span> (1974). A new look at the statistical model identification. *IEEE transactions on automatic control*. **19**(6) 716–723. <span style="font-variant:small-caps;">Banks, H.T. and Joyner, M.L.</span> (2017). AIC under the framework of least squares estimation. *Applied Mathematics Letters*. **74** 33–45.
<span style="font-variant:small-caps;">Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. L.</span> (2004). *Statistics of extremes: theory and applications*. John Wiley & Sons. <span style="font-variant:small-caps;">Bel, L., Bacro, J-N. and Lantu[é]{}joul, C.</span> (2008). Assessing extremal dependence of environmental spatial fields. *Environmetrics*, **19**(2) 163–182.
<span style="font-variant:small-caps;">Brown, B. M. and Resnick, S. I.</span> (1977). Extreme values of independent stochastic processes. *Journal of Applied Probability*. **14**(4) 1732–739. <span style="font-variant:small-caps;">Buhl, S., Davis, R. A., Kl[ü]{}ppelberg, C., and Steinkohl, C.</span> (2016). Semiparametric estimation for isotropic max-stable space-time processes. *submitted for publication- arxiv.org/abs/1609.04967*.
<span style="font-variant:small-caps;">Buhl, S. and Kl[ü]{}ppelberg, C.</span> (2016). Anisotropic Brown-Resnick space-time processes: estimation and model assessment. *Extremes*. **19**(4) 627–660. <span style="font-variant:small-caps;">Castruccio, S., Huser, R. and Genton, M.G.</span> (2016). High-order composite likelihood inference for max-stable distributions and processes. *Journal of Computational and Graphical Statistics*, 25:1212–1229. <span style="font-variant:small-caps;">Coles, S.</span> (2001). *An introduction to statistical modeling of extreme values*. Springer, London. <span style="font-variant:small-caps;">Coles, S., Heffernan, J. and Tawn, J.</span> (1999). Dependence measures for extreme value analyses. *Extremes*. **2**(4) 339-365. <span style="font-variant:small-caps;">Cooley, D., Naveau, P. and Poncet, P.</span> (2006). Variograms for spatial max-stable random fields. *Dependence in probability and statistics*. 373–390.
<span style="font-variant:small-caps;">Davis, R. A., Kl[ü]{}ppelberg, C. and Steinkohl, C.</span> (2013). Max-stable processes for modeling extremes observed in space and time. *Journal of the Korean Statistical Society*. **42**(3) 399–414.
<span style="font-variant:small-caps;">Davis, R. A., Kl[ü]{}ppelberg, C. and Steinkohl, C.</span> (2013). Statistical inference for max-stable processes in space and time. *Journal of the Royal Statistical Society: Series B (Statistical Methodology)*. **75**(5) 791–819.
<span style="font-variant:small-caps;">Davis, R. A. and Mikosch, T.</span> (2009). The extremogram: A correlogram for extreme events. *Bernoulli*. **15**(4) 977–1009. <span style="font-variant:small-caps;">Davis, R. A. and Resnick, S. I.</span> (1989). Basic properties and prediction of max-ARMA processes. *Advances in applied probability*. **21**(4) 781–803. <span style="font-variant:small-caps;">Davison A. C., Gholamrezaee M. M.</span> (2012). Geostatistics of extremes. *Proc. R. Soc. A*. (468) 581–608. <span style="font-variant:small-caps;">Davison, A. C., Huser, R. and Thibaud, E.</span> (2013). Geostatistics of dependent and asymptotically independent extremes. *Mathematical Geosciences*. **45**(5) 511–529. <span style="font-variant:small-caps;">Davison, A. C., Padoan, S. A. and Ribatet, M.</span> (2012). Statistical modeling of spatial extremes. *Statistical science*. 161–186.
<span style="font-variant:small-caps;">De Haan, L.</span> (1984). A spectral representation for max-stable processes. *The annals of probability*. 1194–1204. <span style="font-variant:small-caps;">De Haan, L. and Ferreira, A.</span> (2006). *Extreme value theory: an introduction*. Springer, New York.
<span style="font-variant:small-caps;">Dombry, C., Engelke, S. and Oesting, M.</span> (2016). Exact simulation of max-stable processes. *Biometrika*. **103**(2) 303–317.
<span style="font-variant:small-caps;">Embrechts, P., Koch, E. and Robert, C.</span> (2016). Space-time max-stable models with spectral separability. *Advances in Applied Probability*. **48**(A) 77–97. <span style="font-variant:small-caps;">Fern[á]{}ndez-Avil[é]{}s, G. and Mateu, J.</span> (2015). *Spatial and spatio-temporal geostatistical modeling and kriging*. John Wiley & Sons.
<span style="font-variant:small-caps;">Huser, R. and Davison, A. C.</span> (2014). Space–time modelling of extreme events. *Journal of the Royal Statistical Society: Series B (Statistical Methodology)*. **76**(2) 439–461.
<span style="font-variant:small-caps;">Kabluchko, Z., Schlather, M. and De Haan, L.</span> (2009). Stationary max-stable fields associated to negative definite functions. *The Annals of Probability*. 2042–2065.
<span style="font-variant:small-caps;">Matheron, G.</span> (1987). Suffit-il, pour une covariance, d’[ê]{}tre de type positif. *Sciences de la Terre, s[é]{}rie informatique g[é]{}ologique*. **26** 51–66.
<span style="font-variant:small-caps;">Naveau, P., Guillou, A., Cooley, D. and Diebolt, J.</span> (2009). Modelling pairwise dependence of maxima in space. *Biometrika*. **96**(1) 1–17. <span style="font-variant:small-caps;">Padoan, S. A., Ribatet, M. and Sisson, S. A.</span> (2010). Likelihood-based inference for max-stable processes. *Journal of the American Statistical Association*. **105**(489) 263–277.
<span style="font-variant:small-caps;">Ribatet, M.</span> (2015). SpatialExtremes: R Package, version 2.0–2. Available at . <span style="font-variant:small-caps;">Schlather, M.</span> (2002). Models for stationary max-stable random fields. *Extremes*. **5**(1) 33–44. <span style="font-variant:small-caps;">Schlather, M.</span> (2001). RandomFields, contributed package on random field simulation for R. Available at . <span style="font-variant:small-caps;">Schlather, M. and Tawn, J. A.</span> (2003). A dependence measure for multivariate and spatial extreme values: Properties and inference. *Biometrika*. **90**(1) 139–156.
<span style="font-variant:small-caps;">Smith, R. L.</span> (1990). Max-stable processes and spatial extremes. *Unpublished manuscript*.
<span style="font-variant:small-caps;">Wadsworth, J. L. and Tawn, J. A.</span> (2012). Dependence modelling for spatial extremes. *Biometrika*. **99**(2) 253–272.
|
---
abstract: 'The Shub-Smale $\tau$-Conjecture is a hitherto unproven statement (on integer roots of polynomials) whose truth implies both a variant of ${\mathbf{P}}\!\neq\!{{\mathbf{NP}}}$ (for the BSS model over ${\mathbb{C}}$) and the hardness of the permanent. We give alternative conjectures, some potentially easier to prove, whose truth still implies the hardness of the permanent. Along the way, we discuss new upper bounds on the number of $p$-adic valuations of roots of certain sparse polynomial systems, culminating in a connection between quantitative $p$-adic geometry and complexity theory.'
author:
- Pascal Koiran
- Natacha Portier
- 'J. Maurice Rojas'
title: |
\
Counting Tropically Degenerate Valuations and \[1\][$p$]{}-adic Approaches to the Hardness of the Permanent
---
\
Introduction
============
Deep questions from algebraic complexity, cryptology, and arithmetic geometry can be approached through sufficiently sharp upper bounds on the number of roots of structured polynomials in one variable. (We review four such results in Section \[sub:earlier\] below.) The main focus of this paper is the connection between the number of distinct [*norms*]{} of roots of polynomials, over the $p$-adic rationals ${\mathbb{Q}}_p$, and separations of complexity classes. Our first main theorem motivates the introduction of $p$-adic methods.
\[dfn:sps\] We define ${\mathrm{SPS}}(k,m,t)$ to be the family of polynomials presented in the form $\sum^k_{i=1} \prod^m_{j=1} f_{i,j}$ where, for all $i$ and $j$, $f_{i,j}\!\in\!{\mathbb{Z}}[x_1]\!\setminus\!\{0\}$ and has at most $t$ monomial terms. We call such polynomials [*SPS*]{} (for [ *sum-product-sparse*]{}) polynomials. [$\diamond$]{}
\[thm:koi\] Suppose that there is a prime $p$ with the following property: For all $k,m,t\!\in\!{\mathbb{N}}$ and $f\!\in\!{\mathrm{SPS}}(k,m,t)$, we have that the cardinality of\
$\displaystyle{S_f:=\left\{e\!\in\!{\mathbb{N}}\; : \;
x\!\in\!{\mathbb{Z}},\; f(x)\!=\!0,\; p^e|x, \text{ and }
p^{e+1}\!\!\not|x. \right\}}$\
is $(kmt)^{O(1)}$. Then the permanent of $n\times n$ matrices cannot be computed by constant-free, division-free arithmetic circuits of size $n^{O(1)}$.
The hypothesis of Theorem \[thm:koi\] was the inspiration for this paper, since it is easily implied by the famous Shub-Smale $\tau$-Conjecture (see [@21a; @21b] and Section \[sub:earlier\] below). Theorem \[thm:koi\] can in fact be strengthened further by weakening its hypothesis in various ways: see Remarks \[rem:stronger\] and \[rem:best\] of Section \[sec:koi\] below. [$\diamond$]{}
The special cases of the hypothesis where $k=1$, $t=1$, or $m$ is a fixed constant are easy to prove (see, e.g., Lemma \[lemma:newt\] of Section \[sub:uniup\] below). However, the hypothesis already becomes an open problem for $k=2$ or $t=2$. The greatest $e$ such that $p^e$ divides an integer $x$ is nothing more than the [*$p$-adic valuation*]{} of $x$, hence our focus on $p$-adic techniques.
We now describe certain families of univariate polynomials, and multivariate polynomial systems, where valuation counts in the direction of Theorem \[thm:koi\] can actually be proved. In particular, we give another related hypothesis (in Theorem \[thm:hard\] below), entirely within the realm of $p$-adic geometry, whose truth also implies the hardness of the permanent.
Provable Upper Bounds on the Number of Valuations
-------------------------------------------------
For any commutative ring $R$, we let $R^*\!:=\!R\!\setminus\!\{0\}$. The [*support*]{} of a polynomial $f\!\in\!R\!\left[x^{\pm 1}_1,\ldots,x^{\pm 1}_n
\right]$, denoted ${\mathrm{Supp}}(f)$, is the set of exponent vectors appearing in the monomial term expansion of $f$. For any prime $p$ and $x\!\in\!{\mathbb{Z}}\setminus
\{0\}$ we let ${\mathrm{ord}}_p(x)$ denote the $p$-adic valuation of $x$, and we set ${\mathrm{ord}}_p(0)\!:=\!+\infty$. We then set ${\mathrm{ord}}_p(x/y)\!:=\!
{\mathrm{ord}}_p(x)-{\mathrm{ord}}_p(y)$ to extend ${\mathrm{ord}}_p(\cdot)$ to ${\mathbb{Q}}$, and we let ${\mathbb{Q}}_p$ denote the completion of ${\mathbb{Q}}$ with \[1\][respect to the metric defined by $|u-v|_p\!:=\!p^{-{\mathrm{ord}}_p(u-v)}$. The [*$p$-adic complex numbers*]{}, ${\mathbb{C}}_p$, are]{} then the elements of the completion of the algebraic closure of ${\mathbb{Q}}_p$. Finally, for any polynomials $f_1,\ldots,f_r\!\in\!R\!\left[x^{\pm 1}_1,\ldots,
x^{\pm 1}_n\right]$, we let $Z_R(f_1,\ldots, f_r)$ (resp. $Z^*_R(f_1,\ldots,f_r)$) denote the set of \[1\][roots of $(f_1,\ldots,f_r)$ in $R^n$ (resp. $(R^*)^n$), and we use $\#S$ to denote the cardinality of a set $S$. [$\diamond$]{}]{}
In particular, ${\mathrm{ord}}_p(\cdot)$ and $|\cdot|_p$ extend naturally to ${\mathbb{C}}_p$, and the algebraic closure of ${\mathbb{Q}}$ embeds naturally within ${\mathbb{C}}_p$. [@artin; @weiss; @serre; @schikhof; @robert; @gouvea; @katok] are some excellent sources for further background on $p$-adic fields. What will be most important for our setting is that $p$-adic norms (or, equivalently, $p$-adic valuations) enable new hypotheses — closer to being provable with current techniques — that imply new separations of complexity classes.
We will ultimately focus on counting valuations of roots of polynomial systems with few monomial terms as a means of understanding the valuations of roots of univariate SPS polynomials. For example, a simple consequence of our main multivariate bounds (Theorems \[thm:upper\] and \[thm:mult\] below) is the following univariate bound revealing that at least part of the $k\!=\!t\!=\!2$ case of the hypothesis of Theorem \[thm:koi\] is true.
\[cor:upper\] Suppose $m_1,m_2\!\in\!{\mathbb{N}}$; $\alpha_i,\beta_i\!\in\!{\mathbb{C}}_p$; $\gamma_{i,j}\!\in\!{\mathbb{Z}}$;\
$\displaystyle{f(x_1)\!:=\!\left(\prod^{m_1}_{i=1}(\alpha_{i,1}+
\beta_{i,1}x_1)^{\gamma_{i,1}}\right)+\left(\prod^{m_2}_{i=1}(\alpha_{i,2}+
\beta_{i,2}x_1)^{\gamma_{i,2}}\right)}$\
is not identically zero; and the lower hulls of the $p$-adic Newton polygons (cf. Definition \[dfn:newt\] below) of the two products have no common vertices. Then $\#{\mathrm{ord}}_p\!\left(Z_{{\mathbb{C}}_p}\!\left(f\right)\right)\!\leq\!m_1+m_2$, and this bound is tight. Furthermore, any root of $f$ in ${\mathbb{C}}_p$ not making both products vanish has multiplicity at most $m_1+m_2$, and this bound is tight as well. [$\blacksquare$]{}
Bounds for the number of valuations, independent of the degree, had previously been known only for sparse polynomials, i.e., polynomials in ${\mathrm{SPS}}(k,1,t)$: see, e.g., Lemma \[lemma:newt\] of Section \[sub:uniup\] and [@weiss]. Note in particular that ${\mathrm{SPS}}(2,m,2)$ contains the family of $f$ in our corollary when $\gamma_{i,j}\!=\!1$ for all $i,j$. Also, our valuation count above is independent of the $\alpha_i, \beta_i,\gamma_i$.
(200,65)(5,-100) (380,-102)[ ]{} (0,-100)
For any prime $p$, the system $F\!:=\!(x+y+1,x+y+1+p)$\
\[1\][shows us that having just finitely many roots over ${\mathbb{C}}_p$ need [*not*]{} imply tropical]{}\
genericity. In particular, while $F$ has no roots at all in ${\mathbb{C}}^2_p$, we have that\
${\mathrm{ord}}_p(Z^*_{{\mathbb{C}}_p}(x+y+1)), {\mathrm{ord}}_p(Z^*_{{\mathbb{C}}_p}(x+y+1+p))\!\subset\!{\mathbb{R}}^2$ are identical and\
exactly the set of rational points on the right-hand union of $3$ rays:
[*(So the intersection is non-transversal.) Nevertheless, ${\mathrm{ord}}_p(Z^*_{{\mathbb{C}}_p}(F))$ is empty. [$\diamond$]{}*]{}
Throughout our paper, we abuse notation slightly by setting ${\mathrm{ord}}_p(x_1,\ldots,x_n)\!:=\!({\mathrm{ord}}_p(x_1),\ldots,{\mathrm{ord}}_p(x_n))$. The set ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_1,\ldots,f_r)\right)$ is thus well-defined but, as revealed above, need [*not*]{} be the same as $\bigcap^r_{i=1} {\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_i)\right)$. [$\diamond$]{}
For any finite subsets $A_1,\ldots,A_n\!\subset\!{{\mathbb{Z}}^n}$ we define ${{\overline{{{\mathcal{V}}}}}}_p(A_1,\ldots,A_n)$ (resp. ${{\overline{{{\mathcal{R}}}}}}_p(A_,\ldots,A_n)$) to be the maximum of $\#{\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)$ (resp. ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{Q}}_p}(F)\right)$) over all $F\!:=$\[1\][$(f_1,\ldots,f_n)$ with $f_i\!\in\!{\mathbb{C}}_p[x_1,\ldots,x_n]$ and ${\mathrm{Supp}}(f_i)\!\subseteq\!A_i$ for all $i$, and $Z^*_{{\mathbb{C}}_p}(F)$ finite. We also define]{} \[1\][${{\mathcal{V}}}_p$ to be the corresponding analogue of ${{\overline{{{\mathcal{V}}}}}}_p$ where we restrict further to tropically generic $F$. [$\diamond$]{}]{}
\[1\][Clearly ${{\mathcal{V}}}_p(A_1,\ldots,A_n)\!\leq\!{{\overline{{{\mathcal{V}}}}}}_p(A_1,\ldots,A_n)$ and ${{\overline{{{\mathcal{R}}}}}}_p(A_1,\ldots,A_n)\!\leq\!{{\overline{{{\mathcal{V}}}}}}_p(A_1,\ldots,A_n)$. While [*Smirnov’s*]{}]{} [@smirnov Thm. 3.4] implies that ${{\mathcal{V}}}_p(A_1,\ldots,A_n)$ is well-defined and finite for any fixed $(A_1,\ldots,A_n)$, explicit upper bounds for ${{\overline{{{\mathcal{V}}}}}}_p(A_1,\ldots,A_n)$ appear to be unknown. So we derive such an upper bound for certain $(A_1,\ldots,A_n)$.
\[thm:upper\] Suppose $p$ is any prime, $A_1,\ldots,A_n\!\subset\!{{\mathbb{Z}}^n}$, $A\!:=\#\bigcup_i A_i$, $t\!:=\!\#A$, and $e_i$ denotes the $i{^{\text{\underline{th}}}}$ standard basis vector of ${{\mathbb{R}}^n}$. Then:
1. [ \[1\][ $t\leq n \ \ \ \ \; \Longrightarrow
{{\mathcal{V}}}_p(A_1,\ldots,A_n)={{\overline{{{\mathcal{V}}}}}}_p(A_1,\ldots,A_n)=0$]{}. ]{}
2. [ \[1\][ $t=n+1 \Longrightarrow
{{\mathcal{V}}}_p(A_1,\ldots,A_n)={{\overline{{{\mathcal{V}}}}}}_p(A_1,\ldots,A_n)\leq 1$. In particular, ${{\mathcal{V}}}_p\!\left(\{{\mathbf{O}},e_1\},\ldots,\{{\mathbf{O}},e_n \}\right)\!=\!1$.]{}]{}
3. [ \[1\][\[$t=n+2$ and every collection of $n$ distinct pairs of points of $A$ determines an $n$-tuple]{} of linearly independent vectors\] $\Longrightarrow
{{\overline{{{\mathcal{V}}}}}}_p(A_1,\ldots,A_n)\leq \max\left\{2,{\left\lfloor\frac{n}{2}\right\rfloor}^n+n\right\}$. Also, ${{\mathcal{V}}}_p\!\left(\{{\mathbf{O}},2e_1,e_1+e_2\},\{{\mathbf{O}},2e_1,e_2+e_3\},
\ldots,\{{\mathbf{O}},2e_1,e_{n-1}+e_n\},\{{\mathbf{O}},2e_1,e_n\}\right)\!=\!n+1$.]{}
We conjecture that the upper bound in Assertion (2) can in fact be improved to $n+1$. It is easily shown that the general position assumption on $A$ holds for a dense open set of exponents. For instance, if $A$ has convex hull an $n$-simplex then the hypothesis of Assertion (2) holds automatically.
Whether the equality ${{\mathcal{V}}}_p(A_1,\ldots,A_n)={{\overline{{{\mathcal{V}}}}}}_p(A_1,\ldots,A_n)$ holds beyond the setting of Assertions (0) and (1) is an intriguing open question. In fact, proving just that the growth of orders of ${{\overline{{{\mathcal{R}}}}}}_p(A_1,\ldots,A_n)$ and ${{\mathcal{V}}}_p(A_1,\ldots,A_n)$ differ by a constant has deep implications.
\[thm:hard\] Suppose there is a prime $p$ such that ${{\overline{{{\mathcal{R}}}}}}_p(A_1,\ldots,A_n)\!=\!{{\mathcal{V}}}_p(A_1,\ldots,A_n)^{O(1)}$ for all finite $A_1,\ldots,A_n\!\subset\!{{\mathbb{Z}}^n}$. Then the permanent of $n\times n$ matrices cannot be computed by constant-free, division-free arithmetic circuits of size $n^{O(1)}$.
We thus obtain an entirely tropical geometric statement implying the hardness of the permanent. In Theorem \[thm:hard\] it in fact suffices to restrict to certain families of supports $A_i$ (see Proposition \[prop:red\] below). We are currently unaware of any $(A_1,\ldots,A_n)$ [*not*]{} satisfying the equality ${{\mathcal{V}}}_p(A_1,\ldots,A_n)\!=\!{{\overline{{{\mathcal{V}}}}}}_p(A_1,\ldots,A_n)$ for all primes $p$.
Intersection multiplicity is a key subtlety underlying the counting of valuations.
\[thm:mult\] Suppose $K$ is any algebraically closed field of characteristic $0$ and $A\!\subset\!{{\mathbb{Z}}^n}$ has cardinality at most $n+2$ and no $n+1$ points of $A$ lie in a hyperplane. Suppose also that $f_1,\ldots,
f_n\!\in\!K[x_1,\ldots,x_n]$ have support contained in $A$ and $\#Z^*_{K}(F)$ is finite. Then the intersection multiplicity of any point of $Z^*_{K}(F)$ is at most $n+1$ (resp. $1$) when $\#A\!=\!n+2$ (resp. $\#A\!=\!n+1$), and both bounds are sharp.
The intersection multiplicity considered in our last theorem is the classical definition coming from commutative algebra or differential topology (see, e.g., [@ifulton]).
Earlier Applications of Root Counts for Univariate Structured Polynomials {#sub:earlier}
-------------------------------------------------------------------------
Recall the following classical definitions on the evaluation complexity of univariate polynomials.
For any field $K$ and $f\!\in\!K[x_1]$ let $s(f)$ — the [*SLP complexity of $f$*]{} — denote the smallest $n$ such that $f\!=\!f_n$ identically where the sequence $(f_{-N},\ldots,f_{-1},
f_0,\ldots,f_n)$ satisfies the following conditions: $f_{-1},\ldots,f_{-N}\!\in\!K$, $f_0\!:=\!x_1$, and, for all $i\!\geq\!1$, $f_i$ is a sum, difference, or product of some pair of elements $(f_j,f_k)$ with $j,k\!<\!i$. Finally, for any $f\!\in\!{\mathbb{Z}}[x_1]$, we let $\tau(f)$ denote the obvious analogue of $s(f)$ where the definition is further restricted by assuming $N\!=\!1$ and $f_{-1}\!:=\!1$. [$\diamond$]{}
Note that we always have $s(f)\!\leq\!\tau(f)$ since $s$ does not count the cost of computing large integers (or any constants). One in fact has $\tau(n)\!\leq\!2\log_2 n$ for any $n\!\in\!{\mathbb{N}}$ [@svaiter Prop. 1]. See also [@brauer; @moreira] for further background.
We can then summarize some seminal results of Bürgisser, Cheng, Lipton, Shub, and Smale as follows:
\[thm:tau\]\
I. (See [@bcss Thm. 3, Pg. 127] and [@burgtau Thm. 1.1].) Suppose that for all nonzero $f$\[1\][$\in\!{\mathbb{Z}}[x_1]$ we have $\#Z_{\mathbb{Z}}(f)\!\leq\!\tau(f)^{O(1)}$. Then (a) ${\mathbf{P}}_{\mathbb{C}}\!\neq\!{{\mathbf{NP}}}_{\mathbb{C}}$ and (b) the permanent of $n\times n$ ]{}\[1\][matrices cannot be computed by constant-free, division-free arithmetic circuits of size $n^{O(1)}$.]{}
II\. (Weak inverse to (I) [@lipton].[^1]) If there is an ${\varepsilon}\!>\!0$ and a sequence $(f_n)_{n\in{\mathbb{N}}}$ of polynomialsin ${\mathbb{Z}}[x_1]$ satisfying:\
(a) $\#Z_{\mathbb{Z}}(f_n)\!>\!e^{\tau(f_n)^{\varepsilon}}$ for all $n\!\geq\!1$ and (b) $\deg f_n, \max\limits_{\zeta\in Z_{\mathbb{Z}}(f)}|\zeta|\!\leq\!
2^{(\log \#Z_{\mathbb{Z}}(f_n))^{O(1)}}$\
then, for infinitely many $n$, at least $\frac{1}{n^{O(1)}}$ of the $n$ digit integers that are products of exactlytwo distinct primes (with an equal number of digits) can be factored by a Boolean circuitof size $n^{O(1)}$.
III\. (Number field analogue of (I) implies Uniform Boundedness [@cheng].) Suppose that forany number field $K$ and $f\!\in\!K[x_1]$ we have $\#Z_K(f)\!\leq\!c_1 1.0096^{s(f)}$, with $c_1$ depending onlyon $[K:{\mathbb{Q}}]$. Then there is a constant $c_2\!\in\!{\mathbb{N}}$ depending only on $[K:{\mathbb{Q}}]$ such that for anyelliptic curve $E$ over $K$, the torsion subgroup of $E(K)$ has order at most $c_2$. [$\blacksquare$]{}
The hypothesis in Part (I) is known as the [*(Shub-Smale) $\tau$-Conjecture*]{} and was stated as the fourth problem (still unsolved as of late 2013) on Smale’s list of the most important problems for the $21{^{\text{\underline{st}}}}$ century [@21a; @21b]. Via fast multipoint evaluation applied to the polynomial $(x-1)\cdots(x-m^2)$ [@compualg] one can show that the $O$-constant from the $\tau$-Conjecture should be at least $2$ if the $\tau$-Conjecture is true.
The complexity classes ${\mathbf{P}}_{\mathbb{C}}$ and ${{\mathbf{NP}}}_{\mathbb{C}}$ are respective analogues (for the BSS model over ${\mathbb{C}}$ [@bcss]) of the well-known complexity classes ${\mathbf{P}}$ and ${{\mathbf{NP}}}$. (Just as in the famous ${\mathbf{P}}$ vs. ${{\mathbf{NP}}}$ Problem, the equality of ${\mathbf{P}}_{\mathbb{C}}$ and ${{\mathbf{NP}}}_{\mathbb{C}}$ remains an open question.) The assertion on the hardness of the permanent in Theorem \[thm:tau\] is also an open problem and its proof would be a major step toward solving the [*${\mathbf{VP}}$ vs. ${\mathbf{VNP}}$ Problem*]{} — Valiant’s algebraic circuit analogue of the ${\mathbf{P}}$ vs. ${{\mathbf{NP}}}$ Problem [@valiant; @burgcook; @koiran; @jml]: The only remaining issue to resolve for a complete solution of this problem would then be the restriction to constant-free circuits in Part (I).
The hypothesis of Part (II) (also unproved as of late 2013) merely posits a sequence of polynomials violating the $\tau$-Conjecture in a weakly exponential manner. The conclusion in Part (II) would violate a widely-believed version of the cryptographic hardness of integer factorization.
The conclusion in Part (III) is the famous [*Uniform Boundedness Theorem*]{}, due to Merel [@merel]. Cheng’s conditional proof (see [@cheng Sec. 5]) is dramatically simpler and would yield effective bounds significantly improving known results (e.g., those of Parent [@parent]). In particular, the $K\!=\!{\mathbb{Q}}$ case of the hypothesis of Part (III) would yield a new proof (less than a page long) of Mazur’s landmark result on torsion points [@mazur].
More recently, Koiran has suggested real analytic methods (i.e., upper bounds on the number of real roots) as a means of establishing the desired upper bounds on the number of integer roots [@koiran], and Rojas has suggested $p$-adic methods [@adelic]. In particular, the following variation on the hypothesis from Theorem \[thm:koi\] appears in slightly more refined form in [@adelic]:
For any $k,m,t\!\in\!{\mathbb{N}}$ and $f\!\in\!{\mathrm{SPS}}(k,m,t)$, there is a field\[1\][$L\!\in\!\{{\mathbb{R}},{\mathbb{Q}}_2,{\mathbb{Q}}_3,{\mathbb{Q}}_5,\dots\}$ such that $f$ has no more than $(kmt)^{O(1)}$ distinct roots in $L$.]{}
\[thm:adelic\] If the Adelic SPS-Conjecture is true then the permanent of $n\times n$ matrices cannot be computed by constant-free, division-free arithmetic circuits of size $n^{O(1)}$.
[**Proof of Theorem \[thm:adelic\]:**]{} The truth of the Adelic SPS-Conjecture clearly implies the following special case of the Shub-Smale $\tau$-Conjecture: the number of integer roots of any $f\!\in\!{\mathrm{SPS}}(k,m,t)$ is $(kmt)^{O(1)}$. The latter statement in turn implies the hypothesis of Theorem \[thm:koi\], so by the conclusion of Theorem \[thm:koi\] we are done. [$\blacksquare$]{}
Note that the Adelic SPS-Conjecture can not be simplified to counting just the valuations: [*any*]{} fixed polynomial in ${\mathbb{Z}}[x_1]\setminus\{0\}$ will have exactly [*one*]{} $p$-adic valuation for its roots in ${\mathbb{C}}_p$ for sufficiently large $p$. (This follows easily from, e.g., Lemma \[lemma:newt\] of the next section.) An alternative simplification (and stronger hypothesis) would be to ask for a [*single*]{} field $L\!\in\!\{{\mathbb{R}},{\mathbb{Q}}_2,{\mathbb{Q}}_3,{\mathbb{Q}}_5,\ldots
\}$ where the number of roots in $L$ of any $f\!\in\!{\mathrm{SPS}}(k,m,t)$ is $(kmt)^{O(1)}$. The latter simplification is an open problem, although it is now known that one can not ask for too much more: the stronger statement that the number of roots in $L$ of any $f\!\in\!{\mathbb{Z}}[x_1]$ is $\tau(f)^{O(1)}$ is known to be false. Counter-examples are already known over ${\mathbb{R}}$ (see, e.g., [@bocook]), and over ${\mathbb{Q}}_p$ for any prime $p$ [@adelic Example 2.5 & Sec. 4.5].
The latter examples are much more recent, so for the convenience of the reader we summarize them here: Recall that the $p$-adic [*integers*]{}, ${\mathbb{Z}}_p$, are those elements of ${\mathbb{Q}}_p$ with nonnegative valuation. (So ${\mathbb{Z}}\!\subsetneqq\!{\mathbb{Z}}_p$ in particular.)
\[ex:slp\] Consider the recurrence $h_1\!:=\!x_1(1-x_1)$ and $h_{n+1}\!:=\!\left(p^{3^{n-1}}-h_{n}\right)h_n$ for all $n\!\geq\!1$. Then $h_n$ has degree $2^n$, exactly $2^n$ roots in ${\mathbb{Z}}_p$, and $\tau(h_n)=O(n)$. However, the only integer roots of $h_n$ are $\{0,1\}$ (see [@adelic Sec. 4.5]). Note also that $h_n$ has just $n$ distinct valuations for its roots in ${\mathbb{C}}_p$. The last fact follows easily from Lemma \[lemma:newt\], stated in the next section. [$\diamond$]{}
Note, however, that it is far from obvious if the polynomial $h_n$ above is in ${\mathrm{SPS}}(k,m,t)$ for some triple $(k,m,t)$ of functions growing polynomially in $n$.
From Univariate SPS to Multivariate Sparse {#sub:uniup}
------------------------------------------
Perhaps the simplest reduction of root counts for univariate SPS polynomial to root counts for multivariate sparse polynomial systems is the following.
\[prop:red\] Suppose $f\!\in\!{\mathrm{SPS}}(k,m,t)$ is written $\sum^k_{i=1} \prod^m_{j=1} f_{i,j}$ as in Definition \[dfn:sps\]. Let $F\!:=\!(f_1,\ldots,f_{km+1})$ be the polynomial system defined by $f_{km+1}(x_1,\ldots,y_{i,j},\ldots)\!:=\!
\sum^k_{i=1} \prod^m_{j=1} y_{i,j}$ and $f_{(i-1)m+j}(x_1,y_{i,j})\!:=\!y_{i,j} - f_{i,j}(x_1)$ for all $(i,j)\!\in\!\{1,\ldots,k\}\times \{1,\ldots,m\}$. (Note that $F$ involves exactly $km+1$ variables; $f_1,\ldots,f_{km}$ each have at most $t+1$ monomial terms; and $f_{km+1}$ involves exactly $k$ monomial terms.) Then $f$ not identically zero implies that $F$ has only finitely many roots in ${\mathbb{C}}_p$, and the $x_1$-coordinates of the roots of $F$ in ${\mathbb{C}}_p$ are exactly the roots of $f$ in ${\mathbb{C}}_p$. [$\blacksquare$]{}
Upper bounds for the number of valuations of the roots of multivariate sparse polynomials can then, in some cases, yield useful upper bounds for the number of valuations of the roots of univariate SPS polynomials.
\[lemma:maybetrivial\] Following the notation of Proposition \[prop:red\], suppose $F$ is tropically generic. Then $\#{\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)\!\leq\!
k(k-1)(2km(t-1)+1)/2\!=\!O(k^3mt)$.
The crux of our paper is whether the bound above holds without tropical genericity. We prove Lemma \[lemma:maybetrivial\] in Section \[sec:back\] below.
To prove upper bounds such as Lemma \[lemma:maybetrivial\] (and Corollary \[cor:upper\] and Theorem \[thm:upper\]) we will need to use some polyhedral geometric tricks. So let us first review $p$-adic Newton polygons (see, e.g., [@weiss; @gouvea]).
\[dfn:newt\] Given any prime $p$ and a polynomial $f(x_1)\!:=\!\sum^t_{i=1}c_ix^{a_i}_1$ $\in\!{\mathbb{C}}_p[x_1]$, we define its [*$p$-adic Newton polygon*]{}, ${\mathrm{Newt}}_p(f)$, to be the convex hull of[^2] the points $\{(a_i,{\mathrm{ord}}_p c_i)\; | \;
i\!\in\!\{1,\ldots,t\}\}$. Also, a face of a polygon $Q\!\subset\!{\mathbb{R}}^2$ is called [*lower*]{} if and only if it has an inner normal with positive last coordinate, and the [*lower hull*]{} of $Q$ is simply the union of all its lower edges. Finally, the polynomial associated to summing the terms of $f$ corresponding to points of the form $(a_i,{\mathrm{ord}}_p c_i)$ lying on some lower face of ${\mathrm{Newt}}_p(f)$ is called a [*($p$-adic) lower polynomial*]{}. [$\diamond$]{}
\[ex:newt\] For $f(x_1):=36 -8868x_1 +29305x^2_1 -35310x^3_1 +18240x^4_1
-3646x^5_1+243x^6_1$,
the polygon ${\mathrm{Newt}}_3(f)$ has exactly $3$ lower edges and can easily be verified to resemble the illustration to the right. The polynomial $f$ thus has exactly $2$ lower binomials, and $1$ lower trinomial over ${\mathbb{C}}_3$. [$\diamond$]{}
The $p$-adic Newton polygon is particularly important because it allows us to count valuations (or norms) of $p$-adic complex roots exactly when the monomial term expansion is known.
\[lemma:newt\] (See, e.g., [@weiss Prop. 3.1.1].) The number of roots of $f$ in ${\mathbb{C}}_p$ with valuation $v$, counting multiplicities, is [*exactly*]{} the horizontal length of the lower face of ${\mathrm{Newt}}_p(f)$ with inner normal $(v,1)$. [$\blacksquare$]{}
In Example \[ex:newt\], note that the $3$ lower edges have respective horizontal lengths $2$, $3$, and $1$, and inner normals $(1,1)$, $(0,1)$, and $(-5,1)$. Lemma \[lemma:newt\] then tells us that $f$ has exactly $6$ roots in ${\mathbb{C}}_3$: $2$ with $3$-adic valuation $1$, $3$ with $3$-adic valuation $0$, and $1$ with $3$-adic valuation $-5$. Indeed, one can check that the roots of $f$ are exactly $6$, $1$, and $\frac{1}{243}$, with respective multiplicities $2$, $3$, and $1$. [$\diamond$]{}
To prove Theorem \[thm:hard\] we will need to review the higher-dimensional version of the $p$-adic Newton polygon: the $p$-adic Newton [*polytope*]{}.
Background on $p$-adic Tropical Geometry {#sec:back}
========================================
The definitive extension of $p$-adic Newton polygons to arbitrary dimension (and general non-Archimedean, algebraically closed fields) is due to Kapranov.
\[1\][For any polynomial $f\!\in\!{\mathbb{C}}_p[x_1,\ldots,x_n]$ written $\sum_{a\in A} c_ax^a$ (with $x^a\!=\!x^{a_1}_1 \cdots x^{a_n}_n$]{} understood) we define its [*$p$-adic Newton polytope*]{}, ${\mathrm{Newt}}_p(f)$, to be the convex hull of the point set $\{(a,{\mathrm{ord}}_p(c_a))\; | \;
a\!\in\!A\}$. We also define the [*$p$-adic tropical variety of $f$*]{} (or [*$p$-adic amoeba of*]{} \[1\][$f$), ${\mathrm{Trop}}_p(f)$, to be $\{v\!\in\!{{\mathbb{R}}^n}\; | \; (v,1) \text{ is an inner
normal of a positive-dimensional face of } {\mathrm{Newt}}_p(f)\}$. [$\diamond$]{}]{}
We note that in [@kapranov], the $p$-adic tropical variety of $f$ was defined via a [*Legendre transform*]{} (a.k.a. [*support function*]{} [@ziegler]) of the lower hull of ${\mathrm{Newt}}_p(f)$. It is easy to see that both defintions are equivalent.
[@kapranov] Following the notation above, ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f)\right)\!=\!{\mathrm{Trop}}_p(f)\cap {\mathbb{Q}}^n$. [$\blacksquare$]{}
A simple consequence of Kapranov’s Theorem is that counting valuations is most interesting for zero-dimensional algebraic sets.
\[prop:fin\] Suppose $f_1,\ldots,f_r\!\in\!{\mathbb{C}}_p\!\left[x^{\pm 1}_1,\ldots,x^{\pm 1}_n
\right]$, $F\!:=\!(f_1,\ldots,f_r)$, and $Z^*_{{\mathbb{C}}_p}(F)$ is infinite. Then ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)$ is infinite.
[**Proof:**]{} By the definition of dimension for algebraic sets over an algebraically closed field, there must be a linear projection $\pi : {{\mathbb{C}}^n}_p \longrightarrow I$, for some coordinate subspace $I$ of positive dimension $k$, with $\pi\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)$ dense. Taking valuations, and applying Kapranov’s Theorem, this implies that ${\mathrm{ord}}_p\!\left(\pi\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)\right)$ must be linearly isomorphic to ${\mathbb{Q}}^k$ minus a (codimension $1$) polyhedral complex. In other words, ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)$ must be infinite. [$\blacksquare$]{}
Another consequence of Kapranov’s Theorem is a simple characterization of ${\mathrm{ord}}_p(Z^*_{{\mathbb{C}}_p}(F))$ when $F\!:=\!(f_1,\ldots,f_n)$ is over-determined in a certain sense. This is based on a trick commonly used in toric geometry, ultimately reducing to an old matrix factorization: For any matrix $M=[M_{i,j}]\!\in\!{\mathbb{Z}}^{n\times n}$ and $x\!\in\!({{\mathbb{C}}^*}_p)^n$, we define $x^M\!:=\!\left(x^{M_{1,1}}_1\cdots x^{M_{n,1}},\ldots,
x^{M_{1,n}}_1\cdots x^{M_{n,n}}\right)$. We then call the map $m_M :
({{\mathbb{C}}^*}_p)^n \longrightarrow ({{\mathbb{C}}^*}_p)^n$ defined by $m_M(x)\!:=\!x^M$ a [*monomial change of variables*]{}.
\[lemma:mono\] Given any finite set $A\!=\!\{a_1,\ldots,a_n\}\!\subset\!{{\mathbb{Z}}^n}$ lying in a hyperplane in ${{\mathbb{R}}^n}$, there is a matrix $U\!\in\!{\mathbb{Z}}^{n\times n}$, with determinant $\pm 1$, satisfying the following conditions:
1. [$Ua_i\!\in\!{\mathbb{Z}}^{i}\times\{0\}^{n-i}$ for all $i\!\in\!\{1,\ldots,n\}$.]{}
2. [Left (or right) multiplication by $U$ induces a linear bijection of ${{\mathbb{Z}}^n}$.]{}
3. [$m_U$ is an automorphism of the multiplicative group $({{\mathbb{C}}^*}_p)^n$, with inverse $m_{U^{-1}}$. In particular, the map sending ${\mathrm{ord}}_p(x)\mapsto
{\mathrm{ord}}_p(m_U(x))$ for all $x\!\in\!({{\mathbb{C}}^*}_p)^n$ is a linear automorphism of ${{\mathbb{Q}}^n}$. [$\blacksquare$]{}]{}
Lemma \[lemma:mono\] follows immediately from the existence of [*Hermite factorization*]{} for matrices with integer entries (see, e.g., [@hermite; @storjophd]). In fact, the matrix $U$ above can be constructed efficiently, but this need not concern us here. The characterization of ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)$ for over-determined $F$ is the following statement.
\[prop:over\] Suppose $A_1,\ldots,A_n\!\subseteq\!A\!\subset\!{{\mathbb{Z}}^n}$ and $A$ lies in some $(n-1)$-flat of ${{\mathbb{R}}^n}$. Then ${{\mathcal{V}}}_p(A_1,\ldots,A_n)\!=\!{{\overline{{{\mathcal{V}}}}}}_p
(A_1,\ldots,A_n)\!=\!0$.
[**Proof:**]{} Suppose $F\!:=\!(f_1,\ldots,f_n)$ where ${\mathrm{Supp}}(f_i)\!\subseteq
\!A_i$ for all $i$. By Lemma \[lemma:mono\] we may assume that $A\!\subset\!{\mathbb{Z}}^{n-k}\times \{0\}^k$ for some $k\!\geq\!1$. Clearly then, $Z^*_{{\mathbb{C}}_p}(F)$ is either empty or contains a coordinate $k$-flat. So ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)$ must either be empty or infinite, and we are done. [$\blacksquare$]{}
Another consequence of Kapranov’s Theorem is the following characterization of${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f)\right)$ for certain trinomials. Recall that ${\mathbb{R}}_+$ is the set of positive real numbers and \[1\][that ${\mathbb{R}}_+v$, for any vector $v\!\in\!{\mathbb{R}}^N\setminus\{{\mathbf{O}}\}$, is the [*open ray*]{} generated by all positive multiples of $v$.]{}
\[lemma:vert\] Suppose $g\!\in\!{\mathbb{C}}_p[x_1]$ has exactly $t$ monomial terms, the lower hull of ${\mathrm{Newt}}_p(g)$ consists of exactly $t'$ edges, and $f(x_1,y_i)\!:=\!y_i-g(x_1)$ is considered as a polynomial in ${\mathbb{C}}_p[x_1,y_1,\ldots,y_N]$ with $N\!\geq\!i$. Then ${\mathrm{ord}}_p(Z^*_{{\mathbb{C}}_p}(f))$ is the set of rational points of a polyhedral complex $\Sigma_f$ of the following form: a union of an open $(N-1)$-dimensional half-space parallel to $({\mathbb{R}}_+(-1,\deg(g)))\times {\mathbb{R}}^{N-1}$, $t'$ “vertical” open half-spaces parallel to $({\mathbb{R}}_+(0,1))\times {\mathbb{R}}^{N-1}$, $t'-1$ strips of the form $L\times {\mathbb{R}}^{N-2}$ where $L\!\subset\!{\mathbb{R}}^2$ is a line segment missing one of its vertices, and a closed $(N-1)$-dimensional half-space parallel to $({\mathbb{R}}_+\cup\{0\})\times \{0\} \times {\mathbb{R}}^{N-1}$.
For any prime $p$, the polynomial\
$f(x_1,y_1)\!:=\!y_1-(x^3_1-(1+p+p^2)x^2_1+(p+p^2+p^3)x_1
-p^3)$\
has ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f)\right)$ resembling the diagram to the right. In particular, in the\
(100,5)(5,-5) (360,-100)
notation of Lemma \[lemma:vert\], we have\
$g(x_1)\!:=\!x^3_1-(1+p+p^2)x^2_1+(p+p^2+p^3)x_1-p^3$,\
$N\!=\!1$, $t\!=\!4$, and $t'\!=\!3$. [$\diamond$]{}
[**Proof of Lemma \[lemma:vert\]:**]{} By construction, ${\mathrm{Newt}}(f)$ lies in a $2$-plane in ${\mathbb{R}}^{N+1}$\
\[1\][and thus, thanks to Kapranov’s Theorem, ${\mathrm{ord}}_p\!\left(
Z^*_{{\mathbb{C}}_p}(f)\right)$ is the Minkowski]{}\
\[1\][sum of a $1$-dimensional tropical variety and a complementary subspace of]{}\
dimension $N-1$. In particular, it suffices to prove the $N\!=\!1$ case.
The $N\!=\!1$ case follows easily: the ray of type (a) (resp. (d)) is parallel to the inner normal ray to the edge with vertices $(0,1)$ and $(\deg g,0)$ (resp. $(0,1)$ and $(0,0)$) of ${\mathrm{Newt}}(f)$. The vertical rays correspond to the inner normals corresponding to the lower edges of ${\mathrm{Newt}}_p(g)$ (alternatively, the edges of ${\mathrm{Newt}}_p(f)$ not incident to $(0,1,0)$). Finally, the “strips” are merely the segments connecting the points $v$ with $(v,1)$ a lower facet normal of ${\mathrm{Newt}}_p(f)$. [$\blacksquare$]{}
[**Proof of Lemma \[lemma:maybetrivial\]:**]{} Let $n=km+1$ be the number of variables in the system constructed in Proposition \[prop:red\]. Lemma \[lemma:vert\] (applied to each $y_{i,j}-f_{i,j}$) induces a natural finite partition of ${\mathbb{R}}^n$ into half-open slabs of the form $(-\infty,v_1)\times {\mathbb{R}}^{n-1}$, $[v_\ell,v_{\ell+1})\times {\mathbb{R}}^{n-1}$ for $\ell\!\in\!\{1,\ldots,M-1\}$, or $[v_M,+\infty)\times {\mathbb{R}}^{n-1}$, with $M\!\leq\!km(t-1)-1$. (Note, in particular, that the boundaries of the slabs coming from different $f_{i,j}$ can not intersect, thanks to tropical genericity.) Note also that within the interior of each slab, any non-empty intersection of ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_1)\right),\ldots,
{\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_n)\right)$ must be a transversal intersection of $n$ hyperplanes, thanks to tropical genericity.
In particular, for the left-most (resp. right-most) slab, we obtain a transversal intersection of $n-1$ type (a) (resp. type (d)) $(n-1)$-cells coming from $f_1,\ldots,f_{n-1}$ (in the notation of Lemma \[lemma:vert\]) and an $(n-1)$-cell of ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_n)
\right)$. Each $(n-1)$-cell of ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_n)\right)$, by definition, is dual to an edge of the lower hull of ${\mathrm{Newt}}_p(f_n)$. So there are no more than $\binom{k}{2}$ such $(n-1)$-cells. Thus, there are at most $\binom{k}{2}$ intersections of ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_1)\right),\ldots,{\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_n)
\right)$ occuring in the interior of the left-most (resp. right-most) slab.
Similarly, the number of intersections of ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_1)\right),\ldots,{\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_n)
\right)$ occuring in any other slab interior is $\binom{k}{2}$. Also, within any of the $M+1$ slab boundaries, there are clearly at most $\binom{k}{2}$ intersections of ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_1)\right),\ldots,
{\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_n)\right)$.
So we obtain no more than $\binom{k}{2}(2+M+M+1)
\!\leq\!\binom{k}{2}(2km(t-1)+1)\!=\!O(k^3mt)$ intersections for the underlying $p$-adic tropical varieties and we are done. [$\blacksquare$]{}
Proving Theorem \[thm:koi\] {#sec:koi}
===========================
The proof is a fairly straightforward application of a result from [@koiran], which we paraphrase in Theorem \[thm:hit\] below. However, let us first review some background.
Recall that the [*counting hierarchy*]{} ${{\mathbf{CH}}}$ is a hierarchy of complexity classes built on top of the counting class ${{\mathbf{\# P}}}$; it contains the entire polynomial hierarchy ${\mathbf{PH}}$ and is contained in ${\mathbf{PSPACE}}$. A detailed understanding of ${{\mathbf{CH}}}$ is not necessary here since we will need only one fact (Theorem \[thm:hit\] below) related to ${{\mathbf{CH}}}$. The curious reader can consult [@burgtau; @koiran] and the references therein for more information on the counting hierarchy.
A [*hitting set*]{} $H$ for a family $\mathcal{F}$ of polynomials is a finite set of points such that, for any $f\!\in\!\mathcal F\setminus
\{0\}$, there is at least one $x\!\in\!H$ such that $f(x)\!\neq\!0$. Also, a [*${{\mathbf{CH}}}$-algebraic number generator*]{} is a sequence of polynomials $G\!:=\!(g_i)_{i\in{\mathbb{N}}}$ satisfying the following conditions:
1. [There is a positive integer $c$ such that we can write $g_i(x_1)\!:=\!\sum^{i^c}_{\alpha=0} a(\alpha,i)x^{\alpha}_1$, with $a(\alpha,i)\!\in\!{\mathbb{Z}}$ of absolute value no greater than $2^{i^c}$, for all $i$.]{}
2. [The language $L(G)\!:=\!\left\{(\alpha,i,j,b)\; | \; \text{ the }
j^{\text{\underline{th}}} \text{ bit of } a(\alpha,i) \text{ is equal to }
b\right\}$ is in ${{\mathbf{CH}}}$. [$\diamond$]{}]{}
Hitting sets are sometimes called [*correct test sequences*]{}, as in [@heintzschnorr]. In particular, the deterministic construction of hitting sets is equivalent to the older problem of deterministic identity testing for polynomials given in the [*black-box*]{} model.
The main technical fact we’ll need now is the following:
(See [@koiran Thm. 7].) \[thm:hit\] Let $G\!:=\!(g_{i})$ be a ${{\mathbf{CH}}}$-algebraic number generator and let $Z(G,m)$ be the set of all roots of the polynomials $g_{i}$ for all $i \leq m$. If there is a polynomial $p$ such that $Z(G,p(kmt))$ is a hitting set for ${\mathrm{SPS}}(k,m,t)$ then the permanent of \[1\][$n\times n$ matrices cannot be computed by constant-free, division-free arithmetic circuits of size $n^{O(1)}$. [$\blacksquare$]{}]{}
The last result shows that the construction of explicit hitting sets of polynomial size for sums of products of sparse polynomials implies a lower bound for the permanent. Note that the conclusion of the theorem holds under a somewhat weaker hypothesis (see [@koiran] for details).
[**Proof of Theorem \[thm:koi\]:**]{} By assumption, there is a constant $c\!\geq\!1$ such that any $f\!\in\!{\mathrm{SPS}}(k,m,t)$ has at most $(1+kmt)^c$ integer roots that are powers of $p$. (The hypothesis of Theorem \[thm:koi\] is thus in fact stronger than the preceding statement.) The set $S_f$ therefore forms a polynomial-size hitting set for $f$. By Theorem \[thm:hit\], it just remains to check that the sequence of polynomials $(x_1-p^i)_{i \in {\mathbb{N}}}$ forms a ${{\mathbf{CH}}}$-algebraic number generator. We must therefore show that the following problem belongs to ${{\mathbf{CH}}}$: given two integers $i$ and $j$ in binary notation, compute the $j$-th bit of $p^i$. Note that this problem would be solvable in polynomial time if $i$ was given in unary notation (by performing the $i-1$ multiplications in the most naive way). To deal with the binary notation underlying our setting, we apply Theorem 3.10 of [@burgtau]: iterated multiplication of exponentially many integers can be done within the counting hierarchy. Here we have to multiply together exponentially many (in the binary size of $i$) copies of the same integer $p$. We note that Theorem 3.10 of [@burgtau] applies to a very wide class of integer sequences: the numbers to be multiplied must be computable in the counting hierarchy. In our case we only have to deal with a constant sequence (consisting of $i$ copies of $p$) so the elements of this sequence are computable in polynomial time (and actually in constant time since $p$ is constant). So we are done. [$\blacksquare$]{}
\[rem:stronger\] From our proof we also obtain that the set $S_f$ from the statement of Theorem \[thm:koi\] can be replaced by $S'_f:=\{e\; | \; f\!\left(p^e \right)\!=\!0\}$. Since we clearly have $S'_f\!\subseteq\!S_f$, we thus obtain a strengthening of Theorem \[thm:koi\]. [$\diamond$]{}
It is interesting to note that even a weakly exponential upper bound on the number of valuations would still suffice to prove new hardness results for the permanent: from the development of Sections 5 and 6 of [@koiran], and our development here, one has the following fall-back version of Theorem \[thm:koi\].
\[thm:closer\] Suppose that there is a prime $p$ with the following property: For all $k,m,t\!\in\!{\mathbb{N}}$ and $f\!\in\!{\mathrm{SPS}}(k,m,t)$, we have that the cardinality of\
$\displaystyle{S'_f:=\left\{e\!\in\!{\mathbb{N}}\; | \;
f\!\left(p^e\right)\!=\!0 \right\}}$\
is $2^{(kmt)^{o(1)}}$. Then the permanent of $n\times n$ matrices cannot be computed by polynomial size depth $4$ circuits using polynomial size integer constants. [$\blacksquare$]{}
While the conclusion is weaker than that of Theorem \[thm:koi\], the truth of the hypothesis of Theorem \[thm:closer\] nevertheless yields a hitherto unknown complexity lower bound for the permanent.
\[rem:best\] One can in turn weaken the hypothesis of Theorem \[thm:closer\] even further — by allowing dependence on the coefficients and degrees of the underlying $f_{i,j}$ in the definition of ${\mathrm{SPS}}(k,m,t)$ — and still obtain the same conclusion. This can be formalized via [@adelic Dfn. 2.6] and the development of [@koiran Sec. 3]. [$\diamond$]{}
Proving Theorem \[thm:upper\] {#sec:upper}
=============================
For the sake of disambiguation, let us first recall the following basic definition from linear algebra.
Fix any field $K$. We say that a matrix $E\!=\![E_{i,j}]\!\in\!K^{m\times n}$ is in [*reduced row echelon form*]{} if and only if the following conditions hold:
1. [The left-most nonzero entry of each row of $E$ is $1$, called the [*leading $1$*]{} of the row. ]{}
2. [Every leading $1$ is the unique nonzero element of its column.]{}
3. [\[1\][The index $j$ such that $E_{i,j}$ is a leading $1$ of row $i$ is a strictly increasing function of $i$. [$\diamond$]{}]{} ]{}
Note that in Condition (1), we allow a row to consist entirely of zeroes. Also, by Condition (3), all rows below a row of zeroes must also consist solely of zeroes. For example, the matrix \[.5\][$\begin{bmatrix} \text{\fbox{$1$}} & 0 & 0 & 3 & 0 & 2 \\
0 & 0 & \text{\fbox{$1$}} & 12 & 0 & -5 \\
0 & 0 & 0 & 0 & \text{\fbox{$1$}} & 7 \\
0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$]{} is in reduced row echelon form, and we have boxed the leading $1$s.
By [*Gauss-Jordan Elimination*]{} we mean the well-known classical algorithm that, given any matrix $M\!\in\!K^{m\times n}$, yields the factorization $UM\!=\!E$ with $U\!\in\!{\mathbb{G}\mathbb{L}_m}(K)$ and $E$ in reduced row echelon form (see, e.g., [@prasolov; @strang]). In what follows, we use $(\cdot)^\top$ to denote the operation of matrix tranpose.
\[dfn:gauss\] Given any Laurent polynomials $f_1,\ldots,f_r\!\in\!K\!\left[x^{\pm 1}_1,\ldots,x^{\pm 1}_n\right]$ with supports contained in a set $A\!=\!\{a_1,\ldots,a_t\}\!\subset\!{{\mathbb{Z}}^n}$ of cardinality $t$, [*applying Gauss-Jordan Elimination to*]{} $(f_1,\ldots,f_r)$ means the following: (a) we identify the row vector $(f_1,\ldots,f_r)$ with the vector-matrix product $(x^{a_1},\ldots,x^{a_t})
C$ where $C\!\in\!K^{t\times r}$ and the entries of $C$ are suitably chosen coefficients of the $f_i$, and (b) we replace $(f_1,\ldots,f_r)$ by $(g_1,\ldots,g_r)$ where $(g_1,\ldots,g_r)\!=\!(x^{a_1},
\ldots,x^{a_t})E$ and $E^\top$ is the reduced row echelon form of $C^\top$. [$\diamond$]{}
Note in particular that the ideals $\langle f_1,\ldots,f_r\rangle$ and $\langle g_1,\ldots,g_r\rangle$ are identical. As a concrete example, one can observe that applying Gauss-Jordan Elimination to the pair$(x^3-y-1,x^3-2y+2)$ means that one instead works with the pair $(x^3-4,-y+3)$.
We now proceed with the proof of Theorem \[thm:upper\]. In what follows, we set $A\!:=\!\bigcup_i A_i$, $t\!:=\!\#A$, and let $F\!:=\!(f_1,\ldots,f_n)$ be any polynomial system with $f_i\!\in\!{\mathbb{C}}_p\!\left[x^{\pm 1}_1,\ldots,x^{\pm 1}_n\right]$ and ${\mathrm{Supp}}(f_i)\!\subseteq\!A_i$ for all $i$.
Proving Assertions (0) and (1)
------------------------------
Assume $t\!\leq\!n+1$. If any $f_i$ is a single monomial term then $Z^*_{{\mathbb{C}}_p}(F)$ is empty. Also, if any $f_i$ is identically $0$ then $\#Z^*_{{\mathbb{C}}_p}(F)$ is infinite, so (by Proposition \[prop:fin\]) ${\mathrm{ord}}_p
\!\left(\#Z^*_{{\mathbb{C}}_p}(F)\right)\!=\!+\infty$. So we may assume that no $f_i$ is identically zero or a monomial term. Also, dividing all the $f_i$ by a suitable monomial term, we may assume that ${\mathbf{O}}\!\in\!A$.
Assertion (0) then follows immediately from Proposition \[prop:over\]. So we may now assume that $A$ does [*not*]{} lie in any $(n-1)$-flat (and $t\!=\!n+1$ in particular).
Our remaining case is then folkloric: by Gauss-Jordan Elimination (as in Definition \[dfn:gauss\], ordering so that the last monomial is $x^{{\mathbf{O}}}$), we can reduce to the case where each polynomial has $2$ or fewer terms, and ${\mathrm{Supp}}(f_i)\cap{\mathrm{Supp}}(f_j)\!=\!{\mathbf{O}}$ for all $i\!\neq\!j$. In particular, should Gauss-Jordan Elimination not yield the preceding form, then some $f_i$ is either identically zero or a monomial term, thus falling into one of our earlier cases. So assume $F$ is a binomial system with ${\mathrm{Supp}}(f_i)\cap{\mathrm{Supp}}(f_j)\!=\!{\mathbf{O}}$ for all $i\!\neq\!j$. Since no $n+1$ points of $A$ lie on a hyperplane, ${\mathrm{Newt}}(f_1),\ldots,{\mathrm{Newt}}(f_n)$ define $n$ linearly independent vectors in ${{\mathbb{R}}^n}$. The underlying tropical varieties are then hyperplanes intersecting transversally, and the number of valuations is thus clearly $1$.
So the upper bound from Assertion (1) is proved. The final equality follows immediately from the polynomial system $(x_1-1,\ldots,x_n-1)$. [$\blacksquare$]{}
Note that in our proof, Gauss-Jordan Elimination allowed us to replace any tropically non-generic $F$ by a new, [*tropically generic*]{} system with the same roots over ${\mathbb{C}}_p$. Recalling standard height bounds for linear equations (see, e.g., [@storjophd]), another consequence of our proof is that, when $t\!\leq\!n+1$, we can decide whether $\#{\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)$ is $0$, $1$, or $\infty$ in polynomial-time. [$\diamond$]{}
Proving Assertion (2)
---------------------
\[1\][Let us first see an example illustrating a trick underlying our proof.]{}
\[ex:central\] Consider, for any prime $p\!\neq\!2$, the polynomial system
$F:=(f_1,f_2):=\left\{\begin{matrix}
\mbox{} \ \ \ \ \ \ \ \ \ \ \ px^{21}_2 \ \ \ \ \ \ \ \ -px^{32}_1
\ \ \ \ \ +p \ \ \ \ \ \ \ \ \ \ +x^9_1 x^{10}_2\\
-(p+p^2)x^{21}_2+(p+p^3)x^{32}_1+ p+p^4 + (1+p)x^9_1 x^{10}_2
\end{matrix}\right.$. The tropical varieties ${\mathrm{Trop}}_p(f_1)$ and ${\mathrm{Trop}}_p(f_2)$ turn out to beidentical and equal to a polyhedral complex with exactly $3$$0$-dimensional cells and $6$ $1$-dimensional cells (a truncation of which is shown on the right).
How can we prove that ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)$ in fact has small cardinality?
While it is not hard to apply Bernstein’s Theorem (as in [@bernie]) to see that $F$ has only finitely many roots in $({{\mathbb{C}}^*}_p)^2$, there is a simpler approach to proving ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)$ is finite: First note that via Gauss-Jordan Elimination (and a suitable ordering of monomials), $F$ has the same roots in $({{\mathbb{C}}^*}_p)^2$ as\
$F^{(1,2)}:=\left(f^{(1,2)}_1,f^{(1,2)}_2\right):=
\left\{\begin{matrix}
\mbox{} x^{21}_2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{2+p^2+p^3}{p(p-1)}
+\frac{2+p+p^2}{p^2(p-1)}x^9_1 x^{10}_2\\
\ \ \ \ \ (p+p^3)x^{32}_1 + \frac{2+p+p^3}{p(p-1)} + \frac{2(1+p)}{p^2(p-1)}
x^9_1 x^{10}_2
\end{matrix}\right.$.
\
We then obtain that the tropical varieties ${\mathrm{Trop}}_p\!\left(f^{(1,2)}_1\right)$ and ${\mathrm{Trop}}_p\!\left(f^{(1,2)}_2\right)$ intersect (in a small interval) along a [*single*]{} $1$-dimensional cell, drawn more thickly, as shown to the right of the definition of $F^{(1,2)}$. (The intersection ${\mathrm{Trop}}(f_1)\cap{\mathrm{Trop}}(f_2)\cap
{\mathrm{Trop}}\!\left(f^{(1,2)}_1\right)\cap
{\mathrm{Trop}}\!\left(f^{(1,2)}_2\right)$ is drawn still more thickly.) From the definition of ${\mathrm{Trop}}_p(\cdot)$, it is not hard to check that the degenerately intersecting $1$-cells of ${\mathrm{Trop}}_p\!\left(f^{(1,2)}_1\right)$ and ${\mathrm{Trop}}_p\!\left(f^{(1,2)}_2\right)$ correspond to parallel lower edges of ${\mathrm{Newt}}_p\!\left(f^{(1,2)}_1\right)$ and ${\mathrm{Newt}}_p\!\left(f^{(1,2)}_2\right)$, which in turn correspond to the binomials $\frac{2+p^2+p^3}{p(p-1)}+\frac{2+p+p^2}{p^2(p-1)}x^9_1 x^{10}_2$ and $\frac{2+p+p^3}{p(p-1)} + \frac{2(1+p)}{p^2(p-1)}
x^9_1 x^{10}_2$. (Note that the intersecting $1$-cells of the ${\mathrm{Trop}}\!\left(f^{(1,2)}_i\right)$ are each perpendicular to the resulting Newton polytopes of the preceding binomials.)
So to contend with this remaining degenerate intersection, we simply apply Gauss-Jordan Elimination with the monomials ordered so that the aforementioned pair of binomials becomes a pair of monomials. More precisely, we obtain that $F$ has the same roots in $({{\mathbb{C}}^*}_p)^2$ as\
$F^{(3,4)}:=\left(f^{(3,4)}_1,f^{(3,4)}_2\right):=
\left\{\begin{matrix}
-\frac{2}{p(p-1)}x^{21}_2 \ + \frac{2+p+p^2}{p(p^2-1)} x^{32}_1
+1 \ \ \ \ \ \ \ \ \ \ \ \ \mbox{}\\
\mbox{} \frac{2-p+p^2}{p(p-1)}x^{21}_2 + \frac{2+p^2+p^3}{p^2-1} x^{32}_1
\ \ \ \ \ +x^9_1 x^{10}_2 \end{matrix} \right. .$\
${\mathrm{Trop}}_p\!\left(f^{(3,4)}_1\right)$ and ${\mathrm{Trop}}_p\!\left(f^{(3,4)}_2\right)$ then intersect transversally precisely within the overlapping $1$-cells of ${\mathrm{Trop}}_p(f_1)\cap{\mathrm{Trop}}_p(f_2)$ and ${\mathrm{Trop}}_p\!\left(f^{(1,2)}_1\right)\cap
{\mathrm{Trop}}_p\!\left(f^{(1,2)}_2\right)$. In particular, the intersection of the tropical varieties of the $f_i$, $f^{(1,2)}_i$, and $f^{(3,4)}_i$ consists of exactly $2$ points: $\left(\frac{1}{32},\frac{23}{320}\right)$ and $\left(\frac{11}{189},\frac{1}{21}\right)$. So, thanks to Kapranov’s Theorem, ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)$ in fact has cardinality at most $2$. [$\diamond$]{}
A simple observation used in our example above is the following consequence of the basic ideal/variety correspondence.
\[prop:int\] Given any $f_1,\ldots,f_r\!\in\!{\mathbb{C}}_p\!\left[x^{\pm 1}_1,\ldots,x^{\pm 1}_n
\right]$ and $F\!:=\!(f_1,\ldots,f_r)$, we have ${\mathrm{ord}}_p\left(Z^*_{{\mathbb{C}}_p}(F)\right)\!\subseteq\!\bigcap\limits_{f\in
\langle f_1,\ldots,f_r\rangle} {\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f)\right)$, where $\langle f_1,\ldots,f_r\rangle\!\subseteq\!{\mathbb{C}}_p\!\left[x^{\pm 1}_1,
\ldots,x^{\pm 1}_n\right]$ denotes the ideal generated by $f_1,\ldots,f_r$. [$\blacksquare$]{}
The reverse inclusion also holds, but is far less trivial to prove (see, e.g., [@macsturmf]). In particular, Example \[ex:central\] shows us that restricting the intersection to a [*finite*]{} set of generators can sometimes result in strict containment.
[**Proof of Assertion (2) of Theorem \[thm:upper\]:**]{} The case $n\!=\!1$ is immediate from Lemma \[lemma:newt\] and the example $(A_1,f)\!=\!(\{0,1,2\},(x_1-1)(x_1-2))$. So let us assume henceforth that $n\!\geq\!2$.
Recall from last section that $t\!=\!n+2$ and $A\!:=\!\bigcup_{\ell}A_\ell$. For any distinct $i,j\!\in\!\{1,\ldots,n+2\}$ let us then define $F^{(i,j)}$ by applying Gauss-Jordan Elimination, as in Definition \[dfn:gauss\], where we order monomials so that the last exponents are $a_i$ and $a_j$. In particular, $F$ and $F^{(i,j)}$ clearly have the same roots in $({{\mathbb{C}}^*}_p)^n$ for all distinct $i,j$. We will show that [*every*]{} $F^{(i,j)}$ can be assumed to be a trinomial system of a particular form.
First note that we may assume that ${\mathrm{Supp}}\!\left(f^{(n+1,n+2)}_\ell\right)\!\subseteq\!\{a_{r_\ell},a_{s_\ell},
a_{n+2}\}$ for all $\ell$, where $(r_\ell)_\ell$ is a strictly increasing sequence of integers in $\{1,\ldots,n\}$ satisfying $s_\ell\!\geq\!r_\ell\!\geq\!\ell$ for all $\ell$. This is because, similar to our last proof, we may assume that each $f^{(n+1,n+2)}_\ell$ has at least $2$ monomial terms, thus implying that $r_n\!\in\!\{n,n+1\}$. In particular, no $f^{(n+1,n+2)}_\ell$ can have $4$ or more terms, by the positioning of the leading $1$s in reduced row echelon form.
By dividing by a suitable monomial term, we may assume that $a_{n+2}\!=\!{\mathbf{O}}$. Also, by Lemma \[lemma:mono\], we may assume that $a_\ell\!\in\!{\mathbb{Z}}^{\ell}\times \{0\}^{n-\ell}$ for all $\ell\!\in\!\{1,\ldots,
n\}$. (Our general position assumption on $A$ also implies that the $\ell{^{\text{\underline{th}}}}$ coordinate of $a_\ell$ is nonzero.) Now, should $f^{(n+1,n+2)}_n$ have exactly $2$ monomial terms, then $f^{(n+1,n+2)}_n$ must be of one of the following forms: (a) $x^{a_n}+\alpha_n x^{a_{n+1}}$, (b) $x^{a_n}+\alpha_n x^{a_{n+2}}$, or (c) $x^{a_{n+1}}+\alpha_n x^{a_{n+2}}$, for some $\alpha_n\!\in\!{{\mathbb{C}}^*}_p$. In Case (c), we could then replace all occurences of $x^{a_{n+1}}$ in $f^{(n+1,n+2)}_1,\ldots,
f^{(n+1,n+2)}_{n-1}$ by a nonzero multiple of $x^{a_{n+2}}$. We would thus reduce to the setting of Assertion (1), in which case, the maximal finite number of valuations would be $1$. In Cases (a) and (b), we obtain either that some $x_i$ vanishes, or that ${\mathrm{ord}}_p x_n$ is a linear function of ${\mathrm{ord}}_p x_1,\ldots,
{\mathrm{ord}}_p x_{n-1}$. So we could then reduce to a case one dimension lower.
So we may assume that ${\mathrm{Supp}}\!\left(f^{(n+1,n+2)}_n\right)\!=\!\{a_n,a_{n+1},a_{n+2}\}$, which in turn forces $a_\ell\!\in\!{\mathrm{Supp}}\!\left(f^{(n+1,n+2)}_\ell\right)$ $\subseteq\!\{a_\ell,a_{n+1},a_{n+2}\}$ for all $\ell\!\in\!\{1,\ldots,n-1\}$. Moreover, by repeating the arguments of Cases (a) and (b) above, we may in fact assume ${\mathrm{Supp}}\!\left(f^{(n+1,n+2)}_\ell\right)\!=\!\{a_\ell,a_{n+1},a_{n+2}\}$ for all $\ell\!\in\!\{1,\ldots,n\}$.
Permuting indices, we can then repeat the last $3$ paragraphs and assume further that, for any distinct $i,j\!\in\!\{1,\ldots,n+2\}$, we have
($\star$) ${\mathrm{Supp}}\!\left(f^{(i,j)}_\ell\right)\!=\!\{a_{k_\ell},a_i,a_j\}$ for all $\ell\!\in\!\{1,\ldots,n\}$, where $\{k_\ell\}_\ell\!=\!A\!\setminus
\!\{i,j\}$.
Let us now fix $(i,j)$ and set $G\!=\!(g_1,\ldots,g_n)\!:=\!F^{(i,j)}$. Thanks to ($\star$) and Lemma \[lemma:vert\] (mimicking the proof of Lemma \[lemma:maybetrivial\]), the ${\mathrm{Trop}}_p(g_i)$ each contain a half-plane parallel to a common hyperplane. We then obtain a finite partition of ${\mathbb{R}}^n$ into half-open slabs of a form linearly isomorphic (over ${\mathbb{Q}}$) to $(-\infty,u_1)\times {\mathbb{R}}^{n-1}$, $[u_\ell,u_{\ell+1})\times {\mathbb{R}}^{n-1}$ for $\ell\!\in\!\{1,\ldots,m_{i,j}-1\}$, or $[u_{m_{i,j}},+\infty)\times
{\mathbb{R}}^{n-1}$, with $m_{i,j}\!\leq\!n$. More precisely, the boundaries of the cells of our partition are hyperplanes of the form $H^{(i,j)}_\ell\!:=\!\left\{v\!\in\!{{\mathbb{R}}^n}\; \left|
\; (a_i-a_j)\cdot v\!=\!{\mathrm{ord}}_p\!
\left( \gamma^{(i,j)}_\ell\right)\right. \right\}$ for $\ell\!\in\!\{1,\ldots,m_{i,j}\}$, where $\gamma^{(i,j)}_\ell$ is a ratio of coefficients of $f^{(i,j)}_\ell$.
In particular, Lemma \[lemma:vert\] and our genericity hypothesis tell us that, within any slab, any non-empty intersection of ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_1)\right),\ldots,
{\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(f_n)\right)$ must be a [*transversal*]{} intersection of $n$ hyperplanes, unless it includes the intersection of two or more $H^{(i,j)}_{k}$. So if $G$ is tropically generic, we have by Proposition \[prop:int\] that $\#{\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)
\right)\!\leq\!n+1$.
Otherwise, any non-transversal intersection must occur within an intersection of slab boundaries $H^{(i,j)}_k$. So to finish this case, consider $n-1$ more distinct pairs $(i_2,j_2),\ldots,
(i_n,j_n)$, i.e., $i_\ell\!\neq\!j_\ell$ for all $\ell$ and $\#\{i_\ell,j_\ell,i_{\ell'},j_{\ell'}\}\!\leq\!3$ for all $\ell\!\neq\!\ell'$.
Just as for $G$, the genericity of the exponent set $A$ implies that any non-transversal intersection for ${\mathrm{Trop}}_p\!\left(f^{(i_\ell,j_\ell)}_1\right),
\ldots,{\mathrm{Trop}}_p\!\left(f^{(i_\ell,j_\ell)}_n\right)$ must occur within the intersection of at least two coincident slab boundaries $H^{(i_\ell,j_\ell)}_k$. In particular, we may assume that none of $F^{(i_2,j_2)},\ldots,F^{(i_n,j_n)}$ are tropically generic (for $\#{\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)\!\leq\!n+1$ otherwise).
By our assumption on the genericity of the exponent set $A$, we have that $H^{(i,j)}_{\ell_1},H^{(i_2,j_2)}_{\ell_2},
\ldots,$ $H^{(i_n,j_n)}_{\ell_n}$ intersect transversally, for any choice of $n$-tuples $(\ell_1,\ldots,\ell_n)$. In particular, we have embedded the non-transversal intersections of ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)$ into a (finite) intersection of $m_{i,j}\prod^n_{\ell=2} m_{i_\ell,j_\ell}$ many tropical varieties. In particular, to count the non-transversal intersections, we may assume $m_{i,j},m_{i_2,j_2},\ldots,m_{i_n,j_n}\!\leq\!{\left\lfloor\frac{n}{2}\right\rfloor}$.
From our earlier observations on slab decomposition, there can be at most $n$ intersections occuring away from an intersection of slab boundaries (since we are assuming $G$ and the $F^{(i_\ell,j_\ell)}$ all fail to be tropically generic). The number of distinct points of ${\mathrm{ord}}_p\!\left(Z^*_{{\mathbb{C}}_p}(F)\right)$ lying in intersections of the form $H^{(i,j)}_{\ell_1}\cap H^{(i_2,j_2)}_{\ell_2}
\cap \cdots \cap H^{(i_n,j_n)}_{\ell_n}$ is no greater than ${\left\lfloor\frac{n}{2}\right\rfloor}^n$. So our upper bound is proved.
That ${{\mathcal{V}}}_p\!\left(\{{\mathbf{O}},2e_1,e_1+e_2\},\{{\mathbf{O}},2e_1,e_2+e_3\},\ldots,
\{{\mathbf{O}},2e_1,e_{n-1}+e_n\},\{{\mathbf{O}},2e_1,e_n\}\right)\!\geq\!n+1$ follows directly from [@adelic Thm. 1.6]. To be more precise, the polynomial system\
\[1\][ $\displaystyle{\left(x_1x_2-
p\left(1+\frac{x^2_1}{p}\right),
x_2x_3-\left(1+p x^2_1\right),
x_3x_4-\left(1+p^3x^2_1\right),
\ldots,x_{n-1}x_n-\left(1+p^{2n-5}x^2_1\right),
x_n-\left(1+p^{2n-3}x^2_1\right)\right)}$]{} has exactly $n+1$ valuation vectors for its roots over ${\mathbb{C}}_p$, and tropical genericity follows directly from [@adelic Lemma 3.7]. The reverse inequality then follows from our earlier observations on slab decomposition. In particular, via our earlier reductions, Assertions (0) and (1) easily \[1\][imply that any $F$ with smaller support has no more than $n$ valuation vectors for its roots. [$\blacksquare$]{}]{}
We are currently unaware of any examples where ${{\overline{{{\mathcal{V}}}}}}_p(A_1,\ldots,A_n)$ is larger than $n+1$. In any event, our proof reveals various cases where the number of valuation vectors is at most $n+1$. [$\diamond$]{}
Proving Theorem \[thm:hard\] {#sec:hard}
============================
By Theorem \[thm:koi\] and Proposition \[prop:red\] it is enough to show that, for $n\!:=\!km+1$ and $A_1,\ldots,A_n$ the supports of the polynomial system $F$ from the proposition, we have ${{\mathcal{V}}}_p(A_1,\ldots,A_n)\!=\!(kmt)^{O(1)}$. By Lemma \[lemma:maybetrivial\] we are done. [$\blacksquare$]{}
Proving Theorem \[thm:mult\] {#sec:mult}
============================
First note that we can divide our equations by a suitable monomial term so that ${\mathbf{O}}\!\in\!A$.
The case where $A$ has cardinality $n+1$ can be easily handled just as in the proof of Theorem \[thm:upper\]: $F$ can be reduced to a binomial system via Gauss-Jordan Elimination, and then by Lemma \[lemma:mono\] we can easily reduce to a [*triangular*]{} binomial system. In particular, all the roots of $F$ in $({K^*})^n$ are non-degenerate and thus have multiplicity $1$, so the sharpness of the bound is immediate as well.
So let us now assume that $A$ has cardinality $n+2$. By Gauss-Jordan Elimination and a monomial change of variables again, we may assume that $F$ is of the form\
$\left(x^{a_1} - \alpha_1 - x^{a_{n+1}}/c, \ldots,
x^{a_n} - \alpha_n - x^{a_{n+1}}/c\right)$\
for some $c\!\in\!{K^*}$.
Consider now the matrix $\hat{A}$ obtained by appending a rows of $1$s to the matrix with columns ${\mathbf{O}},a_1,\ldots,a_{n+1}$. By construction, $\hat{A}$ has right-kernel generated by a single vector $b\!=\!(b_0,\ldots,b_{n+1})\!\in\!{\mathbb{Z}}^{n+2}$ with [*no*]{} zero coordinates. So the identity $1^{b_0}\left(x^{a_1}\right)^{b_1}\cdots\left(
x^{a_{n+1}}\right)^{b_{n+1}}\!=\!1$ clearly holds for any $x\!\in\!({K^*})^n$. Letting $u\!:=\!x^{a_{n+1}}$ we then clearly obtain a bijection between the roots of $F$ in $({K^*})^n$ and the roots of\
$g(u)\!:=\!u^{b_{n+1}}\left(\prod^n_{i=1}(\alpha_i +u)^{b_i}
\right)-C$\
where $C\!:=\!c^{b_1+\cdots+b_n}$. Furthermore, intersection multiplicity is preserved under this univariate reduction since each $x_i$ is a radical of a linear function of a root of $g$. We thus need only determine the maximum intersection multiplicity of a root of $g$ in ${K^*}$.
Since the multiplicity of a root $\zeta$ over a field of characteristic $0$ is characterized by the derivative of least order not vanishing at $\zeta$, let us suppose, to derive a contradiction, that $f(\zeta)\!=\!f'(\zeta)\!=\cdots=
\!f^{(n+1)}(\zeta)\!=\!0$, i.e., $\zeta$ is a root of multiplicity $\geq\!n+2$. An elementary calculation then reveals that we must have $$\begin{aligned}
& \frac{b'_1}{\alpha'_{1}+\zeta}+\cdots+\frac{b'_{m+1}}{\alpha'_{m+1}+\zeta}
& = 0\\
& \vdots & \\
& \frac{b'_1}{(\alpha'_{1}+\zeta)^{n+1}}+\cdots+\frac{b'_{m+1}}
{(\alpha'_{m+1}+\zeta)^{n+1}} & = 0 \end{aligned}$$
where $m\!\leq\!n$, the $\alpha'_i$ are distinct and comprise all the $\alpha_i$, $\alpha'_{m+1}\!=\!0$, $b'_i\!:=\!\sum
\limits_{\alpha_{j}=\alpha'_i} b_{j}$, $b'_{m+1}\!:=\!b_{n+1}$, we set $\alpha'_{m+1}\!:=\!0$, and $\zeta\!\not\in\!\{-\alpha_i\}$. In other words, $[b'_1,\ldots,b'_{m+1}]^\top$ is a right-null vector of a Vandermonde matrix with non-vanishing determinant. Since $[b'_1,\ldots,b'_{m+1}]$ has nonzero coordinates, we thus obtain a contradiction. So our upper bound is proved.
To prove that our final bound is tight, let $\zeta_1,\ldots,\zeta_{n+1}$ denote the (distinct) $(n+1){^{\text{\underline{st}}}}$ roots of unity in $K$ and set $g(u)\!:=\!\displaystyle{u\left(\prod^n_{i=1} (u+\zeta_{n+1}-\zeta_i)
\right)+1}$. Since $g(u-\zeta_{n+1})\!=\!u^{n+1}$, it is clear that $g$ has $-\zeta_{n+1}$ as a root of multiplicity $n+1$. Furthermore, $g$ is nothing more than the univariate reduction argument of our proof applied to the system $$\begin{aligned}
\theta x_1 & = & \zeta_{n+1}-\zeta_i +\frac{1}{x_1\cdots x_n} \\
& \vdots & \\
\theta x_n & = & \zeta_{n+1}-\zeta_n +\frac{1}{x_1\cdots x_n} \end{aligned}$$
where $\theta$ is any $n{^{\text{\underline{th}}}}$ root of $-1$. [$\blacksquare$]{}
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Henry Cohn for his wonderful hospitality at Microsoft Research New England (where Rojas presented a preliminary version of Lemma \[lemma:maybetrivial\] on July 30, 2012), and Kiran Kedlaya and Daqing Wan for useful $p$-adic discussions. We also thank Martín Avendano, Bruno Grenet, and Korben Rusek for useful discussions on an earlier version of Lemma \[lemma:vert\], and Jeff Lagarias for insightful comments on an earlier version of this paper.
Most importantly, however, we would like to congratulate Mike Shub on his 70${^{\text{\underline{th}}}}$ birthday: he has truly blessed us with his friendship and his beautiful mathematics. We hope this paper will serve as a small but nice gift for Mike.
[A]{}
Artin, Emil, [*Algebraic Numbers and Algebraic Functions,*]{} Gordon and Breach, New York, 1967.
Bernshtein, David N., [*“The Number of Roots of a System of Equations,"*]{} Functional Analysis and its Applications (translated from Russian), Vol. 9, No. 2, (1975), pp. 183–185.
Blum, Lenore; Cucker, Felipe; Shub, Mike; and Smale, Steve, [*Complexity and Real Computation,*]{} Springer-Verlag, 1998.
Borodin, Alan and Cook, Steve, [*“On the number of additions to compute specific polynomials,”*]{} SIAM Journal on Computing, 5(1):146–157, 1976.
Brauer, Alfred, [*“On addition chains,”*]{} Bull. Amer. Math. Soc. 45, (1939), pp. 736–739.
Bürgisser, Peter, [*“Cook’s versus Valiant’s Hypothesis,”*]{} Theor. Comp. Sci., 235:71–88, 2000.
, [*“On defining integers and proving arithmetic circuit lower bounds,”*]{} Computational Complexity, 18:81–103, 2009.
Bürgisser, Peter; Landsberg, J. M.; Manivel, Laurent; and Weyman, Jerzy, [*“An Overview of Mathematical Issues Arising in the Geometric Complexity Theory Approach to $\mathbf{VP}\neq\mathbf{VNP}$,”*]{} SIAM J. Comput. [**40**]{}, pp. 1179-1209, 2011.
Cheng, Qi, [*“Straight Line Programs and Torsion Points on Elliptic Curves,”*]{} Computational Complexity, Vol. 12, no. 3–4 (sept. 2004), pp. 150–161.
Einsiedler, Manfred; Kapranov, Mikhail; and Lind, Douglas, [*“Non-archimedean amoebas and tropical varieties,”*]{} Journal für die reine und angewandte Mathematik (Crelles Journal), Vol. 2006, no. 601, pp. 139–157, December 2006.
Fulton, William, [*Intersection Theory,*]{} 2${^{\text{\underline{nd}}}}$ ed., Ergebnisse der Mathematik und ihrer Grenzgebiete 3, [**2**]{}, Springer-Verlag, 2008.
von zur Gathen, Joachim and Gerhard, Jürgen, [*“Modern Computer Algebra,”*]{} 2nd ed., Cambridge University Press, 2003.
Gouvêa, Fernando Q., [*$p$-adic Numbers,*]{} Universitext, 2nd ed., Springer-Verlag, 2003.
Heintz, Joos and Schnorr, C.-P., [*“Testing polynomials which are easy to compute,”*]{} in Logic and Algorithmic (international symposium in honor of Ernst Specker), pp. 237–254, monograph no. 30 of L’Enseignement Mathématique, 1982.
Hermite, Charles, [*“Sur le nombres des racines d’une équation algébrique comprisé entre des limites donn’es,"*]{} J.Reine Angew. Math. 52 (1856) 39–51; also: Oeuvres, Vol. I (Gauthier-Villars, Paris, 1905) pp. 397–414; English translation: P. C.Parks, Internat. J. Control 26 (1977), pp. 183–195.
Katok, Svetlana, [*$p$-adic Analysis Compared with Real,*]{} Student Mathematical Library, vol. 37, American Mathematical Society, 2007.
Koiran, Pascal, [*“Shallow Circuits with High-Powered Inputs,”*]{} in Proceedings of Innovations in Computer Science (ICS 2011, Jan. 6–9, 2011, Beijing China), Tsinghua University Press, Beijing.
Lipton, Richard, [*“Straight-line complexity and integer factorization,”*]{} Algorithmic number theory (Ithaca, NY, 1994), pp. 71–79, Lecture Notes in Comput. Sci., 877, Springer, Berlin, 1994.
Maclagan, Diane and Sturmfels, Bernd, [*Introduction to Tropical Geometry,*]{} in progress.
Mazur, Barry, [*“Rational Isogenies of Prime Degree,”*]{} Invent. Math., 44, 1978.
de Melo, W. and Svaiter, B. F., [*“The cost of computing integers,”*]{} Proc. Amer. Math. Soc. (1996), pp. 1377–1378.
Merel, Loic, [*“Bounds for the torsion of elliptic curves over number fields,”*]{} Invent. Math., 124(1–3):437–449, 1996.
T. de Araujo Moreira, Gustavo, [*“On asymptotic estimates for arithmetic cost functions,”*]{} Proccedings of the American Mathematical Society, Vol. 125, no. 2, Feb. 1997, pp. 347–353.
Parent, Philippe, [*“Effective Bounds for the torsion of elliptic curves over number fields,”*]{} J. Reine Angew. Math, 508:65–116, 1999.
Phillipson, Kaitlyn and Rojas, J. Maurice, [*“Fewnomial Systems with Many Roots, and an Adelic Tau Conjecture,”*]{} in proceedings of Bellairs workshop on tropical and non-Archimedean geometry (May 6–13, 2011, Barbados), Contemporary Mathematics, vol. 605, pp. 45–71, AMS Press, to appear.
Prasolov, V. V., [*Problems and Theorems in Linear Algebra,*]{} translations of mathematical monographs, vol. 134, AMS Press, 2004.
Robert, Alain M., [*A course in $p$-adic analysis,*]{} Graduate Texts in Mathematics, 198, Springer-Verlag, New York, 2000.
Schikhof, W. H., [*Ultrametric Calculus, An Introduction to $p$-adic Analysis,*]{} Cambridge Studies in Adv. Math. 4, Cambridge Univ. Press, 1984.
Serre, Jean-Pierre, [*Local fields,*]{} Graduate Texts in Mathematics, 67, Springer-Verlag, New York-Berlin, 1979.
Shub, Mike, [*“Some Remarks on Bézout’s Theorem and Complexity Theory,”*]{} From Topology to Computation: Proceedings of the Smalefest (Berkeley, 1990), pp. 443–455, Springer-Verlag, 1993.
Smale, Steve, [*“Mathematical Problems for the Next Century,”*]{} Math. Intelligencer 20 (1998), no. 2, pp. 7–15.
, [*“Mathematical Problems for the Next Century,”*]{} Mathematics: Frontiers and Perspectives, pp. 271–294, Amer. Math. Soc., Providence, RI, 2000.
Smirnov, A. L., [*“Torus Schemes Over a Discrete Valuation Ring,”*]{} St. Petersburg Math. J. [**8**]{} (1997), no. 4, pp. 651–659.
Storjohann, Arne, [*“Algorithms for matrix canonical forms,”*]{} doctoral dissertation, Swiss Federal Institute of Technology, Zurich, 2000.
Strang, Gilbert, [*Introduction to Linear Algebra,*]{} 4${^{\text{\underline{th}}}}$ edition, Wellesley-Cambridge Press, 2009.
Valiant, Leslie G., [*“The complexity of computing the permanent,”*]{} Theoret. Comp. Sci., 8:189–201, 1979.
Weiss, Edwin, *Algebraic Number Theory*, McGraw-Hill, 1963.
Ziegler, Gunter M., [*Lectures on Polytopes*]{}, Graduate Texts in Mathematics, Springer Verlag, 1995.
[^1]: Lipton’s main result from [@lipton] is in fact stronger, allowing for rational roots and primes with a mildly differing number of digits.
[^2]: i.e., smallest convex set containing...
|
---
author:
- 'Haeran Cho [^1] and Piotr Fryzlewicz [^2]'
title: Multiscale and multilevel technique for consistent segmentation of nonstationary time series
---
**Abstract**
In this paper, we propose a fast, well-performing, and consistent method for segmenting a piecewise-stationary, linear time series with an unknown number of breakpoints. The time series model we use is the nonparametric Locally Stationary Wavelet model, in which a complete description of the piecewise-stationary second-order structure is provided by wavelet periodograms computed at multiple scales and locations. The initial stage of our method is a new binary segmentation procedure, with a theoretically justified and rapidly computable test criterion that detects breakpoints in wavelet periodograms separately at each scale. This is followed by within-scale and across-scales post-processing steps, leading to consistent estimation of the number and locations of breakpoints in the second-order structure of the original process. An extensive simulation study demonstrates good performance of our method.
**keywords:** binary segmentation, breakpoint detection, locally stationary wavelet model, piecewise stationarity, post-processing, wavelet periodogram.
Introduction {#sec:introduction}
============
A stationarity assumption is appealing when analysing short time series. But, it is often unrealistic, for example when observing time series evolving in naturally nonstationary environments. One such example can be found in econometrics, where price processes are considered to have time-varying variance in response to events taking place in the market; @mikosch1999, @leipus2000, and @starica2005, among others, argued in favour of nonstationary modelling of financial returns. For example, given the explosion of market volatility during the recent financial crisis, it is unlikely that the same stationary time series model can accurately describe the evolution of market prices before and during the crisis.
Piecewise stationarity is a well studied and arguably the simplest form of departure from stationarity, and one task of interest is to detect breakpoints in the dependence structure. Breakpoint detection has received considerable attention and the methods that have been developed can be broadly categorized into retrospective (a posteriori) methods and on-line methods. In the interest of space, we do not review on-line breakpoint detection approaches here but refer the reader to @lai2001.
The “a posteriori” approach takes into account the entire set of observations at once and detects breakpoints which occurred in the past. Our interest here lies in the “a posteriori” segmentation category, and we propose a retrospective segmentation procedure that achieves consistency in identifying multiple breakpoints for a class of nonstationary processes. (Note that we use the term “segmentation” interchangeably with “multiple breakpoint detection”.)
Early segmentation literature was mostly devoted to testing the existence of a single breakpoint in the mean or variance of independent observations (@chernoff1964, @sen1975, @hawkins1977, @hsu1977, @worsley1986). When the presence of more than one breakpoint is suspected, an algorithm for detecting multiple breakpoints is needed. In @vostrikova1981, a “binary segmentation” procedure was introduced, a computationally efficient multilevel breakpoint detection procedure that recursively locates and tests for multiple breakpoints, producing consistent breakpoint estimators for a class of random processes with piecewise constant means. However, the critical values of the tests at each stage are difficult to compute in practice due to stochasticity in previously selected breakpoints. @venkatraman1993 employed the same procedure to find multiple breakpoints in the mean of independent and normally distributed variables and showed the consistency of the detected breakpoints with the tests depending on the sample size only, and thus are easier to compute. The binary segmentation procedure was also adopted to detect multiple shifts in the variance of independent observations (@inclan1994, @chen1997).
Various multiple breakpoint detection methods have been proposed for time series of dependent observations. In @lavielle2000, least squares estimators of breakpoint locations were developed for linear processes with changing mean, extending the work of @bai1998. @adak1998 and @ombao2001 proposed methods that divided the time series into dyadic blocks and chose the best segmentation according to suitably tailored criteria. @whitcher2000 [@whitcher2002] and @gabbanini2004 suggested segmenting long memory processes by applying the iterative cumulative sum of squares (ICSS) algorithm (proposed in @inclan1994) to discrete wavelet coefficients of the process which were approximately Gaussian and decorrelated. Davis, Lee, and Rodriguez-Yam (2006) developed the Auto-PARM procedure which found the optimal segmentation of piecewise stationary AR processes via the minimum description length principle, later extended to the segmentation of non-linear processes in @davis2008. In @lavielle2005, a breakpoint detection method was developed for weakly or strongly dependent processes with time-varying volatility that minimises a penalised contrast function based on a Gaussian likelihood. @andreou2002 studied a heuristic segmentation procedure for the GARCH model with changing parameters, based on the work of @lavielle2000.
The aim of our work is to propose a well-performing, theoretically tractable, and fast procedure for detecting breakpoints in the second-order structure of a piecewise stationary time series that is linear but otherwise does not follow any particular parametric model. The nonparametric model we use for this purpose is the Locally Stationary Wavelet (LSW) model first proposed by Nason, von Sachs and Kroisandt (2000) and later studied by @piotr2006a and @bellegem2008. Detailed justification of our model choice is given in Section \[sec:lsw\]. In the LSW model, the piecewise constant second-order structure of the process is completely described by local wavelet periodograms at multiple scales, and it is those basic statistics that we use as a basis of our segmentation procedure.
To achieve the multiple breakpoint detection, we propose a binary segmentation method that is applied to wavelet periodograms separately at each scale, and followed by a within-scale and across-scales post-processing procedure to obtain consistent estimators of breakpoints in the second-order structure of the process. We note that wavelet periodograms follow a multiplicative statistical model, but our binary segmentation procedure is different from previously proposed binary segmentation methods for multiplicative models (@inclan1994, @chen1997) in that it allows for correlated data, which is essential when working with wavelet periodograms. We note that Kouamo, Moulines, and Roueff (2010) proposed a CUSUM-type test for detecting a [*single*]{} change in the wavelet variance at one or several scales that also permits correlation in the data. We emphasise other unique ingredients of our breakpoint detection procedure which lead to its good performance and consistency in probability: the theoretical derivation of our test criterion (which only depends on the length of the time series and is thus fast to compute); the novel across-scales post-processing step, essential in combining the results of the binary segmentation procedures performed for each wavelet periodogram scale separately. We note that our method can simultaneously be termed “multiscale” and “multilevel”, as the basic time series model used for our purpose is wavelet-based and thus is a “multiscale” model, and the core methodology to segment each scale of the wavelet periodogram in the model is based on binary segmentation and is thus a “multilevel” procedure.
The paper is organised as follows. Section \[sec:lsw\] explains the LSW model and justifies its choice. Our breakpoint detection methodology (together with the post-processing steps) is introduced in Section \[sec:second\], where we also demonstrate its theoretical consistency in estimating the total number and locations of breakpoints. In Section \[sec:simulation\], we describe the outcome of an extensive simulation study that demonstrates the good performance of our method. In Section \[sec:real\], we apply our technique to the segmentation of the Dow Jones index and this results in the discovery of two breakpoints: one coinciding with the initial period of the recent financial crisis, and the other coinciding with the collapse of Lehman Brothers, a major financial services firm. The proofs of our theoretical results are provided in the Appendix. Software (an R script) implementing our methodology is available from: <http://personal.lse.ac.uk/choh1/msml_technique.html>.
Locally stationary wavelet time series {#sec:lsw}
======================================
In this section, we define the Locally Stationary Wavelet (LSW) time series model (noting that our definition is a slight modification of that of @piotr2006a), describe its properties, and justify its choice as an attractive framework for developing our time series segmentation methodology.
\[def:lsw\] A triangular stochastic array $\{X_{t,
T}\}_{t=0}^{T-1}$ for $T=1, 2, \ldots,$ is in a class of Locally Stationary Wavelet (LSW) processes if there exists a mean-square representation $$\begin{aligned}
{X_{t, T}}={\sum_{i=-\infty}^{-1}}{\sum_{k=-\infty}^{\infty}}W_i(k/T)\psi_{i, t-k}\xi_{i, k}\label{pswm}\end{aligned}$$ with $i\in\{-1, -2, \ldots\}$ and $k \in \mathbb{Z}$ as scale and location parameters, respectively, the ${\mathbf{\psi}}_{i}=(\psi_{i, 0}, \ldots,
\psi_{i, \mathcal{L}_i-1})$ are discrete, real-valued, compactly supported, non-decimated wavelet vectors with support lengths $\mathcal{L}_i = O(2^{-i})$, and the $\xi_{i, k}$ are zero-mean, orthonormal, identically distributed random variables. For each $i \le -1$, $W_i(z):[0, 1] \rightarrow \mathcal{R}$ is a real-valued, piecewise constant function with a finite (but unknown) number of jumps. If the $L_i$ denote the total magnitude of jumps in $W_i^2(z)$, the variability of functions $W_i(z)$ is controlled so that
- ${\sum_{i=-\infty}^{-1}}W_i^2(z) < \infty$ uniformly in $z$,
- $\sum_{i=-I}^{-1} 2^{-i}L_i = O(\log T)$ where $I=\log_2T$.
The reader unfamiliar with basic concepts in wavelet analysis is referred to the monograph by @vidakovic1999. By way of example, we recall the simplest discrete, non-decimated wavelet system: the Haar wavelets. Here $$\psi_{i,k} = 2^{i/2} \mathbf{I}_{\{0, \ldots, 2^{-j-1}-1\}}(k) - 2^{i/2}\mathbf{I}_{\{2^{-j-1}, \ldots, 2^{-j}-1\}}(k),$$ for $i = -1, -2, \ldots$, $k \in \mathbb{Z}$, where $\mathbf{I}_A(k)$ is 1 if $k\in A$ and 0 otherwise. We note that discrete non-decimated wavelets $\psi_{i, k}$ can be shifted to any location defined by the finest-scale wavelets, and not just to ‘dyadic’ locations (i.e. those with shifts being multiples of $2^{-i}$) as in the discrete wavelet transform. Therefore, discrete non-decimated wavelets are no longer an orthogonal, but an overcomplete collection of shifted vectors (@nason2000).
Throughout, the $\xi_{i, k}$ are assumed to follow the normal distribution; extensions to non-Gaussianity are possible but technically difficult. Comparing the above definition with the Cramér’s representation of stationary processes, $W_i(k/T)$ is a (scale- and location-dependent) transfer function, the wavelet vectors ${\mathbf{\psi}}_{i}$ are analogous to the Fourier exponentials, and the innovations $\xi_{i,k}$ correspond to the orthonormal increment process. Small negative values of the scale parameter $i$ denote “fine” scales where the wavelet vectors are the most localised and oscillatory; large negative values denote “coarser” scales with longer, less oscillatory wavelet vectors. By assuming that $W_i(z)$ is piecewise constant, we are able to model processes with a piecewise constant second-order structure where, between any two breakpoints in $W_i(z)$, the second-order structure remains constant. The Evolutionary Wavelet Spectrum (EWS) is defined as $S_i(z)=W_i(z)^2$, and it is in a one-to-one correspondence with the time-dependent autocovariance function of the process $c(z, \tau):=\lim_{T\to\infty} {\mbox{cov}}(X_{[zT], T}, X_{[zT]+\tau, T})$ (@nason2000). We note that $W_i(z)$ is a valid transfer function; the variance of the resulting time series ${X_{t, T}}$ is uniformly bounded over $t$, and the one-to-one correspondence between the autocovariance function and $S_i(z)$ leads to model identifiability. Our objective is to develop a consistent method for detecting breakpoints in the EWS, and consequently to provide a segmentation of the original time series. The following technical assumption is placed on the breakpoints present in the EWS.
\[assum:one\] The set of locations $z$ where (possibly infinitely many) functions $S_i(z)$ contain a jump, is finite; with $\mathcal{B}=\{z;\quad \exists\, i \ \lim_{u\rightarrow
z-}S_i(u) \neq \lim_{u\rightarrow z+}S_i(u)\}$, then $B=|\mathcal{B}|<\infty$.
We further define the wavelet periodogram of the LSW time series.
Let ${X_{t, T}}$ be an LSW process as in (\[pswm\]). The triangular stochastic array $$\begin{aligned}
{I^{(i)}_{t, T}}=\left|\sum_s X_{s, T}\psi_{i, s-t} \right|^2 \label{def:wp}\end{aligned}$$ is called the wavelet periodogram of ${X_{t, T}}$ at scale i.
With the autocorrelation wavelets $\Psi_i(\tau):=\sum_k\psi_{i, k}\psi_{i, k-\tau}$, the wavelet operator matrix is defined as ${\mathbf{A}}=\left(A_{i, k}\right)_{i, k<0}$ with $A_{i, k}:=\langle \Psi_i, \Psi_k\rangle=\sum_\tau\Psi_i(\tau)\Psi_k(\tau)$. @piotr2006a showed that the expectation of a wavelet periodogram ${\mathbb{E}}{I^{(i)}_{t, T}}$ is “close” (in the sense of the integrated squared bias converging to zero) to the function ${\beta_i(z)}=\sum_{j=-\infty}^{-1}S_j(z)A_{i, j}$, a piecewise constant function with at most $B$ jumps, all of which occur in the set $\mathcal{B}$. Thus, there exists a one-to-one correspondence between EWS, the time-dependent autocovariance function, and the function ${\beta_i(z)}$ (being the asymptotic expectation of the wavelet periodogram). Every breakpoint in the autocovariance structure then results in a breakpoint in at least one of the $\beta_i(z)$’s, and is thus detectable, at least with $T\to\infty$, by analysing the wavelet periodogram sequences. We note that ${\mathbb{E}}{I^{(i)}_{t, T}}$ itself is piecewise constant by definition, except on the intervals of length $C2^{-i}$ around the discontinuities occurring in $\mathcal{B}$ ($C$ denotes an arbitrary positive constant throughout the paper); given a breakpoint $\nu\in\mathcal{B}$, the computation of ${I^{(i)}_{t, T}}$ for $t\in [\nu-C2^{-i}, \nu+C2^{-i}]$ involves observations from two stationary segments, which results in ${\mathbb{E}}{I^{(i)}_{t, T}}$ being “almost” piecewise constant yet not completely so.
The finiteness of $\mathcal{B}$ implies that there exists a fixed index ${I^*}<\lfloor\log_2T\rfloor$ such that each breakpoint in $\mathcal{B}$ can be found in at least one of the functions $S_i(z)$ for $i=-1, \ldots, -{I^*}$. Thus, from the invertibility of $\mathbf{A}$ and the closeness of ${\beta_i(z)}$ and ${\mathbb{E}}{I^{(i)}_{t, T}}$, as noted above, we conclude that every breakpoint is detectable from the wavelet periodogram sequences at scales $i=-1, \ldots, -{I^*}$. Since ${I^*}$ is fixed but unknown, in our theoretical considerations we permit it to increase slowly to infinity with $T$, see the Appendix for further details. A further reason for disregarding the coarse scales $i<-{I^*}$ is that the autocorrelation within each wavelet periodogram sequence becomes stronger at coarser scales; similarly, the intervals on which ${\mathbb{E}}{I^{(i)}_{t, T}}$ is not piecewise constant become longer. Thus, for coarse scales, wavelet periodograms provide little useful information about breakpoints and can safely be omitted.
We end this section by briefly summarising our reasons behind the choice of the LSW model as a suitable framework for developing our methodology:
- The entire piecewise constant second-order structure of the process is encoded in the (asymptotically) piecewise constant sequences ${\mathbb{E}}{I^{(i)}_{t, T}}$.
- Due to the “whitening" property of wavelets, the wavelet periodogram sequences are often much less autocorrelated than the original process. In Section 9.2.2 of @vidakovic1999, the “whitening” property of wavelets is formalised for a second-order stationary time series ${X_{t, T}}$ with a sufficiently smooth spectral density; defining the wavelet coefficient as $r_{i,k}:=\sum_s X_{s, T}\psi_{i, s-k}$, the across-scale covariance of the wavelet coefficients ${\mathbb{E}}(r_{i,k}r_{i',k'})$ vanishes for $|i-i'|>1$, is arbitrarily small for $|i-i'|=1$, and decays as $o(|k-k'|^{-1})$ within each scale, provided the wavelet used is also sufficiently smooth. However, we emphasise that our segmentation method permits autocorrelation in the wavelet periodogram sequences, as described later in Section \[sec:multiplicative\].
- The entire array of the wavelet periodograms at all scales is easily and rapidly computable via the non-decimated wavelet transform.
- The use of the “rescaled time” $z = k/T$ in (\[pswm\]) and the associated regularity assumptions on the transfer functions $W_i(z)$ permit us to establish rigorous asymptotic properties of our procedure.
Binary segmentation algorithm {#sec:second}
=============================
Noting that each wavelet periodogram sequence follows a multiplicative model, as described in Section \[sec:multiplicative\], we introduce a binary segmentation algorithm for such class of sequences. Binary segmentation is a computationally efficient tool that searches for multiple breakpoints in a recursive manner (and can be classed as a “greedy” and “multilevel” algorithm). @venkatraman1993 applied the procedure to a sequence of independent normal variables with multiple breakpoints in its mean and showed that the detected breakpoints were consistent in terms of their number and locations. In the following, we aim at extending these consistency results to the multiplicative model where dependence between observations is permitted.
Generic multiplicative model {#sec:multiplicative}
----------------------------
Recall that each wavelet periodogram ordinate is simply a squared wavelet coefficient of a zero-mean Gaussian time series, is distributed as a scaled $\chi^2_1$ variable, and satisfies ${I^{(i)}_{t, T}}={\mathbb{E}}{I^{(i)}_{t, T}}\cdot
{Z^2_{t, T}}$, where $\{{Z_{t, T}}\}_{t=0}^{T-1}$ are autocorrelated standard normal variables. Hence we develop a generic breakpoint detection tool for multiplicative sequences $$\begin{aligned}
{Y^2_{t, T}}={\sigma^2_{t, T}}\cdot{Z^2_{t, T}}, \ t=0, \ldots, T-1; \label{generic}\end{aligned}$$
${I^{(i)}_{t, T}}$ and ${\mathbb{E}}{I^{(i)}_{t, T}}$ can be viewed as special cases of ${Y^2_{t, T}}$ and ${\sigma^2_{t, T}}$, respectively. We assume additional conditions that are, in particular, satisfied for ${I^{(i)}_{t, T}}$ and ${\mathbb{E}}{I^{(i)}_{t, T}}$ by the assumptions of Theorem 2.
- ${\sigma^2_{t, T}}$ is deterministic and “close” to a piecewise constant function $\sigma^2(t/T)$ in the sense that ${\sigma^2_{t, T}}$ is piecewise constant apart from intervals of length at most $C2^{{I^*}}$ around the discontinuities in $\sigma^2(z)$, and $T^{-1}\sum_{t=0}^{T-1}|{\sigma^2_{t, T}}-\sigma^2(t/T)|^2=o(\log^{-1}T)$, where the latter rate comes from the rate of convergence of the integrated squared bias between $\beta_i(t/T)$ and ${\mathbb{E}}{I^{(i)}_{t, T}}$ (see @piotr2006a for details) and from the fact that our attention is limited to the ${I^*}$ finest scales only. Further, $\sigma^2(z)$ is bounded from above and away from zero, with a finite but unknown number of jumps.
- $\{{Z_{t, T}}\}_{t=0}^{T-1}$ is a sequence of standard Gaussian variables and the function $\rho(\tau)=\sup_{t,T}$ $|{\mbox{cor}}(Z_{t, T}, Z_{t+\tau, T})|$ satisfies $\rho^1_{\infty}<\infty$ where $\rho^p_{\infty}=\sum_{\tau}\left\vert\rho(\tau)\right\vert^p$.
Once the breakpoint detection algorithm for the generic model (\[generic\]) has been established, we apply it to the wavelet periodograms.
Algorithm {#sec:algorithm}
---------
The first step of the binary segmentation procedure is to find the likely location of a breakpoint. We locate such a point in the interval $(0, T-1)$ as the one which maximizes the absolute value of $$\begin{aligned}
{\mathbb{Y}}_{0, T-1}^{\nu}
=\sqrt{\frac{T-\nu}{T\cdot\nu}}\sum_{t=0}^{\nu-1}{Y^2_{t, T}}-
\sqrt{\frac{\nu}{T\cdot(T-\nu)}}\sum_{t=\nu}^{T-1}{Y^2_{t, T}}.
\label{unbalhaar}\end{aligned}$$ Here ${\mathbb{Y}}_{0, T-1}^{\nu}$ can be interpreted as a scaled difference between the partial means of two segments $\{{Y^2_{t, T}}\}_{t=0}^{\nu-1}$ and $\{{Y^2_{t, T}}\}_{t=\nu}^{T-1}$, where the scaling is chosen so as to keep the variance ${\mathbb{Y}}_{0, T-1}^{\nu}$ constant over $\nu$ in the idealised case of $Y_{t,T}^2$ being i.i.d. Once such a $\nu$ has been found, we use ${\mathbb{Y}}_{0, T-1}^{\nu}$ (but not only this quantity; see below for details) to test the null hypothesis of ${\sigma^2(t/T)}$ being constant over $[0, T-1]$. The test statistic and its critical value are established such that when a breakpoint is present, the null hypothesis is rejected with probability converging to $1$. If the null hypothesis is rejected, we continue the simultaneous locating and testing of breakpoints on the two segments to the left and right of $\nu$ in a recursive manner until no further breakpoints are detected. The algorithm is summarised below, where $j$ is the level index and $l$ is the location index of the node at each level. Here the term “level” is used to indicate the progression of the segmentation procedure.
**Algorithm**
Step 1
: Begin with $(j, l)=(1, 1)$. Let ${s_{j, l}}=0$ and ${e_{j, l}}=T-1$.
Step 2
: Iteratively compute ${\mathbb{Y}}^b_{{s_{j, l}}, {e_{j, l}}}$ as in (\[unbalhaar\]) for $b\in({s_{j, l}}, {e_{j, l}})$. Then, find ${b_{j, l}}$ which maximizes its absolute value while satisfying $$\max\left\{\sqrt{({e_{j, l}}-{b_{j, l}})/({b_{j, l}}-{s_{j, l}}+1)}, \sqrt{({b_{j, l}}-{s_{j, l}}+1)/({e_{j, l}}-{b_{j, l}})}\right\} \le c$$ for a fixed constant $c\in(0, \infty)$. Let ${n_{j, l}}={e_{j, l}}-{s_{j, l}}+1$, ${d_{j, l}}={\mathbb{Y}}^{{b_{j, l}}}_{{s_{j, l}}, {e_{j, l}}}$, and ${m_{j, l}}=\sum_{t={s_{j, l}}}^{{e_{j, l}}} {Y^2_{t, T}}/\sqrt{{n_{j, l}}}$.
Step 3
: Perform hard thresholding on $|{d_{j, l}}|/{m_{j, l}}$ with the threshold ${t_{j, l}}=\tau T^{\theta}\sqrt{\log T/{n_{j, l}}}$ so that ${\wh{d}_{j, l}}={d_{j, l}}$ if $|{d_{j, l}}|>{m_{j, l}}\cdot {t_{j, l}}$, and ${\wh{d}_{j, l}}=0$ otherwise. The choice of $\theta$ and $\tau$ is discussed in Section \[sec:parameters\].
Step 4
: If either ${\wh{d}_{j, l}}=0$ or $\max\{{b_{j, l}}-{s_{j, l}}+1, {e_{j, l}}-{b_{j, l}}\}<{\Delta_T}$ for $l$, stop the algorithm on the interval $[{s_{j, l}}, {e_{j, l}}]$; if not, let $(s_{j+1, 2l-1}, e_{j+1, 2l-1})=({s_{j, l}}, {b_{j, l}})$ and $(s_{j+1, 2l}, e_{j+1, 2l})=({b_{j, l}}+1, {e_{j, l}})$, and update the level $j$ as $j \rightarrow j+1$. The choice of ${\Delta_T}$ is discussed in Section \[sec:parameters\].
Step 5
: Repeat Steps 2–4.
The condition imposed on ${b_{j, l}}$ in Step 2 implies that the breakpoints should be sufficiently scattered over time without being too close to each other, and a similar condition is required of the true breakpoints in ${\sigma^2(t/T)}$, see Assumption \[assum:two\] in Section \[sec:consistency\]. The set of detected breakpoints is $\{{b_{j, l}}; {\wh{d}_{j, l}}\ne 0\}$. The test statistic $|{d_{j, l}}|/{m_{j, l}}$ is a scaled version of the test statistics in the ICSS algorithm (@inclan1994). However, the test criteria in that paper are derived empirically under the assumption of independent observations, and there is no guarantee that their algorithm produces consistent breakpoint estimates. @piotr2006a and @fss06 introduced “Haar-Fisz” techniques in different contexts; the former for estimating the time-varying local variance of an LSW time series, and the latter for estimating time-varying volatility in a locally stationary model for financial log-returns. Each Haar-Fisz method has a device (termed the “Fisz transform”) for stabilising the variance of the Haar wavelet coefficients of the data and thereby bringing the distribution of the data close to Gaussianity with constant variance. This is similar to the step in our algorithm where the differential statistic ($d_{j, l}$) is divided by the local mean ($m_{j, l}$), with the convention $0/0 = 0$. However, the Fisz transform was only defined for the case $b = \frac{1}{2}(e_{j,l} + s_{j,l} + 1)$ (meaning the segments were split in half) and it was not used for the purposes of breakpoint detection.
### Post-processing within a sequence {#sec:within}
We equip the procedure with an extra step aimed at reducing the risk of overestimating the number of breakpoints. The ICSS algorithm in @inclan1994 has a “fine-tune” step whereby if more than one breakpoint is found, each breakpoint is checked against the adjacent ones to reduce the risk of overestimation. We propose a post-processing procedure performing a similar task within the single-sequence multiplicative model (\[generic\]). At each breakpoint, the test statistic is re-calculated over the interval between two neighbouring breakpoints and compared with the threshold. Denote the breakpoint estimates as ${\wh{\eta}}_p, p=1, \ldots,
\wh{N}$ and ${\wh{\eta}}_0=0$, ${\wh{\eta}}_{\wh{N}+1}=T$. For each ${\wh{\eta}}_p$, we examine whether $
\left| {\mathbb{Y}}^{{\wh{\eta}}_{p}}_{{\wh{\eta}}_{p-1}+1, {\wh{\eta}}_{p+1}} \right|>\tau
T^{\theta}\sqrt{\log T} \cdot
\sum_{t={\wh{\eta}}_{p-1}+1}^{{\wh{\eta}}_{p+1}}{Y^2_{t, T}}/({\wh{\eta}}_{p+1}-{\wh{\eta}}_{p-1}).
$ If this inequality does not hold, ${\wh{\eta}}_{p}$ is removed and the same procedure is repeated with the reduced set of breakpoints until the set does not change. We emphasise that our within-scale post-processing step is in line with the theoretical derivation of breakpoint detection consistency as (a) the extra checks are of the same form as those done in the original algorithm, (b) the locations of the breakpoints that survive the post-processing are unchanged. The next section provides details of our consistency result.
Consistency of detected breakpoints {#sec:consistency}
-----------------------------------
In this section, we first show the consistency of our algorithm for a multiplicative sequence as in (\[generic\]), which corresponds to the wavelet periodogram sequence at a single scale. Later, Theorem \[thm:two\] shows how this consistency result carries over to the consistency of our procedure in detecting breakpoints in the entire second-order structure of the input LSW process $X_{t,T}$.
Denote the number of breakpoints in ${\sigma^2(t/T)}$ by $N$ and the breakpoints themselves by $0 <\eta_1< \ldots < \eta_N < T-1$, with $\eta_0=0, \ \eta_{N+1}=T-1$.
\[assum:two\] For $\Theta \in (7/8, 1]$ and $\theta\in(5/4-\Theta, \Theta-1/2)$, the length of each segment in $\sigma^2(t/T)$ is bounded from below by $\delta_T=CT^{\Theta}$. Further, there exists some constant $c\in(0, \infty)$ such that, $$\begin{aligned}
\max_{1\le p\le N}\left\{\sqrt{\frac{\eta_p-\eta_{p-1}}{\eta_{p+1}-\eta_{p}}},
\sqrt{\frac{\eta_{p+1}-\eta_{p}}{\eta_{p}-\eta_{p-1}}}\right\} \le c.
\nonumber\end{aligned}$$
\[thm:one\] Suppose that $\{{Y_{t, T}}\}_{t=0}^{T-1}$ follows model (\[generic\]). Assume there exist $M, m>0$ such that $\sup_t|{\sigma^2(t/T)}| \le M$ and $\inf_{1\le i\le N}\left\vert\sigma^2\left((\eta_{i}+1)/T\right)-\sigma^2\left(\eta_{i}/T\right)\right\vert\ge m$. Under Assumption \[assum:two\], the number and locations of the detected breakpoints are consistent. That is, $\mathbf{P}\left\{ \wh{N}=N; \, \left|{\wh{\eta}}_p-\eta_p\right| \le C{{\epsilon}_T}, \ 1\le p \le N \right\}\to1$ as $T \to \infty$, where ${\wh{\eta}}_p, \ p=1, \ldots, \wh{N}$ are detected breakpoints and ${{\epsilon}_T}=T^{5/2-2\Theta}\log T$. (Interpreting this in the rescaled time interval $[0,1]$, ${{\epsilon}_T}/T = T^{3/2-2\Theta}\log T \to 0$ as $T \to 0$.)
### Post-processing across the scales {#sec:across}
We only consider wavelet periodograms ${I^{(i)}_{t, T}}$ at scales $i=-1, \ldots, -{I^*}$, choosing ${I^*}$ to satisfy $2^{{I^*}} \ll {{\epsilon}_T}=T^{5/2-2\Theta}\log T$, so that the bias between ${\sigma^2_{t, T}}$ and ${\sigma^2(t/T)}$ does not preclude the results of Theorem \[thm:one\]. Recall that any breakpoint in the second-order structure of the original process $X_{t,T}$ must be reflected in a breakpoint in at least one of the asymptotic wavelet periodogram expectations ${\beta_i(z)}, \ i = -1,
\ldots, -I^*$, and vice versa: a breakpoint in one of the ${\beta_i(z)}$’s implies a breakpoint in the second-order structure of $X_{t,T}$. Thus, it is sensible to combine the estimated breakpoints across the periodogram scales by, roughly speaking, selecting a breakpoint as significant if it appears in [*any*]{} of the wavelet periodogram sequences. This section provides a precise algorithm for doing this, and states a consistency result for the final set of breakpoints arising from combining them across scales.
The complete across-scales post-processing algorithm follows. Denote the set of detected breakpoints from the sequence ${I^{(i)}_{t, T}}$ as $\wh{\mathcal{B}}_i=\left\{{\wh{\eta}}_p^{(i)}, \ p=1, \ldots,
\wh{N}_i\right\}$. Then the post-processing finds a subset of $\cup_{i=-1}^{-{I^*}}\wh{\mathcal{B}}_i$, say $\wh{\mathcal{B}}$, as follows.
Step 1
: Arrange all breakpoints into groups so that those from different sequences and within the distance of ${\Lambda_T}$ from each other are classified to the same group; denote the groups by $\mathcal{G}_1, \ldots, \mathcal{G}_{\wh{B}}$.
Step 2
: Find $i_0=\max\left\{ \arg\max_{-{I^*}\le k \le -1} \wh{N}_k
\right\}$, the finest scale with the most breakpoints.
Step 3
: Check whether there exists ${\wh{\eta}}^{(i_0)}_{p_0}$ for every ${\wh{\eta}}^{(i)}_{p}$, $i \ne i_0, \ 1\le p \le \wh{N}_i$, which satisfies $\left\vert {\wh{\eta}}^{(i)}_{p}-{\wh{\eta}}^{(i_0)}_{p_0}
\right\vert < {\Lambda_T}$. If so, let $\wh{\mathcal{B}}=\wh{\mathcal{B}}_{i_0}$ and stop the post-processing.
Step 4
: Otherwise let $\wh{\mathcal{B}}=\left\{\wh{\nu}_p, \ p=1, \ldots,
\wh{B} \right\}$ where each $\wh{\nu}_p\in\mathcal{G}_p$ with the maximum $i$.
We set ${\Lambda_T}=\lfloor{{\epsilon}_T}/2\rfloor$ in order to take into account the bias arising in deriving the results of Theorem \[thm:one\]. Breakpoints detected at coarser scales are likely to be less accurate than those detected at finer scales; therefore, our algorithm prefers the latter. The across-scales post-processing procedure preserves the number of “distinct” breakpoints and also their locations as determined by the algorithm. Hence the breakpoints in set $\wh{\mathcal{B}}$ are still consistent estimates of true breakpoints in the second-order structure of the original nonstationary process $X_{t,T}$. Although this is not the only way to combine the breakpoints across scales consistent with our theory, we advocate it due to its good performance.
Denote the set of the true breakpoints in the second-order structure of $X_{t,T}$ by $\mathcal{B}=\left\{\nu_p, \ p=1, \ldots, B
\right\}$, and the estimated breakpoints by $\wh{\mathcal{B}}=\left\{\wh{\nu}_p, \ p=1, \ldots, \wh{B}
\right\}$.
\[thm:two\] Suppose that $X_{t,T}$ satisfies Assumption \[assum:one\] and that $\nu_p, 1\le p \le B$ satisfy the condition required of the $\eta_p$’s in Assumption \[assum:two\]. Further assume that the conditions on $\sigma^2(z)$ in Theorem \[thm:one\] hold for each ${\beta_i(z)}$. Then $
\mathbf{P}\left\{\wh{B}=B; \ \left|\wh{\nu}_p-\nu_p\right|\le C{{\epsilon}_T}, \
1\le p \le B \right\} \to 1
$ as $T \to \infty$.
Choice of ${\Delta_T}$, $\theta$, $\tau$ and ${I^*}$ {#sec:parameters}
----------------------------------------------------
To ensure that each estimated segment is of sufficiently large length so as not to distort our theoretical results, ${\Delta_T}$ is chosen so that ${\Delta_T}\ge C{{\epsilon}_T}$. In practice our method works well for smaller values of ${\Delta_T}$ as well, and in the simulation experiments, ${\Delta_T}=C\sqrt{T}$ is used. As $\theta \in (1/4, 1/2)$, we use $\theta=0.251$ (we have found that the method works best when $\theta$ is close to the lower end of its permitted range) and elaborate on the choice of $\tau$ below. The selection of $\tau$ is not a straightforward task and to get some insight into the issue, a set of numerical experiments was conducted. A vector of random variables $\mathbf{X}\sim\mathcal{N}_T(0, \Sigma)$ was generated, $\mathbf{X}=(X_1, \ldots, X_T)^T$, then transformed into sequences of wavelet periodograms ${I^{(i)}_{t, T}}$. The covariance matrix satisfied $\Sigma=\left({\sigma}_{i, j}\right)_{i, j=1}^T$ where ${\sigma}_{i, j}=\rho^{|i-j|}$. Then we found $b \in (1, T)$ that maximised $$\mathbb{I}^b_i=\left\vert\sqrt{\frac{T-b}{T\cdot b}}\sum_{t=1}^b
{I^{(i)}_{t, T}}-\sqrt{\frac{b}{T(T-b)}}\sum_{t=b+1}^T {I^{(i)}_{t, T}}\right\vert,$$ and computed $\mathbb{U}_{i, \rho, T}=\mathbb{I}^b_i\cdot \{T^{-1}\sum_{t=1}^T
{I^{(i)}_{t, T}}\cdot T^{\theta}\sqrt{\log T}\}^{-1}$. This was repeated with a varying covariance matrix ($\rho=0, 0.3, 0.6, 0.9$) and sample size ($T=512, 1024, 2048$), 100 times for each combination.
The quantity $\mathbb{U}_{i, \rho, T}$ is the ratio between our test statistic and the time-dependent factor $T^{\theta}\sqrt{\log T}$ appearing in the threshold defined in the algorithm of Section \[sec:algorithm\]. $\mathbb{U}_{i, \rho, T}$ is computed under the “null hypothesis” of no breakpoints being present in the covariance structure of $X_{t}$, and its magnitude serves as a guideline as to how to select the value of $\tau$, for each scale $i$, to prevent spurious breakpoint detection in the null hypothesis case. The results showed that the values of $\mathbb{U}_{i, \rho, T}$ and their range tended to increase for coarser scales, this due to the increasing dependence in the wavelet periodogram sequences. In comparison to the scale factor $i$, the parameters $\rho$ or $T$ had relatively little impact on $\mathbb{U}_{i, \rho, T}$.
We thus propose to use different values of $\tau$ in Step 3 of Algorithm of Section \[sec:algorithm\] and in the within-scale post-processing procedure of Section \[sec:within\]. Denoting the former by $\tau_{i, 1}$ and the latter by $\tau_{i, 2}$, we chose $\tau_{i, 1}$ differently for each $i$ as the $95\%$ quantile, and $\tau_{i, 2}$ as the $97.5\%$ quantile of $\mathbb{U}_{i, \rho, T}$ for given $i$ and $T$ and $\rho$ chosen from the set $\{0, 0.3, 0.6, 0.9\}$ with equal probability. The numerical values of $\mathbb{U}_{i, 0, T}$ when $T=1024$ are summarised in Table \[table:tau\].
scale $i$ $-1$ $-2$ $-3$ $-4$
--------------- -------- -------- -------- --------
$\tau_{i, 1}$ $0.39$ $0.46$ $0.67$ $0.83$
$\tau_{i, 2}$ $0.48$ $0.52$ $0.75$ $0.96$
: Values of $\tau$ for each scale $i=-1, \ldots, -4$.[]{data-label="table:tau"}
Finally, we discuss the choice of ${I^*}$. We first detect breakpoints in wavelet periodograms at scales $i=-1, \ldots, -\lfloor \log_2T/3 \rfloor$ and perform the across-scale post-processing as described in Section \[sec:across\], obtaining the set of breakpoints $\wh{\mathcal{B}}=\left\{\wh{\nu}_p, \ p=1, \ldots, \wh{B}
\right\}$. Subsequently, for the wavelet periodogram at the next finest scale, we compute the quantity $\mathbb{V}_p, \ p=1, \ldots,
\wh{B}+1$ as $$\mathbb{V}_p=\max_{\nu \in (\wh{\nu}_{p-1}, \wh{\nu}_p)} \left\vert
\frac{
\sqrt{\frac{\wh{\nu}_{p}-\nu}{(\wh{\nu}_{p}-\wh{\nu}_{p-1})\cdot(\nu-\wh{\nu}_{p-1})}}
\sum_{t=\wh{\nu}_{p-1}+1}^{\nu}{I^{(i)}_{t, T}}-
\sqrt{\frac{\nu-\wh{\nu}_{p-1}}{(\wh{\nu}_{p}-\wh{\nu}_{p-1})\cdot(\wh{\nu}_p-\nu)}}
\sum_{t=\nu+1}^{\wh{\nu}_{p}}{I^{(i)}_{t, T}}}
{\sum_{\wh{\nu}_{p-1}+1}^{\wh{\nu}_p} {I^{(i)}_{t, T}}/(\wh{\nu}_{p}-\wh{\nu}_{p-1})}\right\vert$$ where $\wh{\nu}_0=-1$ and $\wh{\nu}_{\wh{B}+1}=T-1$. Then $\mathbb{V}_p$ is compared to $\tau_{i, 1}\cdot
T^{\theta}\sqrt{\log T}$ to see whether there are any further breakpoints yet to be detected in ${I^{(i)}_{t, T}}$ that have not been included in $\wh{\mathcal{B}}$. (This step is similar to our within-scale post-processing.) If there is an interval $[\wh{\nu}_{p-1}+1, \wh{\nu}_{p}]$ where $\mathbb{V}_p$ exceeds the threshold, ${I^*}$ is updated as ${I^*}:= {I^*}+ 1$ and the above procedure is repeated to update $\wh{\mathcal{B}}$ until either no further changes are made, or ${I^*}\ge \lfloor \log_2 T/2 \rfloor$.
We note that this approach is in line with the theoretical consistency of our breakpoint detection procedure; $\mathbb{V}_p$ is of the same form as the test statistic and Lemma 6 in the Appendix implies that, if there are no more breakpoints to be detected from ${I^{(i)}_{t, T}}$ for $i<-{I^*}$ other than those already chosen ($\wh{\mathcal{B}}$), then $\mathbb{V}_p$ does not exceed the threshold, and vice versa by Lemma 5.
Simulation study {#sec:simulation}
================
In @davis2006, the performance of the Auto-PARM was assessed and compared with the Auto-SLEX (@ombao2001) through simulation in various settings. The Auto-PARM was shown to be superior to Auto-SLEX in identifying both dyadic and non-dyadic breakpoints in piecewise stationary time series. Some examples in the following are adopted from @davis2006 for the comparative study between our method and the Auto-PARM, alongside some other new examples. We also applied the breakpoint detection method proposed in @lavielle2005 to the same simulated processes and, while the performance was found to be good, it was inferior to both Auto-PARM and our method for these particular examples, so we do not report these results. In the simulations, wavelet periodograms were computed using Haar wavelets and both post-processing procedures (Section \[sec:within\] and Section \[sec:across\]) followed the application of the segmentation algorithm. In our examples, $T=1024$ and therefore $I^*$ was set as $3$ at the start of each application of the algorithm, then updated automatically if necessary, as described in Section \[sec:parameters\]. Simulation outcomes are given in Tables \[table:sim:zero\]–\[table:sim\], where the total number of detected breakpoints are summarised over $100$ simulations.
(A) Stationary AR(1) process with no breakpoints
: \
We consider a stationary AR(1) process, $$\begin{aligned}
X_t=aX_{t-1}+{\epsilon}_t &\mbox{ for } 1 \le t \le 1024,
\label{sim:zero}\end{aligned}$$ where ${\epsilon}_t\sim{\mbox{i.i.d.}}$ $\mathcal{N}(0, 1)$ (as in all subsequent examples unless specified otherwise). For a range of values of $a$, we summarise the breakpoint detection outcome in Table \[table:sim:zero\].
(B) Piecewise stationary AR process with clearly observable changes
: \
This example is taken from @davis2006. The target nonstationary process was generated as $$\begin{aligned}
X_t=\left\{\begin{array}{ll}
0.9X_{t-1}+{\epsilon}_t &\mbox{ for } 1 \le t \le 512, \\
1.68X_{t-1}-0.81X_{t-2}+{\epsilon}_t &\mbox{ for } 513 \le t \le 768, \\
1.32X_{t-1}-0.81X_{t-2}+{\epsilon}_t &\mbox{ for } 769 \le t \le 1024.
\end{array}\right.
\label{sim:one}\end{aligned}$$ As seen in Figure \[fig:sim:one\] (a), there is a clear difference between the three segments in the model. Figure \[fig:sim:one\] (b) shows the wavelet periodogram at scale $-4$ and the estimation results, where the lines with empty squares indicate the true breakpoints ($\eta_1=512,\eta_2=768$) while the lines with filled circles indicate the detected ones (${\wh{\eta}}_1=519,{\wh{\eta}}_2=764$). Note that although initially the procedure returned three breakpoints, the within-sequence post-processing successfully removed the false one.
(C) Piecewise stationary AR process with less clearly observable changes
: \
In this example, the piecewise stationary AR model is revisited, but its breakpoints are less clear-cut, as seen in Figure \[fig:sim:two\]. $$\begin{aligned}
X_t=\left\{\begin{array}{ll}
0.4X_{t-1}+{\epsilon}_t & \mbox{ for } 1 \le t \le 400, \\
-0.6X_{t-1}+{\epsilon}_t & \mbox{ for } 401 \le t \le 612, \\
0.5X_{t-1}+{\epsilon}_t & \mbox{ for } 613 \le t \le 1024 \\
\end{array}\right.
\label{sim:two}\end{aligned}$$ Figure \[fig:sim:two\] (b) shows the wavelet periodogram at scale $-1$ for the realisation in the left panel with its breakpoint estimates (${\wh{\eta}}_1=403,{\wh{\eta}}_2=622$). Both procedures achieved good performance.
(D) Piecewise stationary AR process with a short segment
: \
This example is again from @davis2006. A single breakpoint occurs and one segment is much shorter than the other. $$\begin{aligned}
X_t=\left\{\begin{array}{ll}
0.75X_{t-1}+{\epsilon}_t & \mbox{ for } 1 \le t \le 50, \\
-0.5X_{t-1}+{\epsilon}_t & \mbox{ for } 51 \le t \le 1024. \\
\end{array}\right.
\label{sim:three}\end{aligned}$$ A typical realisation of (\[sim:three\]), its wavelet periodogram at scale $-3$, and the estimation outcome are shown in Figure \[fig:sim:three\], where the jump at $\eta_1=50$ was identified as ${\wh{\eta}}_1=49$. Even though one segment is substantially shorter than the other, our procedure was able to detect exactly one breakpoint in $97\%$ of the cases and underestimation did not occur even when it failed to detect exactly one.
(E) Piecewise stationary near-unit-root process with changing variance
: \
Financial time series, such as stock indices, individual share or commodity prices, or currency exchange rates are, for such purposes as pricing of derivative instruments, often modelled by a random walk with a time-varying variance. We generated a piecewise stationary, near-unit-root example following (\[sim:four\]), where the variance has two breakpoints over time and the AR parameter remains constant and very close to 1; a typical realisation is given in Figure \[fig:sim:four\] (a). Note that, within each stationary segment, the process can be seen as a special case of the near-unit-root process of @phillips1988.
$$\begin{aligned}
X_t=\left\{\begin{array}{lll}
0.999X_{t-1}+{\epsilon}_t, & {\epsilon}_t\sim\mathcal{N}(0, 1) & \mbox{ for } 1 \le t \le 400, \\
0.999X_{t-1}+{\epsilon}_t, & {\epsilon}_t\sim\mathcal{N}(0, 1.5^2) & \mbox{ for } 401 \le t \le 750, \\
0.999X_{t-1}+{\epsilon}_t, & {\epsilon}_t\sim\mathcal{N}(0, 1) & \mbox{ for } 751 \le t \le 1024. \\
\end{array}\right.
\label{sim:four}\end{aligned}$$
Recall that the Auto-PARM is designed to find the “best” combination of the total number and locations of breakpoints, and adopts a genetic algorithm to traverse the vast parameter space. However, due to the stochastic nature of the algorithm, it occasionally fails to return consistent estimates. This instability was emphasised here, with each run often returning different breakpoints. For one typical realisation, it detected $t=21$ and $797$ as breakpoints, and then only $t=741$ in the next run on the same sample path. Overall, the performance of Auto-PARM leaves much to be desired for this particular example, whereas our method performed well, though this is not a criticism of Auto-PARM in general, as it performed well in other examples. Note that it was at scale $-1$ of the wavelet periodogram that both breakpoints were consistently identified the most frequently. The computation of the wavelet periodogram at scale $-1$ with Haar wavelets is a differencing operation and naturally “whitens” the near-unit-root process (\[sim:four\]), clearly revealing any changes of variance in the sequence.
(F) Piecewise stationary AR process with high autocorrelation
: \
The features of this AR model are a high degree of autocorrelation and less obvious breakpoints compared to previous examples. A typical realisation is shown in Figure \[fig:sim:five\] (a). $$\begin{aligned}
X_t=\left\{\begin{array}{lll}
1.399X_{t-1}-0.4X_{t-1}+{\epsilon}_t, & {\epsilon}_t\sim\mathcal{N}(0, 0.8^2) & \mbox{ for } 1 \le t \le 400, \\
0.999X_{t-1}+{\epsilon}_t, & {\epsilon}_t\sim\mathcal{N}(0, 1.2^2) & \mbox{ for } 401 \le t \le 750, \\
0.699X_{t-1}+0.3X_{t-1}+{\epsilon}_t, & {\epsilon}_t\sim\mathcal{N}(0, 1) & \mbox{ for } 751 \le t \le 1024. \\
\end{array}\right.
\label{sim:five}\end{aligned}$$ Again, the instability of Auto-PARM was notable here, with the second breakpoint at $t=750$ often left undetected. Our procedure correctly identified both breakpoints in $84\%$ of the cases.
(G) Piecewise stationary ARMA$(1, 1)$ process
: \
We generated piecewise stationary ARMA processes as $$\begin{aligned}
X_t=\left\{\begin{array}{lll}
0.7X_{t-1}+{\epsilon}_t+0.6{\epsilon}_{t-1} & \mbox{ for } 1 \le t \le 125, \\
0.3X_{t-1}+{\epsilon}_t+0.3{\epsilon}_{t-1} & \mbox{ for } 126 \le t \le 532, \\
0.9X_{t-1}+{\epsilon}_t & \mbox{ for } 533 \le t \le 704, \\
0.1X_{t-1}+{\epsilon}_t-0.5{\epsilon}_{t-1} & \mbox{ for } 705 \le t \le 1024.
\end{array}\right.
\label{sim:seven}\end{aligned}$$ As illustrated in Figure \[fig:sim:seven\] (a), the first breakpoint $
t=125$ is less apparent than the other two. Auto-PARM often left this breakpoint undetected, while our procedure found all three in $76\%$ of cases. We note that it was scale $i=-4$ at which $t=125$ was detected most frequently by our procedure. With a time series of length $T=1024$, default scales provided by our algorithm are $i=-1, -2, -3$, and this example demonstrates the effectiveness of the updating procedure for ${I^*}$ described in Section \[sec:parameters\]. That is, after completing the examination of ${I^{(i)}_{t, T}}$ for $i=-1, -2, -3$, our procedure checked if there were more breakpoints to be detected from ${I^{(i)}_{t, T}}$ for the next scale $i=-4$ and, as it was the case, updated ${I^*}$ to $4$. Figure \[fig:sim:seven\] (b) shows the wavelet periodogram at scale $-4$ for the time series example in the left panel.
![[]{data-label="fig:sim:three"}](sim1.eps){width="100.00000%" height=".25\textheight"}
![[]{data-label="fig:sim:three"}](sim2.eps){width="100.00000%" height=".25\textheight"}
![[]{data-label="fig:sim:three"}](sim3.eps){width="100.00000%" height=".25\textheight"}
![[]{data-label="fig:sim:seven"}](sim4.eps){width="100.00000%" height=".25\textheight"}
![[]{data-label="fig:sim:seven"}](sim5.eps){width="100.00000%" height=".3\textheight"}
![[]{data-label="fig:sim:seven"}](sim7.eps){width="100.00000%" height=".25\textheight"}
[ r c c | c c | c c | c c | c c | c c ]{}\
a & & & & & &\
& CF & AP & CF & AP & CF & AP & CF & AP & CF & AP & CF & AP\
0 & **100** & **100** & **100** & **100** & **100** & **100** & **99** & **100** & **99** & **100** & **94** & **100**\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 5 & 0\
$\ge$ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\
total & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100\
[ r c c | c c | c c | c c | c c | c c]{}\
& & & & & &\
& CF & AP & CF & AP & CF & AP & CF & AP & CF & AP & CF & AP\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 42 & 1 & 20 & 0 & 0\
1 & 0 & 0 & 0 & 0 & **97** & **100** & 0 & 31 & 14 & 68 & 1 & 16\
2 & **93** & **99** & **96** & **100** & 3 & 0 & **97** & **16** & **84** & **7** & 6 & 55\
3 & 4 & 1 & 3 & 0 & 0 & 0 & 2 & 9 & 1 & 3 & **76** & **29**\
4 & 3 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 17 & 0\
5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 1 & 0 & 0\
total & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100\
U.S stock market data analysis {#sec:real}
==============================
Many authors, including @starica2005, argue in favour of nonstationary modelling of financial returns. In this analysis, we consider the Dow Jones Industrial Average index and regard it as a process with an extremely high degree of autocorrelation (such as in the near-unit-root model of @phillips1988) and a time-varying variance, similar to the simulation model in Section \[sec:simulation\] (E).
(A) Dow Jones weekly closing values 1970–1975
: \
The time series of weekly closing values of the Dow Jones Industrial Average index between July 1971 and August 1974 was studied in @hsu1979 and revisited in @chen1997. Historical data are available on [www.google.com/finance/historical?q=INDEXDJX:.DJI](www.google.com/finance/historical?q=INDEXDJX:.DJI), where daily and weekly prices can be extracted for any time period. Both papers concluded that there was a change in the variance of the index around the third week of March 1973. For ease of computation of the wavelet periodogram, we chose the same weekly index between 1 July 1970 and 19 May 1975 so that the data size was $T=256$ with the above-mentioned time period was contained in this interval. The third week of March 1973 corresponds to $t=141$ and our procedure detected ${\wh{\eta}}=142$ as a breakpoint, as illustrated in Figure \[fig:real:one\]. The Auto-PARM did not return any breakpoint, while the segmentation procedure proposed in @lavielle2005, when applied to the log-returns ($\log(X_t/X_{t-1})$) of the data rather than the data $X_t$ themselves, returned $t=141$ as a breakpoint, which is very close to ${\wh{\eta}}$.
(B) Dow Jones daily closing values 2007–2009
: \
We further investigated more recent [*daily*]{} data from the same source, between 8 January 2007 and 16 January 2009. Over this period, the global financial market experienced one of the worst crises in history. Our breakpoint detection algorithm found two breakpoints (see Figure \[fig:real:two\]), one in the last week of July 2007 (${\wh{\eta}}_1=135$), and the other in mid-September 2008 (${\wh{\eta}}_2=424$). The Auto-PARM returned three breakpoints on average, although the estimated breakpoints were unstable as in Section \[sec:simulation\] (E): $t=35, 426$ and $488$ were detected most often as breakpoints, while $t=100$ and $t=140$ were detected in place of $t=35$ on other occasions. The segmentation procedure from @lavielle2005, when applied to the log-returns ($\log(X_t/X_{t-1})$) of the data rather than the data $X_t$ themselves, detected $t=127$ and $424$ as breakpoints, which are very close to ${\wh{\eta}}_1$ and ${\wh{\eta}}_2$. The first breakpoint coincided with the outbreak of the worldwide “credit crunch” as subprime mortgage-backed securities were discovered in portfolios of banks and hedge funds around the world. The second breakpoint coincided with the bankruptcy of Lehman Brothers, a major financial services firm, an event that brought even more volatility to the market. Evidence supporting our breakpoint detection outcome is the TED spread (available from <http://www.bloomberg.com/apps/quote?ticker=.tedsp:ind>), an indicator of perceived credit risk in the general economy; it spiked up in late July 2007, remained volatile for a year, then spiked even higher in September 2008. These movements coincide almost exactly with our detected breakpoints.
![[]{data-label="fig:real:two"}](real1.eps){width="100.00000%" height=".3\textheight"}
![[]{data-label="fig:real:two"}](real2.eps){width="100.00000%" height=".3\textheight"}
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Rainer von Sachs for his interesting comments to this work. Also we are grateful to the Editor, an associate editor and two referees for their stimulating reports that led to a significant improvement of the paper.
The proof of Theorem 1 {#sec:proof:one}
======================
The consistency of our algorithm is first proved for the sequence below, $$\begin{aligned}
{{w_{2}}{Y}^2_{t, T}}={\sigma^2(t/T)}\cdot{Z^2_{t, T}}, \ t=0, \ldots, T-1.
\label{bs:generic}\end{aligned}$$ Note that unlike in (3), the above model features the true piecewise constant ${\sigma^2(t/T)}$. Denote $n=e-s+1$ and define $$\begin{aligned}
{{w_{2}}{\mathbb{Y}}}^b_{s, e}&=&{\frac{\sqrt{e-b}}{\sqrt{n}\sqrt{b-s+1}}\sum_{t=s}^b}{{w_{2}}{Y}^2_{t, T}}- {\frac{\sqrt{b-s+1}}{\sqrt{n}\sqrt{e-b}}\sum_{t=b+1}^e}{{w_{2}}{Y}^2_{t, T}}.\end{aligned}$$ ${{w_{2}}{\mathbb{S}}}^b_{s, e}$ and ${\mathbb{S}}^b_{s, e}$ are defined similarly, replacing ${{w_{2}}{Y}^2_{t, T}}$ with ${\sigma^2(t/T)}$ and ${\sigma^2_{t, T}}$, respectively. Note that the above are simply inner products of the respective sequences and a vector whose support starts at $s$, is constant and positive until $b$, then constant negative until $e$, and normalised such that it sums to zero and sums to one when squared. Let $s$, $e$ satisfy $ \eta_{p_0} \le s < \eta_{p_0+1} < \ldots < \eta_{p_0+q} <
e \le \eta_{p_0+q+1} $ for $0 \le p_0 \le B-q$, which will always be the case at all stages of the algorithm. In Lemmas 1–5 below, we impose at least one of following conditions: $$\begin{aligned}
s< \eta_{p_0+r}-C{\delta_T}< \eta_{p_0+r}+C{\delta_T}< e \mbox{ for some } 1 \le r \le q,
\label{lem:cond:one} \\
\{(\eta_{p_0+1}-s)\wedge(s-\eta_{p_0})\} \vee \{(\eta_{p_0+q+1}-e)\wedge(e-\eta_{p_0+q})\} \le {{\epsilon}_T},
\label{lem:cond:two}\end{aligned}$$ where $\wedge$ and $\vee$ are the minimum and maximum operators, respectively. We remark that both conditions (\[lem:cond:one\]) and (\[lem:cond:two\]) hold throughout the algorithm for all those segments starting at $s$ and ending at $e$ which contain previously undetected breakpoints. As Lemma 6 concerns the case when all breakpoint have already been detected, it does not use either of these conditions.
The proof of the theorem is constructed as follows. Lemma \[lem:one\] is used in the proof of Lemma \[lem:two\], which in turn is used alongside Lemma \[lem:three\] in the proof of Lemma \[lem:four\]. From the result of Lemma \[lem:four\], we derive Lemma \[lem:five\] and finally, Lemmas \[lem:five\] and \[lem:six\] are used to prove Theorem 1.
\[lem:one\] Let $s$ and $e$ satisfy (\[lem:cond:one\]), then there exists $1 \le r^* \le q$ such that $$\begin{aligned}
\label{lem:one:eq}
\left|{{w_{2}}{\mathbb{S}}}^{\eta_{p_0+r^*}}_{s, e}\right| =\max_{s<t<e}|{{w_{2}}{\mathbb{S}}}^{t}_{s, e}|
\ge C{\delta_T}/\sqrt{T}. \end{aligned}$$
**Proof.** The equality in (\[lem:one:eq\]) is proved by Lemmas 2.2 and 2.3 of Venkatraman (1993). For the inequality part, we note that in the case of a single breakpoint in $\sigma^2(z)$, $r$ in (\[lem:cond:one\]) coincides with $r^*$ and we can use the constancy of $\sigma^2(z)$ to the left and to the right of the breakpoint to show that $$\begin{aligned}
\left|{{w_{2}}{\mathbb{S}}}^{\eta_{p_0+r}}_{s, e}\right|
=\left|\frac{\sqrt{\eta_{p_0+r}-s+1}\sqrt{e-\eta_{p_0+r}}}{\sqrt{n}}
\left(\sigma^2(\eta_{p_0+r}/T)-\sigma^2((\eta_{p_0+r}+1)/T)\right)\right|,\end{aligned}$$ which is bounded from below by $C{\delta_T}/\sqrt{T}$. In the case of multiple breakpoints, we remark that for any $r$ satisfying (\[lem:cond:one\]), the above order remains the same and thus (\[lem:one:eq\]) follows. $\square$
\[lem:two\] Suppose (\[lem:cond:one\]) holds, and let $\eta \equiv \eta_{p_0+r} \in [s, e]$ for some $r\in\{1, \ldots, q\}$, denote a true change-point. Then there exists $c_0\in(0, \infty)$ such that for $b$ satisfying $|{{w_{2}}{\mathbb{S}}}_{s, e}^b| < |{{w_{2}}{\mathbb{S}}}_{s, e}^\eta|$ and $|\eta-b| \ge c_0{{\epsilon}_T}$ with ${{\epsilon}_T}= T^{5/2-2\Theta}\log\,T$, we have $|{{w_{2}}{\mathbb{S}}}_{s, e}^\eta| - |{{w_{2}}{\mathbb{S}}}_{s, e}^b| \ge 2\log\,T$.
**Proof.** Let both ${{w_{2}}{\mathbb{S}}}_{s, e}^\eta$, ${{w_{2}}{\mathbb{S}}}_{s, e}^b$ $\ge 0$ without loss of generality.
The proof follows directly from the proof of Lemma 2.6 in [@venkatraman1993]. We only consider Case 2 of Lemma 2.6, since adapting the proof of Case 1 (when there is a single change-point within $[s, e]$) to that of the current lemma takes analogous arguments.
Using the notations therein, it is shown that the term $E_{1l}$ is dominant over $E_{2l}$ and $E_{3l}$ in ${{w_{2}}{\mathbb{S}}}_{s, e}^\eta - {{w_{2}}{\mathbb{S}}}_{s, e}^b$, where $l = c_0{{\epsilon}_T}$. Noting further that $i=\eta-s+1$, $h=\delta_T$, $j=e-\eta-h$ and $a = \sum_{t=s}^\eta \sigma^2(t/T) - (e-s+1)^{-1}\sum_{t=s}^e\sigma^2(t/T)$, and that $h \ge 2l$, $$\begin{aligned}
E_{1l} &=& \frac{la\sqrt{i+j+h}}{\sqrt{i}\sqrt{j+h}} \cdot \frac{h-l}{\sqrt{i+l}\sqrt{j+h-l}\{\sqrt{(i+l)(j+h-l)}+\sqrt{i(j+l)}\}}
\\
&\ge& {{w_{2}}{\mathbb{S}}}_{s, e}^\eta \cdot C{{\epsilon}_T}\delta_T T^{-2} \ge 2\log\,T\end{aligned}$$ for large $T$. $\square$
\[lem:three\] $\left|{{w_{2}}{\mathbb{Y}}}^b_{s, e}-{{w_{2}}{\mathbb{S}}}^b_{s, e}\right| \le \log T$ with probability converging to $1$ with $T$ uniformly over $(s, b, e)\in\mathcal{D}$, where, for $c\in(0, \infty)$, $$\mathcal{D}:=\left\{1 \le s <b<e\le T; \ e-s+1 \ge C{\delta_T}, \ \max\left\{\sqrt{\frac{b-s+1}{e-b}},
\sqrt{\frac{e-b}{s-b+1}}\right\} \le c\right\}.$$
**Proof.** We need to show that $$\begin{aligned}
\mathbf{P}\left(\max_{(s, b,
e)\in\mathcal{D}}\frac{1}{\sqrt{n}}\left\vert\sum_{t=s}^e
{\sigma^2(t/T)}({Z^2_{t, T}}-1)\cdot c_t\right\vert>\log T\right) \longrightarrow 0,
\label{lem:three:one}\end{aligned}$$ where $c_t=\sqrt{e-b}/\sqrt{b-s+1}$ for $t\in[s, b]$ and $c_t=\sqrt{b-s+1}/\sqrt{e-b}$ otherwise. Let $\{U_t\}_{t=s}^e$ be i.i.d. standard normal variables, $\mathbf{V}=(v_{i, j})_{i, j=1}^n$ with $v_{i, j}={\mbox{cor}}\left(Z_{i,
T}, Z_{j, T}\right)$, and $\mathbf{W}=(w_{i, j})_{i, j=1}^n$ be a diagonal matrix with $w_{i, i}={\sigma^2(t/T)}\cdot c_t$ where $i=t-s+1$. By standard results (see e.g. @jk1970, page 151), showing (\[lem:three:one\]) is equivalent to showing that $\left\vert\sum_{t=s}^e {\lambda}_{t-s+1}(U_t^2-1)\right\vert$ is bounded by $\sqrt{n}\log T$ with probability converging to 1, where ${\lambda}_i$ are eigenvalues of the matrix $\mathbf{VW}$. Due to the Gaussianity of $U_t$, ${\lambda}_{t-s+1}(U_t^2-1)$ satisfy the Cramér’s condition, i.e., there exists a constant $C>0$ such that $${\mathbb{E}}\left\vert{\lambda}_{t-s+1}(U_t^2-1)\right\vert^p\le C^{p-2}p!
{\mathbb{E}}\left\vert{\lambda}_{t-s+1}(U_t^2-1)\right\vert^2, \ p=3, 4, \ldots.$$ Therefore we can apply Bernstein’s inequality [@bosq1998] and obtain $$\mathbf{P}\left(\left\vert\sum_{t=s}^e
{\sigma^2(t/T)}({Z^2_{t, T}}-1)\cdot c_t\right\vert>\sqrt{n}\log T\right) \le 2\exp\left(-\frac{n\log^2T}{4\sum_{i=1}^n{\lambda}_i^2+2\max_i|{\lambda}_i|C\sqrt{n}\log T}\right).$$ Note that $
\sum_{i=1}^n{\lambda}_i^2={\mbox{tr}}\left(\mathbf{VW}\right)^2\le
c^2\max_z{\sigma}^4(z)n\rho_{\infty}^2.
$ Also it follows that $\max_i|{\lambda}_i| \le c\max_z$ ${\sigma}^2(z)\Vert \mathbf{V} \Vert$ where $\Vert\cdot\Vert$ denotes the spectral norm of a matrix, and $\Vert
\mathbf{V}\Vert\le\rho^1_{\infty}$ since $\mathbf{V}$ is non-negative definite. Then (\[lem:three:one\]) is bounded by $$\begin{aligned}
&&\sum_{(s, b, e)\in\mathcal{D}} 2\exp\left(-\frac{n\log^2T}{4c^2\max_z{\sigma}^4(z)n\rho_{\infty}^2+2c\max_z{\sigma}^2(z)\sqrt{n}\log
T\rho^1_{\infty}}\right) \\
&&\le 2T^3\exp\left(-C\log^2T\right) \rightarrow 0,\end{aligned}$$ as $\rho_{\infty}^p\le C2^{{I^*}}$, which can be made to be of order $\log T$, since the only requirement on $I^*$ is that it converges to infinity but no particular speed is required. Thus the lemma follows. $\square$
\[lem:four\] Assume (\[lem:cond:one\]) and (\[lem:cond:two\]). For $b=\arg\max_{s<t<e}|{{w_{2}}{\mathbb{Y}}}^t_{s, e}|$, there exists $1\le r \le q$ such that $|b-\eta_{p_0+r}|\le C{{\epsilon}_T}$ for a large $T$.
**Proof.** Let ${{w_{2}}{\mathbb{S}}}_{s, e}=\max_{s<t<e}|{{w_{2}}{\mathbb{S}}}^t_{s, e}|$. From Lemma \[lem:three\], ${{w_{2}}{\mathbb{Y}}}^b_{s, e}\ge{{w_{2}}{\mathbb{S}}}_{s, e}-\log T$ and ${{w_{2}}{\mathbb{S}}}^b_{s, e}\ge{{w_{2}}{\mathbb{Y}}}^{b}_{s, e}-\log T$, hence ${{w_{2}}{\mathbb{S}}}^b_{s, e}\ge{{w_{2}}{\mathbb{S}}}_{s, e}-2\log T$. Assume that $|b-\eta_{p_0+r}| > C{{\epsilon}_T}$ for any $r$. From Lemma 2.2 in Venkatraman (1993), ${{w_{2}}{\mathbb{S}}}^t_{s, e}$ is either monotonic or decreasing and then increasing on $[\eta_{p_0+r}, \eta_{p_0+r+1}]$ and ${{w_{2}}{\mathbb{S}}}^{\eta_{p_0+r}}_{s, e}\vee{{w_{2}}{\mathbb{S}}}^{\eta_{p_0+r+1}}_{s, e}>{{w_{2}}{\mathbb{S}}}^{b}_{s, e}$. Suppose that ${{w_{2}}{\mathbb{S}}}^t_{s, e}$ is decreasing and then increasing on the interval. Then from Lemma \[lem:two\], we have $b'=\eta_{p_0+r}+C{{\epsilon}_T}$ satisfying ${{w_{2}}{\mathbb{S}}}^{\eta_{p_0+r}}_{s, e}-2\log T\ge{{w_{2}}{\mathbb{S}}}^{b'}_{s, e}$. Since ${{w_{2}}{\mathbb{S}}}^t_{s, e}$ is locally increasing at $t=b$ (for ${{w_{2}}{\mathbb{S}}}^{b}_{s, e}>{{w_{2}}{\mathbb{S}}}^{b'}_{s, e}$), we have ${{w_{2}}{\mathbb{S}}}^{\eta_{p_0+r+1}}_{s, e}>{{w_{2}}{\mathbb{S}}}^{b}_{s, e}$ and there will again be a $b''=\eta_{p_0+r+1}-C{{\epsilon}_T}$ satisfying ${{w_{2}}{\mathbb{S}}}^{\eta_{p_0+r}}_{s, e}-2\log T\ge{{w_{2}}{\mathbb{S}}}^{b''}_{s, e}$. As $b''>b$, it contradicts that ${{w_{2}}{\mathbb{S}}}^b_{s, e}\ge{{w_{2}}{\mathbb{S}}}_{s, e}-2\log T$. Similar arguments are applicable when ${{w_{2}}{\mathbb{S}}}^t_{s, e}$ is monotonic and therefore the lemma follows. $\square$
\[lem:five\] Under (\[lem:cond:one\]) and (\[lem:cond:two\]), $
\mathbf{P}\left(\left|{{w_{2}}{\mathbb{Y}}}^b_{s, e}\right|<\tau T^{\theta}\sqrt{\log T} \cdot n^{-1}{\sum_{t=s}^e}{{w_{2}}{Y}^2_{t, T}}\right)
\longrightarrow 0
$ for $b=\arg\max_{s<t<e}|{{w_{2}}{\mathbb{Y}}}^t_{s, e}|$.
**Proof.** From Lemma \[lem:four\], there exists some $r$ such that $|b-\eta_{p_0+r}|<C{{\epsilon}_T}$. Denote ${{w_{2}}{d}}={{w_{2}}{\mathbb{Y}}}^b_{s, e}={{w_{2}}{d}_{1}}-{{w_{2}}{d}_{2}}$ and ${{w_{2}}{m}}=n^{-1/2}{\sum_{t=s}^e}{{w_{2}}{Y}^2_{t, T}}=c_1{{w_{2}}{d}_{1}}+c_2{{w_{2}}{d}_{2}}$, where $$\begin{aligned}
{{w_{2}}{d}_{1}}={\frac{\sqrt{e-b}}{\sqrt{n}\sqrt{b-s+1}}\sum_{t=s}^b}{{w_{2}}{Y}^2_{t, T}}, \ \
{{w_{2}}{d}_{2}}={\frac{\sqrt{b-s+1}}{\sqrt{n}\sqrt{e-b}}\sum_{t=b+1}^e}{{w_{2}}{Y}^2_{t, T}}, \ \mbox{ and} \ c_1=c_2^{-1}=\sqrt{\frac{b-s+1}{e-b}}.\end{aligned}$$ For simplicity, let $c_2>c_1$. Further, let ${\mu_{i}}={\mathbb{E}}{{w_{2}}{d}_{i}}$ and ${w_{i}}={\mbox{var}}({{w_{2}}{d}_{i}})$ for $i=1, 2$, and define $\mu={\mathbb{E}}{{w_{2}}{d}}$ and $w={\mbox{var}}({{w_{2}}{d}})$. Finally, ${\mathit{t}_n}$ denotes the threshold $\tau T^{\theta}\sqrt{\log T/n}$. We need to show $
\mathbf{P}(|{{w_{2}}{d}}|\le{{w_{2}}{m}}\cdot{\mathit{t}_n}) \rightarrow 0.
$ Note that $w_i\le c^2\sup_z\sigma^4(z)\rho^2_{\infty}$. Using Markov’s and the Cauchy-Schwarz inequalities, we bound $\mathbf{P}({{w_{2}}{d}}\le{{w_{2}}{m}}\cdot{\mathit{t}_n})$ by $$\begin{aligned}
&&\mathbf{P}\left\{({{w_{2}}{d}_{1}}-{\mu_{1}})(c_1{\mathit{t}_n}-1)+({{w_{2}}{d}_{2}}-{\mu_{2}})(c_2{\mathit{t}_n}+1)+2c_1{\mathit{t}_n}{\mu_{1}}+(c_2-c_1){\mathit{t}_n}{\mu_{2}}\ge(1+c_1{\mathit{t}_n})\mu\right\} \\
&&\le
4\mu^{-2}(1+c_1{\mathit{t}_n})^{-2}\left\{(c_1{\mathit{t}_n}-1)^2{w_{1}}+(c_2{\mathit{t}_n}+1)^2{w_{2}}+4c_1^2{\mathit{t}_n}^2{\mu_{1}}^2+(c_2-c_1)^2{\mathit{t}_n}^2\mu_2^2\right\}
\\
&&\le
O\left\{\mu^{-2}\sup_z\sigma^4(z)\left(\rho^2_{\infty}+\tau^2T^{2\theta}\log T\right)\right\},\end{aligned}$$ and since $
\mu={{w_{2}}{\mathbb{S}}}^b_{s, e}=O\left({\delta_T}/\sqrt{T}\right) > T^{\theta}\sqrt{\log T},
$ the conclusion follows. $\square$
\[lem:six\] For some positive constants $C, \ C'$, let $s$, $e$ satisfy either
- $\exists\, 1 \le p \le B$ such that $s \le \eta_p \le e$ and $[\eta_p-s+1] \wedge [e-\eta_p] \le C{{\epsilon}_T}$ or
- $\exists\, 1 \le p \le B$ such that $s \le \eta_p < \eta_{p+1} \le e$ and $[\eta_p-s+1] \vee [e-\eta_{p+1}] \le C'{{\epsilon}_T}$.
Then for a large $T$, $$\mathbf{P}\left(
\left|{{w_{2}}{\mathbb{Y}}}^b_{s, e}\right|>\tau T^{\theta}\sqrt{\log T} \cdot n^{-1}{\sum_{t=s}^e}{{w_{2}}{Y}^2_{t, T}}\right)\longrightarrow 0,$$ where $b=\arg\max_{s<t<e}|{{w_{2}}{\mathbb{Y}}}^t_{s, e}|$.
**Proof.** First we assume (i). Let $
\mathcal{A}=\left\{
\left|{{w_{2}}{\mathbb{Y}}}^b_{s, e}\right|> \tau T^{\theta}\sqrt{\log T} \cdot n^{-1}{\sum_{t=s}^e}{{w_{2}}{Y}^2_{t, T}}\right\}
$ and $$\begin{aligned}
\mathcal{B}=\left\{
\frac{1}{n}\left|{\sum_{t=s}^e}\left({{w_{2}}{Y}^2_{t, T}}-{\mathbb{E}}{{w_{2}}{Y}^2_{t, T}}\right)\right|<
h=\frac{(\eta_p-s+1)\sigma^2_1+(e-\eta_p)\sigma^2_2}{2n}
\right\},\end{aligned}$$ where $\sigma^2_1=\sigma^2\left(\eta_p/T\right)$ and $\sigma^2_2=\sigma^2\left((\eta_p+1)/T\right)$. We have $
\mathbf{P}\left(\mathcal{A}\right)=
\mathbf{P}\left(\mathcal{A}\cap\mathcal{B}\right)
+\mathbf{P}\left(\mathcal{A}\left|\mathcal{B}^c\right.\right)\mathbf{P}\left(\mathcal{B}^c\right)
\le\mathbf{P}\left(\mathcal{A}\cap\mathcal{B}\right)+\mathbf{P}\left(\mathcal{B}^c\right).
$ The first part is bounded as $$\begin{aligned}
\mathbf{P}\left(\mathcal{A}\cap\mathcal{B}\right)
\le \mathbf{P}\left(\left|{{w_{2}}{\mathbb{Y}}}^b_{s, e}\right|> \tau T^{\theta}\sqrt{\log T} \cdot n^{-1}{\sum_{t=s}^e}\left({\mathbb{E}}{{w_{2}}{Y}^2_{t, T}}-h\right)\right).
\label{lem:six:one}\end{aligned}$$ From Lemma \[lem:three\], we have $\vert {{w_{2}}{\mathbb{Y}}}_{s, e}^b-{{w_{2}}{\mathbb{S}}}_{s, e}^b \vert \le \log T$. Also Lemmas 2.2 and 2.3 of Venkatraman (1993) indicate that $\max_{s<t<e}|{{w_{2}}{\mathbb{S}}}^t_{s, e}|=|{{w_{2}}{\mathbb{S}}}^{\eta_p}|=O(\sqrt{n^{-1}{{\epsilon}_T}(n-C{{\epsilon}_T})})=O(\sqrt{{{\epsilon}_T}})$. Therefore $|{{w_{2}}{\mathbb{Y}}}_{s, e}^b|\le |{{w_{2}}{\mathbb{S}}}^{\eta_p}|+\log T=O(\sqrt{{{\epsilon}_T}})$ and (\[lem:six:one\]) is bounded by $
{\mathbb{E}}\left({{w_{2}}{\mathbb{Y}}}_{s, e}^b\right)^2/(\tau^2 h^2T^{2\theta}\log T)
\le O\left(T^{1/2-2\theta}\right) \longrightarrow 0,
$ by applying Markov’s inequality. Turning our attention to $\mathbf{P}\left(\mathcal{B}^c\right)$, we need to show that $$\mathbf{P}\left(\frac{1}{n}\left\vert\sum_{t=s}^e{\sigma^2(t/T)}({Z^2_{t, T}}-1)\right\vert>h\right)
\longrightarrow 0.$$ This can be shown by applying Bernstein’s inequality as in the proof of Lemma \[lem:three\], and the lemma follows. Similar arguments are applied when (ii) holds. $\square$
We now prove Theorem 1. At the start of the algorithm, as $s=0$ and $e=T-1$, all conditions for Lemma \[lem:five\] are met and it finds a breakpoint within the distance of $C{{\epsilon}_T}$ from the true breakpoint, by Lemma \[lem:four\]. Under Assumption 2, both (\[lem:cond:one\]) and (\[lem:cond:two\]) are satisfied within each segment until every breakpoint in ${\sigma^2(t/T)}$ is identified. Then, either of two conditions (i) or (ii) in Lemma \[lem:six\] is met and therefore no further breakpoint is detected with probability converging to 1.
Next we study how the bias present in ${\mathbb{E}}{I^{(i)}_{t, T}}(={\sigma^2_{t, T}})$ affects the consistency. First we define the autocorrelation wavelet $\Psi_i(\tau)={\sum_{k=-\infty}^{\infty}}\psi_{i, k}\psi_{i, k+\tau}$, the autocorrelation wavelet inner product matrix $A_{i,
j}=\sum_{\tau}\Psi_i(\tau)\Psi_j(\tau)$, and the across-scales autocorrelation wavelets $\Psi_{i, j}(\tau)=\sum_k \psi_{i,
k}\psi_{j, k+\tau}$. Then it is shown in @piotr2006a that the integrated bias between ${\mathbb{E}}{I^{(i)}_{t, T}}$ and $\beta_i(t/T)$ converges to zero.
\[prop:one\] Let ${I^{(i)}_{t, T}}$ be the wavelet periodogram at a fixed scale $i$. Under Assumption 1, $$\begin{aligned}
T^{-1}\sum_{t=0}^{T-1}\left|{\mathbb{E}}{I^{(i)}_{t, T}}-\beta_i(t/T)\right|^2=O(T^{-1}2^{-i})+b_{i, T}, \label{eq:prop:two}\end{aligned}$$ where $b_{i, T}$ depends on the sequence $\{L_i\}_i$. Further, each ${\beta_i(z)}$ is a piecewise constant function with at most $B$ jumps, all of which occur in the set $\mathcal{B}$.
Suppose the interval $[s, e]$ includes a true breakpoint $\eta_{p}$ as in (\[lem:cond:one\]), and denote $b=\arg\max_{t\in(s, e)}
\vert{{w_{2}}{\mathbb{S}}}^t_{s, e}\vert$ and $\wh{b}=\arg\max_{t\in(s,e)}
\left\vert{\mathbb{S}}^t_{s, e}\right\vert$. Recall that ${\mathbb{E}}{I^{(i)}_{t, T}}$ remains constant within each stationary segment, apart from short (of length $C2^{-i}$) intervals around the discontinuities in $\beta_i(t/T)$. Suppose a jump occurs at $\eta_p$ in $\beta_i(t/T)$ yet there is no change in ${\mathbb{E}}{I^{(i)}_{t, T}}$ for $t\in[\eta_p-C2^{-i},\eta_p+C2^{-i}]$. Then the integrated bias is bounded from below by $C{\delta_T}/T$ from Assumption 2, and Proposition \[prop:one\] is violated. Therefore there will be a change in ${\mathbb{E}}{I^{(i)}_{t, T}}$ as well on such intervals around $\eta_p$ and ${\mathbb{E}}I^{(i)}_{t_1, T}\ne{\mathbb{E}}I^{(i)}_{t_2, T}$ for $t_1\le\eta_p-C2^{-i}$ and $t_2\ge\eta_p+C2^{-i}$. Although the bias of ${\mathbb{E}}{I^{(i)}_{t, T}}$ in relation to $\beta_i(t/T)$ may cause some bias between $\wh{b}$ and $b$, we have that $|\wh{b}-b|\le C2^{{I^*}}<{{\epsilon}_T}$ holds for ${I^*}=O(\log\log
T)$, which is an admissible rate for ${I^*}$. Besides, once one breakpoint is detected in such intervals, the algorithm does not allow any more breakpoints to be detected within the distance of ${\Delta_T}$ from the detected breakpoint, by construction. Hence the bias in ${\mathbb{E}}{I^{(i)}_{t, T}}$ does not affect the results of Lemmas \[lem:one\]–\[lem:six\] for wavelet periodograms at finer scales and the consistency still holds for ${Y^2_{t, T}}$ in (3).
Finally, we note that the within-scale post-processing step in Section 3.2.1 is in line with the theoretical consistency of our procedure; (a) Lemma \[lem:five\] implies that our test statistic exceeds the threshold when there is a breakpoint $\eta$ within a segment $[s, e]$ which is of sufficient distance from both $s$ and $e$, and (b) Lemma \[lem:six\] shows that it does not exceed the threshold when $(s, \eta, e)$ does not satisfy the condition in (a).
The proof of Theorem 2 {#sec:proof:two}
======================
From Assumption 1 and the invertibility of the autocorrelation wavelet inner product matrix $A$, there exists at least one sequence of wavelet periodograms among ${I^{(i)}_{t, T}}, \ i=-1,
\ldots, -{I^*}$ in which any breakpoint in $\mathcal{B}$ is detected. Suppose there is only one such scale, $i_0$, for $\nu_q\in\mathcal{B}$ and denote the detected breakpoint as ${\wh{\eta}}^{(i_0)}_{p_0}$. After the across-scales post-processing, ${\wh{\eta}}^{(i_0)}_{p_0}$ is selected as $\wh{\nu}_q$ since no other ${\wh{\eta}}^{(i)}_p$, $i\ne i_0$, is within the distance of ${\Lambda_T}=C{{\epsilon}_T}$ from either $\wh{\nu}_q$ or ${\wh{\eta}}^{(i_0)}_{p_0}$, and $\left\vert\nu_q-{\wh{\eta}}^{(i_0)}_{p_0}\right\vert\le{{\epsilon}_T}$ with probability converging to 1 from Theorem 1. If there are $D(\le{I^*})$ breakpoints detected for $\nu_q$, denote them as ${\wh{\eta}}^{(i_1)}_{p_1}, \ldots, {\wh{\eta}}^{(i_D)}_{p_D}$. Then for any $1\le a<b \le D$, $
\left\vert{\wh{\eta}}^{(i_a)}_{p_a}-{\wh{\eta}}^{(i_b)}_{p_b}\right\vert \le
\left\vert{\wh{\eta}}^{(i_a)}_{p_a}-\nu_q\right\vert +
\left\vert{\wh{\eta}}^{(i_b)}_{p_b}-\nu_q\right\vert \le C{{\epsilon}_T}, $ and only the one from the finest scale is selected as $\wh{\nu}_q$ among them by the post-processing procedure. Hence the across-scales post-processing preserves the consistency for the breakpoints selected as its outcome.
[32]{}
natexlab\#1[\#1]{}
Adak, S. (1998). [[Time-dependent spectral analysis of nonstationary time series]{}.]{} *Journal of the American Statistical Association*. [**93**]{}, 1488–1501.
Andreou, E. and Ghysels, E. (2002). [[Detecting multiple breaks in financial market volatility dynamics]{}.]{} *Journal of Applied Economics*. [**17**]{}, 579–600.
Bai, J. and Perron, P. (1998). [[Estimating and testing linear models with multiple structural changes]{}.]{} *Econometrica*. [**66**]{}, 47–78.
Bosq, D. (1998). *Nonparametric statistics for stochastic process: estimation and prediction.* [Springer]{}.
Chen, J. and Gupta, A. K. (1997). [Testing and locating variance change-points with application to stock prices.]{} *Journal of the American Statistical Association*. [**92**]{}, 739–747.
Chernoff, H. and Zacks, S. (1964). [Estimating the current mean of a normal distribution which is subject to changes in time.]{} *Annals of Mathematical Statistics*. [**35**]{}, 999–1028.
Davis, R. A., Lee, T. C. M., and Rodriguez-Yam, G. A. (2006). [Structural break estimation for non-stationary time series.]{} *Journal of the American Statistical Association*. [**101**]{}, 223–239.
Davis, R. A., Lee, T. C. M., and Rodriguez-Yam, G. A. (2008). [Break detection for a class of nonlinear time series models.]{} *Journal of Time Analysis*. [**29**]{}, 834–867.
Fryzlewicz, P. and Nason, G. (2006). [[H]{}aar-[F]{}isz estimation of evolutionary wavelet spectra.]{} *Journal of the Royal Statistical Society, B*. [**68**]{}, 611–634.
Fryzlewicz, P., Sapatinas, T., and [Subba Rao]{}. S. (2006). [A [H]{}aar-[F]{}isz technique for locally stationary volatility estimation.]{} *Biometrika*. [**93**]{}, 687–704.
Gabbanini, F., Vannucci, M., Bartoli, G., and Moro, A. (2004). [Wavelet packet methods for the analysis of variance of time series with application to crack widths on the [B]{}runelleschi [D]{}ome.]{} *Journal of Computational and Graphical Statistics*. [**13**]{}, 639–658.
Hawkins, D. M. (1977). [Testing a sequence of observations for a shift in location.]{} *Journal of the American Statistical Association*. [**72**]{}, 180–186.
Hsu, D. A. (1977). [Tests for variance shifts at an unknown time point.]{} *Journal of Applied Statistics*. [**26**]{}, 179–184.
Hsu, D. A. (1979). [Detecting shifts of parameters in Gamma sequences with applications to stock price and air traffic flow analysis.]{} *Journal of the American Statistical Association*. [**74**]{}, 31–40.
Inclán, C. and Tiao, G. C. (1994). [Use of cumulative sums of squares for retrospective detection of changes of variance.]{} *Journal of the American Statistical Association*. [**89**]{}, 913–923.
Johnson, N. and Kotz, S. (1970). *[Distributions in Statistics: Continuous Univariate Distributions, Vol. 1.]{}* Houghton Mifflin Company.
Kokoszka, P. and Leipus, R. (2000). [Change-point estimation in [ARCH]{} models.]{} *Bernoulli*. [**6**]{}, 513–539.
Kouamo, O., Moulines, E., and Roueff, F. (2010). [[Testing for homogeneity of variance in the wavelet domain]{}.]{} In *Dependence, with Applications in Statistics and Econometrics* (eds P. Doukhan, G. Lang, D. Surgailis and G. Teyssière). 175–205.
Lai, T. (2001). [[Sequential analysis: some classical problems and new challenges]{}.]{} *Statistica Sinica*. [**11**]{}, 303–350.
Lavielle, M. and Moulines, E. (2000). [Least-squares estimation of an unknown number of shifts in a time series.]{} *Journal of Time Series Analysis*. [**21**]{}, 33–59.
Lavielle, M. and [Teyssière]{}. G. (2005). [[Adaptive detection of multiple change-points in asset price volatility]{}.]{} In *Long Memory in Economics* (eds G. [Teyssière]{} and A. Kirman). 129–156.
Mikosch, T. and Stărică. C. (1999). [[Change of structure in financial time series, long range dependence and the GARCH model]{}.]{} Techinical Report, University of Groningen.
Nason, G. P., von Sachs, R., and Kroisandt, G. (2000). [Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum.]{} *Journal of the Royal Statistical Society, B*. [**62**]{}, 271–292.
Ombao, H. C., Raz, J. A., von Sachs, R., and Malow, B. A. (2001). [Automatic statistical analysis of bivariate nonstationary time series.]{} *Journal of the American Statistical Association*. [**96**]{}, 543–560.
Phillips, P. and Perron, P. (1988). [[Testing for a unit root in time series regression]{}.]{} *Biometrika*. [**75**]{}, 335–346.
Sen, A. and Srivastava, M. S. (1975). [On tests for detecting change in mean.]{} *Annals of Statistics*. [**3**]{}, 98–108.
Stărică. C. and Granger, C. (2005). [[Nonstationarities in stock returns]{}.]{} *Review of Economics and Statistics*. [**87**]{}, 503–522.
Van Bellegem, S. and von Sachs, R. (2008). [Locally adaptive estimation of evolutionary wavelet spectra.]{} *Annals of Statistics*. [**36**]{}, 1879–1924.
Venkatraman, E. S. (1993). [Consistency results in multiple change-point problems.]{} *PhD Thesis, Stanford University*.
Vidakovic, B. (1999). *Statistical Modeling by Wavelets*. Wiley, New York.
Vostrikova, L. J. (1981). [Detecting ‘disorder’ in multidimensional random processes.]{} *Soviet Doklady Mathematics*. [**24**]{}, 55–59.
Whitcher, B., Byers, S. D., Guttorp, P., and Percival, D. B. (2002). [Testing for homogeneity of variance in time series: [L]{}ong memory, wavelets, and the [N]{}ile [R]{}iver.]{} *Water Resources Research*. [**38**]{}, 10–1029.
Whitcher, B., Guttorp, P., and Percival, D. B. (2000). [Multiscale detection and location of multiple variance changes in the presence of long memory.]{} *Journal of Statistical Computation and Simulation*. [**68**]{}, 65–87.
Worsley, K. J. (1986). [Confidence regions and tests for a change-point in a sequence of exponential family random variables.]{} *Biometrika*. [**73**]{}, 91–104.
[^1]: Department of Statistics, London School of Economics, UK. E-mail: [h.cho1@lse.ac.uk]{}, Phone: +44 (0)20 7955 6014, Fax: +44 (0)20 7955 7416
[^2]: Department of Statistics, London School of Economics, UK. E-mail: [p.fryzlewicz@lse.ac.uk]{}
|
---
abstract: 'We consider a simple model for active random walk with general temporal correlations, and investigate the shape of the probability distribution function of the displacement during a short time interval. We find that under certain conditions the distribution is non-monotonic and we show analytically and numerically that the existence of the non-monotonicity is governed by the walker’s tendency to move forward, while the correlations between the timing of its active motion control the magnitude and shape of the non-monotonicity. In particular, we find that in a homogeneous system such non-monotonicity can occur only if the persistence is strong enough.'
author:
- Eial Teomy
- Yael Roichman
- Yair Shokef
bibliography:
- 'references.bib'
title: 'Non-monotonic displacement distribution of active random walkers'
---
Introduction
============
The active and passive motion of biological cells and of their components is a complicated out of equilibrium process, which occurs due to many factors, most of them still far from being well understood [@Groot2005; @Hofling2014; @Bechinger2016; @Metzler2016; @Hakim2017; @Norregaard2017]. This motion has been investigated both at the single-body level [@Hasnain2015; @Detcheverry2015; @Rupprecht2016; @Detcheverry2017; @Sevilla2016; @Ariel2017; @Fedotov2017] and at the many-body level [@Wioland2016; @Stenhammer2017; @Berthier2013; @Viscek1995; @Sepulveda2013; @Grossmann2016; @Liebchen2017; @Zimmermann2016; @Farhadifar2007; @Staple2010; @Sandor2017; @Reichhardt2014; @Zacherson2017; @Reichhardt2014b; @Lam2015; @Graf2017; @Illien2015]. The motions of individual cells or bacteria are modeled in various ways, which include correlations between the motions of a walker at different times. One of the most common models, motivated by experimental observations [@Berg1990], is run and tumble motion [@Detcheverry2015; @Rupprecht2016; @Detcheverry2017; @Reichhardt2014], in which the walker moves in a straight line for some time, and then abruptly changes its direction. A twitching motion [@Zacherson2017] or motion with a self aligning director [@Reichhardt2014b] is captured by a one-step memory term, i.e. the velocity at each step depends on the velocity in the previous step but not on longer term memory. Correlated random walks are also used in other biological processes [@Ghosh2015], such as DNA motors [@Schulz2011] and eye movements [@Hermann2017], as well as in various fields such as polymer chains [@Tchen1952], animal movement [@Kareiva1983], scattering in disordered media [@Boguna1998], artificial microswimmers [@Romanczuk2012; @Ghosh2015b], and motion in ordered media [@Tahir1983; @Tahir1983b]. The origin of the active motion does not have to be the walker itself. It may also come from structural changes in the surrounding medium [@Lettinga2007; @Pouget2011; @Naderi2013; @SonnSegev2017].
In simple Markovian random walks there are no correlations between the motions of the walker at different times. The motion of active walkers, on the other hand, can be correlated in either space or time (or both). *Spatial correlations*, i.e. the correlations between the direction of motion at subsequent points in time, are modeled by persistent motion, in which the walker prefers to continue moving in the same direction as before (or in the opposite direction in the case of anti-persistence). This preference could be either discrete, as in the persistent random walk model [@Goldstein1951], or continuous, as in the Orenstein-Uhlenbeck process [@Uhlenbeck1930]. *Temporal correlations*, i.e. correlations between the time intervals between direction changes, are modeled by non-Poissonian distributed running times, such as in the continuous time random walk model [@Montroll1965] or in Levy walks [@Shlesinger1987] and Levy flights [@Levy1939].
In simple random walks, the probability distribution function (PDF) of the displacements, or the van-Hove distribution function, is Gaussian [@Rayleigh1880]. In many models of active matter, although the PDF might not be Gaussian, it is observed to still be monotonically decreasing with the magnitude of the displacement [@Nigris2017; @Coding2008; @Metzler2014; @Sposini2018; @Jakub2018; @Gnacik2018]. However, there are cases in which the PDF is not monotonic and exhibits peaks, such as an active particle inside a harmonic potential well [@BenIsaac2015; @Razin2019], or an aging Levy walk [@Magdziarz2017]. In the overdamped limit, this is related to peaks that may appear in the velocity distribution due to biological activity [@BenIsaac2011]. We are not aware of any experimental results or other theoretical works which found such non-monotonicity.
In this paper we consider a simple model for active random walk and show that in a homogeneous system, such non-monotonic behaviour is not affected at all by temporal correlations, and it exists only if spatial correlations are positive and strong enough. Temporal correlations do affect the magnitude and the shape of the non-monotonicity, however. We also derive in this paper an explicit expression for the displacement PDF for general temporal correlations and a simple realization of the spatial correlations.
Our results may be used to understand the microscopic processes underlying the overall observed random walk. At long times many of the details of the random motion are averaged out, but they are present in the short time behaviour. Using our results, and expanding the simple model we present here, one may look for non-monotonicity; or, if such non-monotonicity is not observed at short times, this implies a bound on the orientational correlations. By looking for peaks in the displacement distribution in experiments, one may obtain information about the directional correlations of the particles. Moreover, the heights and the separation of the peaks give further insight into the underlying microscopic processes.
The remainder of the paper is organized as follows. In Section \[sec\_model\] we introduce our model, and in Section \[sec\_derpdf\] we solve for the displacement PDF in it. Section \[sec\_proof\] contains our proof that the non-monotonicity may appear only if the persistence is positive, and Section \[sec\_summary\] summarizes the paper.
Model {#sec_model}
=====
We consider a particle moving in $d$ dimensions and are interested in the probability density $P_{r}\left(\vec{r},\tau\right)$ that during a time interval $\tau$ its displacement was $\vec{r}$. We model the movement of the particle as a sum of two independent processes, a *discrete* process representing the active or biological stochastic motion, and a *continuous* process representing the contact of the particle with its thermal environment. Assuming that: 1) the thermal process is isotropic, 2) the directional correlations in the active motion are always relative to the current direction, and 3) the initial condition is isotropic, then when averaging over multiple particles their motion is isotropic, i.e. $P_{r}=P_{r}\left(r,\tau\right)$ depends only on the magnitude $r=|\vec{r}|$ of the displacement.
The probability $P_{r}\left(\vec{r},\tau\right)$ is given by the following convolution of these two sources of fluctuation $$\begin{aligned}
P_{r}\left(\vec{r},\tau\right)=\int P_{d}\left(\vec{r}',\tau\right)P_{c}\left(\vec{r}-\vec{r}',\tau\right)d\vec{r}' ,\label{pr_def}\end{aligned}$$ where $P_{d}\left(\vec{r},\tau\right)$ is the probability that the displacement of the particle due to the discrete process after time $\tau$ is $\vec{r}$, and $P_{c}\left(\vec{r},\tau\right)$ is analogously defined for the continuous process. For simplicity, we assume that the continuous process is Gaussian, $$\begin{aligned}
P_{c}\left(\vec{r},\tau\right)=K_{d}\exp\left(-\frac{r^{2}}{2\alpha^{2}(\tau)}\right) ,\label{pc_def}\end{aligned}$$ with the normalization factor $K_{d}$ in $d$ dimensions given by $$\begin{aligned}
K_{d}=\frac{1}{2^{d/2-1}\alpha^{d}\Gamma\left(\frac{d}{2}\right)\Omega_{d}} ,\label{k_def}\end{aligned}$$ where $\Omega_{d}$ is the surface area of a $d$-dimensional hypersphere, and $\alpha^2(\tau) = \langle r^2(\tau) \rangle$ is the mean squared displacement of the particle during a time interval $\tau$, solely due to the continuous process. Note that the dependence of $\alpha^{2}$ on the time interval $\tau$ could be non-trivial, and not necessarily diffusive. We describe here our model in terms of general spatial dimensionality $d$, but later on we will concentrate on the case $d=3$.
For the discrete process, the time intervals between hops, the distance of each hop and their direction could have any distribution, and can all be correlated in some fashion. In general, the probability $P_{d}\left(\vec{r},\tau\right)$ is given by $$\begin{aligned}
P_{d}\left(\vec{r},\tau\right)=\sum^{\infty}_{n=0}P_{n}\left(\vec{r},\tau\right) ,\label{eq_pr1}\end{aligned}$$ where $P_{n}\left(\vec{r},\tau\right)$ is the probability density that the particle performed $n$ hops during the time interval $\tau$ and that it moved a total distance $\vec{r}$ due to these $n$ hops. Assuming that the time intervals between hops is independent of the details of the hops (i.e. their length and direction), Eq. (\[eq\_pr1\]) may be written as $$\begin{aligned}
P_{d}\left(\vec{r},\tau\right)=\sum^{\infty}_{n=0}q_{n}\left(\tau\right)p_{n}\left(\vec{r}\right) ,\label{pdsum}\end{aligned}$$ where $q_{n}\left(\tau\right)$ is the probability that the particle performed $n$ hops until time $\tau$, and $p_{n}\left(\vec{r}\right)$ is the probability that the particle moved a total distance $\vec{r}$ in these hops given that it performed $n$ hops. Obviously, since the dynamics are isotropic $P_{r}$ and $p_{n}$ depend only on the magnitude $r=\left|\vec{r}\right|$ of the displacement, and not on the direction of $\vec{r}$, but we still keep the notation $\vec{r}$ to emphasize that these are probability densities with respect to the d-dimensional vector $\vec{r}$ and not with respect to the scalar $r$.
We will assume a simple persistent random walk model for the probability $p_{n}\left(\vec{r}\right)$, as schematically shown in Fig. \[schem\]. At each hop, the particle performs with probability $\gamma_{0}\equiv 1-\gamma_{f}-\gamma_{b}$ an uncorrelated hop and moves in a random direction a distance $\ell$ which is drawn from some given distribution $s(\ell)$. With probability $\gamma_{f}$ the particle moves forward and repeats its previous hop (same direction and same magnitude), and with probability $\gamma_{b}$ it moves back, namely it performs the exact opposite of its previous hop (opposite direction and same magnitude).
![Schematic illustration of the model showing a sequence of discrete hops. In the last step the walker reached the position marked in dark gray. In the next step, with probability $\gamma_{f}$ it will perform another step of the same magnitude and in the same direction (orange), with probability $\gamma_{b}$ it will retrace its last step (blue) and with probability $\gamma_{0}$ it will move a random distance drawn from the distribution $s\left(\ell\right)$ in a random direction (green).[]{data-label="schem"}](fig1_yr.pdf){width="0.6\columnwidth"}
In reality, directional correlations are not necessarily sharp as in our model. Namely, if the particle persists in its direction of motion between two subsequent hops, this need not imply that the directions of these two hops are exactly equal. We could extend our work to situations in which there is some general probability distribution function to change the direction of motion by an arbitrary angle. Similarly, the magnitudes of correlated hops are not necessarily identical, and we could allow the length of the hop to be random also if the particle hops in the same direction. However, in this paper we would like to concentrate on the simplest possible version of our model in order to obtain analytical results which remain qualitatively true even in more general cases. In particular, as we will show below, a non-monotonic behaviour of the PDF indicates positive directional correlations.
Derivation of the displacement distribution {#sec_derpdf}
===========================================
In this section we derive the PDF of the displacements. We start by simplifying the convolution representation of the total displacement, Eq. (\[pr\_def\]), for a Gaussian continuous process and a general discrete process. Next, we consider our simple model for the discrete process, still with general step size distribution and general temporal correlations. Lastly, we choose several specific step size distributions and temporal correlations, in order to demonstrate our results.
PDF of the total displacement for general discrete processes
------------------------------------------------------------
Combining Eqs. (\[pr\_def\]-\[k\_def\]) yields $$\begin{aligned}
P_{r}\left(\vec{r},\tau\right)=\frac{1}{2^{d/2-1}\alpha^{d}\Gamma\left(\frac{d}{2}\right)\Omega_{d}}\int P_{d}\left(\vec{r}',\tau\right)\exp\left(-\frac{\left(\vec{r}-\vec{r}'\right)^{2}}{2\alpha^{2}}\right)d\vec{r}' .\end{aligned}$$ Assuming that $P_{d}$ is isotropic, integrating over the angles yields $$\begin{aligned}
P_{r}\left(\vec{r},\tau\right)=\frac{1}{2^{d/2-1}\alpha^{d-2}\Gamma\left(\frac{d}{2}\right)}\int P_{d}\left(\vec{r}',\tau\right)\frac{r'^{d-2}}{r}\exp\left(-\frac{\left(r^{2}+r'^{2}\right)}{2\alpha^{2}}\right)\sinh\left(\frac{rr'}{\alpha^{2}}\right)dr' .\label{pr_2}\end{aligned}$$ We now define the Fourier transform of $P_{d}\left(\vec{r},\tau\right)$, $$\begin{aligned}
&P_{d}\left(\vec{k},\tau\right) = \int d\vec{r} e^{i\vec{k}\cdot\vec{r}} \tilde{P}_{d}\left(\vec{r},\tau\right) .\label{pd_fourier}\end{aligned}$$ Inverting the Fourier transform and integrating over the angles yields $$\begin{aligned}
&P_{d}\left(\vec{r},\tau\right) = \frac{\Omega_{d}}{\left(2\pi\right)^{d}} \int dk k^{d-1}\operatorname{sinc}\left(kr\right)\tilde{P}_{d}\left(\vec{k},\tau\right) .\label{pd_fourier_2}\end{aligned}$$ Combining Eqs. (\[pr\_2\]) and (\[pd\_fourier\_2\]) yields $$\begin{aligned}
&P_{r}\left(\vec{r},\tau\right)=\frac{\Omega_{d}}{2^{3d/2-1}\pi^{d}\alpha^{d-2}r\Gamma\left(\frac{d}{2}\right)}\times\nonumber\\
&\times\int \tilde{P}_{d}\left(\vec{k},\tau\right)k^{d-2}r'^{d-3}\sin(kr')\exp\left(-\frac{\left(r^{2}+r'^{2}\right)}{2\alpha^{2}}\right)\sinh\left(\frac{rr'}{\alpha^{2}}\right)dr'dk .\label{pr_3}\end{aligned}$$ Integrating over $r'$ yields $$\begin{aligned}
&P_{r}\left(\vec{r},\tau\right)=\frac{\Omega(d)\exp\left(-\frac{r^{2}}{2\alpha^{2}}\right)}{2^{d+1}\pi^{d}\left(d-2\right)ir}\times\nonumber\\
&\int^{\infty}_{0}dk P_{d}\left(\vec{k},\tau\right)k^{d-2}\left\{{}_{1}F_{1}\left[\frac{d}{2}-1,\frac{1}{2},-\frac{\left(\alpha^{2}k-ir\right)^{2}}{2\alpha^{2}}\right]-{}_{1}F_{1}\left[\frac{d}{2}-1,\frac{1}{2},-\frac{\left(\alpha^{2}k+ir\right)^{2}}{2\alpha^{2}}\right]\right\} ,\label{pr_4}\end{aligned}$$ where ${}_{1}F_{1}$ is the confluent hypergeometric function [@hypergeometric]. For $d=3$, Eq. (\[pr\_4\]) reduces to $$\begin{aligned}
P_{r}\left(\vec{r},\tau\right)=\frac{1}{2\pi^{2}r}\int^{\infty}_{0}dk P_{d}\left(\vec{k},\tau\right)k\sin\left(kr\right)\exp\left(-\frac{\alpha^{2}k^{2}}{2}\right) .\label{pr_4_d3}\end{aligned}$$ For $d=1$ or $d=2$, Eq. (\[pr\_4\]) is not valid, since its derivation includes integration over variables that exist only in $d\geq3$. However, it is straightforward to check that for $d=1$, the equivalent result to Eq. (\[pr\_4\]) is $$\begin{aligned}
P_{r}\left(r,\tau\right)=\frac{1}{\pi}\int^{\infty}_{0}e^{-\alpha^{2}k^{2}/2}\cos\left(kr\right)\tilde{P}_{d}\left(k,\tau\right)dk ,
\label{pr_4_d1}\end{aligned}$$ and for $d=2$ the equivalent result is $$\begin{aligned}
P_{r}\left(\vec{r},\tau\right)=\frac{1}{2\pi\alpha}\exp\left(-\frac{r^{2}}{2\alpha^{2}}\right)\int^{\infty}_{0}dkP_{d}\left(\vec{k},\tau\right)\int^{\infty}_{0}dxe^{-x^{2}/2}I_{0}\left(\frac{xr}{\alpha}\right)J_{0}\left(\alpha k x\right) .
\label{pr_4_d2}\end{aligned}$$ Therefore, for any stochastic process that can be decomposed into a Gaussian continuous process and an active discrete process, by knowing the distribution of the discrete process, $P_{d}\left(\vec{r},\tau\right)$, we can perform its Fourier transform and substitute $P_{d}\left(\vec{k},\tau\right)$ in Eq. (\[pr\_4\]) to obtain $P_{r}\left(\vec{r},\tau\right)$. We will now obtain $P_{d}$ for our specific simple model for the discrete dynamics, and for it we will calculate $P_{r}$.
The Fourier transform of the model specific discrete process
------------------------------------------------------------
We now consider our specific model for the discrete process, as outlined in Section \[sec\_model\]. We do not yet specify the step size distribution $s\left(\ell\right)$ and the temporal correlations encoded in $q_n(\tau)$. Separating the spatial and temporal correlations of the discrete process as in Eq. (\[pdsum\]), we now write the probability that the particle moved a distance $\vec{r}$ given it performed $n$ hops, $p_{n}\left(\vec{r}\right)$, as an integral over all the possible lengths and directions of the last hop $n$ that the particle performed and which brought it to $\vec{r}$, $$\begin{aligned}
p_{n}\left(\vec{r}\right)=\int d\ell d\hat{r} p_{n,\ell\hat{r}}\left(\vec{r}\right) ,\label{pn_integral}\end{aligned}$$ where $p_{n,\ell\hat{r}}\left(\vec{r}\right)$ is the probability density (in $\vec{r}$, $\ell$ and $\hat{r}$) that the particle is located at $\vec{r}$ after $n$ hops and that in the last hop it moved a distance $\ell$ in direction $\hat{r}$. Clearly $p_{n,\ell\hat{r}}\left(\vec{r}\right)$ includes in it the probability distribution for the direction and magnitude of the previous steps. For our model it obeys the following recursion relation $$\begin{aligned}
p_{n+1,\ell\hat{r}}\left(\vec{r}\right) = \gamma_0 \frac{s\left(\ell\right)}{\Omega_{d}}\int d\ell' d\hat{r}' p_{n,\ell'\hat{r}'}\left(\vec{r}-\ell\hat{r}\right) + \gamma_{f} p_{n,\ell\hat{r}}\left(\vec{r}-\ell\hat{r}\right)+ \gamma_{b} p_{n,-\ell\hat{r}}\left(\vec{r}-\ell\hat{r}\right) . \label{pnrec}\end{aligned}$$ The first term represents the case that hop $n+1$ was chosen with a random length with probability density $s\left(\ell\right)$ and in a random direction in $d$ dimensions with a uniform azimuthal probability density $\frac{1}{\Omega_{d}}$, the second term represents the case that hop $n+1$ was identical in direction and length to hop $n$, and the third term represents the case that hop $n+1$ was the exact opposite of hop $n$. The integration variables $\ell'$ and $\hat{r}'$ in the first term are the magnitude and direction of hop $n$.
Given Eqs. (\[pr\_4\],\[pr\_4\_d3\]), to proceed with the solution we introduce the Fourier transform of $p_{n,\ell\hat{r}}\left(\vec{r}\right)$, $$\begin{aligned}
&\tilde{p}_{n,\ell\hat{r}}\left(\vec{k}\right) = \int d\vec{r} e^{i\vec{k}\cdot\vec{r}} p_{n,\ell\hat{r}}\left(\vec{r}\right) ,\nonumber\\
&p_{n,\ell\hat{r}}\left(\vec{r}\right) = \frac{1}{\left(2\pi\right)^{d}} \int d\vec{k} e^{-i\vec{k}\cdot\vec{r}} \tilde{p}_{n,\ell\hat{r}}\left(\vec{k}\right) , \label{fourier}\end{aligned}$$ Substituting this in the recursion relation (\[pnrec\]) yields $$\begin{aligned}
\tilde{p}_{n+1,\ell\hat{r}}\left(\vec{k}\right) &= \nonumber\\
&=
\int d\vec{r} e^{i\vec{k}\cdot\vec{r}}
\left[
\gamma_0 \frac{s\left(\ell\right)}{\Omega_{d}} \cdot \frac{1}{\left(2\pi\right)^{d}} \int d\ell' d\hat{r}' d\vec{k}' e^{-i\vec{k}'\cdot\left(\vec{r}-\ell\hat{r}\right)} \tilde{p}_{n,\ell'\hat{r}'}\left(\vec{k}'\right) \right.\nonumber\\
&\left.+\gamma_{f} \frac{1}{\left(2\pi\right)^{d}} \int d\vec{k}' e^{-i\vec{k}'\cdot\left(\vec{r}-\ell\hat{r}\right)} \tilde{p}_{n,\ell\hat{r}}\left(\vec{k}'\right) +\gamma_{b} \frac{1}{\left(2\pi\right)^{d}} \int d\vec{k}' e^{-i\vec{k}'\cdot\left(\vec{r}-\ell\hat{r}\right)} \tilde{p}_{n,-\ell\hat{r}}\left(\vec{k}'\right)\right] \nonumber\\
&=
\gamma_0 \frac{s\left(\ell\right)}{\Omega_{d}} \int d\ell' d\hat{r}' d\vec{k}' e^{i\vec{k}'\cdot\ell\hat{r}} \tilde{p}_{n,\ell'\hat{r}'}\left(\vec{k}'\right)\delta\left(\vec{k}-\vec{k}'\right) \nonumber\\
&+ \gamma_{f} \int d\vec{k}' e^{i\vec{k}'\cdot\ell\hat{r}} \tilde{p}_{n,\ell\hat{r}}\left(\vec{k}'\right)\delta\left(\vec{k}-\vec{k}'\right) + \gamma_{b} \int d\vec{k}' e^{i\vec{k}'\cdot\ell\hat{r}} \tilde{p}_{n,-\ell\hat{r}}\left(\vec{k}'\right)\delta\left(\vec{k}-\vec{k}'\right) \nonumber\\
&=
\gamma_0 \frac{s\left(\ell\right)}{\Omega_{d}}\int d\ell' d\hat{r}' e^{i\vec{k}\cdot\ell\hat{r}} \tilde{p}_{n,\ell'\hat{r}'}\left(\vec{k}\right)
+ \gamma_{f} e^{i\vec{k}\cdot\ell\hat{r}} \tilde{p}_{n,\ell\hat{r}}\left(\vec{k}\right)+ \gamma_{b} e^{i\vec{k}\cdot\ell\hat{r}} \tilde{p}_{n,-\ell\hat{r}}\left(\vec{k}\right) ,\end{aligned}$$ where the first equality is substitution of (\[pnrec\]) and (\[fourier\]), in the second equality we integrated over $\vec{r}$, and in the third equality we integrated over $\vec{k}'$.
This may be written in operator form as $$\begin{aligned}
\left|\tilde{p}_{n+1}\left(\vec{k}\right)\right\rangle = \left( \gamma_{0}{\cal M}_{0}+ \gamma_{f}{\cal M}_{f} + \gamma_{b}{\cal M}_{b} \right) \left|\tilde{p}_{n}\left(\vec{k}\right)\right\rangle ,\end{aligned}$$ where $\left|\tilde{p}_{n}\left(\vec{k}\right)\right\rangle$ is an infinite-dimensional vector whose entries are $\tilde{p}_{n,\ell\hat{r}}$, and the linear operators ${\cal M}_{0},{\cal M}_{f}$ and ${\cal M}_{b}$ are defined by the following matrix elements $$\begin{aligned}
&\left[{\cal M}_{0}\right]_{\ell\hat{r},\ell'\hat{r}'} = \frac{s\left(\ell\right)}{\Omega_{d}} e^{i\vec{k}\cdot\ell\hat{r}} ,\nonumber\\
&\left[{\cal M}_{f}\right]_{\ell\hat{r},\ell'\hat{r}'} = e^{i\vec{k}\cdot\ell\hat{r}} \delta\left(\ell-\ell'\right) \delta\left(\hat{r}-\hat{r}'\right) ,\nonumber\\
&\left[{\cal M}_{b}\right]_{\ell\hat{r},\ell'\hat{r}'} = e^{i\vec{k}\cdot\ell\hat{r}} \delta\left(\ell-\ell'\right) \delta\left(\hat{r}+\hat{r}'\right) .\end{aligned}$$ Note that $\left|\tilde{p}_{n}\left(\vec{k}\right)\right\rangle$ is related to $\tilde{p}_{n}\left(\vec{k}\right)$ by the Fourier transform of Eq. (\[pn\_integral\]), or equivalently in bra-ket notation $$\begin{aligned}
\tilde{p}_{n}\left(\vec{k}\right)=\left\langle 1\right|\left.\tilde{p}_{n}\left(\vec{k}\right)\right\rangle .\end{aligned}$$ Thus the evolution of the distribution with the number of hops may be written as $$\begin{aligned}
\left|\tilde{p}_{n}\left(\vec{k}\right)\right\rangle = \left( \gamma_{0}{\cal M}_{0} + \gamma_{f}{\cal M}_{f}+ \gamma_{b}{\cal M}_{b} \right)^{n-1} \left|\tilde{p}_{1}\left(\vec{k}\right)\right\rangle .\end{aligned}$$ We find the distribution $\left|\tilde{p}_{1}\left(\vec{k}\right)\right\rangle$ after the first hop by $$\begin{aligned}
\tilde{p}_{1,\ell\hat{r}}\left(\vec{k}\right) = \int d\vec{r} e^{i\vec{k}\cdot\vec{r}} p_{1,\ell\hat{r}}\left(\vec{r}\right) = \frac{s(\ell)}{\Omega_{d}} \int d\vec{r} e^{i\vec{k}\cdot\vec{r}} \delta\left(\vec{r}-\ell\hat{r}\right) = \frac{s(\ell)}{\Omega_{d}}e^{i\vec{k}\cdot\ell\hat{r}} .\end{aligned}$$ Integrating over all possible lengths $\ell$ and directions $\hat{r}$ of hop $n$ we obtain $$\begin{aligned}
\tilde{p}_{1}\left(\vec{k}\right) = \int d\ell d\hat{r} \tilde{p}_{1,\ell\hat{r}}\left(\vec{k}\right) = \int d\ell d\hat{r} \frac{s(\ell)}{\Omega_{d}} e^{i\vec{k}\cdot\ell\hat{r}} = \int d\ell s(\ell) sinc(k\ell) \equiv f(k), \label{fdef}\end{aligned}$$ where in the last equality we introduce the function $f(k)$, which is a transform of the single step size distribution $s(\ell)$. We note that this function is symmetric, $f(k)=f(-k)$. Using the relations $$\begin{aligned}
&{\cal M}_{0}\left|\tilde{p}_{1}\left(m\vec{k}\right)\right\rangle=f(mk)\left|\tilde{p}_{1}\left(\vec{k}\right)\right\rangle ,\nonumber\\
&{\cal M}_{f}\left|\tilde{p}_{1}\left(m\vec{k}\right)\right\rangle=\left|\tilde{p}_{1}\left[(1+m)\vec{k}\right]\right\rangle ,\nonumber\\
&{\cal M}_{b}\left|\tilde{p}_{1}\left(m\vec{k}\right)\right\rangle=\left|\tilde{p}_{1}\left[(1-m)\vec{k}\right]\right\rangle ,\nonumber\\
&\left\langle 1|\tilde{p}_{1}\left(m\vec{k}\right)\right\rangle=f(mk) ,\label{mb_reduce}\end{aligned}$$ it is a matter of straightforward algebra to find the value of $$\begin{aligned}
\tilde{p}_{n}\left(\vec{k}\right) &= \int d\ell d\hat{r} \tilde{p}_{n,\ell\hat{r}}\left(\vec{k}\right) =
\left\langle 1\right|\left(\gamma_{0}{\cal M}_{0}+\gamma_{f}{\cal M}_{f}+\gamma_{b}{\cal M}_{b}\right)^{n-1}\left|\tilde{p}_{1}\left(\vec{k}\right)\right\rangle ,\label{pngen}\end{aligned}$$ for any finite $n$. The evaluation can be simplified by noting that the operators satisfy $$\begin{aligned}
&{\cal M}^{2}_{b}={\bf 1} ,\nonumber\\
&{\cal M}_{f}{\cal M}_{b}{\cal M}_{f}={\cal M}_{b} ,\nonumber\\
&{\cal M}_{0}{\cal M}_{b}{\cal M}_{0}={\cal M}_{0} ,\nonumber\\
&{\cal M}_{0}{\cal M}_{b}{\cal M}_{f}={\cal M}_{0} ,\nonumber\\
&{\cal M}_{f}{\cal M}_{b}{\cal M}_{0}={\cal M}_{0} ,\end{aligned}$$ where ${\bf 1}$ is the identity operator. For small values of $n$, Eq. (\[pngen\]) yields $$\begin{aligned}
&\tilde{p}_{0}\left(\vec{k}\right)=1 ,\nonumber\\
&\tilde{p}_{1}\left(\vec{k}\right)=f_{1} ,\nonumber\\
&\tilde{p}_{2}\left(\vec{k}\right)=\gamma_{0}f^{2}_{1}+\gamma_{f} f_{2}+\gamma_{b} ,\nonumber\\
&\tilde{p}_{3}\left(\vec{k}\right)=\gamma^{2}_{0}f^{3}_{1}+2\gamma_{0}\gamma_{f}f_{1}f_{2}+\gamma_{f}^{2}f_{3}+\gamma_{b}\left(2-\gamma_{b}\right)f_{1} ,\nonumber\\
&\tilde{p}_{4}\left(\vec{k}\right)=\gamma^{3}_{0}f^{4}_{1}+3\gamma^{2}_{0}\gamma_{f}f^{2}_{1}f_{2}+\gamma_{0}\gamma^{2}_{f}\left(2f_{1}f_{3}+f^{2}_{2}\right)+\gamma^{3}_{f}f_{4}+\nonumber\\
&+\gamma_{b}\gamma_{0}\left(3+\gamma_{f}-\gamma_{b}\right)f^{2}_{1}+2\gamma_{f}\gamma_{b}f_{2}+\gamma_{b}\left(\gamma_{b}+\gamma^{2}_{f}\right) ,\nonumber\\
&\tilde{p}_{5}\left(\vec{k}\right)=\gamma^{4}_{0}f^{5}_{1}+4\gamma^{3}_{0}\gamma_{f}f^{3}_{1}f_{2}+3\gamma^{2}_{0}\gamma^{2}_{f}f_{1}\left(f_{1}f_{3}+f^{2}_{2}\right)+\nonumber\\
&+2\gamma_{0}\gamma^{3}_{f}\left(f_{1}f_{4}+f_{2}f_{3}\right)+\gamma^{4}_{f}f_{5}+\gamma_{b}\left[\gamma_{b}+2\left(1-\gamma_{b}\right)\left(\gamma_{b}+\gamma^{2}_{f}\right)\right]f_{1}+\nonumber\\
&+\gamma_{b}\gamma^{2}_{0}\left(4+2\gamma_{f}-\gamma_{b}\right)f^{3}_{1}+2\gamma_{f}\gamma_{b}\gamma_{0}\left(3+\gamma_{f}\right)f_{1}f_{2}+\gamma_{b}\gamma^{2}_{f}\left(2+\gamma_{b}\right)f_{3} ,
\label{pnsmall}\end{aligned}$$ where we used for brevity $f_{n}\equiv f(nk)$.
Physically, $\gamma_{0}$ is the probability that the particle performs an uncorrelated hop. In real space, we get $$\begin{aligned}
p_{n}\left(\vec{r}\right) = \frac{1}{\left(2\pi\right)^{d}} \int d\vec{k} e^{-i\vec{k}\cdot\vec{r}} \tilde{p}_{n}\left(\vec{k}\right) = \frac{\Omega_{d}}{\left(2\pi\right)^{d}} \int dk \tilde{p}_{n}\left(\vec{k}\right) k^{d-1} sinc(kr) .\label{pnreal}\end{aligned}$$ Note that $\tilde{p}_{n}\left(\vec{k}\right)$ depends only on $k$ and $p_{n}\left(\vec{r}\right)$ depends only on $r$, however we write their arguments as $\vec{k}$ and $\vec{r}$ to emphasize that these are probability densities with respect to the vectors $\vec{k}$ and $\vec{r}$. For $n=0,1$ we get $$\begin{aligned}
&p_{0}\left(\vec{r}\right)=\frac{\delta(r)}{\Omega_{d} r^{d-1}} ,\nonumber\\
&p_{1}\left(\vec{r}\right)=\frac{s(r)}{\Omega_{d} r^{d-1}} .\label{p01}\end{aligned}$$ These results can also be obtained by noting that for $n=0$ the particle does not move, and thus its displacement must be zero, while for $n=1$ it performs one move with a step size distribution $s\left(\ell\right)$.
Since every time the particle performs a backward move it effectively cancels its previous hop, the probability that the particle moved a distance $\vec{r}$ given that it performed $n$ hops, $n_{b}$ of which were backwards, is propotional to the probability that it moved a distance $\vec{r}$ given that it performed $n-2n_{b}$ hops without backward hops, and thus $\tilde{p}_{n}\left(\vec{k}\right)$ for a general value of $\gamma_{b}$ may be written as a linear combination of $\tilde{p}_{n}\left(\vec{k}\right)$ for $\gamma_{b}=0$, $$\begin{aligned}
\tilde{p}_{n}\left(\vec{k},\gamma_{b}\right)=\sum^{\left\lfloor\frac{n}{2}\right\rfloor}_{m=0}A_{m,n}\left(\gamma_{b}\right)\tilde{p}_{n-2m}\left(\vec{k},0\right) ,\end{aligned}$$ where $A_{n,m}(\gamma_{b})$ are some constants. Trivially, for $\gamma_{b}=0$ we have $A_{m,n}=\delta_{m,0}$. Therefore, by linearity, we find that Eq. (\[pdsum\]) for a finite value of $\gamma_{b}$ may be written as $$\begin{aligned}
P_{d}\left(\vec{r},\tau,\gamma_{b}\right)=\sum^{\infty}_{n=0}\tilde{q}_{n}\left(\tau\right)p_{n}\left(\vec{r},\gamma_{b}=0\right) ,\end{aligned}$$ where $\tilde{q}_{n}$ are linear combinations of the original $q_{n}$’s. This means that a system with a finite $\gamma_{b}$ behaves the same as a system with $\gamma_{b}=0$ but with different time correlations. Therefore, for general temporal correlations it is sufficient to discuss only the case $\gamma_{b}=0$. In the limit $\gamma_{b}=0$ we can find an explicit expression for Eq. (\[pngen\]) $$\begin{aligned}
\tilde{p}_{n}\left(\vec{k}\right) &=
\left\langle 1\right|\left[\gamma_0 {\cal M}_{0}+\gamma_{f}{\cal M}_{f}\right]^{n-1}\left|\tilde{p}_{1}\left(\vec{k}\right)\right\rangle = \nonumber\\
&=\sum^{n}_{M=1}\gamma_0^{M-1}\gamma^{n-M}_{f}\sum^{n}_{m_{1}=1}...\sum^{n}_{m_{M}=1}\delta_{n,\sum^{M}_{i=1}m_{i}}\prod^{M}_{i=1}f\left(m_{i}k\right) .\end{aligned}$$
In conclusion, the required stages to evaluate the PDF of the displacements, $P_{r}$, in our model for a given distribution of step length $s\left(\ell\right)$ and temporal correlations encoded in $q_{n}$, are: 1) calculate the function $f(k)$ using Eq. (\[fdef\]), 2) calculate the Fourier transform of the discrete distribution given $n$ steps $\tilde{p}_{n}\left(\vec{k}\right)$ using Eq. (\[pngen\]), 3) evaluate the Fourier transform of the discrete distribution $P_{d}\left(\vec{k}\right)$ using the Fourier transform of Eq. (\[pdsum\]), and 4) calculate the convolution of the discrete and continuous processes using Eq. (\[pr\_4\]).
Specific distributions
----------------------
In this section we show how $P_{r}$ behaves for several specific choices of the step size distribution and of the temporal and orientational correlations. We consider three types of step size distributions, $s(\ell)$, and three types of temporal correlations, encoded in $q_{n}$, and examine all nine combinations. The three different distributions that we consider for the step size are: 1) a Dirac delta distribution $$\begin{aligned}
s_{D}\left(\ell\right)=\delta\left(\ell-a\right) ,\label{s_dist_delta}\end{aligned}$$ 2) a modified Gaussian distribution $$\begin{aligned}
s_{G}\left(\ell\right)=\left(\frac{\ell}{a}\right)^{2\nu}e^{-\nu\ell^{2}/a^{2}}\frac{2\nu^{\nu+1/2}}{a\Gamma\left(\nu+\frac{1}{2}\right)} ,\label{s_dist_gauss}\end{aligned}$$ which in the limit $\nu\rightarrow\infty$ converges to the Dirac distribution, and 3) a Cauchy distribution $$\begin{aligned}
s_{C}\left(\ell\right)=\frac{4a\ell^{2}}{\pi\left(\ell^{2}+a^{2}\right)^{2}} .\label{s_dist_cauchy}\end{aligned}$$ For all three distributions the most likely step size is $a$. The mean step size for the three distributions is $$\begin{aligned}
&\left\langle\ell\right\rangle_{D}=\int^{\infty}_{0}\ell s_{D}\left(\ell\right)d\ell=a ,\nonumber\\
&\left\langle\ell\right\rangle_{G}=\int^{\infty}_{0}\ell s_{G}\left(\ell\right)d\ell=a\frac{\sqrt{\nu}\Gamma\left(\nu\right)}{\Gamma\left(\nu+\frac{1}{2}\right)} ,\nonumber\\
&\left\langle\ell\right\rangle_{C}=\int^{\infty}_{0}\ell s_{C}\left(\ell\right)d\ell=\infty ,\label{means}\end{aligned}$$ and the variance of the step size for the three distributions is $$\begin{aligned}
&\sigma^{2}_{D}=\left\langle\ell^{2}\right\rangle_{D}-\left\langle\ell\right\rangle_{D}^{2}=0 ,\nonumber\\
&\sigma^{2}_{G}=\left\langle\ell^{2}\right\rangle_{G}-\left\langle\ell\right\rangle_{G}^{2}=a^{2}\left(1+\frac{1}{2\nu}-\frac{\nu\Gamma^{2}\left(\nu\right)}{\Gamma^{2}\left(\nu+\frac{1}{2}\right)}\right) ,\nonumber\\
&\sigma^{2}_{C}=\left\langle\ell^{2}\right\rangle_{C}-\left\langle\ell\right\rangle_{C}^{2}=\infty .\end{aligned}$$ We note that for large $\nu$ the asymptotic expansions of the prefactors for the mean and variance of the modified Gaussian distribution are given by $$\begin{aligned}
&\frac{\sqrt{\nu}\Gamma\left(\nu\right)}{\Gamma\left(\nu+\frac{1}{2}\right)}=1+\frac{1}{8\nu}+O\left(\nu^{-2}\right) ,\nonumber\\
&1+\frac{1}{2\nu}-\frac{\nu\Gamma^{2}\left(\nu\right)}{\Gamma^{2}\left(\nu+\frac{1}{2}\right)}=\frac{1}{4\nu}+O\left(\nu^{-2}\right) .\label{asymp}\end{aligned}$$
The Dirac distribution is a natural simplistic choice. The modified Gaussian distribution is numerically almost indistinguishable from a shifted Gaussian with the same mean and variance, with the advantage that many of the calculations are analytically tractable. It is a representative of general distributions with a single peak and finite variance. From Eqs. (\[means\]-\[asymp\]) we see that for $\nu\gg1$ the mean over standard deviation of the Gaussian distribution is $\frac{\left\langle \ell\right\rangle_{G}}{\sigma_{G}}=2\sqrt{\nu}$. The modified Gaussian is a good approximation for every such distribution for which this ratio is large enough. The Cauchy distribution is a simple representative of heavy tail distributions, which are rather common in biological processes [@Coding2008; @Rupprecht2016; @Reynolds2013; @Palyulin2017; @Detcheverry2017; @Johnson2008; @Kurihara2017; @Ariel2017; @Mwaffo2015]. The modified Gaussian distribution for $\nu=4,10,40$ and $a=1$, and the Cauchy distribution with $a=1$ are shown in Fig. \[s\_dist\_fig\].
![The step size distribution $s(\ell)$ for the modified Gaussian Eq. (\[s\_dist\_gauss\]) and the Cauchy Eq. (\[s\_dist\_cauchy\]) distributions, all with most likely step size $a=1$.[]{data-label="s_dist_fig"}](s_dist_fig_sm.pdf "fig:"){width="0.45\columnwidth"} ![The step size distribution $s(\ell)$ for the modified Gaussian Eq. (\[s\_dist\_gauss\]) and the Cauchy Eq. (\[s\_dist\_cauchy\]) distributions, all with most likely step size $a=1$.[]{data-label="s_dist_fig"}](s_dist_fig_loglog_sm.pdf "fig:"){width="0.45\columnwidth"}
The function $f(k)$ defined in Eq. (\[fdef\]) for these three distributions is $$\begin{aligned}
&f_{D}(k)=sinc\left(ak\right) ,\nonumber\\
&f_{G}(k)={}_{1}F_{1}\left(\nu+\frac{1}{2},\frac{3}{2},-\frac{a^{2}k^{2}}{4\nu}\right) ,\nonumber\\
&f_{C}(k)=e^{-ak} .\end{aligned}$$
In the modified Gaussian distribution, we find that for integer values of $\nu$ $$\begin{aligned}
f_{G}(k)=\frac{\left(-1\right)^{\nu-1}\left(\nu-1\right)!\sqrt{\nu}e^{-a^{2}k^{2}/4\nu}H_{2\nu-1}\left(\frac{ak}{2\sqrt{\nu}}\right)}{ak\left(2\nu-1\right)!} ,\label{fint}\end{aligned}$$ where $H$ are the Hermite polynomials [@hermite].
For the Cauchy distribution we find that when $\gamma_{b}=0$ then $$\begin{aligned}
\tilde{p}^{C}_{n}(k)=e^{-nak} ,\end{aligned}$$ for any value of $\gamma_{f}$, and thus $p^{C}_{n}(r)$ is $$\begin{aligned}
&p^{C}_{n}(r)=\frac{\left(d-2\right)!}{r\left[r^{2}+\left(na\right)^{2}\right]^{(d+1)/2}}\times\nonumber\\
&\left\{\left[\left(na\right)^{2}-r^{2}\right]\sin\left[\left(d+1\right)\cot^{-1}\left(\frac{na}{r}\right)\right]-2nar\cdot\cos\left[\left(d+1\right)\cot^{-1}\left(\frac{na}{r}\right)\right]\right\} .\end{aligned}$$ By including the continuous process, we find that in three dimensional systems \[see Eqs. (\[pdsum\]) and (\[pr\_4\_d3\])\], $$\begin{aligned}
&\hat{p}^{C}_{n}(r)=\frac{1}{2\pi^{2}r}\int^{\infty}_{0}dk e^{-nak}k\sin(kr)e^{-\frac{\alpha^{2}k^{2}}{2}}=\nonumber\\
&=\frac{e^{-\frac{r^{2}-n^{2}a^{2}}{2\alpha^{2}}}}{4\sqrt{2}i\pi^{3/2}\alpha^{3}r}\left\{\left(-na+ir\right)e^{\frac{-inar}{\alpha^{2}}}\left[1+Erf\left(\frac{-na+ir}{\sqrt{2}\alpha}\right)\right]-\right.\nonumber\\
&\left.-\left(-na-ir\right)e^{\frac{inar}{\alpha^{2}}}\left[1+Erf\left(\frac{-na-ir}{\sqrt{2}\alpha}\right)\right]\right\} ,\end{aligned}$$ with $Erf(z)$ being the error function [@erf] defined by $$\begin{aligned}
Erf(z)=\frac{2}{\sqrt{\pi}}\int^{z}_{0}e^{-t^{2}}dt .\end{aligned}$$
The three types of temporal correlations that we consider are modeled by three possible distributions $q_{n}$: 1) Poissonian $$\begin{aligned}
q^{Pois}_{n}\left(\tau\right)=\frac{\tau^{n}e^{-\tau}}{n!} ,\end{aligned}$$ 2) binomial $$\begin{aligned}
q^{bin}_{n}\left(\tau\right)=\left(\begin{array}{c}\tau\\n\end{array}\right)p^{n}\left(1-p\right)^{\tau-n} ,\end{aligned}$$ with $p=0.6$, and 3) geometric $$\begin{aligned}
q^{geo}_{n}\left(\tau\right)=\tau^{-1}\left(1-\tau^{-1}\right)^{n} .\end{aligned}$$ In the Poissonian distribution there are no temporal correlations between the hops. The main qualitative difference between the binomial and the geometric distribution is that for the geometric distribution $\frac{\partial q^{geo}_{n}}{\partial n}<0$ always, i.e. more steps are always less likely, while in the binomial distribution the most likely number of steps is finite. Physically, the geometric distribution corresponds to the case where the waiting time between hops grows with time, while the binomial distribution corresponds to the case where only a finite maximal number of steps is allowed at any time interval. In all cases we chose the parameters so that on average the walker performs three discrete steps during the time interval $\tau$, i.e. $$\begin{aligned}
\left\langle n\right\rangle=\sum^{\infty}_{n=0}nq_{n}=3. \end{aligned}$$
Numerical Results
-----------------
We evaluated all the expressions above in order to get $P(r)$. We start by comparing the different step size distributions without temporal correlations. The corresponding results are shown in Fig. \[pr\_dist\]. As expected, the Cauchy distribution shows no peaks in $P_{r}(r)$. For the Gaussian and the Dirac distributions, we find that the peaks, when they are seen, are located at integer multiples of $a$ and are more pronounced as the step size distribution is more narrow.
![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems with a Poissonian distribution of $q_{n}$. We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="pr_dist"}](pr_dist_sm.pdf){width="0.6\columnwidth"}
We now consider the effect of different temporal correlations and of varying the persistence $\gamma_{f}$, see Fig. \[dist\_fig\]. We find that the peaks are more pronounced as the probability to move forward $\gamma_{f}$ is larger.
![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](pn4a01gf03_v2_sm.pdf "fig:"){width="0.3\columnwidth"} ![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](pn4a01gf06_v2_sm.pdf "fig:"){width="0.3\columnwidth"} ![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](pn4a01gf08_v2_sm.pdf "fig:"){width="0.3\columnwidth"}\
![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](pn10a01gf03_sm.pdf "fig:"){width="0.3\columnwidth"} ![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](pn10a01gf06_sm.pdf "fig:"){width="0.3\columnwidth"} ![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](pn10a01gf08_sm.pdf "fig:"){width="0.3\columnwidth"}\
![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](pn40a01gf03_sm.pdf "fig:"){width="0.3\columnwidth"} ![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](pn40a01gf06_sm.pdf "fig:"){width="0.3\columnwidth"} ![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](pn40a01gf08_sm.pdf "fig:"){width="0.3\columnwidth"}\
![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](pda01gf03_sm.pdf "fig:"){width="0.3\columnwidth"} ![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](pda01gf06_sm.pdf "fig:"){width="0.3\columnwidth"} ![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](pda01gf08_sm.pdf "fig:"){width="0.3\columnwidth"}\
![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](pcauchy_sm.pdf "fig:"){width="0.3\columnwidth"} ![The PDF of the walker’s displacement $P(r)$ as a function of $r$ for different distributions in three dimensional systems. The first three rows correspond to a modified Gaussian distributed step size, Eq. (\[s\_dist\_gauss\]), with different values of $\nu$ and $\gamma_{f}$ as noted in each panel, the fourth row shows the results for the Dirac delta step size distribution, Eq. (\[s\_dist\_delta\]), with different values of $\gamma_{f}$, and the last panel shows the results for the Cauchy distribution, Eq. (\[s\_dist\_cauchy\]), for which the value of $\gamma_{f}$ is immaterial. Each line corresponds to a different temporal distribution according to the legend. The dot-dashed black line, which appears only in the top row, is the displacement distribution of the continuous process alone, Eq. (\[pc\_def\]). We set $\gamma_{b}=0$, $a=1$ and $\alpha=0.1$ throughout.[]{data-label="dist_fig"}](plegend_sm.pdf "fig:"){width="0.3\columnwidth"}
For the Poissonian waiting time distribution we find that the height of the peaks decreases exponentially with the displacement, while for the other distributions it does not even necessarily decrease, as shown in the example plotted in Fig. \[bin\_fig\].
![The distribution $P_{r}(r)$ as a function of $r$ for the binomial distribution at $\tau=4$ with $p=0.95$ and a Dirac delta step size distribution. The other parameters are $a=1,\alpha=0.1,\gamma_{b}=0,\gamma_{f}=0.8$.[]{data-label="bin_fig"}](pda01gf08bin95_sm.pdf){width="0.5\columnwidth"}
In conclusion, if peaks are found in a PDF, one should look at several factors. The distance between peaks is the mean size of the discrete step. The height of the peaks correlate with higher values of $\gamma_{f}$. The width of the peak at $r=0$ is mostly due to the continuous process. The width of the peaks is determined by a combination of the continuous process and the variance of the discrete step size distribution. Except for the Dirac delta step size distribution, the width of the peaks increases for larger displacements, and thus only a finite number of peaks is visible. The relative height of the peaks gives information about the temporal correlations. If the heights decrease exponentially with the displacement, then there are no temporal correlations. A faster-than-exponential decrease in the relative height, as in the geometric distribution, is related to negative temporal correlations, i.e. tendency to have fewer moves. Extreme cases, such as that shown in Fig. \[bin\_fig\] are related to a higher probability of having many moves.
No peaks without orientational correlations {#sec_proof}
===========================================
Here we will show that $P_{r}\left(r,\tau\right)$ has at most one peak if there are no positive orientational correlations, i.e. $\gamma_{f}=0$. If $\gamma_{f}=0$, then by Eqs. (\[pngen\]) and (\[pnsmall\]) we find that $\tilde{p}_{n}(k)$ may be written as $$\begin{aligned}
\tilde{p}_{n}(k)=\sum^{n}_{m=0}c_{m}\tilde{p}^{0}_{m}(k) ,\end{aligned}$$ where $\tilde{p}^{0}_{m}(k)$ is the value of $\tilde{p}_{m}(k)$ at $\gamma_{b}=0$, and $c_{m}$ are non-negative coefficients whose exact value is immaterial for the arguments presented in this section.
The continuous Gaussian process, Eq. (\[pc\_def\]), can only smear the peaks, if they exist, and thus we may consider only the case $\alpha=0$. Hence, in order to show that $P_{r}(r)$ has at most one peak, it is enough to show that $p_{1}(r)$ has one peak and that $p_{n\geq2}(r)$ are non-increasing functions of $r$. Furthermore, from the same reasoning, it is enough to consider the step size distribution $s(\ell)=\delta\left(\ell-a\right)$, since for any other step size distribution peaks, if they exist, will be more smeared.
First, let us consider $p_{1}(r)$, which using Eq. (\[p01\]) and the Dirac distribution, is given by $$\begin{aligned}
p_{1}(r)=\frac{\delta\left(\ell-a\right)}{\Omega_{d}r^{d-1}} .\end{aligned}$$ It obviously has a single peak at $r=a$, and is zero elsewhere.
For $n\geq2$, we start from Eq. (\[pnrec\]), and note that in the case $\gamma_f=0$ there is no memory, such that $$\begin{aligned}
&p_{n+1}(r)=\int d\ell d\hat{r}p_{n+1,\ell\hat{r}}\left(\vec{r}\right) = \int d\ell d\hat{r}\frac{\delta(\ell-a)}{\Omega_{d}}\int d\ell' d\hat{r}' p_{n,\ell'\hat{r}'}\left(|\vec{r}-\ell\hat{r}|\right)=\nonumber\\
&=\int d\ell d\hat{r}\frac{\delta(\ell-a)}{\Omega_{d}} p_{n}\left(\sqrt{r^{2}+\ell^{2}-2\ell\vec{r}\cdot\hat{r}}\right) .\end{aligned}$$ Performing the integral over $\ell$, and writing the integral over $\hat{r}$ as an integral over the $d-1$ angles yields $$\begin{aligned}
p_{n+1}(r)=\frac{1}{\Omega_{d}}\int d\vec{\phi}_{d-2}\sin\theta d\theta p_{n}\left(\sqrt{r^{2}+a^{2}-2ar\cos\theta}\right) ,\end{aligned}$$ where $d\vec{\phi}_{d-2}$ contains all the other $d-2$ angles except for $\theta$. Performing the integral over $\vec{\phi}_{d-2}$ and changing the integration variable from $\theta$ to $r'=\sqrt{r^{2}+a^{2}-2ar\cos\theta}$ yields $$\begin{aligned}
p_{n+1}(r)=\frac{1}{2a}\int^{r+a}_{|r-a|}\frac{r'}{r}p_{n}\left(r'\right)dr' .\label{pnrecg0}\end{aligned}$$ We now use Eq. (\[pnrecg0\]) twice, such that $$\begin{aligned}
p_{n+2}(r)=\frac{1}{4a^{2}}\int^{r+a}_{|r-a|}\frac{r'}{r}dr'\int^{r'+a}_{|r'-a|}\frac{r''}{r'}p_{n}\left(r''\right)dr'' .\label{pnrecg02}\end{aligned}$$ We now prove by induction on $n$ that $p_{n}(r)$ is a non-increasing function of $r$. For $n=2$ and $n=3$ we can explicitly calculate $p_{n}(r)$ $$\begin{aligned}
&p_{2}(r)=\frac{1}{2\Omega_{d} a^{d-1}r}\Theta\left(2a-r\right) ,\nonumber\\
&p_{3}(r)=\frac{1}{2\Omega_{d} a^{3}}\left[\Theta\left(a-r\right)+\Theta\left(r-a\right)\Theta\left(3a-r\right)\frac{3a-r}{2r}\right] ,\end{aligned}$$ which are non-increasing functions of $r$. For $n\geq4$ we consider three different cases: $r>2a$, $2a>r>a$, and $a>r$.
$r>2a$
------
If $r>2a$, we first note that $\left|r-a\right|>0$, and thus in the whole integration range over $r'$ in Eq. (\[pnrecg02\]) $r'>r-a>a$. Therefore, $\left|r'-a\right|>0$ as well. Hence, Eq. (\[pnrecg02\]) may be written as $$\begin{aligned}
p_{n+2}(r)=\frac{1}{4a^{2}}\int^{r+a}_{r-a}dr'\int^{r'+a}_{r'-a}dr''\frac{r''}{r}p_{n}\left(r''\right)dr'' .\end{aligned}$$ Changing the order of integration yields $$\begin{aligned}
p_{n+2}(r)=\frac{1}{4a^{2}}\left[\int^{r}_{r-2a}dr''\int^{r''+a}_{r-a}dr'+\int^{r+2a}_{r}dr''\int^{r+a}_{r''-a}dr'\right]\frac{r''}{r}p_{n}\left(r''\right)dr'' .\end{aligned}$$ Performing the integrals over $r'$ yields $$\begin{aligned}
p_{n+2}(r)=\frac{1}{4a^{2}}\left[\int^{r}_{r-2a}dr''\left(r''-r+2a\right)+\int^{r+2a}_{r}dr''\left(r-r''+2a\right)\right]\frac{r''}{r}p_{n}\left(r''\right)dr'' .\end{aligned}$$ The derivative with respect to $r$ is $$\begin{aligned}
\frac{\partial p_{n+2}(r)}{\partial r}=-\frac{1}{4a^{2}r^{2}}\left[\int^{r}_{r-2a}dr''\left(r''+2a\right)+\int^{r+2a}_{r}dr''\left(-r''+2a\right)\right]r''p_{n}\left(r''\right)dr'' .\end{aligned}$$ In the second integral we change the integration variable to $r''-2a$ such that $$\begin{aligned}
\frac{\partial p_{n+2}(r)}{\partial r}=-\frac{1}{4a^{2}r^{2}}\int^{r}_{r-2a}dr''\left(r''+2a\right)r''\left[p_{n}(r'')-p_{n}(r''+2a)\right] .\end{aligned}$$ Since $p_{n}(r)$ is a non-increasing function of $r$ by the induction assumption, the integrand is non-negative, and thus the derivative of $p_{n+2}(r)$ is non-positive as required.
$2a>r>a$
--------
In this case, we find that Eq. (\[pnrecg02\]) reads $$\begin{aligned}
p_{n+2}(r)=\frac{1}{4a^{2}}\left[\int^{r+a}_{a}dr'\int^{r'+a}_{r'-a}dr''+\int^{a}_{r-a}dr'\int^{r'+a}_{a-r}dr''\right]\frac{r''}{r}p_{n}\left(r''\right) .\end{aligned}$$ Changing the order of integration yields $$\begin{aligned}
&p_{n+2}(r)=\nonumber\\
&\frac{1}{4a^{2}}\left[\int^{2a-r}_{0}dr''\int^{r''+a}_{a-r''}dr'+\int^{r}_{2a-r}dr''\int^{r''+a}_{r-a}dr'+\int^{r+2a}_{r}dr''\int^{r+a}_{r''-a}dr'\right]\frac{r''}{r}p_{n}\left(r''\right) .\end{aligned}$$ Performing the integrals over $r'$ yields $$\begin{aligned}
&p_{n+2}(r)=\nonumber\\
&\frac{1}{4a^{2}}\left[\int^{2a-r}_{0}dr''2r''+\int^{r}_{2a-r}dr''\left(r''-r+2a\right)+\int^{r+2a}_{r}dr''\left(r-r''+2a\right)\right]\frac{r''}{r}p_{n}\left(r''\right) .\end{aligned}$$ The derivative with respect to $r$ is $$\begin{aligned}
&\frac{\partial p_{n+2}(r)}{\partial r}=\nonumber\\
&-\frac{1}{4a^{2}r^{2}}
\left[\int^{2a-r}_{0}dr''2r''+\int^{r}_{2a-r}dr''\left(r''+2a\right)+\int^{r+2a}_{r}dr''\left(-r''+2a\right)\right]r''p_{n}\left(r''\right) .\end{aligned}$$ In the third integral we change the integration variable to $r''-2a$ such that $$\begin{aligned}
&\frac{\partial p_{n+2}(r)}{\partial r}=\nonumber\\
&-\frac{1}{4a^{2}r^{2}}
\left\{2\int^{2a-r}_{0}dr''r''^{2}p_{n}(r'')+\int^{r}_{2a-r}dr''\left(r''+2a\right)r''\left[p_{n}\left(r''\right)-p_{n}\left(r''+2a\right)\right]\right\} .\end{aligned}$$ As before, the integrand is positive, and thus $p_{n+2}(r)$ is a non-increasing function of $r$.
$r<a$
-----
In this last case we find that Eq. (\[pnrecg02\]) reads $$\begin{aligned}
p_{n+2}(r)=\frac{1}{4a^{2}}\left[\int^{r+a}_{a}dr'\int^{r'+a}_{r'-a}dr''+\int^{a}_{a-r}dr'\int^{r'+a}_{a-r'}dr''\right]\frac{r''}{r}p_{n}\left(r''\right) .\end{aligned}$$ Changing the order of integration yields $$\begin{aligned}
p_{n+2}(r)=\frac{1}{4a^{2}}\left[\int^{r}_{0}dr''\int^{a+r''}_{a-r''}dr'+\int^{2a-r}_{r}dr''\int^{r+a}_{a-r}dr'+\int^{r+2a}_{2a-r}dr''\int^{r+a}_{r''-a}dr'\right]\frac{r''}{r}p_{n}\left(r''\right) .\end{aligned}$$ Integrating over $r'$ yields $$\begin{aligned}
p_{n+2}(r)=\frac{1}{4a^{2}}\left[2\int^{r}_{0}dr''r''+2r\int^{2a-r}_{r}dr''+\int^{r+2a}_{2a-r}dr''\left(r-r''+2a\right)\right]\frac{r''}{r}p_{n}\left(r''\right) .\end{aligned}$$ The derivative is $$\begin{aligned}
&\frac{\partial p_{n+2}(r)}{\partial r}=-\frac{1}{4a^{2}r^{2}}\left[2\int^{r}_{0}dr''r''+\int^{r+2a}_{2a-r}dr''\left(-r''+2a\right)\right]r''p_{n}\left(r''\right)=\nonumber\\
&=-\frac{1}{4a^{2}r^{2}}\left[2\int^{r}_{0}dr''r''+\int^{r+2a}_{2a}dr''\left(-r''+2a\right)+\int^{2a}_{2a-r}dr''\left(-r''+2a\right)\right]r''p_{n}\left(r''\right) .\end{aligned}$$ In the third integral we change the integration variable to $r''-2a$, and in the fourth integral we change it to $2a-r''$ such that $$\begin{aligned}
&\frac{\partial p_{n+2}(r)}{\partial r}=\nonumber\\
&=-\frac{1}{4a^{2}r^{2}}\int^{r}_{0}dr''\left[2r''^{2}p_{n}(r)-r''\left(r''+2a\right)p_{n}\left(r''+2a\right)+r''\left(2a-r''\right)p_{n}\left(2a-r''\right)\right]=\nonumber\\
&=-\frac{1}{4a^{2}r^{2}}\int^{r}_{0}dr''\left\{r''^{2}\left[p_{n}(r)-p_{n}\left(r''+2a\right)+p_{n}(r)-p_{n}\left(2a-r''\right)\right]+\right.\nonumber\\
&\left.+2ar''\left[p_{n}\left(2a-r''\right)-p_{n}\left(2a+r''\right)\right]\right\} .\end{aligned}$$ Since $r''<r<a$ we find that $2a-r''>r$ and thus the integrand is positive, and as before the derivative is non-positive.
In conclusion, we proved that if $\gamma_{f}=0$, i.e. there are no positive directional correlations, there can be at most one peak in the displacement distribution. Therefore, if there is more than one peak in the displacement distribution, this implies the existence of positive directional correlations.
Summary {#sec_summary}
=======
In this paper we analyzed the motion of a single random walker with a specific type of persistence and external Gaussian noise. At each step it either moves in the exact same fashion (distance and direction) as in its previous move, makes the exact opposite move, or moves in a random direction. The timing of the different steps could be correlated. We derived an exact expression for the PDF of the walker’s location for general time-correlations and general step-length distribution.
We explicitly evaluated the PDF for three types of step-length distributions and three types of time-correlations. We found that under certain conditions the PDF exhibits peaks at specific displacements, and that these peaks are more pronounced for narrower step-length distributions and higher persistence, i.e. propensity to continue in the same direction as before. The heights of these peaks decrease exponentially with the displacement if there are no correlations between the timing of the steps, but if correlations exist the height of the peaks could even increase with the displacement.
Furthermore, we showed analytically that a necessary condition for the appearance of these peaks is a positive persistence, regardless of the time-correlations. Although our model is rather simplistic, we believe that this conclusion can be generalized. Namely, if the PDF exhibits peaks at various values of the displacement then the particle is persistent, i.e. it has a positive velocity autocorrelation.
One way to expand this work would be to consider a more realistic type of persistence, such as an Orenstein-Uhlenbeck process or Levy walks. Investigating more thoroughly the effect of the temporal correlations on the relative height of the peaks would also be an interesting avenue of research.
Our results may be used to understand the microscopic processes underlying the overall observed random walk. At long times many of the details of the random motion are averaged out, but they are present in the short time behaviour. Using our results, and expanding the simple model we present here, one may look for non-monotonicity; or, if such non-monotonicity is not observed at short times, this implies a bound on the orientational correlations. By looking for peaks in the displacement distribution in experiments, one may obtain data about the directional correlations of the particles. Moreover, the heights and the separation of the peaks give further insight into the underlying microscopic processes.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Niv Gov, Ralf Meltzer and Michael Urbakh for helpful discussions. This research was supported by the US-Israel Binational Science Foundation, by the Israel Science Foundation Grant No. 968/16, and by the joint Tel Aviv University - Potsdam University scholarship.
|
---
abstract: 'Recent observations from the Swift gamma-ray burst mission indicate that a fraction of gamma ray bursts are characterized by a canonical behaviour of the X-ray afterglows. We present an effective theory which allows us to account for X-ray light curves of both (short - long) gamma ray bursts and X-ray rich flashes. We propose that gamma ray bursts originate from massive magnetic powered pulsars.'
author:
- Paolo Cea
title: LIGHT CURVES OF SWIFT GAMMA RAY BURSTS
---
INTRODUCTION {#Introduction}
============
The unique capability of the Swift satellite has yielded the discovery of interesting new properties of short and long gamma ray burst (GRB) X-ray afterglows. Indeed, recent observations have provided new informations on the early behavior ( $t \, < \, 10^3 - 10^4 \, sec$) of the X-ray light curves of gamma ray bursts. These early time afterglow observations revealed that [@Chincarini:2005; @Nousek:2005; @O'Brien:2006] a fraction of bursts have a generic shape consisting of three distinct segments: an initial very steep decline with time, a subsequent very shallow decay, and a final steepening (for a recent review, see Piran 2005, Meszaros 2006). This canonical behaviour of the X-ray afterglows of gamma ray bursts is challenging the standard relativistic fireball model, leading to several alternative models (for a recent review of some of the current theoretical interpretations, see Mészáros 2006 and references therein).\
In order to determine the nature of both short and long gamma ray bursts, more detailed theoretical modelling is needed to establish a clearer picture of the mechanism. In particular, it is important to have at disposal an unified, quantitative description of the X-ray afterglow light curves.\
The main purpose of this paper is to present an effective theory which allows us to account for X-ray light curves of both gamma ray bursts and X-ray rich flashes (XRF). In a recent paper [@cea:2006] we set up a quite general approach to cope with light curves from anomalous $X$-ray pulsars (AXP) and soft gamma-ray repeaters (SGR). Indeed, we find that the canonical light curve of the X-ray afterglows is very similar to the light curve after the June 18, 2002 giant burst from AXP 1E 2259+586 (Woods et al. 2004). This suggests that our approach can be extended also to gamma ray bursts.\
The plan of the paper is as follows. In Sect. \[light\] we briefly review the general formalism presented in Cea (2006) to cope with light curves. After that, in Sect. \[050315\] through \[050416A\] we carefully compare our theory with the several gamma ray burst light curves. In Sect. \[origin\] we propose that gamma ray bursts originate from massive magnetic powered pulsars, namely pulsars with super strong dipolar magnetic field and mass $M \sim 10 \, M_{\bigodot}$. Finally, we draw our conclusions in Sect. \[conclusion\].
{#light}
Gamma ray bursts may be characterized by some mechanism which dissipates injected energy in a compact region. As a consequence the observed luminosity is time-dependent. In this section, following Cea (2006), we briefly discuss an effective description that allows us to determine the light curves, i.e. the time dependence of the luminosity. After that, we shall compare our approach with several light curves of Swift gamma ray bursts.\
In general, irrespective of the details of the dissipation process, the dissipated energy leads to the luminosity $L(t) =
- \frac{d E(t)}{dt}$. Actually, the precise behavior of $L(t)$ is determined once the dissipation mechanisms are known. However, we may accurately reproduce the time variation of $L(t)$ without precise knowledge of the microscopic dissipative mechanisms. Indeed, on general grounds we expect that the dissipated energy is given by: $$%
\label{2.1}
%
L(t) \; \; = \; \; - \; \frac{d E(t)}{dt} \; \; = \; \kappa(t) \; E^\eta \; \; , \; \; \eta \; \leq \; 1 \; \; \; ,
%$$ where $\eta$ is the efficiency coefficient. For an ideal system, where the initial injected energy is huge, the linear regime where $\eta = 1$ is appropriate. Moreover, we may safely assume that $\kappa(t) \simeq \kappa_0$ constant. Thus we get: $$%
\label{2.2}
%
L(t) \; \; = \; \; - \; \frac{d E(t)}{dt} \; \; \simeq \; \kappa_0 \; E \; \; .
%$$ It is then straightforward to solve Eq. (\[2.2\]): $$\begin{aligned}
%
\label{2.3}
%
E(t) \; & = & \; E_0 \; \exp(- \frac{t}{\tau_0}) \; \; \; \; , \; \; \;
L(t) \; = \; L_0 \; \exp(- \frac{t}{\tau_0}) \; \; ,
\\
\nonumber
L_0 \; & = &\; \frac{E_0}{\tau_0} \; \; , \; \;
\tau_0 \; = \; \frac{1}{\kappa_0} \; \; \; .
%
\end{aligned}$$ Note that the dissipation time $\tau_0 = \frac{1}{\kappa_0}$ encodes all the physical information on the microscopic dissipative phenomena. Since the injected energy is finite, the dissipation of energy degrades with the decrease in the available energy. Thus, the relevant equation is Eq. (\[2.1\]) with $\eta < 1$. In this case, by solving Eq. (\[2.1\]) we find: $$%
\label{2.4}
%
L(t) \; = \; L_0 \; \left ( 1 - \frac{t}{t_{dis}} \right )^{\frac{\eta}{1-\eta}} \; \; ,
%$$ where we have introduced the dissipation time: $$%
\label{2.5}
%
t_{dis} \; \; = \; \; \frac{1}{\kappa_0} \; \frac{E_0^{1-\eta}}{1-\eta} \; \; .
%$$ Then, we see that the time evolution of the luminosity is linear up to some time $t_{break}$, and after that we have a break from the linear regime $\eta = 1$ to a non linear regime with $\eta < 1$. If we indicate the total dissipation time by $t_{dis}$ , we get: $$\begin{aligned}
%
\label{2.6}
%
L(t) \; & = & \; L_0 \; \exp(- \frac{t}{\tau_0}) \; \; \; \; \; \; , \; \; \; \; \; \; \; \; \; \; \;
\; \; \; \; \; \;\; \; 0 \; < \; t \; < \; t_{break} \; \; , \\
\nonumber
%
L(t) \; & = & \; L(t_{break}) \; \left ( 1 - \frac{t-t_{break}}{t_{dis}-t_{break}} \right )^{\frac{\eta}{1-\eta}} ,
\; t_{break} \; < \; t \; < \; t_{dis} \; \; .
%\end{aligned}$$ Equation (\[2.6\]) is relevant for light curves where there is a huge amount of energy to be dissipated.\
Several observations indicate that after a giant burst there are smaller and more recurrent bursts. According to our approach, we may think about these small bursts as similar to the seismic activity following a giant earthquake (for statistical similarities between bursts and earthquakes, see Cheng et al. 1995). These seismic bursts are characterized by very different light curves from the giant burst light curves.\
During these seismic bursts there is an almost continuous injection of energy, which tends to sustain an almost constant luminosity. This corresponds to an effective $\kappa$ in Eq. (\[2.1\]) which decreases smoothly with time. The simplest choice is: $$%
\label{2.7}
%
\kappa(t) \; = \; \frac{\kappa_0}{1 + \kappa_1 t} \; \; \; .
%$$ Inserting this into Eq. (\[2.1\]) and integrating, we get: $$%
\label{2.8}
%
E(t) \; = \; \left [ E_0^{1-\eta} \; - (1 - \eta) \; \frac{\kappa_0}{\kappa_1} \; \ln (1 + \kappa_1 t)
\right ]^{\frac{\eta}{1-\eta}} \; \; \; ,
%$$ so that the luminosity is: $$%
\label{2.9}
%
L(t) \; = \; \frac{L_0}{(1 + \kappa_1 t)^{\eta}} \; \left [1 \; - (1 - \eta) \; \frac{\kappa_0}{\kappa_1 E_0^{1-\eta}}
\; \ln (1 + \kappa_1 t) \right ]^{\frac{\eta}{1-\eta}} \; \; \; .
%$$ After defining the dissipation time as $$%
\label{2.10}
%
\ln (1 + \kappa_1 \tau_{dis}) \; = \; \frac{\kappa_1}{\kappa_0 }\; \frac{E_0^{1-\eta}}{1 - \eta} \; \; \; ,
%$$ we rewrite Eq. (\[2.9\]) as $$%
\label{2.11}
%
L(t) \; = \; \frac{L_0}{(1 + \kappa_1 t)^{\eta}} \; \left [ 1 \; - \; \frac{\ln (1 + \kappa_1 t)}{\ln (1 + \kappa_1
\tau_{dis})} \right ]^{\frac{\eta}{1-\eta}} \; \; \; .
%$$ Note that the light curve in Eq. (\[2.11\]) depends on two characteristic time constants $\frac{1}{\kappa_1}$ and $\tau_{dis}$. We see that $\kappa_1 \, \tau_{dis}$, which is roughly the number of small bursts that occurred in the given event, gives an estimation of the seismic burst intensity. Moreover, since during the seismic bursts the injected energy is much smaller than in giant bursts, we expect values of $\eta$ which are lower with respect to typical values in giant bursts.\
As we alluded in the Introduction, the canonical light curves of the X-ray afterglows are very similar to the light curve after the 2002 June 18 giant burst from AXP 1E 2259+586. In Cea (2006) we were able to accurately reproduce the puzzling light curve of the June 2002 burst by assuming that AXP 1E 2259+586 has undergone a giant burst, and soon after has entered into intense seismic burst activity. Accordingly, we may parameterize the X-ray afterglow light curves of gamma ray bursts as: $$%
\label{2.12}
%
L_{GRB}(t) \; = \; L_{G}(t) \; + \; L_{S}(t) \; \; \; \; ,
%$$ where, since there are no available data during the first stage of the outbursts, we have for the giant burst’s contribution: $$%
\label{2.13}
%
%
L_{G}(t) \; = \; L_{G}(0) \; \left ( 1 - \frac{t}{t_{diss}} \right )^{\frac{\eta_{G}}{1-\eta_{G}}} \; \; ,
\; \; \; 0 \; < \; t \; < \; t_{diss} \; \; ,
%$$ while $L_{S}(t)$ is given by: $$%
\label{2.14}
%
L_{S}(t) = \frac{L_{S}(0)}{(1 + \kappa t)^{\eta_{S}}} \;
\left [ 1 \; - \; \frac{\ln (1 + \kappa t)}{\ln (1 + \kappa
\tau_{diss})} \right ]^{\frac{\eta_{S}}{1-\eta_{S}}} , \; \;
0 \; < \; t \; < \; \tau_{diss} \; \; .
%$$ Note that, unlike the anomalous $X$-ray pulsars and soft gamma-ray repeaters, we do not need to take care of the quiescent flux since the gamma ray burst sources are at cosmological distances.\
In the following Sections we select a collection of GRBs with the aim to illustrate the variety of displayed light curves. In general, we reproduce the data of light curves from the original figures. For this reason, we display the light curves with the same time intervals as in the original figures. So that, lacking the precise values of data the best fits to our light curves are only indicative. In view of this, a quantitative comparison with different models is not possible. The unique exception is GRB 050801 were the data was taken from Table 1 in Rykoff et al. (2006). In that case (see Sect. \[050801\]) we indicate the reduced chisquare.
{#050315}
On 2005 March 15 the Swift Burst Alert Telescope (BAT) triggered and located on-board GRB 050315 [@Vaughan:2006]. After about $80 \, sec$ the Swift X-ray Telescope (XRT) began observations until about $10 \, days$, providing one of the best-sampled X-ray light curves of a gamma ray burst afterglow.\
In Fig.\[fig\_1\] we display the light curve of GRB 050315 in the $0.2 - 5 \, keV$ band. The data was extracted from Fig. 5 in Vaughan et al.(2006).\
A tentative fit to the X-ray light curve within the standard relativistic fireball model has been proposed in Granot et al. (2006) using a two-component jet model. An alterative description of the light curve of GRB 050315 within the cannonbal model is presented in Dado & De Rujula (2005).\
We fitted the data to our light curve Eq. (\[2.12\]). Indeed, we find a rather good description of the data with the following parameters (see Fig. \[fig\_1\]): $$\begin{aligned}
%
\label{2.15}
%
L_{G}(0) \; & \simeq & \; 5.2 \; 10^{2} \; \frac{count}{ sec} \; \; , \; \; \; \;
\; \; \;
\eta_{G} \; \simeq \;
0.867 \; \; , \; \; t_{diss} \; \simeq \; 380 \; sec \; \; \; \; \; \; \; \; \; \\
%
\nonumber
L_{S}(0) \; & \simeq & \; 0.35 \; \frac{count}{sec} \; \; \; , \; \; \; \; \; \; \;
\; \;
\eta_{S} \; \simeq \; 0.4
\; , \; \tau_{diss} \; \simeq \; 9.0 \; 10^5 \; sec \; , \; \kappa \; \simeq \; 5.0 \; 10^4 \; sec^{-1} \; .
%\end{aligned}$$ A few comments are in order. As discussed in the Sect. \[light\], since we lack the precise values of data, a quantitative comparison of our light curve with data is not possible. Nevertheless, Fig. \[fig\_1\] shows that the agreement with data is rather good. Moreover, our efficiency exponents $\eta_{G}$ and $ \eta_{S}$ are consistent with the values found in giant bursts from anomalous $X$-ray pulsars and soft gamma-ray repeaters [@cea:2006]. Note that, as expected, we have $ \eta_{S} < \eta_{G}$ .
{#050319}
Swift discovered GRB 050319 with the Burst Alert Telescope and began observing after $225 \, s$ after the burst onset [@Cusumano:2006].\
The X-ray afterglow was monitored by the XRT up to 28 days after the burst. In Fig. \[fig\_2\] we display the X-ray light curve in the $0.2 - 10 \, keV$ band. The data are extracted from Fig. 2 in Cusumano et al. (2006). Note that the light curve in the early stage of the outflow has been obtained extrapolating the BAT light curve in the XRT band by using the the best-fit spectral model [@Cusumano:2006].\
An adeguate description of the XRT light curve of GRB 050319 within the cannonbal model is presented in Dado & De Rujula (2005). However, we note that the extrapolation of the best-fit light curve towards the first stage of the outburst overestimates the observed flux by orders of magnitude. On the other hand, we may easily account for the observed flux decay by our light curve. Indeed, in Fig. \[fig\_2\] we compare our light curve Eq. (\[2.12\]) with observational data. The agreement is quite satisfying, even during the early-time of the outburst, if we take: $$\begin{aligned}
%
\label{2.16}
%
L_{G}(0) \; & \simeq & \; 8.5 \; 10^{-8} \; \frac{erg }{ cm^{2} \, sec} \; \; , \; \; \; \;
\;
\eta_{G} \; \simeq \;
0.867 \; \; , \; \; t_{diss} \; \simeq \; 410 \; sec \; \; \; \; \; \\
%
\nonumber
%
L_{S}(0) \; & \simeq & \; 5.5 \; 10^{-9} \; \frac{erg}{ cm^{2} \, sec}\; \; \; , \; \; \;
\; \;
\eta_{S} \; \simeq \; 0.45
\; , \; \tau_{diss} \; \simeq \; 7.5 \; 10^5 \; sec \; , \; \kappa \; \simeq \; 10 \; sec^{-1} \; .
%\end{aligned}$$
{#050406}
On 2005 April 6 BAT triggered on GRB 050406 [@Romano:2006a]. The gamma-ray characteristics of this burst, namely the softness of the observed spectrum and the absence of significant emission above $\sim \, 50 \, keV$, classify the burst as an X-ray flash (XRF 050406).\
In Fig. \[fig\_3\] we display the time decay of the flux. The data was taken from Fig. 2 in Romano et al. (2006a). We fit our light curve Eq. (\[2.12\]) to the available data. Indeed, we find that our light curve, with parameters given by: $$\begin{aligned}
%
\label{2.17}
%
L_{G}(0) \; & \simeq & \; 1.75 \; \; \frac{count}{sec} \; \; , \; \; \;
\eta_{G} \; \simeq \;
0.839 \; \; , \; \; t_{diss} \; \simeq \; 1.5 \; 10^3 \; sec \; \; \; \; \; \; \\
%
\nonumber
%
L_{S}(0) \; & \simeq & \; 0.15 \; \frac{count}{sec}\; \; \; , \; \; \;
\eta_{S} \; \simeq \; 0.40
\; , \; \tau_{diss} \; \simeq \; 8.0 \; 10^6 \; sec \; , \; \kappa \; \simeq \; 0.2 \; sec^{-1} \; ,
%\end{aligned}$$ allows quite a satisfying description of the decline of the flux (see Fig. \[fig\_3\]). Note that in the fit we exclude the bump at $ t \sim
200 \, sec$. For completeness, we also display in Fig. \[fig\_3\] the phenomenological best-fit broken power law [@Romano:2006a]. It is worthwhile to observe that the bump in the flux at $t \, \sim \, 200 \, sec$ is similar to the April 18, 2001 flare from SGR 1900+14 [@Feroci:2003]. Indeed, within our approach we believe that the bump in the flux could naturally be explained as fluctuations in the intense burst activity (see Sect. 5.2 in Cea 2006).
{#050801}
The Swift XRT obtained observations starting at 69 seconds after the burst onset of GRB 050801 [@Rykoff:2006]. In Fig. \[fig\_4\] we display the flux decay, where the data has been extracted from Table 1 in Rykoff et al. (2006).\
In this case we are able to best fit our light curve Eq. (\[2.12\]) to the available data. Since the observations start from $t \, > 74 \, s$, we perform the fit of data to the seismic burst light curve $F_{S}(t)$, Eq. (\[2.14\]). To get a sensible fit we fixed the dissipation time to $\tau_{diss} \, = \, 2.0 \, 10^6 \, s$ and $\kappa \, = \, 10^{-2} \,
s^{-1}$. The best fit of our light curve to data gives: $$%
\label{2.20}
%
L_{S}(0) \; = \; (27.4 \; \pm \; 7.2) \; 10^{-11} \; \frac{erg }{ cm^{2} \, sec} \; , \; \;
\eta_{S} \; = \; 0.748 \; \pm \; 0.026 \; \; ,
%$$ with a reduced $\chi^2/dof \, \simeq \, 0.93$. In Fig. \[fig\_4\] we compare our best-fitted light curve with data. We see that our theory allows a satisfying description of the light curve of GRB 050801. On the other hand, it is difficult to explain the peculiar behaviour of the light curve with standard models of early afterglow emission without assuming that there is continuous late time injection of energy into the afterglow [@Rykoff:2006].
{#051221}
GRB 051221A was detected by the Swift BAT on 2005 December 21. The Swift XRT observations began 88 seconds after the BAT trigger. The late X-ray afterglow of GRB 051221A has been also observed by the Chandra ACIS-S instrument. The combined X-ray light curve, displayed in Fig. \[fig\_5\], was extracted from Fig. 2 in Burrows et al. (2006).\
From Fig. \[fig\_5\], we see that the combined X-ray light curve is similar to those commonly observed in long gamma ray bursts. However, we find that the this peculiar light curve could be interpreted within our approach as the superimposition of two different seismic bursts. Accordingly, we may account for the X-ray afterglow light curve of GRB 051221A by: $$%
\label{2.22}
%
L_{GRB}(t) \; = \; L_{S_1}(t) \; + \; L_{S_2}(t) \; \; \; \; .
%$$ Indeed, we find that our light curve Eq. (\[2.22\]) allows a rather good description of the data once the parameters are: $$\begin{aligned}
%
\label{2.22_bis}
%
L_{S_1}(0) & \simeq & 9.8 \; 10^{-10} \; \frac{erg }{ cm^{2} \, sec} , \;
\eta_{S_1} \simeq 0.74
, \; \tau_{diss_1} \; \simeq \; 7.8 \; 10^3 \; sec \; , \; \kappa_1 \; \simeq \; 2.3 \; 10^{-2} \; sec^{-1}
\; , \\
%
\nonumber
%
L_{S_2}(0) \; & \simeq & \; 8.0 \; 10^{-12} \; \frac{erg }{ cm^{2} \, sec} \; , \;
\eta_{S_2} \; \simeq \; 0.40
\; , \; \tau_{diss_2} \; \simeq \; 1.0 \; 10^7 \; sec \; , \; \kappa_2 \; \simeq \; 1.4 \; 10^{-4} \; sec^{-1}
\; .
%\end{aligned}$$ It is worth mentioning that the data displayed in Fig. \[fig\_5\] start at $ t > 10^2 \, sec$. So that we cannot reliably determine the eventual giant burst contribution. On the other hand, this peculiar light curve is well described by two different seismic bursts, much like the intense burst activity in anomalous $X$-ray pulsars and soft gamma-ray repeaters . Note that the phenomelogical power law fits overestimate the light curve for $ t > 10^6 \, sec$.
{#050505}
On 2005 May 5 the Swift BAT triggered GRB 050505. The X-ray telescope XRT began taking data about 47 minutes after the burst trigger. In Fig. \[fig\_5\] we report the combined XRT and BAT light curve of the afterglow of GRB 050505. The data was extracted from Fig. 5 in Hurkett et al. (2006). The BAT data were extrapolated into the the XRT band using the best fit power law model derived from the BAT data alone [@Hurkett:2006].\
Within the standard models of early afterglows, the light curve is modelled by a broken power law. Nevertheless, we find that our light curve, Eq. (\[2.12\]), with parameters given by: $$\begin{aligned}
%
\label{2.23}
%
L_{G}(0) \; & \simeq & \; 3.0 \; 10^{-8} \; \frac{erg}{ cm^{2} \, sec} \; \; , \; \; \;
\eta_{G} \; \simeq \;
0.867 \; \; , \; \; t_{diss} \; \simeq \; 2.0 \; 10^2 \; sec \; \; \; \; \\
%
\nonumber
%
L_{S}(0) \; & \simeq & \; 2.5 \; 10^{-9} \; \frac{erg}{ cm^{2} \, sec} \; , \; \; \; \; \;
\eta_{S} \; \simeq \; 0.58
\; , \; \tau_{diss} \; \simeq \; 7.0 \; 10^5 \; sec \; , \; \kappa \; \simeq \; 0.13 \; sec^{-1} \; ,
%\end{aligned}$$ is able to descrive quite well the X-ray afterglow (see Fig. \[fig\_6\]).
{#050713}
In Fig. \[fig\_7\] we report the combined XRT and XMM-Newton light curve of the afterglow of GRB 050713A. The data was extracted from Fig. 7 in Morris et al. (2006). The dot-dashed line is the broken-power law best fit of the combined X-ray light curve [@Morris:2006].\
Within our approach we may reproduce the X-ray afterglow of GRB 050713A by our Eq. (\[2.12\]). However, since the giant burst contribution to the light curve $L_G(t)$ lasts up to $t \, \sim 10^2 - 10^3 \, sec$, we need to consider only $L_S(t)$. So that we are lead to: $$%
\label{2.24}
%
L_{GRB}(t) \; = \; L_{S}(t) \; \; \; .
%$$ Indeed, even in this case our light curve, with parameters fixed to: $$%
\label{2.24_bis}
%
L_{S}(0) \, \simeq \, 5.5 \, 10^{-9} \, \frac{erg }{ cm^{2} \, sec} \, , \;
\eta_{S} \, \simeq \, 0.71 \; , \; \kappa \, \simeq \, 7.0 \, 10^{-2} \, sec^{-1} \; , \;
\tau_{diss} \, \simeq \, 9.0 \, 10^6 \, sec \; ,
%$$ reproduces quite accurately the phenomelogical broken-power law best fit (see Fig. \[fig\_7\]).
{#051210}
GRB 051210 triggered the Swift BAT on 2005 December 12. The burst was classified as short gamma ray burst. In Fig. 2 in La Parola et al. (2006) it is presented the XRT light curve decay of GRB 051210. The BAT light curve was extrapolated into the $0.2 - 10 \, keV$ band by converting the BAT count rate with the factor derived from the BAT spectral parameters.\
In Fig. \[fig\_8\] we report the combined BAT and XRT light curve of the afterglow of GRB 051210. The data was extracted from Fig. 2 in La Parola et al. (2006). In Fig. \[fig\_8\] we also display our best fit light curve Eq. (\[2.12\]) with parameters: $$\begin{aligned}
\label{2.25}
%
L_{G}(0) \; & \simeq & \; 4.5 \; 10^{-9} \; \frac{erg}{ cm^{2} \, sec} \; \; , \; \; \;
\eta_{G} \; \simeq \;
0.867 \; \; , \; \; t_{diss} \; \simeq \; 2.8 \; 10^2 \; sec \; \; \; \; \; \; \; \; \\
%
\nonumber
%
L_{S}(0) \; & \simeq & \; 4.0 \; 10^{-9} \; \frac{erg}{ cm^{2} \, sec} \; \; , \; \; \;
\eta_{S} \; \simeq \; 0.63
\; , \; \tau_{diss} \; \simeq \; 1.2 \; 10^3 \; sec \; , \; \kappa \; \simeq \; 0.1 \; sec^{-1} \; .
%\end{aligned}$$ Even in this case the agreement between our light curve and the data is satisfying.
{#060121}
GRB 060121 was detected by HETE-2 on January 21, 2006. Swift performed observations beginning at January 22, 2006 [@Levan:2006]. GRB 060121 was identified as a short and spectrally hard burst.\
In Fig. \[fig\_9\] we report the X-ray light curve in the $0.3 - 10.0 \, keV$ band. The data has been extracted from Fig. 1 in Levan et al. (2006). We also display the phenomenological power-law best fit $L(t) \, \sim \,
t^{-1.18}$ [@Levan:2006].\
Within our approach we may reproduce the X-ray afterglow of GRB 050713A by our Eq. (\[2.12\]). Even in this case we need to consider only the seismic burst contribution $L_S(t)$. Indeed, we find that our Eq. (\[2.24\]) reproduces quite accurately the phenomelogical power law best fit with the following parameters (see Fig. \[fig\_9\]): $$%
\label{2.26}
%
L_{S}(0) \; \simeq \; 9.5 \; 10^{-11} \; \frac{erg }{ cm^{2} \, sec} \; , \;
\eta_{S} \; \simeq \; 0.70 \; , \; \kappa \; \simeq \; 8.0 \; hour^{-1} \;
\tau_{diss} \; \simeq \; 3.0 \; 10^3 \; hour \; .
%$$
{#060124}
Swift BAT triggered on a precursor of GRB 060124 on 2006 January 24, about 570 seconds before the main burst. So that GRB 060124 is the first event for which there is a clear detection of both the prompt and the afterglow emission [@Romano:2006b].\
In Fig. \[fig\_10\] we report the X-ray light curve in the $0.3 - 10.0 \, keV$ band. The data has been extracted from Fig. 9 in Romano et al. (2006b). In Fig. \[fig\_10\] we display our best fit light curve Eq. (\[2.12\]) with parameters: $$\begin{aligned}
%
\label{2.27}
%
%
L_{G}(0) \; & \simeq & \; 5.0 \; 10^{-8} \; \frac{erg}{ cm^{2} \, sec} \; , \;
\eta_{G} \; \simeq \;
0.867 \; \; , \; \; t_{diss} \, + \, t_0 \; \simeq \; 1.0 \; 10^3 \; sec \; \; \; \; \; \\
%
\nonumber
%
L_{S}(0) \; & \simeq & \; 6.5 \; 10^{-9} \; \frac{erg}{ cm^{2} \, sec} \; , \;
\eta_{S} \; \simeq \; 0.60 \; , \; \tau_{diss} \, + \, t_0 \; \simeq \; 1.2 \; 10^6 \; sec \; ,
\; \kappa \simeq 5.0 \; 10^{-2} \; sec^{-1} ,
%\end{aligned}$$ where we assumed that the burst started at $t_0 \, = \, 570 \, s$. Note that our light curve interpolates the X-ray peaks at the early stage of the outflow. On the other hand, our light curve mimics quite well the broken power-law best fit to the XRT data (compare our Fig. \[fig\_10\] with Romano et al. (2006b), Fig. 9).
{#060218}
GRB 060218 was detected with the BAT instrument on 2006 February 18. XRT began observations 159 seconds after the burst trigger [@Campana:2006].\
The XRT light curve is shown in Fig. \[fig\_11\]. The data was extracted from Fig. 2 in Campana et al. (2006). We try to interpret the XRT light curve with our light curve Eq. (\[2.12\]). The result of our best fit is displayed in Fig. \[fig\_11\]. Excluding the data of the bump from $ t \sim 200 \, sec$ to $ t \sim 3000 \, sec$, the parameters for our best fit light curve are: $$\begin{aligned}
%
\label{2.28}
%
L_{G}(0) \; & \simeq & \; 4.1 \; 10^{-9} \; \frac{erg}{ cm^{2} \, sec} \; \; , \; \; \;
\eta_{G} \; \simeq \;
0.867 \; \; , \; \; t_{diss} \; \simeq \; 1.55 \; 10^4 \; sec \; \; \; \; \\
%
\nonumber
%
L_{S}(0) \; & \simeq & \; 8.0 \; 10^{-10} \; \frac{erg}{ cm^{2} \, sec} \; , \; \;
\eta_{S} \; \simeq \; 0.60 \; , \; \tau_{diss} \; \simeq \; 2.0 \; 10^6 \; sec \; ,
\; \kappa \; \simeq \; 1.3 \; 10^{-2} \; sec^{-1} \; ,
%\end{aligned}$$ Indeed, our light curve is able to reproduce quite well the data. However, there is a clear excess in the observed light curve with respect to our light curve in the early-time afterglow. We believe that this excess is due to a component which is not directly related to the burst. Indeed, Campana et al. (2006) pointed out that there was a soft component in the X-ray spectrum, that is present in the XRT starting from 159 s up to about $10^4$ s. This soft component could be accounted for by a black body with an increasing emission radius of the order of $10^{12}$ cm. Moreover, this component is undetected in later XRT observations and it is interpreted as shock break out from a dense wind.
{#050416A}
Swift discovered XRF 050416A with the Burst Alert Telescope on 2005 April 16. After about 76 seconds from the burst trigger, XRT began collecting data [@Mangano:2006]. The X-ray light curve was monitored up to 74 days after the onset of the burst. The very soft spectrum of the burst classifies this event as an X-ray flash.\
In Fig. \[fig\_12\] we show the combined BAT-XRT light curve of XRF 050416A. The BAT light curve was extrapolated into the $0.2 - 10 \, keV$ energy band assuming two different spectral law, the Band best fit model (full squares in Fig. \[fig\_12\]) and a single power law best fit model (full diamonds in Fig. \[fig\_12\]).\
From Fig. \[fig\_12\] we see that the X-ray light curve initially decays very fast and subsequently flattens. It is evident that the XRT light curve decay is not consistent with a single power law. Indeed, Mangano et al. (2006) found that a doubly-broken power law improves considerably the fit of the light curve (dot-dashed line in Fig. \[fig\_12\]). On the other hand, we may adequately reproduce the combined BAT-XRT light curve with our light curve Eq. (\[2.12\]). To this end, we assume that the early light curve is described by the BAT data extrapolated with the Band best fit model. By fitting our Eq. (\[2.12\]) to the data, we find: $$\begin{aligned}
%
\label{2.29}
%
L_{G}(0) \; & \simeq & \; 1.0 \; 10^{-7} \; \frac{erg}{ cm^{2} \, sec} \; \; , \; \;
\eta_{G} \; \simeq \;
0.895 \; \; , \; \; t_{diss} \; \simeq \; 1.8 \; 10^2 \; sec \; \; \; \\
%
\nonumber
%
L_{S}(0) \; & \simeq & \; 2.5 \; 10^{-10} \; \frac{erg}{ cm^{2} \, sec} \; , \;
\eta_{S} \; \simeq \; 0.60 \; , \; \tau_{diss} \; \simeq \; 8.0 \; 10^7 \; sec \; ,
\; \kappa \; \simeq \; 2.0 \; 10^{-2} \; sec^{-1} \; ,
%\end{aligned}$$ Indeed, Fig. \[fig\_12\] shows that our light curve is able to account for the light curve of XRF 050416A.
{#origin}
The results in previous Section show that the light curves of Swift gamma ray bursts can be successfully described by the approach developed in Cea (2006) to quantitatively account for light curves for both soft gamma repeaters and anomalous X-ray pulsars. This leads us to suppose that the same mechanism is responsible for bursts from gamma ray bursts, soft gamma repeaters, and anomalous X-ray pulsars.\
In Cea (2006) we showed that soft gamma repeaters and anomalous X-ray pulsars can be understood within our recent proposal of p-stars, namely compact quark stars in $\beta$-equilibrium with electrons in a chromomagnetic condensate (Cea 2004a,b). In particular, the bursts are powered by glitches, which in our model are triggered by dissipative effects in the inner core. The energy released during a burst is given by the magnetic energy directly injected into the magnetosphere: $$%
\label{3.1}
%
\delta E_{B}^{\emph{ext}} \; \simeq \; \frac{1}{3} \; R^3 \; B_S^2 \; \frac{\delta B_S}{ B_S} \; \; .
%$$ For magnetic powered pulsars with $M \, \sim \, M_{\bigodot}$ and $R \, \sim \, 10 \, Km$, we have $B_S \, \lesssim \,
10^{17} \, Gauss$. So that, from Eq. (\[3.1\]) we get: $$%
\label{3.2}
%
\delta E_{B}^{\emph{ext}} \, \simeq \, 2.6 \, 10^{50} \, ergs \, \left ( \frac{B_S}{10^{17} \, Gauss} \right )^2
\, \frac{\delta B_S}{ B_S} \, \lesssim \, 2.6 \, 10^{50} \, ergs \; \; .
%$$ The gamma-ray energy released in gamma ray bursts is narrowly clustered around $5.0 \, 10^{50} \, ergs$ [@Frail:2001]. Thus, even though it is conceivable that a small fraction of gamma ray bursts could be explained by burst like the 2004 December 27 giant flare from SGR 1806-20, we see that canonical magnetic powered pulsars (canonical magnetars) do not match the required energy budged to explain gamma ray bursts. On the other hand, we find that massive magnetars, namely magnetic powered pulsars with $M \, \sim \, 10 \, M_{\bigodot}$ and $R \, \sim \, 10^2 \, Km$, could furnish the energy needed to fire the gamma ray bursts.\
The possibility to have massive pulsars stems from the fact that our p-stars do not admit the existence of an upper limit to the mass of a completely degenerate configuration. In other words, our peculiar equation of state of degenerate up and down quarks in a chromomagnetic condensate allows the existence of finite equilibrium states for stars of arbitrary mass. In fact, in Fig. \[fig\_13\] we display the gravitational mass versus the radius for p-stars with chromomagnetic condensate $\sqrt{gH} = 0.1 \; GeV$.
Note that the strength of the chromomagnetic condensate of massive magnetars is reduced by less than one order of magnitude with respect to canonical magnetars. Thus, we infer that for massive pulsars $B_S \, \lesssim \, 10^{16} \,
Gauss$. Using Eq. (\[3.1\]) we find: $$%
\label{3.4}
%
\delta E_{B}^{\emph{ext}} \, \simeq \, 2.6 \, 10^{51} \, ergs \, \left ( \frac{B_S}{10^{16} \, Gauss} \right )^2
\, \frac{\delta B_S}{ B_S} \, \lesssim \, 2.6 \, 10^{51} \, ergs \; \; ,
%$$ that, indeed, confirms that massive magnetars are a viable mechanism for gamma ray bursts.\
An interesting consequence of our proposal is that at the onset of the bursts there is an almost spherically symmetric outflow from the pulsar, together with a collimated jet from the north magnetic pole [@cea:2006]. Indeed, following the 2004 27 December giant flare from SGR 1806-20 it has been detected a radio afterglow consistent with a spherical, non relativistic expansion together with a sideways expansion of a jetted explosion. More interestingly, the lower limit of the outflow energy turns out to be $E \gtrsim 10^{44.5} \; ergs$ [@Gelfand:2005]. This implies that the blast wave and the jet may dissipate up to about $10 \, \% $ of the total burst energy. In the case of gamma ray bursts, according to our proposal we see that at the onset of the burst there is a matter outflow with energies up to $\sim \, 10^{50} \, ergs$. We believe that this could explain the association of some gamma ray bursts with supernova explosions.
{#conclusion}
Let us conclude by briefly summarizing the main results of the present paper. We have presented an effective theory which allows us to account for X-ray light curves of both gamma ray bursts and X-ray rich flashes. We have shown that the approach developed to describe the light curves from anomalous $X$-ray pulsars and soft gamma-ray repeaters works successfully even for gamma ray bursts. This leads us the conclusion that the same mechanism is responsible for bursts from gamma ray bursts, soft gamma repeaters, and anomalous X-ray pulsars. In fact, we propose that gamma ray bursts originate by the burst activity from massive magnetic powered pulsars.
Burrows, D.N., et al. 2006, astro-ph/0604320 Campana, S., et al. 2006, astro-ph/0603279, accepted for publication in Nature Cea, P. 2004a, Int. J. Mod. Phys. D, 13, 1917 Cea, P. 2004b, JCAP, 0403011 Cea, P. 2006, A&A, 450, 199 Cheng, B., Epstein, R. I., Guyer, R. A., & Cody Young, A. 1995, Nature, 382, 518 Chincarini, G., SWIFT Collaboration 2005, astro-ph/0511108 Cusumano, G., et al. 2006, ApJ, 639, 316 Dado, S., Dar, A., & De Rujula, A. 2006, astro-ph/0512196 Feroci, M., et al. 2003, ApJ, 596, 470 Frail, D., et al. 2001, ApJ, 562, L55 Gelfand, J.D., et al. 2005, ApJ, 634, L89 Granot, J., Konigl, A., & Piran, T. 2006, astro-ph/0601056 Guetta, D., et al. 2006, astro-ph/0602387 Hurkett, C.P., et al. 2006, MNRAS, 368, 1101 La Parola, V., et al 2006, astro-ph/0602541 Levan, A.J., et al 2006, astro-ph/0603282 Mangano, V., et al 2006, astro-ph/0603738 Mészáros, P. 2006, astro-ph/0605208 Morris, D.C., et al. 2006, astro-ph/0602490 Nousek, J.A., et al. 2006, ApJ, 642, 389 O’Brien, P.T., et al. 2006, astro-ph/0601125, accepted for publication in ApJ Piran, T. 2005, Rev. Mod. Phys. 76, 1143 Romano, P., et al. 2006a, A&A, 450, 59 % Romano, P., et al. 2006b, astro-ph/0602497, accepted for publication in A&A Rykoff, E.S., et al. 2006, ApJ, 638, L5 Soderberg, A.M., et al. 2006, astro-ph/0601455 Vaughan, S., et al. 2006, ApJ, 638, 920 Woods, P.M., et al. 2004, ApJ, 605, 378
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abstract: |
We present the results of lattice QCD calculations of the magnetic moments of the lightest nuclei, the deuteron, the triton and $^3$He, along with those of the neutron and proton. These calculations, performed at quark masses corresponding to $m_\pi\sim 800~{\rm
MeV}$, reveal that the structure of these nuclei at unphysically heavy quark masses closely resembles that at the physical quark masses. In particular, we find that the magnetic moment of $^3$He differs only slightly from that of a free neutron, as is the case in nature, indicating that the shell-model configuration of two spin-paired protons and a valence neutron captures its dominant structure. Similarly a shell-model-like moment is found for the triton, $\mu_{^{3}{\rm H}}\sim \mu_p$. The deuteron magnetic moment is found to be equal to the nucleon isoscalar moment within the uncertainties of the calculations. Furthermore, deviations from the Schmidt limits are also found to be similar to those in nature for these nuclei. These findings suggest that at least some nuclei at these unphysical quark masses are describable by a phenomenological nuclear shell-model.
author:
- 'S.R. Beane'
- 'E. Chang'
- 'S. Cohen'
- 'W. Detmold'
- 'H.W. Lin'
- 'K. Orginos'
- 'A. Parreño'
- 'M.J. Savage'
- 'B.C. Tiburzi'
bibliography:
- 'MM.bib'
title: Magnetic moments of light nuclei from lattice quantum chromodynamics
---
-1.1cm -0.5cm
The electromagnetic interactions of nuclei have been used extensively to elucidate their structure and dynamics. In the early days of nuclear physics, the magnetic moments of the light nuclei helped to reveal that they behaved like a collection of “weakly” interacting nucleons that, to a very large degree, retained their identity, despite being bound together by the strong nuclear force. This feature, in part, led to the establishment of the nuclear shell model as a phenomenological tool with which to predict basic properties of nuclei throughout the periodic table. The success of the shell model is somewhat remarkable, given that nuclei are fundamentally bound states of quarks and gluons, the degrees of freedom of quantum chromodynamics (QCD). The strong nuclear force emerges from QCD as a by-product of confinement and chiral symmetry breaking. The fact that, at the physical values of the quark masses, nuclei are not simply collections of quarks and gluons, defined by their global quantum numbers, but have the structure of interacting protons and neutrons, remains to be understood at a deep level. In this letter, we continue our exploration of nuclei at unphysical quark masses, and calculate the magnetic moments of the lightest few nuclei using lattice QCD. We find that they are close to those found in nature, and also close to the sum of the constituent nucleon magnetic moments in the simplest shell model configuration. This second finding in particular is remarkable and suggests that a phenomenological nuclear shell-model is applicable for at least some nuclei at these unphysical quark masses.
Our lattice QCD calculations were performed on one ensemble of gauge-field configurations generated with a $N_f=3$ clover-improved fermion action [@Sheikholeslami:1985ij] and a Lüscher-Weisz gauge action [@Luscher:1984xn]. The configurations have $L=32$ lattice sites in each spatial direction, $T=48$ sites in the temporal direction, and a lattice spacing of $a\sim0.12~{\rm fm}$. All three light-quark masses were set equal to that of the physical strange quark, producing a pion of mass $m_\pi \sim 806$ MeV. A background electromagnetic ($U_Q(1)$) gauge field giving rise to a uniform magnetic field along the $z$-axis was multiplied onto each QCD gauge field in the ensemble (separately for each quark flavor), and these combined gauge fields were used to calculate up- and down-quark propagators, which were then contracted to form the requisite nuclear correlation functions using the techniques of Ref. [@Detmold:2012eu]. Calculations were performed on $\sim 750$ gauge-field configurations, taken at uniform intervals from $\sim
10,000$ trajectories. On each configuration, quark propagators were generated from 48 uniformly distributed Gaussian-smeared sources for each of four magnetic field strengths (for further details of the production, see Refs. [@Beane:2012vq; @Beane:2013br]).
Background electromagnetic fields have been used extensively to calculate electromagnetic properties of single hadrons, such as the magnetic moments of the lowest-lying baryons [@Bernard:1982yu; @Martinelli:1982cb; @Lee:2005ds; @Lee:2005dq; @Detmold:2006vu; @Aubin:2008qp; @Detmold:2009dx; @Detmold:2010ts; @Primer:2013pva] and electromagnetic polarizabilities of mesons and baryons [@Fiebig:1988en; @Christensen:2004ca; @Lee:2005dq; @Detmold:2009dx; @Detmold:2010ts; @Primer:2013pva; @Lujan:2014kia]. In order that the quark fields, with electric charges $Q_u=+{2\over
3}$ and $Q_{d,s}=-{1\over 3}$ for the up-, down- and strange-quarks, respectively, satisfy spatially-periodic boundary conditions in the presence of a background magnetic field, it is well-known [@tHooft:1979uj] that the lattice links, $U_\mu(x)$, associated with the $U_Q(1)$ gauge field are of the form $$\begin{aligned}
U_\mu (x)&=& e^{i \frac{6\pi Q_q \tilde n}{L^2} x_1\delta_{\mu,2}}
\times e^{-i \frac{6\pi Q_q \tilde n }{L} x_2\delta_{\mu,1}\delta_{x_1,L-1}}
\ ,
\label{eq:bkdgfield}\end{aligned}$$ for quark of flavour $q$, where $\tilde n$ must be an integer. The uniform magnetic field, ${\bf B}$, resulting from these links is $$e {\bf B} =
{6\pi \tilde n \over L^2}\, \hat {\bf z}
\ ,$$ where $e$ is the magnitude of the electric charge and $\hat{\bf z}$ is a unit vector in the $x_3$-direction. In physical units, the background magnetic fields exploited with this ensemble of gauge-field configurations are $e |{\bf B}| \sim 0.046 \ |\tilde n|\ {\rm GeV}^2$. To optimize the re-use of light-quark propagators in the production, calculations were performed for $U_Q(1)$ fields with $\tilde
n=0,1,-2,+4$. Four field strengths were found to be sufficient for this initial investigation. With three degenerate flavors of light quarks, and a traceless electric-charge matrix, there are no contributions from coupling of the ${\bf B}$ field to sea quarks at leading order in the electric charge. Therefore, the magnetic moments presented here are complete calculations (there are no missing disconnected contributions).
The ground-state energy of a non-relativistic hadron of mass $M$, and charge $Q\, e$ in a uniform magnetic field is $$\begin{aligned}
\!\!\!\!\!\!\!E({\bf B}) & = & M\ +\ {| Q\, e\, {\bf B}|\over 2 M}
\ -\ {\bm\mu}\cdot {\bf B}
\nonumber \\ &&
\ -\ 2\pi \beta_{M0}\ |{\bf B}|^2
-2\pi \beta_{M2} T_{ij} B_i B_j+ ... \ ,
\label{eq:energyB}\end{aligned}$$ where the ellipses denote terms that are cubic and higher in the magnetic field, as well as terms that are $1/M$ suppressed [@Hill:2011wy; @Lee:2014iha]. The first contribution in eq. (\[eq:energyB\]) is the hadron’s rest mass, the second is the energy of the lowest-lying Landau level, the third is from the interaction of its magnetic moment, ${\bm\mu}$, and the fourth and fifth terms are from its scalar and quadrupole magnetic polarizabilities, $\beta_{M0,M2}$, respectively ($T_{ij}$ is a traceless symmetric tensor [@Chen:1998vi]). The magnetic moment term is only present for particles with spin, and $\beta_{M2}$ is only present for $j\geq1$. In order to determine ${\bm\mu}$ using lattice QCD calculations, two-point correlation functions associated with the hadron or nucleus of interest in the $j_z=\pm j$ magnetic sub-states, $C^{(B)}_{j_z}(t)$, can be calculated in the presence of background fields of the form given in Eq. (\[eq:bkdgfield\]) with strength $B=\hat{\bf z}\cdot{\bf B}$. The energies of ground-states aligned and anti-aligned with the magnetic field, $E^B_{\pm j}$, will be split by spin-dependent interactions, and the difference, $\delta E^{(B)} =
E^{B}_{+j} - E^{B}_{-j}$, can be extracted from the correlation functions that we consider. The component of $\delta E^{(B)}$ that is linear in ${\bf B}$ determines ${\bm\mu}$ via Eq. (\[eq:energyB\]).
![ The correlator ratios $R(B)$ as a function of time slice for the various states ($p$, $n$, $d$, ${}^3$He, and ${}^3$H) for $\tilde n=+1,-2,+4$. Fits to the ratios are also shown.[]{data-label="fig:EMP"}](figures/ratioplot.pdf "fig:"){width="0.99\columnwidth"}
Explicitly, the energy difference is determined from the large time behaviour of $$\begin{aligned}
R(B) =
\frac{C^{(B)}_{j}(t)\ C^{(0)}_{-j}(t)}{ C^{(B)}_{-j}(t)\ C^{(0)}_{j}(t)}
&\stackrel{t\to\infty}{\longrightarrow} &Z e^{-\delta E^{(B)} t}
\,.
\label{eq:ratio}\end{aligned}$$ Each term in this ratio is a correlation function with the quantum numbers of the nuclear state that is being considered, which we compute using the methods of Ref. [@Detmold:2012eu]. As discussed in Ref. [@Primer:2013pva], subtracting the contribution from the correlation functions calculated in the absence of a magnetic field reduces fluctuations in the ratio, enabling a more precise determination of the magnetic moment. The energy splitting is extracted from a correlated $\chi^2$-minimization of the functional form in Eq. (\[eq:ratio\]) using a covariance matrix generated with the jackknife procedure. Fits are performed only over time ranges where all of the individual correlators in the ratio exhibit single exponential behavior and a systematic uncertainty is assigned from variation of the fitting window. Figure \[fig:EMP\] shows the correlator ratios and associated fits for the various states that we consider: $p$, $n$, $d$, ${}^3$He, and ${}^3$H, for $\tilde
n=+1,-2,+4$.
As mentioned above, the magnetic moments of the proton and neutron have been previously calculated with lattice QCD methods for a wide range of light-quark masses (in almost all cases omitting the disconnected contributions). The present work is the first QCD calculation of the magnetic moments of nuclei.
![ The calculated $\delta E^{(B)}$ of the proton and neutron (upper panel) and light nuclei (lower panel) in lattice units as a function of $|\tilde n|$. The shaded regions corresponds to fits of the form $\delta E^{(B)} =-2 \mu\ |{\bf B}| + \gamma\, |{\bf
B}|^3 $ and their uncertainties. The dashed lines correspond to the linear contribution alone. []{data-label="fig:dEnp"}](figures/allplot.pdf "fig:"){width="0.99\columnwidth"}
In Figure \[fig:dEnp\], we show the energy splittings of the nucleons and nuclei as a function of $|\tilde n|$, and, motivated by Eq. (\[eq:energyB\]), we fit these to a function of the form $\delta
E^{(B)} =-2 \mu\ |{\bf B}| + \gamma\ |{\bf B}|^3 $, where $\gamma$ is a constant encapsulating higher-order terms in the expansion. We find that the proton and neutron magnetic moments at this pion mass are $\mu_p = 1.792(19)(37)$ NM (nuclear magnetons) and $\mu_n =
-1.138(03)(10)$ NM, respectively, where the first uncertainty is statistical and the second uncertainty is from systematics associated with the fits to correlation functions and the extraction of the magnetic moment using the above form. These results agree with previous calculations [@Primer:2013pva] within the uncertainties. The natural units of the system are ${e/
2 M_N^{\rm latt}}$, where $M_N^{\rm latt}$ is the mass of the nucleon at the quark masses of the lattice calculation, which we refer to as natural nuclear magnetons (nNM). In these units, the magnetic moments are $\mu_p =
3.119(33)(64)$ nNM and $\mu_n = -1.981(05)(18)$ nNM. These values at this unphysical pion mass can be compared with those of nature, $\mu_p^{\rm expt} = 2.792847356(23)$ NM and $\mu_n^{\rm expt} =
-1.9130427(05)$ NM, which are remarkably close to the lattice results. In fact, when comparing all available lattice QCD results for the nucleon magnetic moments in units of nNM, the dependence upon the light-quark masses is surprisingly small, reminiscent of the almost completely flat pion mass dependence of the nucleon axial coupling, $g_A$.
In Figure \[fig:dEnp\], we also show $\delta E^{(B)}$ as a function of $|\tilde n|$ for the deuteron, $^3$He and the triton (${}^3$H). Fitting the energy splittings with a form analogous to that for the nucleons gives magnetic moments of $\mu_d = 1.218(38)(87)$ nNM for the deuteron, $\mu_{^3{\rm He}} = -2.29(03)(12)$ nNM for $^3$He and $\mu_{^3{\rm H}} = 3.56(05)(18)$ nNM for the triton. These can be compared with the experimental values of $\mu_d^{\rm expt} =
0.8574382308(72)$ NM, $\mu_{^3{\rm He}}^{\rm expt} = -2.127625306(25)$ NM and $\mu_{^3{\rm H}}^{\rm expt} = 2.978962448(38)$ NM.
![ The magnetic moments of the proton, neutron, deuteron, $^3$He and triton. The results of the lattice QCD calculation at a pion mass of $m_\pi\sim 806~{\rm MeV}$, in units of natural nuclear magnetons (${e}/{2M_N^{\rm latt}}$), are shown as the solid bands. The inner bands corresponds to the statistical uncertainties, while the outer bands correspond to the statistical and systematic uncertainties combined in quadrature, and include our estimates of the uncertainties from lattice spacing and volume. The red dashed lines show the experimentally measured values at the physical quark masses. []{data-label="fig:Nuclei"}](figures/compwithExptplot.pdf){width="0.99\columnwidth"}
The magnetic moments calculated with lattice QCD, along with their experimental values, are presented in Figure \[fig:Nuclei\]. The naive shell-model predictions for the magnetic moments of these light nuclei are $\mu_d^{\rm SM} = \mu_p+\mu_n$, $\mu^{\rm SM}_{^3{\rm He}} =
\mu_n$ (where the two protons in the 1s-state are spin paired to $j_p=0$ and the neutron is in the 1s-state) and $\mu^{\rm SM}_{^3{\rm
H}} = \mu_p$ (where the two neutrons in the 1s-state are spin paired to $j_n=0$ and the proton is in the 1s-state). For these simple s-shell nuclei, the proton and neutron magnetic moments correspond to the Schmidt limits [@schmidt:1937]. In nature, $^3$He is one of the very few nuclei that lie outside the Schmidt limits [@foldy:1950]. In our calculations we also find that $^3$He lies outside the Schmidt limits at this heavier pion mass, with $\delta\mu_{^3{\rm He}} = \mu_{^3{\rm
He}} - \mu_n = -0.340(24)(93)$ nNM (compared to the experimental difference of $\delta\mu_{^3{\rm He}}^{\rm expt} = -0.215$ NM) , and similarly for the triton $\delta\mu_{^3{\rm H}}=\mu_{^3{\rm H}} -
\mu_p = +0.45(04)(16)$ nNM (compared to the experimental difference of $\delta\mu_{^3{\rm H}}^{\rm expt} = +0.186$ NM), corresponding to $\sim 10\%$ deviations from the naive shell-model predictions.
![ The differences between the nuclear magnetic moments and the predictions of the naive shell-model. The results of the lattice QCD calculation at a pion mass of $m_\pi\sim 806~{\rm
MeV}$, in units of natural nuclear magnetons (${e}/{2M_N^{\rm latt}}$), are shown as the solid bands. The inner band corresponds to the statistical uncertainties, while the outer bands correspond to the statistical and systematic uncertainties combined in quadrature, including estimates of the uncertainties from lattice spacing and volume. The red dashed lines show the experimentally measured differences. []{data-label="fig:diffNuclei"}](figures/compShiftwithExptplot.pdf){width="0.99\columnwidth"}
These quantities are summarized in Figure \[fig:diffNuclei\].
At a phenomenological level, it is not difficult to understand why the magnetic moments scale, to a large degree, with the nucleon mass. The success of the non-relativistic quark model (NRQM) in describing the magnetic moments of the lowest-lying baryons as the sum of contributions from three weakly-bound non-relativistic quarks, with up- and down-quark masses of $M_{U,D}\sim 300~{\rm MeV}$ and strange-quark mass of $M_{S}\sim 500~{\rm MeV}$, suggests that naive scaling with the hadron mass should capture most of the quark-mass dependence. From the perspective of chiral perturbation theory ($\chi$PT), the leading contributions to the nucleon magnetic moments are from dimension-five operators, with the leading quark-mass dependence arising from mesons loops that are suppressed in the chiral expansion, and scaling linearly with the mass of the pion. Consistency of the magnetic moments calculated in the NRQM and in $\chi$PT suggests that the nucleon mass scales linearly with the pion mass, which is inconsistent with chiral power counting, but consistent with the results obtained from analysis of lattice QCD calculations [@Walker-Loud:2013yua]. It should be emphasized that the magnetic moments of the light nuclei that we study here are well understood in the context of nuclear chiral effective field theory, where pions and nucleons are the effective degrees of freedom, and heavier meson-exchange-type contributions are included as various contact interactions among nucleons (see, for instance, Ref. [@Piarulli:2012bn]).
The present calculations have been performed at a single lattice spacing and in one lattice volume, and the lack of continuum and infinite volume extrapolations introduces systematic uncertainties into our results. Chiral perturbation theory can be used to estimate the finite volume (FV) effects in the magnetic moments, using the sum of the known [@Tiburzi:2014zva] effects on the constituent nucleons. These contributions are ${\raisebox{-0.7ex}{$\stackrel{\textstyle <}{\sim}$ }}1\%$ in all cases. There may be additional effects beyond the single particle contributions, however the binding energies of light nuclei calculated previously in multiple volumes at this quark mass [@Beane:2012vq] demonstrate that the current lattice volume is large enough for such FV effects to be negligible. In contrast, calculations with multiple lattice spacings have not been performed at this heavier pion mass, and consequently this systematic uncertainty remains to be quantified. However, electromagnetic contributions to the action are perturbatively improved as they are included as a background field in the link variables. Consequently, the lattice spacing artifacts are expected to be small, entering at ${\cal O}(\Lambda_{\rm
QCD}^2a^2)\sim 3$% for $\Lambda_{\rm QCD}=300$ MeV. To account for these effects, we combine the two sources of uncertainty in quadrature and assess an overall multiplicative systematic uncertainty of 3% on all the extracted moments. For the nuclei, this is small compared to the other systematic uncertainties, but for the neutron in particular, it is the dominant uncertainty.
In conclusion, we have presented the results of lattice QCD calculations of the magnetic moments of the lightest nuclei at the flavor SU(3) symmetric point. We find that, when rescaled by the mass of the nucleon, the magnetic moments of the proton, neutron, deuteron, $^3$He and triton are remarkably close to their experimental values. The magnetic moment of $^3$He is very close to that of a free neutron, consistent with the two protons in the 1s-state spin-paired to $j_p=0$ and the valence neutron in the 1s-state. Analogous results are found for the triton, and the magnetic moment of the deuteron is consistent with the sum of the neutron and proton magnetic moments. This work demonstrates for the first time that QCD can be used to calculate the structure of nuclei from first principles. Calculations using these techniques at lighter quark masses and for larger nuclei are ongoing and will be reported in future work. Perhaps even more importantly, these results reveal aspects of the nature of nuclei, not at the physical quark masses, but in a more general setting where Standard Model parameters are allowed to vary. In particular, they indicate that the phenomenological successes of the nuclear shell-model in nature may extend over a broad range of quark masses.
0.25in
We thank D.B. Kaplan and D.R. Phillips for helpful discussions. SRB was supported in part by NSF continuing grant PHY1206498, MJS was supported in part by DOE grant No. DE-FG02-00ER41132, WD was supported by the U.S. Department of Energy Early Career Research Award DE-SC0010495 and the Solomon Buchsbaum Fund at MIT. KO was supported by the U.S. Department of Energy through Grant Number DE- FG02-04ER41302 and through Grant Number DE-AC05-06OR23177 under which JSA operates the Thomas Jefferson National Accelerator Facility. HWL was supported by DOE grant No. DE-FG02-97ER4014. The work of AP was supported by the contract FIS2011-24154 from MEC (Spain) and FEDER. BCT was supported in part by a joint City College of New York–RIKEN/Brookhaven Research Center fellowship, a grant from the Professional Staff Congress of the CUNY, and by the U.S. National Science Foundation, under Grant No. PHY$12$-$05778$. This work made use of high-performance computing resources provided by XSEDE (supported by National Science Foundation Grant Number OCI-1053575), NERSC (supported by U.S. Department of Energy Grant Number DE-AC02-05CH11231), the PRACE Research Infrastructure resource Mare Nostrum at the Barcelona SuperComputing Center, and by the USQCD collaboration. Parts of these calculations were performed using the [chroma]{} lattice field theory library [@Edwards:2004sx].
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abstract: |
Consider a random polynomial $$G_Q(x)=\xi_{Q,n}x^n+\xi_{Q,n-1}x^{n-1}+\dots+\xi_{Q,0}$$ with independent coefficients uniformly distributed on $2Q+1$ integer points $\{-Q, \dots, Q\}$. Denote by $D(G_Q)$ the discriminant of $G_Q$. We show that there exists a constant $C_n$, depending on $n$ only such that for all $Q\ge 2$ the distribution of $D(G_Q)$ can be approximated as follows $$\sup_{-\infty\leq a\leq b\leq\infty}\left|\mathbb{P}\left(a\leq \frac{D(G_Q)}{Q^{2n-2}}\leq b\right)-\int_a^b\varphi_n(x)\, dx\right|\leq\frac{C_n}{\log Q},$$ where $\varphi_n$ denotes the distribution function of the discriminant of a random polynomial of degree $n$ with independent coefficients which are uniformly distributed on $[-1,1]$.
Let $\Delta(G_Q)$ denote the minimal distance between the complex roots of $G_Q$. As an application we show that for any $\varepsilon>0$ there exists a constant $\delta_n>0$ such that $\Delta(G_Q)$ is stochastically bounded from below/above for all sufficiently large $Q$ in the following sense $$\mathbb{P}\left(\delta_n<\Delta(G_Q)<\frac1{\delta_n}\right)>1-\varepsilon .$$
address:
- 'Friedrich Götze, Faculty of Mathematics, Bielefeld University, P. O. Box 10 01 31, 33501 Bielefeld, Germany'
- |
Dmitry Zaporozhets\
St. Petersburg Department of Steklov Institute of Mathematics, Fontanka 27, 191011 St. Petersburg, Russia
author:
- Friedrich Götze
- Dmitry Zaporozhets
bibliography:
- 'bib.bib'
title: Discriminant and root separation of integral polynomials
---
[^1]
Introduction
============
Let $$p(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_0=a_n(x-\alpha_1)\dots(x-\alpha_n)$$ be a polynomial of degree $n$ with real or complex coefficients.
In this note we consider different asymptotic estimates when the degree $n$ is arbitrary but *fixed*. Thus for non-negative functions $f,g$ we write $f\ll g$ if there exists a non-negative constant $C_n$ (depending on $n$ only) such that $f\leq C_ng$. We also write $f\asymp g$ if $f\ll g$ and $f\gg g$.
Denote by $$\Delta(p)=\min_{1\leq i<j\leq n}|\alpha_i-\alpha_j|$$ the shortest distance between any two zeros of $p$.
In his seminal paper [Mahler [@kM64]]{} proved that $$\label{331}
\Delta(p)\geq\sqrt{3}n^{-(n+2)/2}\frac{|D(p)|^{1/2}}{\left(|a_n|+\dots+|a_0|\right)^{n-1}},$$ where $$\label{308}
D(p)=a_n^{2n-2}\prod_{1\leq i<j\leq n}(\alpha_i-\alpha_j)^2$$ [denotes the]{} discriminant of $p(x)$. Alternatively, $D(p)$ [is]{} given by the $(2n-1)$-dimensional determinant $$\begin{gathered}
\label{932}
D(p)=(-1)^{n(n-1)/2}
\\\times
\left|\begin{matrix}
& 1 & a_{n-1} &a_{n-2} & \ldots & a_0 & 0 & \ldots &0&0 \\
& 0 & a_n & a_{n-1} & \ldots & a_1 & a_0 & \ldots &0&0 \\
&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots\\
&0 &0 &0 &\dots &a_{n-2} &a_{n-3}&\dots&a_1 &a_0 \\
&n &(n-1)a_{n-1} &(n-2)a_{n-2} &\dots &0 &0 &\dots &0&0\\
&0&na_n &(n-1)a_{n-1} &\dots &a_1 &0 &\dots &0&0\\
&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots\\
&0 &0 &0 &\dots &(n-1)a_{n-1} &(n-2)a_{n-2}&\dots&2a_2 &a_1 \\
\end{matrix}\right|.\end{gathered}$$
Define the height of the polynomial by $H(p)=\max_{0\leq i\leq n}|a_i|$. It follows immediately from that $$\label{311}
|D(p)|\ll H(p)^{2n-2}.$$
From now on we will always assume that the polynomial $p$ is integral (that is has integer coefficients). Since the condition $D(p)\ne0$ implies $|D(p)|\geq1$ Mahler noted that implies $$\label{445}
\Delta(p)\gg H(p)^{-n+1},$$ provided that $p$ doesn’t have multiple zeros. The estimate [seems to be the best available lower bound up to now.]{} However, for $n\geq 3$ it is still not known how far [it differs from the optimal lower bound.]{} Denote by $\kappa_n$ the infimum of $\kappa$ such that $$\Delta(p)> H(p)^{-\kappa}$$ holds for all integral polynomials of degree $n$ without multiple zeros and large enough height $H(p)$. It is easy to see that is equivalent to $\kappa_n\leq n-1$. Also it is a simple exercise to show that $\kappa_2=1$ (see, e.g., [@BM10]). Evertse [@jE04] showed that $\kappa_3=2$.
For $n\geq 4$ only estimates are known. At first, Mignotte [@mM83] proved that $\kappa_n\geq n/4$ for $n\geq 2$. Later Bugeaud and Mignotte [@BM04; @BM10] have shown that $\kappa_n\geq n/2$ for even $n\geq 4$ and $\kappa_n\geq (n+2)/4$ for odd $n\geq 5$. Shortly after that Beresnivich, Bernik, and Götze [@BBG10], using completely different approach, improved their result in the case of odd $n$: they obtained (as a corollary of more general counting result) that $\kappa_n\geq(n+1)/3$ for $n\geq2$. Recently Bugeaud and Dujella [@BD14] achieved significant progress showing that $\kappa_n\geq(2n-1)/3$ for $n\geq4$ (see also [@BD11] for irreducible polynomials).
Formulated in other terms the above results give answers to the question [“How close to each other can two conjugate algebraic numbers of degree $n$ be?” ]{} Recall that two complex algebraic numbers called conjugate (over $\mathbb Q$) if they are roots of the same irreducible integral polynomial (over $\mathbb Q$). Roughly speaking, if we consider a polynomial $p^*$ which minimizes $\Delta(p)$ among all integral polynomials of degree $n$ having the same height and without multiple zeros, then $\Delta(p^*)$ satisfies the following lower/upper bounds with respect to $H(p^*)$: $$H(p^*)^{-c_1n}\ll \Delta(p^*)\ll H(p^*)^{-c_2n},$$ for some absolute constants $0<c_2\leq c_1$. In this note, instead of considering the extreme polynomial $p^*$, we consider the behaviour of $\Delta(p)$ for a typical integral polynomial $p$. We prove that for ”most“ integral polynomials (see Section \[1049\] for a more precise formulation) we have $$\Delta(p)\asymp 1.$$ We also show that the same estimate holds for ”most“ irreducible integral polynomials (over $\mathbb Q$).
A related interesting problem is to study the distribution of discriminants of integral polynomials. To deal with it is convenient (albeit not necessary) to use probabilistic terminology. Consider some $Q\in\mathbb N$ and consider the class of all integral polynomials $p$ with $\deg(p)\leq n$ and $H(p)\leq Q$. The cardinality of this class is $(2Q+1)^{n+1}$. Consider the uniform probability measure on this class so that the probability of each polynomial is given by $(2Q+1)^{-n-1}$. In this sense, we may consider random polynomials $$G_Q(x)=\xi_{Q,n}x^n+\xi_{Q,n-1}x^{n-1}+\dots+\xi_{Q,0}$$ with independent coefficients which are uniformly distributed on $2Q+1$ integer points $\{-Q, \dots, Q\}$. We are interested in the asymptotic behavior of $D(G_Q)$ when $n$ is fixed and $Q\to\infty$.
Bernik, Götze and Kukso [@BGK08] showed that for $\nu\in[0,1/2]$ $${\mathbb{P}}(|D(G_Q|<Q^{2n-2-2\nu})\gg Q^{-2\nu}.$$ Note that the case $\nu=0$ is consistent with . It has been conjectured in [@BGK08] that this estimate is optimal up to a constant: $$\label{211}
{\mathbb{P}}(|D(G_Q)|<Q^{2n-2-2\nu})\asymp Q^{-2\nu}.$$ The conjecture turned out to be true for $n=2$: Götze, Kaliada, and Korolev [@GKK13] showed that for $n=2$ and $\nu\in (0,3/4)$ it holds $${\mathbb{P}}(|D(G_Q)|<Q^{2-2\nu})=2(\log 2+1)Q^{-2\nu}\left(1+O(Q^{-\nu}\log Q+Q^{2\nu-3/2}\log^{3/2}Q)\right).$$ However, for $n=3$ and $\nu\in [0,3/5)$ Kaliada, Götze, and Kukso [@KGK13] obtained the following asymptotic relation: $$\label{1202}
{\mathbb{P}}(|D(G_Q)|<Q^{4-2\nu})=\kappa Q^{-5\nu/3}\left(1+O(Q^{-\nu/3}\log Q+Q^{5\nu/3-1})\right),$$ where the absolute constant $\kappa$ had been explicitly determined. Recently Beresnevich, Bernik, and Götze [@BBG15] extended the lower bound given by to the full range of $\nu$ and to the arbitrary degrees n. They showed that for $0\leq\nu< n-1$ one has that $${\mathbb{P}}(|D(G_Q|<Q^{2n-2-2\nu})\gg Q^{-n+3-(n+2)\nu/n}.$$ They also obtained a similar result for resultants.
In this note we prove a limit theorem for $D(G_Q)$. As a corollary, we obtain that ”with high probability“ (see Section \[1049\] for details) the following asymptotic equivalence holds: $$|D(P_Q)|\asymp Q^{2n-2}.$$ The same estimate holds ”with high probability“ for irreducible polynomials.
For more comprehensive survey of the subject and a list of references, see [@BBGK13].
Main results {#1049}
============
Let $\xi_0,\xi_1,\dots,\xi_{n}$ be independent random variables [*uniformly*]{} distributed on $[-1,1]$. Consider the random polynomial $$G(x)=\xi_nx^n+\xi_{n-1}x^{n-1}+\dots+\xi_1x+\xi_0$$ and denote by $\varphi$ the distribution function of $D(G)$. It is easy to see that $\varphi$ has compact support and $\sup_{x\in{\mathbb{R}}}\varphi(x)<\infty$.
\[147\] Using the above notations we have $$\label{1133}
\sup_{-\infty\leq a\leq b\leq\infty}\left|{\mathbb{P}}\left(a\leq \frac{D(G_Q)}{Q^{2n-2}}\leq b\right)-\int_a^b\varphi(x)\, dx\right|\ll\frac1{\log Q}.$$
How far is this estimate from being optimal? Relation shows that for $n=3$ the estimate $\log^{-1}Q$ can not be replaced by $Q^{-\varepsilon}$ for any $\varepsilon>0$. Otherwise it would imply that holds for $\nu\leq\varepsilon/2$.
The proof of Theorem \[147\] will be given in Section \[147\]. Now let us derive some corollaries.
Relation means that $|D(G_Q)|\ll Q^{2n-2}$ holds a.s. It follows from Theorem \[147\] that with high probability the lower estimate holds as well.
\[148\] For any $\varepsilon>0$ there exists $\delta>0$ (depending on $n$ only) such that for all sufficiently large $Q$ $$\label{204}
{\mathbb{P}}(|D(G_Q)|>\delta Q^{2n-2})>1-\varepsilon.$$
Since $\sup_{x\in{\mathbb{R}}}\varphi(x)<\infty$, it follows from that $${\mathbb{P}}(|D(G_Q)|<\delta Q^{2n-2})\ll \delta+\frac{1}{\log Q},$$ which completes the proof.
As another corollary we obtain an estimate for $\Delta(G_Q)$.
\[303\] For any $\varepsilon>0$ there exists $\delta>0$ (depending on $n$ only) such that for all sufficiently large $Q$ $$\label{205}
{\mathbb{P}}(\delta<\Delta(G_Q)<\delta^{-1})>1-\varepsilon.$$
For large enough $Q$ we have $${\mathbb{P}}\left(|\xi_{Q,n}|>\frac{\varepsilon}{2}Q\right)>1-\varepsilon.$$ Therefore it follows from and that with probability at least $1-\varepsilon$ $$\Delta(G_Q)\leq\left(\frac2\varepsilon\right)^{2/n},$$ which implies the upper estimate. The lower bound immediately follows from and .
[**Remark on irreducibility.**]{} In order to consider $\Delta(G_Q)$ as distance between the closest conjugate algebraic numbers of $G_Q$ we have to restrict ourselves to irreducible polynomials only. In other words the distribution of the random polynomial $G_Q$ has to be conditioned on $G_Q$ being irreducible. It turns out that the relations and with conditional versions of the left-hand sides still hold. This fact easily follows from the estimate $${\mathbb{P}}(G_Q \text{is irreducible})\asymp1,$$ which was obtained by van der Waerden [@vW36].
Proof of Theorem 2.1 {#1723}
====================
For $k\in\mathbb N$ the moments of $\xi_i$ and $\xi_{i,Q}$ are given by $${\mathbb E}\xi^{2k}_i=\frac1{2k+1},\quad {\mathbb E}\xi^{2k}_{i,Q}=\frac{2}{2Q+1}\sum_{j=1}^Qj^{2k}.$$ Since $$\frac{Q^{2k+1}}{2k+1}=\int_0^Qt^{2k}dt\leqslant \sum_{j=1}^Qj^{2k}\leqslant\int_0^Q(t+1)^{2k}\,dt\leqslant\frac{(Q+1)^{2k+1}}{2k+1},$$ we get $$\begin{gathered}
\left|\frac{2}{2Q+1}\sum_{j=1}^Qj^{2k}-\frac{Q^{2k}}{2k+1}\right|=\frac{2}{2Q+1}\Big|\sum_{j=1}^Qj^{2k}-\frac{2Q+1}{2}\frac{Q^{2k}}{2k+1}\Big|
\\\leqslant\frac{2}{2Q+1}\Big|\sum_{j=1}^Qj^{2k}-\frac{Q^{2k+1}}{2k+1}\Big|+\frac{Q^{2k}}{2Q+1}
\\\leqslant\frac{2}{2Q+1}\cdot\frac{(Q+1)^{2k+1}-Q^{2k+1}}{2k+1}+\frac{Q^{2k}}{2Q+1}\leqslant2^{2k}Q^{2k-1},\end{gathered}$$ which implies $$\label{133}
\left|{\mathbb E}\left(\frac{\xi_{i,Q}}Q\right)^{2k}-{\mathbb E}\xi^{2k}\right|\leqslant\frac{2^{2k}}{Q}.$$ It follows from that for all $k\in\mathbb N$ $$\label{238}
\left|{\mathbb E}\,D^k\left(\frac{G_Q}Q\right)-{\mathbb E}\,D^k(G)\right|\leq
n^{nk}\sum_{k_0,\dots,k_n}\left|\prod_{i=0}^n{\mathbb E}\,\left(\frac{\xi_{i,Q}}Q\right)^{2k_i} -\prod_{i=0}^n{\mathbb E}\,\xi_{i}^{2k_i}\right|,$$ where the summation is taken over at most $((2n-1)!)^k$ summands such that $k_0+\dots+k_n=k(n-1)$. Let us show that $$\label{123}
\left|\prod_{i=0}^n{\mathbb E}\,\left(\frac{\xi_{i,Q}}Q\right)^{2k_i} -\prod_{i=0}^n{\mathbb E}\,\xi_{i}^{2k_i}\right| \leq \frac{2^{2k_0+\dots+2k_n}}{Q}.$$ We proceed by induction on $n$. The case $n=0$ follows from . It holds $$\begin{gathered}
\left|\prod_{i=0}^n{\mathbb E}\,\left(\frac{\xi_{i,Q}}Q\right)^{2k_i} -\prod_{i=0}^n{\mathbb E}\,\xi_{i}^{2k_i}\right|
\\\leq\left|\prod_{i=0}^{n-1}{\mathbb E}\,\left(\frac{\xi_{i,Q}}Q\right)^{2k_i} -\prod_{i=0}^{n-1}{\mathbb E}\,\xi_{i}^{2k_i}\right|{\mathbb E}\,\left(\frac{\xi_{n,Q}}Q\right)^{2k_n}
\\+\prod_{i=0}^{n-1}{\mathbb E}\,\xi_{i}^{2k_i} \left|{\mathbb E}\,\left(\frac{\xi_{n,Q}}Q\right)^{2k_n}-{\mathbb E}\,\xi_{0}^{2k_0}\right|.\end{gathered}$$ Applying the induction assumption and , we obtain .
Thus, using , , and the relation $k_0+\dots+k_n=k(n-1)$ we get $$\label{430}
\left|{\mathbb E}\,D^k\left(\frac{G_Q}Q\right)-{\mathbb E}\,D^k(G)\right|\leq\frac{\gamma^k}{Q},$$ where $\gamma$ depends on $n$ only.
Since $D(G)$ and $D(G_Q/Q)$ are bounded random variables, their characteristic functions $$f(t)={\mathbb E}\,e^{iD(G)},\quad f_Q(t)={\mathbb E}\,e^{iD(G_Q/Q)}$$ are entire functions. Therefore implies that for all real $t$ $$\label{743}
|f_Q(t)-f(t)|=\left|\sum_{k=1}^\infty i^k\frac{{\mathbb E}\,D^k(G_Q/Q)-{\mathbb E}\,D^k(G)}{k!}t^k\right|\leq\frac1Q\sum_{k=1}^\infty\frac{(\gamma |t|)^k}{k!}\leq\frac{\gamma |t|e^{\gamma |t|}}{Q}.$$
Now we are ready to estimate the uniform distance between the distributions of $D(G)$ and $D(G_Q/Q)$ using the closeness of $f(t)$ and $f_Q(t)$. Let $F$ and $F_Q$ be distribution functions of $D(G)$ and $D(G_Q/Q)$. By Esseen’s inequality, we get for any $T>0$ $$\sup_{x}|F_Q(x)-F(x)|\leq\frac{2}{\pi}\int_{-T}^T\,\left|\frac{f_Q(t)-f(t)}{t}\right|\,dt+\frac{24}{\pi}\cdot\frac{\sup_{x\in{\mathbb{R}}}\varphi(x)}{T}.$$ Applying , we obtain that there exists a constant $C$ depending on $n$ only such that for any $T>0$ $$\sup_{-\infty\leq a\leq b\leq\infty}\left|\left({\mathbb{P}}(a\leq D\left(\frac{G_Q}Q\right)\leq b\right)-{\mathbb{P}}(a\leq D(G)\leq b)\right|
\leq C\left(\frac{Te^{\gamma T}}{Q}+\frac{1}{T}\right).$$ Taking $T=\log Q/2\gamma$ completes the poof.
Resultants
==========
Given polynomials $$p(x)=a_n(x-\alpha_1)\dots(x-\alpha_n),\quad q(x)=b_m(x-\beta_1)\dots(x-\beta_m),$$ denote by $R(p,q)$ the resultant defined by $$R(p,q)=a_n^mb_m^n\prod_{i=1}^n\prod_{j=1}^m(\alpha_i-\beta_j).$$ Obviously discriminants are essentially a specialization of resultants via: $$D(p)=(-1)^{n(n-1)/2}a_n^{-1}R(p,p').$$
Repeating the arguments from Section \[1723\] we obtain the following result. Consider the random polynomials $$G_Q(x)=\xi_{Q,n}x^n+\xi_{Q,n-1}x^{n-1}+\dots+\xi_{Q,1}x+\xi_{Q,0},$$ $$F_Q(x)=\eta_{Q,m}x^m+\eta_{Q,m-1}x^{m-1}+\dots+\eta_{Q,1}x+\eta_{Q,0}$$ with independent coefficients uniformly distributed on $2Q+1$ points $\{-Q,\dots,Q\}$ and consider the random polynomials $$G(x)=\xi_nx^n+\xi_{n-1}x^{n-1}+\dots+\xi_1x+\xi_0,$$ $$F(x)=\eta_mx^m+\eta_{m-1}x^{m-1}+\dots+\eta_1x+\eta_0$$ with independent coefficients uniformly distributed on $[-1,1]$. Denote by $\psi$ [the distribution function]{} of $R(G,F)$. We have $$\sup_{-\infty\leq a\leq b\leq\infty}\left|{\mathbb{P}}\left(a\leq \frac{R(G_Q,F_Q)}{Q^{m+n}}\leq b\right)-\int_a^b\psi(x)\, dx\right|\ll\frac1{\log Q}.$$
[**Acknowledgments.**]{} We are grateful to Victor Beresnevich, Vasili Bernik, and Zakhar Kabluchko for [useful]{} discussions. We also would like to thank Andrei Zaitsev for some remarks on notations.
[^1]: The work was done with the financial support of the Bielefeld University (Germany) in terms of project SFB 701.
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abstract: 'We present results from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) photometric redshift methods investigation. In this investigation, the results from eleven participants, each using a different combination of photometric redshift code, template spectral energy distributions (SEDs) and priors, are used to examine the properties of photometric redshifts applied to deep fields with broad-band multi-wavelength coverage. The photometry used includes $U$-band through mid-infrared filters and was derived using the TFIT method. Comparing the results, we find that there is no particular code or set of template SEDs that results in significantly better photometric redshifts compared to others. However, we find codes producing the lowest scatter and outlier fraction utilize a training sample to optimize photometric redshifts by adding zero-point offsets, template adjusting or adding extra smoothing errors. These results therefore stress the importance of the training procedure. We find a strong dependence of the photometric redshift accuracy on the signal-to-noise ratio of the photometry. On the other hand, we find a weak dependence of the photometric redshift scatter with redshift and galaxy color. We find that most photometric redshift codes quote redshift errors (e.g., 68% confidence intervals) that are too small compared to that expected from the spectroscopic control sample. We find that all codes show a statistically significant bias in the photometric redshifts. However, the bias is in all cases smaller than the scatter, the latter therefore dominates the errors. Finally, we find that combining results from multiple codes significantly decreases the photometric redshift scatter and outlier fraction. We discuss different ways of combining data to produce accurate photometric redshifts and error estimates.'
author:
- 'Tomas Dahlen, Bahram Mobasher, Sandra M. Faber, Henry C Ferguson, Guillermo Barro , Steven L. Finkelstein , Kristian Finlator , Adriano Fontana , Ruth Gruetzbauch , Seth Johnson , Janine Pforr , Mara Salvato , Tommy Wiklind , Stijn Wuyts , Viviana Acquaviva, Mark E. Dickinson , Yicheng Guo , Jiasheng Huang , Kuang-Han Huang , Jeffrey A. Newman , Eric F. Bell , Christopher J. Conselice , Audrey Galametz , Eric Gawiser , Mauro Giavalisco , Norman A. Grogin , Nimish Hathi , Dale Kocevski , Anton M. Koekemoer , David C. Koo , Kyoung-Soo Lee , Elizabeth J. McGrath , Casey Papovich , Michael Peth , Russell Ryan , Rachel Somerville , Benjamin Weiner , and Grant Wilson'
title: 'A Critical Assessment of Photometric Redshift Methods: A CANDELS Investigation'
---
Introduction
============
Using photometric redshifts to estimate the distances of faint galaxies has become an integral part of galaxy surveys conducted during recent years. This is driven by the large number of galaxies, and their faint fluxes which have made spectroscopic follow-up infeasible except for a relatively small and bright fraction of the galaxy population. Albeit less precise and less accurate than spectroscopy, photometric redshifts provide a way to estimate distances for galaxies too faint for spectroscopy or samples too large to be practical for complete spectroscopic coverage. Since the early description of using colors to determine distances in Baum (1962), and the important developments over the years described in e.g., Koo (1985), Connolly et al. (1995) and Gwyn (1995), the number of articles describing the method and the number of applications for photometric redshifts have grown rapidly.
The photometric redshift technique is usually divided into two groups, template fitting and empirical fitting. The template fitting technique derives the photometric redshift by minimizing the value $\chi^2$ when comparing an observed SED with the SED computed from a template library that includes spectral-energy distributions for a variety of galaxy types (representing different redshifts, star-formation histories, chemical abundance, and mixtures of dust and stars). The empirical technique uses a training set of galaxies with known spectroscopic redshifts to derive a relation between observed photometry and redshifts. Today, a large number of codes of both techniques exists, many of which are publicly available. Codes based on the template fitting technique include: zphot (Giallongo et al. 1998), HyperZ (Bolzonella et al. 2000), BPZ (Benítez 2000), ImpZ (Babbedge et al. 2004), ZEBRA (Feldmann et al. 2006), SPOC (Finlator et al. 2007), EAZY (Brammer et al. 2008), Low Resolution Template (LRT) Libraries (Assef et al. 2008), GALEV (Kotulla et al. 2009), Rainbow (Barro et al. 2011), GOODZ (Dahlen et al. 2010), LePhare (Ilbert et al. 2006; S. Arnouts & O. Ilbert 2013, in preparation), and SATMC (S. Johnson et al. 2013, in preparation). Empirical codes include: ANNz (Collister & Lahav 2004); Multilayer Perceptron Artificial Neural Network (Vanzella et al. 2004); ArborZ (Gerdes et al. 2010); “Empirical-$\chi^2$” (Wolf 2009); “Random Forests” (Carliles et al. 2010). Certain codes combine the methodology of both techniques (e.g., EAZY, GOODZ, and LePhare) which can use a training set of galaxies to derive corrections to zero-points and/or template SED shapes in order to minimize the scatter between photometric and spectroscopic redshifts in the training sample. These corrections can then be applied to the full set of galaxies without spectroscopy.
The Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS; PIs S. Faber and H. Ferguson; see Grogin et al. 2011 and Koekemoer et al. 2011) is an $HST$ Multi-Cycle Treasury program aimed at imaging distant galaxies in multiple wavebands and detect high redshift supernovae in five sky regions: the GOODS-S, GOODS-N, EGS, UDS, and COSMOS fields. Images and catalogs will be provided to the public for the different fields. Besides photometry, the catalogs will include auxiliary information such as photometric redshifts and stellar masses of galaxies. The CANDELS data include some of the deepest photometry available in both optical and infrared over a wide area and it is important to investigate the behavior of the derived quantities at the faint flux levels typical of the survey. Therefore, the CANDELS team has preformed a series of tests to evaluate how photometric redshift and mass estimates from different codes compare, how well codes reproduce the redshift of objects with spectroscopic redshifts, how well codes reproduce masses from simulated galaxies, and how photometric redshift estimates depend on signal-to-noise, redshift and galaxy color. Furthermore, we investigate how the error estimates determined by the codes compare with the errors expected from either spectroscopic control samples or simulated galaxy catalogs. Finally, we investigate possible ways of combining results from individual codes in order to improve the quality of the photometric redshifts.
While the investigation was performed with the CANDELS data in mind, the questions should be general and the results relevant for any survey targeting distant galaxies. In this paper, we focus the investigation on the photometric redshift technique. A number of collaborators in the CANDELS team were asked to use their preferred photometric redshift code to derive redshifts for a set of photometric catalogs. The results from the different codes and sets of template SEDs were thereafter compared with the aim of deriving the best photometric redshifts possible given the available data set and to minimize possible biases in the derived redshifts. In an accompanying paper, B. Mobasher et al . (2013, in preparation), we discuss estimates of stellar masses using the same catalogs.
This paper is organized as follows: In Section 2, we describe the catalogs used in the testing followed in Section 3 by a presentation of the different codes used. Results are given in Section 4, followed by a discussion on ways to combine data to improve photometric redshifts in Section 5. Section 6 presents a comparison to earlier work. A summary is given in Section 7. Throughout we assume a cosmology with $\Omega_M$=0.3, $\Omega_{\Lambda}$=0.7, and h=0.7. Magnitudes are given in the AB system.
Test Catalogs
=============
Two different catalogs were used to test the photometric redshifts. The first is a near-IR $HST$/WFC3 $H$-band (F160W filter) selected catalog, while the second is an optical $HST$/ACS $z$-band (F850LP filter) selected catalog. Both catalogs cover the GOODS-S area (Giavalisco et al. 2004), with photometry derived using the TFIT method (Laidler et al. 2007). We use two fairly similar catalogs, to investigate possible differences in optical versus near-IR selected photometric redshifts. For both catalogs we provided a test sample with known spectroscopic redshifts for training the photometric redshift codes. Each participant in the CANDELS SED-fitting test was asked to derive photometric redshifts for the objects in each catalog, including both the training sample and a control sample for which the redshifts were not provided. Below we give more details on the different catalogs.
WFC3 $H$-band selected catalog
------------------------------
The primary test catalog includes the $HST$/WFC3 $H$-band selected TFIT multi-band photometry. The catalog contains 20,000 objects in the GOODS-S field and includes photometry in 14 bands: $U$ (VLT/VIMOS), $BViz$ ($HST$/ACS), F098M, F105W, F125W, F160W (WFC3/IR), $K_s$ (VLT/ISAAC) and 3.6, 4.5, 5.8, 8.0 micron ($Spitzer$/IRAC). The total area covered in the catalog is approximately $\sim$100 arcmin$^2$. Note that F098M covers $\sim$40% of the area (data taken from the Early Release Science, Windhorst et al. 2011), while F105W covers most of the remaining $\sim$60%, therefore, 13 band photometry is the maximum for any individual object. Photometry in the ACS and WFC3 bands are measured using SExtractor in dual-image mode with F160W as the detection band. For all other bands, the TFIT method was used. This results in a flux measurement for all objects in all bands that cover the footprint of the F160W data. Both SExtractor and TFIT will provide flux estimates for sources based on prior information on position and shape from the H-band image. Therefore fluxes are provided in every band even for sources that are not formally detected in that band. These fluxes can sometimes be negative due to statistical fluctuations. If the photometric error estimates are corrected, this should not cause problems for the photometric-redshift estimates. We also note that the F160W band photometry includes 4 of the 10 planned epochs of GOODS-S data available at the time of the test. A detailed description of the CANDLES GOODS-S data is given in Guo et al. (2013). The methodology used to derive the photometry is described in Galametz et al. (2013).
ACS z-band selected catalog
---------------------------
As a secondary test catalog, we use an ACS $z$-band selected TFIT catalog of GOODS-S that includes multi-waveband photometry in twelve bands: $U$ (VLT/VIMOS), $BViz$ ($HST$/ACS), $JHKs$ (VLT/ISAAC) and 3.6, 4.5, 5.8, 8.0 micron ($Spitzer$/IRAC). The data is the same as for the primary test catalog except that ISAAC $J$ and $H$ are added and WFC3 IR bands are excluded. The area covered by the $z$-band selected catalog is $\sim$150 arcmin$^2$ and the number of objects included in the catalog is 25,000. We use the secondary catalog to examine the effect of selecting the catalog in the optical vs. near-IR WFC3 when estimating photometric redshifts. Details on the photometry are given in Dahlen et al. (2010).
Spectroscopic comparison sample
-------------------------------
We use a sample of galaxies with known spectroscopic redshifts to evaluate how well the photometric redshifts reproduce the true redshifts as given by the spectra. Our spectroscopic sample is compiled from a set of publicly available data including Cristiani et al. (2000), Croom et al. (2001), Bunker et al. (2003), Dickinson et al. (2004), LeFèvre et al. (2004), Stanway et al. (2004), Strolger et al. (2004), Szokoly et al. (2004), van der Wel et al. (2004), Doherty et al. (2005), Mignoli et al. (2005), Roche et al. (2006), Ravikumar et al. (2007), Vanzella et al. (2008), and Popesso et al. (2009).
When selecting the sources for inclusion in our spectroscopic redshift sample, we specifically include only objects with the highest possible data quality (when available). Furthermore, we exclude all objects with X-ray detection in the Chandra 4Ms sample from Xue et al. (2011) and radio sources in Afonso et al. (2006) and Padovani et al. (2011). Even though there are more than 3000 spectroscopic redshifts in the GOODS-S ACS footprint, we exclude more than half of these to minimize the number of faulty redshifts and AGN contaminants. The latter are excluded since the aim here is to derive and compare photometric redshifts for a population of “normal” galaxies. Photometric redshifts for X-ray sources are discussed in Salvato et al. (2009, 2011, 2013 in preparation). We divide the final set of highest quality spectroscopic redshifts into a training sample provided to each participant in the test. A second control sample is used to evaluate the accuracy of the photometric redshifts. Both catalogs cover the same ranges in magnitude, color, and redshift. The training catalogs include 580 and 640 objects, while the control samples contain 589 and 614 objects for the $H$-selected and $z$-selected catalogs, respectively. The difference in the total number of objects between the different selections is due to the difference in covered area. The redshift and magnitude distributions of the spectroscopic sample are presented in Figure \[fig1\].
Publicly available test catalogs
--------------------------------
The GOODS-S $H$-band selected test catalogs and associated files are available via the STScI Archive High-Level Science Products page for CANDELS[^1]. This includes the 14 band photometry and spectroscopic redshifts for 580 and 589 objects in the training and control samples, respectively.
Participating codes
===================
A total of thirteen submissions to the CANDELS SED-fitting test were received and each participant was given an ID number. Of these thirteen, eleven included calculated photometric redshifts, while the remaining two only presented derived masses (for objects with known spectroscopic redshifts). In Table \[table1\], we list the eleven participants that provided photometric redshifts (participants only producing masses are described in B. Mobasher et al. 2013, in preparation) and the name of the photometric redshift code used. Each different code is assigned a single character code identifier in the range A-I. We hereafter refer the to the combination of participant and photometric redshift code by combining the two identifiers, e.g., 2A, 3B, 4C and so on. This makes it easy to identify the participants that use the same photometric redshift code, i.e., 4C, 7C, and 13C. For simplicity, we refer to the eleven different participant and code combinations as “codes” in the following. The table lists the template set used and shows if emission lines are included. We provide the latter information since emission lines can have a significant effect on broad-band photometry and therefore the template SED fitting (e.g., Atek et al. 2011; Schaerer & de Barros 2012; Stark et al. 2013). Also shown is if the codes use the control sample of spectroscopic redshifts to calculate a “flux shift” to the given photometry or template SEDs. Indicated is also if the code adds an extra error to the provided flux errors when template fitting. The most common way of implementing additional errors is to add (in quadrature) an error corresponding to 2-10% of the flux ($\sim$0.02-0.1 mag) to the given photometric errors. Alternatively, a lower limit to the given errors can be enforced. Finally, the table indicates if the code adjusts the template SEDs based on the training sample and if the code uses interpolations between template SEDs.
[rllclcccccc]{} ID$^a$ & PI & Code & Code ID & Template set & Em lines & Flux shift & $\Delta$err & $\Delta$SED & Inter & ref.\
2 & G. Barro & Rainbow & A & PEGASE$^b$ & yes& yes & no & no & no & $j$\
3 & T. Dahlen & GOODZ & B & CWW$^c$, Kinney$^d$ &yes & yes & yes & yes & yes & $k$\
4 & S. Finkelstein & EAZY & C & EAZY$^e$+BX418$^f$ &yes& no & no & no & yes & $l$\
5 & K. Finlator & SPOC & D & BC03$^g$ &yes& no & no & no & no & $m$\
6 & A. Fontana & zphot & E & PEGASEv2.0$^b$ &yes& yes & yes & no & no & $n,o$\
7 & R. Gruetzbauch & EAZY & C & EAZY$^e$ &yes& yes & yes & no & yes & $l$\
8 & S. Johnson & SATMC & F & BC03$^g$ &no& no & no & no & yes & $p$\
9 & J. Pforr & HyperZ & G & Maraston05$^h$ &no& no & yes & no & no & $q$\
11 & M. Salvato & LePhare & H & BC03$^g$+Polletta07$^i$ &yes& yes & yes & no & no & $r$\
12 & T. Wikind & WikZ & I & BC03$^g$ &no& no & yes & no & no & $s$\
13 & S. Wuyts & EAZY & C & EAZY$^e$ &yes& yes & yes & no & yes & $l$\
\[table1\]
Below we give a short summary of each code participating in the photometric redshift test. For each code we describe if the $\chi^2$ minimization is done in magnitude or flux space and how negative fluxes are treated. We also note the codes using priors in the fitting. We finally comment on any special treatment of the IRAC fluxes in the fitting, such as excluding the 5.8$\mu$m and 8.0$\mu$m channels (hereafter ch3 and ch4) at low redshift where they probe wavelengths where templates may not be as reliable. For details, please see the quoted articles.\
\
A - [*Rainbow*]{} (Pérez-González et al. 2008; Barro et al. 2011)[^2]\
A template fitting code based on $\chi^2$ minimization between observed photometry and a set of $\sim$1500 semi-empirical template SEDs computed from spectroscopically confirmed galaxies modeled with PEGASE stellar population synthesis models (see Pérez-González et al. 2008, Appendix A). The code allows for additional smoothing errors, photometric zero-point corrections and a template error function to down-weight the wavelength ranges where the templates a more uncertain (e.g., the rest frame near-IR). Fitting is done in flux space. Negative fluxes and data points with signal-to-noise $<$2.5 are not included in the fitting. Excludes IRAC bands that probe rest frame $> 5\mu m$ (ch3 at $z\lsim$ 0.15 and ch4 at $z\lsim$ 0.6). Run by G. Barro.\
\
B - [*GOODZ*]{} (Dahlen et al. 2010)\
A template fitting code that minimizes $\chi^2$ between observed photometry and a set of template SEDs. The code allows the use of luminosity function priors. For this test, a rest frame $V$-band luminosity function prior was used. This assignes a low probability at low redshifts where the volume element is small and a low probability for bright objects at high redshifts. The code also calculates and applies shifts to the input photometry based on a training set of galaxies. Can adjust templates using a training sample. Extra smoothing errors were added to existing photometric errors. Includes the option of using interpolations between the provided template SEDs. Fitting is done in flux space. Negative fluxes are included in the fitting. Excludes IRAC bands that probe rest frame $> 5\mu m$ (ch3 at $z\lsim$ 0.15 and ch4 at $z\lsim$ 0.6). Run by T. Dahlen.\
\
C - [*EAZY*]{} (Brammer et al. 2008)[^3]\
Also a template fitting code based on $\chi^2$ minimization between observed photometry and template SEDs. Magnitude priors can be included and the code can apply shifts to the input photometry. Extra smoothing errors can be added to existing errors in the photometric catalogs. Includes the option of using interpolations between the provided template SEDs. Fitting is done in flux space. Run by S. Finkelstein, R. Gruetzbauch, and S. Wuyts. When doing the fitting, SF and SW include negative fluxes, while RG ignores those data points. EAZY includes the option to apply a template error function that down-weights datapoints at rest frame $\gsim$2$\mu$m. This option is used by all three participants running this code. A luminosity function prior can also be included. This assignes a low probability where the volume sampled is small (low redshifts) and a low probability at high redshifts for objects with bright apparent magnitudes. A K-band luminosity function prior was used by SW.\
\
D - [*SPOC*]{} (Finlator et al. 2007)\
A $\chi^2$ minimization code, however the template SEDs are derived directly from cosmological numerical simulations. Numerically simulated SFHs and metallicities are directly adopted, rather than assuming toy-model SFHs (for example, constant or declining), leaving effectively three free parameters, namely M\*, $z$, and A$_V$. However, the code is unlikely to find a match for any galaxy whose stellar mass lies below the mass resolution limit of the simulations from which the template library was extracted (since the code does not scale the luminosity of the galaxies independently). For the version of the code used in this investigation, the mass limit is 1.4$\times10^9M/M_{\odot}$, resulting in matches for only about half of the objects in the spectroscopic sample. Fitting is performed in flux space. Objects with negative fluxes are treated as though the negative flux indicated -1$\times$ the 1$\sigma$ upper limit (i.e., the flux error). In this case, model SEDs that are brighter than 3$\times$ that 1$\sigma$ upper limit in that band are rejected outright, otherwise the band does not contribute to the $\chi^2$. Run by K. Finlator.\
\
E - [*zphot*]{} (Giallongo et al. 1998; Fontana et al. 2000)\
A $\chi^2$ minimization code using template SEDs. It is flexible, allowing the user to adopt a wide variety of templates, including synthetic models taken from BC03, Maraston (2005), and Fioc & Rocca-Volmerange et al. (1997), with various choices of the Star Formation History, as well as observed templates of stars, galaxies and AGNs from a variety of sources. Dust extinction and IGM absorption can also be added. For the redshift determination in this investigation, a library composed of templates taken from the Fioc & Rocca-Volmerange library has been used, with the addition of Calzetti et al. (2000) extinction and Fan et al. (2006) IGM absorption. A minimum photometric errors can be set in each band. zphot can accept both fluxes and magnitudes in the input catalog, and computes the photometric redshifts directly from the best-matching template. The code also computes the error estimate on the fitted values by scanning the probability distribution of the various parameters. For this test, fitting is done in flux space including negative fluxes. When the flux is $<$-flux-error, the flux is set to zero in order to prevent datapoints with unrealistically negative fluxes to inflate the $\chi^2$ value. Excludes IRAC bands that probe rest frame $> 5\mu m$ (ch3 at $z\lsim$ 0.15 and ch4 at $z\lsim$ 0.6). Run by A. Fontana.\
\
F - [*SATMC*]{} (S. Johnson et al. 2013, in prep)\
A general purpose SED fitting routine using Monte Carlo Markov Chain (MCMC) techniques. The SED fits are performed in likelihood space, which are computed in a similar manner to standard $\chi^2$ techniques. During the fits, all available parameters are allowed to vary. Utilizing the BC03 template set, these include galaxy age, E(B-V) extinction, e-folding timescale for the SFH in addition to the photometric redshift and normalization (i.e. stellar mass). Fitting is done in flux space. For negative fluxes, an upper limit (basing the upper limits on 3 times the flux errors) method which follows a one-sided Gaussian distribution is used. Effectively, models with fluxes below the upper limit are always accepted while those with higher flux values are given a proportionally small probability during the fits. Run by S. Johnson.\
\
G - [*HyperZ*]{} (Bolzonella et al. 2000)[^4]\
This is a $\chi^2$ minimization code. Shifts to magnitudes can be added manually. A minimum photometric error can be set, errors smaller than this value will be replaced by the minimum value. Fitting is done in flux space. Negative fluxes are not included in the fitting. Uses a prior that requires the NIR absolute magnitude to be in the range $-30<M<-9$ (Vega mag). Run by J. Pforr.\
\
H - [*LePhare*]{} (S. Arnouts & O. Ilbert 2013, in prep.)[^5]\
Another $\chi^2$ template SED fitting code, which can use a training sample to derive zero-point offsets and to optimize the template SEDs. The code also has the option of using luminosity priors and the possibility of adding extra errors to the given photometry. The code can output both the photometric redshift based on the minimum $\chi^2$ and the median of the photometric redshift probability distribution. The code can be run with or without emission line corrections, as described in Ilbert et al. (2006; 2009). Fitting can be done in either magnitude or flux space. Fitting during this investigation is done in magnitude space and negative fluxes are not included in the fitting. Uses a prior that requires the optical absolute magnitude to be in the range $-24<M<-8$. IRAC ch3 and ch4 are not used in the fitting. Run by. M. Salvato.\
\
I - [*WikZ*]{} (Wiklind et al. 2008)\
A pure template fitting code, minimizing the $\chi^2$ between the observed and template SED photometry. The code has the possibility to add extra smoothing errors to the existing photometric errors. Fitting is done in flux space. For negative fluxes, the data point adds to the $\chi^2$ if the template SED flux is brighter than the 1$\sigma$ upper limit. If the template flux is lower than the upper limit, it does not add to the $\chi^2$. Does not include IRAC ch3 at $z<0.5$ and ch4 at $z<0.7$. Run by T. Wiklind.\
\
Of the eleven submissions including photometric redshifts, nine different photometric redshift codes have been used. Only EAZY has been used by multiple participants, i.e., codes 4C, 7C, and 13C. However, there are some differences in the details of the template sets used in each case, e.g., code 4C includes a template of BX418 (Erb et al. 2010), a metal poor galaxy with strong Ly$\alpha$ emission, code 7C uses the EAZY templates, plus a template with deep Ly$\alpha$ absorption (constructed from an observed high-$z$ galaxy), while finally code 13C uses the six EASY templates after updating them by adding emission lines using the Ilbert et al. (2009) algorithm. Furthermore codes calculate and apply slightly different zero-point shifts and uses different smoothing to the existing photometric errors. In the $\chi^2$ fitting, code 4C and 13C include data points with negative fluxes while code 7C ignores them and only code 13C uses luminosity function priors. Therefore, the codes are sufficiently different to produce independent estimates of the photometric redshifts. In the Section 4, we show that the scatter between the codes using EAZY is similar to the scatter between different codes.
Results
=======
In Table \[table2\] and Table \[table3\], we show the resulting scatter between the photometric redshifts and spectroscopic redshifts for the different codes presented in Table \[table1\]. The scatter is calculated using the control sample only. The tables present the full scatter $$\sigma_F=rms[\Delta z/(1+z_{spec})]$$ where $\Delta$z=$z_{spec}-z_{phot}$. Results are also given in $\sigma_O$, which is the rms after excluding outliers, where an outlier is defined as an object with $|\Delta z|/(1+z_{spec})>0.15$. Since many recent results in photometric redshift present scatter as the normalized median absolute deviation (Ilbert et al. 2009), we also give results as: $$\sigma_{NMAD}=1.48\times median(\frac{|\Delta z|}{1+z_{spec}})$$ Finally, we also calculate the scatter $\sigma_{dyn}$ using a dynamic definition of the outlier fraction. Here outliers are defined as objects with $|\Delta z|/(1+z_{spec})>3\times\sigma_{dyn}$. The scatter and outlier fraction (OLF$_{dyn}$) are here determined iteratively. For a Gaussian distribution of the scatter, the outlier fraction would be constant ($\sim$0.3%) regardless of the width of the distribution. However, since the scatter in the different codes are expected to be highly non Gaussian, the outlier fraction will vary between codes.
Furthermore, to quantify any systematic bias between photometric and spectroscopic redshifts, we define bias$_z$=mean\[$\Delta z/(1+z_{spec})$\], after excluding outliers (using the constant definition).
[rcrccclcc]{} Code & Objects & bias$_z^a$ & OLF$^b$ & $\sigma_F^c$ & $\sigma_O^d$ & $\sigma_{NMAD}^e$ & $\sigma_{dyn}^f$ & OLF$_{dyn}^g$\
2A &589 & -0.010 & 0.092 & 0.167 & 0.041 & 0.038 & 0.038& 0.107\
3B & 589 & -0.007 & 0.036 & 0.099 & 0.035 & 0.034 & 0.033& 0.048\
4C & 589 & -0.009 & 0.051 & 0.114 & 0.044 & 0.040 &0.042 & 0.061\
5D & 408 & -0.030 & 0.147 & 0.197 & 0.073 & 0.097 & 0.098 & 0.034\
6E & 589 & -0.007 & 0.041 & 0.104 & 0.037 & 0.033 & 0.033& 0.065\
7C & 589 & -0.009 & 0.053 & 0.121 & 0.037 & 0.033 & 0.033& 0.070\
8F & 589 & -0.008 & 0.093 & 0.272 & 0.064 & 0.077 & 0.074&0.051\
9G & 589 & 0.013 & 0.078 & 0.189 & 0.050 & 0.045 & 0.053& 0.063\
11H & 589 & -0.008 & 0.048 & 0.132 & 0.038 & 0.033 & 0.030&0.088\
12I & 589 & -0.023 & 0.046 & 0.153 & 0.049 & 0.054 & 0.049&0.046\
13C & 589 & -0.005 & 0.039 & 0.127 & 0.034 & 0.026 & 0.027&0.075\
median(all) & 589 & -0.008& 0.029 & 0.088 & 0.031 & 0.029 & 0.026 & 0.054\
median(5) & 589 & -0.009& 0.031 & 0.079 & 0.029 & 0.025 & 0.024 & 0.056\
\[table2\]
[rcrccclcc]{} ID & Objects & bias$_z$ & OLF & $\sigma_F$ & $\sigma_O$ & $\sigma_{NMAD}$ & $\sigma_{dyn}$ & OLF$_{dyn}$\
2A & 614 & -0.018 & 0.086 & 0.259 & 0.052 & 0.054 & 0.053& 0.083\
3B & 614 & -0.004 & 0.057 & 0.148 & 0.039 & 0.034 & 0.032&0.091\
4C & 614 & -0.011 & 0.077 & 0.197 & 0.046 & 0.045 & 0.045&0.083\
5D & 446 & -0.032 & 0.067 & 0.259 & 0.070 & 0.087 & 0.080&0.029\
6E & 614 & -0.010 & 0.052 & 0.198 & 0.044 & 0.040 & 0.041&0.065\
7C & 614 & -0.008 & 0.046 & 0.149 & 0.039 & 0.038 & 0.036&0.064\
8F & 614 & -0.012 & 0.140 & 0.535 & 0.064 & 0.079 & 0.080&0.073\
9G & 614 & 0.015 & 0.121 & 0.269 & 0.053 & 0.057 & 0.059&0.096\
11H & 614 & -0.009 & 0.042 & 0.131 & 0.040 & 0.036 &0.038 &0.050\
12I & 614 & -0.022 & 0.064 & 0.173 & 0.055 & 0.063 & 0.059&0.042\
13C & 614 & -0.007 & 0.046 & 0.189 & 0.040 & 0.035 & 0.035&0.072\
median(all)& 614 & -0.001& 0.036 & 0.157 & 0.037 & 0.033 & 0.032&0.062\
median(5) & 614 & -0.005& 0.041 & 0.128 & 0.033 & 0.028 & 0.027&0.073\
\[table3\]
Presenting photometric redshift accuracy as the full scatter, $\sigma_F$, gives a non-optimal representation of the scatter since a few objects (i.e., outliers) can drive the scatter to large values. Therefore, the scatter in photometric vs. spectroscopic redshifts is often expressed as the rms after excluding outliers. With this approach there are two quantities that together determine how well a code works, the rms after excluding outliers and the fraction of outliers.
The tables show that most codes produce results that broadly agree. The scatter after excluding outliers is typically in the range $\sigma_O\sim 0.04-0.07$ and the outlier fraction (OLF) is within the range 0.04-0.07 for a majority of the codes. Codes with low $\sigma_O$ tend to have a low OLF. Comparing the scatter $\sigma_O$ using the fixed outlier definition with the scatter $\sigma_{dyn}$ (which uses the dynamic outlier definition), shows a very similar rank between methods; codes with small $\sigma_O$ have small $\sigma_{dyn}$ and codes with high $\sigma_O$ also have high $\sigma_{dyn}$. The outlier fraction is naturally less correlated between the methods. By definition, codes with $\sigma_{dyn}>0.05$ will get a lower dynamic outlier fraction, OLF$_{dyn}$, compared to the fixed outlier fraction OLF and vice versa. Due to the similarity in both the size and rank between the results of the two definitions, we will quote results using $\sigma_O$ and OLF as default, but will also include results from the dynamic definition. The fixed definition allows for comparisons between results and the literature.
In overall performance, there are five codes that have a combination of both low scatter and outlier fraction for both catalogs, i.e., codes 3B, 6E, 7C, 11H, and 13C. Inspecting Table \[table1\], reveals that these five results represent four different photometric redshift codes and four different sets of template SEDs. The result that no particular code gives a significantly better results than others is not surprising since most codes, including the four resulting in the lowest scatter, are based on the same technique, the $\chi^2$ template fitting. Maybe a bit more surprising is that four (or almost five) different SED sets are represented, indicating that there is not a preferred set. We note that all five codes use the training sample of galaxies to derive zero-point shifts and/or corrections to the template SED set used. Furthermore, all five include templates with emission lines and perform additional smoothing of the given flux errors. This suggests that having a code with these options is important for the quality of derived photometric redshifts. Finally, all these codes use template SEDs that include emission lines features, suggesting their importance when deriving photometric redshifts.
At the other end of the spectrum, there are a few codes with an elevated fractions of outliers compared to the other codes. For the $H$-band selected sample (Table \[table2\]), codes 2A, 5D, and 8F have a slightly higher outlier fraction. For code 5D, this should mainly be due to the lack of templates matching low luminosity galaxies. This drives the outlier fraction to high values. Furthermore, it also prevents the code from converging for many fits, resulting in derived photo-z for only a fraction of the objects in the catalog (about 30 % lack a calculated photo-z). For the other two codes, the higher outlier fractions could be due to a combination of not adding smoothing errors, lack of training (i.e., deriving zero-point offsets), or a limited parameter space for constructing the template SED sets. There are two codes, 5D and 8F with a resulting scatter $\sigma_O>0.05$ in the H-band selected catalog and $\sigma_O>0.06$ in the z-band selected catalog. For code 5D, the mass limit on the template SEDs used in this investigation should be the driving factor behind the high scatter. We also note that neither code 5D or code 8F use the spectroscopic training sample to optimize results.
For the three participants that use the EAZY code, the spread in results is comparable to the other codes that also use traditional $\chi^2$ fitting. The scatter should be due to the differences in templates, training of the code, and priors used between the participants running EAZY, which are the main parameters that vary between any $\chi^2$ fitting code, as discussed in Section 3.
From the descriptions of the different codes in Section 3, it is clear that there are many different approaches for treating data points with negative fluxes. For the spectroscopic training sample, the galaxies are relatively bright and there are few data points that are “non-detections” with negative fluxes ($\sim$1%). Therefore, the different treatment of negative fluxes will likely not introduce any extra scatter or biases between codes. At finter limits though, this may lead to systematic differences between the output of the codes.
In Table \[table2\] and \[table3\], we also list the scatter between the photometric redshifts and spectroscopic redshifts where we adopt the median photometric redshift from all codes and the median from the five codes with the lowest scatter. It is very interesting that taking the median in this way produces a lower scatter and outlier fraction than any of the individual codes. This important result is discussed in Section 5, where we investigate different approches of combining results to improve the photometric redshift accuracy.
To illustrate how well the individual codes recover the redshifts of the spectroscopic sample, we plot in Figure \[fig2\] $(z_{phot}-z_{spec})/(1+z_{spec})$ vs. $z_{spec}$ for each code for the $H$-band selected catalog. Also plotted in the right bottom panel in the figure is the scatter after calculating the median of the five codes with the lowest scatter.
To compare the results for all codes, we plot in Figure \[fig3\], the rms, $\sigma_O$, together with the outlier fraction for all codes for both catalogs. In red, we highlight the five codes that produce the lowest scatter and outlier fraction (i.e., are located closest to the origin). Besides the individual results, we also plot the median photometric redshift of all codes (black star symbol) and the median of the five codes with the smallest scatter (red star symbol). This illustrates that taking the median decreases both the scatter and fraction of outliers.
In Figure \[fig4\], we plot the mean bias for the different codes, as well as for the median of all codes and the five selected codes. We find most codes produce photometric redshifts that are slightly shifted by mean\[$\Delta z/(1+z_{spec})$\]$\sim$0.01 in a sense that the photometric redshifts predict higher redshifts compared to the spectroscopic sample. Calculating the error in the mean as $\sigma_{bias_z}/\sqrt{N}$, where $N$ is the number of data points, we find typical errors in the mean of $\sim$0.002. Therefore, all codes have biases inconsistent with zero at a $\gsim$3$\sigma$ level. However, the biases are smaller than the scatter ($\sigma_O$) and will not dominate the overall uncertainties in the photometric redshifts.
Photometric redshift accuracy as a function of selection band
-------------------------------------------------------------
Including NIR data when deriving photometric redshifts is important for photometric redshift accuracy and limiting outliers (e.g., Hogg et al. 1998; Rudnick et al. 2001; Dahlen et al. 2008, 2010). Therefore, having a catalog selected in the NIR should in principle be better than an optically selected catalog since the former assures the availability of NIR data. Of course, having an optically selected catalog that requires NIR coverage should be as close to equivalent to an NIR selected. If we compare the results from the WFC3 $H$-band selected catalog (Table \[table2\]) with the ACS $z$-band selected catalog (Table \[table3\]), we find that the scatter is similar for each code. This is not unexpected since most of the photometry in the two cases are based on the same images, only the NIR bands differ. In more detail, the scatter for 9 of the 11 codes and the outlier fraction for 7 of the 11 codes are lower in the $H$-band selected catalog compared to the $z$-band selected. This slight improvement is consistent with the expected better performance for a NIR selected catalog combined with the extra depth and number of bands when replacing ISAAC $J$ and $H$ by WFC3 F098M/F105W, F125W and F160W.
The bias$_z$ shows similar trends in the two catalogs, with deviations that are statistically inconsistent with being zero, but the absolute values are small compared to the scatter, $\sigma_O$.
Since the CANDELS survey is foremost an infrared survey for which planned catalogs are to be selected in the WFC3 infrared bands, we will concentrate our investigation on the $H$-band selected galaxy sample.
Photometric redshift accuracy as a function of magnitude
--------------------------------------------------------
It is important to note that the photometric redshift accuracy reported for any survey may not be representative of the actual sample of galaxies for which photometric redshifts are derived. The reason being that the scatter is calculated using a subsample of galaxies with spectroscopic redshifts that in most cases are significantly brighter, and in many cases at lower redshift, compared to the full galaxy sample. Since fainter galaxies have larger photometric errors and may be detected in fewer bands, we expect that the errors on the photometric redshifts increase for these objects (e.g., Hildebrandt et al. 2008). As an example of the magnitude and redshift dependence on the photometric redshifts, Ilbert et al. (2009) report for the COSMOS survey $\sigma_{NMAD}$=0.007 and OLF=0.7% for a sample of galaxies at redshift $z<1.5$ brighter than $i^+_{\rm AB}=$22.5. At fainter magnitudes and higher redshift, they report $\sigma_{NMAD}$=0.054 and OLF=20% for galaxies with redshift $1.5<z<3$ brighter than $i^+_{\rm AB}\sim$25, illustrating the significance of this effect.
To quantify the magnitude dependence of the photometric redshifts, we divide the spectroscopic sample from the $H$-band selected catalog into two magnitude bins with equal number of objects, one brighter and one fainter than $m(H)$=22.3. We find that the scatter in the median photometric redshift increases from $\sigma_O$=0.027 to $\sigma_O$=0.034 and the outlier fraction decreases from 3.1% to 2.7% when going from the bright to the faint subsample. The difference is small, reflecting the relative brightness of both subsamples. As a comparison, we find the that faint spectroscopic subsample has a median $m(H)$=23.2, significantly brighter than the median magnitude of the full sample, which is $m(H)$=25.7.
To visualize the behavior of photometric redshifts down to faint magnitudes, we plot in Figure \[fig5\] the scatter between the eleven individual codes and the median of all codes. Each panel shows about $\sim$6000 objects with signal-to-noise $>$10. We do not know how well the median represents the true redshifts at these magnitudes, but the plot illustrates that there are some substantial biases in a number of codes. For example, codes 2A, 5D, and 8F have a fairly prominent population at higher redshift compared to the median. Potentially due to the aliasing between the Lyman and the 4000Å breaks these codes more often chose the higher redshift solution compared to the median. Again, we note that the median we compare to is not necessarily the most correct solution.
To check the magnitude dependence for the full galaxy sample in some more detail, we plot in Figure \[fig6\], the comparison between the five codes with the lowest scatter (3B, 6E, 7C, 11H, and 13C) and the median of all codes in three magnitude bins, $m(H)<$24, 24$<m(H)<$26, and 26$<m(H)<$28. It is clear from the figure that the scatter increases at fainter magnitudes (note that we plot the same number of objects, $\sim$3000, in each panel). To quantify the magnitude dependence, we calculate the mean scatter between the individual codes and the median in the three magnitude bins and find $\sigma_O$=0.040, 0.048, and 0.055, respectively. For the fraction of outliers, we find for the three bins OLF=8%, 16% and 28%, respectively. This increase in scatter, and particularly in the fraction of outliers, further illustrates that the dispersion in the photometric redshifts calculated by different codes becomes significant at faint magnitudes, even though a good agreement is noticeable at brighter magnitudes. Interestingly though, there is a fairly good agreement between code at all magnitudes at redshifts $z\gsim$3-4. This should be due to the strong Lyman-break feature at these redshifts that helps determine the photometric redshift. We select these five particular codes because at bright magnitudes (i.e., typical of the spectroscopic samples) they produce very similar photometric redshifts. This allows us to investigate how results diverge between codes due to mainly the signal-to-noise. We made similar tests using different codes and find results that are consistent.
### Simulating a faint spectroscopic redshift sample
To quantify the difference between the brightness distribution of the sub-sample with spectroscopic redshifts, compared to a full galaxy sample, we plot in Figure \[fig7\] the normalized distributions of the available spectroscopic sample together with the full sample of galaxies for the GOODS-S $H$-band selected catalog. For the full sample, we restrict the selection to galaxies with S/N$>$5 in the $H$-band that are detected in at least six photometric bands. The red line in the figure shows the distribution of the spectroscopic sample while the blue line shows the full sample. Obviously, the spectroscopic sample is significantly brighter than the bulk of the full sample of galaxies. When using the S/N$>$5 limit in the $H$-band, we find that the full sample is on average 3.6 mag fainter than the spectroscopic sub sample.
To better quantify how the brightness of the spectroscopic sample affects the photometric redshift accuracy, we artificially make the spectroscopic sample fainter to resemble the flux distribution expected for a deeper spectroscopic sample. First, we make a catalog consisting of the $\sim$ 1000 objects with highest quality spectroscopic redshifts from the $H$-band selected catalog. The catalog initially has a magnitude distribution according to the red line in Figure \[fig7\]. We thereafter make all fluxes fainter by $\Delta m$=3.6 mag, which is the average difference between the spectroscopic sample and the full sample in Figure \[fig7\]. To each new flux value we assign a photometric error drawn from the original catalog at a flux level matching the new assigned flux. We finally perturb the flux values using the assigned errors, assuming that they are Gaussian and represent 1$\sigma$. The new magnitude distribution of the shifted spectroscopic sample is shown by the gray line in Figure \[fig7\]. This distribution is consistent with the distribution of the full photometric sample. To further quantify the flux dependence of the photometric redshifts, we have also made catalogs where we shift the spectroscopic sample by $\Delta m$=1, 2, 3, and 4 mag, respectively
To show the flux dependence of the photometric redshift accuracy, we plot in Figure \[fig8\] the scatter and outlier fraction for the nominal case and for the five catalogs with perturbed photometry. We illustrate results from one specific code (Code 3B), but we expect a similar behavior for all codes. It is clear that both the scatter and outlier fractions increase as the spectroscopic sample is shifted to fainter flux levels. Particularly, there is a significant increase in outlier fraction at faint magnitudes $\Delta m\gsim 2$. This could be related to the increased risk of misidentifying the Lyman and 4000Å breaks at fainter magnitudes where photometric error are larger.
In a second test using the shifted photometry of the spectroscopic sample, we compare the results from multiple codes run on the same catalog. Here we use the $\Delta$m=3.6 catalog, since this illustrates the difference in photometry between the spectroscopic catalog and the full $H$-band selected catalog used in this investigation. Eight codes participated in this test (codes 3B, 4C, 6E, 7C, 8F, 9G, 11H, and 12I). Results are shown in Figure \[fig9\]. Black dots to the lower left show the photometric redshift scatter and outlier fraction for the original case, while red dots in the upper left show the results after shifting the catalog to fainter fluxes and increased errors. Star symbols represent the results when using the median of all codes. Obviously, both the scatter and outlier fraction increase significantly for all codes when the photometric errors increase. For the median case, the scatter approximately doubles from $\sigma_O$=0.03 to $\sigma_O$=0.06, while the outlier fraction increases from 4% to 15%. At the same time, we note that in the case with shifted photometry, the median produces better results than any of the individual codes.
As a final test, we use data from the simulated catalogs that were made artificially fainter to investigate the reliability of the photometric redshifts as a function of magnitude, using one of the codes (Code 3B), as a representative case. In Figure \[fig10\] we show the scatter ($\sigma_O$) and outlier fraction in magnitude bins with $\Delta m=1$ over the range $19<m(H)<26$. The Figure indicates that both the scatter and outlier fractions are reasonably well behaved and degrade slowly out to magnitudes $m(H)\sim 24$, whereafter both quantities increase more rapidly at $m(H)\gsim 25$ .
Photometric redshift accuracy as a function of redshift
-------------------------------------------------------
To test the redshift dependence of the photometric redshifts, we first divide the spectroscopic control sample in the $H$-band into two bins with equal number of objects. The redshift dividing the bins is $z_{spec}$=0.95 and the median redshift for the two bins are $z_{spec}$=0.7 and $z_{spec}$=1.4, respectively. We find that the scatter in the median photometric redshift increases from $\sigma_O$=0.027 to $\sigma_O$=0.034, while the outlier fraction decreases from 3.4% to 2.4% when going from the low redshift to the high redshift subsamples. This indicates that there is no strong redshift trend in the photometric redshift accuracy. To make a more detailed investigation, we divide the spectroscopic sample into eleven redshift bins and calculate the scatter and outlier fraction in each bin separately. Figure \[fig11\] shows the result for the $H$-band selected catalog when comparing the median photometric redshifts to the spectroscopic redshifts. The scatter, $\sigma_O$, lies at a fairly constant level with redshift, indicating that the redshift-normalized scatter gives a fairly robust indicator of the photometric redshift accuracy almost independent of redshift. The only point that lies significantly above the trend is the $z\sim$2 point. This could be due to the lack of strong spectral features at this redshift. This is also the redshift range where we expect the spectroscopic redshifts to be most uncertain and we cannot rule out some errors in the spectroscopic sample even though we limit our selection to the highest quality spectra. At higher redshifts, the Lyman break moves into the $U$-band, providing an important signal for the photometric redshift determination (e.g., Rafelski et al. 2009). We also note that the VIMOS $U$-band used is redder than the typical $U$-band and therefore starts to probe the break at slightly higher redshifts. Possibly contributing to the relatively high outlier fractions in the $z\sim$2.5 and $z\sim$3.2 data points. However, the tests do not account for high-z galaxies with significantly different SEDs than the moderate-z spec sample. If such a population is common at high redshift and is unrepresented in the template SED libraries, it could affect the accuracy of the photometric redshifts. It is, however, reassuring that for the spectroscopic sample at $z>3\sim 4$, the photometric redshifts agree well with the redshift from the spectra (e.g., Figure \[fig2\]). Contributing to the accuracy of the $z>3$ photometric redshifts is the break due to absorption by intergalactic HI clouds (Madau 1995), which affects the observed signal for all galaxy SED types. In fact, in Figure \[fig11\], there are no outliers in the highest redshift bin ($z>3.7$), indicating that the Lyman break helps to provide robust photometric redshift determinations at these redshifts.
Photometric redshift accuracy as a function of galaxy spectral type
-------------------------------------------------------------------
The most important spectral features for determining photometric redshifts are the Lyman break at $\sim$ 1215Å and the 4000Å break (we let the 4000Å break denote the overall spectral feature caused by the Balmer break at 3646Å and the accumulation of absorption lines of mainly ionized metals around $\sim$4000Å). It is also expected that the size of the break should be important for the accuracy of the photometric redshifts. For example, an old red galaxy with a pronounced 4000Å break should result in more accurate photometric redshift compared to a younger blue galaxy with a more featureless SED. These effects should be most important at lower redshifts ($z\lsim 2-3$), where the redshifted Lyman break has not yet entered the observed $U$-band. At higher redshifts where the break at rest frame wavelengths short of $\sim$1215 Å (Madau 1995) moves into the observed bands, even intrinsically featureless blue galaxies will show a break feature that helps determine the photometric redshift. Galaxies with blue, relatively featureless SED at redshift $z\lsim 2-3$ therefore have the highest risk of being assigned incorrect redshifts.
To investigate the photometric redshift accuracy as a function of galaxy spectral type, we divide our spectroscopic sample into early-types, late-types, and starbursts based on the rest frame $B-V$ color of the galaxy. The colors are calculated using the observed bands that most closely covers the redshifted rest frame $B-V$ together with K-corrections based on the best-fitting template SED following the method in Dahlen et al. (2005). We use a division where galaxies with $B-V <0.34$ are assigned as starbursts and galaxies with $B-V >0.66$ are assigned as early-type galaxies. Galaxies with intermediate colors are assigned as late-type galaxies. This is a single rest frame color definition, we can not rule out that dusty later type galaxies may fall into the early-type category. In Figure \[fig12\], we plot the scatter and outlier fraction for the control sample in six color bins using the median photometric redshift when comparing to the spectroscopic redshift.
From the figure we note that the scatter is not strongly dependent on galaxy color. There is an indication that the early-types have a smaller scatter ($\sigma_O \sim$0.02) compared to the remaining types ($\sigma_O \sim$0.03). However, within the starburst and late-type bins, there is no clear color dependence. If we exclude galaxies at $z>2$ (where the Lyman break may be useful for determining photometric redshifts) there is no significant change in the results. We therefore conclude that there is no strong color dependence in the photometric redshifts, except that we may expect more secure redshifts for early-type galaxies.
Applying zero-point shifts and smoothing errors
-----------------------------------------------
The five codes resulting in the lowest scatter and outlier fraction use the spectroscopic training sample to derive shifts to either the photometry or template SEDs and add extra smoothing errors. The better behavior when applying zero-points shifts could be due to a number of factors. There could be actual errors in the given zero-points used to calculate the photometry, there could also be a mismatch between the template SEDs and the true SEDs of the observed objects. Furthermore, insufficient knowledge of the system throughput given by the filter transmission curves may cause offsets between observed and predicted fluxes. Finally, when photometry from different images are merged to a common catalog, there could be unaccounted aperture corrections contributing to offsets between filters. By using a spectroscopic training sample with sufficiently many objects, a number of codes offer the possibility to calculate zero-point shifts which are thereafter applied to either the photometry or the templates SEDs before deriving photometric redshifts.
Table \[table4\] illustrates the size of the shifts derived by codes 3B, 6E, 7C, 11H, and 13C for both the $H$-band and $z$-band selected catalogs. A positive offset in the table indicates that the observed flux is brighter compared to what is expected from the template SED. For each filter, we also give the median of the available shifts together with the error in the median. There is a noticeable scatter in the size (and sometimes sign) between the corrections derived by the different codes, suggesting that the zero-point shifts depend on the code, implementation and template SED set used. However, there are some common trends among the codes. To highlight this, we have marked in bold face the cases when the mean shift of all codes deviates from zero with at least a 5$\sigma$ significance. There are significant shifts for some of the ACS filters, even though the absolute shifts are small ($\lsim$0.03 mag). More noticeable shifts are noted for some of the NIR ISAAC filters in the $z$-selected catalog, with the $J$-band shift being significant. Most measurements indicate that the IRAC fluxes predicted by template SEDs are too faint compared to the measured fluxes. To some extent, this could be due to the lack of PAH emission at long wavelengths in many template SED libraries.
[lrrrrrr]{}\
Filter &Code 3B & Code 6E & Code 7C & Code 11H & Code 13C & Mean\
VIMOS(U) &0.004 &-0.013 &- &-0.033 &-0.030 &-0.018$\pm$0.007\
ACS(F435W) &-0.004 & 0.028 &- &0.047 &0.030 &0.025$\pm$0.009\
ACS(F606W) & 0.031 &0.008 &- & 0.028 & 0.032 &[**0.025$\pm$0.005**]{}\
ACS(F775W) & 0.010 &0.018 &- &0.002 &0.037 &0.017$\pm$0.006\
ACS(F850LP) & 0.010 &0.025 &- &0.015 &0.040 &0.022$\pm$0.006\
WFC3(F098M) & -0.022&0.001 &- &0.000 &0.016 &-0.001$\pm$0.007\
WFC3(F105W) & -0.011&0.009 &- &0.000 &0.008 &0.002$\pm$0.004\
WFC3(F125W) & -0.062 &-0.009 &-0.100 &-0.022 &-0.011 &-0.041$\pm$0.016\
WFC3(F160W) & -0.091 &-0.010 &0.020 &0.005 &-0.019 &-0.019$\pm$0.017\
ISAAC(Ks) & -0.031& -0.013&0.020 &0.025 &-0.040 &-0.008$\pm$0.012\
IRAC(ch1) & 0.120 &0.117 &0.050&0.106 &0.026 &0.084$\pm$0.017\
IRAC(ch2) & 0.114 &0.098 &- &0.073 &-0.034 &0.063$\pm$0.029\
IRAC(ch3) & 0.236&0.168 &- &- &- &[**0.202$\pm$0.024**]{}\
IRAC(ch4) & 0.455 &0.171 &- &- &- &0.313$\pm$0.100\
\
VIMOS(U) & 0.018&0.029 &- &-0.027 & -0.005&0.004$\pm$0.011\
ACS(F435W) &-0.018 &-0.053 &- &-0.023 & -0.053&-0.037$\pm$0.008\
ACS(F606W) & 0.046&0.004 &- &0.016 & 0.018&0.021$\pm$0.008\
ACS(F775W) & 0.018&0.020 &- &0.024 & 0.025&[**0.022$\pm$0.001**]{}\
ACS(F850LP) & 0.018&0.027 &- &0.032 & 0.013&[**0.022$\pm$0.004**]{}\
ISAAC(J) & -0.095 &-0.057 & -0.050 & -0.054 & -0.094&[**-0.070$\pm$0.009**]{}\
ISAAC(H) & -0.130&-0.060 & - & -0.010 & -0.107&-0.077$\pm$0.023\
ISAAC(Ks) & -0.049 &-0.006 & 0.050 & 0.091 & -0.015&0.014$\pm$0.022\
IRAC(ch1) & 0.101&0.131 & - & 0.175 & 0.023&0.107$\pm$0.028\
IRAC(ch2) & 0.083&0.105 & - & 0.111 & -0.031&0.067$\pm$0.029\
IRAC(ch3) & 0.198 &0.160 & - & 0.148 & -&[**0.169$\pm$0.012**]{}\
IRAC(ch4) & 0.351 &0.179 & - & 0.240 & -&[**0.257$\pm$0.041**]{}\
\[table4\]
Using pair statistics to estimate photometric redshift uncertainties
--------------------------------------------------------------------
As an alternative for estimating the photometric redshift uncertainties at faint magnitudes where spectroscopic redshifts are not available, we use the method outlined in Quadri & Williams (2010) and Huang et al. (2013). This method uses the fact that close pairs have a significant probability of being associated and that they therefore are at similar redshifts. By plotting the distribution of differences in photometric redshifts of close pairs from the photometric redshift catalog, compared to a distribution based on any random two galaxies, the close pair distribution will show excess power at small separations reflecting an elevated probability for close pairs being at similar redshift. Here two objects are considered a close pair if the separation is less than 15 arcsec.
In the top panel of Figure \[fig13\], we show the distribution of differences in photometric redshifts for close pairs as the black line, while the red line shows the distribution for random pairs. In the bottom panel, we show the distribution of differences in photometric redshifts after subtracting out the random pair distribution. The result is shown for code 3B. Evidently, pairs with similar photometric redshifts show an excess in the distribution. Fitting a Gaussian to the excess peak in the bottom panel (red line) results in a width of $\sigma$=0.090. This width includes scatter from both galaxies in the pair for which the difference in photometric redshift is calculated. Therefore, the scatter for individual objects should be 0.090/$\sqrt{2}$=0.064. Note that only galaxies with relatively similar photometric redshifts contribute to the peak, i.e., pairs where one of the objects is an outlier will not be included. The derived width of the peak should be compared to $\sigma_O$, the scatter after excluding outliers. While the derived scatter using the close pair method is larger than the value $\sigma_O$=0.035 derived when comparing to the spectroscopic control sample, the pair method is useful to fainter limits and is not as biased towards brighter fluxes or specific galaxy types as the spectroscopic sample. For the sample shown in Figure \[fig13\], all galaxies with fluxes $>1\mu $Jy (corresponding to $m(H)<23.9$) are used. Going even deeper, using all galaxies with fluxes $>0.5\mu $Jy (corresponding to $m(H)<24.7$), results in a scatter $\sigma$=0.087. These results confirm that the scatter in the photometric redshifts increases at magnitudes fainter than the spectroscopic control sample.
Error estimates for photometric redshifts
-----------------------------------------
Most photometric redshift codes return an estimate of the uncertainty in the derived photometric redshift. This is an estimate of confidence intervals of the photometric redshifts, such as the 68.3% and 95.4% confidence intervals (corresponding to $\pm 1\sigma$ and $\pm 2\sigma$ for a Gaussian distribution). There are also codes that produce full probability distributions, P(z), based on the $\chi^2$ fitting, where P(z) $\propto exp(-\chi^2)$. Ideally, these error estimates should reflect the uncertainties in the derived photometric redshifts. However, there is not necessarily a correlation between how well a photometric redshift code reproduces the spectroscopic redshifts and the accuracy of the error estimates of the photometric redshifts. Hildebrandt et al. (2008) investigated the behavior of a number of photometric redshift codes and found that the error estimates did not correlate tightly with the photometric redshift accuracy. As a test of how well the assigned errors reflect the actual errors, we calculate the fraction of galaxies with known spectroscopic redshifts in the control sample that falls within the 68% and 95% confidence intervals derived by the different codes. If quoted errors in the photometric redshifts are representative of the true redshift errors, then we expect about 68% and 95% of the spectroscopic redshifts fall within the two intervals, respectively. We show results in Table \[table5\].
[ccccc]{} Code & &\
conf. int: & 68.3% & 95.4% & 68.3% & 95.4%\
2A & 46.1 & & 40.9 & \
3B & 81.6 & 92.8 & 76.1 & 89.1\
4C & 64.0 & 88.2 & 58.5 & 85.7\
5D & 2.5 & 4.2 & 2.9 & 5.8\
6E & 52.0 & 84.7 & 48.3 & 81.6\
7C & 65.0 & 87.3 & 62.9 & 89.1\
8F & 15.3 & 15.6 & 14.2 & 14.7\
9G & 16.3 & 44.1 & 15.0 & 39.6\
11H & 35.2 & 54.0$^a$ & 30.9 & 46.9$^a$\
12I & 88.7 & 96.7 & 80.1 & 96.3\
13C & 52.0 & 72.7 & 35.7 & 51.0\
\[table5\]
We find that a majority of codes return underestimated confidence intervals, i.e., fewer than $\sim$68% and 95% of the galaxies with known spectroscopic redshifts fall within the estimated error intervals of the photometric redshifts. There are two main factors affecting the derived $\chi^2$ values, P(z) distributions, and widths of the derived 68% and 95% intervals. First, the size of the quoted photometric errors in the photometric redshift fitting may affect results in the sense that systematically underestimated errors may drive $\chi^2$ to high values and result in narrow P(z) distributions. On the other hand, photometric errors that are unrealistically large decrease the $\chi^2$ values. This could result in seemingly acceptable fits over a larger redshift range and therefore a broad P(z) distribution and an overestimate of the confidence intervals. A difference between the codes compared here is that some have added extra smoothing errors to existing photometric errors (codes shown in Table \[table1\]). Adding extra errors will effectively work as a smoothing of the P(z) distributions and result in relatively larger numbers in Table \[table5\] compared to what the original photometric errors would result in. For example, codes 3B and 12I, which have the largest fractions quoted, are among the codes adding the largest smoothing errors to the existing photometric errors. Secondly, the completeness of the template SED set used affects derived $\chi^2$ values and associated P(z) distributions. Utilizing a coarse set of templates that does not sufficiently cover the true SED distribution, may result in acceptable $\chi^2$ value from only at a very narrow range of redshifts. This could lead to a narrow probability distribution and an underestimate of the confidence intervals. In Table \[table5\], the small values for code 5D is likely due to a relatively coarse grid of template SEDs. Therefore, even if the photometric redshifts agree well with the spectroscopic control sample, one should be cautious when using the errors for photometric redshifts if these are based on the results from the $\chi^2$ fitting. In Section 5.2, we describe a simple method for adjusting the quoted errors so that they better reflect the actual uncertainty suggested by the spectroscopic control sample.
Closer look at outliers
-----------------------
Table \[table2\] shows that the outlier fraction for the $H$-band selected catalog lies in the range $\sim$4-15%, depending on code. When comparing only the five codes with the lowest scatter, the range of outliers is narrowed to 3.6-5.3%. In absolute numbers, this corresponds to 21-31 objects per code of the total 589 objects in the spectroscopic control sample. The number of individual objects flagged as an outlier by at least one of the five codes is 48. Of these, 20 are flagged by one code only, 7 by two codes, 2 by three codes, 8 by four codes, and 11 by all five codes. If we look at the case with the median photometric redshift from the five codes with the lowest scatter, we find an outlier fraction of 3.1%, corresponding to 18 objects. Of these objects, 7 and 11 are flagged as outliers in 4 and 5 codes, respectively. The fact that 18 outliers are flagged by at least 4 of the 5 codes indicates that some feature drives the photometric redshift to an outlier independent of code or template SED used. These objects may have an SED not represented by any of the template SED sets. Otherwise, the spectroscopic redshift could be incorrect or there could be problems with the photometry. To investigate this, we look closer at the spectra for the subsample of 18 objects flagged as outliers by the median method. We find that at least 12 objects have spectroscopic redshifts that most likely are not the highest quality and could therefore be wrong. There are objects with spectra measured by different groups that disagree. A few of the objects also have close companions (within $\sim$1 arcsec) where it is difficult to determine if the correct object in the photometric catalog has been assigned the spectroscopic redshift. So it is possible that the actual outlier fraction for the combined median photometric redshift is significantly less than reported in Table \[table2\] and Table \[table3\], perhaps as low as $\sim$1% when using the median method.
Combining results to improve photometric redshifts
==================================================
We have shown that combining results from multiple codes leads to photometric redshifts with lower scatter and outlier fraction than any individual code. This important result implies that using a combination of outputs from multiple algorithms can significantly improve the quality of photometric redshifts. The fact that the median outperforms any individual method indicates that net systematic errors must go in opposite directions amongst different codes, such that the middle value will have smaller scatter about the true redshift than even the best single technique. We expect systematic errors to vary due to differences in the templates used, priors applied, or fitting algorithms employed. In effect, there is a ’wisdom of crowds’ in combining results from different photometric redshift codes, much like can occur when combining multiple estimates of quantities in other fields (Surowiecki 2005).
Besides deriving accurate photometric redshifts, we are also interested in assigning proper errors to derived photometric redshifts. In this section, we look more in detail into these issues by investigating different ways of combining data when we have results derived independently by different participants. For this particular investigation, we use results from codes number 3B, 6E, 7C, 11H, and 13C. For each code, we have the calculated photometric redshift and the full redshift probability distribution, P(z), tabulated in the range $0<z<7$ in steps of $\Delta z=0.01$. Different codes use different recipes for assigning the photometric redshift based on the P(z). Either the highest peak can be used to determine the photometric redshift, or some kind of weighted photometric redshift can be derived by integrating over the probability distribution. To get a clean comparison between methods, we use below photometric redshifts based on both the peak of the P(z), i.e., $z_{peak}$, as well as the weighted photometric redshift, $z_{weight}$, and compare results separately. We compute the latter by integrating over the main peak of the P(z) distribution. We do not want to integrate over the full P(z) distribution since there are cases with multiple peaks due to e.g., the aliasing between the Lyman and the 4000Å breaks (where the actual P(z) could be basically zero at the reported photometric redshift if it falls between two peaks).
Method 1: Straight median
-------------------------
As already shown above, if we compare the median photometric redshift from multiple codes for each individual object with the spectroscopic control sample, we get a scatter and an outlier fraction lower than any individual code. The resulting scatter and outlier fraction from the straight median is shown in the first two rows of Table \[table6\]. These results indicate that combining results from multiple codes is advantageous. However, using a strict median does not directly produce any useful photometric redshift error estimate. Basing the errors on the scatter between the five codes will not yield a consistent measurement because of the expected highly non-Gaussian shape of the photometric redshift P(z) and the strong possibility that the various photometric redshift estimates are covariant with each other (e.g., they are based on the same photometry), so their scatter will not reflect all measurement uncertainties. We therefore look into a few more ways of combining data that may provide accurate results for both the photometric redshifts and the errors. There is no significant difference between using $z_{peak}$ compared to $z_{weight}$.
Method 2: Adding probability distributions
------------------------------------------
As a second approach we add the full P(z)$_i$ distributions from the different codes to produce a combined P(z). From Table \[table5\] we saw that a number of codes underestimate the errors, i.e., the distributions are too peaked around the derived photometric redshift. This will bias the combined redshift towards the values given by codes that underestimate the errors. At the same time, the photometric redshift of codes that overestimate the error will be given lower weights. To alleviate this, for codes underestimating the errors, we smooth each P(z)$_i$ using a simple recipe where we for each redshift bin $j$ replace the probability with a combination of three adjacent bins P(z$_j$)$_i$=0.25P(z$_{j-1}$)$_i$+0.5P(z$_j$)$_i$+0.25P(z$_{j+1}$)$_i$. We recalculate the fraction of the spectroscopic sample inside the 68.3% interval and iterate this procedure until the correct fraction is recovered. We thereafter apply the same smoothing, individually calculated for each code, to the full sample of galaxies. For the codes that overestimate the errors, we instead use a simple model to sharpen the P(z)$_i$. For each code we set P(z$_j$)$_i$=P(z$_j$)$_i^{1/\alpha}$, adjusting the exponent $\alpha$ so the correct 68.3% of the galaxies in the spectroscopic control sample falls inside the 68.3% confidence interval. After normalizing each P(z)$_i$ to unity, we add all five distributions and renormalize.
To illustrate this procedure, we show in Figure \[fig14\] an example applied to a galaxy with spectroscopic redshift $z$=0.734. The five blue lines show the probability distributions for five individual codes (codes 3B, 6E, 7C, 11H, and 13C). To account for the four codes underestimating the error intervals and one code overestimating them, we apply the smoothing and sharpening described above. This should lead to distributions with more consistent confidence intervals. The resulting individual distributions are shown with red curves in the figure. In this particular case, there is one code that produces a P(z) with a double peak, which turns into a single peak after smoothing. After adding the five individual distributions, the resultant distribution is shown with the black line.
In Table \[table6\], we show the results from adding the probability distributions in rows three and four. Compared to the straight median, the combined P(z) results in slightly higher outlier fraction and $\sigma_F$, but similar $\sigma_O$. Therefore, either method should result in photometric redshifts with no significant difference in accuracy. The advantage with the added P(z)$_i$ method is that it provides an estimate of the full probability distribution, which could be used to calculate e.g., 68.3% confidence intervals. To test how well the combined P(z) distributions reflects the true errors, we repeat the exercise above and calculate the fraction of objects in the control sample that falls within the 68.3% interval of the combined P(z). We find that 85% of the spectroscopically determined redshifts fall within the 68.3% confidence intervals. This suggests that combining the P(z) by adding the individual distributions overestimates the size of the 68.3% confidence intervals. To get a distribution that better represent the errors, we sharpen the distribution to recover 68.3% of the control sample within the 68.3% confidence interval, as described above.
Method 3: Hierarchical Bayesian Approach
----------------------------------------
As an alternative to a straight addition of the probability distributions, we adopt a hierarchical Bayesian approach following the method in Lang & Hogg (2012) (similar methods were employed by Press (1997) and Newman et al. (1999)). We want to determine the consensus P(z) for each object accounting for the measured probability distributions (hereafter P$_m$(z)$_i$) may be wrong. We call the fraction of measurements that are bad $f_{\rm bad}$ and write for each code $i$ $$P(z,f_{\rm bad})_i = P(z | {\rm measurement~is~bad})_i f_{\rm bad} +$$ $ P(z | {\rm measurement~is~good})_i(1-f_{\rm bad})$.\
-0.1truecm Here P(z $\vert$ measurement is bad) (hereafter U(z)) is a redshift probability distribution that we assume in the case where the observed P$_m$(z)$_i$ is wrong. We assume that there is no information on the redshift if the measurement is bad and therefore set U(z) to be uniform for all different codes. For the redshift range $0<z<7$ used, this means U(z)=1/7. We now have $$P(z,f_{\rm bad})_i = \frac{1}{7} f_{\rm bad} + P_m(z)_i(1-f_{\rm bad}).$$ The combined $P(z,f_{\rm bad})$ for all five measurements can be calculated as $$P(z,f_{\rm bad})=\prod_{i=1}^5 P(z,f_{\rm bad})_i ^{1/\alpha}.$$ Here $\alpha$ is a constant reflecting the degree of covariance between the results from the different codes (see below). We finally marginalize over f$_{\rm bad}$ to get the redshift probability distribution for each object $$P(z)=\int_0^1 P(z,f_{\rm bad})df_{\rm bad}$$ From the resulting P(z) we can determine the photometric redshift as either the peak of the distribution, $z_{peak}$, or the integral of the main feature in the distribution, $z_{weight}$. In Table \[table6\], we show the resulting scatter between the photometric redshifts and the spectroscopic control sample. Similar to the methods described in Section 5.1 and 5.2, the Bayesian method produces a scatter that is lower than any of the individual codes. Compared to the straight median and the combined P(z) method, there is no significant difference.
In Equation (5), $\alpha$ can adjust for any covariance between the different individual results. Setting $\alpha$=1 is equivalent to assuming statistical independence between all codes, while setting $\alpha$=5, i.e., the number of codes that are combined, corresponds to assuming full covariance. In this case, we expect some degree of covariance, both because all the photometric redshift estimates are based on identical photometry, and because there are overlaps between the five codes in templates and methods. The peak redshift of the resulting photometric redshift does not depend on the value of $\alpha$; however, the width of the final P(z) distribution does. We find that using $\alpha$=1 underestimates the errors; only 46% of the objects in the spectroscopic control sample fall inside the calculated 68% confidence interval. On the other hand, setting $\alpha$=5 overestimates the errors; 91% of the objects in the spectroscopic control sample fall inside the 68% confidence interval. To make the resulting P(z) distributions consistent with the spectroscopic control sample, we derive the value of $\alpha$ that recovers 68% of the spectroscopic redshifts within the 68% confidence intervals of the derived P(z) distributions. This is achieved for $\alpha$=2.1. Ignoring the impact of priors and $f_{\rm bad}$, setting $\alpha$=5 would be equivalent to averaging the predicted $\chi^2(z)$ curves from each code, as opposed to averaging the $P(z)$ estimates as in Section 5.2. Figure \[fig15\] shows the output P(z) of a single object for a number of cases, as an example the effect the choice of $\alpha$ has on the Bayesian method and sharpening of P(z) distributions in the summation method. For the Bayesian method, we show the results with $\alpha$=1 (thin red line), $\alpha$=5 (dashed red line) and $\alpha$=2.1 (thick red line). It is clear that lower $\alpha$ produces narrower P(z) distributions. The result from the straight summation is shown with the thin blue line, while the result after sharpening the P(z) distribution so that the control sample recovers the expected 68% of the galaxies within the 68% confidence interval is shown with the thick blue line. Although the final P(z) distributions for the two methods are derived using completely different algorithms, they produce very similar results. Note that $\alpha$ and the sharpening are not calculated particularly for this object, but are derived as averages for the full control sample.
Inspecting the bias$_z$ values in Table \[table6\] shows that the shift is small for all methods, mean\[$\Delta z/(1+z_{spec})$\]$<$0.01. The uncertainty in the bias values are typically $\sigma_{bias_z}\le 0.003$, indicating that the bias is statistically non-zero at a $\sim$3$\sigma$ level. However, in every case the bias is significantly smaller than the scatter, so the latter will dominate the statistical risk.
The similarities between the results suggests that either the Bayesian method or the straight adding of the P(z) distributions (after sharpening or smoothing the individual P(z)) could be used to derive the photometric redshifts and probability distributions.
In this example of the hierarchical Bayesian method, we have used a simple assumption for U(z), i.e., that we have no information if the measured P$_m$(z)$_i$ is wrong. Furthermore, we have allowed $ f_{\rm bad}$ in the whole range $ f_{\rm bad}$=\[0.0,1.0\]. Alternatively, we can assume that there is at least some minimum probability that the actual measurement are correct and let the bad fraction vary in the range $ f_{\rm bad}$=\[0.0,x\]. Repeating our analysis after varying x does not change results significantly, however, there is a slight decrease in the outlier fraction and full rms when setting $0.3<x<0.5$, i.e., assuming that the measured P(z)$_i$ are correct at least 50-70% of the times. Setting x=0.0, equivalent to assuming that all measured P(z)$_i$ are always correct, does, however, result in a significant increase in the outlier fraction (from 3.4% to 4.9%) and full rms ($\sigma_F$=0.10 to $\sigma_F$=0.36).
The example above illustrates that the hierarchical Bayesian approach does indeed provide means for improving results. It is possible to assume a more advanced guess for the shape of U(z). For example, if the measurement is bad, one could use a redshift probability following the volume element redshift dependence. Using this assumption, we find that the outlier fraction slightly decreases (from 3.4% to 3.1%), while the full rms show a marginal increase ( $\sigma_F$=0.10 to $\sigma_F$=0.11) and (after excluding outliers) the rms, $\sigma_O$, remains unchanged. Since we do not expect the spectroscopic control sample to follow the distribution of the volume element, we do not expect this example necessarily reflects the true expected effect of the volume element assumption.
A further refinement of the model would be to assume that the redshift distribution of a bad measurement follows the expectations of an assumed luminosity function combined with a magnitude limit appropriate for this particular survey. In addition, it should be possible to let the expected distribution be dependent on, e.g., apparent magnitude or color.
Instead of using a generic form for U(z), another possibility is to dilate the given P(z) and use this for U(z). In this case we assume that the errors are underestimated if the measurement is bad, rather than having no information. There are many possibilities when applying the hierarchical Bayesian method as discussed in Lang & Hogg (2012).
[lccccccc]{} Method & bias$_z$ & OLF & $\sigma_F$ & $\sigma_O$ & $\sigma_{NMAD}$ & $\sigma_{dyn}^f$ & OLF$_{dyn}^g$\
Straight median of $z_{peak}$ & -0.009 & 0.031 & 0.078 & 0.0296 & 0.025 & 0.024 & 0.056\
Straight median of $z_{weight}$ & -0.008 & 0.031 & 0.079 & 0.0296 & 0.025 & 0.024 & 0.056\
Combined P(z), using $z_{peak}$ & -0.006 & 0.044 & 0.108 & 0.0293 & 0.024 & 0.025 & 0.066\
Combined P(z), using $z_{weight}$ & -0.010 & 0.041 & 0.105 & 0.0303 & 0.029 & 0.026 & 0.060\
Bayesian using $z_{peak}$ & -0.007 & 0.034 & 0.099 & 0.0299 & 0.025 & 0.025 & 0.061\
Bayesian using $z_{weight}$ & -0.007 & 0.034 & 0.098 & 0.0296 & 0.026 & 0.025 & 0.058\
\[table6\]
Comparison to earlier work
==========================
Over the years, there has been a number of investigations comparing results from different codes in order to assess the accuracy of and the consistency between different photometric redshift codes. This includes Hogg et al. (1998), Abdalla et al. (2008), and Hildebrandt et al. (2008, 2010). The most comprehensive previous investigation of photometric redshift methods conducted in a similar way to what presented here is described in Hildebrandt et al (2010). In that investigation, the result of twelve different runs, representing eleven codes, are presented. Of these codes, three are common to this investigation (EAZY, LePhare, and HyperZ). Photometric redshifts are calculated using an $R$-filter selected 18-band photometry catalog covering the GOODS-North field. The wavelength range covered is the same as here, i.e., U-band to the IRAC 8.0$\mu$m channel. The spectroscopic sample includes $\sim$2000 objects, of which one quarter was provided as a training sample. The overall scatter after excluding outliers lies in the range $\sigma_O$=0.04-0.08, with a median of the twelve runs of $\sigma_O$=0.059. This is slightly higher than the median found here $\sigma_O$=0.046 (using the $z$-band selected results in Table \[table3\]). More importantly, the outlier fraction in Hildebrandt et al. lies in the range 8-31% and has a median of 18.5%, while our investigation reports outlier fractions 4-14% with a median 6.4%. This significant difference, despite the many similarities in setup, could be due to a number a reasons. We have here used a uniformly produced photometry over the whole wavelength range using the TFIT method, while Hildebrandt et al. used coordinate matching between three different data sets (ground-based optical/NIR, $HST$/ACS, and $Spitzer$/IRAC). This could introduce biases in the photometry due to blending, mismatches and differences in apertures used. Furthermore, we have made an effort to include only the highest quality spectroscopic redshifts and have excluded all known X-ray and radio sources when compiling our training and control samples. This should assure us an unbiased estimate of the scatter and outlier fractions when comparing spectroscopic and photometric redshifts. At the same time, Hildebrandt et al. reports that at least some of the high outlier fraction could be due to X-ray sources or the spectroscopic sample used. We therefore think that the outlier fractions of a few per cent found in our study should be more representative of what is achievable with photometric redshifts when using deep high quality photometry.
Conclusions and summary
=======================
We have used the CANDELS GOODS-S $HST$ WFC3 $H$-band and ACS $z$-band selected catalogs containing uniform TFIT photometry covering the $U$-band to IRAC infrared bands to investigate the behavior of photometric redshifts. Using a control sample with high quality spectroscopic redshifts, we have compared photometric redshifts derived from a number of different codes. We have investigated how the accuracy of the photometric redshifts depends on code and template SED set used. We have also investigated the dependence on redshift, galaxy color and brightness. Finally, we discussed combining results from multiple codes for improving the photometric redshifts and deriving reliable error estimates. Our main conclusions are
- [There is no particular code or template SED set that produces significantly better photometric redshifts compared to others. However, the codes that produce the best photometric redshifts all include training using a spectroscopic sample to calculate offsets or shifts to either the photometric zero-points or the template SEDs.]{}
- [There is a strong magnitude dependence on the accuracy of the photometric redshifts: rms values calculated for a spectroscopic control sample are only valid at the magnitudes probed by that sample. The photometric redshift uncertainty is likely to be significantly larger for a catalog that is deeper than the spectroscopic subsample.]{}
- [We investigated the redshift dependence of the scatter between photometric redshifts and a control sample of spectroscopic redshifts and find that the rms, when normalized to redshift by $\sigma$=rms$[(z_{phot}-z_{spec})/(1+z_{spec})]$, is almost independent of redshift. On the other hand, the fraction of outliers is elevated in the range $2.2<z<3.7$, possibly due to the relatively weak Lyman break signal in the lower part of this range, as well as aliasing between the Lyman and the 4000Å breaks. The outlier fraction at high redshift ($z>3.7$) is low due to the strong Lyman break signal.]{}
- [We find that the rms is only weakly dependent on galaxy color as measured by the rest frame $B-V$ color. Only for the very reddest early-type galaxies is there an indication that the scatter is smaller than the rest of the galaxy population. There is no increase in scatter for the most blue galaxies that should have the smallest 4000Å breaks.]{}
- [The bias$_z$ between the photometric and spectroscopic redshifts, defined as mean\[$(z_{spec}-z_{phot})/(1+z_{spec})$\] after excluding outliers is statistically inconsistent with zero at a significance of $\gsim 3 \sigma$. However, the bias is always smaller than the scatter and the latter therefore dominates the total uncertainty.]{}
- [The photometric redshift codes produce an estimate of the uncertainty in the derived photometric redshift either as a full redshift probability distribution, P(z), or as quoted confidence intervals corresponding to e.g., 68.3% or 95,4% confidence intervals. Using the spectroscopic control sample with known redshifts, we calculate which fraction of the galaxies falls inside the 68.3% or 95.4% confidence intervals for the different codes. We find that a majority of the codes produce confidence intervals that are too narrow compared to expectations, i.e., the errors in the photometric redshifts are most often underestimated. Factors contributing to the narrow distributions could be underestimated photometric errors or too coarse set of template SEDs. We describe a method for adjusting probability distributions so that the correct fraction of galaxies in the control sample falls inside a specified confidence interval.]{}
- [We can derive photo-z with lower scatter and outlier fraction when we combine results from different codes, when compared to any single code. Taking a straight median, using a sum of the individual probability distributions, or using a hierarchical Bayesian method yields very similar results. The two latter methods produce a probability distribution that can be used to assign errors to the photometric redshifts. For our spectroscopic sample, we find an rms of $\sigma_O \sim 0.03$ with an outlier fraction of at most $\sim$3%.]{}
We finally note that the photometric redshifts presented here are based on test catalogs derived from a subset of CANDELS GOODS-S data. After including additional data, particularly the full depth $HST$/WFC3 $J$- and $H$-bands, we expect further improvements in the absolute values of the photometric redshift accuracies. Further improvements are possible by the addition of medium and narrow band data that are available for the CANDELS fields. The CANDELS GOODS-S photometric redshift catalog will be made publicly available and is described in T. Dahlen et al. 2013 (in prep.).
Abdalla, F. B., Banerji, M., Lahav, O., & Rashkov, V. 2008, MNRAS, submitted (astro-ph/0812.3831) Afonso, J., Mobasher, B., Koekemoer, A., Norris, R. P., & Cram, L. 2006, AJ, 131, 1216 Assef, R. J., Kochanek, C. S., Brodwin, M., et al. 2008, ApJ, 676, 286 Atek, H., Siana, B., Scarlata, C., et al. 2011, ApJ, 743, 121 Babbedge, T. S. R., Rowan-Robinson, M., Gonzalez-Solares, E., et al. 2004, MNRAS, 353, 654 Barro, G., Pérez-González, P. G., Gallego, J., et al. 2011, ApJS, 193, 30 Baum, W., A. 1962, IAU Symposium No. 15, 390 Benítez, N. 2000, ApJ, 536, 571 Bolzonella, M., Miralles, J. -M., & Pelló, R. 2000, A&A, 363, 476 Brammer, G. B., van Dokkum, P. G., & Coppi, P. 2008, ApJ, 686, 1503 Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000 (BC03) Bunker, A. J., Stanway, E. R., Ellis, R. S., McMahon, R. G., & McCarthy P. J. 2003, MNRAS, 342, L47 Calzetti, D., Armus, L., Bohlin, R. C., Kinney, A. L., Koornneef, J., & Storchi-Bergmann, T. 2000, ApJ, 533, 682 Carliles, S., Budavári, T., Heinis, S., Priebe, C., & Szalay, S. 2010, ApJ, 712, 511 Coleman, G. D., Wu, C.-C., & Weedman, D. W., 1980, ApJS, 43, 393 Collister, A. A. & Lahav, O. 2004, PASP, 116, 345 Connolly, A. J., Csabai, I., Szalay, A. S., et al. 1995, AJ, 110, 3655 Cristiani, S., Appenzeller, I., Arnouts, S., et al. 2000, A&A, 359, 489 Croom, S. M., Warren, S. J., & Glazebrook, K. 2001, MNRAS, 328, 150 Dahlen, T., Mobasher. B., Somerville, R. S., et al. 2005, ApJ, 631, 126 Dahlen T., Mobasher, B., Jouvel, S, et al. 2008, AJ, 136, 1361 Dahlen T., Mobasher, B., Dickinson, M., et al. 2010, ApJ, 724, 425 Dickinson, M., Stern, D., Giavalisco, M., et al. 2004, ApJ, 600, L99 Doherty, M., Bunker, A. J., Ellis, R. S., & McCarthy, P. J. 2005, MNRAS, 361, 525 Erb, D. K., Pettini, M., Shapley, A. E., et al. 2010, ApJ, 719, 1168 Fan, X., Strauss, M. A., Becker, R. H., et al. 2006, AJ, 132, 117 Feldmann, R., Carollo, C. M., Porciani, C. et al. 2006, MNRAS, 372, 565 Finlator, K., Davé, R., & Oppenheimer, B. D. 2007, MNRAS, 376, 1861 Fioc, M., & Rocca-Volmerange, B. 1997, A&A, 326, 950 Fontana, A., et al. 2000, AJ, 120, 2206 Galametz, A., Grazian, A., Fontana, A., et al. 2013, ApJS, 206, 10 Gerdes, D. W., Sypniewski, A. J., McKay, T. A., et al. 2010, ApJ, 715, 823 Giallongo, E., D’Odorico, S., Fontana, A., et al. 1998, AJ, 115, 2169 Giavalisco, M., Ferguson H. C., Koekemoer A. M., et al. 2004, ApJ, 600, L93 Grogin, N. A., Kocevski, D. D., Faber, S. M., et al. 2011, ApJS, 197, 35 Guo, Y., Ferguson, H. C., Giavalisco, M., et al. 2013, ApJS, 207, 24 Gwyn, S. 1995, Master’s thesis, Univ. Victoria Hildebrandt, H., Wolf, C., & Benítez, N. 2008, A&A, 480, 703 Hildebrandt, H., Arnouts, S., Capak, P., et al. 2010, A&A, 523, 31 Hogg, D. W., Cohen, J. G., Blandford, R., et al. 1998, AJ, 115, 1418 Huang, J. -S., Faber, S. M., Willmer, C. N. A., et al. 2013, ApJ 766, 21 Ilbert, O., Arnouts, S., McCracken, H. J., et al. 2006, A&A, 457, 841 Ilbert, O., Capak, P., Salvato, M., et al. 2009, ApJ, 690, 1236 Kinney, A. L., Calzetti, D., Bohlin, R. C., et al. 1996, ApJ, 467, 38 Koekemoer, A. M., Faber, S. M., Ferguson, H. C., et al. 2011, ApJS, 197, 36 Koo, D., C. 1985, AJ, 90, 148 Kotulla, R., Fritze, U., Weilbacher, P., & Anders, P. 2009, MNRAS, 396, 462 Laidler, V. G., Papovich, C., Grogin, N. A., et al. 2007, PASP, 119, 1325 Lang, D. & Hogg, D. W. 2012, AJ, 144, 46 Le Fèvre, O., Vettolani, G., Paltani, S., et al. 2004, A&A, 428, 1043 Madau, P. 1995, ApJ, 441, 18 Maraston, C. 2005, MNRAS, 362, 799 Mignoli, M., Cimatti, A., Zamorani, G., et al. 2005, A&A, 437, 883 Newman, J. A., Zepf, S. E., Davis, M., et al. 1999, ApJ, 523, 506 Padovani, P., Miller, N., Kellermann, K. I., et al. 2011, ApJ, 740, 20 Pérez-González, P. G., Rieke, G. H., Villar, V., et al. 2008, ApJ, 675, 234 Polletta, M., Tajer, M., Maraschi, L., et al. 2007, ApJ, 663, 81 Popesso, P., Dickinson, M., Nonino, M., et al. 2009, A&A, 494, 443 Press, W. H. 1997, Unsolved Problems in Astrophysics, ed. J. N. Bahcall & J. P. Ostriker (Princeton: Princeton Univ. Press), 49 Quadri, R. F. & Williams, R. J. 2010, ApJ, 725, 794 Rafelski, M., Wolfe, A. M., Cooke, J., et al. 2009, ApJ, 703, 2033 Ravikumar, C. D., Puech, M., Flores, H., et al. 2007, A&A, 465, 1099 Rocca-Volmerange, B., de Lapparent, V., Seymour, N., & Fioc, M. 2007, A&A, 475, 801 Roche, N. D., Dunlop, J., Caputi, K. I., et al. 2006, MNRAS, 370, 74 Rudnick, G., Franx, M., Rix, H.- W., et al. 2001, AJ, 122, 2205 Salvato, M., Hasinger, G., Ilbert, O., et al. 2009, ApJ, 690, 1250 Salvato, M., Ilbert, O., Hasinger, G., et al. 2011, ApJ, 742, 61 Schaerer, D. & de Barros, S. 2012, in Spectral Energy Distribution of Galaxies, Proceedings of the International Astronomical Union, IAU Symp., 284, 20 Stanway, E. R., Glazebrook, K., Bunker, A. J., et al. 2004, ApJ, 604, L13 Stark, D. P., Schenker, M. A., Ellis, R., et al. 2013, ApJ, 763, 129 Strolger, L.-G., Riess, A. G., Dahlen, T., et al. 2004, ApJ, 613, 200 Surowiecki, J. 2005, The wisdom of crowds (New York: Random House) Szokoly, G. P., Bergeron, J., Hasinger, G., et al. 2004, ApJS, 155, 271 van der Wel, A., Franx, M., van Dokkum, P. G., & Rix, H. -W. 2004, ApJ, 601, L5 Vanzella, E., Cristiani, S., Fontana, A., et al. 2004, A&A, 423, 761 Vanzella, E., Cristiani, S., Dickinson, M., et al. 2008, A&A, 478, 83 Wiklind, T., Dickinson, M., Ferguson, H. C., et al. 2008, ApJ, 676, 781 Windhorst, R. A., Cohen, S. H., Hathi, N. P., et al. 2011, ApJS, 193, 27 Wolf, C. 2009, MNRAS, 397, 520 Xue, Y. Q., Luo, B., Brandt, W. N., et al. 2011, ApJS, 195, 10
[^1]: http://archive.stsci.edu/prepds/candels
[^2]: https://rainbowx.fis.ucm.es/Rainbow\_Database/
[^3]: http://www.astro.yale.edu/eazy/
[^4]: http://webast.ast.obs-mip.fr/hyperz/
[^5]: http://www.cfht.hawaii.edu/ arnouts/LEPHARE/lephare.html
|
---
abstract: 'We study the restricted families of projections in vector spaces over finite fields. We show that there are families of random subspaces which admit a Marstrand-Mattila type projection theorem.'
address: 'School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia '
author:
- Changhao Chen
title: Restricted families of projections in vector space over finite fields
---
Introduction
============
A fundamental problem in fractal geometry is to determine how the projections affect dimension. Recall the classical Marstrand-Mattila projection theorem: Let $E\subset {\mathbb{R}}^{n}, n\geq2,$ be a Borel set with Hausdorff dimension $s$.
- (dimension part) If $s\leq m$, then the orthogonal projection of $E$ onto almost all $m$-dimensional subspaces has Hausdorff dimension $s$.
- (measure part) If $s>m$, then the orthogonal of $E$ onto almost all $m$-dimensional subspaces has positive $m$-dimensional Lebesgue measure.
In 1954 J. Marstand [@Marstrand] proved this projection theorem in the plane. In 1975 P. Mattila [@Mattila1975] proved this for general dimension via 1968 R. Kaufman’s [@Kaufman] potential theoretic methods. We refer to the recent survey of K. Falconer, J. Fraser, and X. Jin [@Falconer] for more backgrounds.
Recently there has been a growing interest in studying finite field version of some classical problems arising from Euclidean spaces. For instance, there are finite field Kakeya sets (also called Besicovitch sets), see Z. Dvir [@Dvir]; there are finite field Erdős/ Falconer distance problem, see A. Iosevich, M. Rudnev [@IosevichRudnev], T. Tao [@Tao1]; etc. Motivated by the above works, the author [@ChenP] studied the projections in vector spaces over finite fields, and obtained the Marstrand-Mattila type projection theorem in this setting. In this paper, we turn to the restricted families of projections in the vector spaces over finite fields. For more details on projection in vector space over finite fields see [@ChenP]. For more backgrounds on restricted families of projections in Euclidean spaces, we refer to [@Falconer Section 6], [@FOO], [@KOV] and reference therein.
Let $p$ be a prime number, $\mathbb{F}_{p}$ be the finite field with $p$ elements, and ${\mathbb{F}_{p}^{n}}$ be the $n$-dimensional vector space over this field. We use the same notation as in the Euclidean spaces. Let $G(n,m)$ be the collection of all $m$-dimensional linear subspaces of ${\mathbb{F}_{p}^{n}}$, and ${A(n,m)}$ be the family of all $m$-dimensional planes, i.e., the translation of some $m$-dimensional subspace. In the following we show the definition of projections in ${\mathbb{F}_{p}^{n}}$, see [@ChenP] for more details.
Let $E$ be a subset of ${\mathbb{F}_{p}^{n}}$ and $W$ be a non-trivial subspace of ${\mathbb{F}_{p}^{n}}$. Denoted by $\pi^{W}(E)$ the collection of coset of $W$ which intersects $E$, that is $$\pi^{W}(E)=\{x+W: E\cap (x+W) \neq \emptyset, x\in {\mathbb{F}_{p}^{n}}\}.$$ In this paper we are interested in the cardinality of $\pi^{W}(E)$.
For any set $E \subset {\mathbb{F}_{p}^{n}}$ and $W\in G(n,n-m)$ the Lagrange’s group theorem implies $$|\pi^{W}(E)|\leq \min\{|E|, p^{m}\}.$$ Here and in the following let $|J|$ denote the cardinality of set $J$. The author [@ChenP Corollary 1.3] obtained the following Marstrand-Mattila type projection theorem in ${\mathbb{F}_{p}^{n}}$. In fact the following form is often called the size of the exceptional sets of projections.
\[thm:ChenP\] Let $E \subset {\mathbb{F}_{p}^{n}}$ with $|E|=p^{s}$.
\(a) If $s\leq m$ and $t \in (0, s]$, then $$| \{W \in G(n,n-m) : |\pi^{W} (E)| \leq p^{t}/10 \} \leq \frac{1}{2} p^{m(n - m) -(m - t)}.$$
\(b) If $s> m$, then $$| \{W \in G(n,n-m) : |\pi^{W} (E)| \leq p^{m}/ 10 \}| \leq \frac{1}{2} p^{m(n - m) -(s-m)}.$$
We note that $|G(n,m)|\approx p^{m(n-m)}$, see P. Cameron [@Cameron Theorem 6.3]. We write $f\lesssim g$ if there is a positive constant $C$ such that $f\leq Cg$, $f\gtrsim g$ if $g\lesssim f$, and $f\approx g$ if $f\lesssim g$ and $f\gtrsim g$.
In the following, we formulate finite fields version of restricted families of projections. Let $G$ be a subset of $G(n,k)$, then $(\pi^{W})_{W\in G }$ is called a restricted family of projection. The purpose of this paper is looking for subsets $G\subset G(n,k)$ such that $(\pi^{W})_{W\in G }$ admit a Marstrand-Mattila type projection theorem.
By studying the random subsets of $G(n, n-m)$, we obtain the following result.
\[thm:main\] For any $ \min\{m, n-m\} <\alpha \leq m(n-m)$, there exists a subset $G\subset G(n,n-m)$ with $|G|\approx p^{\alpha}$ such that for any $E\subset {\mathbb{F}_{p}^{n}}$, $$\label{eq:eee}
|\{W\in G: |\pi^{W}(E)|\leq N\}|\lesssim |G|N(|E|^{-1}+p^{-m}).$$
Note that for the case $\alpha=m(n-m)$, Theorem \[thm:main\] follows from Theorem \[thm:ChenP\] by choosing $G=G(n,m)$. Thus we consider the case $\min\{m, n-m\} <\alpha < m(n-m)$ only. We immediately have the following Marstrand-Mattila type projection theorem via the special choice $N$ in Theorem \[thm:main\].
For any $ \min\{m, n-m\} <\alpha \leq m(n-m)$, there exists a subset $G\subset G(n,n-m)$ with $|G|\approx p^{\alpha}$ such the following holds. Let $E\subset {\mathbb{F}_{p}^{n}}$ with $|E|=p^{s}$.
\(1) If $|E|\leq p^{m}$ and $t\in (0,s]$, then $$|\{W\in G: |\pi^{W}(E)|\leq p^{t}\}|\lesssim |G|p^{t-s}.$$
\(2) If $|E|> p^{m}$, then for any small $\varepsilon$ $$|\{W\in G: |\pi^{W}(E)|\leq \varepsilon p^{m}\}|\lesssim |G|\varepsilon.$$
For restricted families of projections in Euclidean spaces, the author [@ChenR] obtained that some random subsets of sphere of ${\mathbb{R}}^{3}$ admit a Marstrand-Mattila type projection theorem. For more details, see [@ChenR].
The structure of the paper is as follows. In Section \[sec:p\], we set up some notation and show some lemmas for later use. We prove Theorem \[thm:main\] in Section \[sec:proof\]. In the last Section we given some examples of restricted families of projections which admit a Marstrand-Mattila type theorem in finite fields setting.
Preliminaries {#sec:p}
=============
In this section we show some lemmas for later use.
Outline of the methods
----------------------
In short words, we take a random subset $G\subset G(n,n-m)$, see the random model in Subsection \[sub:rrr\]. Then we estimate the cardinality of “the exceptional sets”, $$\{W\in G: |\pi^{W} (E)|\leq N\},$$ and show that it satisfies our need. To estimate the “exceptional sets”, we adapt the arguments in [@ChenP] to our setting which is a variant of Orponen’s pairs argument [@OrponenA Estimate (2.1)].
Let $W\in G(n,n-m)$ then Lagrange’s group theorem implies that there are $p^{m}$ cosets of $W$. Let $x_{W, j}+W, 1\leq j\leq p^{m}$ be the different cosets of $W$. Let $E\subset {\mathbb{F}}_{p}^{n}$, then $$|E| =\sum_{j=1}^{p^{m}} |E\cap (x_{W, j}+W)|,$$ and the Cauchy-Schwarz inequality implies $$\label{eq:pairss}
|E|^{2}\leq |\pi^{W}(E)|\sum_{j=1}^{p^{m}} |E\cap (x_{W, j}+W)|^{2}.$$ Note that $|E\cap (x_{W, j}+W)|^{2}$ is the amount of pairs of $E$ inside the coset $x_{W, j}+W$. Let $N\leq p^{m}$, define $$\Theta=\{W\in G: |\pi^{W} (E)|\leq N\}.$$ Summing two sides over $W\in \Theta$ in estimate , we obtain $$\label{eq:argument}
|\Theta| |E|^{2} \leq {\mathcal{E}}(E,\Theta')N$$ where ${\mathcal{E}}(E,\Theta')=\sum_{W\in \Theta}\sum_{j=1}^{p^{m}}|E\cap (x_{W, j}+W)|^{2}.$ Thus the left problem is to estimate ${\mathcal{E}}(E, \Theta')$, and we use the doubling counting argument of Murphy and Petridis [@MurphyPetridis Lemma 1] and the discrete Plancherel identity. The above discusses motivated the following definition.
Let $E\subset {\mathbb{F}_{p}^{n}}$ and $ \mathcal{A} \subset A(n,m)$. Define the energy of $E$ on $\mathcal{A}$ as $${\mathcal{E}}(E, \mathcal{A}) =\sum_{W\in \mathcal{A}} |E\cap W|^{2}.$$
We note that ${\mathcal{E}}(E, \mathcal{A})$ is closely related to the incidence identity of Murphy and Petridis [@MurphyPetridis Lemma 1], and the additive energy in additive combinatorics [@TaoVu Chapter 2].
Discrete Fourier transform
--------------------------
In the following we collect some basic facts about Fourier transformation which related to our setting. For more details on discrete Fourier analysis, see Green [@Green], Stein and Shakarchi [@Stein]. Let $f : {\mathbb{F}_{p}^{n}}\longrightarrow \mathbb{C}$ be a complex value function. Then for $\xi \in {\mathbb{F}_{p}^{n}}$ we define the Fourier transform $$\label{eq:dede}
\widehat{f}(\xi)=\sum_{x\in {\mathbb{F}_{p}^{n}}} f(x)e(-x\cdot \xi),$$ where the dot product $ x\cdot\xi $ is defined as $ x_1\xi_1+\cdots +x_n\xi_n$ and $e(-x \cdot \xi)={e^{-\frac{2\pi i x\cdot\xi}{p}}}$. Recall the following Plancherel identity, $$\sum_{\xi \in {\mathbb{F}_{p}^{n}}}|\widehat{f}(\xi)|^{2}=p^{n}\sum_{x\in {\mathbb{F}_{p}^{n}}} |f(x)|^{2}.$$ Specially for the subset $E\subset {\mathbb{F}_{p}^{n}}$, we have $$\sum_{\xi \in {\mathbb{F}_{p}^{n}}} |\widehat{E}(\xi)|^{2}=p^{n}| E|.$$ Here and in the following we use $E$ as characteristic function of the set $E$.
For $W\in G(n,n-m)$, define $$Per(W):=\{x\in {\mathbb{F}_{p}^{n}}: x\cdot w=0, w\in W\}.$$ Note that if $W$ is some subspace in Euclidean space then $Per(W)$ is the orthogonal complement of $W$. Furthermore, unlike in the Euclidean spaces, here $W\cap Per(W)$ can be some non-trivial subspace. However the rank-nullity theorem of linear algebra implies that for any subspace $W \subset{\mathbb{F}_{p}^{n}}$, $$\label{eq:rank}
\dim W+\dim Per(W)=n.$$
The following Lemma \[lem:fff\] of [@ChenP Lemma 2.3] plays an important role in the proof of Lemma \[lem:abstract\] (2). For more details see [@ChenP Lemma 2.3].
\[lem:fff\] Use the above notation. We have $$\label{eq:kk}
\sum_{j=1}^{p^{m}} | E \cap (x_{j}+W)|^{2}=p^{-m}\sum_{\xi\in Per(W)} |\widehat{E}(\xi)|^{2}.$$
We note that the Lemma \[lem:fff\] is the only place in this paper where the prime field $\mathbb{F}_{p}$ is needed. We do not know if the Lemma \[lem:fff\] also holds for vector spaces over general finite fields.
In the following we extend a result of [@ChenP Lemma 3.1] to general subset of ${G(n,n-m)}$. Let $G\subset G(n,n-m)$, define $$\label{eq:define}
G'=\bigcup_{W\in G}\bigcup_{j=1}^{p^{m}}(x_{j,W}+W)$$ where $x_{W, j}+W, 1\leq j\leq p^{m}$ are the cosets of $W$. For each $W$ we simply use $x_{W, j}+W, 1\leq j\leq p^{m}$ to represent the cosets of $W$.
\[lem:abstract\] Let $G$ be a subset of $G(n, n-m)$ with $|G|\gtrsim p^{\beta}$.
\(1) If for any $\xi\neq 0$, $$\label{eq:l11}
|\{W\in G: \xi \in V\}|\lesssim |G| p^{-\beta},$$ then $${\mathcal{E}}(E,G')\lesssim |E||G|+|E|^{2}|G|p^{-\beta}.$$
\(2) If for any $\xi\neq 0$, $$\label{eq:l22}
|\{W\in G: \xi \in Per(W)\}|\lesssim |G| p^{-\beta},$$ then $${\mathcal{E}}(E,G')\lesssim p^{-m}|G|(|E|^{2}+|E|p^{n-\beta}).$$
The claim $(1)$ follows by doubling counting. Recall that we denote by $F(x)$ the characteristic function of the subset $F\subset {\mathbb{F}_{p}^{n}}$. Then $$\begin{aligned}
{\mathcal{E}}(E, G')&= \sum_{V\in G' }|E \cap V|^{2}\\
&=\sum_{V \in G'} \left(\sum_{x\in E}V(x) \right)^{2}\\
&=\sum_{V \in G'} \left(\sum_{x\in E}V(x)+\sum_{x\neq y \in E} V(x)V(y) \right)\\
&\lesssim |E||G|+|E|(|E|-1)|G|p^{-\beta}.
\end{aligned}$$
To establish $(2)$, the Lemma \[lem:fff\] implies $$\begin{aligned}
{\mathcal{E}}(E, G')
&=\sum_{W\in G} \sum_{j=1}^{p^{m}}|E \cap (x_{W, j}+W)|^{2}\\
& =p^{-m}\sum_{W\in G} \sum_{\xi\in Per(W)}|\widehat{E}(\xi)|^{2}\\
& =p^{-m}(|G||E|^{2}+\sum_{W\in G} \sum_{\xi\in Per(W)\backslash \{0\}}|\widehat{E}(\xi)|^{2})\\
&\lesssim p^{-m}(|G||E|^{2}+p^{n}|E||G|p^{-\beta}).
\end{aligned}$$ Thus we finish the proof.
Proof of Theorem \[thm:main\] {#sec:proof}
=============================
Random subsets of ${G(n,n-m)}$ {#sub:rrr}
------------------------------
We start by a description of these random subsets in ${G(n,n-m)}$. Let $0<\delta<1$. We choose each element of ${G(n,n-m)}$ with probability $\delta$ and remove it with probability $1-\delta$, all choices being independent of each other. Let $G=G^{\omega}$ be the collection of these chosen subspaces. Let $\Omega ({G(n,n-m)}, \delta)$ be our probability space which consists of all the possible outcomes of random subspaces.
For the convenience to our use, we formulate the following large deviations estimate. For more background and details on large deviations estimate, see Alon and Spencer [@Alon Appendix A].
\[lem:law of large numbers\] Let $\{X_j\}_{j=1}^N$ be a sequence independent Bernoulli random variables which takes value $1$ with probability $\delta$ and value $0$ with probability $1-\delta$. Then $${\mathbb{P}}( \sum^N_{j=1} X_j \geq 3N\delta )\leq e^{-N\delta}.$$
We also need the following Lemma of [@ChenP Lemma 2.7].
\[lem:c\] Let $\xi$ be a non-zero vector of ${\mathbb{F}_{p}^{n}}$.
\(1) $|\{W\in G(n, k): \xi\in W\}|=|G(n-1, k-1)|$.
\(2) $|\{W\in G(n, k): \xi\in Per(W)\}|=|G(n-1, k)|$.
\[co:cc\] For any $m<\alpha<m(n-m)$, there exists a subset $G\subset {G(n,n-m)}$ such that $|G|\approx p^{\alpha}$ and for any $\xi\neq 0$, $$|\{W\in G: \xi \in W\}|\lesssim |G|p^{-m}.$$
We consider the random model $\Omega({G(n,n-m)}, \delta)$ where $\delta=|G(n,m)|^{-1}p^{\alpha}$. First observe that $p^{\alpha}/2 \leq |G| \leq 2 p^{\alpha}$ with high probability ($>1/2$) provided large $p$. This follows by applying Chebyshev’s inequality, which says that $$\label{eq:che}
\begin{aligned}
{\mathbb{P}}(||G| - p^{\alpha}|&> \frac{1}{2}p^{\alpha})\leq \frac{4p^{\alpha}(1-\delta)}{p^{2\alpha}}\\
&\leq \frac{4}{p^{\alpha}}\rightarrow 0 \text{ as } p \rightarrow \infty.
\end{aligned}$$
Let $\xi\neq 0$ and $G_{\xi}:=\{W\in {G(n,n-m)}: \xi \in W\}$. Lemma \[lem:c\] (1) implies that $$|G_{\xi}|=|G(n-1,n-m-1)|\approx p^{m(n-m)-m}.$$ Observe that for $G\in \Omega({G(n,n-m)},\delta)$, $$|\{W\in G: \xi \in W\}|=\sum_{W\in G_{\xi}}{\bf 1}_{G}(W).$$ Thus by Lemma \[lem:law of large numbers\], $${\mathbb{P}}(\sum_{W\in G_{\xi}}{\bf 1}_{G}(W)\geq 3|G(n-1,m)|\delta)\leq e^{-Cp^{\alpha-m}}$$ where $C$ is a positive constant. It follows that $$\begin{aligned}
{\mathbb{P}}(\exists \xi\neq 0, s.t. \sum_{W\in G_{\xi}}&{\bf 1}_{G}(W) \geq 3|G(n-1,m)|\delta)\\
&\leq p^{n}e^{-Cp^{\alpha-m}}\rightarrow 0 \text{ as } p \rightarrow \infty.
\end{aligned}$$ Together with the estimate , we conclude that $G\in \Omega({G(n,n-m)}, \delta)$ satisfies our need with high probability (at least one) provided $p$ is large enough.
\[co:ccc\] For any $n-m<\alpha<m(n-m)$, there exists a subset $G\subset {G(n,n-m)}$ such that $|G|\approx p^{\alpha}$ and for any $\xi\neq 0$, $$|\{W\in G: \xi \in Per(W)\}|\lesssim |G|p^{-(n-m)}.$$
We consider the random model $\Omega({G(n,n-m)}, \delta)$ where $\delta=|G(n,m)|^{-1}p^{\alpha}$. For any $\xi\neq 0$, Lemma \[lem:c\] (2) implies that $$|\{W\in G(n, n-m): \xi \in Per(W)\}|=|G(n-1, n-m)|\approx p^{m(n-m)-(n-m)}.$$ Then applying the similar argument to the proof of Corollary \[co:cc\], we obtain that $G\in \Omega({G(n,n-m)}, \delta)$ satisfies our need with high probability provided $p$ is large enough.
Now we intend to apply Lemma \[lem:abstract\] and the above two Corollaries to prove Theorem \[thm:main\].
Suppose $\alpha>m$. By Corollary \[co:cc\] there exists a subset $G\subset {G(n,n-m)}$ such that $|G|\approx p^{\alpha}$ and for any $\xi\neq 0$, $$|\{W\in G: \xi \in W\}|\lesssim |G|p^{-m}.$$ Applying Lemma \[lem:abstract\] (1), we obtain that for any $E\subset {\mathbb{F}_{p}^{n}}$, $${\mathcal{E}}(E,G')\lesssim |G|(|E|+|E|^{2}p^{-m}).$$ By estimate we obtain $$|\{W\in G: |\pi^{W}(E)|\leq N\}|\lesssim |G|N(|E|^{-1}+p^{-m}).$$
For the case $\alpha>n-m$. By Corollary \[co:ccc\] there exists a subset $G\subset G(n,n-m)$ with $|G|\approx p^{\alpha}$ and for any $\xi\neq 0$, $$|\{W\in G: \xi \in Per(W)\}|\lesssim |G|p^{n-m}.$$ Applying Lemma \[lem:abstract\] (2), we obtain that for any $E\subset {\mathbb{F}_{p}^{n}}$, $${\mathcal{E}}(E,G')\lesssim |G|(|E|+|E|^{2}p^{-m}).$$ Again by estimate , we obtain $$|\{W\in G: |\pi^{W}(E)|\leq N\}|\lesssim |G|N(|E|^{-1}+p^{-m}).$$ Thus we complete the proof.
Examples
========
We show two examples in the following. For $D\subset {\mathbb{F}_{p}^{n}}$ let $G_{D}$ be the collection of one dimensional subspaces which intersects $D$, i.e., $$G_{D}=\{kx: x\in D, k\in \mathbb{F}_{p}\}.$$
\[exa:1\] Let $
S_{1}=\{(x_{1}, x_{2}, 1)\in {\mathbb{F}}_{p}^{3}: x_{1}^{2}+x_{2}^{2}=1\}$. Then for any $E\subset \mathbb{F}_{p}^{3}$, $$|\{L\in G_{S_{1}}: |\pi^{L}(E)|\leq N\}|\lesssim |S_{1}|N(p^{-2}+|E|^{-1}).$$
A. Iosevich and M. Rudnev [@IosevichRudnev Lemma 2.2] proved that $|S_{1}|\approx p$, and hence $|G_{S_{1}}|\approx p$. Observe that $|W\cap S_{1}|\lesssim 1$ for any $W\in G(3,2)$.
For $\xi\neq 0$ let $Span(\xi)=\{k\xi: k\in \mathbb{F}_{p}\}$. Then $$\{L\in G_{S_{1}}: \xi \in Per(L)\}=G_{S_{1}}\cap Per(Span(\xi)).$$ The rank-nullity theorem implies that $\dim Per(Span(\xi))=2$. Thus $Per(Span(\xi))\in G(3,2)$, and hence we obtain $$|\{L\in G_{S_{1}}: \xi \in Per(L)\}|\lesssim 1.$$ Applying estimate and Lemma \[lem:abstract\] (2) with $\beta=1, m =2$, we finish the proof.
Note that the above example $S_{1}$ can be considered as a finite fields version of curve $$\Gamma=\{\frac{1}{\sqrt{2}}(\cos t, \sin t, 1): t\in [0, 2\pi])\} \subset {\mathbb{R}}^{3}.$$ For more details on restricted families of projections with respect to $\Gamma$ we refer to [@KOV], [@OV]. In the following, we show a finite fields version of curve $$\{(t,t^{2},\cdots, t^{n}): t\in [0,1]\}\subset {\mathbb{R}}^{n}.$$
\[ex:ex\] Let $S=\{( a, a^{2} \cdots, a^{n}): a \in {\mathbb{F}}_{p}\backslash \{0\}\}$. Then $|G_{S}|=p-1$ and for any subset $E\subset {\mathbb{F}_{p}^{n}}$, $$|\{L\in G_{S}: |\pi^{L}(E)|\leq N\}|\lesssim |G_{S}|N(|E|^{-1}+p^{-(n-1)}).$$
For $n=2$ we have $|G_{S}|\approx |G(2,1)|\approx p$, and the claim follows by applying Theorem \[thm:ChenP\]. In the following we fix $n\geq 3$ and let $p$ be a large prime number.
For any $\xi\neq 0$, $$\{L\in G_{S}:\xi \in Per(L) \}=G_{S}\cap Per(Span(\xi)).$$ The rank-nullity theorem implies that $\dim Per(Span(\xi))=n-1$. Observe that any $n$ elements of $S$ form a nonsingular Vandermonde matrix, and hence these $n$ vectors are linear independent. It follows that for any hyperplane $W\in G(n,n-1)$, $$|W\cap S|\leq n-1\lesssim_{n} 1.$$ Therefore we obtain $$|\{L\in G_{S}: \xi \in Per(L) \}|\lesssim_{n}1.$$ Applying estimate and Lemma \[lem:abstract\] (2) with $\beta=1, m=n-1$, we finish the proof.
By the special choices of $N$ in the above two examples, we conclude that Marstrand-Mattila type projection theorem hold for the restricted families $(\pi^{L})_{L\in G_{S_{1}}}$ and $(\pi^{L})_{L\in G_{S}}$.
N. Alon and J. Spencer. The probabilistic method. New York: WileyInterscience, 2000.
P. Cameron, The art of counting, cameroncounts.files.wordpress.com/2016/04/acnotes1.pdf
C. Chen, Projections in vector spaces over finite fields, to appear in Ann. Acad. Sci. Fenn. Math., arxiv.org/abs/1702.03648
C. Chen, Restricted families of projections and random subspaces, arxiv.org/abs/1706.03456
Z. Dvir, On the size of Kakeya sets in finite fields, J. Amer. Math. Soc. 22 (4) (2009) 1093-1097
K. Falconer, J. Fraser and X. Jin, Sixty Years of Fractal Projections, in Fractal Geometry and Stochastics V, Vol. 70 of Progress in Probability, 3-25
K. Fässler and T. Orponen. On restricted families of projections in ${\mathbb{R}}^{3}$, Proc. London Math. Soc. 109 (2014), 353-381
B. Green, Restriction and Kakeya phonomena, Lecture notes (2003).
A. Iosevich and M. Rudnev, Erdős distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc. 359 (2007), 6127-6142.
R. Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968), 153-155
A. Käenmäki, T. Orponen, and L. Venieri, A Marstrand-type restricted projection theorem in ${\mathbb{R}}^{3}$, preprint (2017), arXiv:1708.04859
J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc.(3), 4, (1954), 257-302
P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 2, 227-244.
P. Mattila, Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, vol. 150, Cambridge University Press, 2015.
B. Murphy and G. petridis, A point-line incidence identity in finite fields, and applications, preprint (2016), available at arxiv.org/abs/1601.03981.
T. Orponen, A discretised projection theorem in the plane, preprint (2014), available at arxiv.org/pdf/1407.6543.pdf
T. Orponen, and L. Venieri, Improved bounds for restricted families of projections to planes in ${\mathbb{R}}^{3}$, preprint (2017), arxiv.org/abs/1711.08934
E. Stein and R. Shakarchi, Fourier Analysis: An Introduction. Princeton and Oxford: Princeton UP, 2003. Print. Princeton Lectures in Analysis.
T. Tao, Finite field analogues of Erdős, Falconer, and Furstenberg problems.
T. Tao and V. Vu, Additive Combinatorics, Cambridge University Press.
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---
abstract: 'We investigate a generalized coupled nonlinear Schrodinger (GCNLS) equation containing Self-Phase Modulation (SPM), Cross-Phase Modulation (XPM) and Four Wave Mixing (FWM) describing the propagation of electromagnetic radiation through an optical fibre and generate the associated Lax-pair. We then construct bright solitons employing gauge transformation approach. The collisional dynamics of bright solitons indicates that it is not only possible to manipulate intensity (energy) between the two modes (optical beams), but also within a given mode unlike the Manakov model which does not have the same freedom. The freedom to manipulate intensity (energy) in a given mode or between two modes arises due to a suitable combination of SPM, XPM and FWM. While SPM and XPM are controlled by an arbitrary real parameter each, FWM is governed by two arbitrary complex parameters. The above model may have wider ramifications in nonlinear optics and Bose-Einstein Condensates (BECs).'
address:
- '$^1$ Centre for Nonlinear Science (CeNSc), PG and Research Department of Physics, Government College for Women (Autonomous), Kumbakonam 612001, India'
- '$^2$ Department of Physics, Pondicherry University, Puducherry-605014, India.'
author:
- 'R.Radha$^{\ast,1}$'
- 'P.S.Vinayagam$^1$'
- 'K. Porsezian$^{\dagger,2}$'
title: Manipulation of light in a generalized coupled Nonlinear Schrödinger equation
---
Coupled Nonlinear Schrödinger system, Bright Soliton, Gauge transformation, Lax pair\
[2000 MSC: 37K40, 35Q51, 35Q55 ]{}
Introduction
============
The potential of solitons to carry information through optical fibres governed by the nonlinear Schrodinger (NLS) equation [@nlseqn; @nlseqn2; @fibre; @fibre2] and the freedom to switch energy between two laser beams in a fibre described by the celebrated Manakov model has made a dramatic turnaround in the field of optical communications. The concept of shape changing collisional dynamics of solitons in the Manakov model is governed by the coupled NLS equation of the following form
$$\begin{aligned}
i q_{1t}+q_{1xx}+2(g_{11}|q_1|^2
+g_{12}|q_2|^2)q_1 =0, \\
i q_{2t}+q_{2xx}+2(g_{21}|q_1|^2 + g_{22} |q_2|^2)q_2 =0,\end{aligned}$$
\[nlsmode1\]
where $q_{i}(x,t)$(i=1,2) corresponds to the envelope of the electromagnetic radiation passing through the optical fibre and $i$ is the imaginary unit. In the above equation, $g_{11}$ and $g_{22}$ correspond to the Self Phase Modulation (SPM) while $g_{12}$ and $g_{21}$ represent the Cross Phase Modulation (XPM). Equations.(\[nlsmode1\]) have been shown to be integrable and admit painlevé property [@painleve] if either (i) $g_{11}=g_{12}=g_{21}=g_{22}$ or (ii) $g_{11}=g_{21}=-g_{12}=-g_{22}$. The first choice corresponds to the celebrated Manakov model [@manakov; @manakov2] while the second choice represents the modified Manakov model [@modifymanakov; @modifymanakov2] and it has been observed that both admit shape changing collisional dynamics of bright solitons [@rkrishnan]. Recently, it was shown [@psv:pre:nls] that one can rotate the trajectories of the bright solitons by varying the system parameters, namely SPM and XPM without violating the integrability of the Manakov (or modified Manakov) model.
It should be mentioned that the inelastic collision of bright solitons which is concerned with redistribution of energy between two modes (optical beams) is brought about by varying the parameters associated with the phase of bright solitons combined with a suitable combination of coupling coefficients. Can one manipulate the intensity (or energy) in a given mode (electromagnetic radiation) or in a given bound state of the electromagnetic radiation $?$. Can one manipulate optical pulses by varying its interaction with the medium rather than changing the parameters associated with the phase of solitons $?$. The answer to these questions assumes tremendous significance as one will have the flexibility of desirably energizing a given mode or a given bound state of the optical pulse as two laser beams propagate through optical fibres. This situation is reminescent of manipulating binary interaction through Feshbach resonance [@FR] in Bose-Einstein Condensates [@review].
In addition to Manakov (or modified Manakov) model, a generalized coupled NLS equation (GCNLS) by including Four Wave Mixing (FWM) with SPM and XPM has been investigated by Park and Shin [@park.shin] which subsequently led to the identification of four different classes of integrable models dealing with the propagation of optical beams through birefringent fibres. Eventhough the variants of the GCNLS equation were investigated recently by Wang and Agalar et al.,[@motopaper; @agalarov] respectively, with three arbitrary parameters (two real parameters corresponding to SPM and XPM and one complex parameter for FWM), the impact of FWM when reinforced with SPM and XPM on the collisional dynamics of solitons has not yet been clearly spelt out. In addition, the impact of the freedom associated with four arbitrary parameters in a GCNLS has not been probed yet.
In this paper, we investigate a GCNLS equation involving four system parameters. The two real arbitrary parameters are associated with SPM and XPM while the two arbitrary complex parameters are associated with Four Wave Mixing (FWM). We then construct the Lax-pair of the GCNLS equation and generate bright solitons. We then show that one can not only manipulate the intensity (or energy) between two laser beams, but also manoeuvre the energy distribution among the bound states of a given laser beam. The freedom to manipulate intensity arises from a suitable combination of nonlinear interaction parameters associated with the system.
Mathematical model and Lax-Pair
===============================
We know that the coupling between co-propagating optical beams in a nonlinear medium determines the application of optical fibres. Considering the propagation of optical pulses through a nonlinear birefringent fibre, the dynamics is governed by the generalized coupled NLS (GCNLS) equation of the following form
$$\begin{aligned}
i \psi_{1t}+\psi_{1xx}+2(a|\psi_1|^2+c|\psi_2|^2+b\psi_1\psi_2^{\ast}+d\psi_2\psi_1^*)\psi_1=0, \\
i
\psi_{2t}+\psi_{2xx}+2(a|\psi_1|^2+c|\psi_2|^2+b\psi_1\psi_2^{\ast}+d\psi_2\psi_1^*)\psi_2=0,\end{aligned}$$
\[gptwo\]
In equations.(\[gptwo\]), $\psi_1$ and $\psi_2$ represent the strengths of electromagnetic beams. The nonlinear co-efficients $a$ and $c$ which are real account for the SPM and XPM respectively while the arbitrary real parameters $b$ and $d$ correspond to FWM. It should be mentioned that eventhough one can allow FWM parameters $b$ and $d$ to be complex, we have retained them to be real in the present model. When FWM effects ($b$ and $d$) are equal to zero and the co-efficients $a = c$, then the above model reduces to the celebrated Manakov or modified Manakov [@manakov; @modifymanakov] model. Equation.(\[gptwo\]) has also been investigated recently for $d= b^*$ [@motopaper] and the impact of FWM on the collisional dynamics of solitons has been investigated. The above equation (\[gptwo\]) admits the following linear eigenvalue problem of the following form,
$$\begin{aligned}
\Phi_x &+ {\cal U} \Phi=0,\\
\Phi_t &+ {\cal V} \Phi=0,\end{aligned}$$
\[lax\] where $\Phi = (\phi_1, \phi_2, \phi_3)^T$ and $$\begin{aligned}
{\cal U} &= \left(%
\begin{array}{ccc}
i\zeta_1 & \psi_{1} & \psi_{2}\\
-R_{1}& -i\zeta_1 & 0 \\
-R_{2}& 0 & -i\zeta_1 \\
\end{array}%
\right),\end{aligned}$$
$$\begin{aligned}
{\cal V}&=\left(%
\begin{array}{ccc}
-i \zeta_1^{2}+ \frac{i}{2} \psi_{1} R_{1}+ \frac{i}{2} \psi_{2}R_{2} & - \zeta_1 \psi_{1}+ \frac{i}{2} \psi_{1x} & - \zeta_1 \psi_{2}+ \frac{i}{2} \psi_{2x} \\
\zeta_{1} R_{1} + \frac{i}{2} R_{1x} & i \zeta_{1}^{2} - \frac{i}{2} \psi_{1} R_{1} & -\frac{i}{2} \psi_{2} R_{1} \\
\zeta_{1} R_{2} + \frac{i}{2} R_{2x} & -\frac{i}{2} \psi_{1} R_{2} & i \zeta_{1}^{2} - \frac{i}{2} \psi_{2} R_{2} \\
\end{array}%
\right),\end{aligned}$$
and $$\begin{aligned}
R_{1}&= -a \psi_1(x,t)^{\ast}-b \psi_2(x,t)^{\ast}, \notag\\
R_{2}&= -d \psi_1(x,t)^{\ast}-c \psi_2(x,t)^{\ast}. \notag\end{aligned}$$
In the above equation, the spectral parameter $\zeta_1$ is isospectral. It is obvious that the compatibility condition $(\Phi_x)_t$=$(\Phi_t)_x$ leads to the zero curvature equation ${\cal U}_t- {\cal V}_x+[{\cal U},{\cal V}]=0$ which yields the integrable generalized coupled NLS equation (\[gptwo\]).
![Elastic collision of solitons in the celebrated Manakov model for $a \equiv c =1$ and $b \equiv d = 0$, $\alpha_{10}=0.15$, $\alpha_{20}=0.15$, $\beta_{10}=0.25$, $\beta_{20}=0.25$, $\chi_{1}=2$, $\chi_{2}=3$, $\delta_{1}=4$, $\delta_{2}=5$, $\varepsilon_1^{(1)}=0.5$, $\varepsilon_1^{(2)}=0.5$ []{data-label="figtwosol1"}](Ntwosolelasticnew.png)
![FWM induced rotation and enhancement of intensities of solitons for $a \equiv c=1$ and $b=d=0.5$ with the other parameters as in fig.(\[figtwosol1\]) []{data-label="figtwosol2"}](Ntwosolelasticrotation.png)
![Enhancement of the intensity $I_{1}$ by manipulating $d=3.5$ keeping the other parameters as in fig.(\[figtwosol2\]) except $\varepsilon_1^{(1)}=0.8$ and $\varepsilon_1^{(2)}=0.5$ []{data-label="figtwosol3"}](Ntwosolelasticfirstmode.png)
![Enhancement of intensity $I_{2}$ by manipulating $b=3.5$ keeping other parameters as in fig.(\[figtwosol2\]) except $\varepsilon_1^{(1)}=0.5$ and $\varepsilon_1^{(2)}=0.8$ []{data-label="figtwosol4"}](Ntwosolelasticsecondmode.png)
Recently Agalarov et.al.,[@agalarov] investigated the GCNLS equation.(\[gptwo\]) for $d = b^{\ast}$ and transformed it to Manakov model. It should be mentioned that equation.(\[gptwo\]) with four independent arbitrary parameters $a,c,b$ and $d$ can also be mapped onto the celebrated Manakov model only for the parametric choice $b=d$ and $ac-bd=\sigma=\pm1$ under the following transformation $$\psi_{1}=q_{1}-d q_{2}, \quad \psi_{2}=a q_{2}.$$ so that we obtain equation.(\[nlsmode1\]) (after suitable algebraic manipulation) with $g_{11}=g_{21}=a$, $g_{12}=g_{22}=a
\sigma$.
We again emphasize that GCNLS equation.(\[gptwo\]) offers the freedom to choose arbitrary $a,b,c$ and $d$ and the Manakov and modified Manakov model only arises as a special case of equation.(\[gptwo\]). We would like to emphasize that any conversion of GCNLS equation.(\[gptwo\]) to either Manakov (or modified Manakov) model deprives us the freedom to choose $a,b,c$ and $d$ arbitrarily.
Bright Solitons and Collisional Dynamics
========================================
It is worth to pointing out at here that the above type of equation such as Davey-Stewartson (DS) equation has been investigated by different methods like first integral method [@Integralmethod], variational iteration method [@variational] and decomposition method [@decomposition], but gauge transformation approach [@llchaw1991] which is employed to investigate the model equations. (\[gptwo\]) is more effective and handy to generate multi soliton solution. Employing gauge transformation approach one obtains the bright soliton solution of the following form $$\begin{aligned}
\psi_{1}^{(1)} &= 2 \varepsilon _{1}^{(1)} \beta_{1}
\mathrm{sech}(\theta
_{1})e^{i(-\xi _{1})}, \label{onesol1} \\
\psi_{2}^{(1)} &= 2 \varepsilon _{2}^{(1)} \beta_{1}
\mathrm{sech}(\theta _{1})e^{i(-\xi _{1})}, \label{onesol2}\end{aligned}$$where $$\begin{aligned}
\theta _{1} &= 2 x \beta _{1}-4 \int (\alpha _{1}\beta
_{1}) dt + 2 \delta_{1},\label{theta} \\
\xi _{1} &= 2 x \alpha _{1}- 2 \int (\alpha _{1}^{2}-\beta
_{1}^{2}) dt -2\chi _{1}\label{phase}.\end{aligned}$$with $~\alpha _{1}=\alpha _{10}(a \tau_1^2 + b \tau_1 \tau_2 + d
\tau_1 \tau_2 +c \tau_2^2)$, $\beta _{1}=\beta _{10}(a \tau_1^2 +
b \tau_1 \tau_2 + d \tau_1 \tau_2 +c \tau_2^2)$ while $\delta
_{1}$, $\chi _{1}$, $\tau_1$ and $\tau_2$ are arbitrary parameters and $\varepsilon _{1}^{(1)}, \varepsilon _{2}^{(1)}$ are coupling parameters, subject to the constraint $|\varepsilon
_{1}^{(1)}|^{2}+|\varepsilon _{2}^{(1)}|^{2}=1$.
It is obvious from the above that the amplitude of the bright solitons depends not only on the SPM and XPM parameters $a$ and $c$, but also on FWM parameters $b$ and $d$. This freedom in the system parameters can be manipulated to switch energy between two light pulses or between two bound states of a given light pulse. The gauge transformation approach [@llchaw1991] can be extended to generate multisoliton solution. For example, the two-solition solution $\psi_{1,2}^{(2)}$ for the two modes can be expressed as
$$\begin{aligned}
\psi_{1}^{(2)} = 2 I \frac{A1}{B},\\
\psi_{2}^{(2)} = 2 I \frac{A2}{B},\end{aligned}$$
\[twosolsolution\]
where $$\begin{aligned}
A1&= M_{121} M_{222} \left(\zeta_2 -\zeta_1 \right) \left(\zeta_1
-\bar{\zeta_1}\right) \left(\zeta_2-\bar{\zeta_2}\right)+M_{122}
M_{221}\left(\zeta_2-\bar{\zeta_1}\right)\left(\bar{\zeta_2}-\zeta_1\right) \left(\zeta_2-\bar{\zeta_2}\right)\notag\\
&+ M_{111} M_{122} \left(\zeta_2-\bar{\zeta_1}\right) \left(\bar{\zeta_2}-\bar{\zeta_1}\right) \left(\zeta_2-\bar{\zeta_2}\right)+M_{112} M_{121} \left(\zeta_1-\bar{\zeta_1}\right) \left(\bar{\zeta_2}-\zeta_1\right) \left(\bar{\zeta_2}-\bar{\zeta_1}\right), \notag\\
A2&= M_{112} M_{211} \left(\zeta_2-\zeta_1\right) \left(\zeta
_1-\bar{\zeta_1}\right) \left(\zeta_2-\bar{\zeta_2}\right)+M_{111}
M_{212} \left(\zeta_2-\bar{\zeta_1}\right)\left(\bar{\zeta_2}-\zeta_1\right) \left(\zeta_2-\bar{\zeta_2}\right)\notag\\
&+ M_{212} M_{221} \left(\zeta_2-\bar{\zeta_1}\right) \left(\bar{\zeta_2}-\bar{\zeta_1}\right) \left(\zeta_2-\bar{\zeta_2}\right)+M_{211} M_{222} \left(\zeta_1-\bar{\zeta_1}\right) \left(\bar{\zeta_2}-\zeta_1\right) \left(\bar{\zeta_2}-\bar{\zeta_1}\right),\notag\\
B&= \left(M_{122} M_{211}+M_{121} M_{212}\right)
\left(\zeta_1-\bar{\zeta_1}\right)\left(\zeta_2-\bar{\zeta_2}\right)+\left(M_{112}
M_{221}+M_{111}M_{222}\right) \left(\zeta_2-\bar{\zeta_1}\right)\notag\\
&\left(\bar{\zeta_2}-\zeta_1\right)
\left(M_{111}M_{112}+M_{221}M_{222}\right)\left(\zeta_2-\zeta_1\right)\left(\bar{\zeta_2}-\bar{\zeta_1}\right),\notag\end{aligned}$$ with $\zeta_2 = \bar{\zeta_2}^* = \alpha_2 + i \beta_2$, $$\begin{aligned}
M_{11j}&= e^{-\theta_j}\sqrt{2};\quad\nonumber
M_{12j}=e^{-i\xi_j}\varepsilon_1^{(j)};\quad\nonumber
M_{13j}=e^{-i\xi_j}\varepsilon_2^{(j)};\nonumber\\
M_{21j}&= e^{i\xi_j}\varepsilon_1^{*(j)};\quad\nonumber
M_{22j}=e^{\theta_j}/\sqrt{2};\quad\nonumber
M_{23j}=0;\nonumber\\
M_{31j}&= e^{i\xi_j}\varepsilon_2^{*(j)};\quad\nonumber
M_{32j}=0;\quad\nonumber M_{33j}=e^{\theta_j}/\sqrt{2},\nonumber\end{aligned}$$ where $$\begin{aligned}
\theta _{j} &= 2 \beta _{j} x - 4\int (\alpha _{j}\beta_{j})dt+2\delta_{j}, \label{thetaj}\\
\xi _{j} &= 2 \alpha _{j} x-
2\int(\alpha_{j}^{2}-\beta_{j}^{2})dt-2\chi _{j},\label{xij}\end{aligned}$$ and $j = 1, 2$
The two soliton solution given by equations.(\[twosolsolution\]-\[xij\]) can be rewritten asymptotically (i.e) at $t=\pm \infty$ in the following form [@motopaper]\
**Before collision:** $$\begin{aligned}
\psi_{(1)}^{(2-)} &= A_1^{(2-)} \varepsilon _{1}^{(1)} [\frac
{{C_1 + C_2 + C_3 + C_4}}{B_1 + B_2}] \rm{e}^{\left(i \xi_1 \right
)},\notag \\
\psi_{(2)}^{(2-)} &= A_2^{(2-)} \varepsilon _{1}^{(1)} [\frac
{{C_1 + C_2 + C_3 + C_4}}{B_1 + B_2}] \rm{e}^{\left(i \xi_1
\right)}, \notag\label{beforecolli} \\\end{aligned}$$
**After collision:** $$\begin{aligned}
\psi_{(1)}^{(2+)}&= A_1^{(2+)} \varepsilon _{1}^{(2)} [\frac {{C_1
+ C_2 + C_3 + C_4}}{B_1 + B_2}] \rm{e}^{\left(i (\xi_2-\xi_1)
\right)},\notag \\
\psi_{(2)}^{(2+)}&= A_2^{(2+)} \varepsilon _{1}^{(2)} [\frac {{C_1
+ C_2 + C_3 + C_4}}{B_1 + B_2}] \rm{e}^{\left(i (\xi_2-\xi_1)
\right)}, \label{aftercolli}\notag\\\end{aligned}$$ In the above expression, the (-) and (+) sign indicates before and after collision and the subscript and superscript depicts the component (mode) and soliton respectively. with, $$\begin{aligned}
A_1^{(2-)} &= \alpha_1 .\Big[
\frac{\zeta_1-\zeta_2^{\ast}}{\zeta_1-\zeta_2}\Big],\quad
A_2^{(2-)} = \beta_1 .\Big[
\frac{\zeta_1-\zeta_2^{\ast}}{\zeta_1-\zeta_2}\Big],\notag\\
A_1^{(2+)} &= \alpha_2.\Big[
\frac{\zeta_2-\zeta_1^{\ast}}{\zeta_2-\zeta_1}\Big],\quad
A_2^{(2+)} =\beta_2.\Big[
\frac{\zeta_2-\zeta_1^{\ast}}{\zeta_2-\zeta_1}\Big].\end{aligned}$$ where $$\begin{aligned}
C_1&= \{-2\beta _2 [(\alpha_2 - \alpha_1 )^2 - (\beta_1^2 -
\beta_2^2 )]- 4i\beta_1 \beta_2 (\alpha_2 - \alpha_1 )\}
\rm{e}^{(\theta_1^{\ast}+\theta_1 + i\xi_2 )},\nonumber\\
C_2&= - 2\beta_2 [(\alpha_2 - \alpha_1 )^2 + (\beta_1^2 +
\beta_2^2 )]\rm{e}^{(\theta_1 - \theta_1^{\ast} + i\xi_2 )},\nonumber\\
C_3&= \{- 2\beta_1 [(\alpha_2 - \alpha_1 )^2 + (\beta_1^2 -
\beta_2^2 )] + 4i\beta_1 \beta_2 (\alpha_2 - \alpha_1 )\}
\rm{e}^{(i\xi_1 + \theta_2^{\ast}+\theta_2 )}, \nonumber \\
C_4&= -4i\beta_1 \beta_2 [(\alpha_2 - \alpha_1 ) - i(\beta_1 -
\beta_2 )]\rm{e}^{(i\xi_1 +\theta_2 - \theta_2^{\ast} )},\nonumber\\
B_1&= -4\beta _1 \beta _2 [\sinh (\chi_1 )\sinh (\chi_2 ) +
\cos (\xi _1 - \xi _2)],\nonumber\\
B_2&=2\;\cosh (\chi_1 )\;\cosh (\chi_2 )\;[(\alpha _2 - \alpha _1
)^2 + (\beta _1^2 + \beta _2^2 )], \nonumber\end{aligned}$$
with $~\alpha _{j}=\alpha _{j0}(a \tau_1^2 + b \tau_1 \tau_2 + d
\tau_1 \tau_2 +c \tau_2^2)$, $\beta _{j}=\beta _{j0}(a \tau_1^2 +
b \tau_1 \tau_2 + d \tau_1 \tau_2 +c \tau_2^2)$ and the notation $\theta_j-\theta_j^{\ast}=2 i \beta_j x - 4 (\alpha_j^2-
\beta_j^2)t$ and $\theta_j^{\ast}+\theta_j= 2 \beta_j x- 4
\alpha_j \beta_j t$ where $j=1,2$.
If we choose $$\frac{\alpha_{10}}{\alpha_{20}}=\frac{\beta_{10}}{\beta_{20}},
\label{elasticasymptcondition}$$ with $\varepsilon _{1}^{(1)}=\varepsilon _{1}^{(2)}$ (or) $\varepsilon _{1}^{(1)}\neq \varepsilon _{1}^{(2)}$ keeping the condition $|\varepsilon _{1}^{(j)}|^2+|\varepsilon
_{2}^{(j)}|^2=1$, $j=1,2$, one observes elastic collision of bright solitons. Any violation of the above condition given by equation.(\[elasticasymptcondition\]) results in inelastic collision of solitons leading to the exchange of energy between two optical beams (modes).
![Inelastic collision of the solitons in the Manakov model for $a \equiv c=1$ and $b\equiv d = 0$ with $\alpha_{10}=0.1$, $\alpha_{20}=0.25$, $\beta_{10}=0.2$, $\beta_{20}=0.3$, $\chi_{1}=0.2$, $\chi_{2}=0.3$, $\delta_{1}=0.1$, $\delta_{2}=0.2$, $\varepsilon_1^{(1)}=0.5$, $\varepsilon_1^{(2)}=0.8$ []{data-label="figie1"}](Ntwosolinelastic.png)
Intramodal collision of bright solitons
---------------------------------------
We first consider the elastic collision of solitons in the Manakov model shown in figure.(\[figtwosol1\]) under the parametric choice given by equation.(\[elasticasymptcondition\]). The corresponding amplitudes of two soliton solution before and after collision can be rewritten as
$$\begin{aligned}
A_j^{(2-)} &= \left(%
\begin{array}{c}
\alpha_1 \\
\beta_1 \\
\end{array}%
\right)(a \tau_1^2 +c \tau_2^2)) \varepsilon _{1}^{(1)} .
\frac{\zeta_1-\zeta_2^{\ast}}{\zeta_1-\zeta_2},\notag\\
A_j^{(2+)} &=\left(%
\begin{array}{c}
\alpha_1 \\
\beta_1 \\
\end{array}%
\right)(a \tau_1^2 +c \tau_2^2))\varepsilon _{1}^{(2)}.
\frac{\zeta_2-\zeta_1^{\ast}}{\zeta_2-\zeta_1}, \quad
j=1,2 \notag\\\end{aligned}$$
When we introduce FWM, the amplitudes of two solitons solution asymptotically take the following form
$$\begin{aligned}
A_j^{(2-)} &= \left(%
\begin{array}{c}
\alpha_1 \\
\beta_1 \\
\end{array}%
\right)(a \tau_1^2 +(b+d)\tau_1 \tau_2 +c \tau_2^2) \varepsilon
_{1}^{(1)} .
\frac{\zeta_1-\zeta_2^{\ast}}{\zeta_1-\zeta_2},\notag\\
A_j^{(2+)} &= \left(%
\begin{array}{c}
\alpha_1 \\
\beta_1 \\
\end{array}%
\right)(a \tau_1^2 +(b+d)\tau_1 \tau_2 +c \tau_2^2)\varepsilon
_{1}^{(2)}. \frac{\zeta_2-\zeta_1^{\ast}}{\zeta_2-\zeta_1},\quad
j= 1,2\end{aligned}$$
From the above, it is obvious that the introduction of FWM parameters $b$ and $d$ contributes to the enhancement of intensities as shown in figure.\[figtwosol2\]. In addition, one observes the rotation of the trajectories of solitons.
Now, to enhance the intensity of a given bound state in a given mode (optical pulse), we manipulate the FWM parameter say $d$ and choose unequal coupling parameter $\varepsilon _{1}^{(1)}=0.8$ & $\varepsilon _{1}^{(2)}=0.5$. The corresponding density profile shown in figure.(\[figtwosol3\]) enhances the intensity of one bound state $(I_1)$ at the expense of the other $(I_2)$ in each mode (optical pulse). By manipulating the real parameter $b$ instead of $d$ and keeping $\varepsilon _{1}^{(1)}=0.5$ & $\varepsilon _{1}^{(2)}=0.8$, the density profile is shown in figure.(\[figtwosol4\]), where one observes the enhancement of $I_2$ at the expense of $I_1$ in each mode (optical beam).
![Rotation of bright solitons and switching of energy for $b=0.5$ and $d=3.5$ keeping the other parameters the same as in fig.\[figie1\]except $\varepsilon_1^{(1)}=0.8$ and $\varepsilon_1^{(2)}=0.5$[]{data-label="figie4"}](Ntwosolinelasticfirstmode.png)
![Reversal of intensity enhancement for $b=3.5$ and $d=0.5$ keeping the other parameters the same as in fig.\[figie1\][]{data-label="figie5"}](Ntwosolinelasticsecondmode.png)
Intermodal collision of bright solitons
---------------------------------------
To manipulate the intensity of a given optical beam (mode), we now begin with the celebrated inelastic collision of solitons in the Manakov model [@kanna] shown in figure.(\[figie1\]) in the absence of FWM. The asymptotic form of two solitions can be written as
$$\begin{aligned}
A_j^{(2-)} &=\left(%
\begin{array}{c}
\alpha_1 \\
\beta_1 \\
\end{array}%
\right)(a \tau_1^2 +c \tau_2^2)) \varepsilon _{1}^{(1)} .
\frac{\zeta_1-\zeta_2^{\ast}}{\zeta_1-\zeta_2},\notag\\
A_j^{(2+)} &=\left(%
\begin{array}{c}
\alpha_2 \\
\beta_2 \\
\end{array}
\right)(a \tau_1^2 +c \tau_2^2))\varepsilon _{1}^{(2)}.
\frac{\zeta_2-\zeta_1^{\ast}}{\zeta_2-\zeta_1},\notag\\\end{aligned}$$
When we introduce unequal FWM now and reverse the choice of coupling parameters (i.e.,) $\varepsilon _{1}^{(1)}=0.8$ & $\varepsilon _{1}^{(2)}=0.5$, the intensity redistribution shown in figure.(\[figie4\]) shows that one can manipulate energy in a given bound state of the beam desirably.
Interchanging the values of the FWM & coupling parameters results in the reversal of intensity distribution of the solitons as shown in fig.\[figie5\].
The above results indicate that one can not only manoeuvre the intensity distribution between the light beams, but also manipulate the intensity distribution of the given bound state in a given mode (optical pulse) and this freedom arises due to a suitable combination of FWM, SPM and XPM. It is also worth pointing out that the introduction of unequal SPM and XPM alongwith FWM in the above interaction of solitons contributes to a marginal increase of intensities besides rotating the trajectories of solitons. From the above, we also observe that the intensity redistribution among the bound states of a given optical beam (mode) or between two optical beams (modes) through manipulation of FWM parameters $b$ & $d$ is always accompanied by rotation to sustain the stability of solitons.
When we neglect SPM and XPM in equation.(\[gptwo\]) (a=c=0), we obtain the following coupled NLS equation $$\begin{aligned}
i \psi_{1t}+\psi_{1xx}+2( b \psi_1 \psi_2^{\ast} + d \psi_2 \psi_1^*)\psi_1=0, \nonumber\\
i \psi_{2t}+\psi_{2xx}+2( b \psi_1 \psi_2^{\ast} + d \psi_2
\psi_1^*)\psi_2=0. \label{gptwonew}\end{aligned}$$ It should be mentioned that equation.(\[gptwonew\]) arises only as a special case of equation.(\[gptwo\]) and the phenomenon of soliton reflection and noninteraction of solitons [@motopaper] can also be obtained as a special case. It should also be mentioned that under suitable transformation, the model governed by GCNLS equation.(\[gptwo\]) can be mapped onto its counterpart in Gross-Pitaevskii (GP) equation which means that one can switch matter wave intensities desirably and this mechanism can be employed for matter wave switching.
Discussion
==========
In this paper, we have derived a generalized coupled NLS (CGNLS) equation containing four arbitrary real parameters with two real parameters corresponding to SPM and XPM and the other two real parameters accounting for FWM. The collisional dynamics of bright solitons shows that one can have the luxury of sustaining desirable intensity in a given bound state of the optical beam or in a given optical pulse and the celebrated Manakov does not have the same freedom. It should be emphasized that the manipulation of light intensities in the above GCNLS equation (\[gptwo\]) explicitly depends on the interaction of light with the medium unlike the Manakov model where the intensity redistribution occurs by changing the parameters associated with the phase of solitons. Our investigation may open the floodgates for optical and matter wave switching in nonlinear optics and BECs. It would be interesting to study the ramifications of complex FWM parameters on the dynamics of bright solitons.
Acknowledgements
================
Authors would like to acknowledge Dr. Telman Gadzhimuradov in sharing his perspective in improving the contents of the paper. PSV wishes to thank Department of Science and Technology (DST) for the financial support. RR wishes to acknowledge the financial assistance received from DST (Ref.No:SR /S2/HEP-26/2012), UGC (Ref.No:F.No 40-420/2011(SR), Department of Atomic Energy -National Board for Higher Mathematics (DAE-NBHM) (Ref.No: NBHM / R.P.16/2014/Fresh dated 22.10.2014) and Council of Scientific and Industrial Research (CSIR) (Ref.No: No.03(1323)/14/EMR-II dated 03.11.2014) dated 4.July.2011) for the financial support in the form Major Research Projects. KP thanks the DST, NBHM, IFCPAR, DST-FCT and CSIR, Government of India, for the financial support through major projects.
[30]{}
Hasegawa A, Tappert F. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett 1973; 23: 142.
Hasegawa A, Tappert F. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion. Appl. Phys. Lett 1973; 23: 171.
Agrawal G P. Nonlinear Fiber Optics. San Diego: Academic Press; 2001.
Hasegawa A, Kodama Y. Solitons in Optical Communication. New York: Oxford University Press; 1995.
Sahadevan R, Tamizhmani K M, Lakshmanan M. Painleve analysis and integrability of coupled non-linear Schrodinger equations. J. Phys. A: Math.Gen 1986; 19: 1783.
Manakov S V. On the theory of two dimensional stationary self-focusing of electromagnetic waves. Zh. Eksp. Teor. Fiz 1973; 65: 505-516.
Kaup D J, Malomed B A. Soliton trapping and daughter waves in the Manakov model. Phys. Rev. A 1993; 48: 599.
Makhankov V G, Makhaldiani N V, Pashaev O K. On the integrability and isotopic structure of the one-dimensional Hubbard model in the long wave approximation. Phys. Lett.A 1981; 81: 161.
Tsoy E N, Akhmediev N. Dynamics and interaction of pulses in the modified Manakov model. Opt. Commun. 2006; 266: 660.
Radhakrishnan R, Lakshmanan M, Hietarinta J. Inelastic collision and switching of coupled bright solitons in optical fibers. Phys. Rev. E 1997; 56: 2213.
Radha R, Vinayagam P S, Porsezian K. Rotation of the trajectories of bright solitons and realignment of intensity distribution in the coupled nonlinear schrödinger equation. Phys. Rev. E 2013; 88: 032903.
Malomed B A. Soliton Management in Periodic Systems. New York: Springer; 2006.
Radha R, Vinayagam P S. An analytical window into the world of ultracold atoms. Rom. Rep. Phys. 2015; 67 (1): 89.
Park Q-H, Shin H J. Painlevé analysis of the coupled nonlinear Schrödinger equation for polarized optical waves in an isotropic medium. Phys. Rev. E 1999; 59: 2737.
Wang D-S, Zhang D-J, Yang J. Integrable properties of the general coupled nonlinear Schrödinger equations. J. Math. Phys 2010; 51: 023510.
Agalarov A, Zhulego V, Gadzhimuradov T. Bright, dark, and mixed vector soliton solutions of the general coupled nonlinear Schr$\rm\ddot{o}$dinger equations. Phys. Rev. E 2015; 91: 042909.
Jafari H, Sooraki A, Talebi Y, Biswas A. The first integral method and traveling wave solutions to Davey–Stewartson equation. Nonlinear Analysis: Modelling and Control. 2012; 17 (2):182.
Jafari H, Kadem A, Baleanu D, Yilmaz T. Solution of the fractional Davey- Stewartson equations with variational iteration method. Rom. Rep Physics. 2012; 64 (2): 337.
Jafari H, Tajadodi H, Bolandtalat A, Johnston S J. A Decomposition Method for Solving the Fractional Davey-Stewartson Equations. Int. J. Appl. Comput. Math. (2015); 1: 559.
Chau L -L , Shaw J C, Yen H C. An alternative explicit construction of N-soliton solutions in 1+1 dimensions. J. Math. Phys. 1991; 32: 1737.
Malomed B A. Inelastic collisions of polarized solitons in a birefringent optical fiber. J. Opt. Soc. Am. B 1992; 9: 2075.
|
---
author:
- Jennifer Paykin Steve Zdancewic
bibliography:
- 'linear.bib'
title: 'A Linear/Producer/Consumer Model of Classical Linear Logic'
---
Introduction {#sec:intro}
============
$ \textsc{LPC} $ Logic {#sec:logic}
======================
Categorical Model {#sec:category}
=================
Examples {#sec:examples}
========
Related work {#sec:related}
============
|
---
abstract: 'After a discussion of the Griesmer and Heller bound for the distance of a convolutional code we present several codes with various parameters, over various fields, and meeting the given distance bounds. Moreover, the Griesmer bound is used for deriving a lower bound for the field size of an MDS convolutional code and examples are presented showing that, in most cases, the lower bound is tight. Most of the examples in this paper are cyclic convolutional codes in a generalized sense as it has been introduced in the seventies. A brief introduction to this promising type of cyclicity is given at the end of the paper in order to make the examples more transparent.'
author:
- 'Heide Gluesing-Luerssen$^*$ and Wiland Schmale[^1]'
bibliography:
- 'literatureAK.bib'
- 'literatureLZ.bib'
title: Distance Bounds for Convolutional Codes and Some Optimal Codes
---
[**Keywords:**]{} Convolutional coding theory, distance bounds, cyclic convolutional codes.
[**MSC (2000):**]{} 94B10, 94B15, 16S36
Introduction
============
The fundamental task of coding theory is the construction of good codes, that is, codes having a large distance and a fast decoding algorithm. This task applies equally well to block codes and [convolutional code]{}s. Yet, the state of the art is totally different for these two classes of codes. The mathematical theory of block codes is highly developed and has produced many sophisticated classes of codes, some of which, like BCH-codes, also come with an efficient decoding algorithm. On the other hand, the mathematical theory of [convolutional code]{}s is still in the beginnings. Engineers make use of these codes since decades, but all [convolutional code]{}s used in practice have been found by systematic computer search and their distances have been found by computer as well, see for instance [@La73] and [@JoZi99 Sec. 8] for codes having the largest distance among all codes with the same parameters. Moreover, in all practical situations decoding of [convolutional code]{}s is done by search algorithms, for instance the Viterbi algorithm or one of the sequential decoding algorithms, e. g. the stack algorithm. It depends on the algorithm how complex a code may be without exceeding the range of the decoding algorithms. However, the important fact about the theory of [convolutional code]{}s is that so far no specific codes are known that allow an [*algebraic decoding*]{} (in the present paper a decoding algorithm will be called algebraic if it is capable to exploit the specific structure of the given code in order to avoid a full search).
Since the seventies quite some effort has been made in order to find algebraic constructions of [convolutional code]{}s that guarantee a large (free) distance [@Ju73; @MCJ73; @Ju75; @SGR01; @GRS03]. The drawbacks of all these constructions are that, firstly, the field size has to be adapted and in general becomes quite large and, secondly, so far no algebraic decoding for these codes is known. A main feature of most of these constructions is that they make use of cyclic block codes in order to derive the desired [convolutional code]{}.
Parallel to these considerations there was an independent investigation of [convolutional code]{}s that have a cyclic structure themselves, which also began in the seventies [@Pi76; @Ro79; @GSS02; @GS02]. It was the goal of these papers to see whether this additional structure has, just like for block codes, some benefit for the error-correcting capability of the code. The first and very important observation of the seventies was the fact that a [convolutional code]{} which is cyclic in the usual sense is a block code. This negative insight has led to a more complex notion of cyclicity for [convolutional code]{}s. The algebraic analysis of these codes has been completed only recently in [@GS02] and yields a nice, yet nontrivial, generalization of the algebraic situation for cyclic block codes. Furthermore, by now plenty of optimal [cyclic convolutional code]{}s have been found in the sense that their (free) distance reaches the Griesmer bound. To the best of our knowledge it was, for most cases of the parameters, not known before whether such optimal codes existed. Many of these codes are over small fields (like the binary field) and are therefore well-suited for the existing decoding algorithms. Along with the algebraic theory of [@GS02] all this indicates that this notion of cyclicity is not only the appropriate one for [convolutional code]{}s but also a very promising one. Yet, the theory of these codes is still in the beginnings. So far, no theoretical results concerning the distance of such a code or its decoding properties are known. But we are convinced that this class of codes deserves further investigation and that the theory developed so far will be a good basis for the next steps.
It is the aim of this paper to present many of these examples in order to introduce the class of [cyclic convolutional code]{}s to the convolutional coding community. The examples are presented via a generator matrix so that no knowledge about cyclicity for [convolutional code]{}s is required from the reader. The (free) distances of all these codes have been obtained by a computer program. A detailed discussion of various distance bounds for convolutional codes over arbitrary fields shows that all the given codes are optimal with respect to their distance. It is beyond the scope of this paper to acquaint the reader with the theory of [cyclic convolutional code]{}s. However, in Section \[S-CCC\] we will give a very brief introduction into this subject so that the reader may see how the examples have been constructed. The details of the theory can be found in [@GS02].
The outline of the paper is as follows. After reviewing the main notions of convolutional coding theory in the next section we will discuss in Section \[S-Bounds\] various bounds for the free distance of a [convolutional code]{}, the Griesmer bound, the Heller bound and the generalized Singleton bound. The first two bounds are well-known for binary convolutional codes and can straightforwardly be generalized to codes over arbitrary fields. It is also shown that for all sets of parameters the Griesmer bound is at least as good as the Heller bound. The generalized Singleton bound is an upper bound for the free distance of a code of given length, dimension, and complexity, but over an arbitrary field. Just like for block codes a code reaching this bound is called an MDS code [@RoSm99]. The Griesmer bound is used for showing how large the field size has to be in order to allow for an MDS code. In Section \[S-Exa\] many examples of codes are presented reaching the respective bound. Most of these examples are [cyclic convolutional code]{}s, but we also include some other codes with the purpose to exhibit certain features of convolutional codes. For instance, we give examples of MDS codes showing that the lower bounds for the field size as derived in Section \[S-Bounds\] are tight. Furthermore, an example is given showing that a code reaching the Griesmer bound may have extreme Forney indices, a phenomenon that does not occur for MDS codes. The paper concludes with a brief account of cyclicity for [convolutional code]{}s.
Preliminaries
=============
We will make use of the following notation. The symbol ${{\mathbb F}}$ stands for any finite field while ${{\mathbb F}}_q$ always denotes a field with $q$ elements. The ring of polynomials and the field of formal Laurent series over ${{\mathbb F}}$ are given by $${{\mathbb F}}[z]=\Big\{\sum_{j=0}^Nf_jz^j\,\Big|\,N\in{{\mathbb N}}_0,\,f_j\in{{\mathbb F}}\Big\}
\ \text{ and }\
{\mbox{${{\mathbb F}}(\!(z)\!)$}}=\Big\{\sum_{j=l}^{\infty}f_jz^j\,\Big|\,l\in{{\mathbb Z}},\,f_j\in{{\mathbb F}}\Big\}.$$
The following definition of a convolutional code is standard.
\[D-CC\] Let ${{\mathbb F}}={{\mathbb F}}_q$ be a field with $q$ elements. An -[*convolutional code*]{} is a $k$-dimensional subspace ${{\mathcal C}}$ of the vector space ${\mbox{${{\mathbb F}}(\!(z)\!)$}}^n$ of the form $${{\mathcal C}}={\mbox{\rm im}\,}G:=\big\{uG\,\big|\, u\in{\mbox{${{\mathbb F}}(\!(z)\!)$}}^k\big\}$$ where $G\in{{\mathbb F}}[z]^{k\times n}$ satisfies
$G$ is [*right invertible*]{}, i. e. there exists some matrix $\tilde{G}\in{{\mathbb F}}[z]^{n\times k}$ such that $G\tilde{G}=I_k$.
$\delta=\max\{\deg\gamma\mid \gamma$ is a $k$-minor of $G\}$.
We call $G$ a [*generator matrix*]{} and $\delta$ the [*complexity*]{} of the code ${{\mathcal C}}$.
The complexity is also known as the [*overall constraint length*]{} [@JoZi99 p. 55] or the [*degree*]{} [@McE98 Def. 3.5] of the code. Notice that a generator matrix is always polynomial and has a polynomial right inverse. This implies that in the situation of Definition \[D-CC\] the polynomial codewords belong to polynomial messages, i. e. $$\label{e-cpoly}
{{\mathcal C}}\cap{{\mathbb F}}[z]^n=\big\{uG\,\big|\,u\in{{\mathbb F}}[z]^k\big\}.$$ In other words, the generator matrix is delay-free and non-catastrophic. As a consequence, a convolutional code is always uniquely determined by its polynomial part. Precisely, if ${{\mathcal C}}={\mbox{\rm im}\,}G$ and ${{\mathcal C}}'={\mbox{\rm im}\,}G'$ where $G,\,G'\in{{\mathbb F}}[z]^{k\times n}$ are right invertible, then $$\label{e-cpolyunique}
{{\mathcal C}}={{\mathcal C}}'\Longleftrightarrow {{\mathcal C}}\cap{{\mathbb F}}[z]^n={{\mathcal C}}'\cap{{\mathbb F}}[z]^n.$$ This follows from[ ]{} and the fact that $\{uG\mid u\in{{\mathbb F}}[z]^k\}=\{uG'\mid u\in{{\mathbb F}}[z]^k\}$ is equivalent to $G'=VG$ for some matrix $V\in{{\mathbb F}}[z]^{k\times k}$ that is invertible over ${{\mathbb F}}[z]$. This also shows that the complexity of a code does not depend on the choice of the generator matrix. From all this it should have become clear that with respect to code construction there is no difference whether one works in the context of infinite message and codeword sequences (Laurent series) or finite ones (polynomials) as long as one considers right invertible generator matrices. Only for decoding it becomes important whether or not one may assume the sent codeword to be finite. The issue whether convolutional coding theory should be based on finite or infinite message sequences, has first been raised and discussed in detail in [@RSY96; @Ro01].
It is well-known [@Fo70 Thm. 5] or [@Fo75 p. 495] that each convolutional code has a minimal generator matrix in the sense of the next definition. In the same paper [@Fo75 Sec. 4] it has been shown how to derive such a matrix from a given generator matrix in a constructive way.
For $v=\sum_{j=0}^N v_jz^j\in{{\mathbb F}}[z]^n$ where $v_j\in{{\mathbb F}}^n$ and $v_N\not=0$ let $\deg v:=N$ be the [*degree*]{} of $v$. Moreover, put $\deg0=-\infty$.
Let $G\in{{\mathbb F}}[z]^{k\times n}$ be a right invertible matrix with complexity $\delta=\max\{\deg\gamma\mid \gamma$ is a $k$-minor of $G\}$ and let $\nu_1,\ldots,\nu_k$ be the degrees of the rows of $G$ in the sense of (1). We say that $G$ is [*minimal*]{} if $\delta=\sum_{i=1}^k\nu_i$. In this case, the row degrees of $G$ are uniquely determined by the code ${{\mathcal C}}:={\mbox{\rm im}\,}G\subseteq{\mbox{${{\mathbb F}}(\!(z)\!)$}}^n$. They are called the [*Forney indices*]{} of ${{\mathcal C}}$ and the number $\max\{\nu_1,\ldots,\nu_k\}$ is said to be the [*memory*]{} of the code. An -code with memory $m$ is also called an -code.
From the above it follows that an -convolutional code has a constant generator matrix if and only if $\delta=0$. In that case the code can be regarded as an $(n,k)_q$-block code.
The definition of the distance of a convolutional code is straightforward. For a constant vector $w=(w_1,\ldots,w_n)\in{{\mathbb F}}^n$ we define its [*(Hamming) weight*]{} as ${\mbox{\rm wt}}(w)=\#\{i\mid w_i\not=0\}$. For a polynomial vector $v=\sum_{j=0}^N v_j z^j\in{{\mathbb F}}[z]^n$, where $v_j\in{{\mathbb F}}^n$, the [*weight*]{} is defined as ${\mbox{\rm wt}}(v)=\sum_{j=0}^N{\mbox{\rm wt}}(v_j)$. Then the [*(free) distance*]{} of a code ${{\mathcal C}}\subseteq{\mbox{${{\mathbb F}}(\!(z)\!)$}}^n$ with generator matrix $G\in{{\mathbb F}}[z]^{k\times n}$ is given as $${\mbox{\rm dist}}({{\mathcal C}}):=\min\big\{{\mbox{\rm wt}}(v)\,\big|\, v\in{{\mathcal C}}\cap{{\mathbb F}}[z]^n,\;v\not=0\big\}.$$ By virtue of[ ]{} this can be rephrased as ${\mbox{\rm dist}}({{\mathcal C}})=\min\{{\mbox{\rm wt}}(uG)\mid u\in{{\mathbb F}}[z]^k,\,u\not=0\}$.
When presenting some optimal codes in Section \[S-Exa\] we will also investigate the column distances of the codes. For each $l\in{{\mathbb N}}_0$ the $l$th [*column distance*]{} of ${{\mathcal C}}$ is defined as $$\label{e-coldist}
d^c_l=\min\Big\{{\mbox{\rm wt}}\big((uG)_{[0,l]}\big)\,\Big|\,
u\in{{\mathbb F}}[z]^k,u_0\not=0\Big\}$$ where for a polynomial vector $v=\sum_{j=0}^N v_jz^j$ we define $v_{[0,l]}=\sum_{j=0}^{\min\{N,l\}} v_jz^j$. It can easily be shown [@JoZi99 Thm. 3.4] that for each code ${{\mathcal C}}$ there exists some $M\in{{\mathbb N}}_0$ such that $$\label{e-coldistfree}
d^c_0\leq d^c_1\leq d^c_2\ldots\leq
d^c_M=d^c_{M+1}=\ldots={\mbox{\rm dist}}({{\mathcal C}}).$$
Distance Bounds {#S-Bounds}
===============
In this section we want to present some upper bounds for the distance of a convolutional code. These bounds are quite standard for binary convolutional codes and can be found in Chapter 3.5 of the book [@JoZi99]. The proof for arbitrary fields goes along the same lines of arguments, but for sake of completeness we wish to repeat the arguments in this paper. We will also compare the numerical values of the bounds with each other.
Let us begin with recalling various distance bounds for block codes. The Plotkin bound as given below can be found in [@Be98 1.4.3], but can also easily be derived from the more familiar formula $$\label{e-Plot}
\text{ if }d>\theta n\text{ where }\theta=\frac{q-1}{q},
\text{ then }
q^k\leq\frac{d}{d-\theta n},$$ see for instance [@vLi99 (5.2.4)]. As for the Singleton and the Griesmer bound we also refer to [@vLi99 Ch. 5.2].
\[T-BCB\] Let ${{\mathcal C}}\subseteq{{\mathbb F}}^n$ be an $(n,k)_q$-block code and let $d={\mbox{\rm dist}}({{\mathcal C}})$. Then $$\begin{array}{ll}
{{\displaystyle}d\leq n-k+1} &\quad \text{(Singleton bound),}\\[1.7ex]
{{\displaystyle}d\leq\Big\lfloor\frac{nq^{k-1}(q-1)}{q^k-1}\Big\rfloor}&\quad
\text{(Plotkin bound),}\\[1.7ex]
{{\displaystyle}\sum_{l=0}^{k-1}\Big\lceil\frac{d}{q^l}\Big\rceil\leq n}&\quad
\text{(Griesmer bound).}
\end{array}$$ An $(n,k)_q$-code ${{\mathcal C}}$ with ${\mbox{\rm dist}}({{\mathcal C}})=n-k+1$ is called an MDS code.
Notice that the Singleton bound does not take the field size into account. As a consequence the question arises as to how large the field size $q$ has to be in order to allow the existence of MDS codes and how to construct such codes. Answers in this direction can be found in [@MS77 Ch. 11].
It is certainly well-known that the Griesmer bound is at least as good as the Plotkin bound. The importance of the Plotkin bound, however, is that it also applies to nonlinear block codes, in which case it is usually given as in[ ]{} and with $M:=|{{\mathcal C}}|$ instead of $q^k$. Since we did not find a comparison of the two bounds for linear block codes in the literature we wish to present a short proof of this statement. We also include the relation between the Griesmer and the Singleton bound.
\[P-GP\] Given the parameters $n,\,k,\,d$, and $q\in{{\mathbb N}}$ where $k<n$ and $q$ is a prime power. Assume $\sum_{l=0}^{k-1}\Big\lceil\frac{d}{q^l}\Big\rceil\leq n$. Then
${{\displaystyle}d\leq\Big\lfloor\frac{nq^{k-1}(q-1)}{q^k-1}\Big\rfloor}$,
$d\leq n-k+1$.
There is no relation between the Plotkin and the Singleton bound in this generality. Roughly speaking, for relatively large values of $q$ the Singleton bound is better than the Plotkin bound while for small values the Plotkin bound is better.
\(a) Assume to the contrary that $d>\big\lfloor\frac{nq^{k-1}(q-1)}{q^k-1}\big\rfloor$. Since $d$ is an integer this implies that $d>\frac{nq^{k-1}(q-1)}{q^k-1}$. Thus $$\sum_{l=0}^{k-1}\Big\lceil\frac{d}{q^l}\Big\rceil
\geq\sum_{l=0}^{k-1}\frac{d}{q^l}
>\sum_{l=0}^{k-1}\frac{n(q-1)}{q^k-1}q^{k-1-l}
=\frac{n(q-1)}{q^k-1}\sum_{l=0}^{k-1}q^l=n.$$ (b) follows from $\sum_{l=0}^{k-1}\big\lceil\frac{d}{q^l}\big\rceil\geq d+k-1$.
One should also recall that the Griesmer bound is not tight. An example is given by the parameters $n=13,\,k=6,\,q=2$ in which case the Griesmer bound shows that the distance is upper bounded by $5$. But it is known that no $(13,6)_2$-code with distance $5$ exists, see [@vLi99 p. 69].
We will now present the generalization of these bounds to convolutional codes. Let us begin with the Singleton bound. The following result has been proven in [@RoSm99 Thm. 2.2].
\[T-MDSC\] Let ${{\mathcal C}}\subseteq{\mbox{${{\mathbb F}}(\!(z)\!)$}}^n$ be an -code. Then
The distance of ${{\mathcal C}}$ satisfies $${\mbox{\rm dist}}({{\mathcal C}})\leq(n-k)\Big(\Big\lfloor\frac{\delta}{k}\Big\rfloor+1\Big)
+\delta+1=:S{\mbox{$(n,k,\delta)$}}.$$ The number $S{\mbox{$(n,k,\delta)$}}$ is called the generalized Singleton bound for the parameters and we call the code ${{\mathcal C}}$ an MDS code if ${\mbox{\rm dist}}({{\mathcal C}})=S{\mbox{$(n,k,\delta)$}}$.
If ${{\mathcal C}}$ is an MDS code and $\delta=ak+r$ where $a\in{{\mathbb N}}_0$ and $0\leq r\leq k-1$, then the Forney indices of ${{\mathcal C}}$ are given by $$\underbrace{a,\ldots,a}_{k-r\text{ times}},
\underbrace{a+1,\ldots,a+1}_{r\text{ times}}.$$ Hence the code is compact in the sense of [@McE98 Cor. 4.3].
Just like for block codes the acronym MDS stands for maximum distance separable. In [@RoSm99 Thm. 2.10] it has been shown that for all given parameters $n,\,k,\,\delta$ and all primes $p$ there exists an MDS code over a suitably large field of characteristic $p$. The proof is non-constructive and, as a consequence, does not give a hint about the field size required. In [@SGR01 Thm. 3.3] a construction of -MDS codes over fields ${{\mathbb F}}_{p^r}$ is given under the condition that $n|(p^r-1)$ and $p^r\geq\frac{n\delta^2}{k(n-k)}$. Notice that this requires $n$ and the characteristic $p$ being coprime. This result gives first information about the field size required in order to guarantee the existence of an MDS code. However, many examples of MDS codes over smaller fields are known. We will present some of them in the next section. Although they all have a certain structure in common (they are cyclic in the sense of Section \[S-CCC\]) we do not know any general construction for cyclic MDS codes yet.
Now we proceed with a generalization of the Plotkin and Griesmer bound to convolutional codes.
\[T-CCB\] Let ${{\mathcal C}}$ be an -convolutional code having distance ${\mbox{\rm dist}}({{\mathcal C}})=d$. Moreover, let $$\hat{{{\mathbb N}}}=\left\{\begin{array}{ll}{{\mathbb N}}:=\{1,2,\ldots\},&\text{if }km=\delta\\[.6ex]
{{\mathbb N}}_0:=\{0,1,2,\ldots\},&\text{if }km>\delta\end{array}\right.$$ Then $$\begin{array}{ll}
d&\leq{{\displaystyle}\min_{i\in\hat{{{\mathbb N}}}}
\Big\lfloor\frac{n(m+i)q^{k(m+i)-\delta-1}(q-1)}{q^{k(m+i)-\delta}-1}\Big\rfloor}=:H_q(n,k,\delta;m)
\hfill \text{(Heller bound)}\\[3ex]
d&\leq{{\displaystyle}\max\Big\{d'\in\{1,\ldots,S{\mbox{$(n,k,\delta)$}}\}\,\Big|\,\sum_{l=0}^{k(m+i)-\delta-1}
\Big\lceil\frac{d'}{q^l}\Big\rceil\leq n(m+i)
\text{ for all }i\in\hat{{{\mathbb N}}}\Big\}}\\
&=:G_q(n,k,\delta;m)\hfill \text{(Griesmer bound)}
\end{array}$$ Moreover, $G_q(n,k,\delta;m)\leq H_q(n,k,\delta;m)$.
In the binary case ($q=2$) both bounds can be found in [@JoZi99 3.17 and 3.22]. In that version the first bound has been proven first by Heller in [@He68]. The Griesmer bound as given above differs slightly from the one given at [@JoZi99 3.22]. We have upper bounded the possible values for $d'$ by the generalized Singleton bound, which is certainly reasonable to do. As a consequence, the Griesmer bound is always less than or equal to the generalized Singleton bound. This would not have been the case had we taken the maximum over all $d'\in{{\mathbb N}}$. This can be seen by taking the parameters ${\mbox{$(n,k,\delta;m)_q$}}=(5,2,3;3)_8$. In this case the generalized Singleton bound is $S{\mbox{$(n,k,\delta)$}}=10$ but the inequalities of the Griesmer bound are all satisfied for the value $d'=12$.
The proof of the inequalities above is based on the same idea as in the binary case as we will show now.
The last statement follows from Proposition \[P-GP\](a). As for the bounds themselves we will see that they are based on certain block codes which appear as subsets of the given convolutional code ${{\mathcal C}}$. This will make it possible to apply the block code bounds of Theorem \[T-BCB\]. The subcodes to be considered are simply the subsets of all codewords corresponding to polynomial messages with an upper bounded degree.\
Let ${{\mathcal C}}={\mbox{\rm im}\,}G$, where $G\in{{\mathbb F}}[z]^{k\times n}$ is right-invertible and minimal with Forney indices $\nu_1,\ldots,\nu_k$. Hence $\delta=\sum_{i=1}^k \nu_i$ and $m=\max\{\nu_1,\ldots,\nu_k\}$. Notice that $km\geq\delta$ and $km=\delta\Longleftrightarrow \nu_1=\ldots=\nu_k=m$. For each $i\in{{\mathbb N}}_0$ define $$U_i=\{(u_1,\ldots,u_k)\in{{\mathbb F}}[z]^k\mid \deg u_l\leq m+i-1-\nu_l\text{ for }
l=1,\ldots,k\}.$$ This implies $u_l=0$ if $\nu_l=m$ and $i=0$. In particular, $U_i=\{0\}\Longleftrightarrow km=\delta\text{ and }i=0$ and this shows that $i=0$ has to be excluded if $km=\delta$. Obviously, the set $U_i$ is an ${{\mathbb F}}$-vector space and $\dim_{{\mathbb F}} U_i=\sum_{l=1}^k(m+i-\nu_l)=k(m+i)-\delta$. Consider now ${{\mathcal C}}_i:=\{uG\mid u\in U_i\}$ for $i\in{{\mathbb N}}_0$. Then ${{\mathcal C}}_i\subseteq{{\mathcal C}}$ and ${{\mathcal C}}_i$ is an ${{\mathbb F}}$-vector space and, by injectivity of $G$, $$\dim_{\mathbb F}{{\mathcal C}}_i=\dim_{{\mathbb F}} U_i=k(m+i)-\delta.$$ Furthermore, minimality of the generator matrix $G$ tells us that $$\deg(uG)=\max_{l=1,\ldots,k}(\deg u_l+\nu_l)\leq m+i-1
\text{ for all }u\in U_i,$$ see [@Fo75 p. 495]. Hence ${{\mathcal C}}_i$ can be regarded as a block code of length $n(m+i)$ and dimension $k(m+i)-\delta$ for all $i\in\hat{{{\mathbb N}}}$. Since ${\mbox{\rm dist}}({{\mathcal C}})\leq{\mbox{\rm dist}}({{\mathcal C}}_i)$ for all $i\in\hat{{{\mathbb N}}}$ we obtain the desired results by applying the Plotkin and Griesmer bounds of Theorem \[T-BCB\] to the codes ${{\mathcal C}}_i$.
The proof shows that the existence of an -code meeting the Griesmer bound implies the existence of $(n(m+i),k(m+i)-\delta)_q$-block codes having at least the same distance for all $i\in\hat{{{\mathbb N}}}$. The converse, however, is not true, since the block codes have to have some additional structure. We will come back to this at the end of this section.
One should note that these bounds do only take the largest Forney index, the memory, into account. More precisely, the proof shows that codewords having degree smaller than $m-1$ are never taken into consideration. As a consequence, codes with a rather bad distribution of the Forney indices will never attain the bound. For instance, for a code with parameters $(n,k,\delta;m)_q=(5,3,4;2)_2$ the Griesmer bound shows that the distance is upper bounded by $6$. This can certainly never be attained if the Forney indices of that code are given by $0,2,2$ since in that case a constant codeword exists. Hence the Forney indices have to be $1,1,2$. In this case a code with distance $6$ does indeed exist, see the first code given in Table I of Section \[S-Exa\]. But also note that, on the other hand, a code reaching the Griesmer bound need not be compact (see Theorem \[T-MDSC\](b)); an example is given by the $(5,2,6;4)_2$-code given in Table I of the next section.
The Griesmer bound as given above has the disadvantage that infinitely many inequalities have to be considered. A simple way to reduce this to finitely many inequalities is obtained by making use of the generalized Singleton bound $S{\mbox{$(n,k,\delta)$}}$. Instead of this bound one could equally well use any of the numbers occurring on the right hand side of the Heller bound.
\[P-Gfinite\] Given the parameters $n,\,k,\,m,\,\delta$ such that $k<n$ and $km\geq\delta$ and let $q$ be any prime power. Define the set $\hat{{{\mathbb N}}}$ as in Theorem \[T-CCB\]. Furthermore, let $i_0\in{{\mathbb N}}$ be such that $q^{k(m+i_0)-\delta}\geq S{\mbox{$(n,k,\delta)$}}$ and put $\hat{{{\mathbb N}}}_{\leq i_0}:=\hat{{{\mathbb N}}}\cap\{0,1,\ldots,i_0\}$. Then $$\label{e-Gfinite}
G_q(n,k,\delta;m)\!=\!\max\Big\{d'\in\{1,\ldots, S{\mbox{$(n,k,\delta)$}}\}\,\Big|\!
\sum_{l=0}^{k(m+i)-\delta-1}\!\Big\lceil\frac{d'}{q^l}\Big\rceil
\leq n(m+i)\text{ for all }i\in\hat{{{\mathbb N}}}_{\leq i_0}\Big\}.$$ Hence the distance of an ${\mbox{$(n,k,\delta;m)_q$}}$-code is upper bounded by the number given in[ ]{}.
We will see in the next section that the Griesmer bound is tight for many sets of parameters.
[Proof:]{} Notice that for $a\geq S{\mbox{$(n,k,\delta)$}}$ we have $\big\lceil\frac{d'}{a}\big\rceil=1$ since $d'\leq S{\mbox{$(n,k,\delta)$}}$. As for[ ]{} it suffices to show that whenever $d'$ satisfies the inequality $\sum_{l=0}^{k(m+i)-\delta-1}\big\lceil\frac{d'}{q^l}\big\rceil\leq n(m+i)$ for some $i\geq i_0$, then it also satisfies the inequality for $i+1$. But this follows easily from $$\sum_{l=0}^{k(m+i+1)-\delta-1}\Big\lceil\frac{d'}{q^l}\Big\rceil
=\sum_{l=0}^{k(m+i)-\delta-1}\Big\lceil\frac{d'}{q^l}\Big\rceil
+\sum_{l=k(m+i)-\delta}^{k(m+i+1)\delta-1}\Big\lceil\frac{d'}{q^l}\Big\rceil
\leq n(m+i)+k\leq n(m+i+1).
\eqno\Box$$
The finite sets for $d'$ and $i$ in[ ]{} are not optimized, but they are good enough for our purposes since they allow for a computation of the Griesmer bound in finitely many steps. Unfortunately,[ ]{} does not reveal the block code case where only the index $i=1$ has to be considered according to Theorem \[T-BCB\]. The consistency of the Griesmer bound for $m=\delta=0$ with that case is guaranteed by the following result.
\[P-Gblock\] Given the parameters $n,\,k$, and $q$. Then $$\max\Big\{d'\in{{\mathbb N}}\,
\Big|\,\sum_{l=0}^{ki-1}\Big\lceil\frac{d'}{q^l}\Big\rceil
\leq ni\text{ for all }i\in{{\mathbb N}}\Big\}
=\max\Big\{d'\in{{\mathbb N}}\,\Big|\,
\sum_{l=0}^{k-1}\Big\lceil\frac{d'}{q^l}\Big\rceil\leq n\Big\}.$$
Let $d'$ be any number satisfying $\sum_{l=0}^{k-1}\big\lceil\frac{d'}{q^l}\big\rceil\leq n$. We have to show that $d'$ satisfies the inequalities given on the left hand side for all $i\in{{\mathbb N}}$. In order to do so, notice that according to Proposition \[P-GP\](a) $$d'\leq\Big\lfloor\frac{nq^{k-1}(q-1)}{q^k-1}\Big\rfloor
\leq\frac{nq^{k-1}}{1+q+\ldots+q^{k-1}}\leq\frac{n}{k}q^{k-1}.$$ But this implies $\frac{d'}{q^l}<\frac{n}{k}$ for all $l\geq k$, thus $\big\lceil\frac{d'}{q^l}\big\rceil\leq\frac{n}{k}$ and $$\sum_{l=0}^{ki-1}\Big\lceil\frac{d'}{q^l}\Big\rceil
=\sum_{l=0}^{k-1}\Big\lceil\frac{d'}{q^l}\Big\rceil
+\sum_{l=k}^{ki-1}\Big\lceil\frac{d'}{q^l}\Big\rceil
\leq n +k(i-1)\frac{n}{k}
=ni.$$ This proves the assertion.
Finally we want to investigate as to how big the field size $q$ has to be in order to allow for an MDS code with parameters ${\mbox{$(n,k,\delta)_q$}}$. A first estimate can be achieved by using the Griesmer bound in combination with the generalized Singleton bound.
\[T-MDSfieldsize\] Let ${{\mathcal C}}\subseteq{\mbox{${{\mathbb F}}(\!(z)\!)$}}^n$ be an ${\mbox{$(n,k,\delta;m)_q$}}$-MDS code, thus $d:={\mbox{\rm dist}}({{\mathcal C}})=S{\mbox{$(n,k,\delta)$}}=(n-k)\big(\big\lfloor\frac{\delta}{k}\big\rfloor+1\big)+\delta+1$. Then the field size $q$ satisfies $$q\geq\left\{\begin{array}{cl}
\frac{d}{n-k+1},&\text{if }[k=1] \text{ or }[k>1\text{ and }km=\delta+1]
\\[1ex]
d,&\text{if }[k>1 \text{ and }km\not=\delta+1].
\end{array}\right.$$
The estimate above also covers the block code case as given in [@MS77 p. 321].
We will consider the various cases separately. In each case we will apply the inequality $$\label{e-GriesMDS}
\frac{d}{q}\leq
n(m+i)-d-\sum_{l=2}^{k(m+i)-\delta-1}\Big\lceil\frac{d}{q^l}\Big\rceil,$$ which is a simple consequence of the Griesmer bound, to the case $d=S{\mbox{$(n,k,\delta)$}}$. Moreover we will make use of the fact that $\big\lceil\frac{d}{q^l}\big\rceil\geq1$ for all $l\in{{\mathbb N}}$.
In this case $m=\delta$ and $d=n(m+1)$. Since $k(m+i)-\delta-1=i-1$ Inequality[ ]{} gives us $$\frac{d}{q}
\leq n(m+i)-n(m+1)-(i-2)
=n(i-1)-i+2$$ for all $i\geq2$. This shows $q\geq\frac{d}{n}$ as desired. Using $i=1$ in the Griesmer bound simply leads to $d\leq n(m+1)$. This is true by assumption and gives no further condition on $q$.\
Now $m=\frac{\delta}{k}$ and thus $d=(n-k)(m+1)+mk+1$. Using $k(m+i)-\delta-1=ki-1$ we obtain from Inequality[ ]{} $$\frac{d}{q}
\leq n(m+i)-(n-k)(m+1)-mk-1-(ki-2)
=(n-k)(i-1)+1$$ for all $i\geq1$. Using $i=1$ leads to $q\geq d$.\
In this case $m=\big\lfloor\frac{\delta}{k}\big\rfloor+1$, see Theorem \[T-MDSC\](b), and $d=(n-k)m+\delta+1$. Therefore Inequality[ ]{} leads to $$\frac{d}{q}\leq
n(m+i)-(n-k)m-\delta-1-\big(k(m+i)-\delta-2\big)
=(n-k)i+1$$ for all $i\geq1$. This shows $q\geq\frac{d}{n-k+1}$. In order to finish the proof we have to consider also $i=0$. In the case $km=\delta+1$ the Griesmer bound applied to $i=0$ simply leads to $d\leq nm$, which is true anyway, and no additional condition on $q$ arises. If $km-\delta>1$ a better bound can be achieved. Since $\big\lfloor\frac{\delta}{k}\big\rfloor=m-1$, we obtain after division with remainder of $\delta$ by $k$ an identity of the form $\delta=(m-1)k+r$ where $0\leq r<k-1$. Thus $d=nm-k+r+1$ and Inequality[ ]{} for $i=0$ leads to $$\frac{d}{q}\leq nm-d-\sum_{l=2}^{k-r-1}\Big\lceil\frac{d}{q^l}\Big\rceil
\leq k-r-1-(k-r-2)=1,$$ hence $q\geq d$.\
This covers all cases, since we always have $km\geq\delta$.
The proof shows that in general the lower bounds on $q$ are not tight since we have estimated $\big\lceil\frac{d}{q^l}\big\rceil$ by $1$ for $l\geq2$ in all cases. For instance, if $(n-k+1)^2>d$, no -MDS code exists for $q=\frac{d}{n-k+1}$ and $k=1$ or $km=\delta+1$. But even if $\big\lceil\frac{d}{q^l}\big\rceil=1$ for all $l\geq2$ there might not exist an -MDS code where $q$ attains the lower bound. The obstacle is that for some $i\in\hat{{{\mathbb N}}}$ there might not exist an $(n(m+i),k(m+i)-\delta)_q$-block code with the appropriate distance as required by the proof of Theorem \[T-CCB\]. Since these block codes have to produce a convolutional code in a very specific way, they even have to have some additional structure. We wish to illustrate this by the following example.
\[E-F3code\] Let ${\mbox{$(n,k,\delta)$}}=(3,2,3)$. The generalized Singleton bound is $d:=S(3,2,3)=6$ and the memory of a $(3,2,3)$-MDS code is $m=2$, see Theorem \[T-MDSC\](b). From Theorem \[T-MDSfieldsize\] we obtain $q\geq3$ for the field size. Taking $q=3$ we have $\big\lceil\frac{d}{q^2}\big\rceil=1$ so that indeed the lower bound for the field size cannot be improved. The existence of a $(3,2,3;2)_3$-MDS code requires the existence of $(3(2+i),1+2i)_3$-block codes with distance at least $6$ for all $i\in{{\mathbb N}}_0$. Such codes do indeed exist[^2]. However, the block codes have to have some additional structure in order to be part of a convolutional code. To see this, let $G\in{{\mathbb F}}_3[z]^{2\times3}$ be a minimal generator matrix of the desired convolutional code ${{\mathcal C}}$. Write $$G=\begin{bmatrix}g_1\\g_2\end{bmatrix}+z\begin{bmatrix}g_3\\g_4\end{bmatrix}
+z^2\begin{bmatrix}g_5\\0\end{bmatrix}\text{ where }g_i\in{{\mathbb F}}_3^3.$$ Recall from the proof of Theorem \[T-CCB\] that our arguments are based in particular on the block code ${{\mathcal C}}_1:=\{(u_1,u_2+u_3z)G\mid u_1,u_2,u_3\in{{\mathbb F}}_3\}$. Comparing like powers of $z$ one observes that this code is isomorphic to $$\hat{{{\mathcal C}}}_1={\mbox{\rm im}\,}\begin{bmatrix}g_1&g_3&g_5\\g_2&g_4&0\\0&g_2&g_4\end{bmatrix}\subseteq{{\mathbb F}}_3^9.$$ Using elementary row operations on the polynomial matrix $G$ we may assume that the entry of $G$ at the position $(1,1)$ is a constant. Furthermore, after rescaling the columns of $G$ we may assume $g_4=(1,1,1)$. Finally, due to non-catastrophicity, the entries of $g_2$ are not all the same and because of ${\mbox{\rm dist}}(\hat{{{\mathcal C}}_1})=6$, all nonzero. This gives us (up to block code equivalence) the two options $${\mbox{\rm im}\,}\begin{bmatrix}a_1&a_2&a_3&0&a_4&a_5&0&a_6&a_7\\1&1&2&1&1&1&0&0&0\\
0&0&0&1&1&2&1&1&1\end{bmatrix}
\text{ or }\quad
{\mbox{\rm im}\,}\begin{bmatrix}a_1&a_2&a_3&0&a_4&a_5&0&a_6&a_7\\1&2&2&1&1&1&0&0&0\\
0&0&0&1&2&2&1&1&1\end{bmatrix}$$ for $\hat{{{\mathcal C}}}_1$. Going through some tedious calculations one can show that no such code in ${{\mathbb F}}_3^9$ with distance $6$ exists. Hence no $(3,2,3)_3$-MDS convolutional code exists.
In the next section we will give examples of MDS codes over fields ${{\mathbb F}}_q$ where $q$ attains the lower bound in all cases except for the case $km=\delta+1$.
Examples of Some Optimal Convolutional Codes {#S-Exa}
============================================
In this section we present some convolutional codes with distance reaching the Griesmer bound. To the best of our knowledge it was for most of the parameters, if not all, not known before whether such codes existed.
In the first column of the tables below the parameters of the given code are listed. In the second column we give the Griesmer bound $g:=G_q(n,k,\delta;m)$ for these parameters. The third column gives a code reaching this bound. In all examples the distance of the code has been computed via a program. In each case the code is given by a minimal generator matrix. Thus, in particular all matrices given below are right invertible. In the forth column we present the index of the first column distance that reaches the free distance, cf.[ ]{}. In the last column we indicate whether the code is a cyclic convolutional code in the sense of Section \[S-CCC\]. At the moment this additional structure is not important. We only want to mention that cyclic convolutional codes do not exist for all sets of parameters, in particular the length and the characteristic of the field have to be coprime (just like for block codes). Moreover, the shortest [*binary*]{} cyclic convolutional codes with complexity $\delta>0$ have length $n=7$ or $n=15$.
The fields being used in the tables are ${{\mathbb F}}_2=\{0,1\},\,{{\mathbb F}}_4=\{0,1,\alpha,\alpha^2\}$ where $\alpha^2+\alpha+1=0$, ${{\mathbb F}}_8=\{0,1,\beta,\ldots,\beta^6\}$ where $\beta^3+\beta+1=0$, and ${{\mathbb F}}_{16}=\{0,1,\gamma,\ldots,\gamma^{14}\}$ where $\gamma^4+\gamma+1=0$.
The generator matrix $\hat{G}_3$ of the $(15,4,12;3)_2$-code in Table I is given by $$\hat{G}_3{\mbox{$\!^{\sf T}$}}=\begin{bmatrix} 1+z^2&1+z+z^3&z+z^2&1+z+z^3 \\ 1+z+z^2&1+z+z^2+z^3&1+z+z^2+z^3&z \\
1+z+z^3&1+z+z^2&1+z+z^2&1+z^2+z^3 \\
z&1+z+z^3&1&1+z+z^2 \\ z&z^2&1+z&1+z^3 \\ z^2&z+z^3&z^3&1+z+z^2+z^3 \\ 1+z+z^3&z^2+z^3&z+z^2+z^3&z \\
z^3&1+z+z^2&z+z^3&z^2 \\ z+z^2+z^3&z+z^2&1+z^3&z^2+z^3 \\ 1+z+z^2+z^3&z^2+z^3&z^2&1+z+z^2 \\
1&1&z+z^2+z^3&z^2 \\ z^2+z^3&1+z&1&0 \\ 1+z&0&1+z^2+z^3&1+z^3 \\ z^2+z^3&1+z^2+z^3&z^3&1+z+z^3 \\
1+z^2+z^3&z^3&1+z+z^2&z+z^2+z^3
\end{bmatrix}.$$
Some additional explanations and remarks will follow the tables.
Table I .4mm
[90]{}
[|c|c|c|c|c|]{} ${\mbox{$(n,k,\delta;m)_q$}}$ & $g$ & code meeting the Griesmer bound & $d^c_i$ & cy\
\
\
$(5,3,4;2)_2$ & $6$ & [$\begin{bmatrix}
1+z^2 & 1+z & z & 1+z^2 & z+z^2 \\ 1+z & z & 1+z & 1 & z \\
z & 1 & 1+z & 1+z & 1\end{bmatrix}$(not even)]{}& $7$ &
------------------------------------------------------------------------
\
\
\
$(5,2,6;3)_2$ & $12$ & [$\begin{bmatrix}z^3+z^2+1\!&\!z^2+z\!&\!z^3+z+1\!&\!z^2+z\!&\!z^3+1\\
z+1\!&\! z^3+z^2+1\!&\!z^3+z^2\!&\!z^3+z+1\!&\!z^2+z
\end{bmatrix}$(even)]{}& $10$ &
------------------------------------------------------------------------
\
$(5,2,6;4)_2$ & $12$ & [$\begin{bmatrix}
1+z^3+z^4\!&\!1+z+z^4\!&\!1+z^3\!&\!1+z^2+z^3\!&\!z+z^3+z^4\\
1+z^2\!&\!1+z\!&\!z^2+z\!&\!z^2+z+1\!&\!z^2+z+1
\end{bmatrix}$(even)]{}& $10$ &
------------------------------------------------------------------------
\
\
\
$(9,3,1;1)_8$ & $8^{*\bullet}$ & [$\begin{bmatrix}z+1&z+\beta&z&z+\beta^2&z+\beta^3&z+\beta^6&z+1&z&z+\beta\\
1&\beta^2&\beta^5&\beta^6&\beta^6&\beta^5&\beta^2&1&0\\
0&1&\beta^2&\beta^5&\beta^6&\beta^6&\beta^5&\beta^2&1
\end{bmatrix}$]{} & $1$ &
------------------------------------------------------------------------
\
\
\
$(3,2,2;1)_5$ & $5^{*\bullet}$ & [$\begin{bmatrix}2+3z&3z&4+4z\\4+2z&1+3z&2z\end{bmatrix}$]{} & $5$ &
------------------------------------------------------------------------
\
\
\
$(7,3,3;1)_2$ & $8$ & [$G_1=\begin{bmatrix} 1&z&1+z&1+z&1&z&0\\z&1+z&0&1+z&1&1&z\\0&z&1&0&1+z&1+z&1+z
\end{bmatrix}$(even)]{} & $2$ & $\times$
------------------------------------------------------------------------
\
$(7,3,6;2)_2$ & $12$ & [$G_2=\begin{bmatrix} 1+z^2&z+z^2&1+z&1+z&1+z^2&z&z^2\\z&1+z+z^2&0&1+z+z^2&1+z^2&1+z^2&z\\
z^2&z+z^2&1+z^2&0&1+z&1+z+z^2&1+z
\end{bmatrix}$(even)]{} & $5$ & $\times$
------------------------------------------------------------------------
\
$(7,3,9;3)_2$ & $16$ & [$\begin{bmatrix}
1+z^2+z^3 & z+z^2 & 1+z+z^3 & 1+z & 1+z^2 & z+z^3 & z^2+z^3\\
z & 1+z+z^2+z^3 & 0 & 1+z+z^2 & 1+z^2+z^3 & 1+z^2+z^3 & z+z^3 \\
z^2+z^3 & z+z^2 & 1+z^2 & z^3 &1+z+z^3 & 1+z+z^2+z^3 & 1+z
\end{bmatrix}$(even?)]{} & $9$ & $\times$
------------------------------------------------------------------------
\
$(7,3,12;4)_2$ & $20$ & [$\begin{bmatrix}
1+z+z^3+z^4 \!\!&\!\! 1+z^3+z^4 \!\!&\!\! 1+z^2 \!\!&\!\! z+z^2+z^4 \!\!&\!\! 1+z^2+z^3 \!\!&\!\!
z \!\!&\!\! z+z^2+z^3+z^4\\
z^2+z^3 \!\!&\!\! 1+z+z^2+z^4 \!\!&\!\! 1+z^4 \!\!&\!\! 1+z+z^2+z^3+z^4 \!\!&\!\! z \!\!&\!\!
1+z+z^3+z^4 \!\!&\!\! z^2+z^3 \\
z^2+z^4 \!\!&\!\! z \!\!&\!\! 1+z+z^3 \!\!&\!\! 1+z+z^2+z^4 \!\!&\!\! 1+z^2+z^3+z^4 \!\!&\!\!
z^2+z^3+z^4 \!\!&\!\! 1+z+z^3
\end{bmatrix}$(doubly even?)]{}& $14$ & $\times$
------------------------------------------------------------------------
\
\
\
$(15,4,4;1)_2$ & $16$ & [$\hat{G}_1=\left[\!\!\begin{array}{ccccccccccccccc}
z& 0&z&1+z&0&0&1+z&1&0&1&z&1+z&1+z&1+z&1\\
1&0&z&0& 1&0&z& 1+z&1+z&z&1&z& 1&1+z&1+z\\
1&1& z& z&z&1+z&0&z&1&1+z&z& 1&0&1+z&1\\
1+z&1+z &1 &z &0&z&1+z&0&0&1+z&1&0 &1 &z&1+z
\end{array}\!\!\right]$(even)]{} & $2$ & $\times$
------------------------------------------------------------------------
\
$(15,4,8;2)_2$ & $24$ & [$\hat{G}_2\!\!=\!\!\left[\!\!\begin{array}{ccccccccccccccc}
1+z^2\!\!&\!\!1+z+z^2\!\!&\!\!1+z\!\!&\!\!z\!\!\!&\!\!\!z\!\!&\!\!z^2\!\!&\!\! 1+z\!\!&\!\!
0\!\!&\!\!z+z^2\!\!&\!\!
1+z+z^2\!\!&\!\!1\!\!&\!\!z^2\!\!&\!\!1+z\!\!&\!\!z^2\!\!&\!\!\!1+z^2\\
1+z\!\!&\!\!1+z+z^2\!\!&\!\!1+z+z^2\!\!\!&\!\!\!1+z\!\!&\!\!z^2\!\!&\!\! z\!\!&\!\!z^2\!\!&\!\!1+z+z^2\!\!&\!\!
z+z^2\!\!&\!\!z^2\!\!&\!\!1\!\!&\!\!1+z\!\!&\!\!0\!\!&\!\!
1+z^2\!\!\!&\!\!0\\
z+z^2\!\!&\!\!1+z+z^2\!\!&\!\!1+z+z^2\!\!\!&\!\!1\!\!\!&\!\!1+z\!\!&\!\!0\!\!&\!\!z+z^2\!\!&\!\!z\!\!&\!\!
1\!\!&\!\!z^2\!\!&\!\!z+z^2\!\!&\!\!1\!\!&\!\!1+z^2\!\!&\!\!0\!\!\!&\!\!1+z+z^2\\
1+z\!\!&\!\!z\!\!&\!\!1+z^2\!\!\!&\!\!\!1+z+z^2\!\!\!&\!\!\!1\!\!&\!\!1+z+z^2 \!\!&\!\!z \!\!&\!\!z^2 \!\!&\!\!
z^2\!\!&\!\!1+z+z^2 \!\!&\!\!z^2\!\!&\!\! 0 \!\!&\!\!1 \!\!&\!\!
1+z \!\!\!&\!\!z+z^2
\end{array}\!\!\!\right]$]{} & $5$ & $\times$
------------------------------------------------------------------------
\
& & (even?) & &
------------------------------------------------------------------------
\
$(15,4,12;3)_2$ & $32$ & [see $\hat{G}_3$ above,(even?)]{} & & $\times$
------------------------------------------------------------------------
\
Table II .4mm
[90]{}
[|c|c|c|c|c|]{} ${\mbox{$(n,k,\delta;m)_q$}}$ & $g$ & code meeting the Griesmer bound & $d^c_i$ &cy\
\
\
$(3,1,1;1)_4$ & $6^*$ & [$[\alpha+\alpha z,\ \alpha^2+\alpha z,\ 1+\alpha z]$]{}& $2^{**}$ & $\times$
------------------------------------------------------------------------
\
$(3,1,2;2)_4$ & $9^*$ & [$[\alpha+\alpha z+z^2,\ \alpha^2+\alpha z+\alpha^2 z^2,\
1+\alpha z+\alpha z^2]$]{}& $5$ & $\times$
------------------------------------------------------------------------
\
$(3,1,3;3)_4$ & $12^{*\bullet}$ & [$[\alpha+\alpha z+z^2+\alpha^2 z^3,\ \alpha^2+\alpha z+\alpha^2 z^2+z^3,\
1+\alpha z+\alpha z^2+\alpha z^3]$]{}& $7$ & $\times$
------------------------------------------------------------------------
\
$(3,1,4;4)_4$ & $14$ & [$[\alpha+\alpha z+z^2+\alpha^2 z^3+\alpha z^4,\
\alpha^2+\alpha z+\alpha^2 z^2+z^3+\alpha z^4,\
1+\alpha z+\alpha z^2+\alpha z^3+\alpha z^4]$]{}& $10$ & $\times$
------------------------------------------------------------------------
\
$(3,1,5;5)_4$ & $16$ & [$[\alpha+\alpha z+z^2+\alpha^2 z^3+\alpha z^4+\alpha z^5,\
\alpha^2+\alpha z+\alpha^2 z^2+z^3+\alpha z^4+z^5,\
1+\alpha z+\alpha z^2+\alpha z^3+\alpha z^4+\alpha^2 z^5]$]{}& $11$ & $\times$
------------------------------------------------------------------------
\
\
\
$(5,2,2;1)_4$ & $8$ & [$\begin{bmatrix}0& \alpha+z& \alpha^2+\alpha^2z& \alpha^2+\alpha^2z& \alpha+z\\
\alpha+\alpha^2z& z& \alpha& \alpha^2+z& \alpha^2+\alpha^2z\end{bmatrix}$]{}& $2$& $\times$
------------------------------------------------------------------------
\
$(5,2,4;2)_4$ & $12$ & [$\begin{bmatrix}
0&\alpha+z+\alpha z^2& \alpha^2+\alpha^2z+\alpha^2z^2&\alpha^2+\alpha^2z+\alpha^2z^2&
\alpha+z+\alpha z^2\\
\alpha+\alpha^2z+\alpha z^2& z+\alpha^2z^2& \alpha+\alpha^2z^2&\alpha^2+z+\alpha z^2&
\alpha^2+\alpha^2z\end{bmatrix}$]{}& $5$ & $\times$
------------------------------------------------------------------------
\
$(5,2,6;3)_4$ & $16$ & [$\begin{bmatrix}
0 &\alpha^2+\alpha^2 z +\alpha z^2+z^3 &1+\alpha z+\alpha^2z^2+\alpha^2z^3 &
1+\alpha z+\alpha^2z^2+\alpha^2z^3 & \alpha^2+\alpha^2z+\alpha z^2+z^3 \\
\alpha^2+\alpha z+\alpha z^2+\alpha^2z^3 & \alpha^2 z+\alpha^2z^2+\alpha^2z^3 &
\alpha^2+\alpha^2z^2+z^3 & 1+\alpha^2 z+\alpha z^2 & 1+\alpha z+z^3
\end{bmatrix}$]{}& $9$ & $\times$
------------------------------------------------------------------------
\
\
\
$(3,2,2;1)_{16}$ &$5^*$& [$\begin{bmatrix}
\gamma^5+\gamma^4 z & \gamma^3+\gamma^8 z & \gamma^9+\gamma^2 z\\
\gamma^9+\gamma^{12} z & \gamma^5+\gamma^{14}z & \gamma^3+\gamma^3 z
\end{bmatrix}$]{} & $3^{**}$ & $\times$
------------------------------------------------------------------------
\
$(3,2,3;2)_{16}$ &$6^*$& [$\begin{bmatrix}
\gamma+\gamma z+z^2 & \gamma^6+\gamma z+\gamma^{10}z^2 & \gamma^{11}+\gamma z+\gamma^{5}z^2\\
1+z & \gamma^{10}+\gamma^{5}z & \gamma^5+\gamma^{10} z
\end{bmatrix}$]{} & $5$ & $\times$
------------------------------------------------------------------------
\
\
\
$(5,1,1;1)_{16}$ &$10^*$& [$[\gamma+\gamma z, \gamma^{13}+\gamma^{10} z, \gamma^{10}+\gamma^4 z,
\gamma^7+\gamma^{13} z, \gamma^4+\gamma^{7}z]$]{} & $2^{**}$ & $\times$
------------------------------------------------------------------------
\
$(5,1,2;2)_{16}$ & $15^*$ & [$[\gamma+\gamma^4 z+\gamma z^2, \gamma^{7}+\gamma z+\gamma^{10}z^2,
\gamma^{13}+\gamma^{13} z+\gamma^4z^2,\gamma^4+\gamma^{10}z+\gamma^{13}z^2,
\gamma^{10}+\gamma^7z+\gamma^7z^2]$]{} & $3^{**}$ & $\times$
------------------------------------------------------------------------
\
$(5,1,3;3)_{16}$ & $20^*$ & [$[\gamma+z+\gamma^2 z^2+z^3, \gamma^{7}+\gamma^{12}z+\gamma^{11}z^2+\gamma^3z^3,
\gamma^{13}+\gamma^{9}z+\gamma^5z^2+\gamma^6z^3,\gamma^4+\gamma^{6}z+\gamma^{14}z^2+\gamma^9z^3,
\gamma^{10}+\gamma^3z+\gamma^8z^2+\gamma^{12}z^3]$]{} & $5$ & $\times$
------------------------------------------------------------------------
\
\
\
$(5,2,2;1)_{16}$ & $9^*$ & [$\begin{bmatrix}
\gamma+\gamma z & \gamma^{13}+\gamma^{10} z & \gamma^{10}+\gamma^4 z & \gamma^7+\gamma^{13}z &
\gamma^4+\gamma^7z\\
1+\gamma^{5} z\ & \gamma^3+\gamma^{11}z\ & \gamma^6+\gamma^2 z & \gamma^9+\gamma^8 z &
\gamma^{12}+\gamma^{14} z
\end{bmatrix}$]{} & $2^{**}$ & $\times$
------------------------------------------------------------------------
\
\
\
$(7,1,1;1)_{8}$ & $14^*$ & [$[\beta+\beta z,\, \beta^3+z,\, \beta^5+\beta^6z,\, 1+\beta^5z,\,
\beta^2+\beta^4z,\, \beta^4+\beta^3z,\, \beta^6+\beta^2z]$]{} & $2^{**}$ & $\times$
------------------------------------------------------------------------
\
$(7,1,2;2)_{8}$ & $21^*$ & [$[\beta^2+\beta z+z^2, \beta^5+\beta^3z+\beta^6z^2, \beta+\beta^5z+\beta^5z^2,
\beta^4+z+\beta^4z^2, 1+\beta^2z+\beta^3z^2, \beta^3+\beta^4z+\beta^2z^2,
\beta^6+\beta^6z+\beta z^2]$]{} & $3^{**}$ & $\times$
------------------------------------------------------------------------
\
$(7,1,3;3)_{8}$ & $28^*$ & [$[1\!+\!\beta z\!+\!\beta^6z^2\!+\!z^3, 1\!+\!\beta^5z\!+\!\beta^5z^2\!+\!\beta^5z^3,
1\!+\!\beta^2z\!+\!\beta^4z^2\!+\!\beta^3z^3,1\!+\!\beta^6z\!+\!\beta^3z^2\!+\!\beta z^3,
1\!+\!\beta^3z\!+\!\beta^2z^2+\beta^6z^3, 1\!+\!z\!+\!\beta z^2+\beta^4z^3,
1\!+\!\beta^4z\!+\!z^2\!+\!\beta^2z^3]$]{} & $5$ & $\times$
------------------------------------------------------------------------
\
$(7,2,3;2)_{8}$ & $14^*$ & [$\begin{bmatrix}
1+z+\beta^4z^2 & \beta^4+\beta^5z+\beta^5z^2 & \beta+\beta^3z+\beta^6z^2 &
\beta^5+\beta z+z^2 & \beta^2+\beta^6 z+\beta z^2 &
\beta^6+\beta^4z+\beta^2z^2 & \beta^3+\beta^2z+\beta^3z^2\\
\beta+\beta z & \beta^3+z & \beta^5+\beta^6z & 1+\beta^5z & \beta^2+\beta^4z &
\beta^4+\beta^3z & \beta^6+\beta^2z
\end{bmatrix}$]{} & $3$ & $\times$
------------------------------------------------------------------------
\
Table III\
[|c|c|c|c|c|]{} ${\mbox{$(n,k,\delta;m)_q$}}$ & $g$ & code meeting the Griesmer bound & $d^c_i$ &cy\
\
\
$(6,3,3;1)_{2}$ & $6$ & [columns 1,2,3,5,6,7 of $G_1$(even)]{} & $3$ &
------------------------------------------------------------------------
\
$(6,3,6;2)_{2}$ & $10$ & [columns 1,2,4,5,6,7 of $G_2$(even)]{} & $3$ &
------------------------------------------------------------------------
\
\
\
$(14,4,4;1)_{2}$ & $14$ & [columns 1 – 14 of $\hat{G}_1$(not even)]{} & $3$ &
------------------------------------------------------------------------
\
$(13,4,4;1)_{2}$ & $13$ & [columns 1,2,4 – 14 of $\hat{G}_1$(not even)]{} & $3$&
------------------------------------------------------------------------
\
$(12,4,4;1)_{2}$ & $12$ & [columns 1,2,4 – 12,14 of $\hat{G}_1$(even)]{} & $3$ &
------------------------------------------------------------------------
\
$(10,4,4;1)_{2}$ & $10$ & [columns 1,2,4,6 – 11,14 of $\hat{G}_1$(even)]{} & $4$ &
------------------------------------------------------------------------
\
$(8,4,4;1)_{2}$ & $8$ & [columns 1,2,4,5,8,11,13,14 of $\hat{G}_1$(not even)]{} & $4$ &
------------------------------------------------------------------------
\
\
\
$(14,4,8;2)_{2}$ & $22$ & [columns 2 – 15 of $\hat{G}_2$(even?)]{} & $6$ &
------------------------------------------------------------------------
\
$(13,4,8;2)_{2}$ & $20$ & [columns 1 – 4,7 – 15 of $\hat{G}_2$ (even?)]{} & $6$ &
------------------------------------------------------------------------
\
$(12,4,8;2)_{2}$ & $18$ & [columns 1,2,4,7 – 15 of $\hat{G}_2$(not even)]{} & $6$ &
------------------------------------------------------------------------
\
$(10,4,8;2)_{2}$ & $16$ & [columns 1,2,4,5,7,8,10,11,13,14 of $\hat{G}_2$ (even?)]{} & $7$ &
------------------------------------------------------------------------
\
$(8,4,8;2)_{2}$ & $12$ & [columns 1,2,6,9,12 – 15 of $\hat{G}_2$ (even?)]{} & $9$ &
------------------------------------------------------------------------
\
It remains to explain some additional notation of the tables. We also make some further comments illustrating the contents of the tables.
\[R-tables\]
A $*$ attached to the bounds in the second column indicate that these numbers are identical to the generalized Singleton bound. Hence the corresponding codes are even MDS codes.
An additional supscript ${\bullet}$ attached to the bound $g$ indicates that the code is an MDS code where the field size reaches the lower bound of Theorem \[T-MDSfieldsize\]. This gives us examples for the three cases $k=1$, $km>\delta+1$, and $km=\delta$. We did not find an example of an -MDS code where $km=\delta+1$ and $q=\frac{d}{n-k+1}$.
In [@GRS03 Prop. 2.3] it has been shown that the $j$th column distance of an -code satisfies $d^c_j\leq(n-k)(j+1)+1$. From this it follows that the earliest column distance of an MDS code that can reach the free distance has index $M:=\big\lfloor\frac{\delta}{k}\big\rfloor+\big\lceil\frac{\delta}{n-k}\big\rceil$, see [@GRS03 Prop. 2.6]. In the same paper an MDS code is called strongly MDS if the $M$th column distance is equal to the free distance. We attached a $^{**}$ to the index of the column distance in the second last column of the tables in order to indicate the strongly MDS codes. As far as we know no upper bound for the column distances is known that also takes the field sizes into account. However, using the estimate $d^c_j\leq(n-k)(j+1)+1$ one observes that the $(5,2,2;1)_4$- and the $(9,3,1;1)_8$-code are also optimal in the sense that no code with the same parameters exists where an earlier column distance reaches the free distance. We did not investigate whether any of the other codes is optimal in this sense.
We investigated the binary codes with respect of being even, that is, whether all codewords have even weight. This can be done by computing the weight distribution (see [@McE98a] or [@JoZi99 Sec. 3.10]). Evenness of a code is indicated by an (even) attached to the generator matrix. Since the computation of the full weight distribution is very complex for larger complexity, we did not fully check the binary codes having complexity bigger than $6$. In those cases we checked the weight of codewords associated with message words of small degree. In case this weight is always even we think there is strong evidence that the code is even and attached an (even?) to the generator matrix. In this sense there is also evidence that the $(7,3,12;4)_2$-code is doubly even, that is, all codewords have weight divisible by $4$. Further investigation is necessary in order to understand whether (and why) all the binary cyclic convolutional codes of length $7$ and $15$ are even.
The second and third code of Table I show that a code meeting the Griesmer bound need not have evenly distributed Forney indices. In other words, such a code need not be compact in the sense of Theorem \[T-MDSC\](b). For both codes in Table I the free distance is attained by the 10th column distance. Only the full weight distribution shows that the code with Forney indices $3,\,3$ is better than the code with indices $4,\,2$. The first one has weight distribution $$W_1(T)=10T^{12}+12T^{14}+71T^{16}+248T^{18}+873T^{20}+\ldots,$$ saying that there are $10$ molecular codewords of weight $12$ and $12$ molecular codewords of weight $14$, etc. (for the definition of molecular codewords, see [@McE98a]; for weight distributions see also [@JoZi99 Sec. 3.10]). The weight distribution of the second code is $$W_2(T)=10T^{12}+27T^{14}+99T^{16}+350T^{18}+1280T^{20}+\ldots.$$
It is worth being mentioned that the codes with parameters $(7,3,3;1)_2,\,(7,3,6;2)_2$, and $(7,3,9;3)_2$ form a sequence in the sense that if one deletes $z^3$ (resp. $z^2$) in the last (resp. second) of the according generator matrices then one obtains the previous code. The same applies to the codes with parameters $(3,1,1;1)_4$, …, $(3,1,5;5)_4$ as well as to the $(5,2,2;1)_4$- and $(5,2,4;2)_4$-codes.
The codes with parameters $(7,3,3;1)_2,\,(7,3,6;2)_2,\,(15,4,4;1)_2$ and $(15,4,8;2)_2$ are extremely robust against puncturing in the sense of cutting columns of the according generator matrix (this is not puncturing in the sense of [@McE98 Sec. 8]). This way we do not only obtain right invertible matrices again, but even minimal matrices and, by doing this appropriately, codes reaching the Griesmer bound. We have cut one column of the codes of length $7$ and up to $7$ columns of the codes of length $15$. The results are given in Table III. The only cases where we did not get codes reaching the Griesmer bound are for $(11,4,4;1)_2$ and for $(9,4,8;2)_2$. We do not know if for these parameters there exist any codes at all that reach the bound. Since $G_2(9,4,4;1)=8=G_2(8,4,4;1)$ and $G_2(11,4,8;2)=16=G_2(10,4,8;2)$ we skipped in both cases the bigger length. Puncturing the code of length $7$ and memory bigger than $2$ did not result in a code meeting the Griesmer bound. We did not puncture the code of length $15$ and memory $3$.
Consider the $(8,4,4;1)_2$-code given in Table III. There are other codes with exactly these parameters given in the literature. Indeed, in [@JSW00] some (doubly-even self-dual) $(8,4,4;1)_2$-codes are presented. Our code is not even, which can easily be seen by writing down the generator matrix. We also computed the weight distribution and obtained $$\begin{aligned}
W(T)=&11T^8+28T^9+39T^{10}+101T^{11}+206T^{12}+565T^{13}+1374T^{14}+3033T^{15}\\
&+7366T^{16}+16984T^{17}+40510T^{18}+95617T^{19}+22348T^{20}+\ldots,
\end{aligned}$$ which is better than the weight distribution of the self-dual code given in [@JSW00 Eq. (10)].
Cyclic Convolutional Codes {#S-CCC}
==========================
The first two tables of the last section list plenty of optimal codes that we have declared as cyclic. Moreover, they gave rise to further sets of optimal codes as listed in Table III. In this section we want to briefly describe the notion of cyclicity for [convolutional code]{}s. The first investigations in this direction have been made in the seventies by Piret [@Pi76] and Roos [@Ro79]. In both papers it has been shown (with different methods and in different contexts) that cyclicity of [convolutional code]{}s must not be understood in the usual sense, i. e. invariance under the cyclic shift, if one wants to go beyond the theory of cyclic block codes (see Theorem \[T-CCCclassic\] below). As a consequence, Piret suggested a more complex notion of cyclicity which then has been further generalized by Roos. In both papers some nontrivial examples of [cyclic convolutional code]{}s in this new sense are presented along with their distances. All this indicates that the new notion of cyclicity seems to be the appropriate one in the convolutional case. Unfortunately, the papers [@Pi76; @Ro79] did not get much attention at that time and the topic came to a halt. Only recently it has been resumed in [@GS02]. Therein, an algebraic theory of [cyclic convolutional code]{}s has been established which goes well beyond the results of the seventies. On the one hand it leads to a nice, yet nontrivial, generalization of the theory of cyclic block codes, on the other hand it gives a very powerful toolbox for constructing [cyclic convolutional code]{}s. We will now give a very brief description of these results and refer to [@GS02] for the details.
Just like for cyclic block codes we assume from now on that the length $n$ and the field size $q$ are coprime. Let ${{\mathbb F}}={{\mathbb F}}_q$ be a field of size $q$. Recall that a block code ${{\mathcal C}}\subseteq{{\mathbb F}}^n$ is called cyclic if it is invariant under the cyclic shift, i. e. $$\label{e-cs}
(v_0,\ldots,v_{n-1})\in{{\mathcal C}}\Longrightarrow
(v_{n-1},v_0,\ldots,v_{n-2})\in{{\mathcal C}}$$ for all $(v_0,\ldots,v_{n-1})\in{{\mathbb F}}^n$. It is well-known that this is the case if and only if ${{\mathcal C}}$ is an ideal in the quotient ring $$\label{e-A}
A:={{\mathbb F}}[x]/_{{\displaystyle}{\mbox{$\langle{x^n-1}\rangle$}}}=\Big\{\sum_{i=0}^{n-1}f_ix^i\;{\mbox{\rm mod}\,}(x^n-1)\,\Big|\,f_0,\ldots,f_{n-1}\in{{\mathbb F}}\Big\},$$ identified with ${{\mathbb F}}^n$ in the canonical way via $${\mbox{$\mathfrak{p}$}}: {{\mathbb F}}^n\longrightarrow A,\quad
(v_0,\ldots,v_{n-1})\longmapsto\sum_{i=0}^{n-1}v_ix^i.$$ At this point it is important to recall that the cyclic shift in ${{\mathbb F}}^n$ translates into multiplication by $x$ in $A$, i. e. $$\label{e-cx}
{\mbox{$\mathfrak{p}$}}(v_{n-1},v_0,\ldots,v_{n-2})=x{\mbox{$\mathfrak{p}$}}(v_0,\ldots,v_{n-1})$$ for all $(v_0,\ldots,v_{n-1})\in{{\mathbb F}}^n$. Furthermore, it is well-known that each ideal $I\subseteq A$ is principal, hence there exists some $g\in A$ such that $I={\mbox{$\langle{g}\rangle$}}$. One can even choose $g$ as a monic divisor of $x^n-1$, in which case it is usually called the [*generator polynomial*]{} of the code ${\mbox{$\mathfrak{p}$}}^{-1}(I)\subseteq{{\mathbb F}}^n$.
It is our aim to extend this structure to the convolutional setting. The most convenient way to do so is by using only the polynomial part ${{\mathcal C}}\cap{{\mathbb F}}[z]^n$ of the [convolutional code]{} ${{\mathcal C}}\subseteq{\mbox{${{\mathbb F}}(\!(z)\!)$}}^n$. Recall from[ ]{} that this uniquely determines the full code. Hence imposing some additional structure on the polynomial part (that is, on the generator matrix) will also impose some additional structure on the full code. In Remark \[R-convcirc\] below we will see from hindsight that one can just as well proceed directly with the full code. The polynomial part of a [convolutional code]{} is always a submodule of the free module ${{\mathbb F}}[z]^n$. Due to the right invertibility of the generator matrix not every submodule of ${{\mathbb F}}[z]^n$ arises as polynomial part of a [convolutional code]{}. It is easy to see [@GS02 Prop. 2.2] that we have
\[R-dirsumm\] A submodule ${{\mathcal S}}\subseteq{{\mathbb F}}[z]^n$ is the polynomial part of some [convolutional code]{} if and only if ${{\mathcal S}}$ is a direct summand of ${{\mathbb F}}[z]^n$, i.e. ${{\mathcal S}}\oplus{{\mathcal S}}'={{\mathbb F}}[z]^n$ for some submodule ${{\mathcal S}}'\subseteq{{\mathbb F}}[z]^n$.
In order to extend the situation of cyclic block codes to the convolutional setting, we have to replace the vector space ${{\mathbb F}}^n$ by the free module ${{\mathbb F}}[z]^n$ and, consequently, the ring $A$ by the polynomial ring $$A[z]:=\Big\{\sum_{j=0}^Nz^ja_j\,\Big|\, N\in{{\mathbb N}}_0,\,a_j\in A\Big\}$$ over $A$. Then we can extend the map ${\mbox{$\mathfrak{p}$}}$ above coefficientwise to polynomials, thus $$\label{e-p}
{\mbox{$\mathfrak{p}$}}: {{\mathbb F}}[z]^n\longrightarrow A[z],\quad
\sum_{j=0}^N z^jv_j\longmapsto \sum_{j=0}^N z^j{\mbox{$\mathfrak{p}$}}(v_j),$$ where, of course, $v_j\in{{\mathbb F}}^n$ and thus ${\mbox{$\mathfrak{p}$}}(v_j)\in A$ for all $j$. This map is an isomorphism of ${{\mathbb F}}[z]$-modules. Again, by construction the cyclic shift in ${{\mathbb F}}[z]^n$ corresponds to multiplication by $x$ in $A[z]$, that is, we have[ ]{} for all $(v_0,\ldots,v_{n-1})\in{{\mathbb F}}[z]^n$. At this point it is quite natural to call a [convolutional code]{} ${{\mathcal C}}\subseteq{\mbox{${{\mathbb F}}(\!(z)\!)$}}^n$ cyclic if it is invariant under the cyclic shift, i. e. if[ ]{} holds true for all $(v_0,\ldots,v_{n-1})\in{\mbox{${{\mathbb F}}(\!(z)\!)$}}^n$. This, however, does not result in any codes other than block codes due to the following result, see [@Pi76 Thm. 3.12] and [@Ro79 Thm. 6]. An elementary proof can be found at [@GS02 Prop. 2.7].
\[T-CCCclassic\] Let ${{\mathcal C}}\subseteq{\mbox{${{\mathbb F}}(\!(z)\!)$}}^n$ be an -convolutional code such that[ ]{} holds true for all $(v_0,\ldots,v_{n-1})\in{{\mathbb F}}[z]^n$. Then $\delta=0$, hence ${{\mathcal C}}$ is a block code.
This result has led Piret [@Pi76] to suggest a different notion of cyclicity for [convolutional code]{}s. We will present this notion in the slightly more general version as it has been introduced by Roos [@Ro79].
In order to do so notice that ${{\mathbb F}}$ can be regarded as a subfield of the ring $A$ in a natural way. As a consequence, $A$ is an ${{\mathbb F}}$-algebra, i. e., a ring and a vector space over the field ${{\mathbb F}}$ and the two structures are compatible. In the sequel the automorphisms of $A$ with respect to this algebra structure will play an important role. Therefore we define $${\mbox{${\rm Aut}_{\mathbb F}$}}(A):=\big\{\sigma:A\rightarrow
A\,\big|\,\sigma|_{{{\mathbb F}}}=\text{id}_{{{\mathbb F}}},\,\sigma\text{ is bijective},\,
\sigma(a{\mbox{\small\raisebox{-.8ex}{$\stackrel{+}{\cdot}$}}}b)
=\sigma(a){\mbox{\small\raisebox{-.8ex}{$\stackrel{+}{\cdot}$}}}\sigma(b)
\text{ for all }a,\,b\in A
\big\}.$$ It is clear that each automorphism $\sigma\in{\mbox{${\rm Aut}_{\mathbb F}$}}(A)$ is uniquely determined by the single value $\sigma(x)\in A$. But not every choice for $\sigma(x)$ determines an automorphism on $A$. Since $x$ generates the ${{\mathbb F}}$-algebra $A$, the same has to be true for $\sigma(x)$ and, more precisely, we obtain for $a\in A$ $$\label{e-sxa}
\left.\begin{array}{l}
\sigma(x)=a\text{ determines an }\\
\text{automorphism on }A
\end{array}\right\}
\Longleftrightarrow
\left\{\begin{array}{l}
1,\,a,\ldots,a^{n-1}\text{ are linearly independent over }{{\mathbb F}}\\
\text{and }a^n=1.
\end{array}\right.$$ Of course, $\sigma(x)=x$ determines the identity map on $A$. It should be mentioned that there is a better way to determine the automorphism group of $A$ by using the fact that the ring is direct product of fields. This is explained in [@GS02 Sec. 3].
The main idea of Piret was to impose a new ring structure on $A[z]$ and to call a code cyclic if it is a left ideal with respect to that ring structure. The new structure is non-commutative and based on an (arbitrarily chosen) automorphism on $A$. In detail, this looks as follows.
\[D-CCC\] Let $\sigma\in{\mbox{${\rm Aut}_{\mathbb F}$}}(A)$.
On the set $A[z]$ we define addition as usual and multiplication via $$\sum_{j=0}^Nz^j a_j\cdot\sum_{l=0}^M z^l b_l=
\sum_{t=0}^{N+M}z^t\sum_{j+l=t}\sigma^l(a_j)b_l
\text{ for all }N,\,M\in{{\mathbb N}}_0\text{ and }a_j,\,b_l\in A.$$ This turns $A[z]$ into a non-commutative ring which is denoted by ${\mbox{$A[z;\sigma]$}}$.
Consider the map ${\mbox{$\mathfrak{p}$}}:{{\mathbb F}}[z]^n\!\rightarrow\!{\mbox{$A[z;\sigma]$}}$ as in[ ]{}, where now the images ${\mbox{$\mathfrak{p}$}}(v)\!=\!\sum_{j=0}^Nz^j{\mbox{$\mathfrak{p}$}}(v_j)$ are regarded as elements of ${\mbox{$A[z;\sigma]$}}$. A direct summand ${{\mathcal S}}\subseteq{{\mathbb F}}[z]^n$ is said to be $\sigma$-[*cyclic*]{} if ${\mbox{$\mathfrak{p}$}}({{\mathcal S}})$ is a left ideal in ${\mbox{$A[z;\sigma]$}}$.
A [convolutional code]{} ${{\mathcal C}}\subseteq{\mbox{${{\mathbb F}}(\!(z)\!)$}}^n$ is said to be $\sigma$-[*cyclic*]{} if ${{\mathcal C}}\cap{{\mathbb F}}[z]^n$ is a direct summand.
A few comments are in order. First of all, notice that multiplication is determined by the rule $$\label{e-az}
az=z\sigma(a)\text{ for all }a\in A$$ along with the rules of a (non-commutative) ring. Hence, unless $\sigma$ is the identity, the indeterminate $z$ does not commute with its coefficients. Consequently, it becomes important to distinguish between left and right coefficients of $z$. Of course, the coefficients can be moved to either side by applying the rule[ ]{} since $\sigma$ is invertible. Multiplication inside $A$ remains the same as before. Hence $A$ is a commutative subring of ${\mbox{$A[z;\sigma]$}}$. Moreover, since $\sigma|_{{{\mathbb F}}}=\text{id}_{{{\mathbb F}}}$, the classical polynomial ring ${{\mathbb F}}[z]$ is a commutative subring of ${\mbox{$A[z;\sigma]$}}$, too. As a consequence, ${\mbox{$A[z;\sigma]$}}$ is a left and right ${{\mathbb F}}[z]$-module and the map ${\mbox{$\mathfrak{p}$}}:{{\mathbb F}}[z]^n\rightarrow{\mbox{$A[z;\sigma]$}}$ is an isomorphism of left ${{\mathbb F}}[z]$-modules (but not of right ${{\mathbb F}}[z]$-modules). In the special case where $\sigma=\text{id}_{A}$ the ring ${\mbox{$A[z;\sigma]$}}$ is the classical commutative polynomial ring and we know from Theorem \[T-CCCclassic\] that no [convolutional code]{}s with nonzero complexity exist.
\[E-binarylength7\] Let us consider the case where ${{\mathbb F}}={{\mathbb F}}_2$ and $n=7$. Thus $A={{\mathbb F}}[x]/_{{\mbox{$\langle{x^7-1}\rangle$}}}$. Using[ ]{} one obtains $18$ automorphisms, also listed at [@Ro79 p. 680, Table II] (containing one typo: the last element of that table has to be $x^2+x^3+x^4+x^5+x^6$ rather than $x+x^3+x^4+x^5+x^6$).\
Let us choose the automorphism $\sigma\in{\mbox{${\rm Aut}_{\mathbb F}$}}(A)$ defined by $\sigma(x)=x^5$. Furthermore, we consider the polynomial $$g := 1+x^2+x^3+x^4+z(x+x^2+x^3+x^5)\in{\mbox{$A[z;\sigma]$}}$$ and denote by ${\mbox{$^{^{\bullet\!\!}}\langle{\, g\, }\rangle$}}:=\{fg\mid f\in{\mbox{$A[z;\sigma]$}}\}$ the left ideal generated by $g$ in ${\mbox{$A[z;\sigma]$}}$. Moreover, put ${{\mathcal S}}:={\mbox{$\mathfrak{p}$}}^{-1}({\mbox{$^{^{\bullet\!\!}}\langle{\, g\, }\rangle$}})\subseteq{{\mathbb F}}[z]^7$. We will show now that ${{\mathcal S}}$ is a direct summand of ${{\mathbb F}}[z]^7$, hence ${{\mathcal S}}={{\mathcal C}}\cap{{\mathbb F}}[z]^7$ for some [convolutional code]{} ${{\mathcal C}}\subseteq{\mbox{${{\mathbb F}}(\!(z)\!)$}}^7$, see Remark \[R-dirsumm\]. In order to do so we first notice that $${\mbox{$^{^{\bullet\!\!}}\langle{\, g\, }\rangle$}}={\mbox{\rm span}\,}_{{{\mathbb F}}[z]}\big\{g,\,xg,\ldots,x^6g\big\}$$ and therefore $${{\mathcal S}}=\big\{u M\,\big|\, u\in{{\mathbb F}}[z]^7\big\}
\text{ where }
M=\begin{bmatrix}{\mbox{$\mathfrak{p}$}}^{-1}(g)\\{\mbox{$\mathfrak{p}$}}^{-1}(xg)\\\vdots\\{\mbox{$\mathfrak{p}$}}^{-1}(x^6g)
\end{bmatrix}.$$ Thus we have to compute $x^ig$ for $i=1,\ldots,6$. Using the multiplication rule in[ ]{} we obtain $$\begin{aligned}
xg&=x+x^3+x^4+x^5+z(1+x+x^3+x^6),
\\
x^2g&=x^2+x^4+x^5+x^6+z(x+x^4+x^5+x^6),
\\
x^3g&=1+x^3+x^5+x^6+z(x^2+x^3+x^4+x^6)\\
&=g+x^2g.\end{aligned}$$ Since $x^3g$ is in the ${{\mathbb F}}$-span of the previous elements, we obtain ${\mbox{$^{^{\bullet\!\!}}\langle{\, g\, }\rangle$}}={\mbox{\rm span}\,}_{{{\mathbb F}}[z]}\big\{g,xg,x^2g\big\}$ and, since ${\mbox{$\mathfrak{p}$}}$ is an isomorphism, $${{\mathcal S}}=\big\{u G\,\big|\, u\in{{\mathbb F}}[z]^3\big\},$$ where $$G=\begin{bmatrix}{\mbox{$\mathfrak{p}$}}^{-1}(g)\\{\mbox{$\mathfrak{p}$}}^{-1}(xg)\\{\mbox{$\mathfrak{p}$}}^{-1}(x^2g)
\end{bmatrix}
=\begin{bmatrix} 1&z&1+z&1+z&1&z&0\\z&1+z&0&1+z&1&1&z\\0&z&1&0&1+z&1+z&1+z
\end{bmatrix}.$$ One can easily check that the matrix $G$ is right invertible. Hence ${{\mathcal S}}$ is indeed a direct summand of ${{\mathbb F}}[z]^7$ and thus we have obtained a [convolutional code]{} ${{\mathcal C}}={\mbox{\rm im}\,}G\subseteq{\mbox{${{\mathbb F}}(\!(z)\!)$}}^7$. This is exactly the $(7,3,3;1)_2$-code given in Table I of the last section.
The other [cyclic convolutional code]{}s in Tables I and II are obtained in a similar way. Since the underlying automorphism cannot easily be read off from the generator matrix of a [cyclic convolutional code]{} we will, for sake of completeness, present them explicitly in the following table. All those codes come from principal left ideals in ${\mbox{$A[z;\sigma]$}}$ and, except for the codes with parameters $(3,2,3;2)_{16},\,(5,2,2;1)_{16},\,(7,2,3;2)_8$, the generator polynomial can be recovered from the given data by applying the map ${\mbox{$\mathfrak{p}$}}$ to the first row of the respective generator matrix. The generator matrices of the remaining three codes are built in a slightly different way. In those cases each row of the given matrix generates a 1-dimensional cyclic code and thus each of those three codes is the direct sum of two 1-dimensional cyclic codes. In each case a generator polynomial of the associated principal left ideal is obtained by applying ${\mbox{$\mathfrak{p}$}}$ to the sum of the two rows of the respective generator matrix.
Table IV\
[|c|c|]{} ${\mbox{$(n,k,\delta;m)_q$}}$-code of Tables I and II & automorphism given by
------------------------------------------------------------------------
\
\
\
$(7,3,3m;m)_2,\,m=1,\ldots,4$ & $\sigma(x)=x^5$
------------------------------------------------------------------------
\
$(15,4,4;1)_2$ & $\sigma(x)=x+x^7+x^{10}$
------------------------------------------------------------------------
\
$(15,4,4m;m)_2,\,m=2,3$ & $\sigma(x)=x^3+x^5+x^7+x^{10}+x^{12}+x^{13}+x^{14}$
------------------------------------------------------------------------
\
$(3,1,\delta;\delta)_4,\, \delta=1,\ldots,5$ & $\sigma(x)=\alpha^2x$
------------------------------------------------------------------------
\
$(5,2,2m;m)_4,\,m=1,2,3$ & $\sigma(x)=x^2$
------------------------------------------------------------------------
\
$(3,2,2;1)_{16}$ and $(3,2,3;2)_{16}$ & $\sigma(x)=\gamma^{10}x$
------------------------------------------------------------------------
\
$(5,1,\delta;\delta)_{16},\,\delta=1,2,3$ and $(5,2,2;1)_{16}$ & $\sigma(x)=x^3$
------------------------------------------------------------------------
\
$(7,1,\delta;\delta)_{8},\,\delta=1,2$ and $(7,2,3;2)_{8}$ & $\sigma(x)=x^5$
------------------------------------------------------------------------
\
$(7,1,3;3)_{8}$ & $\sigma(x)=\beta x+\beta x^2+\beta^3x^3+\beta^3x^4+\beta^3x^5+\beta^2x^6$
------------------------------------------------------------------------
\
The fact that all the [cyclic convolutional code]{}s above come from principal left ideals in ${\mbox{$A[z;\sigma]$}}$ is not a restriction since we have the following important result.
\[T-pli\] Let $\sigma\in{\mbox{${\rm Aut}_{\mathbb F}$}}(A)$. If ${{\mathcal S}}\subseteq{{\mathbb F}}[z]^n$ is a direct summand, then ${\mbox{$\mathfrak{p}$}}({{\mathcal S}})$ is a principal left ideal of ${\mbox{$A[z;\sigma]$}}$, that is, there exists some polynomial $g\in{\mbox{$A[z;\sigma]$}}$ such that ${\mbox{$\mathfrak{p}$}}({{\mathcal S}})={\mbox{$^{^{\bullet\!\!}}\langle{\, g\, }\rangle$}}$. We call $g$ a generator polynomial of both ${{\mathcal S}}$ and the [convolutional code]{}${{\mathcal C}}\subseteq{\mbox{${{\mathbb F}}(\!(z)\!)$}}^n$ determined by ${{\mathcal S}}$, see Remark \[R-dirsumm\] and[ ]{}.
The generator polynomial of a [convolutional code]{} can be translated into vector notation and leads to a generalized circulant matrix. This looks as follows. Let ${{\mathcal S}}\subseteq{{\mathbb F}}[z]^n$ be a direct summand and let ${\mbox{$\mathfrak{p}$}}({{\mathcal S}})={\mbox{$^{^{\bullet\!\!}}\langle{\, g\, }\rangle$}}$. Define $${\mbox{${\cal M}^{\sigma}$}}(g)=\begin{bmatrix}
{\mbox{$\mathfrak{p}$}}^{-1}(g)\\ {\mbox{$\mathfrak{p}$}}^{-1}(xg)\\ \vdots\\
{\mbox{$\mathfrak{p}$}}^{-1}(x^{n-1}g)\end{bmatrix}\in{{\mathbb F}}[z]^{n\times n}.$$ Then it is easy to see that ${\mbox{$\mathfrak{p}$}}\big(u{\mbox{${\cal M}^{\sigma}$}}(g)\big)={\mbox{$\mathfrak{p}$}}(u)g$ for all $u\in{{\mathbb F}}[z]^n$ (see [@GS02 Prop. 6.8(b)]) and therefore, ${{\mathcal S}}=\big\{u{\mbox{${\cal M}^{\sigma}$}}(g)\,\big|\,u\in{{\mathbb F}}[z]^n\big\}$. We call ${\mbox{${\cal M}^{\sigma}$}}(g)$ the $\sigma$-circulant associated with $g$.
\[R-convcirc\] Using the identities above we can now easily see that structure can also be considered without restricting to the polynomial part. Just like the polynomial ring $A[z]$ we can turn the set $A(\!(z)\!)$ of formal Laurent series over $A$ into a non-commutative ring by defining addition as usual and multiplication via[ ]{}. We will denote the ring obtained this way by ${\mbox{$A(\!(z;\sigma)\!)$}}$. Furthermore, we can extend the map ${\mbox{$\mathfrak{p}$}}$ to Laurent series in the canonical way, see also[ ]{}. Then one can easily show that just like in the polynomial case $${\mbox{$\mathfrak{p}$}}\big(u{\mbox{${\cal M}^{\sigma}$}}(g)\big)={\mbox{$\mathfrak{p}$}}(u)g\text{ for all }u\in{\mbox{${{\mathbb F}}(\!(z)\!)$}}^n$$ for each $g\in{\mbox{$A[z;\sigma]$}}$. Using the fact that a code ${{\mathcal C}}\subseteq{\mbox{${{\mathbb F}}(\!(z)\!)$}}^n$ is uniquely determined by its polynomial part (see[ ]{}), and that the latter is a principal left ideal in ${\mbox{$A[z;\sigma]$}}$ due to Theorem \[T-pli\], one can now derive the equivalence $${{\mathcal C}}\subseteq{\mbox{${{\mathbb F}}(\!(z)\!)$}}^n\text{ is $\sigma$-cyclic }
\Longleftrightarrow
{\mbox{$\mathfrak{p}$}}({{\mathcal C}})\text{ is a left ideal in }{\mbox{$A(\!(z;\sigma)\!)$}}.$$ Moreover, if ${{\mathcal C}}$ is $\sigma$-cyclic, a generator polynomial of the ideal ${\mbox{$\mathfrak{p}$}}({{\mathcal C}}\cap{{\mathbb F}}[z]^n)$ in ${\mbox{$A[z;\sigma]$}}$ is also a principal generator of the ideal ${\mbox{$\mathfrak{p}$}}({{\mathcal C}})$ in ${\mbox{$A(\!(z;\sigma)\!)$}}$. This justifies to call $g$ a generator polynomial of the full code ${{\mathcal C}}$ as we did in Theorem \[T-pli\].
At this point the question arises as to how a (right invertible) generator matrix can be obtained from the $\sigma$-circulant ${\mbox{${\cal M}^{\sigma}$}}(g)$. Notice that in Example \[E-binarylength7\] the generator matrix of the code is simply given by the first three rows of the circulant. This is indeed in general the case, but requires a careful choice of the generator polynomial $g$ of the code. Recall that, due to zero divisors in ${\mbox{$A[z;\sigma]$}}$, the generators of a principal left ideal, are highly non unique. The careful choice of the generator polynomial is based on a Gröbner basis theory that can be established in the non-commutative polynomial ring ${\mbox{$A[z;\sigma]$}}$. This is a type of reduction procedure resulting in unique generating sets of left ideals which in turn produce very powerful $\sigma$-circulants. The details of this theory goes beyond the scope of this paper and we refer the reader to [@GS02] for the details, in particular to [@GS02 Thm. 7.8, Thm. 7.18]. Therein it has been shown that a reduced generator polynomial also reflects the parameters of the code, i. e., the dimension and the complexity, and even leads to a minimal generator matrix through $\sigma$-circulants. Only with these results it becomes clear that [cyclic convolutional code]{}s can have only very specific parameters (length, dimension, and complexity) depending on the chosen field ${{\mathbb F}}_q$. Furthermore, the notions of parity check polynomial and associated parity check matrix have been discussed in detail in [@GS02], leading to a generalization of the block code situation.
As for the cyclic codes of the last section we only would like to mention that their generator polynomials obtained as explained right before Table IV are all reduced in the sense above.
So far we do not have any estimates for the distance of a [cyclic convolutional code]{} in terms of its (reduced) generator polynomial and the chosen automorphism. The examples given in the last section have been found simply by trying some promising reduced generator polynomials (using the algebraic theory of [@GS02]). Except for the puncturing in Table III we did not perform a systematic search for optimal codes.
Conclusion {#conclusion .unnumbered}
==========
In this paper we gave many examples of [cyclic convolutional code]{}s that all reach the Griesmer bound. The examples indicate that this class of [convolutional code]{}s promises to contain many excellent codes and therefore deserves further investigation. As one of the next steps the relation between the (reduced) generator polynomial and the automorphism on the one hand and the distance on the other hand should be investigated in detail.
[^1]: Department of Mathematics, University of Oldenburg, 26111 Oldenburg, Germany, email: gluesing@ mathematik.uni-oldenburg.de and wiland.schmale@uni-oldenburg.de
[^2]: For small $i$ these codes can be found in tables listing ternary codes. For the general case we wish to thank H.-G. Quebbemann who pointed out to us a construction of such codes for sufficiently large $i$ using direct products of finitely many “short” MDS-codes over ${{\mathbb F}}_{3^3}$ and mapping them into ternary codes.
|
---
author:
- 'Noah P. Mitchell'
- Vinzenz Koning
- Vincenzo Vitelli
- 'William T. M. Irvine'
title: Fracture in Sheets Draped on Curved Surfaces
---
[^1]
[^2]
**Conforming materials to rigid substrates with Gaussian curvature — positive for spheres and negative for saddles — has proven a versatile tool to guide the self-assembly of defects such as scars, pleats [@irvine_pleats_2010; @bausch_grain_2003; @bowick_two-dimensional_2009; @vitelli_crystallography_2006; @grason_universal_2013], folds, blisters [@holmes_draping_2010; @hure_wrapping_2011], and liquid crystal ripples [@devries_divalent_2007]. Here, we show how curvature can likewise be used to control material failure and guide the paths of cracks. In our experiments, and unlike in previous studies on cracked plates and shells [@Slepyan_cracks_2002; @Folias_Stresses_1965; @amiri_phase-field_2014], we constrained flat elastic sheets to adopt *fixed* curvature profiles. This constraint provides a geometric tool for controlling fracture behavior: curvature can stimulate or suppress the growth of cracks, and steer or arrest their propagation. A simple analytical model captures crack behavior at the onset of propagation, while a two-dimensional phase-field model with an added curvature term successfully captures the crack’s path. Because the curvature-induced stresses are independent of material parameters for isotropic, brittle media, our results apply across scales [@rupich_soft_2014; @Dusseault_Drilling_2004].**
Geometry on curved surfaces defies intuition: ‘parallel’ lines diverge or converge as a consequence of curvature. As a result, when a thin material conforms to such a surface, stretching and compression are inevitable [@bowick_two-dimensional_2009]. As stresses build up, the material can then respond by forming structures such as wrinkles or dislocations, which are themselves of geometric origin. This interplay between curvature and structural response can result in universal behavior, independent of material parameters [@hure_wrapping_2011; @irvine_pleats_2010; @bausch_grain_2003; @vitelli_crystallography_2006; @grason_universal_2013].
A markedly different material response is to break via propagating cracks. While the use of curvature to control the morphology of wrinkles and defects in materials has been recently explored [@irvine_pleats_2010; @hure_wrapping_2011; @bausch_grain_2003], here we investigate the control of cracks by tuning the geometry of a rigid substrate. Can we design the underlying curvature of a substrate to steer paths of cracks in a material draped on that surface, thereby protecting certain regions?
![Gaussian curvature — positive for caps and negative for saddles — governs the behavior of cracks. In the experimental setup, an initially flat PDMS sheet conforms to a curved 3D printed surface. A small incision nucleates the crack.[]{data-label="fig1"}](draft_final_F1.pdf "fig:"){width="\columnwidth"}\
To probe the effect of curvature on cracks, we conform flat PDMS sheets (Smooth-On Rubber Glass II) to 3D-printed substrates (Fig. \[fig1\]). A lubricant ensures that the sheet conforms to the substrate while moving freely along the surface. We consider various geometries having positive and negative Gaussian curvature in both localized and distributed regions: spherical caps, saddles, cones, and bumps. To begin, we focus on the bump as a model surface, as it is a common geometry containing regions of both positive and negative curvature. A typical experimental run can be seen in *Supplementary Videos 1-7*. We seed a crack by cutting a slit in the sheet, with a position and orientation of choice. By successive cuts, we increase their length until they exceed a critical length, known as the Griffith length [@griffith_phenomena_1921; @Rivlin_rupture_1953], and propagate freely.
The Griffith length of a crack in a flat sheet is nearly independent of position and orientation. On our curved geometry, we find that this is not so. On the top of the bump, a shorter slit is necessary to produce a running crack, and on the outskirts of the bump (where the Gaussian curvature is negative), the behavior depends strongly on the orientation of the seed crack: fracture initiation is suppressed for radial cracks, while the Griffith length for azimuthal cracks approaches that of the flat sheet (Fig. \[fig2\]b). Thus curvature can both stimulate and suppress fracture initiation, depending on the position and orientation of the seed crack.
![ a) Gaussian curvature and curvature potential distributions for a bump with height profile $h(\rho) = \alpha x_0 \exp(-\rho^2/2x_0^2)$. b) While the Griffith length for a crack in a flat sheet (dashed line) is nearly constant, curvature modulates the critical length of a seed crack. All samples shown had a 12 cm diameter ($2R$), an aspect ratio $\alpha=1/\sqrt{2}$, bump width $x_0=R/2.35$, and constant radial displacement $u_\rho/R=0.012$. []{data-label="fig2"}](draft_final_F2.pdf "fig:"){width="\columnwidth"}\
![ (a-b) Crack paths kink and curve around a bump. (c-d) Phase-field simulations of cracks on a bump, colored by the phase-modulated energy density so that broken regions are darkened. (e-f) The phase-field crack path predictions (black solid curves) overlie the experimental paths (colored curves). (Inset) Introducing a time delay matching experiment for the right crack tip’s propagation eliminates the discrepancy far from the bump. (g) Analytical prediction (solid black curve) of the kink angle, $\theta_k$, overlies experimental results. (h) Analytical crack path predictions overlie simulations for free (constant stress) boundary conditions. All experiments and simulations had aspect ratio $\alpha=1/\sqrt{2}$ and bump width $x_0=R/2.35$, including the free boundary condition simulations. []{data-label="fig3"}](draft_final_F3.pdf "fig:"){width="\columnwidth"}\
To relate these findings to the curvature distribution, we consider the stresses induced by curvature and their interaction with the crack tip. Stresses generated in the bulk of a material become concentrated near a crack tip. In turn, a crack extends when the intensity of stress concentration exceeds a material-dependent, critical value [@griffith_phenomena_1921; @freund_dynamic_1990]. Expressed mathematically, in the coordinates of the crack tip ($r,\theta$), the stress in the vicinity of the tip takes the form $$\label{stress}
\sigma_{ij} = \frac{K_I}{\sqrt{2 \pi r}} f_{ij}^{I}(\theta) + \frac{K_{II}}{\sqrt{2\pi r}} f_{ij}^{II}(\theta),$$ where $f_{ij}^{I,II}$ are universal angular functions [@freund_dynamic_1990]. The factors $K_{I}$ and $K_{II}$ measure the intensity of tensile and shear stress concentration at the crack tip, respectively, and are known as stress intensity factors (SIFs). Thus, the Griffith length, $a_c$, is the length of the crack at which the intensity of stress concentration reaches the critical value, $K_c$. In curved plates or sheets, the near-tip stress fields display the same singular behavior as in Eqn. \[stress\] [@hui_williams_1998], but the values of the SIFs are governed by curvature.
Curving a flat sheet involves locally stretching and compressing the material by certain amounts at each point. According to the rules of differential geometry, this stretching factor, controlled by the field $\Phi$, is determined by an equation identical to the Poisson equation of electrostatics [@vitelli_anomalous_2004], with the Gaussian curvature, $G$, playing the role of a continuous charge distribution [@bowick_two-dimensional_2009; @vitelli_crystallography_2006]: $$\label{harmonicPhi}
\nabla^2 \Phi({\mathbf{x}}) = - G({\mathbf{x}}).$$ As the sheet equilibrates, its elasticity tends to oppose this mechanical constraint, giving rise to stress. The isotropic stress from curvature is then related to the potential via $\sigma^G_{kk}=E \Phi $, where $E$ is Young’s modulus, and the stress components are determined by integrals of the potential and boundary conditions (see Eqns. 25-26 of the *Supplementary Information*). Our study rests on a general geometric principle: positive (negative) curvature promotes local stretching (compression) of an elastic sheet, leading to the enhancement (suppression) of crack initiation. Variations in the potential $\Phi$ steer the crack path, with the form of $\Phi$ determined nonlocally from the curvature distribution (see Eqn. \[harmonicPhi\] and Eqns. 39-41 of the *Supplementary Information*).
For the bump, the curvature potential, $\Phi$, is large on the cap, where curvature is positive, and decays to zero as the negative curvature ring screens the cap (Fig. \[fig2\]a). As $E\Phi$ is the isotropic stress, crack growth is stimulated where the potential is greatest — on the cap of the bump, resulting in a small Griffith length there (Fig. \[fig2\]b). Moving away from the cap, the potential decays, producing a stress asymmetry. This results in longer Griffith lengths with strong orientation dependence on the outskirts of the bump (see Eqns. 39-41 of the *Supplementary Information*). Fig. \[fig2\]b shows the theoretical results overlying the experimental data, with no fitting parameters. We find that this minimal model is sufficient to capture the phenomenology of our system at the onset of fracture and provides correct qualitative predictions for longer cracks, even in the absence of symmetry.
Curvature not only governs the critical length for fracture initiation, but also the direction of a crack’s propagation. For cracks inclined with respect to the bump, the cracks change direction as they begin to propagate, kinking at the onset of crack growth and curving around the bump, as shown in Fig. \[fig3\]a. Cracks kink and curve towards the azimuthal direction because a decaying curvature potential, $\Phi(\rho)$, creates a local stress asymmetry: $\sigma^G_{\phi \phi} < \sigma^G_{\rho\rho}$. As a result, the crack relieves more elastic energy by deflecting towards the azimuthal direction. Analytical prediction of the kink angle, $\theta_k$, is made by selecting the direction of maximum hoop stress asymptotically near the crack tip (Eqn. 33 of the *Supplementary Information*). Fig. \[fig3\]g shows excellent agreement with experiment.
A purely analytical model is sufficient to capture the long-time behavior of the crack if the stress is fixed at the boundary (see \[fig3\]h). This model extends the first order perturbation theory for slightly curved cracks developed by Cotterell and Rice [@cotterell_slightly_1980] to curved sheets (see the section *Perturbation Theory Prediction of Crack Paths* in the *Supplementary Information*). As shown in Fig. 7 of the *Supplementary Information*, the perturbation theory prediction is also increasingly accurate for constant displacement loading when the system size is large with respect to the crack.
![ (a) Curvature arrests a center crack: as the aspect ratio of the bump increases while the initial stress at the boundary ($\sigma_{\rho\rho}(R) = 0.068\, E$) remains fixed, the final crack length decreases. (b) Simulations reveal that as the aspect ratio of the bump increases, the intensity of stress concentration falls below the critical value at progressively shorter crack lengths. *Inset:* Final crack lengths from spring-lattice (squares) and phase-field simulations (triangles) mimic the arrest behavior seen in experiment (colored circles with error bars marking one standard deviation). The solid line is a guide to the eye. []{data-label="fig4"}](draft_final_F4.pdf){width="\columnwidth"}
{width="\textwidth"}
For modest sample sizes with constant displacement boundary loading, however, a numerical approach is required. To predict the curved fracture trajectories, we adapt the KKL phase-field model [@karma_phase-field_2001; @spatschek_phase_2011] to include curvature by incorporating the height profile of the substrate into the two-dimensional strain field [@Nelson_fluctuations_1987]. This numerical model treats local material damage as a scalar field that evolves if there is both sufficient elastic energy density and a local gradient in the field (see *Supplementary Information*). As depicted in Fig. \[fig3\]c and d, these conditions are met at the tip of a propagating crack. This model captures the full crack paths, as shown by the black curves overlying experimental results in Fig. \[fig3\]e and \[fig3\]f.
A systematic deviation in the extensions of the crack tips further from the bump is evident in Fig. \[fig3\]e. In the experiments, the tip closer to the bump begins its advance first, and the dynamics of the tip are not purely quasistatic. In the phase-field simulation, simply suppressing the tip further from the bump for a short time until the near tip has reached a distance matching experiment eliminates this deviation, as shown in the inset of Fig.\[fig3\]e (see the *Phase-Field Model* section of the *Supplementary Information* for details).
Having seen how curvature affects the initiation and propagation of cracks, we now turn our attention to the ability of curvature to arrest cracks. As seen previously in Fig. \[fig3\], curved cracks can terminate before reaching the sample boundary. We find, moreover, that curvature can arrest cracks even for cases in which the path is undeflected, as shown in Fig. \[fig4\]. In flat sheets, center cracks propagate all the way to the boundary, but if we introduce a bump while holding the initial stress at the boundary fixed, the final crack length decreases.
From the decaying isotropic stress profile, we can infer that curvature generates azimuthal compression, halting the crack’s advance. Using our phase-field model, we indeed find that increasing the aspect ratio of the bump lowers the intensity of stress concentration for larger crack lengths (Fig. \[fig4\]b). A fully 3D spring network simulation using finite element methods provides additional confirmation (open squares in Fig. \[fig4\]b). Thus, curvature decreases the final crack length, despite promoting crack initiation on top of the bump.
Curvature’s influence on the propagation of cracks that we have investigated on the bump is not peculiar to that surface. As shown in Fig. \[fig5\], we demonstrate this generality by testing a number of additional surfaces, including spherical caps (uniform $G>0$), cones ($G=G_0 \,\delta({\mathbf{x}})$), and pseudospherical saddles (uniform $G<0$). A region of positive curvature, such as the tip of a cone, locally stimulates crack growth near the region, but also guides cracks around that region. Conversely, negative curvature of a saddle suppresses crack growth and orients cracks away from the center (see Fig. \[fig5\] and *Supplementary Information*). Thus an opposite curvature source induces an opposite response: the behavior of cracks is tunable by engineering the curvature landscape.
In Fig. \[fig5\]c and d, we demonstrate the robustness of curvature’s effects by considering samples without azimuthal symmetry using the phase-field model. Here, we use a bump to protect a central region from incoming cracks of various orientations, to produce oscillating cracks, and to focus and diverge possible crack paths. For the geometries of Fig. \[fig5\]d, a somewhat reduced critical stress intensity factor compared to our experimental material prevents crack arrest. Though the stress is highest on top of a bump, these regions are protected from approaching cracks (see *Supplementary Video 8*).
The use of substrate curvature to control fracture morphology differs from using existing cracks or inclusions in that our method requires no introduction of pre-existing structure into the fracturing sheets [@ghelichi_modeling_2015; @cheeseman_interaction_2000]. For brittle sheets with isotropic elasticity, curvature-induced stresses are independent of material parameters and only dependent on geometry. Therefore, our results represent the effects of substrate curvature on fracture morphology for a wide range of materials, with potential implications for thin films, monolayers [@rupich_soft_2014; @yuk_high-resolution_2012], geological strata such as near salt diapirs [@price_analysis_1990; @Dusseault_Drilling_2004], and stretchable electronics [@rogers_materials_2010]. Since the results are based on the modulations of the material’s metric, they should also apply beyond conformed sheets, with metrics engineered by other methods — for instance, temperature gradients [@yuse_transition_1993] or differential swelling [@sharon_mechanics_2010].
Acknowledgements
================
The authors thank Efi Efrati, Hridesh Kedia, Dustin Kleckner, Michelle Driscoll, Sid Nagel, Tom Witten, and Ridg Scott for interesting discussions and Jacob Mazor for assistance with some supplementary experiments. Some simulations were carried out on the Midway Cluster provided by the University of Chicago Research Computing Center. We acknowledge the Materials Research and Engineering Centers (MRSEC) Shared Facilities at The University of Chicago for the use of their instruments. This work was supported by the National Science Foundation MRSEC Program at The University of Chicago (Grant DMR-1420709) and a Packard Fellowship. V.K. and V.V. acknowledge funding from FOM and NWO.
Author contributions
====================
W.T.M.I. and V.V. initiated this study. N.P.M. and W.T.M.I. designed experiments. N.P.M. performed and analyzed the experiments and simulations. N.P.M. and V.K. constructed the analytical model. All authors interpreted the data. N.P.M., V.V., and W.T.M.I wrote the manuscript.
Code availability
=================
Custom python codes for phase-field model simulations and analytical crack trajectories are available at https://github.com/irvinelab/fracture, including detailed documentation.
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[^1]: Corresponding author
[^2]: Corresponding author
|
---
author:
- |
$\:^a$, Eike H. Müller$^b$, Lew Khomskii$^a$, Alistair Hart$^b$, Ronald R. Horgan$^a$, Matthew Wingate$^a$\
\
DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK\
SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UK\
\
E-mail:
title: 'Rare *B* decays with moving NRQCD and improved staggered quarks'
---
Introduction
============
Decays of $B$ mesons via the flavour-changing-neutral-current transition $b \rightarrow s$ are particularly sensitive to possible new-physics contributions and provide tests of the CKM mechanism at the loop level. Measurements of exclusive modes like $B\rightarrow K^* \gamma$ have reached a good accuracy, and call for precise theoretical predictions. These are more difficult than for tree-level decays such as $B\rightarrow \pi l \nu$, since a large set of effective electroweak operators contributes and long-distance or spectator effects can be important. Nevertheless, the computation of hadronic form factors in lattice QCD is highly desirable and complements continuum approaches.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Matrix element Form factor Relevant decay(s)
------------------------------------------------------------------------------ ------------------------------------------------ --------------------------------------------------------------------------------
$\langle P|\bar{q}\gamma^\mu b|B\rangle$ $f_+, f_0$ $\left\{\begin{array}{l} B\to\pi\ell\nu\\B\to K\ell^+\ell^-\end{array}\right.$
\[4mm\] $\langle P|\bar{q}\sigma^{\mu\nu}q_\nu b|B\rangle$ $f_T$ $B\to K\ell^+\ell^-$
\[2mm\] $\begin{array}{c}\langle V|\bar{q}\gamma^\mu b|B\rangle $\begin{array}{c}V\\ A_0, A_1, A_2\end{array}$ $\left\{\begin{array}{l} B\to(\rho/\omega)\ell\nu \\
\\ \langle V|\bar{q}\gamma^\mu\gamma^5 b|B\rangle\end{array}$ B\to K^*\ell^+\ell^-\end{array}\right.$
\[5mm\] $\begin{array}{c}\langle V|\bar{q}\sigma^{\mu\nu}q_\nu b|B\rangle \\ $\begin{array}{c}T_1\\T_2, T_3\end{array}$ $\left\{\begin{array}{l} B\to K^*\gamma \\
\langle V|\bar{q}\sigma^{\mu\nu}\gamma^5 q_\nu b|B\rangle\end{array}$ B\to K^*\ell^+\ell^-\end{array}\right.$
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Form factors for semileptonic and radiative $B$ decays.[]{data-label="tab:formfact"}
We are currently working on the calculation of the form factors listed in Table \[tab:formfact\]. The combination of NRQCD and improved staggered actions for heavy-light mesons has already proven very successful in the calculation of form factors [@Dalgic:2006dt]. In order to extend the kinematic range to high recoil (lower $q^2$), we now use a moving-NRQCD (mNRQCD) action for the heavy quark. A brief discussion of our strategy can be found in [@Meinel:2007eh], and a new detailed account of mNRQCD will be given in [@mNRQCDpaper]. Here, we report on the progress in the computation of matrix elements and operator matching coefficients achieved so far.
Lattice methods
===============
The matrix element $\langle F(p')| J | B(p)\rangle$, where $F$ denotes the final pseudoscalar ($P$) or vector ($V$) meson and $J$ is the relevant current in the effective electroweak operator (see Table \[tab:formfact\]), can be extracted from the combination of the Euclidean 3-point correlator $$C_{FJB}(\mathbf{k}_{(\mathbf{q})},\:\mathbf{k}_{(\mathbf{p})},\:x_0,\:y_0,\:z_0)
=\sum_{\mathbf{y}}\sum_{\mathbf{z}}\left\langle \Phi_F(x)\:J^{(\mathrm{lat})}(y)
\:\Phi_B^\dag(z)\right\rangle e^{-i\mathbf{p'}\cdot\mathbf{x}}
e^{-i\mathbf{k}_{(\mathbf{q})}\cdot\mathbf{y}}
e^{i\mathbf{k}_{(\mathbf{p})}\cdot\mathbf{z}} \label{eqn:b3pt}$$
with the two-point functions
$$\begin{aligned}
C_{BB}(\mathbf{k}_{(\mathbf{p})},\:x_0,\:y_0)&=&\sum_{\mathbf{x}}
\left\langle \Phi_B(x)\:\Phi_B^\dag(y)\right\rangle
e^{-i\mathbf{k}_{(\mathbf{p})}\cdot(\mathbf{x}-\mathbf{y})},\\
C_{FF}(\mathbf{p'},\:x_0,\:y_0)&=&\sum_{\mathbf{x}}
\left\langle \Phi_F(x)\:\Phi_F^\dag(y)\right\rangle
e^{-i\mathbf{p'}\cdot(\mathbf{x}-\mathbf{y})}.\end{aligned}$$
Here, $\Phi_B$ and $\Phi_F$ are suitable interpolating fields for the initial and final meson, and $J^{(\mathrm{lat})}$ is a lattice version of the current, obtained by operator matching (see section \[sec:matching\]). For the light quarks, we convert from 1-component staggered to 4-component *naive* fields [@Wingate:2002fh]. Due to the use of mNRQCD for the $b$ quark, the physical momenta $\mathbf{p}$ and $\mathbf{q}$ are related to the lattice momenta $\mathbf{k}_{(\mathbf{p})}$ and $\mathbf{k}_{(\mathbf{q})}$ by $$\begin{aligned}
\nonumber\mathbf{p}&=&\mathbf{k}_{(\mathbf{p})}+Z_p\:\gamma\: m_b \mathbf{v},\\
\mathbf{q}&=&\mathbf{k}_{(\mathbf{q})}+Z_p\:\gamma\: m_b \mathbf{v}\end{aligned}$$ where $Z_p\approx 1$ is the renormalisation of the external momentum, $\gamma =1/ \sqrt{1-\mathbf{v}^2}$ and $\mathbf{v}$ is the boost velocity. One has $\mathbf{p'}=\mathbf{p}-\mathbf{q}=\mathbf{k}_{(\mathbf{p})}-\mathbf{k}_{(\mathbf{q})}$. The physical energy $p_0=E_B$ of the $B$ meson is also shifted, $$\begin{aligned}
E_B(\mathbf{p})=E_\mathbf{v}(\mathbf{k}_{(\mathbf{p})})+\Delta_\mathbf{v} \label{eq:eshift}\end{aligned}$$ where $E_\mathbf{v}(\mathbf{k}_{(\mathbf{p})})$ is the unphysical energy obtained from the fit to the correlator and $\Delta_\mathbf{v}$ is the velocity-dependent energy shift. Writing $t=|x_0-y_0|$ and $T=|x_0-z_0|$, the correlators are fitted by $$\begin{aligned}
C_{FJB}(\mathbf{k}_{(\mathbf{q})},\:\mathbf{k}_{(\mathbf{p})},\:t,\:T)&
\rightarrow&\sum_{k=0}^{K-1}\:\:\sum_{l=0}^{L-1}A_{kl}^{(FJB)}
(-1)^{k\:t}(-1)^{l(T-t)}e^{-E'_k t} e^{-E_l(T-t)},\label{eq:3ptfit}\\
C_{BB}(\mathbf{k}_{(\mathbf{p})},\:t)&\rightarrow&
\sum_{l=0}^{L-1}A_l^{(BB)}(-1)^{l(t+1)}e^{-E_l t},\\
C_{FF}(\mathbf{p'},\:t)&\rightarrow&16
\sum_{k=0}^{K-1}A_k^{(FF)}(-1)^{k(t+1)}e^{-E'_k t}\end{aligned}$$ or equivalent parametrisations. Every other exponential comes with an oscillating pre-factor, as required by the use of naive quarks [@Wingate:2002fh]. The correlator $C_{FF}$ receives an extra factor of 16 due to the trace over a $16\times16$ taste matrix, while the heavy-light correlators $C_{BB}$ and $C_{FJB}$ receive contributions from only one taste [@Wingate:2002fh]. The ground-state fit parameters are related to the matrix elements as follows: $$\begin{aligned}
A_{00}^{(FJB)}&=&\left\{\begin{array}{ll}\displaystyle\frac{\sqrt{Z_V}}{2E_V}
\frac{\sqrt{Z_B}}{2E_B}\sum_s\varepsilon_j(p',s)
\:\langle V\left(p',\varepsilon(p',s)\right)| \:J \:| B(p)\rangle, & F=V,\\
\displaystyle\frac{\sqrt{Z_P}}{2E_P}\frac{\sqrt{Z_B}}{2E_B}
\:\langle P\left(p'\right)| \:J \:| B(p)\rangle, & F=P \end{array} \right.\label{eq:3pt_ampl}\\
A_0^{(BB)}&=&\frac{Z_B}{2E_B} \label{eq:Z1}\\
A_0^{(FF)}&=&\left\{\begin{array}{ll}\displaystyle\sum_s\frac{Z_V}{2E_V}
\:\varepsilon^*_j(p',s)\varepsilon_j(p',s), & F=V, \\
\displaystyle\frac{Z_P}{2E_P} , & F=P. \end{array} \right. \label{eq:Z2}\end{aligned}$$ The amplitudes $\sqrt{Z_B}$, $\sqrt{Z_P}$ and $\sqrt{Z_V}$ in (\[eq:3pt\_ampl\]) depend on the form of the interpolating fields $\Phi_B$, $\Phi_P$ and $\Phi_V$ and can be extracted from (\[eq:Z1\]) and (\[eq:Z2\]).
Operator matching {#sec:matching}
=================
The continuum currents $J$ must be replaced by lattice currents $J^{(\mathrm{lat})}$ containing suitable matching coefficients to correct for the different ultraviolet behaviour of QCD and lattice mNRQCD. As only the high-energetic modes with $E \gtrsim m_b$ differ in the theories and $\alpha_s(m_b) \ll 1$, matching coefficients can be computed perturbatively.
We use tadpole-improved 1-loop lattice perturbation theory. The Feynman rules are generated automatically [@Hart:2004bd] and diagrams are evaluated using the Monte Carlo integrator VEGAS.
The first step is the computation of a set of heavy-quark renormalisation parameters from the self-energy diagrams: the zero-point energy $E_0$, the wavefunction renormalisation $Z_\psi$, the renormalisation of the mass $Z_m$ and the renormalisation of the boost velocity $Z_v$ [@Dougall:2004hw; @mNRQCDpaper]. Results for the full improved $\mathcal{O}(\Lambda_{QCD}^2/m_b^2)$ lattice mNRQCD action will be presented in [@mNRQCDpaper].
Once these parameters are known, one can proceed with the calculation of matching coefficients. For the (axial-)vector currents, these have been computed in the static limit (i.e. neglecting $\mathcal{O}(\Lambda_{QCD}/m_b)$ corrections in $J^{(\mathrm{lat})}$), and the calculation including the $\mathcal{O}(\Lambda_{QCD}/m_b)$ corrections is underway [@LewThesis]. In the following, we focus on the tensor current, which, in the continuum, is given by[^1] $$\begin{aligned}
J_7^{\mu\nu} &=& \frac{e}{16\pi^2} m_b \;\overline{q} \sigma^{\mu\nu} b
\qquad\mathrm{with}\quad\sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu,\gamma^\nu].\end{aligned}$$ We work in the static limit. At this order in the heavy-quark expansion there are two operators with different Dirac structure in lattice mNRQCD. For the $\mu=0$ components one has $$\begin{aligned}
\nonumber J_{7,1}^{0\ell} &=& -\frac{e}{16\pi^2}m_b\sqrt{\frac{\scriptstyle 1+\gamma}{\scriptstyle 2\gamma}}
\left(\overline{q} \sigma_{0\ell}\Psi_v^{(+)}\right),\\
J_{7,2}^{0\ell} &=& i\frac{e}{16\pi^2}m_b\: v\sqrt{\frac{\scriptstyle \gamma}{\scriptstyle 2(1+\gamma)}}
\left(\overline{q} \sigma_{0\ell}\vec{\hat{v}}\cdot\vec{\gamma}\gamma_0\Psi_v^{(+)}\right)\end{aligned}$$
where $\Psi_v^{(+)}$ denotes the mNRQCD field with the antiquark components set to zero. On the lattice, these operators mix under renormalisation; the one-particle irreducible vertex correction that contributes in the static limit is shown in Fig. \[fig:tensor\_matching\_diagram\].
![Matching coefficients for the tensor current. The subtraction point is $\mu=1/a$.[]{data-label="fig:tensor_matching_coefficients"}](vertex_correction.eps)
![Matching coefficients for the tensor current. The subtraction point is $\mu=1/a$.[]{data-label="fig:tensor_matching_coefficients"}](matching_coefficients.eps)
Writing $J_{7,\pm}^{0\ell} = J_{7,1}^{0\ell} \pm J_{7,2}^{0\ell}$, we obtain the lattice operator $$\begin{aligned}
\nonumber J_7^{(\mathrm{lat})0\ell} &=& (1+\alpha_s c_1^{0\ell}) J_{7,1}^{0\ell}
+ (1+\alpha_s c_2^{0\ell}) J_{7,2}^{0\ell}\\
&=& (1+\alpha_s c_+^{0\ell}) J_{7,+}^{0\ell} + \alpha_s c_-^{0\ell} J_{7,-}^{0\ell}.\end{aligned}$$ The matching coefficients $c_{\pm}^{0\ell}$ must be adjusted such that $J_7^{(\mathrm{lat})0\ell}$ has the same one-loop matrix elements as the tensor operator in the continuum theory. They depend on the lattice spacing and contain a logarithmic ultraviolet divergence as the tensor operator is not conserved.
Results using an improved $\mathcal{O}(\Lambda_{QCD}/m_b)$ mNRQCD action at $am=2.8$, $n=2$, the AsqTad action for the light quark and the Lüscher-Weisz gluon action are shown in Fig. \[fig:tensor\_matching\_coefficients\]. The matching coefficient of the operator $J_{7,-}^{0\ell}$, which only arises at 1-loop level, is strongly suppressed and the dependence on the frame velocity is found to be small for all matching coefficients.
Details of the numerical calculations and first results
=======================================================
In our first computations of the 3-point functions (\[eqn:b3pt\]) we used the local interpolating fields $\Phi_B(z)=\bar{q}'(z)\gamma_5b(z)$ and $\Phi_F(x)=\bar{q}'(x)\Gamma_Fq(x)$ with $\Gamma_P=\gamma_5,\:\Gamma_V=\gamma_{1,2,3}$. As in standard NRQCD, the heavy-quark Green function $G_b(y,z)$ can be obtained by solving an initial value problem. Let us consider the case $x_0>y_0>z_0$. Schematically, as initial value at $z_0$ we use the propagator of the light valence quark, $\gamma_5\:G_{q'}(z, x)$, and then evolve the heavy-quark Green function up to the time slice $y_0$ where we perform the contraction with the various gamma matrices and the other light-quark propagator $G_q(x, y)=\gamma_5G_q^\dag(y, x)\gamma_5$. This method only requires light-quark propagators with a fixed origin $x$, and, since the current is inserted only in the final contraction, allows the efficient simultaneous computation of arbitrary currents.
These initial calculations were done on 400 MILC gauge configurations of size $20^3\times64$ with 2+1 flavours of light quarks, at $\beta=6.76$ and $a^{-1}\approx1.6$ GeV. The light sea quark masses were $am_u=am_d=0.007$, $am_s=0.05$ and the light valence quark masses $am_u=am_d=0.007$, $am_s=0.04$ (we used the AsqTad action). On each configuration, we took four different origins $x$, and additionally averaged with the time-reversed process.
Even though we have implemented the full $\mathcal{O}(\Lambda_{QCD}^2/m_b^2)$ mNRQCD action, we only used an $\mathcal{O}(\Lambda_{QCD}/m_b)$ mNRQCD action here to save computer time. This is sufficiently accurate since we only considered currents in the static limit here. The heavy-quark mass was set to $am_b=2.8$ and the stability parameter was $n=2$. All lattice momenta and the boost velocity were always pointing in 1-direction. In this case, 21 combinations of operators/indices and final-state polarisations give non-zero contributions, and all the form factors listed in Table \[tab:formfact\] can be extracted from them.
We performed Bayesian multi-exponential fits in the two variables $T$ and $t$. Gaussian priors for the ground state energies were taken from fit results of the corresponding two-point functions, with widths equal to the error from the fit result. The mNRQCD energy shift $\Delta_\mathbf{v}$ (see eq. (\[eq:eshift\])) was determined non-perturbatively from heavy-heavy meson dispersion relations (for those, the full mNRQCD action accurate to $v_{nr}^4$ in heavy-heavy power counting was used). A bootstrap analysis was used to determine the form factors and their statistical errors.
To give some examples, plots of the 3-point correlators $\langle \:\:\Phi_K \:\:\: \bar s \gamma_0 b \:\:\: \Phi_B^\dag\:\: \rangle$ and $\langle \:\:\Phi_{K^*} \:\:\: \bar s \sigma_{1 3} b \:\:\: \Phi_B^\dag\:\: \rangle$ at the largest $q^2$ (with $\mathbf{v}=0$) are shown in Fig. \[fig:g5\_g0\_sl\_q\_0\_0\_0\_p\_0\_0\_0\_v\_00\] and \[fig:g2\_s13\_sl\_q\_1\_0\_0\_p\_0\_0\_0\_v\_00\]. The results for the tensor current in combination with the vector meson final state are much noisier. As expected, the statistical errors are seen to grow further when the recoil momentum is increased. In Fig. \[fig:g5\_g0\_sl\_q\_2\_0\_0\_p\_1\_0\_0\_v\_04\] and \[fig:g2\_s13\_sl\_q\_2\_0\_0\_p\_1\_0\_0\_v\_04\] we show the corresponding correlators at $\mathbf{v}=(0.4,0,0)$ and $\mathbf{k}_{(\mathbf{p})}=\frac{2\pi}{L}(1,0,0)$, $\mathbf{k}_{(\mathbf{q})}=\frac{2\pi}{L}(2,0,0)$. Note that for the fits shown here, 4..6 timeslices from the source/sink were skipped, so that $K=2$, $L=4$ (for the vector final state) or $K=1$, $L=3$ (for the pseudoscalar final state) was sufficient in (\[eq:3ptfit\]). The results can probably be improved by extending the fitting range and using more exponentials.
Finally, in Fig. \[fig:f0\_f+\_fT\] and \[fig:T2\] we show some first results for the form factors $f_0$, $f_+$, $f_T$ and $T_1$, $T_2$. Note that the momentum of the meson in the final state ($K$ or $K^*$) was exclusively set to the very small values $\mathbf{p'}=0$ or $\mathbf{p'}=\frac{2\pi}{L}(-1,0,0)$. This is made possible by the use of moving NRQCD.
![Three-point correlator $\langle \:\:\Phi_{K^*} \:\:\: \bar s \sigma_{1 3} b
\:\:\: \Phi_B^\dag\:\: \rangle$at $\mathbf{k}_{(\mathbf{p})}=0$, $\mathbf{k}_{(\mathbf{q})}=\frac{2\pi}{L}(1,0,0)$, $\mathbf{v}=0$. The fitting range is $T=8\:...\:20$ and $t=4\:...\:(T-4)$ (not all data shown for legibility).[]{data-label="fig:g2_s13_sl_q_1_0_0_p_0_0_0_v_00"}](g5_g0_sl_q_0_0_0_p_0_0_0_v_00.eps){width="\linewidth"}
![Three-point correlator $\langle \:\:\Phi_{K^*} \:\:\: \bar s \sigma_{1 3} b
\:\:\: \Phi_B^\dag\:\: \rangle$at $\mathbf{k}_{(\mathbf{p})}=0$, $\mathbf{k}_{(\mathbf{q})}=\frac{2\pi}{L}(1,0,0)$, $\mathbf{v}=0$. The fitting range is $T=8\:...\:20$ and $t=4\:...\:(T-4)$ (not all data shown for legibility).[]{data-label="fig:g2_s13_sl_q_1_0_0_p_0_0_0_v_00"}](g2_s13_sl_q_1_0_0_p_0_0_0_v_00.eps){width="\linewidth"}
![Three-point correlator $\langle \:\:\Phi_{K^*} \:\:\: \bar s \sigma_{1 3}
b \:\:\: \Phi_B^\dag\:\: \rangle$at $\mathbf{k}_{(\mathbf{p})}=\frac{2\pi}{L}(1,0,0)$, $\mathbf{k}_{(\mathbf{q})}=\frac{2\pi}{L}(2,0,0)$, $\mathbf{v}=(0.4,0,0)$. No reasonable fit was achieved yet.[]{data-label="fig:g2_s13_sl_q_2_0_0_p_1_0_0_v_04"}](g5_g0_sl_q_2_0_0_p_1_0_0_v_04.eps){width="\linewidth"}
![Three-point correlator $\langle \:\:\Phi_{K^*} \:\:\: \bar s \sigma_{1 3}
b \:\:\: \Phi_B^\dag\:\: \rangle$at $\mathbf{k}_{(\mathbf{p})}=\frac{2\pi}{L}(1,0,0)$, $\mathbf{k}_{(\mathbf{q})}=\frac{2\pi}{L}(2,0,0)$, $\mathbf{v}=(0.4,0,0)$. No reasonable fit was achieved yet.[]{data-label="fig:g2_s13_sl_q_2_0_0_p_1_0_0_v_04"}](g2_s13_sl_q_2_0_0_p_1_0_0_v_04.eps){width="\linewidth"}
![The form factors $T_1$, $T_2$. The points at lowest $q^2$ have $\mathbf{v}=(0.2,0,0)$, $\mathbf{k}_{(\mathbf{p})}=\frac{2\pi}{L}(1,0,0)$, $\mathbf{k}_{(\mathbf{q})}=\frac{2\pi}{L}(2,0,0)$.[]{data-label="fig:T2"}](f0_f+_fT.eps){width="\linewidth"}
![The form factors $T_1$, $T_2$. The points at lowest $q^2$ have $\mathbf{v}=(0.2,0,0)$, $\mathbf{k}_{(\mathbf{p})}=\frac{2\pi}{L}(1,0,0)$, $\mathbf{k}_{(\mathbf{q})}=\frac{2\pi}{L}(2,0,0)$.[]{data-label="fig:T2"}](T1_T2.eps){width="\linewidth"}
Conclusions
===========
Using moving-NRQCD and AsqTad actions, we have calculated matching coefficients for the heavy-light axial-, vector- and tensor currents in the static limit using 1-loop lattice perturbation theory, and performed first computations of form factors for rare $B$ decays. While moving NRQCD significantly reduces discretisation errors at low $q^2$, our initial results suffer from large statistical errors, overshadowing the advantages of the method. However, statistical errors do not constitute a fundamental obstacle and can be reduced further. The first step will be to extend the fitting range and include more exponentials. Then, we plan to work with random-wall sources, which were shown to provide considerable improvement for semileptonic decays at high recoil momentum [@Davies:2007vb]. We will also use smeared interpolating fields to reduce contributions from excited states, thereby improving the fits. Furthermore, note that our initial computations were done with lattice momenta pointing in the 1-direction only. Off-axial momenta and boost velocities will also allow lower values for $q^2$, for example by using the final meson momentum $\mathbf{p'}=\frac{2\pi}{L}(-1,-1,\:\:0)$.
Once statistical errors are under control, we will study the dependence on the lattice spacing and the light quark masses. We also plan to include $\mathcal{O}(\Lambda_{QCD}/m_b)$ operators in the matching calculations.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has made use of the resources provided by the Edinburgh Compute and Data Facility (which is partially supported by the eDIKT initiative), the Fermilab Lattice Gauge Theory Computational Facility and the Cambridge High Performance Computing Facility. We thank the MILC collaboration for making their gauge configurations publicly available.
[99]{}
E. Dalgic, A. Gray, M. Wingate, C. T. H. Davies, G. P. Lepage and J. Shigemitsu, Phys. Rev. D [**73**]{}, 074502 (2006) \[Erratum-ibid. D [**75**]{}, 119906 (2007)\] \[[arXiv:hep-lat/0601021](http://arxiv.org/abs/hep-lat/0601021)\] S. Meinel, R. Horgan, L. Khomskii, L. C. Storoni and M. Wingate, PoS [**LAT2007**]{}, 377 (2007) \[[arXiv:0710.3101 \[hep-lat\]](http://arxiv.org/abs/0710.3101)\]. \[HPQCD and UKQCD Collaborations\], *Moving NRQCD for High Recoil Form Factors in Heavy Quark Physics*, in preparation.
M. Wingate, J. Shigemitsu, C. T. H. Davies, G. P. Lepage and H. D. Trottier, Phys. Rev. D [**67**]{}, 054505 (2003) \[[arXiv:hep-lat/0211014](http://arxiv.org/abs/hep-lat/0211014)\]. A. Hart, G. M. von Hippel, R. R. Horgan and L. C. Storoni, J. Comput. Phys. [**209**]{}, 340 (2005) \[[arXiv:hep-lat/0411026](http://arxiv.org/abs/hep-lat/0411026)\]. A. Dougall, C. T. H. Davies, K. M. Foley and G. P. Lepage \[HPQCD Collaboration and UKQCD Collaboration\], Nucl. Phys. Proc. Suppl. [**140**]{}, 431 (2005) \[[arXiv:hep-lat/0409088](http://arxiv.org/abs/hep-lat/0409088)\]. L. Khomskii, *Perturbation Theory for Quarks and Currents in Moving NRQCD on a Lattice*, PhD thesis, in preparation.
C. T. H. Davies, E. Follana, K. Y. Wong, G. P. Lepage and J. Shigemitsu, PoS [**LAT2007**]{}, 378 (2007) \[[arXiv:0710.0741 \[hep-lat\]](http://arxiv.org/abs/0710.0741)\].
[^1]: For the matching calculation, we treat the light quark as massless; in this limit the matching coefficients for the last two operators in Tab. \[tab:formfact\] are equal due to chiral symmetry.
|
---
abstract: 'Density-functional theory has been one of the most successful approaches ever to address the electronic-structure problem; nevertheless, since its implementations are by necessity approximate, they can suffer from a number of fundamental qualitative shortcomings, often rooted in the remnant electronic self-interaction present in the approximate energy functionals adopted. Functionals that strive to correct for such self-interaction errors, such as those obtained by imposing the Perdew-Zunger self-interaction correction \[Phys. Rev. B [**23**]{}, 5048 (1981)\] or the generalized Koopmans’ condition \[Phys. Rev. B [**82**]{}, 115121 (2010)\], become orbital dependent or orbital-density dependent, and provide a very promising avenue to go beyond density-functional theory, especially when studying electronic, optical and dielectric properties, charge-transfer excitations, and molecular dissociations. Unlike conventional density functionals, these functionals are not invariant under unitary transformations of occupied electronic states, which leave the total charge density intact, and this added complexity has greatly inhibited both their development and their practical applicability. Here, we first recast the minimization problem for non-unitary invariant energy functionals into the language of ensemble density-functional theory \[Phys. Rev. Lett. [**79**]{}, 1337 (1997)\], decoupling the variational search into an inner loop of unitary transformations that minimize the energy at fixed orbital subspace, and an outer-loop evolution of the orbitals in the space orthogonal to the occupied manifold. Then, we show that the potential energy surface in the inner loop is far from convex parabolic in the early stages of the minimization and hence minimization schemes based on these assumptions are unstable, and present an approach to overcome such difficulty. The overall formulation allows for a stable, robust, and efficient variational minimization of non-unitary-invariant functionals, essential to study complex materials and molecules, and to investigate the bulk thermodynamic limit, where orbitals converge typically to localized Wannier functions. In particular, using maximally localized Wannier functions as an initial guess can greatly reduce the computational costs needed to reach the energy minimum while not affecting or improving the convergence efficiency.'
author:
- 'Cheol-Hwan Park$^{1,2}$'
- 'Andrea Ferretti$^{2,3,4}$'
- 'Ismaila Dabo$^{5}$'
- 'Nicolas Poilvert$^{1}$'
- 'Nicola Marzari$^{1,2}$'
title: 'Variational Minimization of Orbital-dependent Density Functionals'
---
I. Introduction
===============
Density functional theory (DFT) [@HohenbergKohn64; @KohnSham65] has become the basis of much computational materials science today, thanks to its predictive accuracy in describing ground-state properties directly from first principles. While DFT is in principle exact, in any practical implementation it requires an educated guess for the exact form of the energy functional. For many years, local or semi-local approximations to the exchange-correlation energy, such as the local density approximation (LDA) [@CeperleyAlder80; @perdew_zunger] or the generalized gradient approximation [@PerdewBurkeErnzerhof96] have been successfully applied to a wealth of different systems [@marzari_mrs]. Still, these approximations lead to some dramatic failures, including the overestimation of dielectric response, incorrect chemical barriers for reactions involving strongly-localized orbitals [@kulik:prl; @zhou:prb], energies of dissociating molecular species, and excitation energies of charge-transfer complexes, to name a few [@cohen_insights_2008].
Key to these failures is the self-interaction error of approximate DFT [@cohen_insights_2008; @perdew_zunger], where the electrostatic and exchange-correlation contributions to the effective energy of the entire charge distribution are not “purified” from this spurious self interaction of an individual electron with itself. To address this issue, Perdew and Zunger (PZ) introduced first an elegant solution to this problem, where a self-interaction correction (SIC) is added to the total energy calculated from approximate DFT (e.g. within the LDA [@CeperleyAlder80; @perdew_zunger]), but practical applications have remained scarce [@svane:prl; @hughes_lanthanide_2007; @stengel_spaldin; @capelle; @kuemmel; @Baumeier; @Filippetti; @ruzsinszky_density_2007; @suraud; @sanvito].
An important property of DFT with local or semi-local exchange-correlation functionals is the invariance of the total energy with respect to unitary transformation of the occupied electronic states. However, SIC-DFT does not have this invariance property, and in fact finding the optimal unitary transformation given a set of orbital wavefunctions is crucial to the numerically consistent minimization of density functionals with SIC [@pederson_local-density_1984; @svane:1996; @svane:2000; @goedecker_umrigar; @vydrov_scuseria; @stengel_spaldin; @klupfel]. In this paper, we focus on the variational minimization of energy functionals that do not satisfy unitary invariance in order to provide a stable, robust, and efficient determination of the electronic structure in this challenging case. In particular, we adopt the formulation of ensemble DFT [@eDFT] to decouple the variational minimization into an inner loop of unitary transformations and an outer loop of evolution for the occupied manifold, and suggest optimal strategies for the dynamics of unitary transformations. In the solid-state limit, this dynamics gives rise to a localized Wannier representation for the electronic states, and we assess their relation with maximally localized Wannier functions (MLWFs) [@marzari_vanderbilt; @souza_marzari_vanderbilt] as obtained in the absence of SIC.
The remainder of the paper is organized as follows. In Sec. II, DFT with SIC is briefly reviewed, the method of inner-loop minimization is explained, and the issue of using MLWFs as an initial guess for the wavefunctions is discussed. In Sec. III, we present and discuss the results. First, we present results on how the total energy varies with the unitary transformation of the occupied electronic states. Second, we discuss the stability and efficiency of our method for inner-loop minimization. Finally, we show how the calculated total energy converges both as a function of the outer-loop iterations and as a function of the CPU time and discuss the optimal scheme for total energy minimization of energy functionals with SIC. We then summarize our findings in Sec. IV.
II. Methodology
===============
A. Background
-------------
For simplicity, we consider in the following the wavefunctions to be real; however, the discussion can straightforwardly be extended to complex wavefunctions. The total energy of the interacting electron system from Kohn-Sham DFT within the LDA is given by [@KohnSham65] $$\begin{aligned}
&&E_{\rm LDA}[\{\psi_{\sigma i}\}]\nonumber\\
&&=-\sum_{\sigma}\sum_{i=1}^N\frac{1}{2}\int\psi_{\sigma i}({\bf r})\nabla^2
\psi_{\sigma i}({\bf r})\,d{\bf r}
+\int V_{\rm ext}({\bf r})\rho({\bf r})d{\bf r}
\nonumber\\
&&+\frac{1}{2}\int\int\frac{\rho({\bf r})\rho({\bf r}')}{|{\bf r}-{\bf r}'|}\,
d{\bf r}\,d{\bf r}'
+\int\epsilon_{\rm xc}^{\rm LDA}(\rho({\bf r}))\,\rho({\bf r})d{\bf r}\,,
\label{eq:E_LDA}\end{aligned}$$ where $\sigma$ is the spin index, the band index $i$ runs through the $N$ occupied electronic states, and $\rho({\bf r})=\sum_{\sigma}\sum_{i=1}^N|\psi_{\sigma i}({\bf r})|^2$ is the total charge density. The first term on the right hand side of Eq. (\[eq:E\_LDA\]) is the kinetic energy, the second term the interaction energy between electrons and the ion cores, the third term the Hartree interaction energy, and the last term the exchange-correlation energy. This energy functional $E_{\rm LDA}[\{\psi_{\sigma i}\}]$ is invariant under the following unitary transformation $$\psi'_{\sigma i}({\bf r}) = \sum_{j=1}^N\psi_{\sigma j}({\bf r})\, O_{\sigma ji}
\label{eq:U}$$ for an arbitrary unitary matrix $O_{\sigma}$ since the total charge density $\rho({\bf r})$ and the kinetic energy \[Eq. (\[eq:E\_LDA\])\] are invariant under this transformation. Given that the wavefunctions are real, we consider $O_\sigma$ to be an orthogonal matrix, i.e., real and satisfying $O_\sigma^{\rm t}O_\sigma=I$ where $I$ is the $N\times N$ identity matrix.
For some density functionals with SIC [@perdew_zunger; @dabo:NK], the total energy $E_{\rm total}[\{\psi_{\sigma i}\}]$ is given by $$E_{\rm total}[\{\psi_{\sigma i}\}]=E_{\rm LDA}[\{\psi_{\sigma i}\}]
+E_{\rm SIC}[\{\rho_{\sigma i}\}]\,,
\label{eq:E_total}$$ where $\rho_{\sigma i}({\bf r})=|\psi_{\sigma i}({\bf r})|^2$. $E_{\rm SIC}[\{\rho_{\sigma i}\}]$ and hence $E_{\rm total}[\{\psi_{\sigma i}\}]$ are in general not invariant under orthogonal transformations because they are dependent not only on the total charge density $\rho({\bf r})$, which is invariant under orthogonal or unitary transformation, but also on the charge densities, $\rho_{\sigma i}({\bf r})$’s, arising from different orbitals.
This can be seen by considering how the SIC energy varies under the orthogonal transformation of Eq. (\[eq:U\]). To this end, it is useful to recall that an orthogonal matrix $O_{\sigma}$ can be written as $$O_{\sigma}=e^{A_\sigma}
\label{eq:O_eA}$$ where $A_\sigma$ is an antisymmetric matrix; if we further consider the case where the norm of $A_\sigma$ is much less than that of an identity matrix, we can assume $$O_{\sigma}\approx I+A_\sigma\,.
\label{eq:O_eA2}$$ Therefore, the transformed wavefunctions are given by $$\psi'_{\sigma j}({\bf r})\approx\psi_{\sigma j}({\bf r})+\sum_{i=1}^N\psi_{\sigma i}({\bf r}) A_{\sigma ij}\,,
\label{eq:A}$$ from which $$\frac{\partial\rho_{\sigma j}({\bf r})}{\partial A_{\sigma ij}}
=2\psi_{\sigma j}({\bf r})\psi_{\sigma i}({\bf r})\,,
\label{eq:drhodA}$$ and (using the antisymmetry of $A_\sigma$) $$\frac{\partial\rho_{\sigma i}({\bf r})}{\partial A_{\sigma ij}}
=-2\psi_{\sigma j}({\bf r})\psi_{\sigma i}({\bf r})\,.
\label{eq:drhodA2}$$ Finally, if we define the SIC potential $$v^{\rm SIC}_{\sigma i}({\bf r})=\frac{\delta E_{\rm SIC}}{\delta \rho_{\sigma i}({\bf r})}\,,
\label{eq:vsic}$$ we obtain the gradient of SIC energy with respect to the transformation matrix elements $$\begin{aligned}
&&G_{\sigma ij}
\equiv\frac{\partial E_{\rm SIC}}{\partial A_{\sigma ij}}\nonumber\\
&&=2\int \psi_{\sigma i}({\bf r})
\left[v^{\rm SIC}_{\sigma j}({\bf r})-v^{\rm SIC}_{\sigma i}({\bf r})\right]
\psi_{\sigma j}({\bf r})\,d{\bf r}\,,
\label{eq:pederson}\end{aligned}$$ which is a result originally obtained by Pederson [*et al.*]{} [@pederson_local-density_1984]. Note that this gradient matrix $G_\sigma$ is also antisymmetric, just like $A_\sigma$. Therefore, at an energy minimum, the wavefunctions satisfy $$0
=\int \psi_{\sigma i}({\bf r})
\left[v^{\rm SIC}_{\sigma j}({\bf r})-v^{\rm SIC}_{\sigma i}({\bf r})\right]
\psi_{\sigma j}({\bf r})\,d{\bf r}\,,
\label{eq:pederson2}$$ which was referred to as the “localization condition” by Pederson [*et al.*]{} [@pederson_local-density_1984].
To date, the most widely used SIC scheme is PZ SIC [@perdew_zunger] (and its few refinements, e.g., Refs. [@Filippetti; @lundin_eriksson; @davezac]). In PZ scheme, the SIC energy is given by $$\begin{aligned}
E^{\rm PZ}_{\rm SIC}[\{\rho_{\sigma i}\}]&=&-\sum_{\sigma}\sum_{i=1}^N
\frac{1}{2}\int\int\frac{\rho_{\sigma i}({\bf r})\rho_{\sigma i}({\bf r}')}{|{\bf r}-{\bf r}'|}\,
d{\bf r}\,d{\bf r}'\nonumber\\
&-&\sum_{\sigma}\sum_{i=1}^N\int\epsilon_{\rm xc}^{\rm LDA}(\rho_{\sigma i}({\bf r}))\,\rho_{\sigma i}({\bf r})d{\bf r}\,.
\label{eq:E_SIC_PZ}\end{aligned}$$ The rationale underlying PZ SIC is both simple and beautiful: correcting the total energy by subtracting the incorrect energy contributions from the interaction of an electron with itself — i.e., the Hartree, exchange, and correlation energies. Hence PZ SIC is exact for one-electron systems, or in the limit where the total charge density can be decomposed into non-overlapping one-electron charge density contributions.
Recently, an alternative scheme suitable for many-electron systems based on the generalized Koopmans condition [@koopmans] was introduced in Ref. [@dabo:NK]. In brief, one could start from Janak’s theorem [@janak] that states that in DFT the orbital energy $\epsilon_{\sigma i}(f)$ with fractional occupation of a state being $f_{\sigma i}=f$ is $$\epsilon_{\sigma i}(f)=\left.\frac{dE_{\sigma i}(f')}{df'}\right|_{f'=f}\,,$$ where $E_{\sigma i}$ is the Kohn-Sham total energy minimized under the constraint $f_{\sigma i}=f$. If there were no self-interaction, the orbital energy of a state $\epsilon_{\sigma i}(f)$ would not change upon varying its own occupation $f$. In other words, for a self-interaction-free functional, $$\epsilon_{\sigma i}(f)={\rm constant}\,\,\,(0\le f\le1)\,.
\label{eq:NK_condition}$$ Alternatively, using Janak’s theorem [@janak], this can be rewritten as $$\begin{aligned}
\Delta E^{\rm Koopmans}_{\sigma i}(f)\equiv
E_{\sigma i}(f_{\sigma i})-E_{\sigma i}(0)
=f_{\sigma i}\,\epsilon_{\sigma i}(f)\nonumber\\
(0\le f\le1)\,,
\label{eq:delta_E}\end{aligned}$$ which is equivalent to the generalized Koopmans theorem [@dabo:NK], telling us that the total energy varies linearly with the fractional occupation $f_{\sigma i}$. In conventional DFT, however, Eq. (\[eq:NK\_condition\]) or Eq. (\[eq:delta\_E\]) does not hold and instead, $$\Delta E_{\sigma i}\equiv
E_{\sigma i}(f_{\sigma i})-E_{\sigma i}(0)
=\int_0^{f_{\sigma i}}\epsilon_{\sigma i}(f')\,df'\,.
\label{eq:delta_E2}$$ From Eqs. (\[eq:delta\_E\]) and (\[eq:delta\_E2\]), the non-Koopmans (NK) energy $\Pi_{\sigma i}(f)$ – i.e., the deviation from the linearity for the energy versus occupation – can be defined as [@dabo:NK] $$\begin{aligned}
\Pi_{\sigma i}(f)&=&
\Delta E^{\rm Koopmans}_{\sigma i}(f)-\Delta E_{\sigma i}\nonumber\\
&=&\int_0^{f_{\sigma i}}
\left[\epsilon_{\sigma i}(f)-\epsilon_{\sigma i}(f')\right]\,df'\,.
\label{eq:Pi}\end{aligned}$$ From this result, the SIC energy term based on the generalized Koopmans theorem has been defined as $$E_{\rm SIC}^{\rm NK}[\{\rho_{\sigma i}\}]=
\sum_\sigma \sum_{i=1}^N\Pi_{\sigma i}(f_{\rm ref})\,,
\label{eq:NKSIC}$$ where $f_{\rm ref}$ is a reference occupation factor (for many-electron systems, $f_{\rm ref}=\frac{1}{2}$ was shown to be the best choice [@dabo:NK]).
The total energy versus (fractional) number of electrons relation calculated by exact DFT should be piecewise linear with slope discontinuities at integral electron occupations [@piecewise]; however, within the LDA, this energy versus occupation relation is piecewise convex [@cohen_insights_2008]. The LDA deviation from the piecewise linearity is the main reason for the failures of approximate DFTs [@cohen_insights_2008]. The new SIC functional \[Eq. (\[eq:NKSIC\])\] is introduced to cure this pathology and to recover the piecewise linearity of exact DFT [@dabo:NK]. The (bare) NK SIC discussed above and its screened version explain some of the most important material properties such as ionization energy and electron affinity better than PZ SIC. We refer the reader to Ref. [@dabo:NK] for the details of NK SIC.
B. Implementation
-----------------
In order to implement a variational minimization of the total energy functional, we adopt the same strategy as the ensemble-DFT approach [@eDFT], decoupling the dynamics of orbital rotations in the occupied subspace and that of orbital evolution in the manifold orthogonal to the occupied subspace. In explicit terms, we minimize the SIC energy through $$\min_{\{\psi'_{\sigma i}\}}E_{\rm SIC}[\{\psi'_{\sigma i}\}]
=\min_{\{\psi_{\sigma i}\}}\left(\min_{\{O_{\sigma}\}}E_{\rm SIC}[\{\psi_{\sigma i}\},\{O_\sigma\}]\right)\,,
\label{eq:twoloop}$$ where $\{\psi'_{\sigma i}\}$ and $\{\psi_{\sigma i}\}$ are connected by an orthogonal transformation $\{O_{\sigma}\}$ \[Eq. (\[eq:U\])\]. Minimization over the basis orbital wavefunctions $\{\psi_{\sigma i}\}$ and that over the orthogonal transformation $\{O_{\sigma i}\}$ – inside the round parenthesis in Eq. (\[eq:twoloop\]) – correspond to the outer-loop minimization and inner-loop minimization, respectively, i.e., given the orbital wavefunctions, an optimal orthogonal transformation is searched and then the orbital wavefunctions are evolved. This process is repeated until convergence. Ensemble-DFT minimization has also been discussed in studying the SIC problem by Stengel and Spaldin [@stengel_spaldin] and by Klüpfel, Klüpfel, and Jónsson [@klupfel].
The main focus here is on inner-loop minimization. The gradient matrix $G_{\sigma ij}=\partial E_{\rm SIC}/\partial A_{\sigma ij}$ in Eq. (\[eq:pederson\]) is antisymmetric and real; hence, $-i\, G_{\sigma }$ is Hermitian (and purely imaginary). Therefore, $-i\,G_\sigma$ can be diagonalized as $$-i\,G_{\sigma}=U_\sigma^\dagger\, D_\sigma\, U_\sigma\,,$$ or, $$G_{\sigma}=i\,U_\sigma^\dagger\, D_\sigma\, U_\sigma\,,
\label{eq:G}$$ where $U_\sigma$ is a unitary matrix and $$D_{\sigma ij}=\lambda_{\sigma i}\,\delta_{ij}
\label{eq:D}$$ a real diagonal matrix. From Eq. (\[eq:G\]), we evolve the matrix $A_\sigma$ along the energy gradient with a step of size $l$ $$\Delta A_{\sigma}=-l\,G_\sigma
=-i\,l\,U_\sigma^\dagger\,D_\sigma\,U_\sigma\,,
\label{eq:dA}$$ calculate the updated orthogonal matrix $$O_\sigma=e^{\Delta A_{\sigma}}=
U_\sigma^\dagger\,e^{-i\,l\,D_\sigma}\,U_\sigma\,,
\label{eq:O_update}$$ and then transform the wavefunctions accordingly.
Here, we use the steepest-descent method for the inner-loop minimization. But one could employ other methods such as damped dynamics or conjugate gradients. In each of the inner-loop steps, we evaluate the SIC energy with two different sets of wavefunctions: first by using the given wavefunctions \[$E_{\rm SIC}(l=0)$\] and second by using the wavefunctions transformed by $O_\sigma$ in Eq. (\[eq:O\_update\]) with a trial step $l=l_{\rm trial}$ \[$E_{\rm SIC}(l=l_{\rm trial})$\]. In addition, the gradient at $l=0$ reads $$\begin{aligned}
\left.\frac{dE_{\rm SIC}(l)}{dl}\right|_{l=0}
&=&\frac{1}{2}\sum_{\sigma i j}\left[\frac{\partial E_{\rm SIC}}{\partial A_{\sigma ij}}\,
\frac{d\Delta A_{\sigma i j}}{dl}\right]_{l=0}\nonumber\\
&=&-\frac{1}{2}\sum_{\sigma ij}|G_{\sigma ij}|^2
\,,
\label{eq:dEdl}\end{aligned}$$ where we have used Eqs. (\[eq:pederson\]) and (\[eq:dA\]), and the fact that only half of the matrix elements of $G_\sigma$ are independent. Thus, knowing $E_{\rm SIC}(l=0)$, $E_{\rm SIC}(l=l_{\rm trial})$, and $dE_{\rm SIC}(l)/dl|_{l=0}$, we can fit a parabola to $E_{\rm SIC}(l)$, yielding the optimal step $l=l_{\rm optimal}$ and the energy minimum $E_{\rm SIC}(l=l_{\rm optimal})$. This completes one inner-loop iteration. We then use the transformed wavefunctions to calculate the gradient \[Eq. (\[eq:pederson\])\] and repeat iterations until the SIC energy converges.
For optimal convergence, we set the step size based on the highest frequency component of the gradient matrix, i.e., $$l=\gamma\,l_{\rm c}\,\,\,\,\,\,\left(l_{\rm c}=\frac{\pi}{\lambda_{\max}}\right)\,,
\label{eq:l_c}$$ where $\gamma$ is a constant of order $\sim 0.1$ and $\lambda_{\rm max}$ the maximum eigenvalue of $D_\sigma$, $$\lambda_{\max}=\,\max_{\sigma i}\, \lambda_{\sigma i}\,.
\label{eq:lambda_max}$$ The critical step $l_{\rm c}$ should be considered as the point when the transformed wavefunctions become appreciably different from the original wavefunctions. Therefore, when we evolve wavefunctions by using a step much larger than $l_{\rm c}$, a fitting of $E_{\rm SIC}$ versus $l$ by a parabola will not be successful. Imposing the constraint $l=\gamma\,l_{\rm c}$ \[Eq. (\[eq:l\_c\])\] when necessary is the key part of our method: (i) We set the trial step of the first iteration of the inner-loop minimization according to Eq. (\[eq:l\_c\]). (In subsequent iterations, the trial step $l_{\rm trial}$ is set based on the optimal step of the previous iteration: we set it to be twice the optimal step of the previous iteration.) By setting the initial trial step based on the eigenspectrum of the gradient matrix, we make the inner-loop process unaffected by the absolute magnitude of the SIC energy gradient with respect to the orthogonal transformation \[Eq. (\[eq:pederson\])\]. (ii) When the calculated optimal step is larger than $\gamma\,l_{\rm c}$, we set $l_{\rm optimal}=\gamma\,l_{\rm c}$. This procedure has proven to be instrumental when $E_{\rm SIC}(l)$ versus $l$ relation cannot be fitted well by a parabola. In such cases, the calculated $l_{\rm optimal}$ can be much larger than $l_{\rm c}$. A similar scaling method based on the highest frequency component of the gradient matrix was used in finding the MLWFs [@marzari_vanderbilt; @Mostofi2008685].
C. MLWFs as an initial guess for the wavefunctions
--------------------------------------------------
SIC tends to localize the orbital wavefunctions \[note e.g., that the Hartree term in Eq. (\[eq:E\_SIC\_PZ\]) will be more negative if the state becomes more localized\]. Therefore, it is natural to consider using some localized basis functions as an initial guess for the wavefunctions of density functionals with SIC. To this end, employing MLWFs [@marzari_vanderbilt; @souza_marzari_vanderbilt] represents a very promising initial-guess strategy. Although the possibility of using MLWFs in this regard was recently suggested [@tsemekhman], no literature is available on the merit of that scheme. We address this issue in conjunction with the inner-loop minimization method discussed in the previous subsection.
D. Computational details
------------------------
We performed DFT calculations with norm-conserving pseudopotentials [@TroullierMartins91] in the LDA [@perdew_zunger] using the Car-Parrinello (CP) code of the Quantum ESPRESSO distribution [@baroni:2006_Espresso] with the inner-loop minimization described in the previous subsections, and a conventional damped dynamics algorithm for the outer-loop minimization. We have performed calculations on both PZ SIC [@perdew_zunger] and NK SIC [@dabo:NK]. Except for the case of investigating the effect of using MLWFs as an initial guess for the wavefunctions, we have used LDA wavefunctions with some arbitrary phases – they are not LDA eigenstates – when we start the calculations.
We performed calculations on a rather big molecule, C$_{20}$ fullerene. A supercell geometry was used with the minimum distance between the carbon atoms in neighboring supercells larger than 6.7 Å. The Coulomb interaction is truncated to prevent spurious interaction between periodic replicas in different supercells [@dabo:Coulomb; @dabo:arXiv].
III. Results and Discussion
===========================
{width="1.6\columnwidth"}
In order to find an optimal strategy for the minimization of SIC DFT, it is important to know how the energy varies with orthogonal transformations. We first show the energy variation along the direction in the orthogonal transformation space parallel to the gradient \[Eq. (\[eq:pederson\])\] of the energy, i.e., $E_{\rm SIC}(l)$ versus $l$, where $l$ is a step representing the amount of orthogonal rotation as defined in Eq. (\[eq:dA\]). Figure \[Fig1\](a) shows the results for PZ SIC at a few different stages during the inner-loop minimization. What we can see is that initially $E_{\rm SIC}^{\rm PZ}(l)$ varies slowly with $l$, and then, in the middle of the inner-loop minimization, varies fast and then, toward the end of the minimization, varies slowly again. There is no good length scale of $l$ which can consistently describe the variation of $E_{\rm SIC}(l)$ during the entire process of an inner-loop minimization. The speed of the energy variation at different stages of the inner-loop minimization with respect to $l$ near $l=0$ can however be very well explained by $\lambda_{\rm max}$ \[Eq. (\[eq:lambda\_max\])\], which is the fastest frequency component of the gradient matrix \[Eq. (\[eq:pederson\])\], as shown in Fig. \[Fig1\](b).
We can draw similar conclusions for NK SIC as shown in Figs. \[Fig1\](c) and \[Fig1\](d). However, there are a few points that are worth mentioning. First, the magnitude of NK SIC energy is several times smaller than that of PZ SIC energy \[Figs. \[Fig1\](a) and \[Fig1\](c)\]. Second, $\lambda_{\rm max}$, or the main driving force for orthogonal transformation near $l=0$, for NK SIC is also much smaller than that for PZ SIC, although eventually both of them converge to zero at energy minima. Because of these differences between different SIC functionals, it is clear that determining the trial step $l_{\rm trial}$ based on $\lambda_{\rm max}$ will be very useful, even more so because $\lambda_{\rm max}$ is also affected by the arbitrary initial phases of the wavefunctions, as will be discussed later (Fig. \[Fig6\]).
![ (a) Unitary variant part of the total energy within PZ SIC, $E_{\rm SIC}^{\rm PZ}$ \[Eq. (\[eq:E\_total\])\], versus $l\,\lambda_{\rm max}/\pi$ \[Eq. (\[eq:lp\])\] for C$_{20}$ at a few inner-loop iteration steps. (b) Similar quantity as in (a) for NK SIC.[]{data-label="Fig2"}](Fig2.eps){width="0.8\columnwidth"}
Based on the previous discussion, we now show, in Fig. \[Fig2\], $E_{\rm SIC}(l)$ as a function of the scaled step $$l_{\rm scaled}\,\equiv\,{l}\,/\,{l_{\rm c}}\,,
\label{eq:lp}$$ i.e., $l$ in units of $l_{\rm c}$. For both PZ SIC and NK SIC, the energy variation length scale near $l=0$ through the entire process of the inner-loop minimization is $\sim0.5$ in units of $l_{\rm scaled}$. The results confirm that indeed a natural length scale for $l$ that should be used in the inner-loop minimization is the $l_{\rm c}$ defined in Eq. (\[eq:l\_c\]). One more thing to note here is that in both PZ SIC and NK SIC, at the initial stages of the inner-loop iterations, the energy profile cannot be well fitted by a parabola. This trend is dramatic especially for NK SIC, where the $E_{\rm SIC}(l)$ versus $l$ (or $l_{\rm scaled}$) relation is concave, not convex, at $l=0$.
![ (a) Unitary variant part of the total energy within PZ SIC, $E_{\rm SIC}^{\rm PZ}$ \[Eq. (\[eq:E\_total\])\], versus $l\,\lambda_{\rm max}/\pi$ \[Eq. (\[eq:lp\])\] for a carbon atom at a few inner-loop iteration steps. []{data-label="Fig3"}](Fig3.eps){width="0.8\columnwidth"}
This can be best understood using a simple system: a carbon atom which has, in our pseudopotential calculations, two orbitals ($2s$ and $2p$), i.e., it is a two-level system. The PZ SIC energy $E_{\rm SIC}^{\rm PZ}(l)$ versus $l_{\rm scaled}$ is shown in Fig. \[Fig3\]. The profile is sinusoidal with a period of 0.5, rather than parabolic for the entire process of minimization. Notably, the period 0.5 in units of $l_{\rm scaled}$ is similar to the previously discussed length scale for C$_{20}$ fullerene. The shape of the curve does not change as we proceed in the inner-loop minimization; the only variation is that the minimum of the curve moves toward the origin ($l_{\rm scaled}=0$).
We can understand this behavior as follows. The gradient matrix in Eq. (\[eq:pederson\]) for a carbon atom is of the form $$G=\left(
\begin{array}{cc}
0 & c\\
-c & 0
\end{array}\right)=i\,c\,\sigma_y
\,,
\label{eq:grad_C}$$ where $c$ is a real constant and $\sigma_y$ the Pauli matrix. (We dropped the spin index for obvious reasons.) Assuming (without losing generality) that $c>0$, the maximum eigenvalue of $G$ is $$\lambda_{\rm max}=c$$ and the orthogonal transformation matrix \[Eqs. (\[eq:dA\]) and (\[eq:O\_update\])\] is given by $$O=e^{-lG}=\cos\,(lc)\,I\,-\,i\,\sin\,(lc)\,\sigma_y\,,$$ or, using $l_{\rm scaled}$ \[Eq. (\[eq:lp\])\], $$O=\cos\,(\pi\,l_{\rm scaled})\,I\,-\,i\,\sin\,(\pi\,l_{\rm scaled})\,\sigma_y\,.$$ In particular, when $l_{\rm scaled}=0.5$, $O=-i\,\sigma_y$, and, under this orthogonal transformation $O$, $\psi_1'=-\psi_2$ and $\psi_2'=\psi_1$, i.e., $O$ just exchanges the two orbital wavefunctions (plus a trivial sign change). When the original wavefunctions $\psi_1$ and $\psi_2$ correspond to the maximum SIC energy configuration, the new set of wavefunctions $\psi_1'$ and $\psi_2'$ will correspond also to the SIC energy maximum. Therefore, the period of $E_{\rm SIC}(l)$ versus $l_{\rm scaled}$ will be 0.5 in agreement with our calculation \[Fig. \[Fig3\]\]. (The shape of the curve is not exactly sinusoidal and varies slightly with the kind of SIC used.)
For this example, which part of the sinusoidal-like curve one starts the inner-loop minimization from depends on the initial orbital wavefunctions (and an arbitrary rotation of them). If we start from the LDA eigenstates, the SIC energy is at its maximum (roughly speaking, the LDA eigenstates are the most delocalized and the SIC energy is highest) and the inner-loop minimization starts from the top of the sinusoidal-like curve, and hence (i) the driving force for the orthogonal transformation is extremely weak (zero at the maximum) and (ii) $E_{\rm SIC}(l)$ versus $l_{\rm scaled}$ is concave. For these reasons, if we do not properly scale $l$, or if we do not constrain $l$ during the inner-loop minimization process, the minimization process based on the assumption that the energy profile is convex parabolic may become unstable or extremely slow. This discussion is also relevant to other systems, as we have seen in the case of C$_{20}$ fullerene.
{width="1.6\columnwidth"}
Figure \[Fig4\](a) compares the performance of the inner-loop minimization for the case PZ SIC. In one case (dashed or blue curve), we take the optimal step size $l_{\rm optimal}$ obtained from fitting $E_{\rm SIC}^{\rm PZ}(l)$ versus $l$ by a parabola from three calculated quantities: $E_{\rm SIC}^{\rm PZ}(l=0)$, $E_{\rm SIC}^{\rm PZ}(l=l_{\rm trial})$, and $dE_{\rm SIC}^{\rm PZ}(l)/dl|_{l=0}$. In the other case (solid or red curve), if the calculated $l_{\rm optimal}$ is larger than $\gamma\,l_{\rm c}$ (with $\gamma=0.1$) \[Eq. (\[eq:l\_c\])\], we set $l_{\rm optimal}=\gamma\,l_{\rm c}$. Apparently, by using this constraint based on $l_{\rm c}$, or, $\lambda_{\rm max}$, the inner-loop minimization process becomes more stable and faster. (In both cases, the trial step of the first iteration was set to $l_{\rm trial}=\gamma\,l_{\rm c}$.) The difference between using and not using this $\lambda_{\rm max}$ constraint is dramatic for NK SIC \[Fig. \[Fig4\](b)\]. This again is due to (i) the small gradient of the SIC energy with respect to the variation of the orthogonal transformation, and (ii) non-concave-parabolic dependence of $E_{\rm SIC}(l)$ on $l$.
{width="1.6\columnwidth"}
Until now, our focus was on the inner-loop minimization. Now we look at the entire minimization procedure including the outer loop. In order to find an optimal minimization strategy, we have performed our calculations by restricting the number of inner-loop minimization iterations per each outer-loop iteration to be less than or equal to $n_{\rm max}$. (However, not every outer-loop iteration will require $n_{\rm max}$ inner-loop iterations because the SIC energy may be converged earlier during inner-loop minimization. We exit the inner loop if the energy difference between consecutive iterations is lower than the energy convergence threshold of $10^{-5}$ Ry.) The case without inner-loop minimization is denoted by $n_{\rm max}=0$. Figure \[Fig5\](a) shows the convergence of PZ SIC energy for various different choices of $n_{\rm max}$. In all cases where the inner-loop minimization routine is used (i.e., $n_{\rm max}>0$), the total number of outer-loop iterations necessary to achieve the same level of convergence is much smaller than that when no inner-loop minimization is used. This, however, does not necessarily mean that the total computation time is reduced. In Fig. \[Fig5\](b), we show the CPU time dependence of the SIC energy (the results include both the inner-loop and outer-loop minimization iterations). Surprisingly, in all cases other than $n_{\rm max}=1$, inner-loop minimization actually slows down the computation for PZ SIC. When we set $n_{\rm max}=1$, i.e., if the number of inner-loop iterations per each outer-loop iteration is restricted to 1, we find about twice improvement over when no inner-loop minimization is performed in terms of the CPU time.
The case of NK SIC is very different. Figures \[Fig5\](c) and \[Fig5\](d) shows that inner-loop minimization reduces not only the required number of outer-loop iterations but also the CPU time significantly. Especially, the CPU time is reduced by $\sim20$ times when we perform inner-loop minimization, and is rather insensitive to $n_{\rm max}$.
These results on PZ SIC and NK SIC support that the presented method works regardless of the absolute magnitude of the SIC energy gradient with respect to the orthogonal transformation \[Eq. (\[eq:pederson\])\]. The method can be applied to density functionals with other kinds of SIC. For example, SIC with screening, for which the total energy is given by $$E_{\rm total}=E_{\rm LDA}+\alpha\,E_{\rm SIC}\,\,\,(\alpha<1)\,,$$ will have the SIC energy gradient lower in magnitude than the unscreened version of SIC ($\alpha=1$), and our method will be more useful.
It has to be noted that the relative CPU time among different calculations shown in Fig. \[Fig5\] at different stages of the minimization is affected only by the ratio of the CPU time for one inner-loop iteration to that for one outer-loop iteration. Therefore, the relative CPU time is rather insensitive to the complexity of the system studied, and in that sense is meaningful. (The absolute CPU time is also affected much by the complexity of the system, the performance and number of processors, etc.) In our case, one inner-loop iteration for PZ SIC takes 3.6 times as long as one outer-loop iteration and one inner-loop iteration for NK SIC takes 2.0 times as long as one outer-loop iteration.
{width="1.6\columnwidth"}
Finally, we discuss how useful it is to use MLWFs [@marzari_vanderbilt; @souza_marzari_vanderbilt] as an initial guess for the wavefunctions [@klupfel]. The following description is relevant for both PZ SIC \[Figs. \[Fig6\](a) and \[Fig6\](b)\] and NK SIC \[Figs. \[Fig6\](c) and \[Fig6\](d)\] and whether or not the inner-loop minimization is employed. Figure \[Fig6\] shows that when MLWFs are used, the initial total energy is lower than when LDA wavefunctions with arbitrary phases is used. On the other hand, the slope of $\log[({\rm current\,\,\,total\,\,\, energy})-({\rm converged \,\,\,total\,\,\, energy})]$ versus either the number of outer-loop iterations or the relative CPU time is not very different in the two cases. Therefore, it is advantageous to use MLWFs as an initial guess for the wavefunctions; however, the lower the energy convergence threshold the smaller the relative advantage.
IV. Conclusions
===============
In summary, we have developed a variational, stable and efficient approach for the total-energy minimization of unitary variant functionals, as they appear in self-interaction corrected formulations, with a focus on properly minimizing the energy by unitary transformations of the occupied manifold. In particular, we have shown that the energy changes along the gradient direction can be very different from being convex parabolic, and suggested the use of the maximum frequency component of the gradient matrix in determining optimal rotations for the inner-loop minimization. When maximally localized Wannier functions are used as an initial guess for the wavefunctions, the initial energy decreases significantly from that corresponding to wavefunctions with arbitrary phases; however, the logarithmic energy convergence rate remains similar in the two cases. We expect that the results will be useful for investigating the physical properties of complex materials and big molecules with self-interaction corrected density functional theory.
We thank fruitful discussions with Peter Klüpfel and Simon Klüpfel. CHP acknowledges financial support from Intel Corporation.
[10]{}
P. Hohenberg and W. Kohn, Phys. Rev. [**136**]{}, B864 (1964).
W. Kohn and L. J. Sham, Phys. Rev. [**140**]{}, A1133 (1965).
D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. [**45**]{}, 566 (1980).
J. P. Perdew and A. Zunger, Phys. Rev. B [**23**]{}, 5048 (1981).
J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. [**77**]{}, 3865 (1996).
N. Marzari, Bull. Mater. Res. Soc. [**31**]{}, 681 (2006).
H. J. Kulik, M. Cococcioni, D. A. Scherlis, and N. Marzari, Phys. Rev. Lett. [**97**]{}, 103001 (2006).
F. Zhou, M. Cococcioni, C. A. Marianetti, D. Morgan, and G. Ceder, Phys. Rev. B [**70**]{}, 235121 (2004).
A. J. Cohen, P. [Mori-Sanchez]{}, and W. Yang, Science [**321**]{}, 792 (2008).
A. Svane and O. Gunnarsson, Phys. Rev. Lett. [**65**]{}, 1148 (1990).
I. D. Hughes, M. Dane, A. Ernst, W. Hergert, M. Luders, J. Poulter, J. B. Staunton, A. Svane, Z. Szotek, and W. M. Temmerman, Nature [**446**]{}, 650 (2007).
M. Stengel and N. A. Spaldin, Phys. Rev. B [**77**]{}, 155106 (2008).
D. Vieira and K. Capelle, J. Chem. Theor. Comput. [**6**]{}, 3319 (2010).
T. Koerzdoerfer, M. Mundt, and S. Kuemmel, Phys. Rev. Lett. [**100**]{}, 133004 (2008).
B. Baumeier, P. Kruger, and J. Pollmann, Phys. Rev. B [**73**]{}, 195205 (2006).
A. Filippetti and N. A. Spaldin, Phys. Rev. B [**67**]{}, 125109 (2003).
A. Ruzsinszky, J. P. Perdew, G. I. Csonka, O. A. Vydrov, and G. E. Scuseria, J. Chem. Phys. [**126**]{}, 104102 (2007).
C. A. Ullrich, P. G. Reinhard, and E. Suraud, Phys. Rev. A [**62**]{}, 053202 (2000).
C. Toher and S. Sanvito, Phys. Rev. Lett. [**99**]{}, 056801 (2007).
M. R. Pederson, R. A. Heaton, and C. C. Lin, J. Chem. Phys. [**80**]{}, 1972 (1984).
A. Svane, Phys. Rev. B [**53**]{}, 4275 (1996).
A. Svane, W. M. Temmerman, Z. Szotek, J. Laegsgaard, and H. Winter, Int. J. Quant. Chem. [**77**]{}, 799 (2000).
S. Goedecker and C. J. Umrigar, Phys. Rev. A [**55**]{}, 1765 (1997).
O. A. Vydrov and G. E. Scuseria, J. Chem. Phys. [**121**]{}, 8187 (2004).
P. Klüpfel, S. Klüpfel, and H. Jónsson, http://vefir.hi.is/para10/extab/para10-paper-150.pdf.
N. Marzari, D. Vanderbilt, and M. C. Payne, Phys. Rev. Lett. [**79**]{}, 1337 (1997).
N. Marzari and D. Vanderbilt, Phys. Rev. B [**56**]{}, 12847 (1997).
I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B [**65**]{}, 035109 (2001).
I. Dabo, A. Ferretti, N. Poilvert, Y. Li, N. Marzari, and M. Cococcioni, Phys. Rev. B [**82**]{}, 115121 (2010).
U. Lundin and O. Eriksson, Int. J. Quantum Chem. [**81**]{}, 247 (2001).
M. dÁvezac, M. Calandra, and F. Mauri, Phys. Rev. B [**71**]{}, 205210 (2005).
T. Koopmans, Physica [**1**]{}, 104 (1934).
J. F. Janak, Phys. Rev. B [**18**]{}, 7165 (1978).
J. P. Perdew, R. G. Parr, and J. L. [Balduz, Jr.]{}, Phys. Rev. Lett. [**49**]{}, 1691 (1982).
A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Comput. Phys. Comm. [**178**]{}, 685 (2008).
K. Tsemekhman, E. Bylaska, and H. Jónsson, http://users.physik.fu-berlin.de/$\sim$ag-gross/oep-workshop/Talks/OEP05Tsemekhman$\_$Talk.pdf.
N. Troullier and J. L. Martins, Phys. Rev. B [**43**]{}, 1993 (1991).
P. Giannozzi [*et al.*]{}, J. Phys.: Cond. Mat. [**21**]{}, 395502 (2009).
I. Dabo, B. Kozinsky, N. E. [Singh-Miller]{}, and N. Marzari, Phys. Rev. B [ **77**]{}, 115139 (2008).
Y. Li and I. Dabo, arXiv:1107.2047, submitted.
|
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abstract: 'The comparison studies of theoretical approaches to the description of the Casimir interaction in layered systems including graphene is performed. It is shown that at zero temperature the approach using the polarization tensor leads to the same results as the approach using the longitudinal density-density correlation function of graphene. An explicit expression for the zero-temperature transverse density-density correlation function of graphene is provided. We further show that the computational results for the Casimir free energy of graphene-graphene and graphene-Au plate interactions at room temperature, obtained using the temperature-dependent polarization tensor, deviate significantly from those using the longitudinal density-density correlation function defined at zero temperature. We derive both the longitudinal and transverse density-density correlation functions of graphene at nonzero temperature. The Casimir free energy in layered structures including graphene, computed using the temperature-dependent correlation functions, is exactly equal to that found using the polarization tensor.'
author:
- 'G. L. Klimchitskaya'
- 'V. M. Mostepanenko'
- 'Bo E. Sernelius'
title: 'Two approaches for describing the Casimir interaction with graphene: density-density correlation function versus polarization tensor'
---
Introduction
============
During the last few years graphene and other carbon-based nanostructures have attracted the particular attention of many experimentallists and theorists due to their remarkable properties [@1; @2]. These investigations have provided further impetus to technological progress. One of the topical subjects, which came to the experimental attention very recently [@3], is the van der Waals and Casimir interaction of graphene deposited on a substrate with the test body made of an ordinary material.
Theorists have already undertook a number of studies of graphene-graphene and graphene-material plate interactions using the Dirac model of graphene [@1], which assumes the linear dispersion relation for the graphene bands at low energies. Specifically, in Ref. [@4] the van der Waals coefficient for two graphene sheets at zero temperature was calculated using the correlation energy from the random phase approximation (in Ref. [@5] the obtained value was improved using the nonlocal dielectric function of graphene). In Ref. [@6], the van der Waals and Casimir forces between graphene and ideal metal plane were calculated at zero temperature using the Lifshitz theory, where the reflection coefficients of the electromagnetic oscillations were expressed via the polarization tensor in (2+1)-dimensions. The important progress was achieved in Ref. [@7], where the force at nonzero temperature between two graphene sheets and between a graphene and a material plate was expressed via the Coulomb coupling between density fluctuations. The density-density correlation function in the random-phase approximation has been used. It was shown [@7] that for graphene the relativistic effects are not essential, and that the thermal effects become crucial at much shorter separations than in the case of ordinary materials. In Ref. [@8], the graphene-graphene interaction was computed under an assumption that the conductivity of graphene can be described by the in-plane optical properties of graphite. It was shown [@8a] that for sufficiently large band gap parameter of graphene the thermal Casimir force can vary several-fold with temperature. In Ref. [@9], the reflection coefficients in the Lifshitz theory were expressed via the polarization tensor at nonzero temperature whose components were explicitly calculated. The detailed computations of graphene-graphene and graphene-real metal Casimir interactions using this method were performed [@9; @10; @11]. Finally, in Ref. [@12] the reflection coefficients of the Lifshitz theory were generalized for the case of planar structures including two-dimensional sheets. The graphene-graphene and graphene-real metal interactions at both zero and nonzero temperature were computed by using the electric susceptibility (polarizability) of graphene expressed via the density-density correlation function. It was argued [@12] that the zero-temperature form of polarizability can be used also at room temperature.
We underline that there is no complete agreement between the results of different papers devoted to the van der Waals and Casimir interactions with graphene (see Ref. [@11] where some of the results obtained are compared). In fact, all the approaches go back to the Lifshitz theory [@13; @14; @15], but with different approximations made and with various forms of the reflection coefficients used. By and large the approaches based on the density-density correlation function used its longitudinal version, i.e., neglected by the role of (small [@7]) relativistic effects. Furthermore, dependence of the correlation function on temperature which was unknown until the present time, was obtained by means of scaling [@7] or even neglected [@12]. By contrast, calculations based on the polarization tensor are fully relativistic and include an explicit dependence of its components on the temperature [@9; @10; @11]. This is the reason why it would be useful to establish a link between the two approaches and to test the validity of the approximations used.
In this paper, we find a correspondence between the reflection coefficients of the electromagnetic fluctuations on graphene expressed in terms of electric susceptibility (polarizability) of graphene and components of the polarization tensor. On this basis, we derive explicit expressions for both longitudinal and transverse electric susceptibilities of graphene, density-density correlation function and conductivities at arbitrary temperature. Then we consider the limiting cases of the obtained expressions at zero temperature and find that the longitudinal version coincides with that derived within the random phase approximation. Furthermore, we compare the computational results for graphene-graphene and graphene-real metal interactions at room temperature obtained using the polarization tensor [@10; @11] with those obtained using the density-density correlation function in Ref. [@12]. In doing so we pay special attention to contributions of the transverse electric susceptibility of graphene and explicit temperature dependence of the longitudinal density-density correlation function to the Casimir free energy.
The paper is organized as follows. In Sec. II we establish a link between the two approaches and derive the density-density correlation functions at nonzero temperature. Section III is devoted to the case of zero temperature. In Sec. IV the computational results for graphene-graphene and graphene-real metal thermal Casimir interactions using the zero-temperature correlation function and the polarization tensor at room temparature are compared. In Sec. V the reader will find our conclusions and discussion.
Comparison between the reflection coefficients in two theoretical approaches
============================================================================
As discussed in Sec. I, all theoretical approaches to the van der Waals and Casimir interaction between two graphene sheets or between graphene and material plate go back to the Lifshitz theory representing the free energy per unit area at temperature $T$ in thermal equilibrium in the form [@13; @14; @15] $$\begin{aligned}
&&
{\cal F}(a,T)=\frac{k_BT}{2\pi}\sum_{l=0}^{\infty}
{\vphantom{\sum}}^{\prime}\int_{0}^{\infty}\!\!
k_{\bot}dk_{\bot}\left\{\ln\left[1-r_{\rm TM}^{(1)}(i\xi_l,k_{\bot})
r_{\rm TM}^{(2)}(i\xi_l,k_{\bot})e^{-2aq_l}\right]
\right.
\nonumber \\
&&~~~~~
\left.
+\ln\left[1-r_{\rm TE}^{(1)}(i\xi_l,k_{\bot})
r_{\rm TE}^{(2)}(i\xi_l,k_{\bot})e^{-2aq_l}\right]\right\}.
\label{eq1}\end{aligned}$$ Here, $k_B$ is the Boltzmann constant, $k_{\bot}$ is the projection of the wave vector on the plane of graphene, $\xi_l=2\pi k_BTl/\hbar$ with $l=0,\,1,\,2,\,\ldots$ are the Matsubara frequencies, $q_l=(k_{\bot}^2+\xi_l^2/c^2)^{1/2}$, and the prime on the summation sign indicates that the term with $l=0$ is divided by two. The reflection coefficients on the two boundary planes separated by the vacuum gap of width $a$ for the two independent polarizations of the electromagnetic field, transverse magnetic (TM) and transverse electric (TE), are notated as $r_{\rm TM,TE}^{(1)}$ and $r_{\rm TM,TE}^{(2)}$.
Let the first boundary plane be the freestanding graphene. There are two main representations for the reflection coefficents $r_{\rm TM,TE}^{(1)}\equiv r_{\rm TM,TE}^{(g)}$ on graphene. We begin with the TM coefficient. Within the first theoretical approach, the longitudinal electric susceptibility (polarizability) of graphene at the imaginary Matsubara frequencies is expressed as $$\alpha^{||}(i\xi_l,k_{\bot})\equiv
\varepsilon^{||}(i\xi_l,k_{\bot})-1=
-\frac{2\pi e^2}{k_{\bot}}\chi^{||}(i\xi_l,k_{\bot}),
\label{eq2}$$ where $\chi^{||}(i\xi_l,k_{\bot})$ is the longitudinal density-density correlation function. The latter is connected with the dynamical conductivity of graphene by [@12] $$\sigma^{||}(i\xi_l,k_{\bot})=
-\frac{e^2\xi_l}{k_{\bot}^2}\chi^{||}(i\xi_l,k_{\bot}),
\label{eq3}$$ where $e$ is the electron charge. Then the TM reflection coefficient of the electromagnetic oscillations on graphene can be expressed as [@12; @16; @17] $$r_{\rm TM}^{(g)}(i\xi_l,k_{\bot})=
\frac{q_l\,\alpha^{||}(i\xi_l,k_{\bot})}{k_{\bot}+
q_l\,\alpha^{||}(i\xi_l,k_{\bot})}.
\label{eq4}$$ The explicit form for $\alpha^{||}$ is discussed below.
Within the second theoretical approach, the TM reflection coefficient is expressed via the 00-component $\Pi_{00}$ of the polarization tensor in (2+1)-dimensional space-time [@9; @10; @11] $$r_{\rm TM}^{(g)}(i\xi_l,k_{\bot})=
\frac{q_l\Pi_{00}(i\xi_l,k_{\bot})}{2\hbar k_{\bot}^2+
q_l\Pi_{00}(i\xi_l,k_{\bot})}.
\label{eq5}$$ The analytic expression for $\Pi_{00}$ is known [@9; @10; @11]. It depends on the temperature both implicitly (through the Matsubara frequencies) and explicitly, as on a parameter. For the pristine (undoped) gapless graphene one has [@9; @10; @11] $$\begin{aligned}
&&
\Pi_{00}(i\xi_l,k_{\bot})=
\frac{\pi\hbar\alpha k_{\bot}^2}{f(\xi_l,k_{\bot})}+
\frac{8\hbar\alpha c^2}{v_F^2}\int_{0}^{1}\!\!\!dx
\left\{
\vphantom{\frac{\xi_l^2\sqrt{x(1-x)}}{c^2f(\xi_l,k_{\bot})}}
\frac{k_BT}{\hbar c}\right.
\label{eq6}\\
&&~\times\ln\left[1+2\cos(2\pi lx)
e^{-\theta_T(\xi_l,k_{\bot},x)}+
e^{-2\theta_T(\xi_l,k_{\bot},x)}\right]
\nonumber \\
&&
~~~
-\frac{\xi_l}{2c}(1-2x)
\frac{\sin(2\pi lx)}{\cosh\theta_T(\xi_l,k_{\bot},x)+
\cos(2\pi lx)}
\nonumber \\
&&
~~~\left.
+\frac{\xi_l^2\sqrt{x(1-x)}}{c^2f(\xi_l,k_{\bot})}
\frac{\cos(2\pi lx)+e^{-\theta_T(\xi_l,k_{\bot},x)}}{\cosh\theta_T(\xi_l,k_{\bot},x)+
\cos(2\pi lx)}\right\},
\nonumber\end{aligned}$$ where $\alpha=e^2/(\hbar c)$ is the fine structure constant, $v_F$ is the Fermi velocity, and the following notations are introduced $$\begin{aligned}
&&
f(\xi_l,k_{\bot})\equiv\left(\frac{v_F^2}{c^2}k_{\bot}^2+
\frac{\xi_l^2}{c^2}\right)^{1/2},
\label{eq7} \\
&&
\theta_T(\xi_l,k_{\bot},x)\equiv
\frac{\hbar c}{k_BT}f(\xi_l,k_{\bot})\sqrt{x(1-x)}.
\nonumber\end{aligned}$$
Now we equate the right-hand sides of Eqs. (\[eq4\]) and (\[eq5\]) and obtain the expression for the longitudinal polarizability of graphene at nonzero temperature via the 00-component of the polarization tensor $$\alpha^{||}(i\xi_l,k_{\bot})=\frac{1}{2\hbar k_{\bot}}
\Pi_{00}(i\xi_l,k_{\bot}).
\label{eq8}$$ Using Eq. (\[eq2\]), for the longitudinal density-density correlation function one obtains $$\chi^{||}(i\xi_l,k_{\bot})=-\frac{1}{4\pi e^2\hbar}
\Pi_{00}(i\xi_l,k_{\bot}),
\label{eq9}$$ where $\Pi_{00}$ is given by Eq. (\[eq6\]). Similar to the polarization tensor, the density-density correlation function depends on $T$ both implicitly and explicitly. The longitudinal conductivity of graphene at any $T$ is given by Eq. (\[eq3\]).
We continue with the TE reflection coefficient. Note that Eqs. (\[eq2\]) and (\[eq3\]) remain valid for the transverse quantities: the polarizability of graphene $\alpha^{\bot}(i\xi_l,k_{\bot})$, the transverse permittivity $\varepsilon^{\bot}(i\xi_l,k_{\bot})$, the density-density correlation function $\chi^{\bot}(i\xi_l,k_{\bot})$, and the conductivity $\sigma^{\bot}(i\xi_l,k_{\bot})$. The TE reflection coefficient on graphene in terms of the transverse polarizability was found in Ref. [@12] $$r_{\rm TE}^{(g)}(i\xi_l,k_{\bot})=
-\frac{\xi_l^2\alpha^{\bot}(i\xi_l,k_{\bot})}{c^2k_{\bot}q_l+
\xi_l^2\alpha^{\bot}(i\xi_l,k_{\bot})}.
\label{eq10}$$ Note that according to our knowledge no explicit expression of $\alpha^{\bot}$ for graphene is available in the published literature.
In terms of the polarization tensor, the TE reflection coefficient takes the form [@9; @10; @11] $$r_{\rm TE}^{(g)}(i\xi_l,k_{\bot})=-
\frac{k_{\bot}^2\Pi_{\rm tr}(i\xi_l,k_{\bot})-
q_l^2\Pi_{00}(i\xi_l,k_{\bot})}{2\hbar k_{\bot}^2q_l+
k_{\bot}^2\Pi_{\rm
tr}(i\xi_l,k_{\bot})-q_l^2\Pi_{00}(i\xi_l,k_{\bot})},
\label{eq11}$$ where the index tr denotes the sum of spatial component $\Pi_1^{\,1}$ and $\Pi_2^{\,2}$. For the undoped gapless graphene the analytic expression for $\Pi_{\rm tr}$ is the following [@9; @10; @11]: $$\begin{aligned}
&&
\Pi_{\rm tr}(i\xi_l,k_{\bot})=\Pi_{00}(i\xi_l,k_{\bot})+
\frac{\pi\hbar\alpha}{f(\xi_l,k_{\bot})}\left[f^2(\xi_l,k_{\bot})+
\frac{\xi_l^2}{c^2}\right]
\label{eq12}\\
&&
+8\hbar\alpha\int_{0}^{1}\!\!\!dx
\left\{
\vphantom{\frac{\xi_l^2\sqrt{x(1-x)}}{c^2f(\xi_l,k_{\bot})}}
\frac{\xi_l}{c}(1-2x)
\frac{\sin(2\pi lx)}{\cosh\theta_T(\xi_l,k_{\bot},x)+
\cos(2\pi lx)}\right.
\nonumber \\
&&
~~~\left.
-\frac{\sqrt{x(1-x)}}{f(\xi_l,k_{\bot})}\left[f^2(\xi_l,k_{\bot})+
\frac{\xi_l^2}{c^2}\right]
\frac{\cos(2\pi lx)+e^{-\theta_T(\xi_l,k_{\bot},x)}}{\cosh\theta_T(\xi_l,k_{\bot},x)+
\cos(2\pi lx)}\right\}.
\nonumber\end{aligned}$$
By equating the right-hand sides of Eqs. (\[eq10\]) and (\[eq11\]), one obtains the expression for the transverse polarizability of graphene at any nonzero temperature $$\alpha^{\bot}(i\xi_l,k_{\bot})=
\frac{c^2}{2\hbar k_{\bot}\xi_l^2}\left[k_{\bot}^2
\Pi_{\rm tr}(i\xi_l,k_{\bot})-
q_l^2\Pi_{00}(i\xi_l,k_{\bot})\right].
\label{eq13}$$ The respective result for the transverse density-density correlation function is found from equation similar to Eq. (\[eq2\]) $$\chi^{\bot}(i\xi_l,k_{\bot})=
-\frac{c^2}{4\pi\hbar e^2\xi_l^2}\left[k_{\bot}^2
\Pi_{\rm tr}(i\xi_l,k_{\bot})-
q_l^2\Pi_{00}(i\xi_l,k_{\bot})\right].
\label{eq14}$$ Then the transverse conductivity of graphene is given by Eq. (\[eq3\]) where the index $||$ is replaced with $\bot$.
We emphasize that Eqs. (\[eq4\]), (\[eq5\]) and (\[eq10\]), (\[eq11\]) are the exact consequencies of the Maxwell equations and electrodynamic boundary conditions imposed on the 2D graphene sheet. For this reason, the obtained connections (\[eq8\]), (\[eq9\]) and (\[eq13\]), (\[eq14\]) between the polarizabilities and density-density correlation functions for graphene, on the one hand, and the components of the polarization tensor, on the other hand, are the exact ones. Keeping in mind that Eqs. (\[eq6\]) and (\[eq12\]) for the polarization tensor are calculated in the one-loop approximation [@9], the specific expressions for the polarizabilities and density-density correlation functions obtained after the substitution of Eqs. (\[eq6\]) and (\[eq12\]) in Eqs. (\[eq8\]), (\[eq9\]) and (\[eq13\]), (\[eq14\]) should be also considered as found in the same approximation. In the next section we compare them with those contained in the literature.
To conclude this section, we present an explicit expression for the quantity $k_{\bot}^2\Pi_{\rm tr}-q_l^2\Pi_{00}$ entering the transverse polarizability, the density-density correlation function and the conductivity of graphene. Substituting $\Pi_{00}$ from Eq. (\[eq6\]) and $\Pi_{\rm tr}$ from Eq. (\[eq12\]), one obtains after identical transformations $$\begin{aligned}
&&
k_{\bot}^2\Pi_{\rm tr}(i\xi_l,k_{\bot})-
q_l^2\Pi_{00}(i\xi_l,k_{\bot})=
\pi\hbar\alpha k_{\bot}^2f(\xi_l,k_{\bot})
\label{eq15}\\
&&~
-\frac{8\hbar\alpha c^2}{v_F^2}\int_{0}^{1}\!\!\!dx
\left\{
\vphantom{\frac{\xi_l^2\sqrt{x(1-x)}}{c^2f(\xi_l,k_{\bot})}}
\frac{k_BT\xi_l^2}{\hbar c^3}
\ln\left[1+2\cos(2\pi lx)
e^{-\theta_T(\xi_l,k_{\bot},x)}+
e^{-2\theta_T(\xi_l,k_{\bot},x)}\right]\right.
\nonumber \\
&&
~~~
-\left[2f^2(\xi_l,k_{\bot})-\frac{\xi_l^2}{c^2}\right]
\frac{\xi_l}{2c}(1-2x)
\frac{\sin(2\pi lx)}{\cosh\theta_T(\xi_l,k_{\bot},x)+
\cos(2\pi lx)}
\nonumber \\
&&
~~~\left.
+\sqrt{x(1-x)}
f^3(\xi_l,k_{\bot})
\frac{\cos(2\pi lx)+e^{-\theta_T(\xi_l,k_{\bot},x)}}{\cosh\theta_T(\xi_l,k_{\bot},x)+
\cos(2\pi lx)}\right\}.
\nonumber\end{aligned}$$ This expression is used in below calculations.
Energy of the Casimir interaction between two graphene sheets at zero temperature
=================================================================================
In the limiting case $T\to 0$ the summation over the discrete Matsubara frequencies in Eq. (\[eq1\]) is replaced with integration over the imaginary frequency axis, and for two graphene sheets one arrives to the Casimir energy per unit area $$\begin{aligned}
&&
E(a,T)=\frac{\hbar}{4\pi^2}\int_{0}^{\infty}\!\!
k_{\bot}dk_{\bot}\int_{0}^{\infty}\!d\xi\left\{
\ln\left[1-{r_{\rm TM}^{(g)}}^2(i\xi,k_{\bot})
e^{-2aq}\right]
\right.
\nonumber \\
&&~~~~~
\left.
+\ln\left[1-{r_{\rm TE}^{(g)}}^2(i\xi,k_{\bot})
e^{-2aq}\right]\right\}.
\label{eq16}\end{aligned}$$ Here, the reflection coefficients are given by either Eqs. (\[eq4\]) and (\[eq10\]) or (\[eq5\]) and (\[eq11\]), where the discrete frequencies $\xi_l$ are replaced with the continuous $\xi$.
We begin from the contribution of the TM mode, $E_{\rm TM}$, to the total energy (\[eq16\]). In terms of the polarization tensor, the reflection coefficient $r_{\rm TM}^{(g)}$ is given by Eqs. (\[eq5\]) and (\[eq6\]) with the notation (\[eq7\]). As can be seen in Eq. (\[eq7\]), the quantity $\theta_T(\xi_l,k_{\bot},x)\to\infty$ when $T\to 0$. Because of this, from Eq. (\[eq6\]) at $T=0\,$K one obtains [@6] $$\Pi_{00}(i\xi,k_{\bot})=
\frac{\pi\hbar\alpha k_{\bot}^2}{f(\xi,k_{\bot})}.
\label{eq17}$$ Using the notation (\[eq7\]), one obtains from Eq. (\[eq8\]) the longitudinal polarizability of graphene at zero temperature $$\alpha^{||}(i\xi,k_{\bot})=
\frac{\pi e^2}{2\hbar}\,
\frac{k_{\bot}}{\sqrt{v_F^2k_{\bot}^2+\xi^2}},
\label{eq18}$$ and from Eq. (\[eq9\]) the respective density-density correlation function $$\chi^{||}(i\xi,k_{\bot})=
-\frac{1}{4\hbar}\,
\frac{k_{\bot}^2}{\sqrt{v_F^2k_{\bot}^2+\xi^2}}.
\label{eq19}$$ The longitudinal conductivity of graphene at $T=0\,$K is obtained from Eqs. (\[eq3\]) and (\[eq19\]).
The density-density correlation function (\[eq19\]) at $T=0\,$K, derived from the polarization tensor, coincides with the classical result [@18; @19] which was used in computations of Ref. [@12]. Then, for the TM reflection coefficient on graphene at $T=0\,$K we obtain one and the same result either from Eqs. (\[eq4\]) and (\[eq18\]) or from Eqs. (\[eq5\]) and (\[eq17\]) $$r_{\rm TM}^{(g)}(i\xi,k_{\bot})=
\frac{\pi e^2 \sqrt{c^2k_{\bot}^2+\xi^2}}{2\hbar c
\sqrt{v_F^2k_{\bot}^2+\xi^2}+\pi e^2 \sqrt{c^2k_{\bot}^2+\xi^2}}.
\label{eq20}$$ This reflection coefficient coincides with that used in Ref. [@12].
We continue by considering the contribution of the TE mode, $E_{\rm TE}$, to the Casimir energy (\[eq16\]). In terms of the polarization tensor, the reflection coefficient $r_{\rm TE}^{(g)}$ is given by Eq. (\[eq11\]). The combination of the components of the polarization tensor, $k_{\bot}^2\Pi_{\rm tr}-q_l^2\Pi_{00}$, entering Eq. (\[eq11\]), is given by Eq. (\[eq15\]). In the limiting case $T\to 0$, one obtains from Eq. (\[eq15\]) $$k_{\bot}^2\Pi_{\rm tr}(i\xi,k_{\bot})-q^2\Pi_{00}(i\xi,k_{\bot})=
\pi\hbar\alpha k_{\bot}^2f(\xi,k_{\bot}).
\label{eq21}$$
Substituting Eq. (\[eq21\]) in Eq. (\[eq13\]), we find the transverse polarizability of graphene at zero temperature $$\alpha^{\bot}(i\xi,k_{\bot})=
\frac{\pi e^2k_{\bot}}{2\hbar\xi^2}\,
\sqrt{v_F^2k_{\bot}^2+\xi^2}.
\label{eq22}$$ In a similar way, substituting Eq. (\[eq21\]) in Eq. (\[eq14\]), we find the transverse density-density correlation function at $T=0\,$K $$\chi^{\bot}(i\xi,k_{\bot})=
-\frac{k_{\bot}^2}{4\hbar\xi^2}\,
\sqrt{v_F^2k_{\bot}^2+\xi^2}.
\label{eq23}$$
The TE reflection coefficient at $T=0\,$K is obtained either substituting Eq. (\[eq21\]) in Eq. (\[eq11\]) or Eq. (\[eq22\]) in Eq. (\[eq10\]). The result is $$r_{\rm TE}^{(g)}(i\xi,k_{\bot})=
-\frac{\pi e^2 \sqrt{v_F^2k_{\bot}^2+\xi^2}}{2\hbar c
\sqrt{c^2k_{\bot}^2+\xi^2}+\pi e^2 \sqrt{v_F^2k_{\bot}^2+\xi^2}}.
\label{eq24}$$ As is seen from the comparison of Eqs. (\[eq20\]) and (\[eq24\]), the reflection coefficient $r_{\rm TE}^{(g)}$ has the opposite sign, as compared with $r_{\rm TM}^{(g)}$, and its magnitude is obtained from the latter by the interchanging of $c$ and $v_F$.
Now we compare the computational results for the Casimir energy per unit area of two parallel graphene sheets at zero temperature obtained in Ref. [@12] by means of the density-density correlation function and here using the polarization tensor. In both cases the Fermi velocity $v_F=8.73723\times 10^{5}\,$m/s is employed [@12; @20; @21]. In Fig. \[fg1\] the computational results of Ref. [@12] for $E(a)$ normalized for the Casimir energy per unit area of two parallel ideal-metal planes $$E_{im}(a)=-\frac{\pi^2}{720}\,\frac{\hbar c}{a^3}
\label{eq25}$$ are shown as black dots over the separation region from 10nm to $5\,\mu$m. In making computations it was assumed [@12] that $\chi^{\bot}(i\xi,k_{\bot})=\chi^{||}(i\xi,k_{\bot})$. The gray line shows our computational results for $E(a)/E_{im}(a)$ using the polarization tensor at $T=0\,$K given by Eqs. (\[eq17\]) and (\[eq21\]). In this case the contribution of the TE mode was calculated precisely.
As can be seen in Fig. \[fg1\], both sets of computational results are in a very good agreement. This is explained by the fact that $E_{\rm TM}(a)$ contributes 99.6% of $E(a)$ and $E_{\rm TE}(a)=0.004E(a)$ at all separation distances. Furthermore, the relative differences between the computational results of Ref. [@12] for $E_{\rm TE}(a)$ (obtained under the assumption that $\chi^{\bot}=\chi^{||}$) and our results here computed with the exact reflection coefficient $r_{\rm TE}^{(g)}$ are of about 0.1%. Thus, the role of the TE contribution to the Casimir energy of two graphene sheets is really negligibly small [@7], and it is not critical what form of the transverse density-density correlation function is used in computations. Physically this is connected with the fact that the TE contribution is missing in the nonrelativistic limit, whereas the relativistic effects contain additional small factors of the order of $v_F/c$.
Casimir interaction with graphene at nonzero temperature
========================================================
In this section we compare the computational results for the Casimir free energy of two graphene sheets and a freestanding graphene sheet interacting with an Au plate obtained using the approach of Ref. [@12] and using the polarization tensor. All computations here are done at room temperature $T=300\,$K. In this way we find the role of explicit dependence of the density-density correlation function and polarization tensor on the temperature.
Two graphene sheets
-------------------
The free energy of the Casimir interaction between two sheets of undoped graphene was computed at $T=300\,$K using Eq. (\[eq1\]) with $r_{\rm TM,TE}^{(1)}=r_{\rm TM,TE}^{(2)}=r_{\rm TM,TE}^{(g)}$. All computations were performed using the following two approaches: the approach of Ref. [@12] using the reflection coefficients (\[eq4\]) and (\[eq10\]), expressed via the zero-temperature longitunidal density-density correlation function (\[eq19\]) and the approach of Ref. [@11] using the reflection coefficients (\[eq5\]) and (\[eq11\]) expressed via the components of the polarization tensor (\[eq6\]) and (\[eq12\]). Within the approach Ref. [@12], the dependence of the free energy on $T$ is determined by the $T$-dependent Matsubara frequencies, whereas in the approach of Ref. [@11] there is also explicit dependence of the polarization tensor on $T$ as a parameter.
In Fig. \[fg2\] we present the computational results for the Casimir free energy of two graphene sheets at $T=300\,$K as functions of separation over the interval from 10nm to $1\,\mu$m. The results obtained using the polarization tensor at $T=300\,$K are shown as the upper solid line, and the results obtained using the longitudinal density-density correlation function (\[eq19\]) defined at $T=0\,$K are shown as dots. From Fig. \[fg2\] it is seen that the upper solid line deviates from dots significantly even at short separations. This is explained by the dependence of the polarization tensor on $T$ as a parameter in addition to the implicit $T$-dependence through the Matsubara frequencies. The lower (gray) solid line in Fig. \[fg2\] shows the computational results obtained by means of the polarization tensor (\[eq17\]) and (\[eq21\]) at $T=0\,$K. This solid line is in a very good agreement with dots computed using the formalism of Ref. [@12], as it should be according to the results of Secs. II and III.
Note that the dominant contribution to the free energy of graphene-graphene interaction plotted in Fig. \[fg2\] is given by the TM mode. Thus, at $a=10\,$nm ${\cal F}_{\rm TM}=0.9965{\cal F}$ and ${\cal F}_{\rm TE}=0.0035{\cal F}$. Computations show that the contribution of the TM mode to the total free energy increases with the increase of separation. As a result, at $a=100\,$nm it holds ${\cal F}_{\rm TM}=0.9992{\cal F}$ and at $a=1\,\mu$m ${\cal F}_{\rm TM}=0.9999{\cal F}$.
It should be stressed also that the deviation between the upper and lower lines in Fig. \[fg2\] is explained entirely by the thermal dependence of the polarization tensor at zero Matsubara frequency. As was shown in Refs. [@9; @10; @11] (see also Ref. [@22]) contributions of all Matsubara terms with $l\geq 1$ are nearly the same irrespective of weather the polarization tensor at $T=0\,$K or at $T\neq 0\,$K is used in computations.
In this respect we remind that the contribution of the zero-frequency term, ${\cal F}_{l=0}$, to the total free energy of two graphene sheets ${\cal F}$ is increasing with the increase of separation, for example, ${\cal F}_{l=0}=0.32{\cal F}$ at $a=10\,$nm, ${\cal F}_{l=0}=0.946{\cal F}$ at $a=100\,$nm, and ${\cal F}_{l=0}=0.9994{\cal F}$ at $a=1\,\mu$m. The classical limit is already achieved at $a=400\,$nm where ${\cal F}_{l=0}=0.996{\cal F}$. At $a\geq 400\,$nm the Casimir free energy shown by the upper line in Fig. \[fg2\] is given to high accuracy by the asymptotic expression [@9; @11; @22a] $${\cal F}(a,T)\approx -\frac{k_BT\zeta(3)}{16\pi a^2}
\left[1-\frac{1}{4\alpha \ln2}\left(
\frac{v_F}{c}\right)^2\frac{\hbar c}{ak_BT}\right],
\label{eq26}$$ where $\zeta(z)$ is the Riemann zeta function.
This should be compared with the asymptotic free energy $${\cal F}(a,T)\approx -\frac{k_BT}{16\pi a^2}
{\rm Li}_3\left({r_0^{(g)}}^2\right),
\label{eq27}$$ where ${\rm Li}_3(z)$ is the polylogarithm function. Equation (\[eq27\]) is obtained using the approach of Ref. [@12] where the TM reflection coefficient at $\xi_0=0$ is defined by Eqs. (\[eq4\]) and (\[eq18\]) $$r_{\rm TM}^{(g)}(0,k_{\bot})\equiv r_0^{(g)}=
\frac{\pi e^2}{2\hbar v_F+\pi e^2}.
\label{eq28}$$ The asymptotic expression (\[eq27\]) is in a very good agreement with the lower solid line (and dots) in Fig. \[fg2\] at $a\geq 400\,$nm. We also notice that the contribution of the TE mode at $l=0$ to the free energy, ${\cal F}_{{\rm TE},\,l=0}$, is negligibly small, as compared with contribution of the TM mode at $l=0$ and with the total free energy: $|{\cal F}_{{\rm TE},\,l=0}|<2\times 10^{-9}
|{\cal F}_{{\rm TM},\,l=0}|$ and $|{\cal F}_{{\rm TE},\,l=0}|<1.2\times 10^{-9}|{\cal F}|$ over the entire region of separations.
Now we present a more informative comparison between the approaches using the polarization tensor at $T=300\,$K and the longitudinal density-density correlation function at $T=0\,$K, avoiding the use of the logarithmic scale. For this purpose we plot in Fig. \[fg3\](a,b) the ratios of the obtained results for the free energy to the asymptotic free energy of two ideal metal planes at high temperature defined as [@15] $${\cal F}_{im}(a,T)=-\frac{k_BT\zeta(3)}{8\pi a^2}.
\label{eq29}$$ The upper and lower solid lines are computed by Eq. (\[eq1\]) using the polarization tensor at $T=300\,$K and $T=0\,$K, respectively. The dots indicate the computational results of Ref. [@12] obtained at $T=300\,$K using the longitudinal density-density correlation function defined at $T=0\,$K. From Fig. \[fg3\](a) it becomes clear that even at the shortest separations from 10 to 50nm, where in the logarithmic scale of Fig. \[fg2\] the computational results using the two approaches might seem to be very close, there are in fact large deviations illustrating the role of explicit thermal dependence of the polarization tensor. In Fig. \[fg3\](b) plotted for the separation region from 50nm to $1\,\mu$m it is seen that the high-temperature limits predicted by the two approaches also differ significantly. Note that the computational results shown by the upper solid lines agree with those of Ref. [@7] where the temperature dependence of the longitudinal density-density correlation function was found by scaling.
Finally, in Fig. \[fg4\] we plot by the lower solid line the relative deviation between the free energies of two graphene sheets computed using the longitudinal density-density correlation function at $T=0\,$K (${\cal F}_{dd}$) and the polarization tensor at $T=300\,$K (${\cal F}_{pt}$) $$\delta{\cal F}(a,T)=\frac{{\cal F}_{dd}(a,T)-
{\cal F}_{pt}(a,T)}{{\cal F}_{pt}(a,T)}.
\label{eq30}$$ As is seen in Fig. \[fg4\], at the shortest separation $a=10\,$nm the magnitude of the relative deviation $|\delta{\cal F}|=8.5$%, then it achieves the value of $|\delta{\cal F}|=41.2$% at $a=400\,$nm, and does not exceed 41.8% at all larger separations.
To conclude the consideration of two graphene sheets, we stress that the calculation approach using the temperature-dependent density-density correlation functions (\[eq9\]) and (\[eq14\]) found in Sec. II and the reflection coefficients (\[eq4\]) and (\[eq10\]) lead to precisely the same results as the temperature-dependent polarization tensor.
Graphene sheet and a gold plate
-------------------------------
We have calculated the Casimir free energy at $T=300\,$K for a graphene sheet interacting with an Au plate using two theoretical approaches discussed above. For this purpose Eq. (\[eq1\]) was used where the reflection coefficients $r_{\rm TM,TE}^{(1)}=r_{\rm TM,TE}^{(g)}$ are defined in Secs. II and III and $r_{\rm TM,TE}^{(2)}=r_{\rm TM,TE}^{(\rm Au)}$ are defined as $$\begin{aligned}
&&
r_{\rm TM}^{(\rm Au)}(i\xi_l,k_{\bot})=
\frac{\varepsilon(i\xi_l)q_l-k_l}{\varepsilon(i\xi_l)q_l
+k_l},
\nonumber \\
&&
r_{\rm TE}^{(\rm Au)}(i\xi_l,k_{\bot})=
\frac{q_l-k_l}{q_l+k_l}.
\label{eq31}\end{aligned}$$ Here, $\varepsilon(\omega)$ is the frequency-dependent dielectric permittivity of Au and $$k_l\equiv k_l(i\xi_l,k_{\bot})=\left[k_{\bot}^2+
\varepsilon(i\xi_l)\frac{\xi_l^2}{c^2}\right]^{1/2}.
\label{eq32}$$
The dielectric permittivity of Au at the imaginary Matsubara frequencies was obtained from the experimental optical data [@23] for the imaginary part of the dielectric function by means of the Kramers-Kronig relation. The data were previously extrapolated to lower frequencies by means of the Drude model. In this paper the data for $\varepsilon(i\xi_l)$ from Ref. [@12] have been used in computations. The alternative extrapolation of the optical data by means of the plasma model leads to a maximum relative deviation in the obtained free energy equal to 0.8% at the shortest separation $a=10\,$nm and to smaller deviations at larger separations. As noted in Ref. [@10], for graphene-metal interaction the Casimir free energy and pressure do not depend on what model of metal (Drude or plasma) is used to describe the metal. For two metallic plates there are large differences in the results obtained using the Drude or plasma models [@24] due to the contribution of the TE mode which is negligibly small for a graphene sheet.
In Fig. \[fg5\] the computational results for the Casimir free energy of a graphene sheet interacting with an Au plate at $T=300\,$K are presented as functions of separation in the region from 10nm to $1\,\mu$m. The upper and lower solid lines indicate the results obtained using Eq. (\[eq1\]) and the polarization tensor at $T=300\,$K and $T=0\,$K, respectively. The dots show the results [@12] computed from the longitudinal density-density correlation function (\[eq19\]) at $T=0\,$K. As is seen in Fig. \[fg5\], dots are in agreement with the lower solid line, but deviate significantly from the upper one. This demonstrates important role of the explicit dependence of the polarization tensor on the temperature.
As in the case of two graphene sheets, the dominant contribution to ${\cal F}$ is given by the TM mode. Here, however, the ratio ${\cal F}_{\rm TM}/{\cal F}$ is not a monotonous function of $a$. Thus, at $a=10\,$nm ${\cal F}_{\rm TM}=0.983{\cal F}$ and at $a=100\,$nm the TM contribution achieves its minimum value ${\cal F}_{\rm TM}=0.961{\cal F}$. With further increase of separation ${\cal F}_{\rm TM}$ increases to $0.983{\cal F}$, $0.992{\cal F}$, and $0.998{\cal F}$ at $a=500\,$nm, $1\,\mu$m, and $2\,\mu$m, respectively. Similar to the case of two graphene sheets, the difference between the upper and lower solid lines is explained by the explicit thermal dependence of the polarization tensor at zero Matsubara frequency [@9; @10].
For a graphene sheet interacting with an Au plate, the contribution of the zero Matsubara frequency to the total free energy increases with separation slower than for two graphene sheets. Thus, at $a=10\,$nm and 100nm one obtains ${\cal F}_{l=0}=0.11{\cal F}$ and $0.58{\cal F}$, respectively. At $a=1\,\mu$m ${\cal F}_{l=0}=0.97{\cal F}$, and at $a=1.6\,\mu$m the classical limit is achieved: ${\cal F}_{l=0}=0.99{\cal F}$. At this and larger separations the Casimir free energy per unit area is given by [@9; @11] $${\cal F}(a,T)\approx -\frac{k_BT\zeta(3)}{16\pi a^2}
\left[1-\frac{1}{8\alpha \ln2}\left(
\frac{v_F}{c}\right)^2\frac{\hbar c}{ak_BT}\right].
\label{eq33}$$
The calculation approach using the longitudinal density-density correlation function defined at zero temperature describes the reflection coefficient on graphene at $\xi_0=0$ by Eq. (\[eq28\]). Taking into account that for Au $r_{\rm TM}^{(\rm Au)}(0,k_{\bot})=1$, the classical limit is obtained in the form $${\cal F}(a,T)\approx -\frac{k_BT}{16\pi a^2}
{\rm Li}_3\left({r_0^{(g)}}\right),
\label{eq34}$$ This asymptotic expression is in a very good agreement with computational results of Ref. [@12] at $a\geq 1.6\,\mu$m.
To avoid the use of the logarithmic scale, in Fig. \[fg6\](a,b) we plot the ratios of the computed free energies of graphene-Au plate interaction to the asymptotic free energy of two ideal-metal planes (\[eq29\]). The upper and lower solid lines are computed by Eq. (\[eq1\]) using the polarization tensor defined at $T=300\,$K and $T=0\,$K, respectively. The dots are computed in Ref. [@12] at $T=300\,$K using the longitudinal density-density correlation function defined at $T=0\,$K. From Figs. \[fg6\](a) and \[fg6\](b) it is seen that there are significant deviations between the upper line, on the one hand, and the lower line and dots, on the other hand, at both short and relatively large separations. At $a\geq 1.6\,\mu$m the descrepancy between the theoretical predictions of the two approaches is illustrated by Eqs. (\[eq33\]) and (\[eq34\]).
The relative deviation (\[eq30\]) between the Casimir free energies of graphene-Au plate interaction computed using the two approaches is shown by the upper line in Fig. \[fg4\]. Here, the magnitude of the relative deviation is equal to $|\delta{\cal F}|=1.5$% at $a=10\,$nm , achieves $|\delta{\cal F}|=24$% at $a=1.5\,\mu$m, and does not exceed 24.5% at larger separations. This means that large thermal effects inherent to graphene are less pronounced in the graphene-Au plate configuration, as compared to the case of two graphene sheets.
Similar to two graphene sheets, for a graphene interaction with an Au plate the theoretical predictions using the polarization tensor at $T\neq 0$ are in full agreement with respective predictions using the $T$-dependent density-density correlation function defind in Eqs. (\[eq9\]) and (\[eq14\]).
Conclusions and discussion
==========================
In the foregoing, we have performed the comparison studies of two approaches used to calculate the van der Waals and Casimir interaction between two graphene sheets and between a graphene sheet and a metal plate. One of these approaches is based on the use of the polarization tensor. All its components are found [@9] at any temperature. The other approach is based on the use of the density-density correlation function. Only the longitudinal version of this function was available in the literature and only at zero temperature. Because of this, previous calculations estimated the TE contribution to the free energy as negligibly small and either modeled the temperature dependence of the correlation function by means of scaling between two asymptotic regimes [@7] or argued that this dependence is not essential [@12].
We have shown that at zero temperature the approaches using the polarization tensor and the standard longitudinal density-density correlation function lead to almost coinciding computational results. The coincidence becomes exact if in the calculation of negligibly small contribution of the TE mode one replaces the longitudinal density-density correlation function at $T=0\,$K for the transverse one. We have provided an explicit expression for this function.
Computations at nonzero temperature using the polarization tensor with an explicit thermal dependence demonstrate significant deviations from the computational results using the density-density correlation function at $T=0\,$K. The latter include only an implicit dependence of the Casimir free energy on the temperature through the Matsubara frequencies. It was shown that for graphene-graphene and graphene-Au plate interactions the free energies obtained using this approach deviate from those calculated using the temperature-dependent polarization tensor up to 41.8% and 24.5%, respectively. However, the computational results obtained using the zero-temperature density-density correlation function are reproduced when the polarization tensor defined at $T=0\,$K is used. Similar to the case of zero temperature, at $T\neq 0\,$K the contribution of the TE mode of the electromagnetic field to the Casimir free energy is shown to be negligibly small for both graphene-graphene and graphene-Au plate systems.
We have performed a comparison between the exact TM and TE reflection coefficients expressed via the components of the polarization tensor, on the one hand, and via the longitudinal and transverse density-density correlation functions, on the other hand. In this way we have found explicit expressions for both longitudinal and transverse density-density correlation functions at any nonzero temperature. In the limiting case of vanishing temperature, our temperature-dependent longitudinal density-density correlation function goes into the well known classical result. The computational results for graphene-graphene and graphene-Au plate Casimir interactions obtained using the temperature-dependent density-density correlation functions found by us are exactly coinciding with those obtained using the temperature-dependent polarization tensor.
One can conclude that an equivalence of the two approaches to calculation of the van der Waals and Casimir forces in layered systems including graphene demonstrated in this paper provides a reliable foundation for the comparison between experiment and theory.
[99]{} A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. [**81**]{}, 109 (2009). M. S. Dresselhaus, Physica Status Solidi (b) [**248**]{}, 1566 (2011). A. A. Banishev, H. Wen, J. Xu, R. K. Kawakami, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, Phys. Rev. B [**87**]{}, 205433 (2013). J. F. Dobson, A. White, and A. Rubio, [Phys. Rev. Lett.]{} [**96**]{}, 073201 (2006). Bo E. Sernelius, Europhys. Lett. [**95**]{}, 57003 (2011). M. Bordag, I. V. Fialkovsky, D. M. Gitman, and D. V. Vassilevich, [Phys. Rev. B]{} [**80**]{}, 245406 (2009). G. Gómez-Santos, [Phys. Rev. B]{} [**80**]{}, 245424 (2009). D. Drosdoff and L. M. Woods, Phys. Rev. B [**82**]{}, 155459 (2010). V. Svetovoy, Z. Moktadir, M. Elvenspoek, and H. Mizuta, Europhys. Lett. [**96**]{}, 14006 (2011). I. V. Fialkovsky, V. N. Marachevsky, and D. V. Vassilevich, [Phys. Rev. B]{} [**84**]{}, 035446 (2011). M. Bordag, G. L. Klimchitskaya, and V. M. Mostepanenko, [Phys. Rev.]{} B [**86**]{}, 165429 (2012). G. L. Klimchitskaya and V. M. Mostepanenko, [Phys. Rev.]{} B [**87**]{}, 075439 (2013). Bo E. Sernelius, [Phys. Rev.]{} B [**85**]{}, 195427 (2012). E. M. Lifshitz and L. P. Pitaevskii, [*Statistical Physics*]{}, Pt.II (Pergamon Press, Oxford, 1980). Bo E. Sernelius, [*Surface Modes in Physics*]{} (Wiley-VCH, Berlin, 2001). M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, [*Advances in the Casimir Effect*]{} (Oxford University Press, Oxford, 2009). L. A. Falkovsky and S. S. Pershoguba, Phys. Rev. B [**76**]{}, 153410 (2007). T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B [**78**]{}, 085432 (2008). K. W.-K. Shung, Phys. Rev. B [**34**]{}, 979 (1986). J. Gonzáles, F. Guinea, and M. A. H. Vozmediano, Nucl. Phys. [**424**]{}, 595 (1994). B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New J. Phys. [**8**]{}, 318 (2006). N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Phys. Rev. B [**73**]{}, 125411 (2006). M. Chaichian, G. L. Klimchitskaya, V. M. Mostepanenko, and A. Tureanu, Phys. Rev. A [**86**]{}, 012515 (2012). G. L. Klimchitskaya and V. M. Mostepanenko, [Phys. Rev.]{} B [**89**]{}, 035407 (2014). , ed. E. D. Palik (Academic, New York, 1985). G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Rev. Mod. Phys. [**81**]{}, 1827 (2009).
![\[fg1\] The Casimir energy per unit area of two graphene sheets at zero temperature normalized to that of two ideal-metal planes is computed using the longitudinal density-density correlation function (dots) and by the polarization tensor (solid line), as functions of separation. ](figKMS-1.ps)
 The magnitude of the Casimir free energy per unit area for two graphene sheets at $T=300\,$K is shown as a function of separation in the logarithmic scale. The upper and lower solid lines are computed using the polarization tensor at $T=300\,$K and at $T=0\,$K, respectively, whereas dots indicate the computational results using the longitudinal density-density correlation function at $T=0\,$K. ](figKMS-2.ps)
 The Casimir free energy per unit area of two graphene sheets at $T=300\,$K normalized to that of two ideal-metal planes in the limit of high $T$ is computed using the longitudinal density-density correlation function at $T=0\,$K (dots), by the polarization tensor at $T=300\,$K (the upper solid line) and by the polarization tensor at $T=0\,$K (the lower solid line) over the separation regions (a) from 10 to 50nm and (b) from 50nm to $1\,\mu$m. ](figKMS-3.ps)
 The solid lines show the relative deviations between the Casimir free energies of graphene-graphene (the lower line) and graphene-Au plate (the upper line) interactions computed using the longitudinal density-density correlation function at $T=0\,$K and the polarization tensor at $T=300\,$K. ](figKMS-4.ps)
 The magnitude of the Casimir free energy per unit area for a graphene sheet and an Au plate at $T=300\,$K is shown as a function of separation in the logarithmic scale. The upper and lower solid lines are computed using the polarization tensor at $T=300\,$K and at $T=0\,$K, respectively, whereas dots indicate the computational results using the longitudinal density-density correlation function at $T=0\,$K. ](figKMS-5.ps)
 The Casimir free energy per unit area of a graphene sheet and an Au plate at $T=300\,$K normalized to that of two ideal-metal planes in the limit of high $T$ is computed using the longitudinal density-density correlation function at $T=0\,$K (dots), by the polarization tensor at $T=300\,$K (the upper solid line) and by the polarization tensor at $T=0\,$K (the lower solid line) over the separation regions (a) from 10 to 50nm and (b) from 50nm to $1\,\mu$m. ](figKMS-6.ps)
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§[S-6pt/]{} H-7.5pt/ \#1[[Re]{}\#1]{} \#1[[Im]{}\#1]{} \#1[[Tr]{}\#1]{} ‘=11 \#1\#2 \#1[\#1|]{} \#1[| \#1]{} \#1[\#1]{} \#1[\#1]{} versim\#1\#2 \#1[$\bf#1$]{} \#1[$\bf\overline{#1}$]{} 1[[1]{}]{} \#1\#2 /\#1[\#1-6pt/]{} \#1\#2\#3[Nucl. Phys. B [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[Phys. Lett. B [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[B [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[Phys. Rev. D [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[Phys. Rev. Lett. [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[Phys. Rep. [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[Mod. Phys. Lett. A [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[Int. J. Mod. Phys. A [**\#1**]{} (19\#2) \#3]{} \#1[Texas A & M University preprint CTP-TAMU-\#1]{} \#1\#2\#3[Ann. Rev. Astron. Astrophys. [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[Ann. Rev. Nucl. Part. Sci. [**\#1**]{} (19\#2) \#3]{}
6.0in 8.5in -0.25truein 0.30truein 0.30truein
\
[CTP-TAMU-68/93]{}\
[ACT-24/93]{}\
0.3cm [**New Precision Electroweak Tests of\
**]{} 0.5cm [JORGE L. LOPEZ$^{(a),(b)}$, D. V. NANOPOULOS$^{(a),(b),(c)}$, GYE T. PARK$^{(a),(b)}$,\
]{} [and A. ZICHICHI$^{(d)}$\
]{} 0.4cm [*$^{(a)}$Center for Theoretical Physics, Department of Physics, Texas A&M University\
*]{} [*College Station, TX 77843–4242, USA\
*]{} [*$^{(b)}$Astroparticle Physics Group, Houston Advanced Research Center (HARC)\
*]{} [*Mitchel Campus, The Woodlands, TX 77381, USA\
*]{} [*$^{(c)}$CERN, Theory Division, 1211 Geneva 23, Switzerland\
*]{} [*$^{(d)}$CERN, 1211 Geneva 23, Switzerland\
*]{}
0.4cm [ABSTRACT]{}
-0.2cm [ We explore the one-loop electroweak radiative corrections in $SU(5)\times U(1)$ supergravity via explicit calculation of vacuum-polarization and vertex-correction contributions to the $\epsilon_1$ and $\epsilon_b$ parameters. Experimentally, these parameters are obtained from a global fit to the set of observables $\Gamma_{l}, \Gamma_{b}, A^{l}_{FB}$, and $M_W/M_Z$. We include $q^2$-dependent effects, which induce a large systematic negative shift on $\epsilon_{1}$ for light chargino masses ($m_{\chi^\pm_1}\lsim70\GeV$). The (non-oblique) supersymmetric vertex corrections to $\Zbb$, which define the $\epsilon_b$ parameter, show a significant positive shift for light chargino masses, which for $\tan\beta\approx2$ can be nearly compensated by a negative shift from the charged Higgs contribution. We conclude that at the 90%CL, for $m_t\lsim160\GeV$ the present experimental values of $\epsilon_1$ and $\epsilon_b$ do not constrain in any way $SU(5)\times U(1)$ supergravity in both no-scale and dilaton scenarios. On the other hand, for $m_t\gsim160\GeV$ the constraints on the parameter space become increasingly stricter. We demonstrate this trend with a study of the $m_t=170\GeV$ case, where only a small region of parameter space, with $\tan\beta\gsim4$, remains allowed and corresponds to light chargino masses ($m_{\chi^\pm_1}\lsim70\GeV$). Thus $SU(5)\times U(1)$ supergravity combined with high-precision LEP data would suggest the presence of light charginos if the top quark is not detected at the Tevatron.]{}
\
[CTP-TAMU-68/93]{}\
[ACT-24/93]{}\
October 1993
Introduction
============
Since the advent of LEP, precision electroweak tests have become rather deep probes of the Standard Model of electroweak interactions and its challengers. These tests have demonstrated the internal consistency of the Standard Model, as long as the yet-to-be-measured top-quark mass ($m_t$) is within certain limits, which depend on the value assumed for the Higgs-boson mass ($m_H$): $m_t=135\pm18\GeV$ for $m_H\sim60\GeV$ and $m_t=174\pm15\GeV$ for $m_H\sim1\TeV$ (for a recent review see , Ref. [@Altlecture]). In the context of supersymmetry, such tests have been performed throughout the years within the Minimal Supersymmetric Standard Model (MSSM) [@OldEW; @BFC; @ABC; @ABCII]. The problem with such calculations is well known but usually ignored – there are too many parameters in the MSSM (at least twenty) – and therefore it is not possible to obtain precise predictions for the observables of interest.
In the context of supergravity models, on the other hand, any observable can be computed in terms of at most five parameters: the top-quark mass, the ratio of Higgs vacuum expectation values ($\tan\beta$), and three universal soft-supersymmetry-breaking parameters $(m_{1/2},m_0,A)$ [@Erice93]. This implies much sharper predictions for the various quantities of interest, as well as numerous correlations among them. Of even more experimental interest is $SU(5)\times U(1)$ supergravity where string-inspired ansätze for the soft-supersymmetry-breaking parameters allow the theory to be described in terms of only three parameters: $m_t$, $\tan\beta$, and $m_{\tilde g}$ [@EriceDec92]. Precision electroweak tests in the no-scale [@LNZI] and dilaton [@LNZII] scenarios for $SU(5)\times U(1)$ supergravity have been performed in Refs. [@ewcorr; @bsg-eps], using the description in terms of the $\epsilon_{1,2,3}$ parameters introduced in Refs. [@AB; @ABJ]. In this paper we extend these tests in two ways: first, we include for the first time the $\epsilon_b$ parameter [@ABC] which encodes the one-loop corrections to the $Z\to b\bar b$ vertex, and second we perform the calculation of the $\epsilon_1$ parameter in a new scheme [@ABC], which takes full advantage of the latest experimental data.
The calculation of $\epsilon_b$ is of particular importance since in the Standard Model, of the four parameters $\epsilon_{1,2,3,b}$ at present only $\epsilon_b$ falls outside the 1$\sigma$ experimental error (for $m_t>120\GeV$) [@ABC; @BV]. This discrepancy is not of great statistical significance, although the trend should not be overlooked, especially in the light of the much better statistical agreement for the other three parameters. Within the context of the Standard Model, another reason for focusing attention on the $\epsilon_{b}$ parameter is that, unlike the $\epsilon_1$ parameter, $\epsilon_b$ provides a constraint on the top-quark mass which is [*practically independent*]{} of the Higgs-boson mass. Indeed, at the 95% CL, the limits on $\epsilon_b$ require $m_t<185\GeV$, whereas those from $\epsilon_1$ require $m_t<177-198\GeV$ for $m_H\sim100-1000\GeV$ [@BV].
In supersymmetric models, the weakening of the $\epsilon_1$-deduced $m_t$ upper bound for large Higgs-boson masses does not occur (since the Higgs boson must be light) and both $\epsilon_1$ and $\epsilon_b$ are expected to yield comparable constraints. In this context it has been pointed out [@ABCII] that if certain mass correlations in the MSSM are satisfied, then the prediction for $\epsilon_b$ will be in better agreement with the data than the Standard Model prediction is. However, the opposite situation could also occur (, worse agreement), as well as negligble change relative to the Standard Model prediction (when all supersymmetric particles are heavy enough). We show that this three-way ambiguity in the MSSM prediction for $\epsilon_b$ disappears when one considers $SU(5)\times U(1)$ supergravity in both no-scale and dilaton scenarios. The $SU(5)\times U(1)$ supergravity prediction is practically always in better statistical agreement with the data (compared with the Standard Model one).
This study shows that at the 90%CL, for $m_t\lsim160\GeV$ the present experimental values of $\epsilon_1$ and $\epsilon_b$ do not constrain $SU(5)\times U(1)$ supergravity in any way. On the other hand, for $m_t\gsim160\GeV$ the constraints on the parameter space become increasingly stricter. We demonstrate this trend with a study of the $m_t=170\GeV$ case, where only a small region of parameter space, with $\tan\beta\gsim4$, remains allowed and corresponds to a light supersymmetric spectrum, and in particular light chargino masses ($m_{\chi^\pm_1}\lsim70\GeV$). Thus $SU(5)\times U(1)$ supergravity combined with high-precision LEP data would suggest the presence of light charginos if the top quark is not detected at the Tevatron.
SU(5)xU(1) Supergravity
=======================
Our study of one-loop electroweak radiative corrections is performed within the context of $SU(5)\times U(1)$ supergravity [@EriceDec92]. Besides the several theoretical string-inspired motivations that underlie this theory, of great practical importance is the fact that only three parameters are needed to describe all their possible predictions. This fact has been used in the recent past to perform a series of calculations for collider [@collider; @LNPWZh] and rare [@rare; @ewcorr; @bsg-eps] processes within this theory. The constraints obtained from all these analyses should help sharpen even more the experimental predictions for the remaining allowed points in parameter space.
In $SU(5)\times U(1)$ supergravity, gauge coupling unification occurs at the string scale $10^{18}\GeV$ [@EriceDec92], because of the presence of a pair of , representations with intermediate-scale masses. The three parameters alluded to above are: (i) the top-quark mass ($m_t$), (ii) the ratio of Higgs vacuum expectation values ($\tan\beta$), which satisfies $1\lsim\tan\beta\lsim40$, and (iii) the gluino mass, which is cut off at 1 TeV. This simplification in the number of input parameters is possible because of specific string-inspired scenarios for the universal soft-supersymmetry-breaking parameters ($m_0,m_{1/2},A$) at the unification scale. These three parameters can be computed in specific string models in terms of just one of them [@IL]. In the [*no-scale*]{} scenario one obtains $m_0=A=0$, whereas in the [*dilaton*]{} scenario the result is $m_0=\frac{1}{\sqrt{3}}m_{1/2}, A=-m_{1/2}$. After running the renormalization group equations from high to low energies, at the low-energy scale the requirement of radiative electroweak symmetry breaking introduces two further constraints which determine the magnitude of the Higgs mixing term $\mu$, although its sign remains undetermined. Finally, all the known phenomenological constraints on the sparticle masses are imposed (most importantly the chargino, slepton, and Higgs mass bounds). This procedure is well documented in the literature [@aspects] and yields the allowed parameter spaces for the no-scale [@LNZI] and dilaton [@LNZII] scenarios.
These allowed parameter spaces in the three defining variables ($m_t,\tan\beta,m_{\tilde g}$) consist of a discrete set of points for three values of $m_t$ ($m_t=130,150,170\GeV$), and a discrete set of allowed values for $\tan\beta$, starting at 2 and running (in steps of two) up to 32 (46) for the no-scale (dilaton) scenario. The chosen lower bound on $\tan\beta$ follows from the requirement by the radiative breaking mechanism of $\tan\beta>1$, and because the LEP lower bound on the lightest Higgs boson mass ($m_h\gsim60\GeV$ [@LNPWZh]) is quite constraining for $1<\tan\beta<2$.
------------------------------------------------------------------------
no-scale dilaton
--------------------------------- ------------- -------------
$\tilde e_R,\tilde \mu_R$ $0.18$ $0.33$
$\tilde\nu$ $0.18-0.30$ $0.33-0.41$
$2\chi^0_1,\chi^0_2,\chi^\pm_1$ $0.28$ $0.28$
$\tilde e_L,\tilde \mu_L$ $0.30$ $0.41$
$\tilde q$ $0.97$ $1.01$
$\tilde g$ $1.00$ $1.00$
: The approximate proportionality coefficients to the gluino mass, for the various sparticle masses in the two supersymmetry breaking scenarios considered.[]{data-label="Table1"}
------------------------------------------------------------------------
In the models we consider all sparticle masses scale with the gluino mass, with a mild $\tan\beta$ dependence (except for the third-generation squark and slepton masses). In Table \[Table1\] we give the approximate proportionality coefficient (to the gluino mass) for each sparticle mass. Note that the relation $2m_{\chi^0_1}\approx m_{\chi^0_2}\approx m_{\chi^\pm_1}$ holds to good approximation. The third-generation squark and slepton masses also scale with $m_{\tilde g}$, but the relationships are smeared by a strong $\tan\beta$ dependence. From Table \[Table1\] one can (approximately) translate any bounds on a given sparticle mass on bounds on all the other sparticle masses.
One-loop electroweak radiative corrections and the new $\epsilon$ parameters
============================================================================
There are different schemes to parametrize the electroweak (EW) vacuum polarization corrections [@Kennedy; @PT; @efflagr; @AB]. It can be shown, by expanding the vacuum polarization tensors to order $q^2$, that one obtains three independent physical parameters. Alternatively, one can show that upon symmetry breaking three additional terms appear in the effective lagrangian [@efflagr]. In the $(S,T,U)$ scheme [@PT], the deviations of the model predictions from the SM predictions (with fixed SM values for $m_t,m_{H_{SM}}$) are considered as the effects from “new physics". This scheme is only valid to the lowest order in $q^2$, and is therefore not applicable to a theory with new, light $(\sim M_Z)$ particles. In the $\epsilon$-scheme[@ABJ; @ABC], on the other hand, the model predictions are absolute and also valid up to higher orders in $q^2$, and therefore this scheme is more applicable to the EW precision tests of the MSSM [@BFC] and a class of supergravity models [@ewcorr].
There are two different $\epsilon$-schemes. The original scheme[@ABJ] was considered in our previous analyses [@ewcorr; @bsg-eps], where $\epsilon_{1,2,3}$ are defined from a basic set of observables $\Gamma_{l},
A^{l}_{FB}$ and $M_W/M_Z$. Due to the large $m_t$-dependent vertex corrections to $\Gamma_b$, the $\epsilon_{1,2,3}$ parameters and $\Gamma_b$ can be correlated only for a fixed value of $m_t$. Therefore, $\Gamma_{tot}$, $\Gamma_{hadron}$ and $\Gamma_b$ were not included in Ref. [@ABJ]. However, in the new $\epsilon$-scheme, introduced recently in Ref. [@ABC], the above difficulties are overcome by introducing a new parameter $\epsilon_b$ to encode the $\Zbb$ vertex corrections. The four $\epsilon$’s are now defined from an enlarged set of $\Gamma_{l}$, $\Gamma_{b}$, $A^{l}_{FB}$ and $M_W/M_Z$ without even specifying $m_t$. In this work we use this new $\epsilon$-scheme. Experimentally, including all LEP data allows one to determine the allowed ranges for these parameters [@Altlecture] \^[exp]{}\_1=(-0.33.2)10\^[-3]{},\^[exp]{}\_b=(3.15.5)10\^[-3]{} . Since among $\epsilon_{1,2,3}$ only $\epsilon_1$ provides constraints in supersymmetric models at the 90%CL [@ewcorr; @ABCII], we discuss below only $\epsilon_1$ and $\epsilon_b$.
The expression for $\epsilon_1$ is given as [@BFC] \_1=e\_1-e\_5-[G\_[V,B]{}G]{}-4g\_A,\[eps1\] where $e_{1,5}$ are the following combinations of vacuum polarization amplitudes $$\begin{aligned}
e_1&=&{\alpha\over 4\pi \sin^2\theta_W M^2_W}[\Pi^{33}_T(0)-\Pi^{11}_T(0)],
\label{e1}\\
e_5&=& M_Z^2F^\prime_{ZZ}(M_Z^2),\label{e5}\end{aligned}$$ and the $q^2\not=0$ contributions $F_{ij}(q^2)$ are defined by \^[ij]{}\_T(q\^2)=\^[ij]{}\_T(0)+q\^2F\_[ij]{}(q\^2). The $\delta g_A$ in Eqn. (\[eps1\]) is the contribution to the axial-vector form factor at $q^2=M^2_Z$ in the $Z\to l^+l^-$ vertex from proper vertex diagrams and fermion self-energies, and $\delta G_{V,B}$ comes from the one-loop box, vertex and fermion self-energy corrections to the $\mu$-decay amplitude at zero external momentum. These non-oblique SM corrections are non-negligible, and must be included in order to obtain an accurate SM prediction. As is well known, the SM contribution to $\epsilon_1$ depends quadratically on $m_t$ but only logarithmically on the SM Higgs boson mass ($m_H$). In this fashion upper bounds on $m_t$ can be obtained which have a non-negligible $m_H$ dependence: up to $20\GeV$ stronger when going from a heavy ($\approx1\TeV$) to a light ($\approx100\GeV$) Higgs boson. It is also known (in the MSSM) that the largest supersymmetric contributions to $\epsilon_1$ are expected to arise from the $\tilde t$-$\tilde b$ sector, and in the limiting case of a very light stop, the contribution is comparable to that of the $t$-$b$ sector. The remaining squark, slepton, chargino, neutralino, and Higgs sectors all typically contribute considerably less. For increasing sparticle masses, the heavy sector of the theory decouples, and only SM effects with a [*light*]{} Higgs boson survive. (This entails stricter upper bounds on $m_t$ than in the SM, since there the Higgs boson does not need to be light.) However, for a light chargino ($m_{\chi^\pm_1}\to{1\over2}M_Z$), a $Z$-wavefunction renormalization threshold effect can introduce a substantial $q^2$-dependence in the calculation, , the presence of $e_5$ in Eq. (\[eps1\]) [@BFC]. The complete vacuum polarization contributions from the Higgs sector, the supersymmetric chargino-neutralino and sfermion sectors, and also the corresponding contributions in the SM have been included in our calculations [@ewcorr].
Following Ref. [@ABC], $\epsb$ is defined from $\Gamma_b$, the inclusive partial width for $\Zbb$, as follows
$$\Gamma_b=3 R_{QCD} {G_FM^3_Z\over 6\pi\sqrt 2}\left(
1+{\alpha\over 12\pi}\right)\left[ \beta _b{\left( 3-\beta
^2_b\right)\over 2}(g^b_V)^2+\beta^3_b (g^b_A)^2\right] \;,$$
with $$\begin{aligned}
R_{QCD} &\cong&\left[1+1.2{\alpha_S\left(
M_Z\right)\over\pi}-1.1{\left(\alpha_S\left(
M_Z\right)\over\pi\right)}^2-12.8{\left(\alpha_S\left(
M_Z\right)\over\pi\right)}^3\right] \;,\\
\beta_b&=&\sqrt {1-{4m_b^2\over M_Z^2}} \;, \\
g^b_A&=&-{1\over2}\left(1+{\epsilon_1\over2}\right)\left(
1+{\epsb}\right)\;,\\
{g^b_V\over{g^b_A}}&=&{{1-{4\over3}{\ov s}^2_W+\epsb}\over{1+\epsb}}\;.\end{aligned}$$ Here ${\ov s}^2_W$ is an effective $\sin^2\theta_W$ for on-shell $Z$, and $\epsb$ is closely related to the real part of the vertex correction to $\Zbb$, denoted in the literature by $\nabla_b$ and defined explicitly in Ref. [@BF]. In the SM, the diagrams for $\nabla_b$ involve top quarks and $W^\pm$ bosons [@RbSM], and the contribution to $\epsb$ depends quadratically on $m_t$. In supersymmetric models there are additional diagrams involving Higgs bosons and supersymmetric particles. The charged Higgs contributions have been calculated in Refs. [@Denner; @Rbbsg2HD; @epsb2HD] in the context of a non-supersymmetric two Higgs doublet model, and the contributions involving supersymmetric particles in Refs. [@BF; @Rb2HD]. Moreover, $\epsilon_b$ itself has been calculated in Ref. [@epsb2HD]. The additional supersymmetric contributions are: (i) a negative contribution from charged Higgs–top exchange which grows as $m^2_t/\tan^{2}\beta$ for $\tan\beta\ll{m_t\over{m_b}}$; (ii) a positive contribution from chargino-stop exchange which in this case grows as $m^2_t/\sin^{2}\beta$; and (iii) a contribution from neutralino(neutral Higgs)–bottom exchange which grows as $m^2_b\tan^{2}\beta$ and is negligible except for large values of $\tan\beta$ (, $\tan\beta\gsim{m_t\over{m_b}}$) (the contribution (iii) has been neglected in our analysis).
Results and discussion
======================
In Figures 1–4 we show the results of the calculation of $\epsilon_1$ and $\epsilon_b$ (as described above) for all the allowed points in $SU(5)\times
U(1)$ supergravity in both no-scale and dilaton scenarios. Since all sparticle masses nearly scale with the gluino mass (or the chargino mass), it suffices to show the dependences of these parameters on, for example, the chargino mass. Table 1 can be used to deduce the dependences on any of the other masses. We only show the explicit dependence on the chargino mass (in Figs. 1,3) for the case $m_t=170\GeV$, since for $m_t=130,150\GeV$ there are no constraints at the 90%CL. However, in the correlated $(\epsilon_1,\epsilon_b)$ plots (Figs. 2,4) we show the results for all three values of $m_t$.
The qualitative results for $\epsilon_1$ are similar to those obtained in Refs. [@ewcorr; @bsg-eps] using the old definition of $\epsilon_1$. That is, for light chargino masses there is a large negative shift due to a threshold effect in the $Z$-wavefunction renormalization for $m_{\chi^\pm_1}\to
{1\over2}M_Z$ (as first noticed in Ref. [@BFC]). As soon as the sparticle masses exceed $\sim100\GeV$ the result quickly asymptotes to the Standard Model value for a light Higgs boson mass ($\lsim100\GeV$). Quantitatively, the enlarged set of observables in the new $\epsilon$-scheme shifts the experimentally allowed range somewhat and the bounds become slightly weaker than in Refs. [@ewcorr; @bsg-eps]. These remarks apply to both no-scale and dilaton scenarios.
In the case of $\epsilon_b$, the results also asymptote to the Standard Model values for large sparticle masses as they should. Two competing effects are seen to occur: (i) a positive shift for light chargino masses, and (ii) and negative shift for light charged Higgs masses and small values of $\tan\beta$. In fact, the latter effect becomes evident in Figures 1,3 (bottom rows) as the solid curve corresponding to $\tan\beta=2$. What happens here is that the charged Higgs contribution nearly cancels the chargino contribution [@BF], making $\epsilon_b$ asymptote much faster to the SM value.
We also notice from Figure 3 (bottom row) that there are lines of points far below the solid curve corresponding to $\tan\beta=2$ in the dilaton scenario. These correspond to [*large*]{} $\tan\beta(\gsim {m_t\over{m_b}})$ for which the charged Higgs diagram gets a significant contribution $\sim
m^2_b\tan^{2}\beta$ coming from the charged Higgs coupling to $b_R$. Such large values of $\tan\beta$ are not allowed in the no-scale scenario. It must be emphasized that for such large values of $\tan\beta$, the neglected neutralino–neutral Higgs diagrams will also become significant [@BF] and since especially neutralino diagrams give a positive contribution, their effect could compensate the large negative charged Higgs contributions.
For $m_t=170\GeV$ at the 90%CL one can safely exclude values of $\tan\beta\lsim2$ in the no-scale and dilaton (except for just one point for $\mu<0$) scenarios. Moreover, as Figs. 1,3 show, there are excluded points for all values of $\tan\beta$. In the dilaton scenario, large values of $\tan\beta$ (, $\tanb\gsim32$ for $\mu>0$ and $\tanb\gsim24$ for $\mu<0$) are also constrained, and even perhaps excluded if the neutralino–neutral-Higgs contributions are not large enough to compensate for these values.
It is seen that for light chargino masses and not too small values of $\tan\beta$, the fit to the $\epsilon_b$ data is better in $SU(5)\times
U(1)$ supergravity than in the Standard Model, although only marginally so. To see the combined effect of $\epsilon_{1,b}$ for increasing values of $m_t$, in Figs. 2,4 we show the calculated values of these parameters for $m_t=130,150,170\GeV$, as well as the $1\sigma$ experimental ellipse (from Ref. [@ABCII]). Clearly smaller values of $m_t$ fit the data better.
Conclusions
===========
We have computed the one-loop electroweak corrections in the form of the $\epsilon_1$ and $\epsilon_b$ parameters in the context of $SU(5)\times U(1)$ supergravity in both no-scale and dilaton scenarios. The new $\epsilon$-scheme used allows to include in the experimental constraints all of the LEP data. In addition, the minimality of parameters in $SU(5)\times U(1)$ supergravity is such that rather precise predictions can be made for these observables and this entails strict constraints on the parameter spaces of the two scenarios considered.
In agreement with our previous analysis, we find that for $m_t\lsim160\GeV$, at the 90%CL these constraints are not restricting at present. However, their quadratic dependence on $m_t$ makes them quite severe for increasingly large values of $m_t$. We have studied explicitly the case of $m_t=170\GeV$ and shown that most points in parameter space are excluded. The exceptions occur for light chargino masses which shift $\epsilon_1$ down and $\epsilon_b$ up. However, for $\tan\beta\lsim2$ the $\epsilon_b$ constraint is so strong that no points are allowed in the no-scale scenario.
In the near future, improved experimental sensitivity on the $\epsilon_b$ parameter is likely to be a decisive test of $SU(5)\times U(1)$ supergravity. In any rate, the trend is clear: lighter values of the top-quark mass fit the data much better than heavier ones do. In addition, supesymmetry seems to always help in this statistical agreement. Finally, if the top quark continues to remain undetected at the Tevatron, high-precision LEP data in the context of $SU(5)\times U(1)$ supergravity would suggest the presence of light charginos.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been supported in part by DOE grant DE-FG05-91-ER-40633. The work of G.P. has been supported by a World Laboratory Fellowship. G.P. thanks Michael Boulware and Donnald Finnell for very helpful discussions.
[99]{} G. Altarelli, CERN-TH.6867/93 (April 1993). E. Eliasson, ; S. Lim, , ; J. Grifols and J. Sola, ; B. Lynn, , in Physics at LEP, eds. J. Ellis and R. Peccei, CERN Yellow Report CERN86-02, Vol. 1; R. Barbieri et al., ; A. Bilal, J. Ellis, and G. Fogli, ; M. Drees and K. Hagiwara, ; M. Drees, K. Hagiwara, and A. Yamada, . R. Barbieri, M. Frigeni, and F. Caravaglios, . G. Altarelli, R. Barbieri, and F. Caravaglios, . G. Altarelli, R. Barbieri, and F. Caravaglios, . For a recent review and extensive references see , Erice 93 Subnuclear Physics School Lecture, . For a recent review see , , and A. Zichichi, CERN-TH.6926/93 and . , , and A. Zichichi, CERN-TH.6667/92, (to appear in Phys. Rev. D). , , and A. Zichichi, CERN-TH.6903/93, . , , G. T. Park, H. Pois, and K. Yuan, . , , G. T. Park, and A. Zichichi, (to appear in Phys. Rev. D). G. Altarelli and R. Barbieri, G. Altarelli, R. Barbieri, and S. Jadach, . A. Blondel and C. Verzegnassi, . , , X. Wang, and A. Zichichi, ; , , H. Pois, X. Wang, and A. Zichichi, ; , , X. Wang, and A. Zichichi, . , , H. Pois, X. Wang, and A. Zichichi, . , , and G. T. Park, ;, , and X. Wang, (to appear in Phys. Rev. D). See , L. Ibáñez and D. Lüst, ; V. Kaplunovsky and J. Louis, ; A. Brignole, L. Ibáñez, and C. Muñoz, FTUAM-26/93 (August 1993). See , S. Kelley, , , H. Pois, and K. Yuan, . D. Kennedy and B. Lynn, ; D. Kennedy, B. Lynn, C. Im, and R. Stuart, . M. Peskin and T. Takeuchi, ; W. Marciano and J. Rosner, ; D. Kennedy and P. Langacker, . B. Holdom and J. Terning, ; M. Golden and L. Randall, ; A. Dobado, D. Espriu, and M. Herrero, . M. Boulware and D. Finnell, . J. Bernabeu, A. Pich, and A. Santamaria, ; W. Beenaker and W. Hollik, Z. Phys. C40, 141(1988); A. Akhundov, D. Bardin, and T. Riemann, ; F. Boudjema, A. Djouadi, and C. Verzegnassi, . A. Denner, R. Guth, W. Hollik, and J. Kühn, Z. Phys. C51, 695(1991). The neutral Higgs contributions to $\Zbb$ were also calculated here. G. T. Park, (September 1993). G. T. Park, (October 1993). A. Djouadi, G. Girardi, C. Verzegnassi, W. Hollik, and F. Renard, .
[**Figure Captions**]{}
Figure 1: The predictions for the $\epsilon_1$ (top row) and $\epsilon_b$ (bottom row) parameters versus the chargino mass in the no-scale $SU(5)\times
U(1)$ supergravity scenario for $m_t=170\GeV$. In the top (bottom) row, points between (above) the horizontal line(s) are allowed at the 90% CL. The solid curve (bottom row) represents the $\tanb=2$ line.
Figure 2: The correlated predictions for the $\epsilon_1$ and $\epsilon_b$ parameters in $10^{-3}$ in the no-scale $SU(5)\times U(1)$ supergravity scenario. The ellipse represents the $1\sigma$ contour obtained from all LEP data. The values of $m_t$ are as indicated.
Figure 3: The predictions for the $\epsilon_1$ (top row) and $\epsilon_b$ (bottom row) parameters versus the chargino mass in dilaton $SU(5)\times U(1)$ supergravity scenario for $m_t=170\GeV$. In the top (bottom) row, points between (above) the horizontal line(s) are allowed at the 90% CL. The solid curve (bottom row) represents the $\tanb=2$ line.
Figure 4: The correlated predictions for the $\epsilon_1$ and $\epsilon_b$ parameters in $10^{-3}$ in the dilaton $SU(5)\times U(1)$ supergravity scenario. The ellipse represents the $1\sigma$ contour obtained from all LEP data. The values of $m_t$ are as indicated.
|
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abstract: 'Recent studies adopting $\lambda_{\rm Re}$, a proxy for specific angular momentum, have highlighted how early-type galaxies (ETGs) are composed of two kinematical classes for which distinct formation mechanisms can be inferred. With upcoming surveys expected to obtain $\lambda_{\rm Re}$ from a broad range of environments (e.g. SAMI, MaNGA), we investigate in this numerical study how the $\lambda_{\rm Re}$-$\epsilon_{\rm e}$ distribution of fast-rotating dwarf satellite galaxies reflects their evolutionary state. By combining N-body/SPH simulations of progenitor disc galaxies (stellar mass $\simeq$10$^{\rm 9}$ M$_{\odot}$), their cosmologically-motivated sub-halo infall history and a characteristic group orbit/potential, we demonstrate the evolution of a satellite ETG population driven by tidal interactions (e.g. harassment). As a general result, these satellites remain intrinsically fast-rotating oblate stellar systems since their infall as early as $z=2$; mis-identifications as slow rotators often arise due to a bar/spiral lifecycle which plays an integral role in their evolution. Despite the idealistic nature of its construction, our mock $\lambda_{\rm Re}$-$\epsilon_{\rm e}$ distribution at $z<0.1$ reproduces its observational counterpart from the [ATLAS]{}$^{\rm 3D}$/SAURON projects. We predict therefore how the observed $\lambda_{\rm Re}$-$\epsilon_{\rm e}$ distribution of a group evolves according to these ensemble tidal interactions.'
author:
- |
C. Yozin[^1] and K. Bekki\
\
ICRAR, M468, The University of Western Australia, 35 Stirling Highway, Crawley Western Australia, 6009, Australia
bibliography:
- 'bib.bib'
date: 'Accepted 2015 January. Received 2015 January; in original form 2015 January'
title: The global warming of group satellite galaxies
---
\[firstpage\]
\[3\][\#2]{}
galaxies: interactions – galaxies: dwarf
Introduction
============
In a major step beyond classifying early-type galaxies (ETGs) according to their morphology alone, the spatially-resolved velocity fields arising from recent volume-limited IFS (integral-field spectroscopy) surveys (SAURON, [[ATLAS]{}$^{\rm 3D}$]{}, CALIFA) have provided deeper insight into their formation. By consolidating this kinematic data into a proxy ($\lambda_{\rm Re}$) for the specific angular momentum, a metric tightly correlated with morphology [@obre14], it has become possible to distinguish two classes of ETGs: fast and slow rotators [FR and SR respectively; @emse07]. The pressure-supported SRs are predominantly round, old and massive (stellar mass [M$_{\ast}$]{}$>$[10$^{\rm 10.5}$]{} [M$_{\odot}$]{}), whilst FRs appear continuous (in both kinematics and shape) with late-type galaxies [LTGs; @emse11; @kraj11; @weij14].
This dichotomy has parallels with the red-blue classification of galaxies, and likewise, SR ETGs are conceived to be the natural products of their hierarchical assembly [@khoc06]. Myriad numerical studies have addressed this scenario; those recently motivated by [[ATLAS]{}$^{\rm 3D}$]{} have established that both binary major mergers and successive minor mergers can reproduce the kinematic peculiarities of SRs [@bour07; @jess09; @bois11; @mood14]. In practice, the cosmological assembly of a SR galaxy follows a non-trivial evolution in $\lambda_{\rm R}$ depending on merger mass ratios/gas fractions [@naab14].
As IFS surveys broaden their sample size, more focus will shift to the influence of different environments on forming SR/FRs. Early results in this context suggest SRs are preferentially located in dense locations [@capp13], with a clear radial trend in ellipticity among clusters [@deug15]. Mounting evidence also suggests the formation of FR ETGs [such as S0s, the underlying driver of the morphology-density relation; @dres80] most commonly follows the transformation of LTGs in groups [@wilm09; @just10].
The distinct kinematics between LTGs and S0s is a strong argument against fading alone in this transition, although it remains to be established which of the associated group mechanisms dominate. In a precursor to this paper, @bekk11 demonstrated that group tides/harassment can faciliate spiral-to-S0 transformation, and @foga15 showed how the corresponding evolution in $\lambda_{\rm Re}$ is consistent with those of the pilot SAMI survey. On the other hand, @quer15 proposed major mergers as a viable formation pathway, in the context of a group host, for those S0s observed in the CALIFA survey.
To address this issue (and build upon the success of adopting $\lambda_{\rm Re}$ as a discriminant of formation mechanisms), we introduce in this paper a new concept. Based on the premise that environmental mechanisms acting within a group or cluster can be imprinted on the [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distribution of its satellite galaxies, we describe here the appearance of this [*global warming*]{} from LTGs to ETGs as a function of redshift.
Although tested against the SAURON and [[ATLAS]{}$^{\rm 3D}$]{} samples (the latter comprising 260 galaxies), the work is orchestrated for the purpose of comparison with the upcoming Sydney-AA0 Multi-object IFS survey [SAMI; @croo12]. The complete survey will catalogue 3400 galaxies in the local universe ($z$$<$0.1), with excellent coverage anticipated for galaxies of stellar mass [M$_{\ast}$]{}$=$[10$^{\rm 9-10}$]{} [M$_{\odot}$]{} in group hosts of dynamical mass [10$^{\rm 13}$]{} [M$_{\odot}$]{}. This particular combination incorporates some of the most common galaxies in the universe [@wein09], the most dominant environment to shape cosmic SFH since [$z=1$]{} [@pope15], and is therefore the focus in this present study. Additionally, this builds upon our previous work [@yozi15b hereafter YB15] concerning the long ($\sim$7 Gyr) quenching/transformation timescales of Magellanic Cloud-type dwarf group-satellites as inferred from near-field observations. The paper is organised as follows: in Section 2, we describe the numerical simulations and the post-processing that permits comparison with the [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distributions of [[ATLAS]{}$^{\rm 3D}$]{}/SAMI galaxies; in Section 3, we describe the results of a parameter study with these models, and Section 4 concludes this study with a discussion on the merits of our proposed concept for understanding the dynamical state of a group galaxy population.
Method
======
![(Top panel) Probabilility distribution of lookback times at which a [M$_{\ast}$]{}$=$[10$^{\rm 9}$]{} [M$_{\odot}$]{} first becomes a satellite [reproduced from @delu12]; (Bottom panel) A schematic of the group model with the virial radius (r$_{\rm vir}$, shown as grey dashed line) bounding the group halo potential (with grey colour scale representing mass density), characteristic galaxy orbit (black solid line) and other satellites (red circles, with size$\propto$mass); (inset panel) The characterstic galaxy orbit, expressed with the group-centric radius as a function of time.](p1.eps){width="1.\columnwidth"}
Our methodology is comprised of two key steps, the first of which is the simulation of a self-consistent N-Body galaxy model with an interstellar medium (ISM) treated with smoothed particle hydrodynamics (SPH). This model is initially placed on a characteristic orbit within a fixed potential representing the group dark matter halo, as defined in YB15 for satellites with stellar mass ([M$_{\ast}$]{}) of [10$^{\rm 9-9.5}$]{} [M$_{\odot}$]{}.
The second step involves the statistical sampling of this simulated galaxy model evolution, such that we can construct a mock satellite population at a given redshift comprised of galaxies with nominally different infall times. We assume here that infall time ($t_{\rm inf}$) refers to the lookback time at which a satellite first crossed the virial radius of its host halo. Furthermore, our choice of $t_{\rm inf}$ is cosmologically motivated insofar as it is drawn from a probability distribution $p$($t_{\rm inf}$) of satellite infall times (with [M$_{\ast}$]{}$\simeq$[10$^{\rm 9}$]{} [M$_{\odot}$]{}; top panel of Fig. 1) corresponding to merger trees constructed from the Millennium Simulation [@delu12].
The main results of this paper, the mock [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distribution at a given redshift (lookback time $t$), are thus constructed using the Monte-Carlo sampling of $p$($t_{\rm inf}$). Repeated for a minimum of 1000 samplings, we incorporate the simulated galaxy model properties into the mock distribution at a simulation timestep $t_{\rm i} = t_{\rm inf}-t$ if that timestep $t_{\rm i}\geq$0.
These steps are repeated for several galaxy models, differing in properties such as their gas fraction or orbital orientation. This scenario is also compared with those in which other group satellites, represented as point masses, are added to the group model to establish the influence of satellite-satellite tidal interactions. We emphasize therefore that our methodology is based on an idealised group model comprising a fixed dark halo and its satellite population (bottom panel, Fig. 1); the adopted orbit is sufficiently energetic (e.g. the pericentre radius or distance to the group halo centre is sufficiently large) that the explicit modelling of a central galaxy (which by definition lies at the halo centre) is not considered to be necessary.
Numerical models
----------------
Parameter Fiducial value
------------------------------------------------ -----------------------------
N$^o$. DM particles [10$^{\rm 6}$]{}
N$^o$. Stellar particles 3$\times$[10$^{\rm 5}$]{}
N$^o$. Gas particles 2.5$\times$[10$^{\rm 5}$]{}
Stellar mass/[M$_{\odot}$]{} (M$_{\rm d}$) 10$^{\rm 9.25}$
Gas/stellar mass (M$_{\rm g}$/M$_{\rm d}$) 2
DM halo/stellar mass (M$_{\rm h}$/M$_{\rm d}$) 85
Stellar disc scalelength (r$_{\rm d}$/kpc) 1.6
Stellar disc scaleheight (z$_{\rm d}$/kpc) 0.32
Gas disc scalelength (r$_{\rm g}$/kpc) 4.2
Gas disc scaleheight (z$_{\rm g}$/kpc) 0.84
DM Halo virial radius (r$_{\rm vir}$/kpc) 142
DM Halo concentration ($c$) 6.5
Metallicity (\[Fe/H\]) -0.72
Metallicity gradient ($\Delta$\[Fe/H\]/kpc) -0.04
SN expansion timescale ($\tau_{\rm SN}$/yr) [10$^{\rm 5.5}$]{}
: A summary of fiducial galaxy model properties
We present results from 22 galaxy model simulations, each run with an original parallelised chemodynamical code GRAPE-SPH [@bekk09], combining gravitational dynamics and an ISM with radiative cooling/stellar processing. Our numerical satellite/group models follow that of YB15 closely, to which we refer the interested reader.
To summarise, the galaxy commences at apocentre on a characteristic orbit (constrained to the $x-y$ plane) which is derived from simulations of collisionless satellite infall with a live model of the dark matter (DM) halo [@vill12]. The orientation of the galaxy model is specified by $\theta$ and $\phi$, corresponding to the angle between the $z-$axis and the vector of the angular momentum of the disc, and the azimuthal angle measured from the $x-$axis to the projection of the angular momentum vector of the disc on to the $x-y$ plane, respectively. Our fiducial model adopts $\theta$=45$^{\circ}$ and $\phi$=30$^{\circ}$.
Fig. 1 conveys this orbit and a schematic of the group model which consists of a spherically-symmetric potential alone in the fiducial case (Sections 3.1-2), and incorporates $\sim$50 group satellites (represented in the simulation as point masses) when considering the additional effect of tidal harassment (Section 3.3).
The group potential has total mass of [10$^{\rm 13}$]{} [M$_{\odot}$]{} and structural properties characteristic of [$z=1$]{} [virial radius 380 kpc and a NFW density profile with concentration 4.9; @vill12]. We use Monte-Carlo sampling to fit the point mass satellites to a Schecter function (slope -1.07) function, limited to a luminosity range 0.01 to 2.5 solar and with the assumption of a mean mass-to-light ratio $\sim$40. Their initial orbits are assigned according to a NFW profile with concentration 3.0.
### Fiducial galaxy model
Table 1 summarises the principle properties of our fiducial galaxy model, which consists of a gas-rich bulgeless (bulge-to-total ratio, B/T=0) galaxy as in YB15. The initial gas and halo mass with respect to the stellar mass is selected according to best-fit relations from the ALFALFA survey [@huan12] and @mill14 respectively, while DM halo properties are obtained from mass-redshift-concentration relations of @muno11. The stellar disc scalelength r$_{\rm d}$ is observationally motivated from the size-mass relations of @ichi12, while the gas disc scalelength r$_{\rm g}$ is assumed to be a factor $\sim$3 larger [@krav13]; stellar and gas scaleheights z$_{\rm d}$ and z$_{\rm g}$ are assigned as 0.2r$_{\rm d}$ and 2z$_{\rm d}$ respectively.
While we do not consider the star formation (SF) history here, gas inflow and bulge growth are crucial components to our model evolution, and therefore subgrid SF parametrisations and the efficiency of supernovae (SNe, controlled primarily with the adiabatic expansion timescale $\tau_{\rm SN}$) are constrained according to the Kennicutt-Schmidt law. To offset the idealistic nature of our scenario, we emphasise that our minimum baryonic resolution of $\sim$[10$^{\rm 4}$]{} [M$_{\odot}$]{} is two orders of magnitude less than recent cosmological simulations [e.g. Illustris; @gene15].
### Alternate galaxy models
The simulations constituting our parameter study are summarised in Table 2.
- To account for pre-processing prior to group infall, we construct [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distributions for a galaxy with an initial B/T of 0.2, where the bulge has a Hernquist density profile, isotropic velocity dispersion (with radial velocity dispersion assigned according the Jeans equation of a spherical system).
- To further account for the possibility of efficient gas stripping prior to or upon infall to the group, we consider the evolution of a gas-poor galaxy (gas mass$=$$0.2$[M$_{\ast}$]{}).
- The dependence on orbital orientation is addressed in three scenarios, including: Orbit:Face-on ($\theta$=90$^{\circ}$, $\phi$=0$^{\circ}$), Orbit:Edge-on ($\theta$=0$^{\circ}$, $\phi$=0$^{\circ}$) and Orbit:Retrograde ($\theta$=-45$^{\circ}$, $\phi$=-30$^{\circ}$).
- The influence of satellite-satellite interactions (in a satellite population whose mean velocity dispersion is $\sim$150 kms$^{\rm -1}$) is established for the fiducial and B/T=0.2 models with two suites of simulations, each comprising 8 runs where the live galaxy is placed randomly on the locus defined by the group virial radius. This method accounts for the quasi-stochastic influence of the point-mass group satellites on the live galaxy model.
Simulation Deviation from fiducial
--------------------- ------------------------------------
B/T=0.2 Initial B/T of 0.2
Orbit:Edge-on Edge-on disc w.r.t. orbit plane
Orbit:Face-on Face-on disc w.r.t. orbit plane
Orbit:Retrograde Retrograde w.r.t. orbit plane
Gas fraction:Low Initial gas fraction of 0.2
Fiducial+Harassment Group satellites added to fiducial
B/T=0.2+Harassment Group satellites added to B/T=0.2
: Summary of 21 non-fiducial models and their deviations from the fiducial; the two entries incorporating harassment are each comprised of eight simulations.
Analysis parameters
-------------------
We deduce both intrinsic and apparent specific angular momenta and ellipticities of our galaxy models. Intrinsic parameters are calculated from the 3D spatial/velocity data of all stellar particles within the half-mass radius, where the intrinsic ellipticity is established from decomposition of the inertial tensor of all stars (from which the ellipticity is 1-$c$/$a$ where $c$ and $a$ are the minor and major axes respectively).
The apparent specific angular momentum is computed according to the relation defined by @emse07: $$\lambda_{\rm R} = \frac{ \sum_{i=1}^{N_{\rm b}} F_{\rm i} R_{\rm i} |V_{\rm i}| } { \sum_{i=1}^{N_{\rm b}} F_{\rm i} R_{\rm i} \sqrt{ V_{\rm i}^{\rm 2} + \sigma_{\rm i}^{\rm 2} } }$$ where F$_{\rm i}$, R$_{\rm i}$, V$_{\rm i}$ and $\sigma_{\rm i}$ are the flux (computed here from the integrated surface brightness), radius, projected stellar velocity and stellar velocity dispersion respectively, in the $i$th spatial bin.
As in YB15, we adopt synthesis models for stellar evolution and an assumed gas-to-dust ratio of 5$\times$[10$^{\rm 21}$]{} cm$^{\rm -2}$A$_{\rm V}$ to establish stellar luminosities from the stellar age/mass of simulation particles. Surface brightness distributions of the galaxy models can be therefore constructed on 2D spatial grids. Ellipticities are calculated according to the [Kinemetry]{} method [@kraj06], in which concentric ellipses are fitted to binned line-of-sight kinematic maps.
@emse07 showed that the apparent properties taken at the effective radius, r$_{\rm e}$ (enclosing half the projected light), are sufficient to characterise inner disc kinematics. From the radial profile of $\lambda_{\rm R}$ and the ellipse ellipticities expressed as a function of their mean radii, we compute $\lambda_{\rm Re}$ and $\epsilon_{\rm e}$ respectively by linear interpolation at r$_{\rm e}$.
In general, our method in obtaining these values is consistent with those of previous studies [e.g. @emse07; @bois11; @naab14; @mood14]; however, we have not adopted their Voronoi binning in construction of the kinematic maps, as our simulation resolution is sufficient to provide the requisite signal-to-noise ratio up to r$_{\rm e}$ [i.e. at least $\sqrt{N}$$\ge$20 per bin; @capp03].
{width="1.\columnwidth"}
Our mock [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distributions (at a given lookback time) combine the $\lambda_{\rm Re}$ and $\epsilon_{\rm e}$ values deduced from the corresponding galaxy model data when inclined and viewed at 100 equispaced increments of $i$, where $i$ is the stellar disc inclination $0^{\circ}<i<90^{\circ}$ with respect to face-on. This approach is based on the premise that observed galaxy populations have no preferential viewing inclination.
Furthermore, to compare this data with [[ATLAS]{}$^{\rm 3D}$]{} galaxies (and those of the SAMI pilot survey), whose criterion for ETGs is predicated on the absence of spiral arms, we exclude those $\lambda_{\rm Re}$ and $\epsilon_{\rm e}$ computed at timesteps where the face-on surface brightness distribution has a mean second Fourier amplitude (A$_{\rm 2}$, within 2r$_{\rm e}$) exceeding 0.2. We have confirmed that our criterion does indeed correspond to an apparent absence of spiral structure in mock B-band surface brightness imaging of our galaxy models down to $\sim$28 mag arcsec$^{\rm 2}$.
Fig. 2 compares this A$_{\rm 2}$ threshold with the computed A$_{\rm 2}$ of the fiducial model as a function of time. Also shown is the stellar bar radius, computed as in YB15. We find the model to be unstable to $m=2$ perturbations following tidal interactions at successive orbital pericentres. Subsequent gas inflow and angular momentum exchange to the bar can destroy it [@frie93; @bour05a], and therefore, the satellite undergoes multiple cycles between ETG and LTG classification.
Concurrently, the bulge-to-total ratio (which we calculate from the excess light with respect to a fitted exponential disc; see YB15 for more details) grows monotonically with time (Fig. 2, bottom panel). We assert that this constitutes a pseudo-bulge for the initially bulgeless Fiducial model because the corresponding kinematic fields strongly imply its rotational-support [@korm04].
Results
=======
![A mock distribution of apparent specific angular momentum, $\lambda_{\rm Re}$ vs. apparent ellipticity, $\epsilon_{\rm e}$ (both measured at at r$_{\rm e}$) for the Fiducial galaxy model at [$z=1.0$]{} (inset panel) and $<0.1$ (main panel). The 2D probability density plot in each panel shows the normalised distribution of ETGs (as classified with A$_{\rm 2}$). In the bottom panel, this is compared with the full LTG+ETG population (green contours), together with those FRs identified in [[ATLAS]{}$^{\rm 3D}$]{} [@emse11] to appear as SRs (white symbols). The white dashed line shows an empirical boundary between SR/FRs ($\lambda_{\rm Re}=0.31\epsilon_{\rm e}^{\rm 0.5}$). The red solid line shows show an isotropic stellar system viewed edge-on would appear in this plot; the red dashed lines show how this varies with viewing inclination (where face-on lies at the origin).](p3.eps){width="1.\columnwidth"}
Fig. 3 shows the mock probability distribution of [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} for our fiducial simulation at redshifts [$z=1$]{} and $<$0.1 (the priority coverage of the full SAMI survey). This distribution is comprised of galaxies with stellar masses in the range [10$^{\rm 9-9.5}$]{} [M$_{\odot}$]{} due to the combined effect of secular/triggered SF and gas stripping acting on a disc with initial mass [M$_{\ast}$]{}$=$[10$^{\rm 9.25}$]{} [M$_{\odot}$]{}. Overlaid upon each density plot is the theoretical properties of an edge-on stellar system (solid red line) assuming a two-integral dynamical Jeans model [@capp07; @emse11], and its corresponding projection towards face-on (dashed lines, towards the origin).
We find at an early epoch that our (predominantly Sc/d) satellite population is spread along along the upper-most dashed arcs (corresponding to an intrinsic $\epsilon_{\rm e}=0.7-0.8$). At lower redshifts, this distribution becomes more concentrated with a clear peak forming by $z<0.1$ at $\epsilon_{\rm e}=0.5$, $\lambda_{\rm Re}=0.5$. Fig. 2 (top panel) shows that in the intervening time the B/T ratio grows from 0 to 0.25; the corresponding (pseudo-)bulge is rotationally-supported and dominated by young stars.
An outlying proportion of the ETG population falls below an empirical criterion from [[ATLAS]{}$^{\rm 3D}$]{} that denotes slow rotation ($\lambda_{\rm Re}=0.31\sqrt{\epsilon_{\rm e}}$). Since we enforce a consistent definition of [$\lambda_{\rm Re}$]{} based on apparent properties at r$_{\rm e}$, we expect some apparent SR classifications to arise due to a stellar bar. This bar, the size of which diminishes with time after a peak at first pericentre (Fig. 2), distorts the ellipicity metric in viewing inclinations close to face-on [@capp07].
Our autonomous ETG criterion based on A$_{\rm 2}$ filters out many more such SR classifications arising from the full LTG+ETG population (Fig. 3, green contours); this distribution shows a [*tendril*]{} at $\epsilon_{\rm e}=0.6, \lambda_{\rm Re}$=0.3 which lies outside the theoretical bounds for an oblate axisymmetric system, and corresponds to strong tidal features formed during the first orbital pericentre.
The capacity for group tidal interactions to induce non-equilibrium kinematics in the satellite is demonstrated here also in terms of intrinsic properties (Fig. 4). Here we find in general a close concordance between our simulation of a single fiducial galaxy model (with infall at 10 Gyr ago) and the best-fit relation for a two-integral Jeans dynamical system [@capp07; @emse11]. This applies across a wide range of $\epsilon_{\rm e}$, albeit with some significant transient deviations at orbital pericentres (e.g. $\epsilon_{\rm e}=0.5, \lambda_{\rm Re}$=0.5).
The existence of apparent SRs are therefore traced to transient perturbations to an otherwise persistently disc-like kinematic field [@kraj11; @barr15], and also projection effects including that due to bars which we note are undetected by photometry in as many as 10 percent of galaxies in the [CALIFA]{} survey [@holm15]. The location of apparent SRs (at $\epsilon_{\rm e}=0.2$, $\lambda_{\rm Re}=0.1$; Fig. 3) coincides with the simulated remnants of late mergers [@naab14], and observed [[ATLAS]{}$^{\rm 3D}$]{} galaxies whose line-of-sight velocity distributions show kinematically-decoupled components (KDCs), non-regular rotation (NRR) or double-sigma components (2$\sigma$) as opposed to pressure-supported kinematics [white symbols; @emse11].
![The intrinsic evolution of the Fiducial simulation (black filled circles) and B/T=0.2 (red open squares)in the [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} plane; black labels give an approximate time since the start of the simulation. The approximate best-fit relation ($\lambda_{\rm Re}=0.93\epsilon_{\rm e}^{\rm 0.5}$) to the expected evolution of an isotropic stellar system is conveyed with the grey dashed line.](p4.eps){width="1.\columnwidth"}
Parameter dependencies
----------------------
Fig. 5 (top panel) shows how the $z<0.1$ [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distribution depends on our initial galaxy model parameters. The key results are summarised as follows:
- Our B/T=0.2 model displays an concentration lying closer to the origin than the fiducial case. This is to be expected from an inner disc hosting a classical (pressure-supported) bulge in addition to the rotation-supported pseudo-bulge formed also in the fiducial case (with a final B/T=0.4; Fig. 2). Though not shown in this figure, we note that fewer apparent SRs are detected due in part to an inner disc less favourable for the radial ($x_{\rm 1}$) stellar orbits that constitute a stellar bar [@norm96].
- Gas fraction:Low does not form a pseudo-bulge as efficiently as the fiducial case, where in the latter case the loss in gas angular momentum by successive tidal torques fuels bulge growth [@barn91]. In spite of this, the [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distribution shows a small deviation towards higher $e_{\rm e}$, a result similar to that found among the simulated remnants of dissipative/collisonless mergers [@jess09]. Of further note is that the relatively small dissipative component allows the triggered bar (and its associated distortion of $\epsilon_{\rm e}$) to persist on order of a Hubble time causing a higher incidence of SR mis-identifications.
- The [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distribution shows a clear dependence on the satellite orbit, a proxy to the dependence on tidal torque efficiency. Orbit:Face-on is not sufficiently perturbed to facilitate much morphological/kinematic evolution, and there exists negligible variation between prograde or retrograde orbits. In general, we find [$\lambda_{\rm Re}$]{} and $\epsilon_{\rm e}$ drop sharply at orbital pericentres, as a function of time, by values that scale with ${\sim}cos(i)$ (where $i$ is galaxy inclination to its orbit).
Dependence on tidal harassment
------------------------------
Fig. 5 (Bottom panel) shows the [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distribution for Fiducial + Harassment at $z<0.1$. The addition of point-mass group satellites causes the dispersion of the highly concentrated distribution associated with our fiducial galaxy model at an intermediate orbital inclination (top panel). This dispersion, while largely remaining within theoretical limits (red lines), is due to the quasi-stochasticity of satellite-satellite interaction impact parameters and inclinations. As in the case of simulated minor mergers wherein kinematically-distinct stellar components often arise [@bois11], the proportion of SR misidentifications is more significant than in our fiducial case (Fig. 3).
![(Top panel) As Fig. 3, but where the [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distribution of the fiducial simulation at $z<0.1$ (2D probability density plot) is compared with simulations commencing with differing galaxy model conditions (contours shown at $p=0.75$ and 0.875); (bottom panel) As Fig. 3, but showing the ensemble [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distribution of Harassment at $z<0.1$. Green contours represent the mock distribution of 50,000 FR ETGs [@emse11] based on the Monte-Carlo method of @capp07.](p5.eps){width="1.\columnwidth"}
Discussion and conclusions
==========================
Although $\lambda_{\rm Re}$ has already proven a useful metric for distinguishing the formation paths of ETGs, galaxy evolution on the [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} plane remains a complex, non-linear process within a $\Lambda$CDM cosmology [@naab14]. In this study, we demonstrate how it is instructive to consider the global [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distribution of many galaxies, analysed in a consistent manner and including projection effects. In environments like groups, a target for the upcoming SAMI survey among others, this method can overcome the non-monotonic evolution of individual satellites. Instead, we can describe the effective redshift-evolution of the group itself, which we euphemistically describe as undergoing a process of [*global warming*]{}.
This demonstration was performed with a suite of idealised simulations derived from those of @yozi15b and thus shares the same limitations, including a fixed spherical group potential. We argue that since the intrinsic evolution of our satellites generally follow that of an oblate FR (Fig. 4), they will tend towards a similar [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distribution albeit at a rate determined by orbital parameters. This particular evolution is predicated on an absence of major mergers, which we deem rare for the [M$_{\ast}$]{}$=$[10$^{\rm 9}$]{} [M$_{\odot}$]{} satellites considered here, in accordance with the ILLUSTRIS simulation which predicts $\sim$0.1 such events per satellite since [$z=2$]{} [@rodr15].
Significant minor mergers (e.g. mass ratios of 1:10 to 1:6) will be more frequent, however; previous numerical studies [e.g. @bour05b] have suggested these events yield remnants analogous to S0s, and comparable therefore to the FR ETGs produced by our methodology (where bulge growth is likewise promoted by in-situ SF). However, this parallelism should be explicitly challenged in a future study.
Hydrodynamical interactions between the satellite and the intra-group medium are also ignored; our results and those of @bois11 have suggested that the extant gas fraction has only a small impact on [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{}, although smooth accretion has been shown to raise [$\lambda_{\rm Re}$]{} if suitably aligned with the disc [@chri15]. We do not expect, however, the environmental mechanism of ram pressure stripping will significantly influence a kinematics-driven evolution of [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{}; in fact, this can be exploited in efforts to discern galaxy formation mechanisms [e.g. S0s, see also @quer15].
Our models also lack an explicit model of galactic outflow, which recent cosmological simulations demonstrate as a key actor in regulating angular momentum [@gene15]. Our preliminary simulations suggest the effect of stronger feedback in our model (applied with a longer adiabatic timescale, e.g. $\tau_{\rm SN}={\rm 10}^{\rm 6}$ yr) merely controls the rate of [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} evolution rather than its shape, the latter being the focus of this present study.
Lastly, several studies have noted that SR formation is highly resolution dependent [@bois10], insofar as convergence requires a baryonic mass resolutions of at least $\sim$[10$^{\rm 4}$]{} [M$_{\odot}$]{}; indeed, the merger simulations of @mood14 adopt this threshold and find noise fluctuations cause $\lambda_{\rm Re}$ to deviate by only $\sim$0.01. Since our simulations also lie at this threshold, we maintain our claim that the group mechanisms modelled here produce only FR ETGs.
![As Fig. 3, but for mock [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distributions derived from Fiducial + Harassment and B/T=0.2 + Harassment at [$z=1.0$]{} and $<0.1$. White contours are chosen at $p$=0.7 and 0.85 to illustrate the distinct evolutions associated with disc-dominated FR ETGs (B/T=0 and B/T=0.2) over $\sim$8 Gyr.](p6.eps){width="1.\columnwidth"}
In an attempt to verify our idealised numerical methodology, we compare our results in Fig. 5 (bottom panel) to the mock distribution of 50,000 FR ETGs galaxies obtained by @emse11. Their Monte-Carlo method follows from a linear relation found between intrinsic $\epsilon$ and anisotropy among [SAURON]{} galaxies [@capp07], for which the associated scatter is modelled as a Gaussian [see also @weij14]. The resulting mock distribution (green contours) shows qualitative agreement with both the [[ATLAS]{}$^{\rm 3D}$]{} sample of FRs (which we note is drawn from volume-limited surveys and therefore comprises satellite and central galaxies), and our $z<0.1$ snapshot of Fiducial + Harassment.
Specifically, Fiducial + Harasment reproduces the spread of [[ATLAS]{}$^{\rm 3D}$]{} galaxies and their concentration around $e_{\rm e}=0.5, \lambda_{\rm Re}=0.5$. Since the majority of the [[ATLAS]{}$^{\rm 3D}$]{} galaxies were identified by morphological type $T$ as S0s, we can claim therefore that our simulations capture the fundamental mechanisms that yield a S0 morphology. Fig. 6 (bottom panels) illustrates how those FR ETGs galaxies hosting an isotropic bulge (classically the remnant of merger events) will appear to be more concentrated towards the origin; this [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distribution is therefore qualitatively consistent with that of the pilot-SAMI ETGs identified as cluster members by @foga15.
Of course, a comparison to the [[ATLAS]{}$^{\rm 3D}$]{}/SAURON and pilot-SAMI catalogs is not strictly correct (given their generally more massive stellar masses with respect to that represented by our models) and thus we look forward to the full SAMI survey for future comparisons.
However, this cursory verification of our low-redshift mock [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distributions of ETG FRs allows us to propose, firstly, that the observed data (for a given stellar mass) represents the superposition of [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distributions specific to particular morphological groups. @capp13b have already shown the bulge fraction can be more reliably inferred from kinematics than bulge-disc decompositions; our broader assertion is that the decomposition of [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} will in principle provide insight as to the evolutionary processes involved in those satellite populations [e.g. pre-processing, and the relative role of classical and pseudo-bulge building mechanisms; @delu12; @wetz13].
This concept complements that of the fundamental/mass plane (FP), a tight scaling relation for ETGs, which the SAMI survey has already proven capable in its construction [@scot15]. Unlike our concept, however, it is not conclusive if properties of the FP (e.g. slope, offset) vary with environment [e.g. @hou15], and indeed its utility as a distance indicator would suffer if it did [@mago12].
A second proposition is that the tidal-driven evolution of a satellite population drives an increasing spheroidal fraction with time, which manifests qualitatively as an increasing concentration of the [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} distribution. This can therefore be exploited to parametrise the [*global warming*]{} of the environment. While admittedly a non-trivial exercise given a cosmological assembly bias, the well-known environment-driven evolution of dwarf satellites since [$z=1$]{} makes them the ideal targets for this concept [@geha12]. Indeed, we advocate the comparison of [$\lambda_{\rm Re}$-$\epsilon_{\rm e}$]{} from high redshift IFU surveys [e.g. the KMOS Redshift One Spectroscopic Survey, presently conducted to compare star forming conditions within 800 star forming galaxies at $z=1$; @stot16] with local counterparts to verify the redshift-dependent behaviour illustrated in Fig. 6.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the anonymous referee for their comments which improved this paper. Part of this work was supported by the Australian Postgraduate Award Scholarship. The authors also wish to thank members of the Computational Theory Group at ICRAR for their useful comments.
\[lastpage\]
[^1]: [**E-mail**]{} 21101348@student.uwa.edu.au; kenji.bekki@uwa.edu.au
|
---
abstract: 'A mechanical electroscope based on a change in the resonant frequency of a cantilever one micron in size in the presence of charge has recently been fabricated. We derive the decoherence rate of a charge superpositon during measurement with such a device using a master equation theory adapted from quantum optics. We also investigate the information produced by such a measurement, using a quantum trajectory approach. Such instruments could be used in mesoscopic electronic systems, and future solid state quantum computers, so it is useful to know how they behave when used to measure quantum superpositions of charge.'
author:
- 'R. E. S. Polkinghorne'
- 'G. J. Milburn'
title: Single electron measurements with a micromechanical resonator
---
[^1]
Introduction
============
As devices for processing and storing information become smaller the demands on the readout technology become ever greater. This is especially true for proposed solid state quantum computers which store information in various quantum degrees of freedom (qubits): in quantum dots[@ld98], nuclear spin[@k98; @pvk98], superconducting islands[@npt99] and persistent currents[@fpctl2000], to cite just a small sample.
Kane has proposed storing a qubit in the spin of a single phosphorous nucleus implanted in silicon. In his original readout scheme, this was coupled by the hyperfine interaction to the spin of the donor electron bound weakly to the nucleus. A surface gate would then draw the electron towards an adjacent ancilla donor, to which it might tunnel, producing a doubly charged $D^-$ state. Under appropriate bias conditions, this transfer can only occur if the nuclear spin of the qubit is oriented opposite the ancilla.
A spin measurement is thus reduced to detecting the transfer of a single electron charge to the ancilla. This can be done by a sensitive electroscope such as a single electron transistor[@kmdcmsw2000]. However, the techniques used for fabricating microelectronics have recently been adapted to build mechanical structures at micron and even nanometre scales[@r2000], and mechanical electroscopes sensitive to small numbers of electrons have been constructed[@cr98]. We will consider how effectively such devices might perform the measurements required for quantum information processing.
Classical treatments of measurement sensitivity assume that the observable being measured has a definite value, which influences the measuring instrument in a definite way. The only question is how much data we must gather to reliably distinguish this effect from other influences on the apparatus, which produce noise. Once we know the size of the effect we wish to distinguish, and the level of noise in the system, some elementary statistics tell us the integration time required for a reliable measurement.
This assumption does not hold when we measure an observable of a quantum system. If the system is in a superposition state, the observable will not have a definite value until some sort of measurement is carried out. Any interesting quantum information device will produce such superpositions. The process by which the superposition is reduced so that the observable has a certain value imposes a minimum level of noise in the measurement, which might be increased by the same sources of technical noise that affect measurements of classical systems.
In the proposed readout scheme for the Kane computer, a donor electron is induced to tunnel between two phosphorous nuclei, depending on the state of the nuclear spins. In general, the nuclei are in a superposition of a state which would permit tunneling, and one which would prevent it. After this tunneling has occured, the electon is left in a superpostion of two position states, each localised on one nucleus. It then interacts with the electroscope, and in general with other degrees of freedom in the crystal lattice, with the result that we see it become localised on one nucleus or the other, so that the electroscope gives a definite signal that the charge is present or absent.
Note that we are not discussing an ensemble of quantum systems subject to a single measurement, but rather a single quantum system subject to a dynamical measurement process. In such a situation we need to be able to describe the instantaneous conditional state of the measured system as the measurement results accumulate. This is quite different from the usual situation that prevails in condensed matter systems, where typically a measurement is made on a large number of (almost) identical constituents undergoing quantum dynamics, and the measurement results are already an average over an ensemble. Fortunately mathematical techniques (known as quantum trajectory methods) are available to describe the conditional dynamics of a single quantum system subject to measurement with added noise, and these methods have been applied with considerable success to experiments in quantum optics and ion traps[@pk98]. Recently such methods have been applied to mesoscopic electronic systems[@wusmkdc2001; @gmws2001; @k99a].
The Mechanical Electroscope {#sec:electroscope}
===========================
The operation of a micro-mechanical electroscope is shown schematically in [Figure \[fig:diagram\]]{}. The active part is an electrode, mounted on a cantilever no longer than $1\,\mu\text{m}$, which is set in motion near the charge to be measured. The electrode is held at constant potential, so that its motion with respect to the unknown charge induces a flow of charge between it and its voltage source. The induced charge gives the electroscope electric potential energy as well as elastic, and changes its resonant frequency. If we envision the electroscope being used to readout a qubit in a quantum computer, there will be two charge states we wish to distinguish. We will denote the difference between the resonant frequencies of the cantilever in these two states by ${\delta\omega}$; it is determined by geometry and the mutual capacitance between the electrode and the measured charge distribution.
We will assume the mechanical motion of the cantilever is elastic and treat it as a simple harmonic oscillator. Then its motion, including the capacitive coupling to the target charge, is described by a harmonic oscillator Hamiltonian $$H = \hbar ( \omega_0 + {\delta\omega}\, n_1 ) {{c}^\dagger}c$$ where $\omega_0$ is the resonant frequency of the cantilever in the absence of surface charge, and $c$ the annihilation operator for its oscillation. The observable $n_1$ will be defined shortly.
During readout of a Kane computer, a single donor electron may occupy a bound state around either of two adjacent nuclei. We will denote these distinct spatial states by ${|{\psi}\rangle}$ and ${|{\phi}\rangle}$. Only one state (suppose ${|{\psi}\rangle}$) couples to the electroscope - this is how we can distinguish them.
During readout, the surface gates will be configured to produce tunneling between the two nuclei, depending on the state of the nuclear spin qubits. This entangles the charge states with the qubit states ${|{\uparrow}\rangle}$ and ${|{\downarrow}\rangle}$. We will denote the combined states by $ {|{{\text{\sf 0}}}\rangle} = {|{\uparrow}\rangle}\otimes{|{\phi}\rangle} $, and $ {|{{\text{\sf 1}}}\rangle} = {|{\downarrow}\rangle}\otimes{|{\psi}\rangle} $, according to the number of electrons interacting with the electroscope, which we will represent by the operator $ n_1 = { {|{{\text{\sf 1}}}\rangle}{\langle{{\text{\sf 1}}}|} } $. In general the measured qubit will be in a superposition state, so the total state will take the form $${|{\Psi}\rangle} = a{|{{\text{\sf 0}}}\rangle} + b{|{{\text{\sf 1}}}\rangle}$$
Table \[tab:data\] gives numerical parameters for a cantilever electroscope fabricated in 1998. The frequency and operating temperature of this electroscope meant that themal noise completely dominated any quantum effects. Besides lowering the temperature, this could be changed by using a cantilever with a higher resonant frequency, and such devices have been fabricated. However, the sensitivity of the electroscope depends on the frequency changing significantly when change is present, and this might not be the case in higher frequency cantilevers.
We note that the interaction Hamiltonian commutes with the number operator $\hat{n}_1$. Furthermore, in the absence of tunnelling, the free Hamiltonian for the charge state itself is proportional to the square of the charge (capacitive electrostatic energy) and itself commutes with the charge number operator. In the presence of the measurement the number operator is thus a constant of motion. Such a measurement is known as a quantum nondemolition measurement[@wam]. Number eigenstates are not changed by the coupling to the apparatus, and moments of the number operator are constant in time. On the other hand any state that is initially a coherent superposition in the number basis will be reduced to a mixture diagonal in this basis, a process known as decoherence. In an ideal quantum nondemolition measurement, the probability distribution for observed results at the conclusion of the measurement should accurately reflect the intrinsic probability distributions of the quantum nondemolition variable in the quantum state at the start of the measurement.
This model, where the electroscope performs a QND measurement of the coupled charge, is idealised. If such an electroscope was used to measure any interesting device, the motion of the cantilever would disrupt the distribution of charge being measured. The nature and extent of this disruption would depend on the electrical properties of the system being measured; for the Kane computer, determining these is an unsolved problem in atomic physics. In general, back action (and interference from sources unrelated to the measurement) imposes a time limit on the measurement, after which the charge state will have been disrupted and the results will be meaningless. The results of this paper determine whether the electroscope can measure the charge with the necessary precision within that time.
To detect the change in resonant frequency, we must set the cantilever in motion with some driving mechanism. In the device described in Table \[tab:data\], this was supplied by driving an alternating current through a wire on the cantilever in the presence of a magnetic field. The current induced by the field in another wire was used to monitor the response of the cantilever to the driving.
However, the details of the driving are not important. As long as the cantilever is coupled weakly to the driving system and is not damped so strongly that its state changes significantly over the period of its vibration (in other words, it has high finesse), the effect can be described by a Hamiltonian. In the interaction picture this takes the form $ \hbar\mathcal{E}({{\hat{c}}}+{{{{\hat{c}}}}^\dagger}) $, where $\mathcal{E}$ is the strength of the driving in units of frequency. If the finess of the cantilever is low, noise from the driving system affects its motion significantly, and the dynamics due to the driving can not be approximated by a Hamiltonian.
The frequency shift could be detected in a number of ways. We could sweep the driving frequency and monitor the amplitude of the oscillations. Or else we could drive the oscillator at a constant frequency $\omega$, and then detect the change in phase of the oscillation due to the shift in resonance frequency when a small charge is coupled; this is the method analysed in this paper. We will assume that if the charge state is ${|{{\text{\sf 0}}}\rangle}$, the cantilever will be driven on resonance; if it is ${|{{\text{\sf 1}}}\rangle}$, the change $\delta\omega$ in its resonant frequency will cause its phase to differ from that of the driving force. The rate of change of the phase of the output current with frequency of the driving is greatest when the cantilever is driven near its resonant frequency.
We will measure time by the inverse damping rate $\gamma^{-1}$. Then, defining a dimensionless driving strength $ E = \mathcal{E}/\gamma $ and a detuning $ \Delta = {\delta\omega}/\gamma $, the Hamiltonian for the coherent driving, in the interaction picture, is $$\label{eq:hamiltonian}
{{\hat{H}}}_D = \hbar E \left( {{\hat{c}}}+ {{{{\hat{c}}}}^\dagger} \right)
+ \hbar \Delta n_1\,{{{{\hat{c}}}}^\dagger}{{\hat{c}}}$$
In reality of course the mechanical oscillations of the cantilever will be subject to frictional damping, and accompanying mechanical noise. The rate of energy dissipation is specified by the quality factor, $Q$, which is the ratio of the resonance frequency to the width of the resonance. For linear response, this gives $Q=\omega_0/\gamma_M$, where $\gamma_M$ is the decay rate of energy due to mechanical dissipation. Roukes et al.[@cr99] have measured quality factors up to $2 \times 10^4$. With such quality factors and resonance frequencies approaching GHz, these devices are approaching low quality optical resonators. So we will treat the effect of mechanical damping with the master equation methods of quantum optics. These methods assume that the coupling of the resonator to the dissipative degrees of freedom is sufficiently weak[@hr85; @hrsw86]. Specifically we assume that $ \gamma_M \ll \omega_0,\ kT/\hbar $.
Under these assumptions the coupling between the oscillator and the thermal mechanical reservoir is[@gardiner] $$H_M= \sqrt{\gamma_M}(c\, a^\dagger(t)+c^\dagger\, a(t))
\label{eq:thermal-noise}$$ where $a(t),\ a^\dagger(t)$ are bosonic reservoir operators. The state of the reservoir will be taken to be that of a Planck thermal equilibrium density operator with temperature $T_M$.
We now consider in more detail the mechanism by which the small changes in resonance frequency induced by the proximity of a target charge are transduced. This may be done[@cr98] by fabricating a wire loop on the mechanical oscillator and placing the whole apparatus in a strong magnetic field. As the mechanical oscillator moves, an induced EMF is set up in the loop and we may measure the induced current. When the current for the driving circuit is such as to drive the mechanical oscillator at its resonance frequency, the induction current is out of phase with the driving current. However when a small target charge shifts the resonance frequency of the oscillator, the induced current shifts in phase with respect to the driving current. We can detect this phase shift by an electrical comparison of the driving current and induction current. This is essentially homodyne detection in which the driving current plays the role of a local oscillator. Unfortunately this electrical transduction of the mechanical motion introduces another source of noise for the measurement.
The induction current is coupled into an external amplifier circuit which can be treated as a bosonic reservoir, with some non zero noise temperature[@louisell], $T_E$. The readout circuit variable coupled to the cantilever is the current operator $i(t)$ in the readout circuit. We will assume that the coupling is linear in the current and coordinate degree of freedom of the cantilever. Under standard assumptions the interaction between the mechanical oscillator and the readout circuit is described by the interaction picture Hamiltonian, $$H_R = i \sqrt{\gamma_E}\,
\left( c^\dagger \Gamma(t) - c \Gamma^\dagger(t) \right)
\label{eq:elec-noise}$$ where $\Gamma(t)=b(t)e^{i\omega_0 t}$ with the actual current in the circuit given by $ i(t) = \sqrt{\hbar\omega_0 / 2Lz_0}
( b(t) + {{b}^\dagger}(t) ) $ , $L$ being the inductance per unit length of the transmission line, and $z_0$ the quantisation length. We will assume that the readout circuit reservoir is bosonic and also in thermal equilibrium at some temperature $T_E$.
Using the interaction Hamiltonians for the reservoir coupling (Equations \[eq:thermal-noise\] and \[eq:elec-noise\]), we may obtain the Heisenberg equations of motion for the oscillator and reservoir variables. Using standard techniques[@gardiner], the reservoir variables may be eliminated to give a quantum Langevin stochastic differential equation describing the dynamics of the oscillator amplitude $$\frac{da}{dt} =
-i {\delta\omega}\, a
-i\mathcal{E} - \frac{\gamma_M}{2}\,a - \frac{\gamma_E}{2}\,a
+ \sqrt{\gamma_M}\,a_{\text{in}}(t)
+\sqrt{\gamma_E}\,b_{\text{in}}(t)
\label{eq:qsde}$$ where $a_{\text{in}}(t)$, $b_{\text{in}}(t)$ are the quantum noise sources for the mechanical and electrical reservoirs respectively. These noise terms are defined by correlation functions, which are Fourier transforms of $$\begin{aligned}
{\left\langle{a_{\text{in}}(t)}\right\rangle} & = & \langle b_{\text{in}}(t)\rangle = 0 \\
{\left\langle{a^\dagger_{\text{in}}(\omega) a_{\text{in}}(\omega)^\prime}\right\rangle}
& = & \bar{n}(\omega,T_M) \, \delta(\omega - \omega^\prime) \\
{\left\langle{a_{\text{in}}(\omega) a^\dagger_{\text{in}}(\omega)^\prime}\right\rangle}
& = & ( \bar{n}(\omega,T_M) + 1 ) \, \delta(\omega - \omega^\prime) \\
{\left\langle{b^\dagger_{\text{in}}(\omega) b_{\text{in}}(\omega)^\prime}\right\rangle}
& = & \bar{n}(\omega,T_E) \, \delta(\omega - \omega^\prime) \\
{\left\langle{b_{\text{in}}(\omega) b^\dagger_{\text{in}}(\omega)^\prime}\right\rangle}
& = & ( \bar{n}(\omega,T_E) + 1 ) \, \delta(\omega - \omega^\prime)\end{aligned}$$ where $$\bar{n}(\omega,T) = \frac12
\left( \coth \left( \hbar\omega / 2 k_B T \right) - 1 \right)$$ Note the equation explicitly includes a friction term (proportional to $\gamma_E$) that arises form the electrical coupling to the readout circuit. The steady state average amplitude $\alpha_n=\langle
a(t)\rangle_{t\rightarrow \infty}$, is given by $$\label{eq:alpha}
\alpha_n = \frac{-2i \mathcal{E}} {(\gamma_M+\gamma_E)+2i\,\delta\omega\,n_1}
\label{eq:ss-amp}$$
The actual measured quantity is the current in the readout circuit, that is to say the readout variable is an electrical bath variable, $b_{\text{out}}$ at the output from the system interaction. The output amplitudes for both the mechanical and electrical baths are related to the input variables for these two baths and the amplitude of the mechanical oscillator by[@wam] $$\begin{aligned}
a_{\text{out}}(t) & = & \sqrt{\gamma_M}\,a(t) - a_{\text{in}}(t)\\
b_{\text{out}}(t) & = & - i \sqrt{\gamma_E}\,a(t) - b_{\text{in}}(t)
\label{eq:input-output}\end{aligned}$$ The average value of the electrical readout amplitude in the steady state is then found using equations [Equation \[eq:qsde\]]{} and [Equation \[eq:input-output\]]{}. $$\langle b_{\text{out}}\rangle = \sqrt{\gamma_E}\,\alpha_n
\label{eq:output-amp}$$ where $\alpha_n$ is given in [Equation \[eq:ss-amp\]]{}. We see that the steady state amplitude of the cantilever, and hence the output electrical signal undergoes a change in phase and amplitude, see [Figure \[fig:amplitudes\]]{}. If we monitor the component in the imaginary direction (that is, in quadrature with the driving signal, $E$) we will have maximum sensitivity to this change in phase. Furthermore it is desirable to have $E$ as large as possible so that small changes in phase translate into large changes in the quadrature.
We can now proceed to calculating the noise power spectrum for the measured current. The calculation is analogous to that for a double-sided cavity given in reference[@wam]. We now do not work in the rotating frame but return to the laboratory frame. The Fourier component of the output operator for the current is given by $$b_{\text{out}}(\omega) =
\frac{\left [\left (\frac{\gamma_E-\gamma_M}{2}\right )-i(\omega_0-
\omega)-i \,\delta\omega\, n_1 \right]b_{\text{in}}(\omega)
-i\sqrt{\gamma_E}{\cal
E}(\omega)+\sqrt{\gamma_E\gamma_M}a_{\text{in}}(\omega)}{\left [\left
(\frac{\gamma_E+\gamma_M}{2}\right )+i(\omega_0-\omega)+i \,\delta\omega\, n_1\right]}$$ where ${\cal E}(\omega)$ is the Fourier component of the driving amplitude. If the driving is noiseless and monochromatic, ${\cal
E}(\omega)=\mathcal{E}\delta(\omega-\omega_d)$. However in reality there would be some noise in the driving amplitude derived from the electrical noise in the driving circuit. We will treat this as entirely classical.
Equations \[eq:ss-amp\] and \[eq:output-amp\] suggest that the signal will appear in the quadrature of the current out of phase with the driving force, defined by $$\label{eq:quadrature}
X_{2,out}(t) = i ( b^\dagger_{\text{out}}(t) - b_{\text{out}}(t) )$$ with Fourier components $X_{2,out}(\omega)$. The measured power spectrum is then given by the correlation function, $$S_{2,out}(\omega,\omega^\prime)=\langle
X_{2,out}(\omega),X_{2,out}(\omega^\prime)\rangle$$ Using the specified states for the electronic and mechanical noise operators, we find, $$S_{2,out}(\omega,\omega^\prime)=\left [|{\cal
B}(\omega)|^2(2 \bar{n}(\omega,T_E)+1)+|{\cal A}(\omega)|^2(2\bar{n}(\omega,T_M)+1)\right
]\delta(\omega-\omega^\prime)$$ where $$\begin{aligned}
{\cal B}(\omega) & = & \frac{\frac{\gamma_E-\gamma_M}{2}-i\left ((\omega-
\omega_0)+ \,\delta\omega\, n_1\right )}
{\frac{\gamma_E+\gamma_M}{2}+i\left ((\omega-\omega_0)+
\,\delta\omega\, n_1\right)}\\
{\cal A}(\omega) & = &
\frac{\sqrt{\gamma_E\gamma_M}}{\frac{\gamma_E+\gamma_M}{2}+i\left((\omega-\omega_0)+
\,\delta\omega\, n_1\right )}\end{aligned}$$ To estimate the signal to noise ratio (SNR) we evaluate the spectrum at the driving frequency (that is to say, at the central Fourier component of the coherent driving); $$S(\omega_0) = \frac{ \left[
\left(\frac{\gamma_E-\gamma_M}{2} \right)^2 + (\delta\omega\, n_1)^2
\right] (2 \bar{n}(\omega,T_E)+1)
+ \gamma_E\gamma_M(2 \bar{n}(\omega,T_M)+1)}
{\left (\frac{\gamma_M+\gamma_E}{2}\right)^2 + (\delta\omega\, n_1)^2}$$ Equations \[eq:alpha\], \[eq:output-amp\] and \[eq:quadrature\] show that the magnitude of the Fourier component of the mean signal at the driving frequency is given by $$|\langle X_{2,out}(\omega_D)\rangle| =
\frac{8 \sqrt{\gamma_E} \mathcal{E} \,\delta\omega\, n_1} {(\gamma_M + \gamma_E)^2 + 4 \,\delta\omega^2\, n_1}$$ The signal is a sharp peak at $ \omega = \omega_d = \omega_0 $, in which there is a noise power $ S(\omega_0) $ per root Hertz. So the SNR per root Hertz is $ |\langle X_{2,out}(\omega_D)\rangle|^2 /
S(\omega_0) $, or $$\text{SNR} = \frac
{16 \,\gamma_E\, \mathcal{E}^2 \,\delta\omega^2\, n_1}
{ \left[ (\gamma_M + \gamma_E)^2 + 4 \,\delta\omega^2\, n_1 \right]
\left[
( \gamma_E-\gamma_M )^2 + 4 \,\delta\omega^2\, n_1
\right] (2 \bar{n}(\omega,T_E)+1)
+ \gamma_E\gamma_M(2 \bar{n}(\omega,T_M)+1)}$$ If the SNR required for the measurement is $\text{SNR}_r$, then we must average over noise for a time $t$ such that $ \text{SNR}_r =
\text{SNR}/\sqrt{t} $. If we set $ n_1 = 1 $, so we are measuring the charge on one electron, the sensitivity is then $ e \sqrt{t} = e \,
\text{SNR}_r / \text{SNR} $.
Unconditional description of the measurement.
=============================================
When we measure a quantum system, we bring an extremely large set of independent observables of our instrument and its environment into correlation with the measured system observable. The environment of the electroscope has two distinct components. Firstly there is the environment associated with the mechanical oscillator, which is responsible for mechanical damping and noise. Secondly there is the environment associated with the electrical readout, which is responsible for Johnson-Nyquist noise in the electrical circuit, and ultimately provides the measured result. However, we are interested in what the measurement tells us about the system, not in the exact quantum state of the instrument and its environment. Useful instruments must operate independently of the detailed state of their environments.
There are two ways to describe the partial state of the charge and oscillator. Firstly we can ignore the results of the measurement and average over states of the environment completely. In this case the evolution of the charge and oscillator is described by a master equation. Effectively we are averaging over the ensemble of partial states distinguished by different measurement records
Secondly, we can ask for the conditional states of the charge and oscillator, given a particular measurement record. Each member of the ensemble of partial states is associated with a distinct measurement record of the instrument. For it to be an effective measurement, observers must be able to distinguish the states of the instrument. In other words the charge must end up correlated with some simple macroscopic quantity, like the current in a wire or the position of a pointer on a scale. It is then possible to ask for the particular partial state of the measured system that is correlated with a known pointer value. In other words we need to be able to specify the conditional state of the system given a readout of the instrument variable that distinguishes different charge states. This is the conditional, or selective, description of the measured system. Of course if we average over the readout variables, we must obtain the unconditional description of the system.
We begin with the unconditional description of the measurement. The dominant sources of excess noise that limit the quality of the measurement are the thermal mechanical noise and thermal electrical noise on the readout circuit. Under certain Markoff and rotating wave assumptions[@carmichael1; @gardiner], the explicit states of the mechanical and electrical reservoirs may be traced out. This leaves the following master equation for the density operator of the composite system of charge and cantilever, $$\begin{aligned}
\dot\rho(t) & = &
- i [ \mu (e n_1)^2 \, {{{{\hat{c}}}}^\dagger}{{\hat{c}}}, \rho]
- i [ \mathcal{E} \left( {{\hat{c}}}+ {{{{\hat{c}}}}^\dagger} \right) , \rho] \\
& & + \sum_{i = M,E} \gamma_i({\bar{n}}_i + 1) {\cal D}[c] \rho
+ \gamma_i{\bar{n}}_i {\cal D}[c^\dagger]\rho\end{aligned}$$ where the superoperator $\cal D$ is defined by $${\cal D}[c]\rho =
c\rho{{c}^\dagger} - \frac{1}{2} ( {{c}^\dagger}c\rho + \rho{{c}^\dagger}c )$$
This can be written in a more standard form $$\begin{aligned}
\label{eq:master}
\dot\rho & = &
- i [ \mu (e n_1)^2 \, {{{{\hat{c}}}}^\dagger}{{\hat{c}}}, \rho]
- i [ \mathcal{E} \left( {{\hat{c}}}+ {{{{\hat{c}}}}^\dagger} \right) , \rho] \\
& & + \gamma ({\bar{n}}+ 1) {\cal D}[c] \rho
+ \gamma {\bar{n}}{\cal D}[{{c}^\dagger}] \rho\end{aligned}$$ where $ \gamma \equiv \gamma_M + \gamma_E $, and $ {\bar{n}}\equiv ( \gamma_M \bar{n}(\omega,T_M) + \gamma_E \bar{n}(\omega,T_E) )/\gamma $.
We will begin solving this master equation by separating the dynamics of the cantilever and the charge. As before, we assume there is only one charge in the system, and consider the charge states ${|{{\text{\sf 0}}}\rangle}$ and ${|{{\text{\sf 1}}}\rangle}$. We can decompose $\rho$ into a $2 \times 2$ matrix of cantilever operators $$\rho = {{\hat{A}}}{{ {|{{\text{\sf 0}}}\rangle}{\langle{{\text{\sf 0}}}|} }} + {{\hat{B}}}{{ {|{{\text{\sf 1}}}\rangle}{\langle{{\text{\sf 1}}}|} }}
+ {{\hat{Z}}}{ {|{{\text{\sf 0}}}\rangle}{\langle{{\text{\sf 1}}}|} } + {{{{\hat{Z}}}}^\dagger}{ {|{{\text{\sf 1}}}\rangle}{\langle{{\text{\sf 0}}}|} }
\label{eq:matrix}$$ Since $\rho$ is Hermititan, we need only three cantilever operators, ${{\hat{A}}}$, ${{\hat{B}}}$ and ${{\hat{Z}}}$. We can now decompose [Equation \[eq:master\]]{} into three independent equations involving only cantilever operators: $$\label{eq:no}
{\frac{d{{\hat{A}}}}{dt}} = -i [ E({{\hat{c}}}+{{{{\hat{c}}}}^\dagger}) , {{\hat{A}}}] + { ({\bar{n}}+ 1) {\cal D}[c] {{{\hat{A}}}}
+ {\bar{n}}{\cal D}[{{c}^\dagger}] {{{\hat{A}}}}}$$ $$\label{eq:yes}
{\frac{d{{\hat{B}}}}{dt}} =
-i [ E({{\hat{c}}}+{{{{\hat{c}}}}^\dagger}) + \Delta {{{{\hat{c}}}}^\dagger}{{\hat{c}}}, {{\hat{B}}}] + { ({\bar{n}}+ 1) {\cal D}[c] {{{\hat{B}}}}
+ {\bar{n}}{\cal D}[{{c}^\dagger}] {{{\hat{B}}}}}$$ $$\label{eq:maybe}
{\frac{d{{\hat{Z}}}}{dt}} =
- i [ E({{\hat{c}}}+{{{{\hat{c}}}}^\dagger}) , {{\hat{Z}}}]
+ i \Delta {{\hat{Z}}}{{{{\hat{c}}}}^\dagger}{{\hat{c}}}+ { ({\bar{n}}+ 1) {\cal D}[c] {{{\hat{Z}}}}
+ {\bar{n}}{\cal D}[{{c}^\dagger}] {{{\hat{Z}}}}}$$ As before, we are now measuring time relative to the damping time $1/\gamma$.
If we measured the state of the charge by means other than the cantilever, the state of the cantilever immediately after the measurement would be ${{\hat{B}}}$ if the charge were present, or ${{\hat{A}}}$ if it were absent. Hence ${{\hat{A}}}$ and ${{\hat{B}}}$ must be density operators, and Equations \[eq:no\] and \[eq:yes\] have the form of master equations for a damped harmonic oscillator. Such equations, and their solutions, are familiar to quantum opticians. The stable solution is a displaced thermal state, which can be written $$\rho = \left( 1 - e^{-\lambda(t)} \right) \, D(\alpha(t)) \,
e^{-\lambda(t){{{{\hat{c}}}}^\dagger}{{\hat{c}}}} \, {{D}^\dagger}(\alpha(t))$$ , where $D(\alpha)$ is a displacement operator $ \exp(\alpha{{{{\hat{c}}}}^\dagger} -
{{\alpha}^\ast}{{\hat{c}}}) $, and in the steady state $ \lambda =
\hbar\omega_0/k_b T $. In the limit of low temperature, $ kT \ll
\hbar\omega_0 $, this becomes a coherent state $ {{ {|{\alpha}\rangle}{\langle{\alpha}|} }} $. In the steady state, the cantilever has as many thermal phonons as a resevoir mode with the same frequency, i.e. $ e^{-\lambda} =
{\bar{n}}/({\bar{n}}+1) $. Its coherent amplitude $\alpha_0$ reaches a balance with the driving and damping after a time around $2/\gamma$: $$\label{eq:amplitudes}
\alpha(t) = \alpha_0 e^{-\kappa t/2}
-\frac{2iE}\kappa (1 - e^{-\kappa t/2} )$$ $$\label{eq:cases}
\kappa = \left\{
\begin{array}{cc}
1 & n = 0 \\
1+2i\Delta & n = 1
\end{array} \right.$$ During measurement, the cantilever states ${{\hat{A}}}$ and ${{\hat{B}}}$ are displaced thermal states with distinct coherent amplitudes.
As the measurement proceeds, we expect the charge state to evolve from a coherent superpostion of ${|{{\text{\sf 0}}}\rangle}$ and ${|{{\text{\sf 1}}}\rangle}$ to an incoherent mixture; in terms of our decomposition, we expect the off-diagonal term $Z$ to decay with time. An operator of the form $$\label{eq:Z}
Z = {z}(t) D(\alpha) \exp(-\lambda{{{{\hat{c}}}}^\dagger}{{\hat{c}}}) {{D}^\dagger}(\beta)$$ where ${z}(t)$ is a (possibly complex) amplitude, solves [Equation \[eq:maybe\]]{} if $\alpha$, $\beta$, $\lambda$ and $z$ obey the following differential equations: $$\label{eq:l}
{\frac{dl}{dt}} = ({\bar{n}}+1) l^2 - (2{\bar{n}}+ 1 + i\Delta) l + {\bar{n}}$$ $$\label{eq:a}
{\frac{da}{dt}} =
\left( -i\Delta + ({\bar{n}}+1)l - {\bar{n}}- {{\textstyle\frac12}}\right) a
- iE(1-l)$$ $$\label{eq:b}
{\frac{db}{dt}} =
\left( ({\bar{n}}+1)l - {\bar{n}}- {{\textstyle\frac12}}\right) b
+ iE(1-l)$$ $$\label{eq:k}
{\frac{dk}{dt}} = -iE(a - b) + ({\bar{n}}+1)(l + ab - 1) + 1$$ Here $ l = \exp(-\lambda) $, $ a = \alpha - l\beta
$, $ b = {{\beta}^\ast} - l{{\alpha}^\ast} $, and $ k = \log
z + l{{\alpha}^\ast}\beta - {{\textstyle\frac12}}({|\alpha|}^2 + {|\beta|}^2) $
In general, these equations can be solved numerically. However, there are some special cases where we can get interesting information analytically. First we consider the zero temperature limit, where the off diagonal term ${{\hat{Z}}}$ is a projector $ z { {|{\alpha}\rangle}{\langle{\beta}|} } $. The amplitudes $\alpha$ and $\beta$ are the amplitudes of the diagonal terms given by [Equation \[eq:cases\]]{}, and $z$ is a complex amplitude. Once $\alpha$ and $\beta$ have reached their steady state, the trace of the off-diagonal term, decays exponentially with a rate $
{|\alpha-\beta|}^2/2 $.
If we assume the detuning $\Delta$ is small, and hence $
{|\beta|} \approx {|\alpha|} = 2E $, The difference between the steady state amplitudes of [Equation \[eq:cases\]]{} is $${|\alpha - \beta|}^2
= \frac{16 E^2 \Delta^2}{1+4\Delta^2}
\approx 4 {|\alpha|}^2 \Delta^2$$ Cleland and Roukes give enough information about their devices for us to calculate this explicitly [@cr98]. Using the data in Table \[tab:data\], we can calculate $\alpha$ from the definition of the annihilation operator for a torsional pendulum $$\alpha = {\left\langle{{{\hat{c}}}}\right\rangle}
= \sqrt{\frac{\kappa}{2\hbar\omega_0}} {\left\langle{\theta_{\text{max}}}\right\rangle}
= 5.3 \times 10^6$$
The normalised detuning can be calculated from the frequency shift per electron and the measured quality factor: $$\Delta = \delta\omega /\gamma = \frac{2 \pi \, \delta\nu \, Q}{\omega_0}
= 2.4 \times 10^{-4}$$ The decoherence rate is then $ 3.2 \times 10^{6} \, \gamma $, or $
8.1 \times 10^{9} \, \text{s}^{-1} $.
As ${\bar{n}}$ increases from zero, the amplitudes $\alpha$ and $\beta$ for the off diagonal operator ${{\hat{Z}}}$ are reduced, as shown in [Figure \[fig:amplitudes\]]{}. The initial decay of $z(t)$ is shown in [Figure \[fig:coherencetime\]]{}, and the steady decay rate, i.e. the limit of $ |
z^\prime(t)/z(t) | $ when $ t \gg 1/\gamma $, in [Figure \[fig:graph\]]{}. At low temperatures (below $130\,\text{mK}$), the increased thermal noise from the bath causes ${{\hat{Z}}}$ to decay more rapidly as the temperature of the bath is increased. Contrary to expectations, the steady decoherence rate of the charge superposition decreases as the bath temperature increases above $130\,\text{mK}$. The extra thermal noise increases the overlap between the oscillator states corresponding to the presence and absence of charge.
Conditional description {#sec:trajectories}
=======================
We now turn to the correlations between the charge and the resevoir system. These are important because we must be able to distinguish the results corresponding to different charges to make a measurement of the charge at all. They can be studied most simply using quantum trajectory theory, which associates charge states with possible observed states of the apparatus [@carmichael-lect].
We will assume we monitor the current in the electrical resevoir; this is equivalent to an optical homodyne measurement [@sa96]. The inferred state of the charge as such a measurement proceeds is governed by a Wiener process, which is generated by a stochastic increment $dW$. The average of $dW$ over the ensemble of possible measurement results is zero. Since the deviation of the Wiener process represented by $dW$ increases proportional to $\sqrt t$, the average of $(dW)^2$ is $dt$. The simplest way to manipulate such differentials is to modify the chain rule, to give what is know as Ito calculus.
Given a particular measurement result, labelled by a Wiener increment $dW$, the evolution of the charge and cantilever is $$d{|{\psi}\rangle} = \left( \frac1{i\hbar} {{\hat{H}}}dt
- \frac\gamma2
\left( {{c}^\dagger}c - 2 {\left\langle{\frac{x}2}\right\rangle}c + {\left\langle{\frac{x}2}\right\rangle}^2 \right)dt
+ \sqrt\gamma \left( c - {\left\langle{\frac{x}2}\right\rangle} \right)dW \right) {|{\psi}\rangle}$$ When we insert the charge and cantilever Hamiltonian, and normalise time by the damping rate as before, this becomes $$\label{eq:trajectory}
d{|{\psi}\rangle} = \left( - i( E(c+{{c}^\dagger}) + \Delta n{{c}^\dagger}c )dt
- \frac12\left( {{c}^\dagger}c - 2 {\left\langle{\frac{x}2}\right\rangle}c + {\left\langle{\frac{x}2}\right\rangle}^2 \right)dt
+ \left( c - {\left\langle{\frac{x}2}\right\rangle} \right)dW \right) {|{\psi}\rangle}$$ When a particular function $dW$ is selected from the Wiener ensemble, this can be solved to show the evolution of a pure state ${|{\psi}\rangle}$. These states form an ensemble with density operator $\rho$. Of course $\rho$ can be decomposed into many ensembles, so the evolution generated by [Equation \[eq:trajectory\]]{} is not unique. The details are given in Carmichael[@carmichael-lect].
Mixed states of the cantilever and charge must be written in the form of [Equation \[eq:matrix\]]{}. However pure states can always be written as $${|{\psi}\rangle} = {|{A}\rangle} \otimes {|{{\text{\sf 0}}}\rangle} + {|{B}\rangle} \otimes {|{{\text{\sf 1}}}\rangle}$$ as before we will assume the state of the cantilever is initially coherent, so that $${|{\psi}\rangle} = p {|{\alpha 0}\rangle} + {q}{|{\beta 1}\rangle}$$ The differential of a scaled coherent state ${q}(t) {|{\beta(t)}\rangle}$ is $$d ({q}{|{\beta}\rangle})
= \left(d {q}- \frac12 {q}d{|\beta|}^2\right) {|{\beta}\rangle}
+ {q}\dot\beta \, dt \, {{c}^\dagger} {|{\beta}\rangle}$$ comparison with [Equation \[eq:trajectory\]]{} gives Equations \[eq:amplitudes\] for the evolution of $\alpha$ and $\beta$ as before. Some Ito calculus manipulations show that $$d {|{q}|}^2 =
{|{p}{q}|}^2 ({\left\langle{x}\right\rangle}_\alpha - {\left\langle{x}\right\rangle}_\beta) dW$$ where $ {\left\langle{x}\right\rangle}_\alpha $ is the expectation value of the amplitude quadrature $x$ in a coherent state ${|{\alpha}\rangle}$, which is just $ 2
\text{Re}\,\alpha $. The normalisation of ${|{\psi}\rangle}$ requires that $ d {|{p}|}^2 = - d{|{q}|}^2 $.
We need to compare the gain in knowledge shown by this trajectory picture to the decay of coherence modelled by the master equation. The results of the measurement are the probabilities ${|{p}|}^2$ and ${|{q}|}^2$; the pure state which the observer will infer from these has a density operator $$\label{eq:inferredrho}
\rho = {|{p}|}^2 { {|{0}\rangle}{\langle{0}|} } + {|{q}|}^2 { {|{1}\rangle}{\langle{1}|} }
+ {|{p}{q}|} ( { {|{1}\rangle}{\langle{0}|} } + { {|{0}\rangle}{\langle{1}|} } )$$ The off-diagonal terms in this have magnitude ${|{p}{q}|}$; we can average over $dW$ to see the behaviour of the density operator for the ensemble of measurement results.
Some more routine Ito calculus gives the evolution of this: $$d {|{p}{q}|} = - {|{p}{q}|}
\left( \frac18 \left( {\left\langle{x}\right\rangle}_\beta - {\left\langle{x}\right\rangle}_\alpha \right)^2 dt
+ \frac12 \left( {|{p}|}^2 - {|{q}|}^2 \right)
\left( {\left\langle{x}\right\rangle}_\beta - {\left\langle{x}\right\rangle}_\alpha \right) dW \right)$$ Since the average of $dW$ over different measurement results is zero, on average $$\label{eq:sensitivity}
d {|{p}{q}|} = \
- \frac18 ( {\left\langle{x}\right\rangle}_\beta - {\left\langle{x}\right\rangle}_\alpha )^2 {|{p}{q}|} dt$$ If the difference between the charges associated with states ${|{0}\rangle}$ and ${|{1}\rangle}$ is $e$, then in the state $ {p}{|{\alpha 0}\rangle} + {q}{|{\beta 1}\rangle} $, the uncertainty in the charge is given by $$\left({\left\langle{(ne)^2}\right\rangle} - {\left\langle{ne}\right\rangle}^2\right)^{{\textstyle\frac12}}= e {|{p}{q}|}$$ From [Equation \[eq:sensitivity\]]{}, this decreases exponentially as the measurement progresses, at a rate $$\frac18 ( {\left\langle{x}\right\rangle}_\beta - {\left\langle{x}\right\rangle}_\alpha )^2
= \frac{ 8 \Delta^2 {E}^2 }{ ( 1 + 4\Delta^2 )^2 }$$ This differs from the square root decay of classical uncertainty as measurements are averaged over time, but exponential decay is what we would expect for decay of coherence[@wam]. For the device described in [@cr98], this is almost equal to the decoherence rate. In real devices, thermal noise will cause the trajectory states to be mixed, however the evolution of such mixed states is much harder to calculate.
Discussion {#sec:discussion}
==========
To estimate the time required for our measurement, we have calculated how long it takes for an initially pure superposition of charge states to be reduced to a mixture, and how long (in some sense) it takes us to find out which charge eigenstate we have been left with. While these questions are interesting in their own right (they composed the deepest mystery of physics for the best part of a century), it could be argued that they don’t reflect the way measurements would be used in a real computer.
The most that we could do with measurements on pure states is state preparation. In a coherent quantum computer this would be rather pointless though, since if we know the initial state we could just rotate it into the eigenstate we want. We carry out measurements to find out something we don’t know: in other words we apply them to mixed states, with a view to finding out which of the possibilities is real.
Information theory provides tools to quantify this, such as conditional entropy and mutual information. Unfortunately calculating any of these requires knowledge of the ensemble of trajectories generated by each component of the mixture, and the overlaps between them. In general it is hard to find the probability distribution of trajectories; we usually just calculate averages. It might be worth doing this numerically, however.
There is a more straightforward limitation to our analysis: in present day devices the thermal effects that we have neglected in the trajectory treatment utterly dominate the vacuum noise we have considered. Hence the measurement time will be limited by the need to average classical fluctuations. It is possible that future devices operating at higher frequencies will reduce the level of thermal noise so that quantum effects will be important. This presents the remarkable prospect of a solid cantilever with position and momentum known to the limit allowed by the uncertainty principle.
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------------------------------------------ --------------------------------- --------------------------------------------------
Operating temperature $T$ $4.2\,\text{K}$ $k_{\text{B}} T = 3.6 \times 10^{-4}\,\text{eV}$
Resonant frequency $\omega_0/2\pi$ $2.6\,\text{MHz}$ $\hbar\omega_0 = 1.1 \times 10^{-8}\,\text{eV}$
Torsional spring constant $\kappa$ $1.1\times 10^{-10}\,\text{Nm}$
Amplitude $\theta_{\text{max}}$ $30\,\text{mrad}$
Frequency shift per electron $\delta\nu$ $0.1\,\text{Hz}$
Quality factor $\omega_0/\gamma$ $6.5\times 10^3$
------------------------------------------ --------------------------------- --------------------------------------------------
: Data for an electroscope fabricated by Cleland and Roukes [@cr98].[]{data-label="tab:data"}
![ Operation of the mechanical electroscope. A charge trapped near the surface of some material is coupled to a cantilever suspended above the surface, as explained in the text. The cantilever is driven at a rate ${E}$ and damped by a combination of mechanical friction and reaction from the electronic readout loop at a rate $\gamma$. If an excess charge is present on the surface, the frequency of the pendulum is increased by ${\delta}$. For simplicity, the figure shows a simple pendulum, but in practice the cantilever would be a torsional pendulum, oscillating due to strain in the material.[]{data-label="fig:diagram"}](bw-cant.eps)
![ The off-diagonal element $Z$ of the density operator is a thermal state displaced by amplitudes $\alpha$ and $\beta$, which depend on the temperature (see [Equation \[eq:Z\]]{}). When the cantilever is coupled to a hot bath, these coherent amplitudes decrease, and $Z$ approaches a purely thermal state. The values these amplitudes take in the Cleland and Roukes electroscope at temperatures from absolute zero up to $10\,\text{K}$ are plotted in the complex plane, in units of the ground state fluctuations. The amplitudes of the diagonal elements $A$ and $B$ do not vary with temperature, but remain at the $0\,\text{K}$ values.[]{data-label="fig:amplitudes"}](plot1.eps)
![ When the measurement has been running for a time around $1/\gamma$, and the cantilever amplitudes have reached their steady state, any remaining coherence between the two charge states decays exponentially. Here the rate of this decay is plotted as a function of temperature, for the device described in Table \[tab:data\]. The maximum decay rate of $-3.2 \times
10^9\,\gamma$ occurs at $130\,\text{mK}$. Beyond this point the decay rate decreases with temperature, possibly because the increased thermal noise makes the coherent amplitude of the cantilever harder to distinguish. []{data-label="fig:graph"}](plot2.eps)
![ The state of the cantilever takes some time to become entangled with the charge after they begin to interact, as the cantilever state moves towards its steady value. After this the charge state decoheres rapidly. Here the coherence between the two charge states is plotted as a function of time for an array of temperatures. The cantilever is initially in a thermal state at the appropriate temperature. Note the the charge state has decohered long before the cantilever reaches its steady amplitude, which occurs after a time $1/\gamma$.[]{data-label="fig:coherencetime"}](plot3.eps)
[^1]: Work supported by the SRC for Quantum Computing Technology
|
---
author:
- |
Rémi Bardenet\
Université de Lille, CNRS, Centrale Lille\
UMR 9189 - CRIStAL, Villeneuve d’Ascq, France\
`remi.bardenet@gmail.com`\
Subhroshekhar Ghosh\
National University of Singapore , Dept of Math\
10 Lower Kent Ridge Road, Singapore 119076\
`subhrowork@gmail.com`\
bibliography:
- 'Sampling.bib'
- 'zonotope.bib'
- 'guillaume-thesis.bib'
title: |
Learning from DPPs via Sampling:\
Beyond HKPV and symmetry[^1]
---
Introduction
============
Determinantal point processes and their sampling {#s:determinantal}
================================================
The Laplace transform and sampling {#s:laplace}
==================================
Our algorithm {#s:algorithm}
=============
Experiments {#s:experiments}
===========
Discussion {#s:discussion}
==========
Acknowledgments {#acknowledgments .unnumbered}
===============
RB acknowledges support from ERC grant <span style="font-variant:small-caps;">Blackjack</span> (ERC-2019-STG-851866). SG acknowledges support from MOE grant R-146-000-250-133.
[^1]: Authors listed in alphabetical order
|
---
abstract: 'The aim is to present the ability of a network of transitions as a nonlinear tool providing a graphical representation of a time series. This representation is used for cardiac RR-intervals in follow-up observation of changes in heart rhythm of patients recovering after heart transplant.'
author:
- 'Danuta Makowiec, Stanisław Kryszewski, Beata Graff, Joanna Wdowczyk-Szulc, Marta Żarczyńska- Buchnowiecka, Marcin Gruchała and Andrzej Rynkiewicz [^1][^2]'
title: 'Network representation of cardiac interbeat intervals for monitoring restitution of autonomic control for heart transplant patients. '
---
[Makowiec : Network representation of cardiac interbeat intervals for monitoring restitution of autonomic control for heart transplant patients. ]{}
heart transplant, heart rate variability, graphical representation
Introduction
============
The patient’s life is usually endangered before the decision to transplant the heart is taken. On the other hand, in many cases it is amazing how the patient’s organism recovers after heart transplant (HTX) [@Toledo2002].
It is generally believed that the time intervals between subsequent heart contractions (so-called RR-intervals) carry the information about the cardiac control system mainly driven by autonomic nervous system [@TASK]. However, heart transplantation interrupts the possibility of autonomic control over the heart beating. Therefore, heart rate variability (HRV) in patients after HTX is low, regardless of the time elapsed since surgery. It is controversial whether the cardiac reinnervation occurs after HTX [@Toledo2002]. But it is expected that progressive reinnervation sets in and it is a good prognosis for survival.
We believe that the changes that occur in heart rhythm may provide the first signals of the recovery of cardiac control. Furthermore, these signals should be connected with the increasing influence of the sympathetic nervous system. Therefore we hope that observing these changes, with the help of carefully selected tools, we can describe the process of cardiac reinnervation.
The considerable success of the network theory in various fields of research (see, e.g., [@Donner; @Costa]) motivated us to explore these ideas in the analysis of HRV time series. Tools of the complex networks allow one to resolve important and complementary properties of a dynamical system. For example, it is possible to study spatial dependencies between individual observations instead of temporal correlations. In the following we continue our earlier studies (see [@Makowiec2011]) on networks of transitions applied to study recordings of time intervals between subsequent cardiac contractions in patients after HTX.
We will also raise the question whether these transitions build a monotonic sequence of accelerations or decelerations. We are of opinion that sequences of monotonic accelerations or decelerations may indicate response of the cardiac system to some special needs of the organism. Hence, these studies may have a chance to offer additional insights into the emergence of the heart regulatory control.
It must be emphasized that all graphical representations of the discussed networks are produced with the [*Pajek*]{} software package [@pajek].
Methods
=======
The patient group
-----------------
We analyze 24-hour sequences of 23 ECG signals comprising of the intervals between two successive R waves of sinus rhythm. These signals were taken from 11 patients recovering after heart transplantation in the First Cardiology Clinic of Gdańsk Medical University. The recordings were taken from the same patients within different periods after surgery. Therefore, we have 3 recording from two patients, 2 recordings from 8 patients and 1 recording from one patient. We considered signals taken from two weeks to 38 months after HTX.
From each RR-signal we have carefully selected a sequence of 15,500 points corresponding to nocturnal rest of a patient. There are two reasons why we investigate the nocturnal heart rhythm. The first one is that during the sleep the central nervous system is less dependent on the patient’s intentions, and therefore we may have a more direct insight into reflexes regulating the cardiac rhythm. Moreover, the nocturnal recordings appear to be less perturbed by artifacts what enables us to study sufficiently long and consistent signals.
To avoid influence of artifacts (errors in detecting the R-wave) the consecutive RR-intervals were thoroughly reviewed. The parts, which consist of at least 500 normal-to-normal intervals, were identified. If two such parts were separated by artifacts or ectopic beats of the length smaller than 10, then the gap was edited manually. To preserve time chronology, the corresponding RR-intervals were interpolated by the value of median from the last normal-to-normal seven events. Additionally, the value of median was confronted with the total length of the edited gap.
Network representation of RR-signal
-----------------------------------
In general, the construction of the transition network [@Donner] is based on the concept of phase space. The phase space represents all possible values of studied dynamical system partitioned into mutually disjoint sets. Since the recorded values of RR-intervals have well-separated magnitudes then partitioning of the value space of RR-intervals is natural. Assuming these values as vertices of a network, we represent each pair of consecutive in time RR-intervals as a transition between these vertices.
![Construction of a network from a time series. [*Left*]{}: space values from a time series discretized at $\Delta =10msec$. [*Right*]{}: a time series as a path on a network of values of phase space.[]{data-label="Fig1"}](phase_space.JPG "fig:"){height="23em"} ![Construction of a network from a time series. [*Left*]{}: space values from a time series discretized at $\Delta =10msec$. [*Right*]{}: a time series as a path on a network of values of phase space.[]{data-label="Fig1"}](network.JPG "fig:"){height="19em"}
More precisely, the $N$ values of a recording with RR-intervals $ \{ RR_1, RR_2, \dots,RR_N \} $ are uniformly discretized (rounded) with accuracy $\Delta = 10$ ms. Then, in order to determine the phase space, a set of ordered distinct values $RR^{MIN}\!=\!RR^{(1)}\! <\! RR^{(2)}\! < \! \dots \! < \! RR^{(K)}
\! = \! RR^{MAX}$ is extracted. These $K$ different values label different vertices in the network. This is depicted in Fig. \[Fig1\] left where the first column gives the sequence of RR intervals (already discretized), the second column gives the ordered values from $RR^{MIN}$ to $RR^{MAX}$, thus constituting the phase space of the signal. The right panel shows the studied time series as a transition network. Here, the labels of vertices correspond to values from the phase space. Vertices are arranged from $RR^{MIN}$(top) to $RR^{MAX}$(bottom). An arrow between two vertices is plotted if the corresponding two RR-intervals represent a pair of the consecutive values in the time series.
It appears that neighbors in time are often also neighbors in values what results that the transition network takes the linear shape. But to improve the visualization of the transition properties we propose a ladder presentation of the network. Namely, vertices are placed alternatively in the left and right columns.
Moreover, to classify differences among the transitions we use the following coloring scheme:\
– no change in the value, i.e., $RR_i=RR_{i+1}$ : violet;\
– to a nearest neighbor, i.e., $|RR_i-RR_{i+1}|= \Delta $ : green;\
– to a next neighbor, i.e., $|RR_i-RR_{i+1}|= 2\Delta $ : blue;\
– to a second neighbor, i.e., $|RR_i-RR_{i+1}|= 3\Delta $ : red;\
– to other neighbors, i.e., $|RR_i-RR_{i+1}|> 3\Delta $ : black.\
![Typical networks obtained for a patient (here, called [*loj*]{} ) when the patient was 4 (left) and 38 (right) months after HTX. For better readability the transitions with probability less than 0.1% are omitted but all vertices which correspond to all recorded RR-intervals are plotted. The widths of arrows represent logarithms of counts of the particular transitions.[]{data-label="Fig2"}](004_loj.JPG "fig:"){height="17.2em"} ![Typical networks obtained for a patient (here, called [*loj*]{} ) when the patient was 4 (left) and 38 (right) months after HTX. For better readability the transitions with probability less than 0.1% are omitted but all vertices which correspond to all recorded RR-intervals are plotted. The widths of arrows represent logarithms of counts of the particular transitions.[]{data-label="Fig2"}](038_loj.JPG "fig:"){height="17.2em"}
Fig. \[Fig2\] shows two transition networks obtained for a patient called [*loj*]{}. The left network represents heart rhythm after 4 months elapsed after HTX, the rigth one 34 months later. The important message of this example is that when the time after HTX increases the number of transitions other than to nearest neighbors also increases (in the right panel there are more red and black arrows). We believe that it is a good symptom.
It should be explained that widths of the transition arrows in Fig. \[Fig2\] and all further network plots are determined by logarithms of the frequencies of particular transitions.
Study of the transitions between RR-intervals can be compared to investigations of RR increments – the popular measure of HRV. However, the adopted scheme additionally serves as a classification with respect to the size of increments. Therefore, due to the network representation we learn not only what kind of increments dominates in a signal but also when these events take place.
Results
=======
Group study: healthy versus patients after HTX
----------------------------------------------
The classification of transitions scheme introduced in Sec. II.B leads to clear distinction between the heart rhythm of healthy individuals and those of people after HTX. This is shown in Fig. \[Fig3\]. For example, the transitions to the second and farther neighbors, in the case of healthy young persons, occur with probability $0.5$ (upper part of the left bar), while in case of people after HTX these events are quite rare.
![Distributions of changes in consecutive RR-intervals for healthy young people (of age: 18–26, 21 women and 14 men) and the group of considered patients after HTX. Abbreviations [*acc*]{} and [*dcc*]{} refer to acceleration and deceleration. For other details, see the main text.[]{data-label="Fig3"}](distribution.JPG){height="19em"}
The majority of transitions in patients after HTX can be described as no change events. Moreover the decelerating or accelerating transitions occur rarely. Sequences of monotonic increases of length 3, (i.e., sequences in which $RR_{i} \! < \! RR_{i+1} \! < \! RR_{i+2} \! < \! RR_{i+3}$) occurred, on average, once in the whole signal. Similar statistics holds for decreases. Therefore, only monotonic sequences of the length equal to 1 (called one-step mono-transitions) or equal to 2 (two-step mono-transitions) describe the short-time dynamics of the heart contractions of the HTX patients.
Transitions in individuals
--------------------------
Statistics of one-step mono-transitions found for each recording separately are presented in Fig. \[Fig4\]. They are compared to the corresponding values calculated for young, healthy young persons (the first entry on horizontal axis).
![Probability to find a given type of transition in a signal for different patients after HTX: [*top*]{} – no change or transitions to nearest neighbors; [ *bottom*]{} – transitions to the next neighbors.[]{data-label="Fig4"}](nearest_nghb.JPG "fig:"){height="14em"} ![Probability to find a given type of transition in a signal for different patients after HTX: [*top*]{} – no change or transitions to nearest neighbors; [ *bottom*]{} – transitions to the next neighbors.[]{data-label="Fig4"}](next_nghb.JPG "fig:"){height="14em"}
The first observation from Fig. \[Fig4\] top is that plotted values seem to be independent of the time elapsed after the HTX. As already observed, the no-change or transitions to the nearest neighbor dominate. It means that almost all increments are less then 10 ms. The transitions to the next-neighbors shown in Fig. \[Fig4\] bottom (the increments are grater than 10 ms but lower than 20ms) do not provide any regular picture. Moreover also differences between number of accelerations and decelerations are statistically not significant. However, concentrating on characteristics for each patient individually, we get hints whether the changes are evolving towards the healthy people characteristics or not.
The results obtained for the carefully selected two-step mono-transitions are reported in Fig. \[Fig5\]. We classified the monotonic changes with respect to the total size of the transition. Namely, the consecutive three RR-intervals $RR_i$, $RR_{i+1}$ and $RR_{i+2}$ are quantified as:\
double loop:\
$RR_i=RR_{i+1}=RR_{i+2}$;\
slow deceleration:\
$RR_i< RR_{i+1} < RR_{i+2}
\mbox{~and~}
|RR_{i+2} - RR_{i}| = 2\Delta $;\
mid deceleration:\
$RR_i \! < \! RR_{i+1} \! < \! RR_{i+2}
\mbox{~and~}
|RR_{i+2} \! - \! RR_{i}| = 3
\mbox{~or~} 4 \Delta $.\
The classification of the corresponding acceleration events goes with the changed directions of the above inequalities.
![Log plots of the probability to observe double loop, slow ([*top*]{}) and mid ([*bottom* ]{}) transitions in signals of patients after HTX. To observe restitution of cardiac control in a patient, the results are collected by a patient name. In the case of [*sit*]{} patient after a month after HTX no mid transitions occur.[]{data-label="Fig5"}](double_loops.JPG "fig:"){height="13em"} ![Log plots of the probability to observe double loop, slow ([*top*]{}) and mid ([*bottom* ]{}) transitions in signals of patients after HTX. To observe restitution of cardiac control in a patient, the results are collected by a patient name. In the case of [*sit*]{} patient after a month after HTX no mid transitions occur.[]{data-label="Fig5"}](mid_trans.JPG "fig:"){height="13em"}
Probabilities of the occurrence of such two-step mono-transitions, shown in Fig. \[Fig5\], are ordered with respect to the patient name, and restricted to recordings from patients being less than about 12 months after the surgery. This ordering helps to track alternations in the cardiac rhythm in the particular patient. The entry corresponding to healthy young people is added for comparison. Note that the plots are in log-scale.
The data presented in Figs \[Fig4\] and \[Fig5\] give a total picture of changes in heart rhythm of a patient. Both analysis: one-step mono-transitions and two-step mono-transitions may be useful in assessing the progress of patient’s recovery, and possibly to produce an alarming signal that the progress is not satisfactory.
For example, Figs \[Fig4\] and \[Fig5\] does not provide the clear picture for the direction of changes of patients [*daw*]{} and [*boc*]{}. Therefore, via the network presentation we get the eye catching and easily readable additional information on quality and quantity of changes. In Figs \[Fig6\], \[Fig7\] we present network representation for these two patients.
![Networks for [*daw*]{} patient obtained from signals recorded after 1 and 2 months months after HTX. Transitions with probability less than 0.1% are omitted but all vertices which correspond to all recorded RR-intervals are plotted. The widths of arrows represent logarithms of counts of the particular transitions.[]{data-label="Fig6"}](001_daw.JPG "fig:"){height="17.2em"} ![Networks for [*daw*]{} patient obtained from signals recorded after 1 and 2 months months after HTX. Transitions with probability less than 0.1% are omitted but all vertices which correspond to all recorded RR-intervals are plotted. The widths of arrows represent logarithms of counts of the particular transitions.[]{data-label="Fig6"}](002_daw.JPG "fig:"){height="17.2em"}
![Networks for [*boc*]{} patient obtained from signals recorded after 2 and 6 months after HTX. []{data-label="Fig7"}](002_boc.JPG "fig:"){height="17.2em"} ![Networks for [*boc*]{} patient obtained from signals recorded after 2 and 6 months after HTX. []{data-label="Fig7"}](006_boc.JPG "fig:"){height="17.2em"}
The networks of [*daw*]{} patient (Fig. \[Fig6\]) show the rhythm after 1 and 2 months after surgery. We see that networks are quite similar, what indicates that the picture is stable. On the other hand, the networks obtained for [*boc*]{} patient (Fig. \[Fig7\]) show gradual simplification of the cardiac rhythm structure. It is worth noting that our predictions based on the analysis of the constructed networks (Figs. \[Fig6\] and \[Fig7\]) coincide with the clinical state of these two patients.
Conclusions
===========
The network of transitions provides yet another way to assess the heart rhythm. It appears that RR-series leads to the transition network with the specific shape. Therefore one can classify typical properties of these networks, and then construct a measure of heart rhythm changes. The network representation, first of all, offers a total assessment of increments between the consecutive RR-intervals by quantifications and qualification of their values. Additionally, it gives the eye-catching picture of RR-intervals as a map from which one can read at what RR-interval and how frequently the particular increase occur. In this work we used this approach to observe the restitution of cardiac control in patients after heart transplantation.
The arguments supporting the decision to perform HTX were different for each patient. The clinical state of every patient was specific and, therefore, each time series should be analyzed individually, and also the progress in the process of the acceptance of the graft had to be evaluated for each patient separately. This evaluation was attempted due to quantification and classification of transitions in the phase space of his/her RR-intervals. We propose to interpret the alternations in the number of particular type transitions, here towards corresponding values found in the healthy people rhythms, in the prognosis for individual patient. Moreover, since sequences of accelerations and decelerations of heart rate are considered as a sign of autonomic control [@Piskorski2011], then consecutive intervals with fixed acceleration or deceleration rates give us additional insights into the activity of the control mechanisms.
[1]{} R. Toledo, I. Pinhas, D. Aravot, Y. Almog and S. Akselrod *t Functional restitution of cardiac control in heart transplant patients*1em plus 0.5em minus 0.4em Am. J. Physiol. Reg. Integrative Comp.Physiol. 282:R900–R908, 2002. 1em plus 0.5em minus 0.4em Eur. Heart J. 17:354–81, 1996. RV. Donner, Y. Zou, J. F. Donges, N. Marwan and J. Kurths *Recurrence networks -a novel paradigm for nonlinear time series analysis* 1em plus 0.5em minus 0.4em New J. of Phys. 12: 033025, 2010. LDaF. Costa, FA. Rodrigues, G. Travieso and PR. Villad Boas, *t Characterization of comp\[lex networks: a survey of measurements* 1em plus 0.5em minus 0.4em Advances in Physics 56:167–242, 2007. D. Makowiec, J. Wdowczyk-Szulc, M. Żarczyñska-Buchowiecka, A. Rynkiewicz and M. Gruchała *t Study heart rate by tools from complex networks* 1em plus 0.5em minus 0.4em Acta Phys. Pol. B Proc. Suppl. 4:139-153, 2011 J. Piskorski and P. Guzik, *Structure of heart rate asymmetry: deceleration and acceleration runs* 1em plus 0.5em minus 0.4em Physiol.Meas. 32:1–13, 2011. *http://pajek.imfm.si/*
[^1]: Danuta Makowiec and Stanisław Kryszewski are with Gdańsk University, Poland, e-mail: fizdm@univ.gda.pl.
[^2]: Beata Graff, Joanna Wdowczyk-Szulc, Marta Żarczyńska-Buchnowiecka, Marcin Gruchała and Andrzej Rynkiewicz are with Gdańsk Medical University, Poland.
|
---
author:
- 'Richard P. Kent IV[^1]'
bibliography:
- 'join.bib'
date: 'August 14, 2008'
title: '**Intersections and joins of free groups**'
---
*The possible ranks higher than the actual.*
—common paraphrase of M. Heidegger.
Introduction
============
Let $F$ be a free group. If $H$ and $K$ are subgroups of $F$, we let $H\vee K = \langle H, K \rangle$ denote the **join** of $H$ and $K$.
We study the relationship between the rank of $H \cap K$ and that of $H\vee K$ for a pair of finitely generated subgroups $H$ and $K$ of $F$. In particular, we have the following particular case of the Hanna Neumann Conjecture, which has also been obtained by L. Louder [@louder] using his machinery for folding graphs of spaces [@louderKrull1; @louderKrull2; @louderfolding]. For detailed discussions of the Hanna Neumann Conjecture, see [@hanna; @hannaaddendum; @walter; @stallings; @gersten; @dicks].
\[particulartheorem\] Let $H$ and $K$ be nontrivial finitely generated subgroups of a free group of ranks $h$ and $k$, respectively. If $${\mathrm{rank}}(H\vee K) - 1 \geq \frac{h+k-1}{2}$$ then $${\mathrm{rank}}(H\cap K) - 1 \leq (h-1)(k-1).$$
We also give a new proof of R. Burns’ theorem [@burns]:
Let $H$ and $K$ be nontrivial finitely generated subgroups of a free group with ranks $h$ and $k$, respectively. Then $${\mathrm{rank}}(H\cap K) - 1 \leq 2(h - 1)(k - 1) - \min\big\{(h - 1), (k - 1)\big\}.$$
(In fact, we obtain W. Neumann’s form of this inequality [@walter], see Section \[burnssection\].)
Our main theorem is the following strong form of Burns’ inequality:
\[strongburns\] Let $H$ and $K$ be nontrivial finitely generated subgroups of $F$ of ranks $h$ and $k \geq h$, respectively, that intersect nontrivially. Then $${\mathrm{rank}}(H\cap K) -1 \ \leq \ 2(h -1)(k-1) - (h -1)\big({\mathrm{rank}}(H\vee K) - 1\big).$$
This theorem, with an additional hypothesis, is claimed by W. Imrich and T. Müller in [@imrichmuller]. Unfortunately, their proof contains an error—see the end of the Section \[backgroundsection\] for a detailed discussion. Note that the hypothesis on the intersection cannot be dispensed with entirely, for when $h=k \geq 3$, the inequality will fail if ${\mathrm{rank}}(H\vee K) = 2k$—but this is the only situation in which it fails.
We were brought to Theorem \[strongburns\] by the following question of M. Culler and P. Shalen.
If $H$ and $K$ are two rank–$2$ subgroups of a free group and $H \cap K$ has rank two, must their join have rank two as well?
An affirmative answer follows immediately from Theorem \[strongburns\], and we record this special case as a theorem—this has also been derived using Louder’s folding machine by Louder and D. B. McReynolds [@louder], independently of the work here.[^2]
\[main\] Let $H$ and $K$ be rank–2 subgroups of a free group $F$. Then $${\mathrm{rank}}(H\cap K) \leq 4 - {\mathrm{rank}}(H\vee K).$$
In [@loudermcreynolds], Louder and McReynolds also give a new proof of W. Dicks’ theorem [@dicks] that W. Neumann’s strong form of the Hanna Neumann Conjecture is equivalent to Dicks’ Amalgamated Graph Conjecture.
Theorem \[main\] allows Culler and Shalen to prove the following, see [@cullershalen]. Recall that a group is **$k$–free** if all of its $k$–generator subgroups are free.
Let $G$ be a $4$–free Kleinian group. Then there is a point $p$ in $\mathbb H^3$ and a cyclic subgroup $C$ of $G$ such that for any element $g$ of $G-C$, the distance between $p$ and $g p$ is at least $\log 7$.
This has the following consequence, see [@cullershalen].
Let $M$ be a closed orientable hyperbolic $3$–manifold such that $\pi_1(M)$ is $4$–free. Then the volume of $M$ is at least $3.44$.
Theorem \[main\] is sharp in that, given nonnegative integers $m$ and $n$ with $n\geq 2$ and $m \leq 4-n$, there are $H$ and $K$ of rank two with ${\mathrm{rank}}(H\cap K) = m$ and ${\mathrm{rank}}(H\vee K) = n$.
To see this, note that, by Burns’ theorem, the rank of $H \cap K$ is at most two.
If $H \vee K$ has rank four, then, since finitely generated free groups are Hopfian, we have $H \vee K = H * K$, and hence $H \cap K = 1$.
If the join has rank two, $H \cap K$ may have rank zero, one, or two. For completeness, we list examples. If $H=K$, then $H\cap K = H = H\vee K$. If $H=\langle a, bab \rangle$ and $K=\langle b,a^2 \rangle$, the join is $\langle a,b \rangle$ and the intersection is $\langle a^2 \rangle$. If $H=\langle a, bab \rangle$ and $K = \langle b, ab^{-1}aba^{-1} \rangle$, then $H \cap K =1$ and the join is $\langle a, b \rangle$.
Finally, there are rank two $H$ and $K$ whose join has rank three and whose intersection is trivial. For example, consider the free group on $\big\{a,b,c\}$ and let $H = \langle c, a^{-1} b a\rangle$ and $K = \langle a, b^{-1}cb \rangle$. Of course, there are rank two $H$ and $K$ whose intersection is infinite cyclic and whose join has rank three, like $\langle a, b \rangle$ and $\langle b, c \rangle$ in a free group on $\{a,b,c\}$.
Perspective {#perspective .unnumbered}
-----------
The heart of the work here lies in the study of a certain pushout and the restraints it places on the rank of the intersection $H \cap K$. The pictures that emerge here and in the work of Louder and McReynolds [@loudermcreynolds] share a common spirit, and both are akin to the work of W. Dicks [@dicks]. The arguments here are chiefly combinatorial; those of [@loudermcreynolds] more purely topological. Whilst having the same theoretical kernel, the two discussions each have their own merits, and the authors have decided to preserve them in separate papers.
**Acknowledgments.** The author thanks Warren Dicks, Cameron Gordon, Wilfried Imrich, Lars Louder, Joe Masters, Ben McReynolds, Walter Neumann, and Alan Reid for lending careful ears. He thanks Ben Klaff for bringing Culler and Shalen’s question to his attention. The author also extends his thanks to the referee for many thoughtful comments that have improved the exposition tremendously.
When the author first established Theorem \[main\], he used the pushout of ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$ along the component of ${\mathcal G_{H\cap K}}$ carrying the group $H \cap K$, rather than the pushout ${\mathcal T}$ along the core ${\Gamma_{\! H\cap K}}$—the former is somewhat disagreeable, and may possess special vertices. In correspondence with Louder, it was Louder’s use of the core ${\Gamma_{\! H\cap K}}$ that prompted the author’s adoption of the graph ${\mathcal T}$. The author thus extends special thanks to Louder.
Graphs, pullbacks, and pushouts {#backgroundsection}
===============================
We may assume that $F$ is free on the set $\{a, b\}$, and we do so. We identify $F$ with the fundamental group of a wedge ${\mathcal X}$ of two circles based at the wedge point, and we orient the two edges of ${\mathcal X}$.
We have distilled here the notions of [@stallings] and [@gersten] into a form that is convenient for our purpose.
Given a subgroup $H$ of $F$, there is a covering space $\widetilde {\mathcal X}_H$ corresponding to $H$. There is a unique choice of basepoint $*$ in $\widetilde {\mathcal X}_H$ so that $\pi_1(\widetilde {\mathcal X}_H, *)$ is identical to $H$. We let ${\Gamma_{\! H}}$ denote the smallest subgraph of $\widetilde {\mathcal X}_H$ containing $*$ that carries $H$. The graph ${\Gamma_{\! H}}$ comes naturally equipped with an **oriented labeling**, meaning that each edge is oriented and labeled with an element of $\{a, b\}$. The orientation of a given edge $e$ yields an **initial vertex** $\iota(e)$ and a **terminal vertex** $\tau(e)$, which may or may not be distinct.
The graphs so far discussed are labeled **properly**, meaning that if edges $e$ and $f$ have the same labeling and either $\iota(e) = \iota(f)$ or $\tau(e) = \tau(f)$, then the two edges agree.
The **star** of a vertex $v$, written ${\mathrm{star}}(v)$, is the union of the edges incident to $v$ equipped with the induced oriented labeling. The **valence** of a vertex $v$ is the number of edges incident to $v$ counted with multiplicities. All of the above graphs are at most $4$–valent, meaning that their vertices have valence at most four. A vertex is a **branch vertex** if its valence is at least $3$. We say that a vertex is **extremal** if its valence is less than or equal to one. We say that a graph is **$k$–regular** if all of its *branch* vertices have valence $k$.
A **map of graphs** between two oriented graphs is a map that takes vertices to vertices, edges to edges, and preserves orientations. A map of graphs is an **immersion** if it is injective at the level of edges on all stars. A **labeled map of graphs** between two labeled oriented graphs is a map of graphs that preserves labels. A **labeled immersion** is a labeled map of graphs that is also an immersion.
Two $k$–valent vertices of labeled oriented graphs are of the **same type** if there is a labeled immersion from the star of one to the star of the other.
J. Stallings’ category of oriented graphs is the category whose objects are oriented graphs (without labelings), and whose morphisms are maps of graphs—S. Gersten’s category has the same objects, but more maps [@gersten]. The collection of all oriented graphs with labels in $\{a, b\}$ together with all labeled maps of graphs form a category that we call the **category of labeled oriented graphs**—there is an obvious forgetful functor into Stallings’ category.
We will also consider the **category of properly labeled oriented graphs**, whose objects are properly labeled oriented graphs and whose morphisms are labeled immersions.
Given a graph $\Gamma$, let $V(\Gamma)$ be its set of vertices.
We define a graph ${\mathcal G_{H\cap K}}$ as follows. Its set of vertices is the product $V({\Gamma_{\! H}}) \times V({\Gamma_{\! K}})$ and there is an edge labeled $x$ joining $(a,b)$ to $(c,d)$ oriented from $(a,b)$ to $(c,d)$ if and only if there is an edge in ${\Gamma_{\! H}}$ labeled $x$ joining $a$ to $c$ oriented from $a$ to $c$ *and* an edge in ${\Gamma_{\! K}}$ labeled $x$ joining $b$ to $d$ oriented from $b$ to $d$. The graph ${\mathcal G_{H\cap K}}$ is the **fiber product** of the maps ${\Gamma_{\! H}}\to {\mathcal X}$ and ${\Gamma_{\! K}}\to {\mathcal X}$—in other words, the pullback of the diagram $$\xymatrix{ & {\Gamma_{\! H}}\ar[d] \\
{\Gamma_{\! K}}\ar[r] & {\mathcal X}}$$ in the category of oriented graphs—it is also the pullback in the category of properly labeled oriented graphs, and in this category, it is in fact the direct product ${\Gamma_{\! H}}\times {\Gamma_{\! K}}$.
The graph ${\Gamma_{\! H\cap K}}$ is a subgraph of ${\mathcal G_{H\cap K}}$, and carries the fundamental group [@stallings].
Note that there are projections $\Pi_H{\colon\thinspace}{\mathcal G_{H\cap K}}\to {\Gamma_{\! H}}$ and $\Pi_K{\colon\thinspace}{\mathcal G_{H\cap K}}\to {\Gamma_{\! K}}$ and that a path $\gamma$ from $(*,*)$ to $(u,v)$ in ${\mathcal G_{H\cap K}}$ projects to paths $\Pi_H(\gamma)$ and $\Pi_K(\gamma)$ with the same labeling from $*$ to $u$ and $*$ to $v$, respectively. Conversely, given two pointed paths $\gamma_H$ and $\gamma_K$ with identical oriented labelings from $*$ to $u$ and $*$ to $v$ respectively, there is an identically labeled path $\gamma$ in ${\mathcal G_{H\cap K}}$ from $(*,*)$ to $(u,v)$ that projects to $\gamma_H$ and $\gamma_K$.
Given a graph $\Gamma$ with an oriented (nonproper) labeling, a **fold** is the following operation: if $e_1$ and $e_2$ are two edges of $\Gamma$ with the same label and $\iota(e_1) = \iota(e_2)$ or $\tau(e_1)=\tau(e_2)$, identify $e_1$ and $e_2$ to obtain a new graph.
The properly labeled graph ${\Gamma_{\! H\vee K}}$ is obtained from ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$ by forming the wedge product of ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$ at their basepoints and folding until no more folding is possible.
In what follows, we identify ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$ with their images in ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$ whenever convenient.
The graph ${\Gamma_{\! H\vee K}}$ is the pushout in the category of *properly labeled* oriented graphs of the diagram $$\xymatrix{ \ast \ar[r] \ar[d]
& {\Gamma_{\! H}}\\
{\Gamma_{\! K}}}$$ where the single point $\ast$ maps to the basepoints in ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$. This category is somewhat odd in that ${\Gamma_{\! H\vee K}}$ is also the pushout of $$\xymatrix{{\Gamma_{\! H\cap K}}\ar[r] \ar[d]
& {\Gamma_{\! H}}\\
{\Gamma_{\! K}}}$$
We will make use of a labeled oriented graph that is not properly labeled. This is the **topological pushout** ${\mathcal T}$ of the diagram $$\xymatrix{{\Gamma_{\! H\cap K}}\ar[r] \ar[d]
& {\Gamma_{\! H}}\\
{\Gamma_{\! K}}}$$ The letters $x$ and $y$ will denote points in ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$, respectively. The graph ${\mathcal T}$ is the quotient of ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$ by the equivalence relation $\mathfrak{R}$ *generated* by the relations $x \sim y$ if $x \in \Pi_H\big({\Pi_K}^{\! \! \! -1}(y)\big)$ or $y \in \Pi_K\big({\Pi_H}^{\! \! \! -1}(x)\big)$.
So, points $a,b \in {\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$ map to the same point in ${\mathcal T}$ if and only if there is a sequence $\{(x_i,y_i)\}_{i=1}^n$ in ${\Gamma_{\! H\cap K}}$ such that $a$ is a coordinate of $(x_1,y_1)$, $b$ is a coordinate of $(x_n,y_n)$, and for each $i$ either $x_i = x_{i+1}$ or $y_i = y_{i+1}$. We call such a sequence a **sequence for $a$ and $b$**. Note that a minimal sequence for $x$ and $y$ will not have $x_i = x_j=x_k$ or $y_i = y_j=y_k$ for any pairwise distinct $i$, $j$, and $k$.
We warn the reader that the equivalence relation on ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$ whose quotient is ${\Gamma_{\! H\vee K}}$ is typically coarser than the one just described. For instance, in the example in Figure \[counterexample\], ${\Gamma_{\! H\vee K}}$ is ${\mathcal X}$, but ${\mathcal T}$ is not.
![Above, the graph ${\Gamma_{\! K}}$ is at the bottom, ${\Gamma_{\! H}}$ at the right. Their basepoints are encircled. White arrows correspond to $a$, black arrows $b$. Writing $x^g = gxg^{-1}$, we have $K = \langle a^2ba, (b^2a^2)^{ab^2} \rangle$, $H = \langle ab^{-2}a, (ba^{-2})^{ab}, a^{a^{-1}b^{-1}}\rangle$, ${H \cap K} = \langle ab^2a^{-2}b^{-4}aba\rangle$, and $H\vee K = \langle a, b \rangle$. The graph ${\mathcal T}$ is to the right. Notice that $\chi({\mathcal T}) = -3 < -1 = \chi({\Gamma_{\! H\vee K}})$. []{data-label="counterexample"}](close.pdf "fig:")
The graph ${\mathcal T}$ is also the pushout in the category of labeled oriented graphs, but not necessarily the pushout in the category of *properly* labeled oriented graphs—again, see Figure \[counterexample\].
Though not equal to ${\Gamma_{\! H\vee K}}$ in general, ${\mathcal T}$ does fit into the commutative diagram $$\xymatrix{{\Gamma_{\! H\cap K}}\ar[r]^{\Pi_H} \ar[d]_{\Pi_K} & {\Gamma_{\! H}}\ar[d] \ar[ddr] \\
{\Gamma_{\! K}}\ar[r] \ar[drr] & {\mathcal T}\ar[dr] \\
& & {\Gamma_{\! H\vee K}}}$$ where the map ${\mathcal T}\to {\Gamma_{\! H\vee K}}$ factors into a series of folds. As a fold is surjective at the level of fundamental groups, see [@stallings], we have $\chi({\mathcal T}) \leq \chi({\Gamma_{\! H\vee K}})$.
Confusing ${\mathcal T}$ and ${\Gamma_{\! H\vee K}}$ can be hazardous, and we call ${\mathcal T}$ the topological pushout to prevent such confusion. This is the source of the error in [@imrichmuller], which we now discuss.
The proof of the lemma on page 195 of [@imrichmuller] is incorrect. The error lies in the last complete sentence of that page:
> In order that both $x$ and $y$ be mapped onto $z$ there must be a sequence $$x = x_0, x_1, x_2, \ldots, x_n = y$$ of vertices of $\Gamma_0$ such that for every $i$ $x_i$ and $x_{i+1}$ have the same image in either $\Gamma_1$ or $\Gamma_2$ (and all are mapped to $z$ in $\Delta$).
Here $\Gamma_0$ is our graph ${\Gamma_{\! H\cap K}}$, the graphs $\Gamma_1$ and $\Gamma_2$ are our graphs ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$, and the graph $\Delta$ is our ${\Gamma_{\! H\vee K}}$. Here is a translation of this into our terminology:
> Let $z$ be a vertex in ${\Gamma_{\! H\vee K}}$ and let $a$ and $b$ be vertices of ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$ that map to $z$. In order that both $a$ and $b$ be mapped onto $z$, there must be a sequence for $a$ and $b$.
This statement is false. The example in Figure \[counterexample\] is a counterexample: the graph ${\Gamma_{\! H\vee K}}$ is the wedge of two circles with a vertex $z$, say, and so all vertices in ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$ map to $z$ under the quotient map ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}\to {\Gamma_{\! H\vee K}}$; on the other hand, the basepoints for ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$ are the only vertices in their $\mathfrak{R}$–equivalence class—as is easily verified by sight.
The statement *is* correct once ${\Gamma_{\! H\vee K}}$ has been replaced by ${\mathcal T}$, but, unfortunately, the arguments in [@imrichmuller] rely on the fact that ${\Gamma_{\! H\vee K}}$ is $3$–regular, a property that ${\mathcal T}$ does not generally possess.
The lemma in [@imrichmuller] would be quite useful, and though its proof is incorrect, we do not know if the lemma actually fails.[^3]
Estimating the Euler characteristic of ${\mathcal T}$
=====================================================
Let $H$ and $K$ be subgroups of $F$ of ranks $h$ and $k$. Suppose that $H \cap K \neq 1$.
For simplicity, we reembed $H\vee K$ into $F$ so that all branch vertices in ${\Gamma_{\! H\vee K}}$ are $3$–valent and of the same type: we replace $H\vee K$ with its image under the endomorphism $\varphi$ of $F$ defined by $\varphi(a) = a^2$, and $\varphi(b) = [a,b] = aba^{-1}b^{-1}$. Note that this implies that all branch vertices of ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$ are $3$–valent and of the same type.
If a restriction of a covering map of graphs fails to be injective on an edge, then the edge must descend to a cycle of length one. So our normalization above guarantees that the restriction of the quotient ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}\to {\Gamma_{\! H\vee K}}$ to any edge is an embedding (as the target has no unit cycles), and hence *the restriction of the quotient ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}\to {\mathcal T}$ to any edge is an embedding*.
We claim that it suffices to consider the case where neither ${\Gamma_{\! H}}$ nor ${\Gamma_{\! K}}$ possess extremal vertices. It is easy to see that by conjugating $H\vee K$ in $F$, one may assume that ${\Gamma_{\! H}}$ has no extremal vertices, and we assume that this is the case. Let $p$ and $q$ be the basepoints of ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$, respectively. Suppose that $q$ is extremal. Let $\gamma$ be the shortest path in ${\Gamma_{\! K}}$ starting at $q$ and ending at a branch vertex. Suppose that $\gamma$ is labeled with a word $w$ in $F$. Since $H \cap K$ is not trivial, the graph ${\Gamma_{\! H\cap K}}$ contains a nontrivial loop based at $(p,q)$, and so there is a path $\delta$ in ${\mathcal G_{H\cap K}}$ starting at $(p,q)$ labeled $w$. Now $\delta$ projects to a path in ${\Gamma_{\! H}}$ starting at $p$ that is labeled $w$. This means that if we conjugate $H\vee K$ by $v = w^{-1}$, the graphs $\Gamma_{\! H^v}$ and $\Gamma_{\! K^v}$ have no extremal vertices, and of course, ${\mathrm{rank}}\big(H^v \cap K^v\big) = {\mathrm{rank}}\big((H \cap K)^v\big) = {\mathrm{rank}}(H\cap K)$, and ${\mathrm{rank}}\big((H\vee K)^v\big) = {\mathrm{rank}}(H\vee K)$.
We assume these normalizations throughout. Note that since ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$ have no extremal vertices, neither does ${\Gamma_{\! H\cap K}}$.
Stars
-----
If $\Gamma$ is a graph, let $\mathbf{b}(\Gamma)$ denote the number of branch vertices in $\Gamma$. If $\Gamma $ is $3$–regular, then $-\chi(\Gamma) = {\mathrm{rank}}\big(\pi_1(\Gamma)\big) -1=\mathbf{b}(\Gamma)/2$.
Consider the topological pushout ${\mathcal T}$ of ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$ along ${\Gamma_{\! H\cap K}}$, and the equivalence relation $\mathfrak{R}$ on ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$ that defines it. Again, $-\chi({\mathcal T}) \geq {\mathrm{rank}}(H\vee K) -1$.
This section is devoted to the proof of the following theorem—compare Lemma 5.3 of [@louderfolding]. We estimate the Euler characteristic of ${\mathcal T}$ by studying the set of $\mathfrak{R}$–equivalence classes of stars. The equivalence class of the star of a vertex $b$ in ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$ is denoted by $[{\mathrm{star}}(b)]_\mathfrak{R}$. If $X$ is a set, $\#X$ will denote its cardinality.
\[eulertheorem\] $$\label{euler}
-\chi({\mathcal T}) \leq \frac{1}{2} \# \big\{ [{\mathrm{star}}(b)]_\mathfrak{R}\ \big | \ b \in {\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}\ \mathrm{and}\ \mathrm{valence}(b)=3 \big\}.$$
In the following, we will denote the type of a $2$–valent vertex in ${\Gamma_{\! H}}$, ${\Gamma_{\! K}}$, or ${\Gamma_{\! H\cap K}}$ by a Roman capital.
We say that a vertex $z$ is **special** if it is a branch vertex of ${\mathcal T}$ that is not the image of a branch vertex in ${\Gamma_{\! H}}$ or ${\Gamma_{\! K}}$—we will show that there are no such vertices.
\[notallsame\] Let $z$ be a special vertex of ${\mathcal T}$. Then there are vertices $a$ and $b$ in ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$ that have different types and get carried to $z$.
Suppose to the contrary that any $a$ and $b$ that get carried to $z$ have the same type.
Let $a$ and $b$ be such a pair and let $\{(x_i,y_i)\}$ be a sequence for $a$ and $b$. Since $z$ is special, all of the $x_i $ and $y_i $ are $2$–valent. By our assumption, all of the $x_ i $ and $y_ i $ have the same type. But this means that the $(x_i,y_i)$ are all $2$–valent and of the same type. This means, in turn, that the stars of all the $x_i $ and $y_i $ get identified in ${\mathcal T}$.
This contradicts the fact that $z$ was a branch vertex.
\[nospecials\] There are no special vertices in ${\mathcal T}$.
Let $z$ be a special vertex.
By Lemma \[notallsame\], there are vertices $a$ and $b$ of types $A$ and $B \neq A$ that map to $z$.
Let $\{v_i\}_{i=1}^n$ be a sequence for $a$ and $b$. The vertex $v_1$ has type $A$, and $v_n$ has type $B$. Somewhere in between, the types must switch, and by the definition of sequence, we find a $v_j$ with a coordinate of type $A$, and a coordinate of type $X \neq A$. This implies that $v_j$ is extremal. But ${\Gamma_{\! H\cap K}}$ has no extremal vertices.
\[staridentification\] Let $z$ be a branch vertex in ${\mathcal T}$. Let ${\mathcal G}^z$ be the subgraph of ${\mathcal T}$ obtained by taking the union of the images of the stars of all branch vertices in ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$ mapping to $z$. If ${\mathrm{valence}}_{{\mathcal G}^z}(z)$ is the valence of $z$ in ${\mathcal G}^z$, then $${\mathrm{valence}}_{{\mathcal G}^z}(z) \leq 2 + \#\big\{ [{\mathrm{star}}(b)]_\mathfrak{R}\ \big | \ b \in {\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}\mathrm{,}\ \mathrm{valence}(b)=3 \mathrm{,\ and}\ b \mapsto z
\big\}.$$
Let $$n = \#\big\{ [{\mathrm{star}}(b)]_\mathfrak{R}\ \big | \ b \in {\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}\mathrm{,}\ \mathrm{valence}(b)=3 \mathrm{,\ and}\ b \mapsto z
\big\}$$ and let $b_1, \ldots, b_n$ be a set of branch vertices whose stars form a set of representatives for the set $
\big\{ [{\mathrm{star}}(b)]_\mathfrak{R}\ \big | \ b \in {\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}\mathrm{,}\ \mathrm{valence}(b)=3 \mathrm{,\ and}\ b \mapsto z
\big\}.
$
For $1\leq j \leq n$, let ${\mathcal G}_j$ be the union of the images in ${\mathcal T}$ of the stars of $b_1, \ldots, b_j$. So, ${\mathcal G}_n = {\mathcal G}^z$.
Since the restriction of ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}\to {\mathcal T}$ to any edge is an embedding, the valence of $z$ in ${\mathcal G}_1$ is $2 + 1 = 3$.
Now let $m \geq 1$ and assume that the valence of $z$ in ${\mathcal G}_{m-1}$ is less than or equal to $2 + m-1$.
After rechoosing our representatives and reordering the vertices $b_1, \ldots, b_{m-1}$, as well as the $b_m, \ldots, b_n$, we may assume that there is a sequence $\{v_i\}_{i =1}^{\ell}$ for $b_{m-1}$ and $b_m$ where $b_{m-1}$ and $b_m$ are the only branch vertices appearing as coordinates in the sequence and each appears only *once*. To see this, take $\{v_ i\}$ to be a sequence shortest among all sequences between vertices $a$ and $b$ such that ${\mathrm{star}}(a)$ is identified with the star of one of $b_1, \ldots, b_{m-1}$ and ${\mathrm{star}}(b)$ is identified with the star of one of $b_m, \ldots, b_{n}$.
Now, all of the $v_i$ are $2$–valent and of the same type. It is now easy to see that $z$ is at most $4$–valent in the image of ${\mathrm{star}}(b_{m-1}) \cup {\mathrm{star}}(b_m)$ in ${\mathcal G}_m$—again we are using the fact that each edge of ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$ embeds in ${\mathcal T}$. This means that $z$ is at most $(2 + m)$–valent in ${\mathcal G}_m$, and we are done by induction.
Suppose that there is a $2$–valent vertex $a$ in ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$ carried to a branch vertex in ${\mathcal T}$ whose star is not carried into the star of any branch vertex.
Let $\{v_i\}_{i=1}^n$ be a sequence for $a$ and a branch vertex $x$ that is minimal among all sequences for $a$ and branch vertices—such a sequence exists by Corollary \[nospecials\]. So $x$ is the only branch vertex that appears as a coordinate in the sequence and it only appears once, in $v_n$.
Let $A$ be the type of $a$. If there were a $2$–valent vertex of type $B \neq A$ appearing as a coordinate in the sequence, then there would be a term in the sequence with a coordinate of type $A$ and a $2$–valent coordinate of type $X \neq A$, making this term in the sequence extremal, which is impossible. So every $2$–valent coordinate in the sequence is of type $A$. It follows that the stars of all of the $2$–valent coordinates in the sequence are identified in ${\mathcal T}$.
But $v_n$ is a $2$–valent vertex of ${\Gamma_{\! H\cap K}}$, as only one of its coordinates is a branch vertex. So the star of the $2$–valent coordinate of $v_n$ is carried into the image of the star of $x$. We conclude that the star of $a$ is carried into the image of the star of $x$, a contradiction.
It follows from this and Corollary \[nospecials\] that for each branch vertex $z$ in ${\mathcal T}$, we have $${\mathrm{valence}}_{\mathcal T}(z) = {\mathrm{valence}}_{{\mathcal G}^z}(z).$$ So, by Lemma \[staridentification\], we have $${\mathrm{valence}}_{\mathcal T}(z) \leq 2 + \#\big\{ [{\mathrm{star}}(b)]_\mathfrak{R}\ \big | \ b \in {\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}\mathrm{,}\ \mathrm{valence}(b)=3 \mathrm{,\ and}\ b \mapsto z
\big\}.$$ We conclude that $$\begin{aligned}
-\chi({\mathcal T}) & = \frac{1}{2} \sum_{z\ \mathrm{vertex}} ({\mathrm{valence}}_{\mathcal T}(z) - 2) \\
& \leq \frac{1}{2} \# \big\{ [{\mathrm{star}}(b)]_\mathfrak{R}\ \big | \ b \in {\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}\ \mathrm{and}\ \mathrm{valence}(b)=3 \big\}. \qedhere\end{aligned}$$
Matrices
--------
Let $X =\{x_1, \ldots, x_{2h-2}\}$ and $Y=\{y_1, \ldots, y_{2k-2}\}$ be the sets of branch vertices of ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$, respectively. Define a function $f {\colon\thinspace}X \times Y \to \{0,1\}$ by declaring $f(x_i,y_j) =1$ if $(x_i,y_j)$ is a branch vertex of ${\Gamma_{\! H\cap K}}$, zero if not.
Consider the $(2h-2) \times (2k-2)$–matrix $M = \big(f(x_i,y_j)\big)$. Note that $\sum_{i,j} f(x_i,y_j) = \mathbf{b}({\Gamma_{\! H\cap K}})$. In particular, H. Neumann’s inequality [@hanna; @hannaaddendum] $${\mathrm{rank}}(H\cap K) - 1 \leq 2(h - 1)(k - 1)$$ becomes the simple statement that the entry–sum of $M$ is no more than $(2h-2)(2k-2)$.
\[normalformlemma\] After permuting its rows and columns, we may assume that $M$ is in the block form $$\label{normalform}
\left( \begin{array}{cccccc}
M_1 & & &\\
& \ddots & &\\
&& M_\ell &\\
&& & \mbox{\large $0$}_{p\times q}
\end{array} \right)$$ where every row and every column of every $M_i$ has a nonzero entry and $$\ell + p + q = \# \big\{ [{\mathrm{star}}(b)]_\mathfrak{R}\ \big | \ b \in {\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}\ \ \mathrm{and}\ \mathrm{valence}(b)=3 \big\}.$$
When $p$ or $q$ is zero, the notation means that $M$ possesses $q$ zero–columns at the right or $p$ zero–rows at the bottom, respectively.
Let $$\{e_1,\ldots, e_s\} = \big\{ [{\mathrm{star}}(b)]_\mathfrak{R}\ \big | \ b \in {\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}\ \ \mathrm{and}\ \mathrm{valence}(b)=3 \big\},$$ let $\{r_{i,j}\}_{j=1}^{m_i}$ be the set of rows corresponding to branch vertices in ${\Gamma_{\! H}}$ of class $e_i$, and let $\{c_{i,t}\}_{t=1}^{n_i}$ be the set of columns corresponding to branch vertices in ${\Gamma_{\! K}}$ of class $e_i$.
By permuting the rows we may assume that the $r_{1,j}$ are the first $m_1$ rows, the $r_{2,j}$ the next $m_2$ rows, and so on. Now, by permuting columns, we may assume that the $c_{1,k}$ are the first $n_1$ columns, the $r_{2,k}$ the next $n_2$ columns, and so forth. Moving all of the zero–rows to the bottom, and all of the zero–columns to the right, we obtain our normal form .
To see that the stated equality holds, first notice that the normal form and the definition of the equivalence relation $\mathfrak{R}$ together imply that: there are precisely $p$ branch vertices in ${\Gamma_{\! H}}$ whose stars are not $\mathfrak{R}$–equivalent to that of *any* other branch vertex in ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$, corresponding to the the $p$ zero–rows at the bottom; and there are precisely $q$ branch vertices in ${\Gamma_{\! K}}$ whose stars are not $\mathfrak{R}$–equivalent to that of *any* other branch vertex in ${\Gamma_{\! H}}\sqcup {\Gamma_{\! K}}$, corresponding to the the $q$ zero–columns at the right. After reordering the $\mathfrak{R}$–equivalence classes, we may thus list them as $$e_1, \ldots, e_L; \ e_{L + 1}, \ldots, e_{L + p}; \ e_{L + p + 1}, \ldots, e_{L + p + q}$$ where $e_{L + 1}, \ldots, e_{L + p}$ are the classes of the branches corresponding to the last $p$ rows, and $e_{L + p + 1}, \ldots, e_{L + p + q}$ are the classes corresponding to the last $q$ columns.
By construction of the normal form $M$, each block represents an $\mathfrak{R}$–equivalence class of stars: if the entries $(a,b)$ and $(c,d)$ of $M$ lie in a block $M_i$, then the vertices $x_a$, $y_b$, $x_c$, and $y_d$ all represent the same $\mathfrak{R}$–equivalence class. Furthermore, distinct blocks represent distinct classes. So the number $L$ is at least $\ell$.
Finally, as an equivalence class either corresponds to a block (the equivalence class has representatives in ${\Gamma_{\! H}}$ *and* ${\Gamma_{\! K}}$), a zero–row (the equivalence class has a unique representative in ${\Gamma_{\! H}}$), or a zero–column (the equivalence class has a unique representative in ${\Gamma_{\! K}}$), we conclude that $L = \ell$.
We will make repeated use of the following lemma.
\[entrysum\] The entry–sum of $M$ is less than or equal to the entry–sum of the $(2h-2) \times (2k-2)$–matrix $$\left( \begin{array}{ccc}
\mbox{\large $1$}_{m\times n} & & \\
& {\vspace{4pt}\! \! \! \! {{}_{\mbox{$1$}}}_{\, \ddots_{\mbox{\! \! $1$}}}}& \\
& &\! \! \! \! \mbox{\large $0$}_{p\times q}
\end{array} \right)$$ where $m = 2h-2 - p - (\ell -1)$, $n = 2k-2 - q - (\ell-1)$, and $\mbox{\large $1$}_{m\times n}$ is the $m \times n$–matrix all of whose entries are $1$.
We perform a sequence of operations to $M$ that do not decrease the entry–sum and which result in the matrix displayed in the lemma.
First replace each block in $M$ with a block of the same dimensions and whose entries are all $1$. Of course, this does not decrease the entry–sum.
Now, reorder the blocks in order of nonincreasing entry–sum. If there are only $1 \times 1$–blocks, we are done. If all but one of the blocks are $1 \times 1$, we are again done. So we may assume that at least two blocks have more than one entry.
Let $M_t$ be the last block with more than one entry. Say that $M_{t -1}$ and $M_t$ are $a \times b$ and $c \times d$ matrices, respectively. We now replace $M_{t - 1}$ with an $(a + c - 1) \times (b + d -1)$–block all of whose entries are $1$, and replace $M_t$ with a $1 \times 1$–block whose entry is $1$. That this does not decrease the entry–sum is best understood using a diagram, which we have provided in Figure \[countingfigure\].
Repeating this procedure eventually terminates in the matrix displayed in the lemma, and the proof is complete.
Burns’ theorem {#burnssection}
==============
We record here a proof of Burns’ theorem that requires only the matrix $M$ and a simple count—we recommend B. Servatius’ [@servatius] and P. Nickolas’ [@nickolas] proof of this theorem, which involves a clever consideration of a minimal counterexample, as do we recommend the discussion of said argument in [@walter]. To our knowledge, the argument here is new.
\[noPandQ\] If $p = q = 0$, then $\ell > 1$.
If $p=q=0$ and $\ell =1$, then ${\mathcal T}$ has a single branch vertex of valence $3$, which is impossible, as it has no extremal vertices.
Let $H$ and $K$ be nontrivial finitely generated subgroups of a free group with ranks $h$ and $k$, respectively. Then $${\mathrm{rank}}(H\cap K) - 1 \leq 2(h - 1)(k - 1) - \min\big\{(h - 1), (k - 1)\big\}.$$
Let $h$ and $k$ be the ranks of $H$ and $K$ with $h \leq k$.
If one of $p$ or $q$ is nonzero, then $M$ has a zero–row or a zero–column, by . Since $M$ is a $(2h-2) \times (2k-2)$–matrix with entries in $\{0,1\}$ and entry–sum $\mathbf{b}({\Gamma_{\! H\cap K}}) = -2\chi({\Gamma_{\! H\cap K}})$, we are done.
So, by Lemma \[noPandQ\], we may assume that $\ell \geq 2$, and the comparison of entry–sums in Lemma \[entrysum\] yields [$$\begin{aligned}
\mathbf{b}({\Gamma_{\! H\cap K}}) & \leq \ell -1 + \big(2h-2 - (\ell-1)\big)\big(2k-2 - (\ell-1)\big)\\
& = \ell - 1 + (2h-2)(2k-2) - (\ell-1)(2h-2) - (\ell-1)(2k-2) + (\ell-1)^2 \\
&\leq 4(h - 1)(k - 1) - (2h-2) + \big(\ell - (2k-2)\big)(\ell-1)\\
& \leq 4(h-1)(k-1) - (2h-2),\end{aligned}$$ ]{}as desired—the inequality is again more easily understood using a diagram, which we have provided in Figure \[BurnsCount\].
Notice that nothing prevents us from considering pushouts along disconnected graphs, and so we in fact obtain W. Neumann’s [@walter] strong form of Burns’ inequality: $$\sum_{\overset{g\in H \backslash F \slash K}{H \cap K^g \neq 1}} \! \! \big( {\mathrm{rank}}(H\cap K^g) - 1\big)
\leq 2(h - 1)(k - 1) - \min\big\{(h - 1), (k - 1)\big\}.$$
Strengthening Burns’ inequality
===============================
\[joinburns\] Let $H$ and $K$ be nontrivial finitely generated subgroups of $F$ of ranks $h$ and $k \geq h$ that intersect nontrivially. Then $${\mathrm{rank}}(H\cap K) -1 \leq 2(h -1)(k-1) - (h -1)\big({\mathrm{rank}}(H\vee K) - 1\big).$$
First suppose that ${\mathrm{rank}}(H\cap K) = 1$. The desired inequality is then $$0 \leq (h -1)\big(2k - {\mathrm{rank}}(H\vee K) - 1\big).$$ If $$2k - {\mathrm{rank}}(H\vee K) - 1 \geq 0,$$ then we are done. If this is not the case, then we must have ${\mathrm{rank}}(H\vee K) = 2k$, and hence $h=k$, since ${\mathrm{rank}}(H\vee K) \leq h + k$. Since finitely generated free groups are Hopfian, we must conclude that ${\mathrm{rank}}(H\cap K) = 0$, which contradicts our assumption.
So we assume as we may that ${\mathrm{rank}}(H\cap K) \geq 2$. As every branch vertex of ${\Gamma_{\! H\cap K}}$ is associated to a block of our normal form $M$, this implies that $\ell \geq 1$.
First note that $2h-2 > p + \ell -1$ and $2k-2 > q + \ell -1$. By Lemma \[entrysum\], we have $$\begin{aligned}
\mathbf{b}({\Gamma_{\! H\cap K}}) & \leq \ell - 1 + \big(2h-2 - (p + \ell-1)\big)\big(2k-2 - (q+\ell-1)\big) \notag \\
& = \ell -1 + (2h-2)(2k-2) - (p + \ell-1)(2k-2) \notag \\
& \quad \quad \quad \ + \big[ (p + \ell-1)(q + \ell-1) - (2h-2)(q + \ell-1) \big] \notag \\
& \leq \ell -1 + (2h-2)(2k-2) - (p + \ell-1)(2k-2) \notag \\
& \quad \quad \quad \ - [\ell -1] \notag \\
& = (2h-2)(2k-2) - (p + \ell-1)(2k-2).\label{estimate1}\end{aligned}$$ The proof of the inequality is illustrated in Figure \[StrongCount\].
Similarly, $$\label{estimate2}
\mathbf{b}({\Gamma_{\! H\cap K}}) \leq (2h-2)(2k-2) - (q + \ell-1)(2h-2).$$
Since $\ell \geq 1$, the inequality provides the theorem unless $$q < {\mathrm{rank}}(H\vee K) - 1 \leq -\chi({\mathcal T}).$$ So we assume that $q \leq -\chi({\mathcal T}) - 1$, the rest of the argument proceeding as in [@imrichmuller].
By Theorem \[eulertheorem\] and Lemma \[normalformlemma\], we also have $\ell + p + q \geq -2\chi({\mathcal T})$ and so $$\ell + p \geq -\chi({\mathcal T}) + 1 \geq {\mathrm{rank}}(H\vee K).$$ By , we now have $$\mathbf{b}({\Gamma_{\! H\cap K}}) \leq (2h-2)(2k-2) - \big({\mathrm{rank}}(H\vee K) -1\big)(2k-2),$$ and since $k \geq h$, the proof is complete.
A particular case of the Hanna Neumann Conjecture {#particular}
=================================================
\[bigjoin\] Let $H$ and $K$ be nontrivial finitely generated subgroups of a free group of ranks $h$ and $k$, respectively. If $${\mathrm{rank}}(H\vee K) - 1 \geq \frac{h+k-1}{2}$$ then $${\mathrm{rank}}(H\cap K) - 1 \leq (h-1)(k-1).$$
Note that if $q \geq k$, then the $(2h-2)\times(2k-2)$–matrix $M$ has at least $k$ zero–columns. As $\mathbf{b}({\Gamma_{\! H\cap K}})$ is the entry–sum of $M$, we have $$\mathbf{b}({\Gamma_{\! H\cap K}}) \leq (2k-2 - k)(2h-2) = (k-2)(2h-2),$$ and so $${\mathrm{rank}}(H\cap K) - 1 \leq (h-1)(k-2),$$ which is better than desired.
So assume that $q \leq k-1$. Then, by assumption, Theorem \[eulertheorem\], Lemma \[normalformlemma\], and the fact that $-\chi({\mathcal T}) \geq {\mathrm{rank}}(H\vee K) - 1$, we have $$\label{hypoineq}
\ell -1 + p + q \geq h + k - 2.$$ Note that $2h-2 > p + \ell -1$. So, by Lemma \[entrysum\] and , we have $$\begin{aligned}
\mathbf{b}({\Gamma_{\! H\cap K}}) & \leq \ell - 1 + \big(2h-2 - (p + \ell-1)\big)\big(2k-2 - (q+\ell-1)\big) \\
& = \ell -1 + (2h-2)(2k-2) - (p + \ell-1)(2k-2)\\
& \quad \quad \quad \ + \big[ (p + \ell-1)(q + \ell-1) - (2h-2)(q + \ell-1) \big] \\
& \leq \ell -1 + (2h-2)(2k-2) - (h+k-2-q)(2k-2)\\
& \quad \quad \quad \ - [\ell -1] \\
& \leq (2h-2)(2k-2) - (h-1)(2k-2)\\
& = 2(h-1)(k-1). \qedhere\end{aligned}$$
We do not obtain the stronger inequality $$\sum_{\overset{g\in H \backslash F \slash K}{H \cap K^g \neq 1}} \! \! \big( {\mathrm{rank}}(H\cap K^g) - 1\big)
\leq (h - 1)(k - 1)$$ here, nor the analogous inequality in Theorem \[joinburns\], as the pushout along a larger graph could have Euler characteristic dramatically smaller in absolute value than $-\chi({\Gamma_{\! H\vee K}})$. For example, it is easy to find $H$ and $K$ and $u$ and $v$ such that the pushouts ${\mathcal T}^{\, uv}$ and ${\mathcal T}$ of ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$ along $\Gamma_{\! H^u \cap K^v}$ and ${\Gamma_{\! H\cap K}}$, respectively, satisfy $-\chi({\mathcal T}^{\, uv}) \geq -\chi(\Gamma_{\! H^u \vee K^v}) \gg -\chi({\mathcal T})$. As a consequence, the pushout along ${\Gamma_{\! H\cap K}}\sqcup \Gamma_{\! H^u \cap K^v}$ will have Euler characteristic much smaller in absolute value than $-\chi(\Gamma_{\! H^u \vee K^v})$.
If the reader would like a particular example of this phenomenon, she may produce one as follows.
Begin with subgroups $A$ and $B$ of large rank so that the topological pushout of $\Gamma_{\! A}$ and $\Gamma_{\! B}$ is the wedge of two circles: take $A$ and $B$ to be of finite index in $F$, the subgroup $A$ containing $a$, the subgroup $B$ containing $b$.
Now consider the endomorphism $F \to F$ that takes $a$ and $b$ to their squares. Let $H$ and $K$ be the images of $A$ and $B$ under this endomorphism, respectively. It is a simple exercise to see that the pushout ${\mathcal T}$ of ${\Gamma_{\! H}}$ and ${\Gamma_{\! K}}$ along ${\Gamma_{\! H\cap K}}$ is homeomorphic to that of $\Gamma_{\! A}$ and $\Gamma_{\! B}$ along $\Gamma_{\! A\cap B}$—it is a wedge of two circles labeled $a^2$ and $b^2$. It is also easy to see that the pullback ${\mathcal G_{H\cap K}}$ contains an isolated vertex $(x,y)$, where $x$ is the $2$–valent center of a segment labeled $a^2$ and $y$ is the $2$–valent center of a segment labeled $b^2$.
We may conjugate $H$ and $K$ by elements $u$ and $v$ in $F$, respectively, so that $\Gamma_{\! H^u}= {\Gamma_{\! H}}$, $\Gamma_{\! K^v} = {\Gamma_{\! K}}$, and $\Gamma_{\! H^u \cap K^v}$ is our isolated point. So the pushout ${\mathcal T}^{\, uv}$ of $\Gamma_{\! H^u}$ and $\Gamma_{\! K^v}$ along $\Gamma_{\! H^u \cap K^v}$ is the wedge of $\Gamma_{\! H^u}$ and $\Gamma_{\! K^v}$. In fact, by our choice of isolated point, the pushout ${\mathcal T}^{\, uv}$ will be equal to $\Gamma_{\! H^u \vee K^v}$, as the former admits no folds.
In such a case, the pushout of $\Gamma_{\! H^u}$ and $\Gamma_{\! K^v}$ along $\Gamma_{\! H^u \cap K^v} \sqcup {\Gamma_{\! H\cap K}}$ has Euler characteristic small in absolute value (being a quotient of ${\mathcal T}$, it has no more than four edges), despite the fact that the graph $\Gamma_{\! H^u \vee K^v}$ has Euler characteristic very large in absolute value.
See Section \[walterineq\] for what can be said about the general situation.
Remarks.
========
A bipartite graph
-----------------
Estimating $-\chi({\mathcal T})$ may be done from a different point of view, suggested to us by W. Dicks—compare [@dicks].
Given our subgroups $H$ and $K$, define a bipartite graph $\Delta$ with $2h-2$ black vertices $x_1, \ldots, x_{2h-2}$ and $2k-2$ white vertices $y_1, \ldots, y_{2k-2}$ where $x_i$ is joined to $y_j$ by an edge if and only if the $i,j$–entry of $M$ is $1$. It is easy to see that the number $c$ of components of $\Delta$ is equal to $\ell + p + q$, and that its edges are $2 \, {\mathrm{rank}}(H\cap K)-2$ in number.
One may estimate the number of edges of $\Delta$, and hence ${\mathrm{rank}}(H\cap K)$, by counting the maximum number of edges possible in a bipartite graph with $2h-2$ black vertices and $2k-2$ white vertices whose number of components is equal to $c$.
It may be that a direct study of $\Delta$ would produce the inequalities given here, but we have not investigated this.
Walter Neumann inequalities {#walterineq}
---------------------------
Let $X$ be a set of representatives for the double coset space $H \backslash F \slash K$ and let $Y$ be the subset of $X$ consisting of those $g$ such that $H \cap K^g$ is nontrivial. As mentioned at the end of Section \[particular\], other than in our treatment of Burns’ theorem, we have not estimated the sum $$\sum_{g\in Y} \! \! \big( {\mathrm{rank}}(H\cap K^g) - 1\big)$$ using hypotheses on ${\mathrm{rank}}(H\vee K)$. However, we are free to replace ${\mathrm{rank}}(H\cap K) -1$ with this sum throughout provided we replace ${\mathrm{rank}}(H\vee K)$ with ${\mathrm{rank}}\langle H, K, Y \rangle$.
To see this, note that we may replace ${\mathcal T}$ with the pushout $\mathcal S$ of the diagram $$\xymatrix{\displaystyle{\bigsqcup}\ \Gamma_{H\cap K^g} \ar[r] \ar[d]_<{\overset{}{g \in Y}\ \ }
& {\Gamma_{\! H}}\\
{\Gamma_{\! K}}}$$ to obtain a diagram $$\xymatrix{\displaystyle{\bigsqcup}\ \Gamma_{H\cap K^g} \ar[r] \ar[d]_<{\overset{}{g \in Y}\ \ } & {\Gamma_{\! H}}\ar[d] \ar[ddr] \\
{\Gamma_{\! K}}\ar[r] \ar[drr] & \mathcal S \ar[dr] \\
& & \Gamma_{\! \langle H, K, Y \rangle}
}$$ where the map $\mathcal S \to \Gamma_{\! \langle H, K, Y \rangle}$ factors into a series of folds.
Department of Mathematics, Brown University, Providence, RI 02912 `rkent@math.brown.edu`
[^1]: Work supported by a Donald D. Harrington Dissertation Fellowship and a National Science Foundation Postdoctoral Fellowship.
[^2]: This theorem was proven by both parties before Theorems \[particulartheorem\] and \[strongburns\] were proven.
[^3]: *History of the error:* In the Fall of 2005, the author produced a faulty proof of Theorem \[main\]. Following this, he discovered the paper [@imrichmuller], from which Theorem \[main\] would follow. Unable to prove the existence of the sequence $x = x_0, x_1, x_2, \ldots, x_n = y$ in the quoted passage, the author contacted Imrich. Amidst the resulting correspondence, the author found the example in Figure \[counterexample\].
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